Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 587841 14 pageshttpdxdoiorg1011552013587841
Research ArticleTwo-Layer Predictive Control of a Continuous BiodieselTransesterification Reactor
Hongyan Shi12 Dingding Wang3 Decheng Yuan2 and Tianran Wang1
1 Shenyang Institute of Automation Chinese Academy of Sciences Shenyang 110016 China2 College of Information Engineering Shenyang University of Chemical Technology Shenyang 110142 China3 College of Information Engineering Zhejiang University of Technology Hangzhou 310023 China
Correspondence should be addressed to Hongyan Shi shihongyansiacn
Received 19 July 2013 Accepted 29 August 2013
Academic Editor Baocang Ding
Copyright copy 2013 Hongyan Shi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A novel two-layer predictive control scheme for a continuous biodiesel transesterification reactor is presented Based on a validatedmechanistic model the least squares (LS) algorithm is used to identify the finite step response (FSR) process model adapted inthe controller The two-layer predictive control method achieves the steady-state optimal setpoints and resolves the multivariabledynamic control problems synchronously Simulation results show that the two-layer predictive control strategy leads to a significantimprovement of control performance in terms of the optimal set-points tracking and disturbances rejection as compared toconventional PID controller within a multiloop framework
1 Introduction
With the depletion of fossil fuels and global environmen-tal degradation the development of alternative fuels fromrenewable resources has received considerable attentionBiodiesel has become the foremost alternative fuel to thoserefined from petroleum products It can be produced fromrenewable sources such as vegetable and animal oils as wellas from wastes such as used cooking oil Transesterificationis the primary method of converting these oils to biodiesel[1ndash3] A block diagram for a biodiesel production process bytransesterification is shown in Figure 1
A modern transesterification plant is continuous insteadof batch A continuous biodiesel production leads to betterheat economization better product purity from phase sepa-ration by removing only the portion of the layer furthest fromthe interface better recovery of excess methanol in order tosave on methanol cost and regulatory issues minimal oper-ator interference in adjusting plant parameters and lowercapital costs per unit of biodiesel produced The same tech-nology can also be applied to other biofuels production [4ndash6]
Biodiesel transesterification reactor is the most cru-cial operation unit to be controlled because any drift instandard operating condition may lead to significant changesin process variable and production quality specification [4ndash7]These reactors have complicated dynamics and heat trans-fer characteristics Moreover they are inherently concernedwith nonlinearity which arises from fluctuations of reactantconcentration reactant temperature coolant temperatureand instrumentationnoise or complexmicrobial interactionsThe complicated nonlinear multivariable and coupling innature are the fundamental control problems involved inbiodiesel reactor [8 9]
Recently a number of reports have appeared on thecontroller design and dynamic optimization in continuousand batch biodiesel reactorsMjalli et al developed a rigorousmechanistic model of a continuous biodiesel reactor and pro-posed a multimodel adaptive control strategy which realizedthe set-point tracking and disturbance rejection [4] Ho et alfurther adopted adaptive generalized predictive control strat-egy to handle multivariable problems of a biodiesel reactor[8] Wali et al proposed an artificial intelligence techniqueto design online genetic-ANFIS temperature control based
2 Journal of Applied Mathematics
Triglyceridebase catalyst
methanol
Biodiesel
Transesterificationreactor
Glycerol recovery
Glycerol
Settler Washer Dryer
Methanol recovery
Figure 1 Biodiesel production by transesterification
on LabVIEW for a novel continuous microwave biodieselreactor [10] Benavides and Diwekar realized the optimalcontrol of a batch biodiesel reactor involved optimization ofthe concentration based on maximum principle [11]
This work considers the advanced control strategy ofbiodiesel continuous transesterification reactor Model pre-dictive control (MPC) is one of the most popular advancedcontrol strategies It is a class of model-based control algo-rithm which has become a complex standard process indus-try solving complicated constrained multivariable controlproblems and widely used in the chemical and petrochem-ical processes [12] The main technical characteristics ofMPC include using mathematical models and history inputand output data to predict future output combined withthe established control objectives to calculate the optimalfeedback rate Compared with the traditional multiloopPID controllers MPC takes into account simultaneouslythe effects of all manipulated variables to all controlledvariables Usually successfully put into operation MPC cansignificantly reduce the standard deviation of the controlledvariable and then through the card edge operations improvethe overall efficiency of the control system
In recent years there has been an integrated steady-state optimization of the two-layer predictive control strategyin MPC industry technology [13ndash15] Two-layer predictivecontrol is divided into upper steady-state optimization (SSO)layer and lower dynamic control layer SSO can achieve real
time optimization (RTO) objectives tracking asymptoticallyindependently complete local economic optimization of thecorresponding MPC procedure Specifically the upper SSOuses steady-state gain of MPC dynamic mathematical modelas the mathematical model and searches the optimum valuewithin the constraints space of MPC Part steady-state valuesof the operating or output variables will be in the position ofldquocard edgerdquo The calculation results of the SSO layer will be asthe set- points to the lower MPC layer
Although two-layer predictive control strategy has beenwidely used in many applications of chemical reactorshardly any work was done on the biodiesel transesterifica-tion reactor In this paper a two-layer predictive controlstrategy is designed tested and simulated on a continuousbiodiesel transesterification reactor The scheme can amplifythe advantages of both technologies in terms of processstability and optimal and improved performances Section 2discusses the transesterification mechanism which uses avalidatedmechanisticmodel ofMjalli et al [4]Then the two-layer predictive control strategy is developed in Section 3Section 4 gives the control system design based on two-layer predictive control theory Section 5 discusses modelidentification results and the performances of the controlstrategy
2 Mathematical Models
The modeling of transesterification reactors starts withunderstanding the complex reaction kinetic mechanismThestoichiometry of vegetable oil methanolysis reaction requiresthree mol of methanol (A) and one mol of triglyceride (TG)to give three mol of fatty acid methyl ester (E) and one mol ofglycerol (G) [16]Theoverall reaction scheme for this reactionis
TG + 3A larrrarr 3E + G (1)
The methanolysis in turn consists of three consecutivereversible reactions where a mole of fatty acid methyl esteris released in each step and monoglycerides (MG) anddiglycerides (DG) are intermediate products The stepwisereactions are
CHOCO CHOCO
CHOCO CHOCO
CHOCO
CH2OCO CH2OCOCH3OCO
CH3OCO
CH3OCO
CH2OCOCH2OCO
CH2OH
CH2OHCH2OH
CH2OH
CH2OHCH2OHCH2OH
CH3OH
CH2OH
CH2OH
R1
R1R1
R1
R2
R2
R2
R2
R2
R2
R3
R3K1
K2
K3
K4
K5
K6
+ +
+
+
CH3OH+
CH3OH+
(2)
Journal of Applied Mathematics 3
The stepwise reactions can be termed as pseudo-homo-geneous catalyzed reactions following second-order kineticsThe second-order kineticmodel can be explained through thefollowing set of differential equations [17]
119889119862TG119889119905
= minus1198961015840
1119862TG119862A + 119896
1015840
2119862DG119862E
119889119862DG119889119905
= 1198961015840
1119862TG119862A minus 119896
1015840
2119862DG119862E minus 119896
1015840
3119862DG119862A + 119896
1015840
4119862MG119862E
119889119862MG119889119905
= 1198961015840
3119862DG119862A minus 119896
1015840
4119862MG119862E minus 119896
1015840
5119862MG119862A + 119896
1015840
6119862GL119862E
119889119862E119889119905
= 1198961015840
1119862TG119862A minus 119896
1015840
2119862DG119862E + 119896
1015840
3119862DG119862A minus 119896
1015840
4119862MG119862E
+ 1198961015840
5119862MG119862A minus 119896
1015840
6119862GL119862E
119889119862A119889119905
= minus
119889119862E119889119905
119889119862GL119889119905
= 1198961015840
5119862MG119862A minus 119896
1015840
6119862GL119862E
(3)
where 119862TG 119862DG 119862MG 119862E 119862A and 119862GL are concentrationsof triglyceride diglyceride monoglyceride methyl estermethanol and glycerol respectively 1198961015840
1 1198961015840
3 and 119896
1015840
5are the
effective rate constants for the forward reactions and 1198961015840
2 1198961015840
4
and 1198961015840
6are the effective rate constants for the reverse reactions
The previously selected kinetic model can be formulatedin terms of a general reaction equation
119903119895= 119896
1015840
119895[119862
119894]2
(4)
The catalyst concentration remained constant because thesidereactions that consume the catalyst were supposed tobe negligible Therefore each effective rate constant includesthe catalyst concentration (119862cat) and the corresponding rateconstant for the catalyzed reaction [18]
1198961015840
119895= 119896
119895119862cat (5)
The temperature influence on the reaction rate wasstudied from the Arrhenius equation (6) that shows thetemperature dependency of the reaction rate constant
119896119895= 119896
0119890
(minus119864119886119877119879)
(6)
where 1198960is a constant called the preexponential factor 119864
119886is
the activation energy of the reaction and119877 is the gas constantIn order to realize the optimization and control of
continuous biodiesel production process the model used inthe paper on the basis of the second-order kinetic modeljointing the material and energy balance equations as wellas the dynamic equation of the coolant temperature Thematerial balance for each component is expressed as follows[4]
119881
119889119862119894
119889119905
= 1198651198940119862
1198940minus 119865
119894119862
119894minus
119899
sum
119895=1
119903119895119881 (7)
SSO(local economic
optimization mode)
RTO
SSO(tracking mode)
PID
MPC
Figure 2 Framework of two-layer predictive control of industrialprocesses
The reactor energy balance is expressed as
119881
119899119894
sum
119894=1
119862119894119862
119901119894
119889119879
119889119905
= (119865TG0119862TG0
119862119875TG
+ 119865A0119862A0119862119875A) (119879
0minus 119879)
minus (119881
119899
sum
119894=119895
119903119895Δ119867
119895) minus (119880119860
119867Δ119879)
(8)
The coolant fluid energy balance is expressed as
119889119879119862
119889119905
=
1198651198620
119881119862
(1198791198620
minus 119879119862) +
119880119860119867Δ119879
120588119862119881
119862119862
119875119862
(9)
The function equation of heat transfer coefficient is approxi-mately expressed as
119880 = 120572119865119862
120573119873
120574= 7355119865
1095
119862119873
0405 (10)
3 Theory of Two-Layer Predictive Control
In modern process industries the MPC controller is part ofa multilevel hierarchy of optimization and control functionsTypically it is three-layer structure that is an RTO block isat the top layer a MPC block is at the middle and a PIDblock is at the bottom [19] Therefore under this multilevelhierarchy control system structure the primary task of theMPC is to dynamic track the computational target calculatedby the RTO RTO layer should be optimized for the wholedevice
Reference [20] proposed the framework of two-layerpredictive control shown in Figure 2 SSO is added betweenRTO and MPC Left branch the SSO layer is used forrecalculating the results of RTO layer make the outputsteady-state target be located in the steady state gain matrixcolumn space so as tomeet the compatibility and consistency
4 Journal of Applied Mathematics
conditions of steady state solution Right branch the role ofSSO is to conduct local optimization to further improve theMPC steady-state performance which can effectively resolvethe nonparty system setpoints in the given problem
Mathematical description of the two-layer predictive con-trol include establishing steady-state mathematical modelsteady-state target calculation and a dynamic controllerdesign [21]
31 Establish Steady-State Mathematical Model Assume anMIMO plant with 119898 control input and 119901 controlled outputand the coefficients of the corresponding step responsemodelbetween control input 119906
119895and output 119910
119894are given the model
vector is
119886119894119895(119905) = [119886
119894119895(1) 119886
119894119895(119873)]
119879
(11)
where 119894 = 1 119901 119895 = 1 119898119873 in (11) denotes modelinghorizon of step response model Thus a multistep predictivemodel can be obtained
119879 are the steady-state outputincrement and input increment respectively and 119860
119873is the
steady-state step response coefficients matrix
119860119873=[[
[
11988611(119873) sdot sdot sdot 119886
1119898(119873)
119886
1199011(119873) sdot sdot sdot 119886
119901119898(119873)
]]
]
(18)
To meet the requirements of steady-state target calcula-tion model (17) can also be written as
Δ119910infin(119896) = 119860
119873Δ119906
infin(119896) (19)
32 Steady-State Target Calculation
321 Basic Problem Description Steady-state target calcula-tion is to maximize economic benefits for the purpose ofself-optimization under MPC existing configuration modeaccording to the process conditions According to the pro-duction process characteristics and objectives the basicproblem of steady-state target calculation is the optimizationprocess which controlled input as cost variables controlledoutput as steady-state variables A commondescription of theobjective function is as follows [21]
minΔ119906infin
(119896)Δ119910infin
(119896)
119869 = 120572119879Δ119906
infin(119896) + 120573
119879Δ119910
infin(119896) (20)
Since Δ119906infinand Δ119910
infinare linearly related the input output
variation of objective function can be unified to control theinput change The formula (20) can be unified as
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896) (21)
where 119888119879= [119888
1 119888
119898] is the cost coefficient vector con-
structed by the normalized benefit or cost of each input var-iable Δ119906
infin(119896) = [Δ119906
1
infin Δ119906
119898
infin]119879 is the steady-state change
value of every input at time 119896Given the steady-state constraints of input and output
variables global-optimization problem of steady-state target
Journal of Applied Mathematics 5
calculation can be described as the following linear program(LP) problem
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896)
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(22)
where 119866119906 119866
119891are the steady-state gain matrices of control
input and disturbance variables and 119890 is the model bias119906min 119906max are low limit and upper limit of steady-state inputvariables 119910min 119910max are low limit and upper limit of steadystate output variables
The global-optimization problem of steady-state targetcalculation can be described as the following quadraticprogram (QP) problem
minΔ119906infin
(119896)
119869 = 119888119879(Δ119906
infin(119896) minusMaxprofit)2
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(23)
where Maxprofit is the potential maximum economic profit
322 Feasibility Judgment and Soft Constraint AdjustmentMathematically optimization feasibility is the existence prob-lem of the optimal solution Feasibility of steady-state targetcalculation means that optimal steady state of input-outputshould meet their operating constraints if feasible solutiondoes not exist the optimization calculation has no solutionThe solving process is as follows first judge the existence ofspace domain formed by the constraints and if there is init for optimization if does not exist then through the softconstraints adjustment to obtain the feasible space domainand then to solve
Soft constraints adjustment is an effective way to solveinfeasible optimization [22 23] By relaxing the outputconstraints within the hard constraints increasing the opti-mization problem feasible region that feasible solution to beoptimized Hard constraints refer to unalterable constraintslimited by the actual industrial process
Engineering standards of the priority strategy of softconstraints adjustment are the following give priority tomeetthe highly important operating constraints and allow less
important operating constraints to be violated appropriatelyunder the premise of satisfying the engineering constraints
Considering the following constraints (24) constituted bysteady-state model input constraints and output constraintscontaining slack variables the priority rank is ldquo119873rdquo where
Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906119871119871le 119906
infin(119896) + Δ119906
infin(119896) le 119906
119867119871
119910119895
119871119871minus 120576
119895
2le 119910
infin(119896) + Δ119910
infin(119896) le 119910
119895
119867119871+ 120576
119895
1
120576119895
1ge 0 120576
119895
2ge 0
120576119895
1le 119910
119867119867119871minus 119910
119867119871
120576119895
2le 119910
119871119871minus 119910
119871119871119871
119895 = 1 119873
(24)
The algorithm steps of feasibility judgment and soft con-straint adjustment based on the priority strategy are asfollows
Step 1 Initialization according to the characteristics of theoutput variables and process conditions set the upper andlower output constraints priority ranks the same priorityrank setting adjustments according to actual situation con-straint weights
Step 2 According to the priority ranks judge the feasibilityand adjust the soft constraints in accordance with the ranksfrom large to small Under a larger priority rank if cannotfind a feasible solution the constraints of the rank will berelaxed to hard constraints and then consider less priorityrank constraints until we find a feasible solution
Step 3 Then the steady-state target calculation entered thestage of economy optimization or target tracking
For Step 2 constraints of the highest priority rank119873 areadjusted first by solving the following optimization problem
Solving (25) may appear in three different cases respectivelyif (25) is feasible and the optimum solution is 120576119873
= 0 subjectto 119869 = 0 that is no need for soft constraints adjustmentdirectly solve the original problem (22) if (25) is feasible but120576
119873= 0 just need to relax constraints of priority ranks119873 and
further optimization solution if (25) is infeasible not get afeasible solution to soft constraints adjustment of the priorityrank119873 relaxing the constraints of the priority rank119873 to hardconstraints that is
120576119873
1= 119910
119873
119867119867119871minus 119910
119873
119867119871
120576119873
2= 119910
119873
119871119871minus 119910
119873
119871119871119871
(27)
Go to the procedure of judging rank119873 minus 1 constraints
min120576119873minus1
119869 = (119882119873minus1
)
119879
120576119873minus1
(119882119873minus1
)
119879
= [119882119873minus1
1 119882
119873minus1
2times119899119873minus1
]
st Θ119873minus1
119885119873minus1
= 119887119873minus1
Ω119873minus1
119885119873minus1
le Ψ119873minus1
(28)
For (28) the matrix form is the same with priority rank 119873only in the corresponding position of 120576119873minus1 to replace 120576119873 119887119873minus1
matrix is adjusted
119887119873minus1
=
[[[[[[[[[[[[[[[[[
[
119906119867119871
minus 119906119871119871
1198661
119906119906
infin(119896) minus 119866
1
119906119906
119871119871(119896) + 119910
1
119867119871minus 119910
1
infin(119896) minus 119866
1
119891Δ119891
infin(119896) minus 119890
1
119866
119873
119906119906
infin(119896) minus 119866
119873
119906119906
119871119871(119896) + 119910
119873
119867119871minus 119910
119873
infin(119896) minus 119866
119873
119891Δ119891
infin(119896) minus 119890
119873+ (119910
119873
119867119867119871minus 119910
119873
119867119871)
119879
1198661
119906119906
119867119871minus 119866
1
119906119906
infin(119896) + 119910
1
infin(119896) + 119866
1
119891Δ119891
infin(119896) minus 119910
1
119871119871+ 119890
1
119866
119873
119906119906
119867119871minus 119866
119873
119906119906
infin(119896) + 119910
119873
infin(119896) + 119866
119873
119891Δ119891
infin(119896) minus 119910
119873
119871119871+ 119890
119873+ (119910
119873
119871119871minus 119910
119873
119871119871119871)
119879
0
0
]]]]]]]]]]]]]]]]]
]
(29)
Journal of Applied Mathematics 7
119873minus1 rank and119873 rank are the same for the soft constraintsadjustment processing until the end of constraint adjustmentof the priority rank 1 If all ranks of constraints are relaxed tothe hard constrain and a feasible solution still canrsquot be foundthen the original problem of soft constraints adjustment isinfeasible and needs to be redesigned
33 Dynamic Controller Design In engineering applicationsdynamic matrix control (DMC) algorithm is one of the mostwidely used MPC algorithms based on the step responsemodel of the plant This paper adopts DMC and steady-statetarget calculation integration strategy
The difference is that the general DMC algorithms haveno requirements on the steady-state position of the controlinput and they only require the controlled output as close aspossible to arrive at its set point However the integrationstrategy DMC requires both input and output variables toapproach their steady-state targets (u
119904 y
119904) as far as possible
The algorithm has three basic characteristics predictivemodel receding horizon optimization and feedback correc-tion [24]
331 Predictive Model Based on system process step re-sponse model at the current time 119896 the future 119875-stepprediction output can be written as follows
y119875119872
(119896) = y1198750(119896) + AΔu
119872(119896) (30)
where 119875 denotes the prediction horizon 119872 is the con-trol horizon A is the prediction matrix composed by thecorresponding step response coefficients y
1198750is the initial
output prediction value when control action starting fromthe present time does not change Δu
119872(119896) is the prediction
incremental in119872 control horizon and y119875119872(119896) is the future119875-
step prediction output under 119872-step control action changeAmong them
y119875119872
(119896) =[[
[
1199101119875119872
(119896)
119910
119901119875119872(119896)
]]
]
y1198750(119896) =
[[
[
11991011198750
(119896)
119910
1199011198750(119896)
]]
]
Δu119872(119896) =
[[
[
Δ1199061119872
(119896)
Δ119906
119898119872(119896)
]]
]
A =[[
[
11986011
sdot sdot sdot 1198601119898
d
1198601199011
sdot sdot sdot 119860119901119898
]]
]
(31)
332 Receding Horizon Optimization In the receding hori-zon optimization process control increment can be obtainedin every execution cycle by minimizing the following perfor-mance index
minΔu119872
(119896)
119869 (119896) =1003817100381710038171003817w (119896) minus y
119875119872(119896)
1003817100381710038171003817
2
Q + 120576 (119896)2
S
+1003817100381710038171003817u
119872(119896) minus u
infin
1003817100381710038171003817
2
T +1003817100381710038171003817Δu
119872(119896)
1003817100381710038171003817
2
R
(32)
Subject to the modely
119875119872(119896) = y
1198750(119896) + AΔu
119872(119896) (33)
Subject to bound constraintsymin minus 120576 le y
119875119872(119896) le ymax + 120576
umin le u119872le umax
Δumin le Δu119872(119896) le Δumax
(34)
where 120576 denotes the slack variables guaranteeing the feasibil-ity of theDMCoptimization and119908(119896) = [119908
1(119896) 119908
119901(119896)]
119879
is the setpoint of controlled output obtained from upper SSOlayer Q R are the weight coefficient matrix
Q = block-diag (1198761 119876
119901)
Q119894= diag (119902
119894(1) 119902
119894(119875)) 119894 = 1 119901
R = block-diag (1198771 119877
119898)
R119895= diag (119903
119894(1) 119903
119894(119872)) 119895 = 1 119898
(35)
Through the necessary conditions of extreme value120597119869120597Δ119906
119872(119896) = 0 the optimal increment of control input can
be obtained
Δu119872(119896) = (A119879QA + R)
minus1
A119879Q [w (119896) minus y1198750(119896)] (36)
The instant increment can be calculated as follows
Δu (119896) = LD [w (119896) minus y1198750(119896)] (37)
whereD = (A119879QA +R)minus1A119879Q remark the operation of onlythe first element with
119871 = [
[
1 0 sdot sdot sdot 0 0
d0 1 0 sdot sdot sdot 0
]
]
(38)
333 Feedback Correction The difference between the pro-cess sample values by the present moment 119896 and predictionvalues of (30) is
119890 (119896 + 1) =[[
[
1198901(119896 + 1)
119890
119901(119896 + 1)
]]
]
=[[
[
1199101(119896 + 1) minus 119910
11(119896 + 1 | 119896)
119910
119901(119896 + 1) minus 119910
1199011(119896 + 1 | 119896)
]]
]
(39)
where 1199101198941(119896+1 | 119896) is the first element of 119910
119894119875119872(119896+1 | 119896) and
the corrected output prediction value can be obtained usingthe error vector that is
119873 moment initial prediction value when all of the inputremained unchanged at the time 119896 y
1198731(119896) is the future
119873 moment output prediction value under one-step controlinput action 119867 is the error correct matrix Then using ashift matrix 119878 next time the initial prediction value can beobtained which is
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Figure 1 Biodiesel production by transesterification
on LabVIEW for a novel continuous microwave biodieselreactor [10] Benavides and Diwekar realized the optimalcontrol of a batch biodiesel reactor involved optimization ofthe concentration based on maximum principle [11]
This work considers the advanced control strategy ofbiodiesel continuous transesterification reactor Model pre-dictive control (MPC) is one of the most popular advancedcontrol strategies It is a class of model-based control algo-rithm which has become a complex standard process indus-try solving complicated constrained multivariable controlproblems and widely used in the chemical and petrochem-ical processes [12] The main technical characteristics ofMPC include using mathematical models and history inputand output data to predict future output combined withthe established control objectives to calculate the optimalfeedback rate Compared with the traditional multiloopPID controllers MPC takes into account simultaneouslythe effects of all manipulated variables to all controlledvariables Usually successfully put into operation MPC cansignificantly reduce the standard deviation of the controlledvariable and then through the card edge operations improvethe overall efficiency of the control system
In recent years there has been an integrated steady-state optimization of the two-layer predictive control strategyin MPC industry technology [13ndash15] Two-layer predictivecontrol is divided into upper steady-state optimization (SSO)layer and lower dynamic control layer SSO can achieve real
time optimization (RTO) objectives tracking asymptoticallyindependently complete local economic optimization of thecorresponding MPC procedure Specifically the upper SSOuses steady-state gain of MPC dynamic mathematical modelas the mathematical model and searches the optimum valuewithin the constraints space of MPC Part steady-state valuesof the operating or output variables will be in the position ofldquocard edgerdquo The calculation results of the SSO layer will be asthe set- points to the lower MPC layer
Although two-layer predictive control strategy has beenwidely used in many applications of chemical reactorshardly any work was done on the biodiesel transesterifica-tion reactor In this paper a two-layer predictive controlstrategy is designed tested and simulated on a continuousbiodiesel transesterification reactor The scheme can amplifythe advantages of both technologies in terms of processstability and optimal and improved performances Section 2discusses the transesterification mechanism which uses avalidatedmechanisticmodel ofMjalli et al [4]Then the two-layer predictive control strategy is developed in Section 3Section 4 gives the control system design based on two-layer predictive control theory Section 5 discusses modelidentification results and the performances of the controlstrategy
2 Mathematical Models
The modeling of transesterification reactors starts withunderstanding the complex reaction kinetic mechanismThestoichiometry of vegetable oil methanolysis reaction requiresthree mol of methanol (A) and one mol of triglyceride (TG)to give three mol of fatty acid methyl ester (E) and one mol ofglycerol (G) [16]Theoverall reaction scheme for this reactionis
TG + 3A larrrarr 3E + G (1)
The methanolysis in turn consists of three consecutivereversible reactions where a mole of fatty acid methyl esteris released in each step and monoglycerides (MG) anddiglycerides (DG) are intermediate products The stepwisereactions are
CHOCO CHOCO
CHOCO CHOCO
CHOCO
CH2OCO CH2OCOCH3OCO
CH3OCO
CH3OCO
CH2OCOCH2OCO
CH2OH
CH2OHCH2OH
CH2OH
CH2OHCH2OHCH2OH
CH3OH
CH2OH
CH2OH
R1
R1R1
R1
R2
R2
R2
R2
R2
R2
R3
R3K1
K2
K3
K4
K5
K6
+ +
+
+
CH3OH+
CH3OH+
(2)
Journal of Applied Mathematics 3
The stepwise reactions can be termed as pseudo-homo-geneous catalyzed reactions following second-order kineticsThe second-order kineticmodel can be explained through thefollowing set of differential equations [17]
119889119862TG119889119905
= minus1198961015840
1119862TG119862A + 119896
1015840
2119862DG119862E
119889119862DG119889119905
= 1198961015840
1119862TG119862A minus 119896
1015840
2119862DG119862E minus 119896
1015840
3119862DG119862A + 119896
1015840
4119862MG119862E
119889119862MG119889119905
= 1198961015840
3119862DG119862A minus 119896
1015840
4119862MG119862E minus 119896
1015840
5119862MG119862A + 119896
1015840
6119862GL119862E
119889119862E119889119905
= 1198961015840
1119862TG119862A minus 119896
1015840
2119862DG119862E + 119896
1015840
3119862DG119862A minus 119896
1015840
4119862MG119862E
+ 1198961015840
5119862MG119862A minus 119896
1015840
6119862GL119862E
119889119862A119889119905
= minus
119889119862E119889119905
119889119862GL119889119905
= 1198961015840
5119862MG119862A minus 119896
1015840
6119862GL119862E
(3)
where 119862TG 119862DG 119862MG 119862E 119862A and 119862GL are concentrationsof triglyceride diglyceride monoglyceride methyl estermethanol and glycerol respectively 1198961015840
1 1198961015840
3 and 119896
1015840
5are the
effective rate constants for the forward reactions and 1198961015840
2 1198961015840
4
and 1198961015840
6are the effective rate constants for the reverse reactions
The previously selected kinetic model can be formulatedin terms of a general reaction equation
119903119895= 119896
1015840
119895[119862
119894]2
(4)
The catalyst concentration remained constant because thesidereactions that consume the catalyst were supposed tobe negligible Therefore each effective rate constant includesthe catalyst concentration (119862cat) and the corresponding rateconstant for the catalyzed reaction [18]
1198961015840
119895= 119896
119895119862cat (5)
The temperature influence on the reaction rate wasstudied from the Arrhenius equation (6) that shows thetemperature dependency of the reaction rate constant
119896119895= 119896
0119890
(minus119864119886119877119879)
(6)
where 1198960is a constant called the preexponential factor 119864
119886is
the activation energy of the reaction and119877 is the gas constantIn order to realize the optimization and control of
continuous biodiesel production process the model used inthe paper on the basis of the second-order kinetic modeljointing the material and energy balance equations as wellas the dynamic equation of the coolant temperature Thematerial balance for each component is expressed as follows[4]
119881
119889119862119894
119889119905
= 1198651198940119862
1198940minus 119865
119894119862
119894minus
119899
sum
119895=1
119903119895119881 (7)
SSO(local economic
optimization mode)
RTO
SSO(tracking mode)
PID
MPC
Figure 2 Framework of two-layer predictive control of industrialprocesses
The reactor energy balance is expressed as
119881
119899119894
sum
119894=1
119862119894119862
119901119894
119889119879
119889119905
= (119865TG0119862TG0
119862119875TG
+ 119865A0119862A0119862119875A) (119879
0minus 119879)
minus (119881
119899
sum
119894=119895
119903119895Δ119867
119895) minus (119880119860
119867Δ119879)
(8)
The coolant fluid energy balance is expressed as
119889119879119862
119889119905
=
1198651198620
119881119862
(1198791198620
minus 119879119862) +
119880119860119867Δ119879
120588119862119881
119862119862
119875119862
(9)
The function equation of heat transfer coefficient is approxi-mately expressed as
119880 = 120572119865119862
120573119873
120574= 7355119865
1095
119862119873
0405 (10)
3 Theory of Two-Layer Predictive Control
In modern process industries the MPC controller is part ofa multilevel hierarchy of optimization and control functionsTypically it is three-layer structure that is an RTO block isat the top layer a MPC block is at the middle and a PIDblock is at the bottom [19] Therefore under this multilevelhierarchy control system structure the primary task of theMPC is to dynamic track the computational target calculatedby the RTO RTO layer should be optimized for the wholedevice
Reference [20] proposed the framework of two-layerpredictive control shown in Figure 2 SSO is added betweenRTO and MPC Left branch the SSO layer is used forrecalculating the results of RTO layer make the outputsteady-state target be located in the steady state gain matrixcolumn space so as tomeet the compatibility and consistency
4 Journal of Applied Mathematics
conditions of steady state solution Right branch the role ofSSO is to conduct local optimization to further improve theMPC steady-state performance which can effectively resolvethe nonparty system setpoints in the given problem
Mathematical description of the two-layer predictive con-trol include establishing steady-state mathematical modelsteady-state target calculation and a dynamic controllerdesign [21]
31 Establish Steady-State Mathematical Model Assume anMIMO plant with 119898 control input and 119901 controlled outputand the coefficients of the corresponding step responsemodelbetween control input 119906
119895and output 119910
119894are given the model
vector is
119886119894119895(119905) = [119886
119894119895(1) 119886
119894119895(119873)]
119879
(11)
where 119894 = 1 119901 119895 = 1 119898119873 in (11) denotes modelinghorizon of step response model Thus a multistep predictivemodel can be obtained
119879 are the steady-state outputincrement and input increment respectively and 119860
119873is the
steady-state step response coefficients matrix
119860119873=[[
[
11988611(119873) sdot sdot sdot 119886
1119898(119873)
119886
1199011(119873) sdot sdot sdot 119886
119901119898(119873)
]]
]
(18)
To meet the requirements of steady-state target calcula-tion model (17) can also be written as
Δ119910infin(119896) = 119860
119873Δ119906
infin(119896) (19)
32 Steady-State Target Calculation
321 Basic Problem Description Steady-state target calcula-tion is to maximize economic benefits for the purpose ofself-optimization under MPC existing configuration modeaccording to the process conditions According to the pro-duction process characteristics and objectives the basicproblem of steady-state target calculation is the optimizationprocess which controlled input as cost variables controlledoutput as steady-state variables A commondescription of theobjective function is as follows [21]
minΔ119906infin
(119896)Δ119910infin
(119896)
119869 = 120572119879Δ119906
infin(119896) + 120573
119879Δ119910
infin(119896) (20)
Since Δ119906infinand Δ119910
infinare linearly related the input output
variation of objective function can be unified to control theinput change The formula (20) can be unified as
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896) (21)
where 119888119879= [119888
1 119888
119898] is the cost coefficient vector con-
structed by the normalized benefit or cost of each input var-iable Δ119906
infin(119896) = [Δ119906
1
infin Δ119906
119898
infin]119879 is the steady-state change
value of every input at time 119896Given the steady-state constraints of input and output
variables global-optimization problem of steady-state target
Journal of Applied Mathematics 5
calculation can be described as the following linear program(LP) problem
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896)
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(22)
where 119866119906 119866
119891are the steady-state gain matrices of control
input and disturbance variables and 119890 is the model bias119906min 119906max are low limit and upper limit of steady-state inputvariables 119910min 119910max are low limit and upper limit of steadystate output variables
The global-optimization problem of steady-state targetcalculation can be described as the following quadraticprogram (QP) problem
minΔ119906infin
(119896)
119869 = 119888119879(Δ119906
infin(119896) minusMaxprofit)2
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(23)
where Maxprofit is the potential maximum economic profit
322 Feasibility Judgment and Soft Constraint AdjustmentMathematically optimization feasibility is the existence prob-lem of the optimal solution Feasibility of steady-state targetcalculation means that optimal steady state of input-outputshould meet their operating constraints if feasible solutiondoes not exist the optimization calculation has no solutionThe solving process is as follows first judge the existence ofspace domain formed by the constraints and if there is init for optimization if does not exist then through the softconstraints adjustment to obtain the feasible space domainand then to solve
Soft constraints adjustment is an effective way to solveinfeasible optimization [22 23] By relaxing the outputconstraints within the hard constraints increasing the opti-mization problem feasible region that feasible solution to beoptimized Hard constraints refer to unalterable constraintslimited by the actual industrial process
Engineering standards of the priority strategy of softconstraints adjustment are the following give priority tomeetthe highly important operating constraints and allow less
important operating constraints to be violated appropriatelyunder the premise of satisfying the engineering constraints
Considering the following constraints (24) constituted bysteady-state model input constraints and output constraintscontaining slack variables the priority rank is ldquo119873rdquo where
Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906119871119871le 119906
infin(119896) + Δ119906
infin(119896) le 119906
119867119871
119910119895
119871119871minus 120576
119895
2le 119910
infin(119896) + Δ119910
infin(119896) le 119910
119895
119867119871+ 120576
119895
1
120576119895
1ge 0 120576
119895
2ge 0
120576119895
1le 119910
119867119867119871minus 119910
119867119871
120576119895
2le 119910
119871119871minus 119910
119871119871119871
119895 = 1 119873
(24)
The algorithm steps of feasibility judgment and soft con-straint adjustment based on the priority strategy are asfollows
Step 1 Initialization according to the characteristics of theoutput variables and process conditions set the upper andlower output constraints priority ranks the same priorityrank setting adjustments according to actual situation con-straint weights
Step 2 According to the priority ranks judge the feasibilityand adjust the soft constraints in accordance with the ranksfrom large to small Under a larger priority rank if cannotfind a feasible solution the constraints of the rank will berelaxed to hard constraints and then consider less priorityrank constraints until we find a feasible solution
Step 3 Then the steady-state target calculation entered thestage of economy optimization or target tracking
For Step 2 constraints of the highest priority rank119873 areadjusted first by solving the following optimization problem
Solving (25) may appear in three different cases respectivelyif (25) is feasible and the optimum solution is 120576119873
= 0 subjectto 119869 = 0 that is no need for soft constraints adjustmentdirectly solve the original problem (22) if (25) is feasible but120576
119873= 0 just need to relax constraints of priority ranks119873 and
further optimization solution if (25) is infeasible not get afeasible solution to soft constraints adjustment of the priorityrank119873 relaxing the constraints of the priority rank119873 to hardconstraints that is
120576119873
1= 119910
119873
119867119867119871minus 119910
119873
119867119871
120576119873
2= 119910
119873
119871119871minus 119910
119873
119871119871119871
(27)
Go to the procedure of judging rank119873 minus 1 constraints
min120576119873minus1
119869 = (119882119873minus1
)
119879
120576119873minus1
(119882119873minus1
)
119879
= [119882119873minus1
1 119882
119873minus1
2times119899119873minus1
]
st Θ119873minus1
119885119873minus1
= 119887119873minus1
Ω119873minus1
119885119873minus1
le Ψ119873minus1
(28)
For (28) the matrix form is the same with priority rank 119873only in the corresponding position of 120576119873minus1 to replace 120576119873 119887119873minus1
matrix is adjusted
119887119873minus1
=
[[[[[[[[[[[[[[[[[
[
119906119867119871
minus 119906119871119871
1198661
119906119906
infin(119896) minus 119866
1
119906119906
119871119871(119896) + 119910
1
119867119871minus 119910
1
infin(119896) minus 119866
1
119891Δ119891
infin(119896) minus 119890
1
119866
119873
119906119906
infin(119896) minus 119866
119873
119906119906
119871119871(119896) + 119910
119873
119867119871minus 119910
119873
infin(119896) minus 119866
119873
119891Δ119891
infin(119896) minus 119890
119873+ (119910
119873
119867119867119871minus 119910
119873
119867119871)
119879
1198661
119906119906
119867119871minus 119866
1
119906119906
infin(119896) + 119910
1
infin(119896) + 119866
1
119891Δ119891
infin(119896) minus 119910
1
119871119871+ 119890
1
119866
119873
119906119906
119867119871minus 119866
119873
119906119906
infin(119896) + 119910
119873
infin(119896) + 119866
119873
119891Δ119891
infin(119896) minus 119910
119873
119871119871+ 119890
119873+ (119910
119873
119871119871minus 119910
119873
119871119871119871)
119879
0
0
]]]]]]]]]]]]]]]]]
]
(29)
Journal of Applied Mathematics 7
119873minus1 rank and119873 rank are the same for the soft constraintsadjustment processing until the end of constraint adjustmentof the priority rank 1 If all ranks of constraints are relaxed tothe hard constrain and a feasible solution still canrsquot be foundthen the original problem of soft constraints adjustment isinfeasible and needs to be redesigned
33 Dynamic Controller Design In engineering applicationsdynamic matrix control (DMC) algorithm is one of the mostwidely used MPC algorithms based on the step responsemodel of the plant This paper adopts DMC and steady-statetarget calculation integration strategy
The difference is that the general DMC algorithms haveno requirements on the steady-state position of the controlinput and they only require the controlled output as close aspossible to arrive at its set point However the integrationstrategy DMC requires both input and output variables toapproach their steady-state targets (u
119904 y
119904) as far as possible
The algorithm has three basic characteristics predictivemodel receding horizon optimization and feedback correc-tion [24]
331 Predictive Model Based on system process step re-sponse model at the current time 119896 the future 119875-stepprediction output can be written as follows
y119875119872
(119896) = y1198750(119896) + AΔu
119872(119896) (30)
where 119875 denotes the prediction horizon 119872 is the con-trol horizon A is the prediction matrix composed by thecorresponding step response coefficients y
1198750is the initial
output prediction value when control action starting fromthe present time does not change Δu
119872(119896) is the prediction
incremental in119872 control horizon and y119875119872(119896) is the future119875-
step prediction output under 119872-step control action changeAmong them
y119875119872
(119896) =[[
[
1199101119875119872
(119896)
119910
119901119875119872(119896)
]]
]
y1198750(119896) =
[[
[
11991011198750
(119896)
119910
1199011198750(119896)
]]
]
Δu119872(119896) =
[[
[
Δ1199061119872
(119896)
Δ119906
119898119872(119896)
]]
]
A =[[
[
11986011
sdot sdot sdot 1198601119898
d
1198601199011
sdot sdot sdot 119860119901119898
]]
]
(31)
332 Receding Horizon Optimization In the receding hori-zon optimization process control increment can be obtainedin every execution cycle by minimizing the following perfor-mance index
minΔu119872
(119896)
119869 (119896) =1003817100381710038171003817w (119896) minus y
119875119872(119896)
1003817100381710038171003817
2
Q + 120576 (119896)2
S
+1003817100381710038171003817u
119872(119896) minus u
infin
1003817100381710038171003817
2
T +1003817100381710038171003817Δu
119872(119896)
1003817100381710038171003817
2
R
(32)
Subject to the modely
119875119872(119896) = y
1198750(119896) + AΔu
119872(119896) (33)
Subject to bound constraintsymin minus 120576 le y
119875119872(119896) le ymax + 120576
umin le u119872le umax
Δumin le Δu119872(119896) le Δumax
(34)
where 120576 denotes the slack variables guaranteeing the feasibil-ity of theDMCoptimization and119908(119896) = [119908
1(119896) 119908
119901(119896)]
119879
is the setpoint of controlled output obtained from upper SSOlayer Q R are the weight coefficient matrix
Q = block-diag (1198761 119876
119901)
Q119894= diag (119902
119894(1) 119902
119894(119875)) 119894 = 1 119901
R = block-diag (1198771 119877
119898)
R119895= diag (119903
119894(1) 119903
119894(119872)) 119895 = 1 119898
(35)
Through the necessary conditions of extreme value120597119869120597Δ119906
119872(119896) = 0 the optimal increment of control input can
be obtained
Δu119872(119896) = (A119879QA + R)
minus1
A119879Q [w (119896) minus y1198750(119896)] (36)
The instant increment can be calculated as follows
Δu (119896) = LD [w (119896) minus y1198750(119896)] (37)
whereD = (A119879QA +R)minus1A119879Q remark the operation of onlythe first element with
119871 = [
[
1 0 sdot sdot sdot 0 0
d0 1 0 sdot sdot sdot 0
]
]
(38)
333 Feedback Correction The difference between the pro-cess sample values by the present moment 119896 and predictionvalues of (30) is
119890 (119896 + 1) =[[
[
1198901(119896 + 1)
119890
119901(119896 + 1)
]]
]
=[[
[
1199101(119896 + 1) minus 119910
11(119896 + 1 | 119896)
119910
119901(119896 + 1) minus 119910
1199011(119896 + 1 | 119896)
]]
]
(39)
where 1199101198941(119896+1 | 119896) is the first element of 119910
119894119875119872(119896+1 | 119896) and
the corrected output prediction value can be obtained usingthe error vector that is
119873 moment initial prediction value when all of the inputremained unchanged at the time 119896 y
1198731(119896) is the future
119873 moment output prediction value under one-step controlinput action 119867 is the error correct matrix Then using ashift matrix 119878 next time the initial prediction value can beobtained which is
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
The stepwise reactions can be termed as pseudo-homo-geneous catalyzed reactions following second-order kineticsThe second-order kineticmodel can be explained through thefollowing set of differential equations [17]
119889119862TG119889119905
= minus1198961015840
1119862TG119862A + 119896
1015840
2119862DG119862E
119889119862DG119889119905
= 1198961015840
1119862TG119862A minus 119896
1015840
2119862DG119862E minus 119896
1015840
3119862DG119862A + 119896
1015840
4119862MG119862E
119889119862MG119889119905
= 1198961015840
3119862DG119862A minus 119896
1015840
4119862MG119862E minus 119896
1015840
5119862MG119862A + 119896
1015840
6119862GL119862E
119889119862E119889119905
= 1198961015840
1119862TG119862A minus 119896
1015840
2119862DG119862E + 119896
1015840
3119862DG119862A minus 119896
1015840
4119862MG119862E
+ 1198961015840
5119862MG119862A minus 119896
1015840
6119862GL119862E
119889119862A119889119905
= minus
119889119862E119889119905
119889119862GL119889119905
= 1198961015840
5119862MG119862A minus 119896
1015840
6119862GL119862E
(3)
where 119862TG 119862DG 119862MG 119862E 119862A and 119862GL are concentrationsof triglyceride diglyceride monoglyceride methyl estermethanol and glycerol respectively 1198961015840
1 1198961015840
3 and 119896
1015840
5are the
effective rate constants for the forward reactions and 1198961015840
2 1198961015840
4
and 1198961015840
6are the effective rate constants for the reverse reactions
The previously selected kinetic model can be formulatedin terms of a general reaction equation
119903119895= 119896
1015840
119895[119862
119894]2
(4)
The catalyst concentration remained constant because thesidereactions that consume the catalyst were supposed tobe negligible Therefore each effective rate constant includesthe catalyst concentration (119862cat) and the corresponding rateconstant for the catalyzed reaction [18]
1198961015840
119895= 119896
119895119862cat (5)
The temperature influence on the reaction rate wasstudied from the Arrhenius equation (6) that shows thetemperature dependency of the reaction rate constant
119896119895= 119896
0119890
(minus119864119886119877119879)
(6)
where 1198960is a constant called the preexponential factor 119864
119886is
the activation energy of the reaction and119877 is the gas constantIn order to realize the optimization and control of
continuous biodiesel production process the model used inthe paper on the basis of the second-order kinetic modeljointing the material and energy balance equations as wellas the dynamic equation of the coolant temperature Thematerial balance for each component is expressed as follows[4]
119881
119889119862119894
119889119905
= 1198651198940119862
1198940minus 119865
119894119862
119894minus
119899
sum
119895=1
119903119895119881 (7)
SSO(local economic
optimization mode)
RTO
SSO(tracking mode)
PID
MPC
Figure 2 Framework of two-layer predictive control of industrialprocesses
The reactor energy balance is expressed as
119881
119899119894
sum
119894=1
119862119894119862
119901119894
119889119879
119889119905
= (119865TG0119862TG0
119862119875TG
+ 119865A0119862A0119862119875A) (119879
0minus 119879)
minus (119881
119899
sum
119894=119895
119903119895Δ119867
119895) minus (119880119860
119867Δ119879)
(8)
The coolant fluid energy balance is expressed as
119889119879119862
119889119905
=
1198651198620
119881119862
(1198791198620
minus 119879119862) +
119880119860119867Δ119879
120588119862119881
119862119862
119875119862
(9)
The function equation of heat transfer coefficient is approxi-mately expressed as
119880 = 120572119865119862
120573119873
120574= 7355119865
1095
119862119873
0405 (10)
3 Theory of Two-Layer Predictive Control
In modern process industries the MPC controller is part ofa multilevel hierarchy of optimization and control functionsTypically it is three-layer structure that is an RTO block isat the top layer a MPC block is at the middle and a PIDblock is at the bottom [19] Therefore under this multilevelhierarchy control system structure the primary task of theMPC is to dynamic track the computational target calculatedby the RTO RTO layer should be optimized for the wholedevice
Reference [20] proposed the framework of two-layerpredictive control shown in Figure 2 SSO is added betweenRTO and MPC Left branch the SSO layer is used forrecalculating the results of RTO layer make the outputsteady-state target be located in the steady state gain matrixcolumn space so as tomeet the compatibility and consistency
4 Journal of Applied Mathematics
conditions of steady state solution Right branch the role ofSSO is to conduct local optimization to further improve theMPC steady-state performance which can effectively resolvethe nonparty system setpoints in the given problem
Mathematical description of the two-layer predictive con-trol include establishing steady-state mathematical modelsteady-state target calculation and a dynamic controllerdesign [21]
31 Establish Steady-State Mathematical Model Assume anMIMO plant with 119898 control input and 119901 controlled outputand the coefficients of the corresponding step responsemodelbetween control input 119906
119895and output 119910
119894are given the model
vector is
119886119894119895(119905) = [119886
119894119895(1) 119886
119894119895(119873)]
119879
(11)
where 119894 = 1 119901 119895 = 1 119898119873 in (11) denotes modelinghorizon of step response model Thus a multistep predictivemodel can be obtained
119879 are the steady-state outputincrement and input increment respectively and 119860
119873is the
steady-state step response coefficients matrix
119860119873=[[
[
11988611(119873) sdot sdot sdot 119886
1119898(119873)
119886
1199011(119873) sdot sdot sdot 119886
119901119898(119873)
]]
]
(18)
To meet the requirements of steady-state target calcula-tion model (17) can also be written as
Δ119910infin(119896) = 119860
119873Δ119906
infin(119896) (19)
32 Steady-State Target Calculation
321 Basic Problem Description Steady-state target calcula-tion is to maximize economic benefits for the purpose ofself-optimization under MPC existing configuration modeaccording to the process conditions According to the pro-duction process characteristics and objectives the basicproblem of steady-state target calculation is the optimizationprocess which controlled input as cost variables controlledoutput as steady-state variables A commondescription of theobjective function is as follows [21]
minΔ119906infin
(119896)Δ119910infin
(119896)
119869 = 120572119879Δ119906
infin(119896) + 120573
119879Δ119910
infin(119896) (20)
Since Δ119906infinand Δ119910
infinare linearly related the input output
variation of objective function can be unified to control theinput change The formula (20) can be unified as
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896) (21)
where 119888119879= [119888
1 119888
119898] is the cost coefficient vector con-
structed by the normalized benefit or cost of each input var-iable Δ119906
infin(119896) = [Δ119906
1
infin Δ119906
119898
infin]119879 is the steady-state change
value of every input at time 119896Given the steady-state constraints of input and output
variables global-optimization problem of steady-state target
Journal of Applied Mathematics 5
calculation can be described as the following linear program(LP) problem
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896)
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(22)
where 119866119906 119866
119891are the steady-state gain matrices of control
input and disturbance variables and 119890 is the model bias119906min 119906max are low limit and upper limit of steady-state inputvariables 119910min 119910max are low limit and upper limit of steadystate output variables
The global-optimization problem of steady-state targetcalculation can be described as the following quadraticprogram (QP) problem
minΔ119906infin
(119896)
119869 = 119888119879(Δ119906
infin(119896) minusMaxprofit)2
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(23)
where Maxprofit is the potential maximum economic profit
322 Feasibility Judgment and Soft Constraint AdjustmentMathematically optimization feasibility is the existence prob-lem of the optimal solution Feasibility of steady-state targetcalculation means that optimal steady state of input-outputshould meet their operating constraints if feasible solutiondoes not exist the optimization calculation has no solutionThe solving process is as follows first judge the existence ofspace domain formed by the constraints and if there is init for optimization if does not exist then through the softconstraints adjustment to obtain the feasible space domainand then to solve
Soft constraints adjustment is an effective way to solveinfeasible optimization [22 23] By relaxing the outputconstraints within the hard constraints increasing the opti-mization problem feasible region that feasible solution to beoptimized Hard constraints refer to unalterable constraintslimited by the actual industrial process
Engineering standards of the priority strategy of softconstraints adjustment are the following give priority tomeetthe highly important operating constraints and allow less
important operating constraints to be violated appropriatelyunder the premise of satisfying the engineering constraints
Considering the following constraints (24) constituted bysteady-state model input constraints and output constraintscontaining slack variables the priority rank is ldquo119873rdquo where
Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906119871119871le 119906
infin(119896) + Δ119906
infin(119896) le 119906
119867119871
119910119895
119871119871minus 120576
119895
2le 119910
infin(119896) + Δ119910
infin(119896) le 119910
119895
119867119871+ 120576
119895
1
120576119895
1ge 0 120576
119895
2ge 0
120576119895
1le 119910
119867119867119871minus 119910
119867119871
120576119895
2le 119910
119871119871minus 119910
119871119871119871
119895 = 1 119873
(24)
The algorithm steps of feasibility judgment and soft con-straint adjustment based on the priority strategy are asfollows
Step 1 Initialization according to the characteristics of theoutput variables and process conditions set the upper andlower output constraints priority ranks the same priorityrank setting adjustments according to actual situation con-straint weights
Step 2 According to the priority ranks judge the feasibilityand adjust the soft constraints in accordance with the ranksfrom large to small Under a larger priority rank if cannotfind a feasible solution the constraints of the rank will berelaxed to hard constraints and then consider less priorityrank constraints until we find a feasible solution
Step 3 Then the steady-state target calculation entered thestage of economy optimization or target tracking
For Step 2 constraints of the highest priority rank119873 areadjusted first by solving the following optimization problem
Solving (25) may appear in three different cases respectivelyif (25) is feasible and the optimum solution is 120576119873
= 0 subjectto 119869 = 0 that is no need for soft constraints adjustmentdirectly solve the original problem (22) if (25) is feasible but120576
119873= 0 just need to relax constraints of priority ranks119873 and
further optimization solution if (25) is infeasible not get afeasible solution to soft constraints adjustment of the priorityrank119873 relaxing the constraints of the priority rank119873 to hardconstraints that is
120576119873
1= 119910
119873
119867119867119871minus 119910
119873
119867119871
120576119873
2= 119910
119873
119871119871minus 119910
119873
119871119871119871
(27)
Go to the procedure of judging rank119873 minus 1 constraints
min120576119873minus1
119869 = (119882119873minus1
)
119879
120576119873minus1
(119882119873minus1
)
119879
= [119882119873minus1
1 119882
119873minus1
2times119899119873minus1
]
st Θ119873minus1
119885119873minus1
= 119887119873minus1
Ω119873minus1
119885119873minus1
le Ψ119873minus1
(28)
For (28) the matrix form is the same with priority rank 119873only in the corresponding position of 120576119873minus1 to replace 120576119873 119887119873minus1
matrix is adjusted
119887119873minus1
=
[[[[[[[[[[[[[[[[[
[
119906119867119871
minus 119906119871119871
1198661
119906119906
infin(119896) minus 119866
1
119906119906
119871119871(119896) + 119910
1
119867119871minus 119910
1
infin(119896) minus 119866
1
119891Δ119891
infin(119896) minus 119890
1
119866
119873
119906119906
infin(119896) minus 119866
119873
119906119906
119871119871(119896) + 119910
119873
119867119871minus 119910
119873
infin(119896) minus 119866
119873
119891Δ119891
infin(119896) minus 119890
119873+ (119910
119873
119867119867119871minus 119910
119873
119867119871)
119879
1198661
119906119906
119867119871minus 119866
1
119906119906
infin(119896) + 119910
1
infin(119896) + 119866
1
119891Δ119891
infin(119896) minus 119910
1
119871119871+ 119890
1
119866
119873
119906119906
119867119871minus 119866
119873
119906119906
infin(119896) + 119910
119873
infin(119896) + 119866
119873
119891Δ119891
infin(119896) minus 119910
119873
119871119871+ 119890
119873+ (119910
119873
119871119871minus 119910
119873
119871119871119871)
119879
0
0
]]]]]]]]]]]]]]]]]
]
(29)
Journal of Applied Mathematics 7
119873minus1 rank and119873 rank are the same for the soft constraintsadjustment processing until the end of constraint adjustmentof the priority rank 1 If all ranks of constraints are relaxed tothe hard constrain and a feasible solution still canrsquot be foundthen the original problem of soft constraints adjustment isinfeasible and needs to be redesigned
33 Dynamic Controller Design In engineering applicationsdynamic matrix control (DMC) algorithm is one of the mostwidely used MPC algorithms based on the step responsemodel of the plant This paper adopts DMC and steady-statetarget calculation integration strategy
The difference is that the general DMC algorithms haveno requirements on the steady-state position of the controlinput and they only require the controlled output as close aspossible to arrive at its set point However the integrationstrategy DMC requires both input and output variables toapproach their steady-state targets (u
119904 y
119904) as far as possible
The algorithm has three basic characteristics predictivemodel receding horizon optimization and feedback correc-tion [24]
331 Predictive Model Based on system process step re-sponse model at the current time 119896 the future 119875-stepprediction output can be written as follows
y119875119872
(119896) = y1198750(119896) + AΔu
119872(119896) (30)
where 119875 denotes the prediction horizon 119872 is the con-trol horizon A is the prediction matrix composed by thecorresponding step response coefficients y
1198750is the initial
output prediction value when control action starting fromthe present time does not change Δu
119872(119896) is the prediction
incremental in119872 control horizon and y119875119872(119896) is the future119875-
step prediction output under 119872-step control action changeAmong them
y119875119872
(119896) =[[
[
1199101119875119872
(119896)
119910
119901119875119872(119896)
]]
]
y1198750(119896) =
[[
[
11991011198750
(119896)
119910
1199011198750(119896)
]]
]
Δu119872(119896) =
[[
[
Δ1199061119872
(119896)
Δ119906
119898119872(119896)
]]
]
A =[[
[
11986011
sdot sdot sdot 1198601119898
d
1198601199011
sdot sdot sdot 119860119901119898
]]
]
(31)
332 Receding Horizon Optimization In the receding hori-zon optimization process control increment can be obtainedin every execution cycle by minimizing the following perfor-mance index
minΔu119872
(119896)
119869 (119896) =1003817100381710038171003817w (119896) minus y
119875119872(119896)
1003817100381710038171003817
2
Q + 120576 (119896)2
S
+1003817100381710038171003817u
119872(119896) minus u
infin
1003817100381710038171003817
2
T +1003817100381710038171003817Δu
119872(119896)
1003817100381710038171003817
2
R
(32)
Subject to the modely
119875119872(119896) = y
1198750(119896) + AΔu
119872(119896) (33)
Subject to bound constraintsymin minus 120576 le y
119875119872(119896) le ymax + 120576
umin le u119872le umax
Δumin le Δu119872(119896) le Δumax
(34)
where 120576 denotes the slack variables guaranteeing the feasibil-ity of theDMCoptimization and119908(119896) = [119908
1(119896) 119908
119901(119896)]
119879
is the setpoint of controlled output obtained from upper SSOlayer Q R are the weight coefficient matrix
Q = block-diag (1198761 119876
119901)
Q119894= diag (119902
119894(1) 119902
119894(119875)) 119894 = 1 119901
R = block-diag (1198771 119877
119898)
R119895= diag (119903
119894(1) 119903
119894(119872)) 119895 = 1 119898
(35)
Through the necessary conditions of extreme value120597119869120597Δ119906
119872(119896) = 0 the optimal increment of control input can
be obtained
Δu119872(119896) = (A119879QA + R)
minus1
A119879Q [w (119896) minus y1198750(119896)] (36)
The instant increment can be calculated as follows
Δu (119896) = LD [w (119896) minus y1198750(119896)] (37)
whereD = (A119879QA +R)minus1A119879Q remark the operation of onlythe first element with
119871 = [
[
1 0 sdot sdot sdot 0 0
d0 1 0 sdot sdot sdot 0
]
]
(38)
333 Feedback Correction The difference between the pro-cess sample values by the present moment 119896 and predictionvalues of (30) is
119890 (119896 + 1) =[[
[
1198901(119896 + 1)
119890
119901(119896 + 1)
]]
]
=[[
[
1199101(119896 + 1) minus 119910
11(119896 + 1 | 119896)
119910
119901(119896 + 1) minus 119910
1199011(119896 + 1 | 119896)
]]
]
(39)
where 1199101198941(119896+1 | 119896) is the first element of 119910
119894119875119872(119896+1 | 119896) and
the corrected output prediction value can be obtained usingthe error vector that is
119873 moment initial prediction value when all of the inputremained unchanged at the time 119896 y
1198731(119896) is the future
119873 moment output prediction value under one-step controlinput action 119867 is the error correct matrix Then using ashift matrix 119878 next time the initial prediction value can beobtained which is
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
conditions of steady state solution Right branch the role ofSSO is to conduct local optimization to further improve theMPC steady-state performance which can effectively resolvethe nonparty system setpoints in the given problem
Mathematical description of the two-layer predictive con-trol include establishing steady-state mathematical modelsteady-state target calculation and a dynamic controllerdesign [21]
31 Establish Steady-State Mathematical Model Assume anMIMO plant with 119898 control input and 119901 controlled outputand the coefficients of the corresponding step responsemodelbetween control input 119906
119895and output 119910
119894are given the model
vector is
119886119894119895(119905) = [119886
119894119895(1) 119886
119894119895(119873)]
119879
(11)
where 119894 = 1 119901 119895 = 1 119898119873 in (11) denotes modelinghorizon of step response model Thus a multistep predictivemodel can be obtained
119879 are the steady-state outputincrement and input increment respectively and 119860
119873is the
steady-state step response coefficients matrix
119860119873=[[
[
11988611(119873) sdot sdot sdot 119886
1119898(119873)
119886
1199011(119873) sdot sdot sdot 119886
119901119898(119873)
]]
]
(18)
To meet the requirements of steady-state target calcula-tion model (17) can also be written as
Δ119910infin(119896) = 119860
119873Δ119906
infin(119896) (19)
32 Steady-State Target Calculation
321 Basic Problem Description Steady-state target calcula-tion is to maximize economic benefits for the purpose ofself-optimization under MPC existing configuration modeaccording to the process conditions According to the pro-duction process characteristics and objectives the basicproblem of steady-state target calculation is the optimizationprocess which controlled input as cost variables controlledoutput as steady-state variables A commondescription of theobjective function is as follows [21]
minΔ119906infin
(119896)Δ119910infin
(119896)
119869 = 120572119879Δ119906
infin(119896) + 120573
119879Δ119910
infin(119896) (20)
Since Δ119906infinand Δ119910
infinare linearly related the input output
variation of objective function can be unified to control theinput change The formula (20) can be unified as
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896) (21)
where 119888119879= [119888
1 119888
119898] is the cost coefficient vector con-
structed by the normalized benefit or cost of each input var-iable Δ119906
infin(119896) = [Δ119906
1
infin Δ119906
119898
infin]119879 is the steady-state change
value of every input at time 119896Given the steady-state constraints of input and output
variables global-optimization problem of steady-state target
Journal of Applied Mathematics 5
calculation can be described as the following linear program(LP) problem
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896)
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(22)
where 119866119906 119866
119891are the steady-state gain matrices of control
input and disturbance variables and 119890 is the model bias119906min 119906max are low limit and upper limit of steady-state inputvariables 119910min 119910max are low limit and upper limit of steadystate output variables
The global-optimization problem of steady-state targetcalculation can be described as the following quadraticprogram (QP) problem
minΔ119906infin
(119896)
119869 = 119888119879(Δ119906
infin(119896) minusMaxprofit)2
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(23)
where Maxprofit is the potential maximum economic profit
322 Feasibility Judgment and Soft Constraint AdjustmentMathematically optimization feasibility is the existence prob-lem of the optimal solution Feasibility of steady-state targetcalculation means that optimal steady state of input-outputshould meet their operating constraints if feasible solutiondoes not exist the optimization calculation has no solutionThe solving process is as follows first judge the existence ofspace domain formed by the constraints and if there is init for optimization if does not exist then through the softconstraints adjustment to obtain the feasible space domainand then to solve
Soft constraints adjustment is an effective way to solveinfeasible optimization [22 23] By relaxing the outputconstraints within the hard constraints increasing the opti-mization problem feasible region that feasible solution to beoptimized Hard constraints refer to unalterable constraintslimited by the actual industrial process
Engineering standards of the priority strategy of softconstraints adjustment are the following give priority tomeetthe highly important operating constraints and allow less
important operating constraints to be violated appropriatelyunder the premise of satisfying the engineering constraints
Considering the following constraints (24) constituted bysteady-state model input constraints and output constraintscontaining slack variables the priority rank is ldquo119873rdquo where
Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906119871119871le 119906
infin(119896) + Δ119906
infin(119896) le 119906
119867119871
119910119895
119871119871minus 120576
119895
2le 119910
infin(119896) + Δ119910
infin(119896) le 119910
119895
119867119871+ 120576
119895
1
120576119895
1ge 0 120576
119895
2ge 0
120576119895
1le 119910
119867119867119871minus 119910
119867119871
120576119895
2le 119910
119871119871minus 119910
119871119871119871
119895 = 1 119873
(24)
The algorithm steps of feasibility judgment and soft con-straint adjustment based on the priority strategy are asfollows
Step 1 Initialization according to the characteristics of theoutput variables and process conditions set the upper andlower output constraints priority ranks the same priorityrank setting adjustments according to actual situation con-straint weights
Step 2 According to the priority ranks judge the feasibilityand adjust the soft constraints in accordance with the ranksfrom large to small Under a larger priority rank if cannotfind a feasible solution the constraints of the rank will berelaxed to hard constraints and then consider less priorityrank constraints until we find a feasible solution
Step 3 Then the steady-state target calculation entered thestage of economy optimization or target tracking
For Step 2 constraints of the highest priority rank119873 areadjusted first by solving the following optimization problem
Solving (25) may appear in three different cases respectivelyif (25) is feasible and the optimum solution is 120576119873
= 0 subjectto 119869 = 0 that is no need for soft constraints adjustmentdirectly solve the original problem (22) if (25) is feasible but120576
119873= 0 just need to relax constraints of priority ranks119873 and
further optimization solution if (25) is infeasible not get afeasible solution to soft constraints adjustment of the priorityrank119873 relaxing the constraints of the priority rank119873 to hardconstraints that is
120576119873
1= 119910
119873
119867119867119871minus 119910
119873
119867119871
120576119873
2= 119910
119873
119871119871minus 119910
119873
119871119871119871
(27)
Go to the procedure of judging rank119873 minus 1 constraints
min120576119873minus1
119869 = (119882119873minus1
)
119879
120576119873minus1
(119882119873minus1
)
119879
= [119882119873minus1
1 119882
119873minus1
2times119899119873minus1
]
st Θ119873minus1
119885119873minus1
= 119887119873minus1
Ω119873minus1
119885119873minus1
le Ψ119873minus1
(28)
For (28) the matrix form is the same with priority rank 119873only in the corresponding position of 120576119873minus1 to replace 120576119873 119887119873minus1
matrix is adjusted
119887119873minus1
=
[[[[[[[[[[[[[[[[[
[
119906119867119871
minus 119906119871119871
1198661
119906119906
infin(119896) minus 119866
1
119906119906
119871119871(119896) + 119910
1
119867119871minus 119910
1
infin(119896) minus 119866
1
119891Δ119891
infin(119896) minus 119890
1
119866
119873
119906119906
infin(119896) minus 119866
119873
119906119906
119871119871(119896) + 119910
119873
119867119871minus 119910
119873
infin(119896) minus 119866
119873
119891Δ119891
infin(119896) minus 119890
119873+ (119910
119873
119867119867119871minus 119910
119873
119867119871)
119879
1198661
119906119906
119867119871minus 119866
1
119906119906
infin(119896) + 119910
1
infin(119896) + 119866
1
119891Δ119891
infin(119896) minus 119910
1
119871119871+ 119890
1
119866
119873
119906119906
119867119871minus 119866
119873
119906119906
infin(119896) + 119910
119873
infin(119896) + 119866
119873
119891Δ119891
infin(119896) minus 119910
119873
119871119871+ 119890
119873+ (119910
119873
119871119871minus 119910
119873
119871119871119871)
119879
0
0
]]]]]]]]]]]]]]]]]
]
(29)
Journal of Applied Mathematics 7
119873minus1 rank and119873 rank are the same for the soft constraintsadjustment processing until the end of constraint adjustmentof the priority rank 1 If all ranks of constraints are relaxed tothe hard constrain and a feasible solution still canrsquot be foundthen the original problem of soft constraints adjustment isinfeasible and needs to be redesigned
33 Dynamic Controller Design In engineering applicationsdynamic matrix control (DMC) algorithm is one of the mostwidely used MPC algorithms based on the step responsemodel of the plant This paper adopts DMC and steady-statetarget calculation integration strategy
The difference is that the general DMC algorithms haveno requirements on the steady-state position of the controlinput and they only require the controlled output as close aspossible to arrive at its set point However the integrationstrategy DMC requires both input and output variables toapproach their steady-state targets (u
119904 y
119904) as far as possible
The algorithm has three basic characteristics predictivemodel receding horizon optimization and feedback correc-tion [24]
331 Predictive Model Based on system process step re-sponse model at the current time 119896 the future 119875-stepprediction output can be written as follows
y119875119872
(119896) = y1198750(119896) + AΔu
119872(119896) (30)
where 119875 denotes the prediction horizon 119872 is the con-trol horizon A is the prediction matrix composed by thecorresponding step response coefficients y
1198750is the initial
output prediction value when control action starting fromthe present time does not change Δu
119872(119896) is the prediction
incremental in119872 control horizon and y119875119872(119896) is the future119875-
step prediction output under 119872-step control action changeAmong them
y119875119872
(119896) =[[
[
1199101119875119872
(119896)
119910
119901119875119872(119896)
]]
]
y1198750(119896) =
[[
[
11991011198750
(119896)
119910
1199011198750(119896)
]]
]
Δu119872(119896) =
[[
[
Δ1199061119872
(119896)
Δ119906
119898119872(119896)
]]
]
A =[[
[
11986011
sdot sdot sdot 1198601119898
d
1198601199011
sdot sdot sdot 119860119901119898
]]
]
(31)
332 Receding Horizon Optimization In the receding hori-zon optimization process control increment can be obtainedin every execution cycle by minimizing the following perfor-mance index
minΔu119872
(119896)
119869 (119896) =1003817100381710038171003817w (119896) minus y
119875119872(119896)
1003817100381710038171003817
2
Q + 120576 (119896)2
S
+1003817100381710038171003817u
119872(119896) minus u
infin
1003817100381710038171003817
2
T +1003817100381710038171003817Δu
119872(119896)
1003817100381710038171003817
2
R
(32)
Subject to the modely
119875119872(119896) = y
1198750(119896) + AΔu
119872(119896) (33)
Subject to bound constraintsymin minus 120576 le y
119875119872(119896) le ymax + 120576
umin le u119872le umax
Δumin le Δu119872(119896) le Δumax
(34)
where 120576 denotes the slack variables guaranteeing the feasibil-ity of theDMCoptimization and119908(119896) = [119908
1(119896) 119908
119901(119896)]
119879
is the setpoint of controlled output obtained from upper SSOlayer Q R are the weight coefficient matrix
Q = block-diag (1198761 119876
119901)
Q119894= diag (119902
119894(1) 119902
119894(119875)) 119894 = 1 119901
R = block-diag (1198771 119877
119898)
R119895= diag (119903
119894(1) 119903
119894(119872)) 119895 = 1 119898
(35)
Through the necessary conditions of extreme value120597119869120597Δ119906
119872(119896) = 0 the optimal increment of control input can
be obtained
Δu119872(119896) = (A119879QA + R)
minus1
A119879Q [w (119896) minus y1198750(119896)] (36)
The instant increment can be calculated as follows
Δu (119896) = LD [w (119896) minus y1198750(119896)] (37)
whereD = (A119879QA +R)minus1A119879Q remark the operation of onlythe first element with
119871 = [
[
1 0 sdot sdot sdot 0 0
d0 1 0 sdot sdot sdot 0
]
]
(38)
333 Feedback Correction The difference between the pro-cess sample values by the present moment 119896 and predictionvalues of (30) is
119890 (119896 + 1) =[[
[
1198901(119896 + 1)
119890
119901(119896 + 1)
]]
]
=[[
[
1199101(119896 + 1) minus 119910
11(119896 + 1 | 119896)
119910
119901(119896 + 1) minus 119910
1199011(119896 + 1 | 119896)
]]
]
(39)
where 1199101198941(119896+1 | 119896) is the first element of 119910
119894119875119872(119896+1 | 119896) and
the corrected output prediction value can be obtained usingthe error vector that is
119873 moment initial prediction value when all of the inputremained unchanged at the time 119896 y
1198731(119896) is the future
119873 moment output prediction value under one-step controlinput action 119867 is the error correct matrix Then using ashift matrix 119878 next time the initial prediction value can beobtained which is
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
calculation can be described as the following linear program(LP) problem
minΔ119906infin
(119896)
119869 = 119888119879Δ119906
infin(119896)
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(22)
where 119866119906 119866
119891are the steady-state gain matrices of control
input and disturbance variables and 119890 is the model bias119906min 119906max are low limit and upper limit of steady-state inputvariables 119910min 119910max are low limit and upper limit of steadystate output variables
The global-optimization problem of steady-state targetcalculation can be described as the following quadraticprogram (QP) problem
minΔ119906infin
(119896)
119869 = 119888119879(Δ119906
infin(119896) minusMaxprofit)2
st Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906min le 119906infin(119896) + Δ119906
infin(119896) le 119906max
119910min le 119910infin(119896) + Δ119910
infin(119896) le 119910max
(23)
where Maxprofit is the potential maximum economic profit
322 Feasibility Judgment and Soft Constraint AdjustmentMathematically optimization feasibility is the existence prob-lem of the optimal solution Feasibility of steady-state targetcalculation means that optimal steady state of input-outputshould meet their operating constraints if feasible solutiondoes not exist the optimization calculation has no solutionThe solving process is as follows first judge the existence ofspace domain formed by the constraints and if there is init for optimization if does not exist then through the softconstraints adjustment to obtain the feasible space domainand then to solve
Soft constraints adjustment is an effective way to solveinfeasible optimization [22 23] By relaxing the outputconstraints within the hard constraints increasing the opti-mization problem feasible region that feasible solution to beoptimized Hard constraints refer to unalterable constraintslimited by the actual industrial process
Engineering standards of the priority strategy of softconstraints adjustment are the following give priority tomeetthe highly important operating constraints and allow less
important operating constraints to be violated appropriatelyunder the premise of satisfying the engineering constraints
Considering the following constraints (24) constituted bysteady-state model input constraints and output constraintscontaining slack variables the priority rank is ldquo119873rdquo where
Δ119910infin(119896) = 119866
119906Δ119906
infin(119896) + 119866
119891Δ119891
infin(119896) + 119890
119906119871119871le 119906
infin(119896) + Δ119906
infin(119896) le 119906
119867119871
119910119895
119871119871minus 120576
119895
2le 119910
infin(119896) + Δ119910
infin(119896) le 119910
119895
119867119871+ 120576
119895
1
120576119895
1ge 0 120576
119895
2ge 0
120576119895
1le 119910
119867119867119871minus 119910
119867119871
120576119895
2le 119910
119871119871minus 119910
119871119871119871
119895 = 1 119873
(24)
The algorithm steps of feasibility judgment and soft con-straint adjustment based on the priority strategy are asfollows
Step 1 Initialization according to the characteristics of theoutput variables and process conditions set the upper andlower output constraints priority ranks the same priorityrank setting adjustments according to actual situation con-straint weights
Step 2 According to the priority ranks judge the feasibilityand adjust the soft constraints in accordance with the ranksfrom large to small Under a larger priority rank if cannotfind a feasible solution the constraints of the rank will berelaxed to hard constraints and then consider less priorityrank constraints until we find a feasible solution
Step 3 Then the steady-state target calculation entered thestage of economy optimization or target tracking
For Step 2 constraints of the highest priority rank119873 areadjusted first by solving the following optimization problem
Solving (25) may appear in three different cases respectivelyif (25) is feasible and the optimum solution is 120576119873
= 0 subjectto 119869 = 0 that is no need for soft constraints adjustmentdirectly solve the original problem (22) if (25) is feasible but120576
119873= 0 just need to relax constraints of priority ranks119873 and
further optimization solution if (25) is infeasible not get afeasible solution to soft constraints adjustment of the priorityrank119873 relaxing the constraints of the priority rank119873 to hardconstraints that is
120576119873
1= 119910
119873
119867119867119871minus 119910
119873
119867119871
120576119873
2= 119910
119873
119871119871minus 119910
119873
119871119871119871
(27)
Go to the procedure of judging rank119873 minus 1 constraints
min120576119873minus1
119869 = (119882119873minus1
)
119879
120576119873minus1
(119882119873minus1
)
119879
= [119882119873minus1
1 119882
119873minus1
2times119899119873minus1
]
st Θ119873minus1
119885119873minus1
= 119887119873minus1
Ω119873minus1
119885119873minus1
le Ψ119873minus1
(28)
For (28) the matrix form is the same with priority rank 119873only in the corresponding position of 120576119873minus1 to replace 120576119873 119887119873minus1
matrix is adjusted
119887119873minus1
=
[[[[[[[[[[[[[[[[[
[
119906119867119871
minus 119906119871119871
1198661
119906119906
infin(119896) minus 119866
1
119906119906
119871119871(119896) + 119910
1
119867119871minus 119910
1
infin(119896) minus 119866
1
119891Δ119891
infin(119896) minus 119890
1
119866
119873
119906119906
infin(119896) minus 119866
119873
119906119906
119871119871(119896) + 119910
119873
119867119871minus 119910
119873
infin(119896) minus 119866
119873
119891Δ119891
infin(119896) minus 119890
119873+ (119910
119873
119867119867119871minus 119910
119873
119867119871)
119879
1198661
119906119906
119867119871minus 119866
1
119906119906
infin(119896) + 119910
1
infin(119896) + 119866
1
119891Δ119891
infin(119896) minus 119910
1
119871119871+ 119890
1
119866
119873
119906119906
119867119871minus 119866
119873
119906119906
infin(119896) + 119910
119873
infin(119896) + 119866
119873
119891Δ119891
infin(119896) minus 119910
119873
119871119871+ 119890
119873+ (119910
119873
119871119871minus 119910
119873
119871119871119871)
119879
0
0
]]]]]]]]]]]]]]]]]
]
(29)
Journal of Applied Mathematics 7
119873minus1 rank and119873 rank are the same for the soft constraintsadjustment processing until the end of constraint adjustmentof the priority rank 1 If all ranks of constraints are relaxed tothe hard constrain and a feasible solution still canrsquot be foundthen the original problem of soft constraints adjustment isinfeasible and needs to be redesigned
33 Dynamic Controller Design In engineering applicationsdynamic matrix control (DMC) algorithm is one of the mostwidely used MPC algorithms based on the step responsemodel of the plant This paper adopts DMC and steady-statetarget calculation integration strategy
The difference is that the general DMC algorithms haveno requirements on the steady-state position of the controlinput and they only require the controlled output as close aspossible to arrive at its set point However the integrationstrategy DMC requires both input and output variables toapproach their steady-state targets (u
119904 y
119904) as far as possible
The algorithm has three basic characteristics predictivemodel receding horizon optimization and feedback correc-tion [24]
331 Predictive Model Based on system process step re-sponse model at the current time 119896 the future 119875-stepprediction output can be written as follows
y119875119872
(119896) = y1198750(119896) + AΔu
119872(119896) (30)
where 119875 denotes the prediction horizon 119872 is the con-trol horizon A is the prediction matrix composed by thecorresponding step response coefficients y
1198750is the initial
output prediction value when control action starting fromthe present time does not change Δu
119872(119896) is the prediction
incremental in119872 control horizon and y119875119872(119896) is the future119875-
step prediction output under 119872-step control action changeAmong them
y119875119872
(119896) =[[
[
1199101119875119872
(119896)
119910
119901119875119872(119896)
]]
]
y1198750(119896) =
[[
[
11991011198750
(119896)
119910
1199011198750(119896)
]]
]
Δu119872(119896) =
[[
[
Δ1199061119872
(119896)
Δ119906
119898119872(119896)
]]
]
A =[[
[
11986011
sdot sdot sdot 1198601119898
d
1198601199011
sdot sdot sdot 119860119901119898
]]
]
(31)
332 Receding Horizon Optimization In the receding hori-zon optimization process control increment can be obtainedin every execution cycle by minimizing the following perfor-mance index
minΔu119872
(119896)
119869 (119896) =1003817100381710038171003817w (119896) minus y
119875119872(119896)
1003817100381710038171003817
2
Q + 120576 (119896)2
S
+1003817100381710038171003817u
119872(119896) minus u
infin
1003817100381710038171003817
2
T +1003817100381710038171003817Δu
119872(119896)
1003817100381710038171003817
2
R
(32)
Subject to the modely
119875119872(119896) = y
1198750(119896) + AΔu
119872(119896) (33)
Subject to bound constraintsymin minus 120576 le y
119875119872(119896) le ymax + 120576
umin le u119872le umax
Δumin le Δu119872(119896) le Δumax
(34)
where 120576 denotes the slack variables guaranteeing the feasibil-ity of theDMCoptimization and119908(119896) = [119908
1(119896) 119908
119901(119896)]
119879
is the setpoint of controlled output obtained from upper SSOlayer Q R are the weight coefficient matrix
Q = block-diag (1198761 119876
119901)
Q119894= diag (119902
119894(1) 119902
119894(119875)) 119894 = 1 119901
R = block-diag (1198771 119877
119898)
R119895= diag (119903
119894(1) 119903
119894(119872)) 119895 = 1 119898
(35)
Through the necessary conditions of extreme value120597119869120597Δ119906
119872(119896) = 0 the optimal increment of control input can
be obtained
Δu119872(119896) = (A119879QA + R)
minus1
A119879Q [w (119896) minus y1198750(119896)] (36)
The instant increment can be calculated as follows
Δu (119896) = LD [w (119896) minus y1198750(119896)] (37)
whereD = (A119879QA +R)minus1A119879Q remark the operation of onlythe first element with
119871 = [
[
1 0 sdot sdot sdot 0 0
d0 1 0 sdot sdot sdot 0
]
]
(38)
333 Feedback Correction The difference between the pro-cess sample values by the present moment 119896 and predictionvalues of (30) is
119890 (119896 + 1) =[[
[
1198901(119896 + 1)
119890
119901(119896 + 1)
]]
]
=[[
[
1199101(119896 + 1) minus 119910
11(119896 + 1 | 119896)
119910
119901(119896 + 1) minus 119910
1199011(119896 + 1 | 119896)
]]
]
(39)
where 1199101198941(119896+1 | 119896) is the first element of 119910
119894119875119872(119896+1 | 119896) and
the corrected output prediction value can be obtained usingthe error vector that is
119873 moment initial prediction value when all of the inputremained unchanged at the time 119896 y
1198731(119896) is the future
119873 moment output prediction value under one-step controlinput action 119867 is the error correct matrix Then using ashift matrix 119878 next time the initial prediction value can beobtained which is
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Solving (25) may appear in three different cases respectivelyif (25) is feasible and the optimum solution is 120576119873
= 0 subjectto 119869 = 0 that is no need for soft constraints adjustmentdirectly solve the original problem (22) if (25) is feasible but120576
119873= 0 just need to relax constraints of priority ranks119873 and
further optimization solution if (25) is infeasible not get afeasible solution to soft constraints adjustment of the priorityrank119873 relaxing the constraints of the priority rank119873 to hardconstraints that is
120576119873
1= 119910
119873
119867119867119871minus 119910
119873
119867119871
120576119873
2= 119910
119873
119871119871minus 119910
119873
119871119871119871
(27)
Go to the procedure of judging rank119873 minus 1 constraints
min120576119873minus1
119869 = (119882119873minus1
)
119879
120576119873minus1
(119882119873minus1
)
119879
= [119882119873minus1
1 119882
119873minus1
2times119899119873minus1
]
st Θ119873minus1
119885119873minus1
= 119887119873minus1
Ω119873minus1
119885119873minus1
le Ψ119873minus1
(28)
For (28) the matrix form is the same with priority rank 119873only in the corresponding position of 120576119873minus1 to replace 120576119873 119887119873minus1
matrix is adjusted
119887119873minus1
=
[[[[[[[[[[[[[[[[[
[
119906119867119871
minus 119906119871119871
1198661
119906119906
infin(119896) minus 119866
1
119906119906
119871119871(119896) + 119910
1
119867119871minus 119910
1
infin(119896) minus 119866
1
119891Δ119891
infin(119896) minus 119890
1
119866
119873
119906119906
infin(119896) minus 119866
119873
119906119906
119871119871(119896) + 119910
119873
119867119871minus 119910
119873
infin(119896) minus 119866
119873
119891Δ119891
infin(119896) minus 119890
119873+ (119910
119873
119867119867119871minus 119910
119873
119867119871)
119879
1198661
119906119906
119867119871minus 119866
1
119906119906
infin(119896) + 119910
1
infin(119896) + 119866
1
119891Δ119891
infin(119896) minus 119910
1
119871119871+ 119890
1
119866
119873
119906119906
119867119871minus 119866
119873
119906119906
infin(119896) + 119910
119873
infin(119896) + 119866
119873
119891Δ119891
infin(119896) minus 119910
119873
119871119871+ 119890
119873+ (119910
119873
119871119871minus 119910
119873
119871119871119871)
119879
0
0
]]]]]]]]]]]]]]]]]
]
(29)
Journal of Applied Mathematics 7
119873minus1 rank and119873 rank are the same for the soft constraintsadjustment processing until the end of constraint adjustmentof the priority rank 1 If all ranks of constraints are relaxed tothe hard constrain and a feasible solution still canrsquot be foundthen the original problem of soft constraints adjustment isinfeasible and needs to be redesigned
33 Dynamic Controller Design In engineering applicationsdynamic matrix control (DMC) algorithm is one of the mostwidely used MPC algorithms based on the step responsemodel of the plant This paper adopts DMC and steady-statetarget calculation integration strategy
The difference is that the general DMC algorithms haveno requirements on the steady-state position of the controlinput and they only require the controlled output as close aspossible to arrive at its set point However the integrationstrategy DMC requires both input and output variables toapproach their steady-state targets (u
119904 y
119904) as far as possible
The algorithm has three basic characteristics predictivemodel receding horizon optimization and feedback correc-tion [24]
331 Predictive Model Based on system process step re-sponse model at the current time 119896 the future 119875-stepprediction output can be written as follows
y119875119872
(119896) = y1198750(119896) + AΔu
119872(119896) (30)
where 119875 denotes the prediction horizon 119872 is the con-trol horizon A is the prediction matrix composed by thecorresponding step response coefficients y
1198750is the initial
output prediction value when control action starting fromthe present time does not change Δu
119872(119896) is the prediction
incremental in119872 control horizon and y119875119872(119896) is the future119875-
step prediction output under 119872-step control action changeAmong them
y119875119872
(119896) =[[
[
1199101119875119872
(119896)
119910
119901119875119872(119896)
]]
]
y1198750(119896) =
[[
[
11991011198750
(119896)
119910
1199011198750(119896)
]]
]
Δu119872(119896) =
[[
[
Δ1199061119872
(119896)
Δ119906
119898119872(119896)
]]
]
A =[[
[
11986011
sdot sdot sdot 1198601119898
d
1198601199011
sdot sdot sdot 119860119901119898
]]
]
(31)
332 Receding Horizon Optimization In the receding hori-zon optimization process control increment can be obtainedin every execution cycle by minimizing the following perfor-mance index
minΔu119872
(119896)
119869 (119896) =1003817100381710038171003817w (119896) minus y
119875119872(119896)
1003817100381710038171003817
2
Q + 120576 (119896)2
S
+1003817100381710038171003817u
119872(119896) minus u
infin
1003817100381710038171003817
2
T +1003817100381710038171003817Δu
119872(119896)
1003817100381710038171003817
2
R
(32)
Subject to the modely
119875119872(119896) = y
1198750(119896) + AΔu
119872(119896) (33)
Subject to bound constraintsymin minus 120576 le y
119875119872(119896) le ymax + 120576
umin le u119872le umax
Δumin le Δu119872(119896) le Δumax
(34)
where 120576 denotes the slack variables guaranteeing the feasibil-ity of theDMCoptimization and119908(119896) = [119908
1(119896) 119908
119901(119896)]
119879
is the setpoint of controlled output obtained from upper SSOlayer Q R are the weight coefficient matrix
Q = block-diag (1198761 119876
119901)
Q119894= diag (119902
119894(1) 119902
119894(119875)) 119894 = 1 119901
R = block-diag (1198771 119877
119898)
R119895= diag (119903
119894(1) 119903
119894(119872)) 119895 = 1 119898
(35)
Through the necessary conditions of extreme value120597119869120597Δ119906
119872(119896) = 0 the optimal increment of control input can
be obtained
Δu119872(119896) = (A119879QA + R)
minus1
A119879Q [w (119896) minus y1198750(119896)] (36)
The instant increment can be calculated as follows
Δu (119896) = LD [w (119896) minus y1198750(119896)] (37)
whereD = (A119879QA +R)minus1A119879Q remark the operation of onlythe first element with
119871 = [
[
1 0 sdot sdot sdot 0 0
d0 1 0 sdot sdot sdot 0
]
]
(38)
333 Feedback Correction The difference between the pro-cess sample values by the present moment 119896 and predictionvalues of (30) is
119890 (119896 + 1) =[[
[
1198901(119896 + 1)
119890
119901(119896 + 1)
]]
]
=[[
[
1199101(119896 + 1) minus 119910
11(119896 + 1 | 119896)
119910
119901(119896 + 1) minus 119910
1199011(119896 + 1 | 119896)
]]
]
(39)
where 1199101198941(119896+1 | 119896) is the first element of 119910
119894119875119872(119896+1 | 119896) and
the corrected output prediction value can be obtained usingthe error vector that is
119873 moment initial prediction value when all of the inputremained unchanged at the time 119896 y
1198731(119896) is the future
119873 moment output prediction value under one-step controlinput action 119867 is the error correct matrix Then using ashift matrix 119878 next time the initial prediction value can beobtained which is
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
119873minus1 rank and119873 rank are the same for the soft constraintsadjustment processing until the end of constraint adjustmentof the priority rank 1 If all ranks of constraints are relaxed tothe hard constrain and a feasible solution still canrsquot be foundthen the original problem of soft constraints adjustment isinfeasible and needs to be redesigned
33 Dynamic Controller Design In engineering applicationsdynamic matrix control (DMC) algorithm is one of the mostwidely used MPC algorithms based on the step responsemodel of the plant This paper adopts DMC and steady-statetarget calculation integration strategy
The difference is that the general DMC algorithms haveno requirements on the steady-state position of the controlinput and they only require the controlled output as close aspossible to arrive at its set point However the integrationstrategy DMC requires both input and output variables toapproach their steady-state targets (u
119904 y
119904) as far as possible
The algorithm has three basic characteristics predictivemodel receding horizon optimization and feedback correc-tion [24]
331 Predictive Model Based on system process step re-sponse model at the current time 119896 the future 119875-stepprediction output can be written as follows
y119875119872
(119896) = y1198750(119896) + AΔu
119872(119896) (30)
where 119875 denotes the prediction horizon 119872 is the con-trol horizon A is the prediction matrix composed by thecorresponding step response coefficients y
1198750is the initial
output prediction value when control action starting fromthe present time does not change Δu
119872(119896) is the prediction
incremental in119872 control horizon and y119875119872(119896) is the future119875-
step prediction output under 119872-step control action changeAmong them
y119875119872
(119896) =[[
[
1199101119875119872
(119896)
119910
119901119875119872(119896)
]]
]
y1198750(119896) =
[[
[
11991011198750
(119896)
119910
1199011198750(119896)
]]
]
Δu119872(119896) =
[[
[
Δ1199061119872
(119896)
Δ119906
119898119872(119896)
]]
]
A =[[
[
11986011
sdot sdot sdot 1198601119898
d
1198601199011
sdot sdot sdot 119860119901119898
]]
]
(31)
332 Receding Horizon Optimization In the receding hori-zon optimization process control increment can be obtainedin every execution cycle by minimizing the following perfor-mance index
minΔu119872
(119896)
119869 (119896) =1003817100381710038171003817w (119896) minus y
119875119872(119896)
1003817100381710038171003817
2
Q + 120576 (119896)2
S
+1003817100381710038171003817u
119872(119896) minus u
infin
1003817100381710038171003817
2
T +1003817100381710038171003817Δu
119872(119896)
1003817100381710038171003817
2
R
(32)
Subject to the modely
119875119872(119896) = y
1198750(119896) + AΔu
119872(119896) (33)
Subject to bound constraintsymin minus 120576 le y
119875119872(119896) le ymax + 120576
umin le u119872le umax
Δumin le Δu119872(119896) le Δumax
(34)
where 120576 denotes the slack variables guaranteeing the feasibil-ity of theDMCoptimization and119908(119896) = [119908
1(119896) 119908
119901(119896)]
119879
is the setpoint of controlled output obtained from upper SSOlayer Q R are the weight coefficient matrix
Q = block-diag (1198761 119876
119901)
Q119894= diag (119902
119894(1) 119902
119894(119875)) 119894 = 1 119901
R = block-diag (1198771 119877
119898)
R119895= diag (119903
119894(1) 119903
119894(119872)) 119895 = 1 119898
(35)
Through the necessary conditions of extreme value120597119869120597Δ119906
119872(119896) = 0 the optimal increment of control input can
be obtained
Δu119872(119896) = (A119879QA + R)
minus1
A119879Q [w (119896) minus y1198750(119896)] (36)
The instant increment can be calculated as follows
Δu (119896) = LD [w (119896) minus y1198750(119896)] (37)
whereD = (A119879QA +R)minus1A119879Q remark the operation of onlythe first element with
119871 = [
[
1 0 sdot sdot sdot 0 0
d0 1 0 sdot sdot sdot 0
]
]
(38)
333 Feedback Correction The difference between the pro-cess sample values by the present moment 119896 and predictionvalues of (30) is
119890 (119896 + 1) =[[
[
1198901(119896 + 1)
119890
119901(119896 + 1)
]]
]
=[[
[
1199101(119896 + 1) minus 119910
11(119896 + 1 | 119896)
119910
119901(119896 + 1) minus 119910
1199011(119896 + 1 | 119896)
]]
]
(39)
where 1199101198941(119896+1 | 119896) is the first element of 119910
119894119875119872(119896+1 | 119896) and
the corrected output prediction value can be obtained usingthe error vector that is
119873 moment initial prediction value when all of the inputremained unchanged at the time 119896 y
1198731(119896) is the future
119873 moment output prediction value under one-step controlinput action 119867 is the error correct matrix Then using ashift matrix 119878 next time the initial prediction value can beobtained which is
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Figure 3 Two-layer predictive framework of biodiesel process
where
119878 =
[[[[[
[
0 1 0
0 1
d d0 1
0 1
]]]]]
]119873lowast119873
(42)
4 Control System Design
In the biodiesel reactor control multiloops are necessaryto stabilize the plant One loop is needed to maintain theset point of specifying the product purity and another loopis needed to ensure an optimal yield of biodiesel and tominimize the generation of unwanted by-products even inthe presence of disturbances
To achieve these goals the control loop configurationsanalysis is meaningful Based on the analysis of Mjalli etal [4] the favorable pairings are as follows the biodieselconcentration (119862
119864) is maintained by manipulating reactant
flow rate (119865119900) the reactor temperature (119879) is maintained
by manipulating coolant flow rate (119865119888) respectively and the
effect of stirred rotational speed on the reactor output isinsignificant and it would be regarded as one of disturbancesto the control system The relative gain array (RGA) showsthat there are some interactions among the controlled andmanipulated variables which make two-layer predictive con-troller better qualified
Consequently the two-layer predictive controller isdesigned to handle a 2 times 2 system of inputs and outputs Thecontrolled output variables include biodiesel concentration(119862
119864) and reactor temperature (119879) the manipulated variables
include reactant flow rate (119865119900) and coolant flow rate (119865
119888) It
is very important for a reactor to handle the disturbancesin the feed concentration and initial temperatures as thesedisturbances heavily change the system performance
The design of the control loop based on the two-layerpredictive control strategy for the biodiesel reactor is shownin Figure 3 The SSO layer searches the optimal output set-points 119862
119864119904119904and 119879
119904119904according to the economic optimization
goal of the actual production process The MPC layer selectsthe real-time control actions Δ119906 to complete the dynamictracking control
5 Simulation Results and Analysis
51 Model Identification For the two-layer predictive controlscheme to be successful process modeling plays a key rolein capturing the varying dynamics of the system Section 4shows that the biodiesel process is a two-input two-outputmultivariable process The process nonlinear model was pro-grammed and simulated in Matlab as a function Simulationresults show system is open stable process
Firstly generalized binary noise (GBN) signal is selectedas the excitation signal GBN signals switch between 119886 and minus119886according to the following rules
119875 [119906 (119905) = minus119906 (119905 minus 1)] = 119901119904119908
119875 [119906 (119905) = 119906 (119905 minus 1)] = 1 minus 119901119904119908
(43)
where 119901119904119908
is transition probability 119879min is defined as thesampling time of the signal held constant 119879
119904119908is time interval
of twice conversion The average conversion time and powerspectrum are respectively
119864119879119904119908=
119879min119901
119904119908
Φ119906(120596) =
(1 minus 1199022) 119879min
1 minus 2119902 cos119879min120596 + 1199022 119902 = 1 minus 2119901
119904119908
(44)
Next least squares (LS) identification method is used toestimate the process model parameters Suppose an MIMOplant with 119898 input 119901 output for the 119894th output of the finiteimpulse response (FIR) model is described as
119910119894(119896) =
119898
sum
119895=1
119873
sum
119897=1
ℎ119894119895119897119906
119895(119896 minus 119897) (45)
Consider experimental tests of collecting input sequence
1199061(1) 119906
1(2) sdot sdot sdot 119906
1(119871)
119906119898(1) 119906
119898(2) sdot sdot sdot 119906
119898(119871)
(46)
and output sequence
1199101(1) 119910
1(2) sdot sdot sdot 119910
1(119871)
119910119901(1) 119910
119901(2) sdot sdot sdot 119910
119901(119871)
(47)
Journal of Applied Mathematics 9
0 100 200 300 400 500
0
005
01
Samples
minus005
minus01
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500Samples
0
01
02
03
04
Rela
tive e
rror
(b)
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Figure 4 Biodiesel concentration prediction result and relative error under reactor flow rate 119865119900action
0 100 200 300 400 500
0
2
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
05
1
Samples
minus05
Rela
tive e
rror
(b)
Figure 5 Reactor temperature prediction result and relative error under reactor flow rate 119865119900action
Consider matching between data and models the intro-duction of residuals for each output can be independentlyexpressed as follows
119910119894(119896) = 120593 (119896) 120579
119894+ 119890 (119896) (48)
Matrix form is written as
119910119894= Φ120579
119894+ 119890 (49)
where
119910119894=
[[[[
[
119910119894(119873 + 1)
119910119894(119873 + 2)
119910
119894(119871)
]]]]
]
119890 =
[[[[
[
119890 (119873 + 1)
119890 (119873 + 2)
119890 (119871)
]]]]
]
Φ =
[[[[
[
1199061(119873) 119906
1(119873 minus 1) sdot sdot sdot 119906
1(1) 119906
119898(119873) 119906
119898(119873 minus 1) sdot sdot sdot 119906
119898(1)
1199061(119873 + 1) 119906
1(119873) sdot sdot sdot 119906
1(2) 119906
119898(119873 + 1) 119906
119898(119873) sdot sdot sdot 119906
119898(2)
sdot sdot sdot
119906
1(119871 minus 1) 119906
1(119871 minus 2) 119906
1(119871 minus 119873) 119906
119898(119871 minus 1) 119906
119898(119871 minus 2) 119906
119898(119871 minus 119873)
]]]]
]
(50)
Minimize the squared residuals
min 119869 = 119890119879119890 = [119910 minus Φ120579]
119879
[119910 minus Φ120579] (51)
Obtain the optimal estimate
120579 = [Φ
119879Φ]
minus1
Φ119879119910 (52)
For themodel predictive controller design the FIRmodelof system identification needs to be further converted intofinite step response (FSR) model The relationship betweenFSR coefficients and FIR coefficients is as follows
119892119895=
119895
sum
119894=1
ℎ119895 (53)
10 Journal of Applied Mathematics
0 100 200 300 400 500
0
001
002
Samples
minus002
minus001
Con
cent
ratio
nCE
(km
olm
3)
Predictive valueActual value
(a)
0 100 200 300 400 500
0
02
04
Samples
minus02
minus04
Rela
tive e
rror
(b)
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Figure 6 Biodiesel concentration prediction result and relativeerror under reactor flow rate 119865
119888action
Coefficients matrix of FSR is
119866119906
119897=
[[[[
[
11990411119897
11990412119897
sdot sdot sdot 1199041119898119897
11990421119897
11990422119897
sdot sdot sdot 1199042119898119897
d
1199041199011119897
1199041199012119897
sdot sdot sdot 119904119901119898119897
]]]]
]
(54)
Finally (11)ndash(19) are used to create a steady-state mathe-matical model of two-layer prediction control The concretesimulation process is as follows
In the work GBN as the excitation signal was added tothe model input to produce output data The parameters ofGBN signal applied to the first input are 119879
119904119908= 65 119886119898119901 = 01
the parameters of GBN applied to the second input are 119879119904119908=
65 119886119898119901 = 0005 both the conversion probabilities are takento be 119875
119904119908= 1119879
119904119908 Simulation time 119905 = 2000 s and sample
time equals 2 s under each input excitation correspondingto two sets of output data each set of data capacity is 1000Among them the former 500 data as model identificationthe remaining data are used as model validations and FSRmodel length value is taken as 200
Under the action of two inputs reactant flow rate 119865119900
and coolant flow rate 119865119888 respectively predicted value actual
value and the relative error of two outputs biodiesel concen-tration 119862
119864and reactor temperature 119879 were shown in Figures
4 5 6 and 7 Figures 4ndash7 show that relative error is smallenough and the model can describe 119862
119864and 119879 change trends
under 119865119900and 119865
119888
Figures 8 and 9 give the two output step response curvesunder two input 119865
119900 119865
119888action respectively further shows the
multiple-input multiple-output system is open-loop stable
0 100 200 300 400 500
0
2
4
Samples
minus4
minus2
Reac
tor t
empe
ratu
reT
(K)
Predictive valueActual value
(a)
0 100 200 300 400 5000
01
02
03
04
SamplesRe
lativ
e err
or(b)
Figure 7 Reactor temperature prediction result and relative errorunder reactor flow rate 119865
119888action
0 50 100 150 200
0
Samples
minus15
minus1
minus05
Step
resp
onse
g11
(a)
0 50 100 150 2000
20
40
60
80
Samples
Step
resp
onse
g21
(b)
Figure 8 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119900action
and the step response model has been identified successfullyThe FSRmodel will be utilized to represent the actual processin latter optimization and controller design
52 Dynamic Simulation To validate the effectiveness andimmunity in two-layer predictive control the models ob-tained in Section 51 are used in the simulations
Journal of Applied Mathematics 11
0 50 100 150 200
0
Samples
minus8
minus6
minus4
minus2
Step
resp
onse
g12
(a)
0 50 100 150 200
0
Samples
minus1000
minus800
minus600
minus400
minus200
Step
resp
onse
g22
(b)
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Figure 9 Step response curve of biodiesel concentration and reactortemperature respectively under 119865
119888action
The reaction rate constants come from [18] under thecommon industrial conditions of 6 1 methanoloil moleratio 10 wt catalyst KOH and 600 rpm stirrer rotationalspeed These kinetics parameters can be considered as con-stants The initial operating conditions refer to the literature[4] the validated data According to these parameters andreaction conditions the simulation of biodiesel transesteri-fication reactor can be carried out
The economic optimization method described in (22)is adopted as SSO whose main parameters are selected asfollows the cost coefficients of control input in steady-stateoptimization are set to [1 minus1] the input 119865
119900is constrained
between 0 and 02m3s the input 119865119888is constrained between
0 and 01m3s and the output 119862119864is constrained between
30536 kmolm3 and 3196 kmolm3 the output 119879 is con-strained between 33777 K and 33825 K
The parameters of the dynamic control layer adopted theunconstrained DMC algorithm the modeling time domain119873 = 200 prediction horizon 119875 = 200 control horizon119872 =
20 The weight coefficient values of weight matrix 119876 and 119877equal to 10 and 1000 respectively
Conventional PID controller has also been designed inthis simulation for comparison of performance to two-layerpredictive controllerThe parameters of PID controller for119862
119864
with 119865119900control loop are 119896119901 = minus6119890minus5 119896119894 = minus005 and 119896119889 = 0
the parameters for 119879 with 119865119888control loop are 119896119901 = minus002
119896119894 = minus0001 and 119896119889 = 0 The simulations of general PIDcontroller and two-layer predictive controller are comparedto validate the performance of the latter algorithm whoseresults are shown in Figures 10 and 11
As Figures 10 and 11 show the two-layer predictivecontroller starts running at the time 119905 = 0 The results ofsteady state optimization are
119910119904119904= [3196 33777] 119906
119904119904= [0073 00062] (55)
0 500 1000 1500 2000312
314
316
318
32
322
324
326
328
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
0 500 1000 1500 2000
0
002
004
006
008
01
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
minus002
(b)
Figure 10 Biodiesel concentration and controller moves of two-layer predictive controller and PID controller
The optimized values as the setpoints were send to thelower layer DMC In the beginning the closed loop responseof the two-layer predictive controller was a little sluggishin bringing the biodiesel concentration back the optimumsteady-state values this is because that the algorithm enterthe constraint adjustment stage based on the priority strategywhich adjusting the upper limit and lower limit to be handledAbout At the time 119905 = 400 the response gradually becomesstable It can be seen that the two-layer predictive controllerpreceded the PID controller in terms of the ability to attainlower overshoot smaller oscillation and faster response time
Considering the actual application the control input isalso an important indicator of good or bad controller FromFigures 10 and 11 the two-layer predictive controller hasmuchmore stable controller moves than does PID that meets thepractical implementation constrains
12 Journal of Applied Mathematics
0 500 1000 1500 2000
33775
3378
33785
3379
33795
338
33805
3381
33815
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
0 500 1000 1500 20003
4
5
6
7
8
9
10
Time (s)
Con
trolle
r mov
es
Two-layer predictive controllerPID controller
times10minus3
(b)
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Figure 11 Reactor temperature and controller moves of two-layerpredictive controller and PID controller
To challenge the stability of two-layer predictive con-troller some disturbances were exerted alone and at the sametimeThe chosen disturbance variables include coolant inputtemperature (119879
1198880) feed temperature (119879
0) triglyceride initial
concentration (119862TG0) and stirrer rotational speed (119873) After
the system has attained the steady state The nominal valuesof 119879
1198880 119879
0were increased 3K respectively and 119862TG0
119873 wereincreased 5 respectively at the time 119905 = 1000 s Figures12 and 13 show the biodiesel concentration and reactortemperature profiles when these disturbance variables wereintroduced
Figures 12 and 13 showed satisfactory rejection of alldisturbances Two-layer predictive controller was able tobring back the controlled variables to their setpoints in lessthan 1000 s and overshoot was within the acceptable rangeFor the biodiesel concentration loop the initial concentration
800 1000 1200 1400 1600 1800 2000316
317
318
319
32
321
322
323
324
Time (s)
Biod
iese
l con
cent
ratio
n (k
mol
m3)
(a)
800 1000 1200 1400 1600 1800 2000004
006
008
01
012
014
016
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 12 Biodiesel concentration and controller moves of fourindividual disturbance variables effects
119862TG0has the highest effect with an overshoot of less than
001 kmolm3 For the reactor temperature loop the feedtemperature 119879
0has the largest effect with an overshoot of
less than 033 K For the two loops the stirrer rotational speedalmost has no effect on the controlled variables
6 Conclusions
Biodiesel transesterification reactor control has become veryimportant in recent years due to the difficulty in controllingthe complex and highly nonlinear dynamic behavior Inthis paper a novel two-layer predictive control scheme fora continuous biodiesel transesterification reactor has beenproposed The SSO layer achieved optimal output setpointsaccording to the local economic optimization goal of theactual production process and the MPC layer realized the
Journal of Applied Mathematics 13
800 1000 1200 1400 1600 1800 20003376
33765
3377
33775
3378
33785
3379
33795
338
33805
3381
Time (s)
Reac
tor t
empe
ratu
re (K
)
(a)
800 1000 1200 1400 1600 1800 20000004
0006
0008
001
0012
0014
0016
0018
002
Time (s)
Con
trolle
r mov
es
Tc0 increase 3KT0 increase 3K
CTG0 increase 5N increase 5
(b)
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
Figure 13 Reactor temperature and controller moves of fourindividual disturbance variables effects
dynamic tracking controlThemain aim was to optimize andcontrol the biodiesel concentration and reactor temperaturein order to obtain the product of the highest quality at thelower cost With steady-state optimum target calculation andDMCalgorithm implement the performance of the two-layerpredictive controller was superior to that of a conventionalPID controller The two-layer predictive control is not onlystable but also tracks set points more efficiently with minimalovershoots and shorter settling times Moreover it exhibitsgood disturbance rejection characteristics
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (61034008) and the Science Research
Foundation of Liaoning Provincial Department of Education(L2012145)
References
[1] D Y C Leung X Wu and M K H Leung ldquoA reviewon biodiesel production using catalyzed transesterificationrdquoApplied Energy vol 87 no 4 pp 1083ndash1095 2010
[2] S Shahla N G Cheng and R Yusoff ldquoAn overview ontransesterification of natural oils and fatsrdquo Biotechnology andBioprocess Engineering vol 15 no 6 pp 891ndash904 2010
[3] N N A N Yusuf S K Kamarudin and Z Yaakub ldquoOverviewon the current trends in biodiesel productionrdquo Energy Conver-sion and Management vol 52 no 7 pp 2741ndash2751 2011
[4] F S Mjalli L K San K C Yin and M A Hussain ldquoDynamicsand control of a biodiesel transesterification reactorrdquo ChemicalEngineering and Technology vol 32 no 1 pp 13ndash26 2009
[5] T Eevera K Rajendran and S Saradha ldquoBiodiesel produc-tion process optimization and characterization to assess thesuitability of the product for varied environmental conditionsrdquoRenewable Energy vol 34 no 3 pp 762ndash765 2009
[6] C S Bildea andA A Kiss ldquoDynamics and control of a biodieselprocess by reactive absorptionrdquo Chemical Engineering Researchand Design vol 89 no 2 pp 187ndash196 2011
[7] L Zong S Ramanathan and C-C Chen ldquoFragment-basedapproach for estimating thermophysical properties of fats andvegetable oils for modeling biodiesel production processesrdquoIndustrial and Engineering Chemistry Research vol 49 no 2 pp876ndash886 2010
[8] Y K Ho F S Mjalli and H K Yeoh ldquoMultivariable adaptivepredictive model based control of a biodiesel transesterificationreactorrdquo Journal of Applied Sciences vol 10 no 12 pp 1019ndash10272010
[9] H Y Kuen F SMjalli andYHKoon ldquoRecursive least squares-based adaptive control of a biodiesel transesterification reactorrdquoIndustrial and Engineering Chemistry Research vol 49 no 22pp 11434ndash11442 2010
[10] W A Wali A I Al-Shamma K H Hassan and J D CullenldquoOnline genetic-ANFIS temperature control for advancedmicrowave biodiesel reactorrdquo Journal of Process Control vol 22pp 1256ndash1272 2012
[11] P T Benavides and U Diwekar ldquoOptimal control of biodieselproduction in a batch reactormdashpart I deterministic controlrdquoFuel vol 94 pp 211ndash217 2012
[12] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquoControl Engineering Practice vol11 no 7 pp 733ndash764 2003
[13] T A Johansen and A Grancharova ldquoApproximate explicitconstrained linear model predictive control via orthogonalsearch treerdquo IEEE Transactions on Automatic Control vol 48no 5 pp 810ndash815 2003
[14] T Zou B C Ding and D Zhang Model Predictive ControlEngineering Applications Introduction Chemical Industry PressBeijing China 2010
[15] A Nikandrov and C L E Swartz ldquoSensitivity analysis of LP-MPC cascade control systemsrdquo Journal of Process Control vol19 no 1 pp 16ndash24 2009
[16] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997
14 Journal of Applied Mathematics
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
[17] A-F Chang and Y A Liu ldquoIntegrated process modeling andproduct design of biodiesel manufacturingrdquo Industrial andEngineering Chemistry Research vol 49 no 3 pp 1197ndash12132010
[18] G Vicente M Martınez and J Aracil ldquoKinetics of Brassicacarinata oil methanolysisrdquo Energy and Fuels vol 20 no 4 pp1722ndash1726 2006
[19] R Scattolini ldquoArchitectures for distributed and hierarchicalModel Predictive Controlmdasha reviewrdquo Journal of Process Controlvol 19 no 5 pp 723ndash731 2009
[20] T ZouHQ Li B CDing andDDWang ldquoCompatibility anduniqueness analyses of steady state solution for multi-variablepredictive control systemsrdquo Acta Automatica Sinica vol 39 pp519ndash529 2013
[21] D E Kassmann T A Badgwell and R B Hawkins ldquoRobuststeady-state target calculation for model predictive controlrdquoAIChE Journal vol 46 no 5 pp 1007ndash1024 2000
[22] Y G Xi and H Y Gu ldquoFeasibility analysis of constrainedmulti-objective multi-degree-of-freedom optimization controlin industrial processesrdquoActaAutomatica Sinica vol 24 pp 727ndash732 1998
[23] T Zou H Q Li X X Zhang Y Gu and H Y Su ldquoFeasibilityand soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systemsrdquo in Proceedings ofthe International Symposium on Advanced Control of IndustrialProcesses (ADCONIP rsquo11) pp 524ndash529 May 2011
[24] Y G Xi Predictive Control National Defense Industry PressBeijing China 1993
