Abstract Disturbance rejection is one of the most chal-lenging issues when the system under control is nonlinearand little a priori information is available about the system.Internal model control (IMC) has been extensively used fordisturbance rejection but has certain drawbacks. Most of thework reported in literature deals with additive output distur-bance. The main focus of this study is on multiplicative inputdisturbance. In this work, fuzzy model-free adaptive control(FMAC) is used to reject the disturbance in an uncertain non-linear plant. Different schemes have been investigated forrejecting the disturbance. It is demonstrated that the particu-lar type of disturbance cannot be completely rejected usingthe IMC. The second methodology used to reject the dis-turbance is feedforward of the measured disturbance. Feed-forward of the input disturbance is used which is able tocounteract the effect of the disturbances but resulting in anincrease in the control activity. The control activity is relatedto the noise on the sensor measuring the input disturbance.The FMAC is modelled as a fuzzy relational model (FRM)which is able to represent the noise level in the fuzzy controlsignal. Conditional defuzzification is applied on the result-ing fuzzy control signal; which is able to reduce the controlactivity while maintaining the controlled output at the desiredlevel. FMAC is tested with a modified version of the Ham-merstein Model. The control performance demonstrates theeffectiveness of the proposed novel methodology in reject-ing the input multiplicative disturbance while reducing thecontrol activity.
M. B. Kadri (B)Electronics and Power Engineering Department, Pakistan NavyEngineering College, National University of Sciences and Technology,Habib Ibrahim Rehmatullah Road, 75350 Karachi, Pakistane-mail: [email protected]; [email protected]
Nonlinear uncertain plants pose a challenging problem incontrol engineering. Uncertainty can be either in the relation-ship of the system variables or in the measurements. Fuzzytheory introduced by Zadeh [1] has been used to quantifyuncertainties such that it can be effectively utilized to con-trol nonlinear processes.
123
2382 Arab J Sci Eng (2014) 39:2381–2392
Model-free control is synonymous to direct control, simi-larly model-based control is also referred to as indirect con-trol in the adaptive control literature. As the name implies thedirect controller tries to control the plant directly, i.e. withoutdeveloping a model of the plant or introducing any interme-diate blocks between the plant and the controller. The termmodel free does not imply that there is absolutely no modelwhatsoever in the whole system but it just signifies that thereis no model of the plant in the system. A model of the plant isneither available a priori nor does the system try to developa model of the plant under control during the learning phase.Developing a model is usually not suitable for online pur-poses due to extra computation incurred by the process [2].Adaptive model-free control algorithms have just one goal toachieve, i.e. to minimize the error between the setpoint andthe plant output if it is a tracking problem or to maintain thesetpoint at a constant level for regulator problems.
Takagi-Sugeno Models (TS) and fuzzy relational models(FRM) have been extensively used for modelling and controlof nonlinear plants [3,4]. Although the FRM and Takagi-Sugeno (TS) models are mathematically equivalent undercertain conditions [5] FRMs are best suited when the controlloop is affected by severe noise due to uncertainties in themeasurements. The major advantage of FRM as comparedto the TS models is the representation of the uncertainty inthe output produced by the controller [6]. The paper dis-cusses the impact of uncertainty in the measurement of thecontrolled variable on the control performance. The controlactivity is directly related to the amount of senor noise in thecontrol loop. Conditional defuzzification has been applied toreduce the control activity. Novel methodologies based onconditional defuzzification have been proposed to reduce thecontrol activity.
Most of the control strategies reported in literature dealwith additive output disturbance. A plethora of control strate-gies have been reported in literature which guarantees robustcontrol performance in the presence of noise and additive dis-turbance [7–9]. Graebe et al. [10] has discussed input distur-bance rejection with a nonlinear IMC control setting but theinput disturbance considered is additive. This work focuseson multiplicative input disturbance which changes the gin ofthe system as the disturbance varies. The continuous changein the gain of the system poses a very challenging problemand FMAC can offer tight control performance in the face ofvarying input disturbance. Control techniques, such as inter-nal model control (IMC), which are best suited for additivedisturbance rejection, are unable to offer good control per-formance due to model plant mismatch.
This work does not address explicitly stability and robust-ness specifications, i.e. no design specifications are specifiedconcerning these features by themselves. These issues are,however, implicitly considered in the control structure byincorporating feedback error learning. Hence, only perfor-
mance specifications such as RMSE of the control error andMAE of the control activity are explicitly treated in this work.
The first section of the paper discusses the internal modelcontrol scheme. Model-free control along with the FRM andlearning scheme (RSK) used to train the controller parame-ters is elaborated in the next section. Conditional defuzzifi-cation and its application to reduce the control activity areexplained in the third section. Simulation results are pre-sented in the fourth section whereas conclusion and futureresearch directions are discussed in the fifth section.
2 Internal Model Control (IMC)
Internal model control (IMC) is based on the idea that, if aperfect representation of the plant to be controlled is avail-able (referred to as internal model), and the controller is anexact inverse of the plant then perfect disturbance rejectionis achievable [11,12]. Internal model control (IMC) can beused to reject unmeasured disturbances but the control per-formance largely depends on the availability of a good plantmodel which can be used as an internal model and a con-troller which ideally should be a perfect inverse of the plant.IMC control structure is shown in Fig. 1. To analyze IMC,a linear plant and controller model is considered. In the setof Eqs. (1), (2) and (3), it is assumed that the controller is anexact inverse of the plant (Eq. (1)) and the internal model isan ideal model of the plant (Eq. (2)).
C(s) = G−1(s) (1)
where ‘G(s)’ represents the plant model and ‘C(s)’ is theLaplace transform of the controller. ‘M(s)’ represents theinternal model.
M(s) = G(s) (2)
‘Dout’ and ‘Din’ represent the Laplace transform of the out-put and input disturbances acting on the plant. Since the plantis linear therefore input disturbance can be represented at theoutput. ‘Y ’ and ‘Y ’ represent the output from the plant andthe internal model, respectively. ‘Eimc’ is the error generatedby the IMC loop which has the representation of both theinput and output disturbances. This ‘Eimc’ is deducted fromthe reference signal ‘R’ to compensate for the disturbances.
Y = Dout + G[U + Din]Y = GU
Eimc = Y − Y = Dout + G Din
U = C[R − Dout − G Din]Y = Dout + G[C[R − Dout − G Din] + Din]Y = R
(3)
123
Arab J Sci Eng (2014) 39:2381–2392 2383
G(s)
M(s)
C(s)
din(t)
u(t) y(t)
+
+
+
-
Internal Modeleimc(t)
Controller Plant
+
-
r(t)
dout(t)
+
+
(t)
Fig. 1 IMC with both additive input and output disturbance
The results derived from Eq. (3) are valid for steady-stateand with zero initial conditions. It is evident from the setof equations that to guarantee total disturbance rejection anideal internal model is required [13]. Many strategies havebeen reported in the literature to obtain a good plant model,though this can be a tedious task. A range of solutions havebeen proposed which can be categorized into modelling byfirst principle (physical models) [14–16] and the black boxmodels. Black box models cover a wide range in which fuzzymodelling [17–22], and neural network modelling [23–25]have been recently applied to model a number of processes.Obtaining an exact plant model is unrealistic in most ofthe situations. With inaccurate model, complete disturbancerejection is not guaranteed. Since the internal model mimicsthe behaviour of the plant to some extent, it has been shown[26] that steady-state errors can still be removed. If the plantunder consideration is nonlinear and the input disturbance ismultiplicative then results of Eq. (3) does not hold true. Themultiplicative disturbance changes the gain of the plant whenthe input disturbance is varied. This case often arises in airconditioning systems where the air mass flow rate which isacting at the plant input changes the gain of the system. Multi-plicative input disturbances cannot be rejected by IMC loop.
3 Model-Free Adaptive Control
The fuzzy model-free adaptive controller (FMAC) [5,27–29,42–48] is a feedback controller. A self-learning con-trol scheme (feedback error learning) is used to obtain aninverse model of the plant [30]. A block diagram of thecontrol scheme is given in Fig. 2. The self-learning fuzzycontrol scheme has four major blocks: namely the Refer-ence Model, Proportional Controller, Online Fuzzy Identifi-cation Scheme and the Feedforward Controller. The Refer-
ence Model is used to make the inverse model and setpointtrajectory achievable. In this control scheme, the referencemodel is assumed to be of first order to match with the dynam-ics of the plant. The Fuzzy Identification Scheme is one ofthe most important component of the whole control strategy.If the learning scheme is good then the system will achieveperfect control after sufficient training. The Fuzzy Identifi-cation Scheme is used to train the controller. The learningscheme has dual purpose, firstly it estimates the desired con-trol action and secondly it determines the controller parame-ters. The purpose of the Proportional Controller is to cancelthe effect of unmeasured disturbances and to make the FMACmore robust during the initial training phase, when the rulebase is empty and the Feedforward Controller is unable tocontrol the plant [30].
3.1 Fuzzy Relational Models
The feedforward fuzzy controller can be modelled as aTakagi-Sugeno (TS) [31] model as well as a fuzzy relationalmodel (FRM). The advantage of FRM over TS modellingtechnique is the fuzzy output generated by the FRM. Thefeedforward controller is modelled as a FRM to generate afuzzy control signal. The fuzzy control signal will be defuzzi-fied using two different defuzzification schemes. FRM [32]have rules equal to all the possible different combinationsof the fuzzy sets defining inputs and outputs. A rule con-fidence ∈ [0, 1] represents the possibility of obtaining anoutput ‘y’ in the fuzzy output set q (q is less than or equalto r) from inputs x = [x1, x2, . . . , xn]T in the fuzzy inputsets A1, A2, . . . An . The output of a multi input single output(MISO) fuzzy relational model can be described [29] by:
Y = X1 ◦ X2 ◦ · · · ◦ Xn ◦ R (4)
123
2384 Arab J Sci Eng (2014) 39:2381–2392
Fuzzy Identification
Algorithm
ProportionalController
Plant
FeedforwardController
e(t)
r(t) u(t)ub(t)
uf(t)
+
-
+
+
MeasurableDisturbances
UnmeasuredDisturbances
Reference Model
Fig. 2 Block diagram of fuzzy model-free adaptive control (FMAC)
y(x) =∑n
i=1 { f Ai (xi )[RAi ,B1U1 + · · · + RAi ,Bn Un]}∑n
i=1 { f Ai (xi )[RAi ,B1 + · · · + RAi ,Bn ]}(5)
where, Y and y(x) are the fuzzified and defuzzified outputsof the fuzzy model, respectively. Y is a N × 1 array whoseelements are the membership grade of the output in the ref-erence sets B1, B2,…,Br . The crisp output y(x) can be com-puted from Eq. (5). Ui is the position of the apex of the i thoutput set. x = [x1, x2, . . . , xn]T are the crisp inputs to thefuzzy model. f Ai (xi ) = μAi (xi ) is the membership gradeof the input xi , in the multi-dimensional fuzzy set Ai whichdescribe the i th input space. ‘R’ is the fuzzy relational arraycontaining N × n elements (rule confidences) and ‘o’ is thefuzzy composition operator. Sum product is used as fuzzycomposition operator and height defuzzification is used tocalculate the crisp output.
3.2 Learning Scheme
The scheme was proposed by [33] and is one of the simplestschemes for estimating the rule confidences in the presenceof noise. It can be used online effectively due to its computa-tional simplicity [34,35]. The algorithm is defined for a multi-input single-output (MISO) system. Consider a MISO rela-tional model consisting of ‘n’ inputs x1(t), x2(t), . . . , xn(t)and one output y(t). The inputs and output are characterizedby A1s1 , . . . , Ansn and By fuzzy reference sets, respectively.The entry (rule confidence) RA1s1 ,...,Ansn ,B j (t) in the fuzzyrelational array represents the possibility of obtaining an out-put ‘y(t)’ in the fuzzy output set ‘B j ’ ( j ≤ y) from inputsx1(t), x2(t), . . . , xn(t) in sets A1s1, . . . , Ansn , respectively,[34]. The input space of each variable is divided into ‘r ’ ref-
erential sets using ‘r ’ fuzzy membership functions. The RSKalgorithm estimates the rule confidence by taking a weightedaverage over all the training data:
RA1s1 ,...,Ansn ,B j (t) =∑N
t=1 f A1s1 ,...,Ansn(x(t))μB j (y(t))
∑Nt=1 f A1s1 ,...,Ansn
(x(t))
(6)
where, f A1s1 ,...,Ansn(x(t)) is the product μA1s1
(x1(t)), . . . ,μAnsn
(xn(t)) and the summation runs over the relevant obser-vations ‘N ’. The recursive form of the RSK algorithm canbe written as:
The RSK scheme can be modified to weight the data expo-nentially [36]. This will enable the recent data to have moreimpact on the rule confidences and to forget the old data.
RA1s1 ,...,Ansn ,B j (t) =∑m
t=1 λm−t f A1s1 ,...,Ansn(x(t))μB j (y(t))
∑mt=1 λm−t f A1s1 ,...,Ansn
(x(t))
(9)
where 0 < λ ≤ 1 is the forgetting factor and ‘m’ is thenumber of times a particular combination of input sets hasbeen fired. The recursive form of the modified RSK is:
Feedback error learning is a an adaptive control technique[37,38]. The feedback error learning law estimates thedesired control action uf(t), which is then used to update therule confidences. The usage of the feedback error learninglaw depends on the nature of the plant and the requirementsput on the controller. The feedback error learning law gov-erns the behaviour of the controller during the learning phase.The following feedback error learning law is used:
uf(t − td) = uf(t − td) + γ e(t) (12)
where uf(t − td) is the estimate of the desired control action,uf(t−td) is the actual control action from the fuzzy controller,e(t) = r(t − td)− y(t), γ is the online learning rate and td isthe time delay of the plant. The updating of uf(t − td) stopswhen e(t) is zero. This ensures that there will be no moreupdates to the controller parameters and the controller willbe fully trained.
4 Conditional Defuzzification
Conditional defuzzification [28,39] is based on the simpleidea of only defuzzifying the fuzzy control signal when the
value of the previous control signal is no longer considered tobe optimal enough. As shown in Fig. 3, the membership gradeof the previous crisp control signal u(n − 1) is determinedfrom the current fuzzy control signal U (n). If the intersectionpoint (*) lies below the threshold level (α), a new value of thecontrol signal is calculated using height defuzzification. Inthis example, the value of the threshold is 0.8 h where h is themaximum height of the fuzzy control signal. If the intersec-tion point lies above the threshold value, indicating that theprevious value of the crisp control signal is optimal enough toproduce satisfactory control performance, then the previousvalue of the control signal is used. The threshold level α isdependent on the particular application. The resulting con-trol activity depends on the value of the threshold and thepossibility distribution of the fuzzy control signal. The pos-sibility distribution depends on the amount of sensor noiseentering the control loop as well as the amount of fuzzinessin the controller inputs. In the control problem discussed inthe paper, the noise due to the sensor measuring the con-trolled output is considered. Conditional defuzzification isonly applied when the controller has been trained at all theoperating points, hence the fuzziness in the control signal isonly due to the sensor noise. To demonstrate the effective-ness of conditional defuzzification two separate cases can beconsidered which are discussed in the following paragraphs:
5 Simulation Setup
The system under consideration in this paper is a cooling coilof an air handling unit. A simplified version of a cooling coil[27] can be represented by the Hammerstein model given inEqs. (13) and (14).
123
2386 Arab J Sci Eng (2014) 39:2381–2392
y(s) = L( f (u))e−Tds
τ s + 1(13)
f (u) = 1
3.433ln(30u + 1) (14)
In Eq. (14), the control signal ‘u’ is normalized by scalingit with a factor of 3.433. The purpose of the cooling coil isto reduce the temperature of the air flowing over it. Changesin the air mass flow rate vary the gain of the system. Whenthe air mass flow rate is higher, there is less amount of cool-ing (of the inlet air) due to a lower value of the gain. On thecontrary, if the air mass flow rate is decreased then the gainof the system increases hence the outlet air is colder. In thiswork, air mass flow rate is used as an input disturbance. Theoriginal Hammerstein model is modified to incorporate theeffect of air mass flow rate. In the modified Hammersteinmodel the input disturbance is represented by ‘din’. Mod-ified Hammerstein model is represented by the followingequation:
y(s) = L( f (u))
[1
0.9din + 0.8
]e−Tds
τ s + 1(15)
The constants 0.8 and 0.9 are calculated from a detailed SIM-BAD simulation [40] of the cooling coil. The air mass flowrate in the SIMBAD model was varied between 100 and 50 %of its design value (0.29 kg/s). The inlet air temperature andchilled water temperature were held constant at 30 and 8 ◦C,respectively. The control signal was also held constant at 1,i.e. the valve which controls the amount of water flowingthrough the cooling coil pipes is fully open. When the airmass flow rate changes from 100 to 50 % of the design value,the gain of the SIMBAD model changes by approximately10 %. The time constant (τ) and dead time (Td) of the mod-ified Hammerstein model is 200 and 10 s, respectively. Toincorporate the effect of air mass flow rate in the Hammer-stein model, the SIMABAD model was used to determinethe coefficients of Eq. (16):
1
adin + b(16)
where ‘a’ and ‘b’ are the coefficients need to be calcu-lated and ‘din’ is the input disturbance (air mass flow rate).The coefficients can be found by solving the simultaneousEqs. (17) and (18):
1
a0.2 + b= 1.0 (17)
1
a0.1 + b= 1.1 (18)
On solving the above equations, the values of the coefficients‘a’ and ‘b’ are found to be 0.9 and 0.8, respectively.
The parameter setting for the FMAC is discussed in thefollowing paragraph. The value of the forgetting factor (λ) forthe learning scheme (RSK) is 0.99. Feedback error learningwith a learning rate (γ ) of 0.5 is used. The F array (used inrecursive RSK, Eq. (11)) values are initialized to zero andall of the rule consequents, i.e. ‘R(t − 1)’ are initialized to0.5. Input to the controller is x(t) = [r(t + td), y(t)] withFMAC–IMC whereas it is x(t) = [r(t + td), y(t), din(t)]when feedforward of the disturbance is used. All the inputshave five membership functions with apexes at 0.0, 0.25,…,1.0. The reference model has a time constant of 200 s. Thenormalized reference setpoint is held constant at 0.85. Thesensor noise is uniformly distributed. The standard deviationof the noise is 0.8 and the sensor bias is 0.05. The sensortime constant is 300 s. The input disturbance is a sinusoidwith amplitude of 0.1 and a mean of 0.1. The sinusoid has aperiod of 900 samples. The sampling time of the controlleris 10 samples. The integration time of the simulation is set to1 second. Conditional defuzzification is only used after theinitial learning stage. The learning is continued till 1.9×105.All the values of root mean squared error (RMSE) of thecontrol error and mean absolute error (MAE) of the controlactivity are calculated in the testing stage, i.e. from 1.9×105
to 2.0 × 105.
6 Control Performance
6.1 Disturbance Rejection Using IMC
To demonstrate the fact that the IMC is unable to reject themultiplicative input disturbance, control performance is pre-sented with ideal internal model as well as ideal measurementof the disturbance. Figure 4 shows the control performancein the presence of IMC loop. It can be seen that, even withaccurate measurement of the input disturbance and an idealinternal model, IMC is unable to eliminate the effect of thedisturbance. Since the internal model is producing the sameoutput as that of the plant model, ‘eimc(t)’ is zero all thetime. The error ‘eimc(t)’ does not represent the error due tothe disturbance hence the controller has no knowledge aboutthe existence of the disturbance. The IMC loop is ineffectiveand is providing no useful information and the controller hasdegenerated into a pure feedback control [7]. It can be con-cluded that for this specific class of nonlinear systems, wherethe input disturbance is changing the relationship between thecontrol signal and the controlled variable, IMC is not a viableoption.
6.2 Disturbance Rejection Using FMAC–IMC
If the disturbance cannot be measured then internal modelcontrol (IMC) can be used to achieve better control perfor-
123
Arab J Sci Eng (2014) 39:2381–2392 2387
Fig. 4 Control performance ofIMC with accurately measureddisturbance and accurateinternal model
mance [7,41]. FMAC with the IMC loop is shown in Fig. 5.The generation of an exact plant model is unrealistic in prac-tice. The control performance of FMAC–IMC is shown inFig. 6. It should be noted that, with ideal internal model,FMAC is unable to reject the disturbance. The reason for thepoor control performance is the inaccurate internal modelwhich is not replicating the true plant behaviour. The inputdisturbance is changing the gain of the plant whereas thesame effect cannot be reproduced in the plant due to theunavailability of the measurement of the disturbance. It can
be argued that with multiplicative input disturbance IMC isnot able to reject the effect of disturbance. The control signalis also varying sinusoidally.
6.3 Disturbance Rejection Using Feedforwardof the Measured Disturbance
The second methodology used to reject the input multi-plicative disturbance is the feedforward of the measureddisturbance. The control structure is shown in Fig. 7. The
Fig. 7 FMAC withfeedforward of the measureddisturbance
PlantMFACr(t) u(t)
y(t)Reference
Signal
Sensor
din(t)
measured disturbance is one of the inputs to the controller.When the measured disturbance is fed into the controllerthen the learning scheme (RSK in this case) will try todevelop the correct relation between r(t), d(t) and u(t).Without incorporating the disturbance, inverse of the plantis not achievable. The control performance, i.e. perfect dis-turbance rejection is dependent on the availability of goodplant inverse. It can be observed from Fig. 8 that FMAC isable to reduce the impact of disturbance as compared to theFMAC–IMC, but the resulting control activity is high. Thereason for the greater control activity is the noisy measure-ment of the disturbance. This noisy measurement is fed intothe controller consequently the learning scheme keeps onupdating the controller parameters (rule confidences). The
rule confidences are updated due to the inconsistencies inthe measurement of the disturbance. The constant update inthe rule confidences translates into the control activity beinghigher. Since height defuzzification is used the fuzzy con-trol signal is defuzzified at every sampling instant. The nextsection discusses how conditional defuzzification in place ofheight defuzzification can be used to reduce the control activ-ity without compromising the RMSE of the control error.
6.4 Reduction of the Control Activity Using ConditionalDefuzzification
Figure 9 shows the effect of using conditional and heightdefuzzification on the mean absolute error (MAE) of the con-
123
Arab J Sci Eng (2014) 39:2381–2392 2389
Fig. 8 Control performance ofFMAC with feedforward of themeasured disturbance
Fig. 9 MAE of the controlactivity with height andconditional defuzzification
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
4.1
4.15
4.2
4.25
4.3
4.35x 10
-3
standard deviation (σ) of the sensor noise
MA
E
Conditional Defuzzification
Height Defuzzification
trol activity. It can be observed from Fig. 9 that when condi-tional defuzzification is used the MAE of the control activityreduces with the increasing noise level on the measurement
of the input disturbance. The reason for the reduced controlactivity can be understood from Fig. 10. The spread of thefuzzy control signal is directly related to the amount of sensor
123
2390 Arab J Sci Eng (2014) 39:2381–2392
Increasing amount of fuzziness in the fuzzy control signal control U(n)
Previous value of the crisp control signal u(n-1)Conditional defuzzification threshold (α)
CASE 1 CASE 2 CASE 3 CASE 4
Fig. 10 Conditional defuzzification cases to understand the impact of noise on control activity
Fig. 11 RMSE of the controlerror with height andconditional defuzzification
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.009
0.01
0.011
0.012
0.013
0.014
0.015
0.016
standard deviation (σ) of the sensor noise
RM
SE
Conditional Defuzzification
Height Defuzzification
noise. The greater the amount of sensor noise, the larger willbe the spread of the fuzzy control signal. As can be observedfrom Fig. 10, the defuzzification of the fuzzy control signal isinversely related to the amount of fuzziness of the fuzzy con-trol signal. If the threshold (α) for conditional defuzzificationis constant, the fuzzy control signal will not be defuzzifiedat every sample, hence the MAE of the control activity willreduce. On the contrary, when height defuzzification is usedthe MAE of the control activity slightly increases. The slightincrease in the control activity is related to the amount ofnoise entering the control loop. Since height defuzzificationdefuzzifies the fuzzy control signal at every step hence theMAE of the control activity increases as shown in Fig. 9.It can be argued that the additional control activity is inresponse to the noise and does not help to improve the con-trol performance as evident from Fig. 11. The RMSE of the
control error is shown in Fig. 11 for both the cases, i.e. con-ditional and height defuzzification. The RMSE of the controlerror is similar in both the cases. It might be expected that,using conditional defuzzification will have an adverse effecton the control performance. Since conditional defuzzifica-tion reduces the control activity the RMSE should increase ascompared to the case when height defuzzification is used. Theeffectiveness of conditional defuzzification in maintaining asimilar control performance can be observed from Fig. 11.
7 Conclusion
Fuzzy model-free adaptive control was used to reject themultiplicative input disturbance. It was demonstrated thatthe disturbance cannot be rejected using internal model con-
123
Arab J Sci Eng (2014) 39:2381–2392 2391
trol. Feedforward of the measured disturbance was used withFMAC, which was able to reduce the impact of input distur-bance. Although the control performance was satisfactory,the sensor noise associated with the measurement of the inputdisturbance resulted in an increase of the control activity. Itwas successfully demonstrated that the control activity canbe reduced by employing conditional defuzzification withoutcompromising the control error.
Acknowledgments The research work was completed at ControlLaboratory, Department of Engineering Science, University of Oxfordunder the supervision of Professor Arthur Dexter. The research projectwas funded by National University of Sciences and Technology, Islam-abad, Pakistan.
References
1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)2. Karr, C.L.; Gentry, E.J.: Fuzzy control of pH using genetic algo-
rithms. IEEE Trans Fuzzy Syst. 1, 46–53 (1993)3. Angelov, P.; Filev, D.: An approach to online identification of
4. Demircan, M.; Camurdan, M.C.; Postlethwaite, B.E.: On-linelearning fuzzy relational model based dynamic matrix control ofan open-loop unstable process. Trans. IChemE 77, 421–428 (1999)
5. Kadri, M.B.: Model free adaptive fuzzy control: beginnersapproach. VDM Verlag Dr. Müller, Saarbrücken (2009)
6. Scherer, R.; Starczewski, J.; Gaweda, A.: New methods for uncer-tainty representation in neuro-fuzzy systems. In: Parallel Process-ing and Applied Mathematics, vol. 3019/2004. Lecture Notes inComputer Science: Springer Berlin, pp. 659–667 (2004)
7. Nitsche, R.; Schwarzmann, D.; Hanschke, J.: Nonlinear internalmodel control of diesel air systems. Oil Gas Sci. Technol. 62, 501–512 (2007)
8. Economou, C.G.; Morari, M.: Internal model control. 5. Extensionto nonlinear systems. Ind. Eng. Chem. Process Des. 25, 430–411(1986)
9. Boukezzoula, R.; Galichet, S.; Foulloy, L.: Nonlinear internalmodel control: application of inverse model based fuzzy control.IEEE Trans. Fuzzy Syst. 11, 814–829 (2003)
10. Graebe, S.F.; Seron, M.M.; Goodwin, G.C.: Nonlinear trackingwith input disturbance rejection with application to pH control. J.Process Control 6, 195–202 (1996)
11. Burns, R.S.: Advanced Control Engineering. Butterworth-Heinemann, Oxford (2001)
12. Garcia, C.E.; Morari, M.: Internal model control. 1. A unifyingreview and some new results. Ind. Eng. Chem. Proc. Des. Dev. 21,308–323 (1982)
13. Xie, W. F.; Rad, A. B.: Fuzzy adaptive internal model control. IEEETrans. IE 47, 193–202 (2000)
14. Johansson, R.: System Modeling and Identification. Prentice Hall,New Jersey (1993)
15. Seborg, D.E.; Edgar, T.F.; Mellichamp, D.A.: Process Dynamicsand Control. Wiley, New York (1989)
16. Smith, O.J.M.: Closer control of loops with dead time. Chem. Eng.Prog. 53, 217–219 (1957)
18. Jang, M.J.; Chen, C.L.: Fuzzy successive modelingand control fortime-delay systems. Int. J. Syst. Sci. 27, 1483–1490 (1996)
19. Lu, Y.Z.; He, M.; Xu, C.W.: Fuzzy modeling and expert optimiza-tion control for industrial processes. IEEE Trans. Control Syst.Technol. 5, 2–12 (1997)
20. Oliveira, J.V.D.; Lemos, J.M.: Long range predictive adaptive fuzzyrelational control. Fuzzy Sets Syst. 70, 337–357 (1995)
21. Wang, L.X.: Design and analysis of fuzzy identifier of nonlineardynamic systems. IEEE Trans. Autom. Control 40, 11–23 (1995)
22. Xu, C.W.; Lu, Y.Z.: Fuzzy model identification and self-learningfor dynamic system. IEEE Trans. Syst. Man Cybern. SMC-17,683–689 (1987)
23. Bha, N.; Mcavoy, T.J.: Use of neural nets for dynamic modelingand control of chemical process systems. Comput.Chem. Eng. 14,573–583 (1990)
25. Li, Y.; Rad, A.B.Y.; Wong, K.; Chan, H.S.: Model based controlusing artificial neural networks. In: Proceedings 1996 IEEE Inter-national Symposium Intelligent Control, pp. 283–288 (1996)
26. Henson, M.A.; Seborg, D.E.: An internal model control strategyfor nonlinear systems. AIChE J. 37, 1065–1081 (1991)
27. Gongsheng, H.; Dexter, A.L.: Predictive control based on TS fuzzymodel with uncertain parameters. In: Fuzzy Control Robotics andIntelligent Control Conference, IFSA 2005 Beijing, pp. 812–817(2005)
28. Thompson, R.; Dexter, A.L.: A fuzzy decision-making approachto temperature control in air-conditioning systems. Control Eng.Pract. 13, 689–698 (2005)
31. Tan, W.-W.: An on-line modified least-mean-square algorithm fortraining neurofuzzy controllers. ISA Trans. 46, 181–188 (2007)
32. Czogala, E.; Pedrycz, W.: On identification in fuzzy systems and itsapplications in control problems. Fuzzy Sets Syst. 6, 73–83 (1981)
33. Ridley, J.N.; Shaw, I.S.; Kruger, J.J.: Probabilistic fuzzy model fordynamic systems. Electron. Lett. 24(14), 890–892 (1988)
34. Wu, Y.: The design and application of a new type of adaptive fuzzymodel based controller. In: Department of Engineering Science.University of Oxford, Oxford (2005)
36. Postlethwaite, B.E.: Empirical comparisons of methods of a fuzzyrelational identification. IEE Proc. 138, 199–206 (1991)
37. Sabahi, K.; Teshnehlab, M.; Shoorhedeli, M.A.: Recurrent fuzzyneural network by using feedback error learning approaches forLFC in interconnected power system. Energy Convers. Manag. 50,938–946 (2009)
39. Kadri, M.B.: Disturbance Rejection using Neuro-Fuzzy Con-trollers: Special Emphasis on HVAC Application. VDM Verlag,Saarbrücken (2009)
40. Husaunndee, A.; Lahrech, R.; Riederer, P.; Nejad, H.V.: SIMBADBuilding and HVAC Toolbox. CSTB, Sustainable DevelopmentDepartment, Automation and Energy Management Group, France(2001)
41. Alvarez, J.: An internal model control for non-linear systems.In: Proceedings 3rd European Control Conference, Rome, Italy,pp. 301–306 (1995)
42. Kadri, M.B.: System identification of a cooling coil using recurrentneural networks. Arab. J. Sci. Eng. 37, 2193–2203 (2012)
43. Kadri, M.B.; Dexter, A.L.: Disturbance rejection in information-poor systems using an adaptive model-free fuzzy controller. Pre-
123
2392 Arab J Sci Eng (2014) 39:2381–2392
sented at annual meeting of the North American fuzzy informationprocessing society, NAFIPS (2009)
44. Kadri, M.B., Hussain, S.: Model Free Adaptive Control based onFRM with an approach to reduce the control activity. Presented at2010 IEEE international conference on systems man and cybernet-ics (SMC). Istanbul, Turkey (2010)
45. Kadri, M.B.: Robust model free fuzzy adaptive controller withfuzzy and crisp feedback error learning schemes. Presented at 12thinternational conference on control, automation systems (ICCAS).Jeju, South Korea (2012)
46. Kadri, M.B.: Disturbance rejection in information poor systemsusing model free neurofuzzy control. Dissertation, University ofOxford (2009)
47. Kadri, M.B.: Disturbance rejection in nonlinear uncertain systemsusing feedforward control. Arab. J. Sci. Eng. 38, 2439–2450 (2013)
48. Kadri, M.B.: Disturbance rejection in model free adaptive controlusing feedback. Presented at modelling, identification, and control.Innsbruck, Austria (2010)
