ARTICLE IN PRESS

0967-0661/$ - se

doi:10.1016/j.co

�CorrespondE-mail addr

fernando@esi.u

rubio@esi.us.es

Control Engineering Practice 15 (2007) 487–500

www.elsevier.com/locate/conengprac

Multivariable robust control of a rotary dryer: Analysis and design

M.G. Ortega�, F. Castano, M. Vargas, F.R. Rubio

Depto. Ingenierıa de Sistemas y Automatica, Universidad de Sevilla, Escuela Superior de Ingenieros, Camino de los Descubrimientos sn, 41092 Sevilla, Spain

Received 27 April 2005; accepted 7 September 2006

Available online 13 November 2006

Abstract

This paper describes the analysis and control of an industrial process usually controlled in manual mode: a continuous rotary dryer.

Two process variables are controlled simultaneously: the outlet moisture of the dried material and the output temperature of the exhaust

air. To do this, the flow of wet material and the flow of fuel are used as control variables. Thus, this constitutes a MIMO system, with two

inputs and two outputs. The system is identified at several operating points, and a controllability analysis is performed in order to find

any constraints in its performance. A multivariable robust control based on the H1 mixed sensitivity problem is proposed. Simulation

and experimental results are included, using different adjustments of the weighting matrices based on the performed experiments. Results

attained agree with the controllability analysis conclusions.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Process control; Industrial processes; Rotary dryer; Controllability analysis; Robust control; H1 Mixed sensitivity problem

1. Introduction

Despite the importance of drying processes for theindustry, the automatic control of this kind of system is stillin a precarious state, since many installations are usuallyoperated in manual mode. This paper offers an analysisand study of a continuous rotary dryer, showing a numberof problems connected to these kinds of processes.

Basically, the drying process in a rotary installationconsists in the reduction of the outlet moisture content of aproduct to a desired value. This process uses a continuousrotary drum in which the wet material is tumbled, ormechanically turned over, usually in the same direction,while extremely hot air is continuously flowing.

The main control objective is to regulate the outletmoisture level, but it is also interesting to control theoutput temperature of the exhaust air, since it may beemployed as supply for other processes. These require-ments impose a strict and permanent control of the plant,which is not easy to achieve in manual mode.

e front matter r 2006 Elsevier Ltd. All rights reserved.

nengprac.2006.09.005

ing author. Tel.: +349 54486037; fax: +34 9 54487540.

esses: mortega@esi.us.es (M.G. Ortega),

s.es (F. Castano), vargas@esi.us.es (M. Vargas),

(F.R. Rubio).

The difficulty in controlling this type of plants lies in thephysical properties of the product to be dried, as well as inthe change of these properties with the different moisturepercentages. The movement of the material throughout thedrum, the adhesion of the particles to one another or to thedrum blades, the evaporation speed, etc., are examples offactors strongly influenced by the moisture level. Thisimplies a quite different system behavior depending on theoperating conditions.In fact, there is not a complete knowledge of the drying

process, since too many phenomena all connected to eachother, such as heat and mass exchange, movement ofsolids, evaporation, capillarity, superficial tension, diffu-sion, and so on, are involved in it. This implies a highlycomplex behavior that is quite difficult to model.The scientific community has been interested in the

modelling problem of these kinds of processes, and so far,some mathematical models have been developed (see, forinstance, Brasil and Seckler, 1988; Deich and Stalskii, 1975;Douglas et al., 1993; Duchesne et al., 1997; Reay, 1979).Most agree that it is very difficult to carry out a fine tuningof the model parameters. Furthermore, the tuning proce-dure is different for each specific material, and it is validonly for very limited operating conditions.

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500488

This paper describes the control of a rotary dryer thatuses sand as a working material. The output temperature ofthe exhaust air and the outlet moisture of the sand areregulated simultaneously, making a multivariable controlapproach suitable.

From the point of view of control, the application ofseveral control techniques can be found in the literaturefor these types of systems, ranging from elementaryPID control (Arjona et al., 2005) to predictive control(Didriksen, 2002), fuzzy control (Yliniemi, 1999) or others(Courtois, 1997; Savaresi et al., 2001), though they allimplement monovariable control loops.

Being aware of the complexity of the drying process, itcould be appropriate to implement a controller which cancope with the coupling and strong uncertainties thatmay exist when modelling these systems. Accordingto this, the development of a robust controller seemssuitable. This paper describes one in particular: a multi-

variable H1 controller, based on the mixed sensitivityproblem.

The remainder of this article is organized as follows: inSection 2, a description of the process is presented, andits model is obtained in Section 3. This model is validatedwith real data obtained from the process, and employedfor deriving some transfer matrices at different workingpoints. A controllability analysis of the linearized systemis carried out in Section 4, in order to show the behaviorlimitations of the plant. The synthesis of the H1 robustcontroller is described in Section 5. Simulation andexperimental results attained by means of this robustcontroller are presented in Sections 6 and 7, and,finally, the main conclusions to be drawn are given inSection 8.

Fig. 1. Co-current

2. Plant description

The system considered corresponds to a co-current rotary

dryer (see Fig. 1), which is located on the terrace roof of thelaboratories of the Dept. Ingenierıa de Sistemas y Auto-

matica of the University of Seville. The plant makes use of adistributed control system that allows the control andintegral monitoring of the drying process.A simplified diagram of the rotary dryer is depicted in

Fig. 2, basically showing three different areas:

�

ro

The feeding area, which consists of a burner and twohoppers. The burner uses gas as fuel to heat theincoming flow of air. This current of hot air is employedto dry the wet matter. The hoppers contain the wetmaterial, which is carried into the drum by means of aworm gear and a conveyor belt. This configurationallows us to control both the flow of fuel and the inletflow of wet matter.

� The rotary drum area. The wet material from the feeding

area is continuously hauled by the rotation of the drum,and dropped into the hot air stream circulatingthroughout the drum. The cylinder (4m in length and0.8m in width) has inside a number of continuousblades, so while it is turning, these blades take thematerial and propels it into the gaseous current. Thedrum usually turns at a speed between 3 and 10 rpm,moved by a 3 kW electrical motor coupled to a gearreduction unit. The air speed ranges between 1.5 and4m/s, depending on the size of the particles to be driedand on the quantity of fine powder produced during theprocess. The speed of rotation, pitch angle and airvelocity determine the material delay time.

tary dryer.

ARTICLE IN PRESS

FEEDING AREA ROTARY DRUM AREA OUTPUT AREA

Fuel

Air

Hot Air

Wet material Exhaust air

Dried materialwater

water

water

motor

Air filter

Fig. 2. Diagram of the co-current rotary dryer.

Wi

Fgi-1+Fsti-1

Fsi-1+Fwi-1

Fsi+Fwi

Fgi+Fsti

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500 489

For the sake of efficiency, it is essential to set up the flowof solid materials helped by the air stream (see Fig. 2),since a significant amount of moisture must be evapo-rated in the first stages of the drying process. This makesit possible for high temperatures to be reached in theincoming air without reaching dangerously high tem-peratures in the material to be dried. Since thetemperatures of the air and of the solid materials tendto converge when both flows reach the drum outlet, thetemperature of the solids that leave the cylinder can becontrolled to reach their maximum value, while main-taining the advantage of having a wide range oftemperatures in the first stages.

�

Fig. 3. ith element of the dryer drum.

The output area. In this area, the dried product and theexhaust air are extracted from the drum and takenoutside the process. On one end, the solid material dropsinto a small hopper, and from this one to anotherconveyor belt. On the other end, the exhaust air is guidedthrough a dust filter, which is located before the suctionfan used to create the air stream.

From this output area, the exhaust air could be used forfeeding another process, provided that the air currentmeets the adequate conditions. Therefore, it may beimportant to control not only the moisture of the driedproduct, but also the temperature of the exhaust air.

3. Process model

3.1. Thermodynamic model

In rotary dryers, water is removed from a material bymeans of a hot air stream. This transfers heat to the solidand reduces its humidity content. Heat transmission takesplace mainly by conduction and convection in adiabaticconditions.

Thus, the system can be modelled by a set of nonlinearequations describing energy and mass balances (Rubio

et al., 2000; Rubio et al., 2001). Due to the drum’sgeometry, strong gradients of concentration and tempera-ture appear. Therefore, the application of mass andenergy balances gives rise to partial derivatives anddifferential equations, which can be avoided dividing thedryer drum into a finite number, n, of elements in series(Savaresi et al., 2001). Fig. 3 shows a generic element, towhich balance equations are applied. In this figure, Fg isthe mass flow of dry gas, F st is the mass flow of steamin the gas, F s is the mass flow of dry solid, F w is themass flow of water in the solid, and W is the flow ofevaporated water.Balance equations come from the analysis of the flows

that go across the ith section. Assuming that the conditionsat the dryer input are known, the outputs of each sectionare computed and used as the input values for the nextsection. Many variables appear in each element, such asspecific heat of water and solid, specific heat at constantpressure of the dry gas and water steam, latent heatof vaporization, volumetric heat transfer coefficient,among others.

ARTICLE IN PRESS

Rotary dryer

To

Mo

Fc

Fp

Fig. 5. MIMO system.

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500490

In the same way, it was necessary to develop a model ofthe combustion process in the feeding area, which is alsogoverned by differential equations. A complete model of aburner should include a combustion performance study, aheat transmission analysis, along with other information.The variables involved in the combustion process are quitesimilar to those mentioned in the previous paragraph.

For the sake of completeness, a more detailed butconcise description of the equations and physical variablesinvolved in both parts of the nonlinear model is given inAppendix A.

3.2. Nonlinear model validation

This model has been validated using data obtained fromthe real plant. These data were used to estimate many of theparameters that appear in the model which are not perfectlyknown, since they depend on the operating conditions.

Fig. 4 shows a comparison between real and simulatedoutputs obtained from real input data. It can be seen thatthe nonlinear model is able to reproduce the fundamentaldynamics, with a relatively small steady-state error undersignificant changes on either output: the output tempera-ture of the exhaust air (To), and the outlet moisture contentof the dried product (Mo).

The data used as input for these model validationexperiments come from independent PRBS-type (pseudor-andom binary sequence) variations in the inputs around theconstant DC values of the corresponding operating points.

3.3. Linear models

Linear models have been developed in order tosynthesize the robust controller. Thus, the plant has beenmodelled as a multivariable system (see Fig. 5) whosecontrol variables are the mass flow of fuel (F c) and the

0 50 100 150 200 250 30040

45

50

55

60

time (minutes)

To:

Exh

aust

air

tem

pera

ture

(°C

)

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

time (minutes)

Mo

: Out

let

moi

stur

e (

%)

real

simulated

real

simulated

Fig. 4. Measured and simulated outputs. Left: significant chan

mass flow of wet material (Fp). The controlled variables arethe outlet moisture content of the dried product (Mo) andthe output temperature of the exhaust air (To).Several models were obtained at different operating

points. Fig. 6a shows the working space in terms of controlvariables, whilst Fig. 6b shows it in terms of outputvariables. The nominal operating point, denoted as NP, aswell as other extreme points are marked. These values wereobtained from the real plant, where the variation of the fuelmass flow is around 70% and the variation of the mass flowof wet material is approximately 33%, both with respect tonominal conditions.It should be noticed that the plant does not work

properly if the flow of fuel is low and the solid flow is highor vice versa. In the first case, a small amount of heat isprovided for too much wet material, which causes tosignificant bonding of the material. In the second extremecondition, too much heat is supplied for a small amount ofwet material and, therefore, there may be risk of a fire.In order to describe the fundamental dynamic behavior of

the system, some linear models have been identified atdifferent operating points by means of classical identifica-tion techniques (Camacho and Bordons, 1995; Ljung, 1986).In particular, multivariable ARX models were computed

applying pseudorandom binary sequences at the non-linearmodel inputs. As a result, the identified nominal model was

GðsÞ ¼1

dðsÞ

n11ðsÞ n12ðsÞ

n21ðsÞ n22ðsÞ

!, (1)

0 50 100 150 200 250 30040

45

50

55

60

65

time (minutes)

To:

Exh

aust

air

tem

pera

ture

(°C

)

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

time (minutes)

Mo:

Out

let

moi

stur

e (

%)

real

simulated

real

simulated

ges in output To. Right: significant changes in output Mo

�ðs� 2:7544� 10 Þðsþ 5:5013� 10 Þ,

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500 491

where

n11ðsÞ ¼ � 3:7724ðs� 3:3333� 10�2Þðsþ 5:8870� 10�3Þ

�ðsþ 5:4580� 10�4Þ

�ðsþ 4:2980� 10�4 � 2:4298� 10�4 jÞ,

n12ðsÞ ¼ 6:1057� 10�3ðs� 4:7402� 10�2Þ

�ðs� 3:3333� 10�2Þ

�ðsþ 4:8541� 10�3Þðs� 1:3053� 10�3Þ

�ðsþ 5:2376� 10�4Þ,

Fc (Kg/h)

F p (K

g/m

in)

3

2

4

2.6 40.7

NP

P1

P2

P3

P4

Control variables(a) (

Fig. 6. Working area of the rotary dryer: (a

0 200 400 600 800 10000

2

4

6

8

10

12

time (minutes)

To

Step in Fc

0 200 400 600 800 1000−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

time (minutes)

Mo

Fig. 7. Comparison of the nominal (thick line) and non-nominal time res

n21ðsÞ ¼ 2:8079� 10�2ðs� 3:3333� 10�2Þ

�ðsþ 1:0052� 10�2Þðsþ 2:7027� 10�3Þ

�ðsþ 5:5607� 10�4 � 3:1026� 10�4Þ,

n22ðsÞ ¼ � 1:8574� 10�4ðs� 3:3333� 10�2Þ

�ðsþ 2:0368� 10�2Þ

�ðs� 5:8427� 10�3Þ

�3 �3

To(°C)

Mo(

%)

1.08

1.79

37.6°27.8°

NP

P1P2

P3

P4

57°28.4°

0.43

1.62

56°

0.30

Output variablesb)

) control variables; (b) output variables.

0 200 400 600 800 1000−1.5

−1

−0.5

0

0.5

time (minutes)

Step in Fp

0 200 400 600 800 1000−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time (minutes)

ponses under step changes in inputs F c, Fp for each operation point.

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500492

dðsÞ ¼ ðsþ 3:3049� 10�2Þðsþ 5:1752� 10�3Þ

�ðsþ 5:5221� 10�4Þ

�ðsþ 3:9957� 10�4 � 2:2141� 10�4 jÞ.

These continuous-time transfer matrices have been ob-tained for each operation point after conversion fromdiscrete-time domain using a bilinear inverse transforma-tion (sampling time equal to 60 s). In the next section, acontrollability analysis is made based on the continuousnominal model, though keeping in mind that the results areno longer valid for frequencies near the sampling rate.

In order to help the reader perceive the differences of thefive transfer functions, Fig. 7 shows the corresponding stepresponses under changes in either input.

4. Controllability analysis

In this section, a basic input–output controllability

analysis (Skogestad and Postlethwaite, 1996) on thenominal linear model is made in order to check limitationsof the closed-loop performance. This analysis was alsocarried out on the other operating points, and the attainedresults were similar to those presented next.

First, the system has to be adjusted in scale, in agreementwith the maximum allowed deviation of each input andoutput. In this application, the following values have beenchosen as maximum variations:

DF c max ¼ 1:4 kg=h; DTo max ¼ 5 �C,

DFp max ¼ 0:5 kg=min; DMo max ¼ 0:3.

Taking these values into account, the system can be scaledby means of the following expression:

GðsÞ ¼ D�1e GðsÞDu,

where GðsÞ is the original linear model obtained in thepreceding section and GðsÞ is the scaled system. The twoscaling matrices De and Du are built as follows:

De ¼DTo max 0

0 DMo max

" #¼

5:0 0

0 0:3

� �,

Du ¼DF c max 0

0 DFp max

" #¼

1:4 0

0 0:5

� �.

The first step of the analysis consists in studying the poles

and zeros of the system (see Skogestad and Postlethwaite,1996, for instance). Table 1 shows these values in thecontinuous time domain.

Table 1

Poles and zeros of the nominal model

Poles: �3:305� 10�2, �5:175� 10�3, �3

Zeros: �7:065� 10�3, 2:287� 10�3 � 2:04

It can be seen that the system is stable, which does notimply any strong constraint in the closed-loop behavior.However, the system has some zeros in the right half plane(RHP), which impose some limitations on the systemperformance (Astrom, 2000). The most restrictive RHP-zeros are those with the smallest magnitudes. In ourcase, such zeros are located at zRHP ¼ 2:287� 10�3�2:041� 10�3j. This yields an approximate upper boundfor the control bandwidth oc equal to (Skogestad andPostlethwaite, 1996):

ocojzRHPj

2:8’ 1:09� 10�3 rad=s

which gives the following approximate lower bound for therise time, at least for one output:

tr �p

2oc41434 s ’ 24 min .

However, since the system is a multivariable one, it is alsoimportant to consider the direction of the RHP-zeros, atleast their output directions. In our case, zRHP has thefollowing output direction:

To

Mo

!¼

�0:145þ 0:250 j

�0:156� 0:94457 j

!.

It can be seen that the first component (its absolute value,equals to 0:289) is much smaller than the second one(whose absolute value is equal to 0:957). Therefore,although the degrading effect of a RHP-zero can be movedto a given output (Holt and Morari, 1985), its natural effectis focused on the second output variable. This way, thecontrol of the outlet moisture of the dried product is morelimited than the control of the output temperature of theexhaust air. Therefore, if a tight control on the outletmoisture is imposed, the behavior of the exhaust airtemperature would be strongly degraded.Another important item is the analysis of the singular

values of the system as a function of the frequency. Inparticular, the minimum singular value is useful as acontrollability index. Singular values of the nominal modelare depicted in Fig. 8 and Table 2 shows their directions(see Skogestad and Postlethwaite, 1996) at low frequencies.It can be observed that the minimum singular value issmaller than one for all frequencies. This implies thatindependent output changes cannot be made. In particular,as the output direction of the minimum singular value ismainly in the direction of Mo, some troubles may beexpected if tight control of the outlet moisture is requiredusing the allowed variations of the control variables.

:995� 10�4 � 2:214� 10�4j, �5:522� 10�4

1� 10�3j , 3:333� 10�2, 3:333� 10�2

ARTICLE IN PRESS

10−5 10−4 10−3 10−2 10−1

10−4

10−2

100

mag

nitu

de

frequency (rad/s)

Minimum singular values

Maximum singular values

Fig. 8. Singular values of the nominal model.

Table 2

Directions of the singular values at low frequencies

Singular value Input direction ½F c;Fp�T Output direction ½To;Mo�

T

Maximum ½0:495� 0869�T ½0:997� 0:080�T

Minimum ½�0:869� 0:495�T ½�0:080� 0:997�T

K(s)

z

vu

P(s)ω

Fig. 9. General formulation of the control problem.

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500 493

From Fig. 8 it can be inferred that the condition number

corresponding to the scaled nominal plant is slightly highat low frequencies, and it experiences a significant increasestarting from frequencies around 5� 10�4 rad=s, mainlydue to the drop-off in the minimum singular value at thesefrequencies. Therefore, the system can be considered ill-conditioned from these frequencies on, making it stronglysensitive to uncertainties.

This fact gives an additional approximate lower boundfor the rise time. In this case, this bound is the following:

tr �p

2� 5� 10�443141 s ’ 52 min .

Other factors have been analyzed, like the relative gain

array (RGA) along the frequency, but no other noticeablerestrictions have been found.

5. Controller synthesis

As stated in the Introduction, the main control objectiveof this process is to regulate the moisture percentage ofthe dried product, as well as to control the outputtemperature of the exhaust air. To do this, the wet solidflow and the fuel flow of fuel are used as controlvariables. Therefore, as already mentioned, this constitutesa multivariable control problem, with two inputs and twooutputs.

A centralized multivariable robust controller is pro-posed, taking into account that the system must workproperly at different operating points. It is difficult to findout the way variations in each model parameter affect thesystem outputs. Therefore, it is not reasonable tosynthesize a controller based on parametric uncertainty,

as is usual in very complex systems. Thus, an H1controller, based on unstructured uncertainty, is designedin this section, using the mixed-sensitivity problem ap-proach.The feedback H1 controller design can be formulated as

an optimization problem, which can be posed under thegeneral configuration shown in Fig. 9. In this figure, PðsÞ isthe generalized plant, KðsÞ is the controller, u represents thecontrol signals, v the measured variables, o the exogenoussignals and z stands for the so-called error variables.The optimal H1 control problem with this configuration

consists in computing a controller in such a way that theratio g between the energy of the error vector z and theenergy of the exogenous signals o is minimized.This optimal problem is still open, but a solution exists

for the suboptimal case (see Doyle et al., 1989 for adescription in the continuous time domain and Iglesiasand Glover, 1991 for discrete time). Therefore, the value ofthe energy ratio g is decreased as much as possible bymeans of an iteration procedure. This is the synthesismethod which has been implemented in well-knownsoftware packages such as Balas et al. (1995) or Chiangand Safonov (1998).The configuration used for building up the generalized

plant is given by the S=KS=T mixed sensitivity problem(for example, see Zhou et al., 1996), which is shown inFig. 10. In this case, the expression of the resulting closed-loop transfer function, TzoðsÞ, is as follows:

TzoðsÞ ¼

W SðsÞSoðsÞ

W KSðsÞKðsÞSoðsÞ

W T ðsÞToðsÞ

264

375,

ARTICLE IN PRESS

G(s)

WS(s)

WKS(s)

e

u

y

z1

z2

ω = r

-

+

v

P(s)

z

WT(s) z3

K(s)

Fig. 10. S=KS=T mixed sensitivity configuration. 10−5 10−4 10−3 10−2−10

−5

0

5

10

15

20

25

30

35

Max

. S.V

. (dB

)

WTdiag

σmax(ÊoP2)

σmax(ÊoP1)

σmax(ÊoP3)

σmax(ÊoP4)

frequency (rad/s)

Fig. 11. W T ðsÞ matrix as an upper bound of the maximum singular values

of the multiplicative output uncertainty.

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500494

where SoðsÞ is the output sensitivity transfer matrix, ToðsÞ isthe output complementary sensitivity transfer matrix, andKðsÞSoðsÞ is the so-called control sensitivity transfer matrix:

SoðsÞ ¼ ðI þ GðsÞKðsÞÞ�1,

KðsÞSoðsÞ ¼ KðsÞðI þ GðsÞKðsÞÞ�1,

ToðsÞ ¼ GðsÞKðsÞðI þ GðsÞKðsÞÞ�1.

The factors W SðsÞ, W KSðsÞ and W T ðsÞ constitute theirrespective weighting matrices, which allow to specify therange of frequencies of relevance for the correspondingclosed-loop transfer matrix.

As it is known, proper shaping of ToðsÞ is desirable fortracking problems, for noise attenuation, and for robuststability with respect to multiplicative output uncertainties.On the other hand, a convenient shaping of SoðsÞ will allowto improve the performance of the system. In addition,matrix W KSðsÞ helps us avoid some numerical problems inthe synthesis algorithms.

Therefore, since the controller is obtained from thegeneralized plant, the synthesis problem with this config-uration is reduced to the design of some appropriateweighting matrices which will impose the control specifica-tions. Based on this, the generalized plant can be built up,and consequently the controller can be calculated on acomputer by using a synthesis algorithm.

The selection of the weighting matrices W SðsÞ and W T ðsÞ

has been accomplished following the design rules fromOrtega and Rubio (2004). Thus, once the scaled nominalmodel is available (see Eq. (1)), the multiplicative outputuncertainty can be estimated as follows:

EoPiðsÞ ¼ ðGPiðsÞ � GðsÞÞGðsÞ�1,

where GðsÞ is the nominal model and GPiðsÞ stands for eachnon-nominal identified model at the extreme operatingpoints. Maximum singular values of these estimateduncertainties are shown in Fig. 11.

Next, matrix W T ðsÞ is designed as a square diagonalmatrix with all its diagonal elements with the same transferfunction, that is

W T ðsÞ ¼W TdiagðsÞI2�2,

where the transfer function W TdiagðsÞ must be stable,

minimum phase, with a high gain at high frequencies,and with magnitude greater than the maximum singularvalue of the uncertainty previously computed, for eachnon-nominal model and frequency, that is,

jW TdiagðjoÞjXsmaxðEoPi

ðjoÞÞ 8o; 8Pi.

In this application, W TdiagðsÞ has been chosen as follows:

W TdiagðsÞ ¼

100sþ 0:01

sþ 0:0125.

The magnitude of this transfer function is plotted inFig. 11. It can be seen that it satisfies all the requirementsimposed on it.Matrix W SðsÞ is taken as a square diagonal matrix of

transfer functions:

W SðsÞ ¼W STo

ðsÞ 0

0 W SMoðsÞ

" #,

where each diagonal element W SiðsÞ is designed with thefollowing expression:

W SiðsÞ ¼aisþ 10ðki�1ÞoT

sþ bi10ðki�1ÞoT

; i ¼ To;Mo,

where oT is the crossover frequency of W TdiagðsÞ (see Fig. 11),

and whose value is about 7:75� 10�5 rad=s. Parameters ai

and bi are the transfer function gains at high and lowfrequencies, respectively. According to the design rulesstated in Ortega and Rubio (2004), the following valueshave been chosen: aTo

¼ aMo¼ 0:5 and bTo

¼ bMo¼ 10�4.

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500 495

Finally, dimensionless parameters kToand kMo

determine thecorresponding transfer function bandwidth, and their valuesmust be higher as the desired response speed of thecorresponding output increases. An initial value equal tozero is recommended, which yields slow and rarely oscillatoryresponses. The final adjustment of these parameters must becarried out on the real plant.

Then, the weighting design is completed after makingweight W KSðsÞ equal to the identity matrix (i.e.W KSðsÞ ¼ I2�2) in order to avoid numerical problems inthe synthesis algorithm.

Once the weighting matrices have been designed, thegeneralized plant (see Fig. 10) can be built up, andtherefore, the controller can be computed. At this point,it is important to remember that the controller has beensynthesized from a scaled model. To figure out thecontroller to be implemented in the real application it isnecessary to carry out a reverse scaling procedure, that is

KðsÞ ¼ DuKðsÞD�1e ,

where matrices De and Du were introduced in Section 4 forthe scaling of the plant.

6. Simulation results

Several tests were carried out on the nonlinear model ofthe plant in order to adjust controller parameters kTo

andkMo

. Fig. 12 shows the responses obtained for a set ofvalues of kTo

and kMo.

0 50 100 150 20035

36

37

38

39

40

41

42

time

To

(o C )

0 50 100 150 2000.95

1

1.05

1.1

1.15

1.2

1.25

time

Mo

( %

)

Controller 2: κTo=3.5 κMo=2.5

Controller 1: κTo=3 κMo=2.5

Cont

Controller 3: κTo=3 κMo

Controller 1: κTo=3 κMo=2.5

Controller 2: κTo=3.5 κMo=

Fig. 12. Simula

A nominal controller (labelled as Controller 1 in Fig. 12)was synthesized using values for kTo

and kMoequal to 3

and 2.5, respectively. It can be observed that these valuesyield an appropriate response for both outputs, with atrade-off between the time responses of To and Mo, andwith no substantial overshoot in any response.Two other controllers were synthesized from slight

variations in the nominal values of these parameters. Oneof them, marked as Controller 2 in Fig. 12, aims to obtain abetter response speed of To by increasing the value of kTo

from 3 to 3.5. It can be observed that, as expected from thecontrollability analysis (see Section 4), an improvement inthe behavior of To does not significantly affect theperformance of Mo. The other one, noted as Controller 3

in Fig. 12, attempts to improve the response speed of Mo

by increasing the value of kMofrom 2.5 to 3. This

implies that the natural degrading effect of the mostrestrictive RHP-zero is moved to the output Mo. In thiscase, as expected from the controllability analysis, thereis a significant worsening of the transient response ofvariable To.Finally, Fig. 13 shows a comparison between the

performance achieved with the previous nominal controllerand that attained by means of a finely tuned multivariablePID controller. A similar performance can be observedin the response of Mo, while the evolution of To is faster(and almost without any overshoots) in the case of theH1 controller. However, implementing a multivariablePID had, in our case, led to an annoying 12-parametertuning procedure, while using the proposed methodology,

250 300 350 400 450 500

(minutes)

250 300 350 400 450 500

(minutes)

roller 3: κTo=3 κMo=3

=3

2.5

tion results.

ARTICLE IN PRESS

0 50 100 150 200 250 300 350 400 450 50035

36

37

38

39

40

41

42

time (minutes)

To

(°C

)

0 50 100 150 200 250 300 350 400 450 5000.95

1

1.05

1.1

1.15

1.2

1.25

time (minutes)

Mo

( %

)Controller 1: κTo=3 κMo=2.5

Controller 1: κTo=3 κMo=2.5

Multivariable PID controller

Multivariable PID controller

Fig. 13. Simulation results.

0 20 40 60 80 100 120 140 160 18030

35

40

45

50

55

time (minutes)

To

(°C

)

0 20 40 60 80 100 120 140 160 1800.3

0.35

0.4

time (minutes)

Mo

(%)

0 20 40 60 80 100 120 140 160 18040

45

50

55

time (minutes)

To

(°C

)

0 20 40 60 80 100 120 140 160 1800.2

0.25

0.3

0.35

0.4

0.45

time (minutes)

Mo

(%)

Fig. 14. Experimental results with Controller 1. Step change in Mo reference (left). Step change in To reference (right).

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500496

only two parameters had to be tuned (once oTdiagis chosen

from Fig. 11) following a few intuitive rules in a systematic way.

7. Experimental results

In this section, step responses of the output variables onthe real plant have been obtained to evaluate theperformance provided by the controllers.

Unlike the previous experiments performed on thesystem model, the experiments reported in this sectionhave been made at non-nominal operating points.The step responses of the real system obtained using the

nominal controller (Controller 1 with kTo¼ 3 and

kMo¼ 2:5) are presented in Fig. 14. From these responses,

two main issues can be pointed out. First, the achieved risetimes are similar to those attained by simulations using thenonlinear model. These times are approximately equal to70 and 100min for To and Mo, respectively. In addition,

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500 497

with this controller, changes at the reference of each outputvariable have no noticeable effect on the other output. Thecorresponding control inputs are plotted in Fig. 15.

Fig. 16 shows the experimental responses obtained bymeans of Controller 3 (with kTo

¼ 3 and kMo¼ 3) under

step change in the reference of Mo. It can be observed thatthe response of Mo is faster (with a rise time about 50min)than the one of Fig. 14, as expected from the increment ofthe value of kMo

. This faster response is achieved by meansof an excessive control effort, which makes the system to

0 20 40 60 80 100 120 140 160 1802

2.22.42.62.8

33.23.43.63.8

4

time (minutes)

Fp

(Kg/

min

)

0 20 40 60 80 100 120 140 160 1802.5

3

3.5

4

time (minutes)

Fc

(Kg/

h)

Fig. 15. Experimental control variables with Controller 1. Step cha

0 10 20 330

35

40

45

50

55

60

65

time

To

(°C

)

0 10 20 30.35

0.4

0.45

0.5

time

Mo

(%)

Fig. 16. Experimental results with Contr

reach its upper saturation level for the Fp control variable(see Fig. 17). Furthermore, it must be pointed out thedegradation of the response of To (more evident than theone corresponding to the simulation experiments) due tothe natural deteriorating effect of the most restrictive RHP-zero (see Section 4).It should also be noticed that in this experiment no

steady-state data have been recorded as the transientperiod of the experiment is long enough to show the maineffects of Controller 3.

0 20 40 60 80 100 120 140 160 1802

2.22.42.62.8

33.23.43.63.8

4

time (minutes)F

p (K

g/m

in)

0 20 40 60 80 100 120 140 160 1802.5

3

3.5

4

time (minutes)

Fc

(Kg/

h)

nge on Mo reference (left). Step change in To reference (right).

0 40 50 60

(minutes)

0 40 50 60

(minutes)

oller 3. Step changeon Mo reference.

ARTICLE IN PRESS

0 10 20 30 40 50 602

2.22.42.62.8

33.23.43.63.8

4

time (minutes)

Fp

( K

g/m

in)

0 10 20 30 40 50 602.5

3

3.5

4

time (minutes)

Fc

( K

g/h)

Fig. 17. Experimental control variables with Controller 3: step change on Mo reference.

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500498

Finally, it is interesting to note that the rise time isgreater than 52min for an appropriate performance,as stated in the controllability analysis performed inSection 4.

8. Conclusions

This article has presented the analysis and robust controlof a rotary dryer. A multivariable system has beenconsidered, with the outlet moisture of the dried productand the temperature of the exhaust air as output variables.The flow of the incoming wet material and the flow of fuelhave been used as control variables. Linear models havebeen obtained at different operating points from a testednonlinear simulator of the plant. A controllability analysisof the linear system has been carried out to show someconstraints in its performance. A multivariable robustcontrol, based on the H1 mixed sensitivity problem, hasbeen designed for the rotary dryer. Finally, simulation andexperimental results have been presented, which are inagreement with the conclusions provided by the controll-ability analysis.

Acknowledgements

The authors wish to acknowledge CICYT for foundingthis work under Grants DPI2004-06419 and DPI2003-00429.

Appendix A. Nonlinear dryer model

The set of physical parameters and variables involved inthe nonlinear model of the rotary dryer are listed below:Set of parameters and their numerical values:

Cs specific heat of solid (1.5 kJ/kg 1C )Cw specific heat of water (4.180 kJ/kg 1C)Cg specific heat at constant pressure of dry gas

(1.007 kJ/kg 1C)Cst specific heat at constant pressure of water steam

(1.890 kJ/kg 1C)R constant of perfect gas (8314 J/kmol 1C)l latent heat of vaporization at 0 �C (2502 kJ/kg)U volumetric heat transfer coefficient

ð0:3 kW=m3 �CÞV volume of each section ð0:201m3Þ

rs density of solid ð1400 kg=m3Þ

mw molecular weight of water (18 kg/kmol)mg molecular weight of gas in the output (29 kg/kmol)

The set of physical variables are given now. Most of themwill appear with the subscript i, meaning that they aredefined for every drum’s slice: i ¼ 0 n, where i ¼ 0represents variables at drum’s input (input to the first slice),while i ¼ n means the drum’s output (output from the lastslice).

Fsimass flow of dry solid (kg/s)

Fwimass flow of water in the solid (kg/s)

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500 499

Fgimass flow of dry gas (kg/s)

Fstimass flow of steam in the gas (kg/s)

msimass of dry solid (kg)

mwimass of water in the solid (kg)

mgimass of dry gas (kg)

mstimass of steam in the gas (kg)

Tsitemperature of solid (1C)

Tgitemperature of gas (1C)

X i moisture of solid in dry basis (kg water/kg drysolid)

Y i humidity of gas in dry basis (kg water/kg dry gas)Mi moisture of solid in wet basis (kg water/kg wet

solid)W i mass flow of evaporated water (kg/s)hsti

steam enthalpy (kJ)vvapi

vaporization speed (1/s)P gas pressure (Pa)

In order to determine typical operation points for thesevariables, the relations of the system inputs (F c and Fp)and outputs (To and Mo) with some of them have to beconsidered:

Fp ¼ F s0 þ Fw0� 0:05 kg=s,

To ¼ Tgn� 37 �C,

Mo ¼Mn � 0:01.

The set of equations relating these variables and para-meters are given next:

�

Dry solid balance (msiand F si

are unknowns):

qmsi

qt¼ Fsi�1

� F si.

�

Water balance in solid and gas (where mwi, Fwi

, msti, F sti

and W i are unknowns):

qmwi

qt¼ F wi�1

� Fwi�W i,

qmsti

qt¼ Fsti�1

� F stiþW i.

�

Dry gas balance:

qmgi

qt¼ Fgi�1

� F gi.

�

Energy balance in solid (the overall heat transfercoefficient is known):

q½ðCsmsiþ Cwmwi

ÞTsi�1�

qt¼ UV ðTgi

� TsiÞ

þ ðF si�1Cs þ Fwi�1

CwÞTsi�1

� ðF siCs þ Fwi

CwÞTsi�W ihsti

.

Considering that the enthalpy of the vaporized water is

hsti¼ CwTsi

þ lTsi¼ CstTsi

þ l

and using the previous mass balance equations in thesolid:

ðCsmsiþ Cwmwi

ÞqTsi

qt¼ UV ðTgi

� TsiÞ

þ ðF si�1Cs þ F wi�1

CwÞðTsi�1� Tsi

Þ

�W i½ðCst � CwÞTsiþ l�.

�

The mass balance equation of the gas:

ðCstmstiþ Cgmgi

ÞqTgi

qt¼ �UV ðTgi

� TsiÞ

þ ðFgi�1Cg þ Fsti�1

CstÞðTgi�1� Tgi

Þ �W iCstðTgi� Tsi

Þ.

�

There are also other static equations as the unknownfactors relate to one another. It is clear that:

X i ¼Fwi

F si

; X i ¼mwi

msi

,

Y i ¼Fsti

F gi

; Y i ¼msti

mgi

,

Mi ¼X i

1þ X i

.

�

The other static equations are given by the relationbetween the mass of dry gas and the mass of dry solid,which can be expressed as

mgi¼ V �

msi

rs

� �mgmw

mgY i þ mw

P

RðTgiþ 273Þ

.

�

The relation involving the vaporization speed of thesolid:

W i ¼ vvapmsi.

�

Finally, the equation that links mass and flow ofdry solid, using the concept of residence time, thatis, the mean time employed by the solid to cross thedrum:

msi¼ tresFsi

.

This time has been estimated through a correlationproposed in Saeman and Mitchell (1954), as a value ofapproximately 25min.

Therefore, it can be observed that there are 13 equationsand 13 unknowns.On the other hand, the dynamics of the combustion

system is quite fast compared to the drum’s dynamics.

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487–500500

A number of new parameters and variables appear. The listof parameters is:

Hcomb combustible heating power (39083 kJ/kg comb.)Zcomb combustion performance (0.95)Cgenv specific heat of inlet air (1.006 kJ/kg 1C)

The variables involved in the combustion model are:

Tenv input temperature (1C)Y env input humidity (kg water/kg dry air)Fgenv mass flow of inlet dry air (kg dry air/s)Tg0 output temperature from the combustion stage

(input to the first drum’s slice, i ¼ 0) (1C)Y 0 humidity of combustion gas (kg steam/kg dry gas)Fg0 mass flow of dry air from the combustion section

(input to the first drum’s slice) (kg dry gas/s)Qg0

flow of dry gas from the combustion sectionðm3=sÞ

F comb mass flow of combustible (equal to controlvariable Fc, but time units are seconds) (kg/s)

Pat atmospheric pressure (Pa)

Again some of these variables of the input and outputsignals are related. In particular, in this case it can bewritten as

F c ¼ F comb � 0:00072 kg=s.

The new set of equations, governing the combustionprocess is:

�

Gas balance:

Fg0 ¼ Fgenv þ F comb.

�

Energy balance:

FgenvCgenvT env þ F combHcombZcomb ¼ CgTg0F g0 .

�

Equation of perfect gas:

Fg0 ¼Qg0

mgPat

RðTg0 þ 273Þ.

�

Humidity of gas:

Y 0 ¼ Y env.

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