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215: 365 2001Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering

V Bobál, J Macháckek and P DostálMultivariable adaptive decoupling temperature control of a thermo-analyser

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365

Multivariable adaptive decoupling temperature controlof a thermo-analyser

V Bobal1*, J MachacÊek2 and P Dostal11Department of Control Theory, Institute of Information Technologies, Tomas Bata University in ZlÌn, Czech Republic2Department of Process Control and Computer Technique, Faculty of Chemical Technology, University of Pardubice,Czech Republic

Abstract: This paper deals with multivariable self-tuning temperature control of a thermo-analyser.The decoupling method is used and the parameters of two independent controllers are derived usingthe deadbeat approach with reference signal tracking. The recursive least-squares method with LDdecomposition and directional forgetting is used to estimate the parameters of the process model.This multivariable controller has been designed specially for technological processes where processoutput variables without overshoot are desired and reference signals are changed with steps and linearincreasing.

Keywords: multivariable control, decoupling, self-tuning controller, recursive identi�cation, dead-beat method, thermo-analyser

NOTATION Gij

(i, j=1, 2) polynomials, elements of the matrixG

G (q) matrix of the systemajki

model parameter estimatesH

ii(i=1) polynomials, elements of the matrixA

ij(i, j=1, 2) matrix polynomials

HA(qÕ1) matrix of the dynamics systemH (q ) transfer matrix function (systemA

1 , A2 polynomial matrices

with compensator)bjkci(i=1, 2, 3, j, k=1, 2)

I unit matrixpolynomial coeYcientsJ(£) minimization criterion of thebjk

imodel parameter estimates

recursive identi�cationBij

(i, j=1, 2) matrix polynomialsk discrete timeB(qÕ1) matrix of the dynamics systemK integrative termB

1 , B2 , B

c polynomial matricesM auxiliary polynomialC

ij(i, j=1, 2) polynomials, elements of the matrix

n(k) non-measurable random componentCN auxiliary polynomialC(k) covariance matrixp1 , p

2 controller parametersD, D1 matrices in the polynomial equationP, Q, R controller polynomialse errorq shift operatore(k) prediction errorq0 , q1 , q2 controller parametersE statistical averaging operatorr0 , r

Õ1 controller parametersE matrix in the polynomial equationr, r1 vectors in the polynomial equationF

ij(i, j=1, 2) polynomials, elements of the matrix

s0 , s

1 , s2 vector elementsF

S auxiliary polynomialF(q) compensator matrix functionu(k) input vector of the systemtransfer functionU (q) transform input vector of the systemv(k) output vector of the controllerV(q ) transform output vector of theThe MS was received on 25 October 2000 and was accepted after revision

for publication on 22 January 2001. controller* Corresponding author: Department of Control Theory, Institute of

w(k) vector of the reference signalsInformation Technologies, Tomas Bata University in ZlÌn, Nan. T GMasaryka 275, 762 72 ZlÌn, Czech Republic. y(k) output vector of the system

I07500 © IMechE 2001 Proc Instn Mech Engrs Vol 215 Part I



366 V BOBAL, J MACHACÊ EK AND P DOSTAL

Y(q) transform output vector of the trol algorithm is presented in detail in Section 3, exper-imental results are given in Section 4 and Section 5systemconcludes the paper.

á vector in the polynomial equationâ vector in the polynomial equationå(k), ú(k), è(k), ì(k), î (k)

auxiliary variables2 A DESCRIPTION OF THE THERMO-£(k) vector of the parameter estimates

ANALYSERr constantQ(k) adaptive forgetting factor

The properties of polymer materials depend on the tem-¼(k) regression vectorperature and mechanical conditions used during pro-cessing. Therefore it is necessary to determine thetemperature and time dependencies of the polymer mat-erial’s behaviour. Various thermo-analysers have been1 INTRODUCTIONdeveloped to measure this [14]. A material sample is putinto the heat chamber of such a thermo-analyser and theMany technological processes require that there be eithermaterial parameters of the sample are controlled follow-a prescribed or no overshoot at all on the process vari-ing a predetermined schedule. Specialized real-timeables from the reference signals. This requirement cansoftware is used for measurement, control and data pro-be ful�lled using the deadbeat or pole placementcessing. A special unit connects the mechanical appar-approach for a digital controller design. The thermo-atus to the controlling personal computer. The computeranalyser is a piece of equipment for testing polymerperforms the following tasks:materials. A multivariable controller has been developed

to control the temperature in its dual furnace. (a) measuring and controlling the temperature in theVarious methods have been suggested for the control heated chamber in a range of up to 400 ßC,

of multivariable systems. Controllers for a multivariable (b) setting the position of the measuring tip (measure-system can either have a multivariable design which ment of the shift),works with all input and output variables (see, for (c) controlling the rate and position of the tip duringexample, references [1 ] and [2 ]) or be a series of one- its dynamic actuating on the material sample,variable controllers, where coupling is suppressed, for (d) setting the frequency and form of the loading testexample, by the appropriate tuning of the controller par- signals,ameters (see, for example, reference [3 ]) or by one of (e) displaying the measurements taken from the vari-the decoupling methods. The decoupling methods sup- ables in digital and graphics form andpress undesirable interactions between input and output (f ) processing data on the material in a databasevariables so that each input aVects only one controlled system.variable. The controller can then be designed for inde-pendent one-variable loops. A comparison of several The main requirement of all these measurements is an

accurate temperature control program (most often linearmethods for multivariable control has been given in ref-erences [4 ] and [5 ]. On the basis of this comparison, temperature increase and then isothermal control with-

out overshoot ) with quick warming-through of thedecoupling by precompensator was chosen. This prob-lem is solved in many studies, such as those made by sample. Microsamples no more than 1 mm thick are used

to ensure that the material warms through quickly. TheyLuyben [6 ], Waller [7 ], Tade et al. [8 ], Chien et al. [9 ],Kinnaert et al. [10], Wittenmark et al. [11], Peng [12 ] are put into a miniature measurement furnace, consisting

of two parts. Both parts are heated by electrical resist-and Krishnaswamy et al. [13]. The main problem withthe decoupling methods is achieving internal stability, ance while the temperature is measured by two Pt 100

resistance thermometers (one for control, and the otherwhich is ensured when no pole zeros are cancelled.The control synthesis of the adaptive multivariable for temperature recording). Both parts of the furnace

are controlled independently and the true temperaturecontrollers designed below is based on the theoreticalmethods introduced in references [1 ] to [13 ]. This paper is kept as close as possible to the temperature set up by

the program. It is very important to ensure there is noconsists in applying these theoretical methods to thedesign of adaptive modi�cations for these approaches overshoot in temperature during the transition from

linear increase to isothermal control. The thermo-and their practical implementation in real-time con-ditions. The controller algorithms are given in the form analyser is connected to the personal computer by

the measurement and signal transformation unit. Theof the �nal resulting relations and may easily be appliedby other users. personal computer is connected to the interface unit

provided with the PCL-812PG acquisition card. A sche-The paper is organized as follows: Section 2 containsa description of the thermo-analyser, the self-tuning con- matic diagram of the mechanical apparatus of a thermo-

I07500 © IMechE 2001Proc Instn Mech Engrs Vol 215 Part I



367MULTIVARIABLE ADAPTIVE DECOUPLING TEMPERATURE CONTROL

analyser is shown in Fig. 1. The schematic diagram ofthe electrical parts is shown in Fig. 2.

A self-tuning single-input single-output controller basedon the pole placement method has been proposed to con-trol the temperature in both parts of the furnace [15].

The heating part of the thermo-analyser is a multivari-able interacting system with two input and two outputsignals. The input signals are electrical voltages in therange 0–10 V, which are converted to pulses of heat bya triac power unit. The output signals are the tempera-tures in both parts of the furnace converted to the samerange of voltages as the inputs. This corresponds to atemperature range 0–400 ßC.

The dynamic behaviour of this control system can bedescribed in a near steady state by the discrete linearmodel

A(qÕ1)Y(q)=B(qÕ1)U(q) (1)

where U(q ) is the (261) input vector, Y(q) is the (261)output vector, A(qÕ1) and B(qÕ1) are (262) leftcoprime polynomial matrices and q is the shift operator.The transfer function can be written in the form of theleft matrix fraction

G (qÕ1)=AÕ1(qÕ1)B(qÕ1) (2)

From the experiments it follows that each transferfunction can be approximated by the models withsecond-order polynomials without time delay: Fig. 2 Schematic diagram of the electrical part of a thermo-

analyserA(qÕ1)=I+A1qÕ1+A2qÕ2

B(qÕ1)=B1qÕ1+B

2qÕ2

(3)The initial state of the system output is supposed to

be zero. Further it is assumed, without loss of generality,that the matrix A is diagonal. Individual matrices in thepolynomials (3) then have the following structure:

A1=Ca11

10

0 a221D , A2

= Ca112

0

0 a222D

B1= Cb11

1b121

b211

b221D , B

2=Cb11

2b122

b212

b222D

(4)

3 CONTROL ALGORITHM

The adaptive controller described in this paper belongsto a class of self-tuning controllers based on:

(a) recursive identi�cation of the process model param-eters and

(b) controller synthesis using constantly recalculatedestimates of the process parameters.

The individual types of self-tuning controllers diVer inthe combination of methods for parameter identi�cationFig. 1 Schematic diagram of the mechanical part of a

thermo-analyser and controller synthesis.





The identi�cation part of the designed controller is wherecarried out by a recursive least-squares method; numeri- ú(kÕ1)=¼T(kÕ1)C(kÕ1)¼(kÕ1) (13)cal stability is improved by means of LD �lters, and

is an auxiliary scalar and�exibility is supported by directional forgetting [16, 17].The control strategy used here is a decoupling method e(k)=y(k)Õ£T(k)¼(kÕ1) (14)

with two independent deadbeat controllers tracking theis a prediction error in step k. If ú(kÕ1)>0, thenrequired temperature that the reference signal uses.a rectangular covariance matrix is computed by therecurrent algorithm

3.1 Recursive identi�cation algorithm

C(k)=C(kÕ1)ÕC(kÕ1)¼(kÕ1)¼T(kÕ1)C(kÕ1)

åÕ1(kÕ1)+ú(kÕ1)The autoregressive model with exogenous inputs [18] inthe form

(15)y(k)=£T(k)¼(kÕ1)+n(k) (5)

whereis applied in the identi�cation part of the designed con-troller, where £(k) is the vector of the parameter esti- å(kÕ1)=Q(k) Õ

1ÕQ(k)ú(kÕ1)

(16)mates and ¼(kÕ1) is the regression vector with processinput and output values. The non-measurable random

When ú(kÕ1)=0, thencomponent n(k) is assumed to have zero mean valueE [n(k)]=0 and constant covariance (dispersion) R= C(k)=C(kÕ1) (17)E [n2(k)].

The value of the adaptive forgetting factor Q(k) is cal-The identi�cation of the 262 system was divided into

culated in each sampling period by the relationtwo parts. A model with two inputs and one output wasused in both. The initial model (5) has a vector of the

Q(k)=A1+(1+r){ ln [1+ú(kÕ1)]}parameter estimates

£T1(k)=[a11

1a112

b111

b121

b112

b122

] (6)+G [u(kÕ1)+1]è(kÕ1)

1+ú(kÕ1)+è(k Õ1)Õ1Hand regression vector

¼T1(kÕ1)= [Õy1(kÕ1) Õy1(kÕ2) u1(kÕ1)

6ú(kÕ1)

1+ú(kÕ1)BÕ1(18)u2(kÕ1) u1(kÕ2) u2(kÕ2)] (7)

Subsequently identi�cation is repeated with the follow- whereing vectors:

£T2(k)=[a22

1a222

b211

b221

b212

b222

] (8) è(k)=e2(k)ì(k)

, u(k)=Q(k)[u(kÕ1)+1] (19)

¼T2(kÕ1)= [Õy2(kÕ1) Õy2(kÕ2) u1(kÕ1)

andu2(kÕ1) u

1(kÕ2) u2(kÕ2)] (9)

ì(k)=Q(k) Cì(kÕ1)+e2(kÕ1)

1+ú(kÕ1)D (20)The task of the recursive identi�cation is to �nd theparameter estimate vector £(k) which minimizes thegiven criterion are auxiliary variables and r is a constant (0¢r<1).

The start-up conditions for the most commonly usedJ(£ )= ~

k

i=k0

QkÕ i(k)e2(i ), Q(k)¢1 (10) identi�cation methods are the initial parameter estimatesand their covariance matrix. Although most users under-

where stand the importance of the initial parameter estimatesand, with a certain amount of eVort, are usually able toe(i )=y(i )Õ£T(i )¼(iÕ1)assign realistic values using their technical expertise, theimportance of the covariance matrix is often neglected= [1 Õ£T(i )] C y(i )

¼(iÕ1)D (11)and its value diYcult to assess. Although the issue of apriori information in the selection of start-up conditionsis the prediction error, Q(k) is the adaptive directionalhas been discussed in reference [18], it is suitable toforgetting factor and k0 is the initial identi�cation time.choose the following conditions for the start of the algor-The vector of the parameter estimates is then calculatedithm [16 ]; the elements of the main diagonal of theaccording to the square root version (LD decompositioncovariance matrix should be C(0)=103I, and the startof the covariance matrix) of the recursive relationsvalues for the directional forgetting factor Q(0)=1,ì(0)=0.001, u(0)=10Õ6 and r=0.99.£(k)=£(kÕ1)+

C(kÕ1)¼(kÕ1)

1+ú(kÕ1)e(kÕ1) (12)

When the forgetting factor is Q(k)=1, this algorithm





is equivalent to the well-known recursive least-squaresF=BÕ1 CC11 0

0 C22Dmethod. Estimated parameters were used for the compu-

tation of the control law in each sampling period.

=adj Bdet B CC11 0

0 C22D

=1

B11B22ÕB12B22 C B22C11

ÕB12C22ÕB21C11 B11C22

D (24)3.2 Decoupling method

All decoupling methods suppress to a certain extent the Various recommendations for choosing the poly-undesirable interactions between input and output vari- nomials C11 and C22 have been published. In theables so that each input aVects only one controlled vari- so-called ideal decoupling [7 ], the diagonal elements ofable. One way to achieve this aim is to use a method the matrix H are the same as in matrix G; therefore C

ii=

where the dynamic compensator is placed ahead of the Bii, i=1, 2. In this case the non-diagonal elements of

system (Fig. 3). the matrix H which have parallels in the matrix G areThe resulting transfer function H (q ) is then given by consequently fully suppressed.

The simplest matrix H is obtained when the poly-nomials C are chosen to be C11

=C22=1. When theH(q)=

Y(q)

V(q)=G(q)F(q) (21)

polynomials C are taken to be

C11=C22

=q det B (25)where F (q ) is the compensator transfer function. Thedecoupling condition is ful�lled when matrix H (q ) is the compensator transfer function takes the form of thediagonal. For a two-variable system, equation (21) takes adjoint matrix B:the form

F=q adj B (26)

CH11 0

0 H22D= CG

11G

12G21 G22

D CF11

F12

F21 F22D (22) The two-element compensator matrix F is another

candidate for compensator design. Usually two elementsare equal to unity, known as simplicity decoupling [7 ].

where all elements of the matrices are polynomial func- There are four variations of this method. For example,tions (the backward shift operator qÕ1 will be omitted where F11

= F22=1, the matrix F is given as

for simpli�cation). Only the parameters of system trans-fer matrix G, where the structure is given in equations(2), are estimated in the identi�cation algorithm. The

F= C 1 ÕB

12B11

ÕB21B22

1 D (27)solution of equation (22) is not unique as there are onlyfour conditions for the determination of six unknownfunctions (H

iiand F

ij). Two of these functions must be

somehow chosen, but not arbitrarily. The main con- and the resulting transfer functiondition is to ensure the internal stability of the resultingtransfer functions H. This stability may be compromisedif an unstable pole zero is cancelled.

H=det B C 1A11B22

0

01

A22

B11D (28)One possible solution to this problem is to assume

that matrix H takes the form

Only the compensator given by equation (26) ensuresinternal stability for an arbitrary process transfer func-H=CC

11A11

0

0C22A22

D (23)tion G. As this transfer function is calculated by on-lineidenti�cation, no assumptions about the stability ofzeros and poles of its partial transfer functions can bemade. The compensator (26) was therefore used in theThe compensator transfer function F is derived usingdesigned controller.equation (22):

The resulting transfer function of a system with pre-compensator is given by equations (2), (21) and (26):

H=GF=AÕ1Bq adj B=AÕ1q det B (29)

After denotation,

Fig. 3 System with a compensator Bc=q det BI (30)





The symmetrical diagonal matrix Bc has the structure at zero; therefore the parameters of the polynomials Pand Q are determined as the minimum-degree solutionB

c=B

c1qÕ1+B

c2qÕ2+B

c3qÕ3 (31)

of the equationwhere

AKP+BcQ=1 (38)

b11c3

=b22c3

=b112

b222

Õb122

b212

(32)The degrees of the polynomials are given by equations

The resulting transfer function is then given byqP=qB

cÕ1, qQ=qA+qKÕ1 (39)

H=AÕ1Bc (33)In this case, qP=2 and qQ=2.

The controller parameters are calculated from the par- The reference signals are created according to Fig. 5ameters of the resulting transfer function H. with step and linear increase parts. The reference

The compensator transfer function (26) is used to cal- sequence of each part can be modelled by the equationculate process input signals u from controller output v

w=MÕ1N (40)according to

In the case of unit step changes, N=1, M=1ÕqÕ1 and,Cu1u2D=C qB22

ÕqB12ÕqB

21qB

11D Cv1

v2D (34) in the case of linear ramps with increment ¢, N=¢qÕ1,

M=1Õ2qÕ1+qÕ2.After substituting from equations (38) and (40), equa-

tion (37) can be rewritten in the form3.3 Computation of controller parametersy=BcRMÕ1N (41)

As the matrix H is diagonal, the system can be dividedThe polynomial R is calculated from the condition thatinto two independent single-input single-output loops.the sequence of errorThe block diagram of the two controllers is shown in

Fig. 4. Each controller consists of three parts: feedbacke=wÕy=MÕ1N(1ÕBcR) (42)with transfer function PÕ1Q (indices will not be used),

feedforward with transfer function PÕ1R, and integral is of minimal length. This is true whenaction KÕ1, which ensures zero steady state error under

BcR+SM=1 (43)the in�uence of disturbance. P, Q and R are polynomials,

K=1ÕqÕ1 and the variables w1 and w2 are reference The degree of polynomial, R, is modi�ed according tosignals. the shape of reference signal sequence

The equation for the input signals results directly fromqR=qMÕ1 (44)Fig. 4:

and the degree of auxiliary polynomial, S, isv=KÕ1(PÕ1RwÕPÕ1Qy) (35)

qS=qBcÕ1 (45)The output variable using equation (35) is given by

The sequence of error is theny=AÕ1Bcv=AÕ1Bc

KÕ1(PÕ1RwÕPÕ1Qy) (36)

Equation (36) can be rewritten, after slight rearrange- e=NS (46)ment, in the form

The controller equation is given byy=(AKB+BcQ)Õ1BcRw (37)

v=KÕ1(PÕ1RwÕPÕ1Qy) (47)As the temperatures y should track the reference sig-

nals w, the deadbeat method based upon the polynomialtheory [1, 2 ] was used. The denominator poles are placed

Fig. 5 Time behaviour of the reference signalFig. 4 Block diagram of the two-variable control system





where particular polynomials are given by From equation (53),

P=1+p1qÕ1+p

2qÕ2, Q=q

0+q

1qÕ1+q

2qÕ2

r0+r

1=

1bc1

+bc2

+bc3

(61)R=r0

+rÕ1qÕ1, K=1ÕqÕ1

(48) and, from equation (56),

for a reference signal with a linear increase or R= r0 r0=

1bc1

+bc2+bc3

(62)for a step increase.The solution of equations (38) and (43) gives the

unknown controller parameters. The polynomial equa- The control law in the formtion (38) leads to the matrix equation

v(k)=(1Õp1)v(kÕ1)+( p1Õp2)v(kÕ2)

Ep=á (49)+p2v(kÕ3)+r0w(k)+r1w(kÕ1)Õq0y(k)

whereÕq1y(kÕ1)Õq2y(kÕ2) (63)

can then be used for both the linear increase and theconstant value of the reference signal. The output vari-able only overshoots at the point where the linear refer-

E=C 1 0 0 0 0 0a1

Õ1 1 0 bc1 0 0a2

Õa1 a1Õ1 1 bc2 bc1 0

Õa2 a2Õa1 a1

Õ1 bc3 bc2 bc10 Õa2 a2

Õa1 0 bc3 bc20 0 Õa2 0 0 bc3

D ence signal changes to constant. To eliminate thisovershoot the following condition is used for the control-ler output u(k): if w(k)=w(k+1), then

v(k)=(1Õp1)v(kÕ1)+( p1Õp2)v(kÕ2)(50)

+p2v(kÕ3)+r0w(k)+r1w(k)Õq0y(k)p= [1 p1 p2 q0 q1 q2 ]T (51)Õq1y(kÕ1)Õq2y(kÕ2) (64)á= [1 0 0 0 0 0]T (52)

else use equation (63).For a reference signal with a linear increase, the poly-The control algorithm then has following computingnomial identity (43) gives the matrix equation

steps:Dr=â (53)Step 1. New process parameter estimates: equationswhere

(11) to (20); repeat for both outputs.

Step 2. Matrix Bc according to equation (32).

Step 3. Controller parameters from equations (49)D=C 1 0 0 0 0

Õ2 1 0 bc1 0

1 Õ2 1 bc2 bc10 1 Õ2 b

c3bc2

0 0 1 0 bc3D (54)

and (53).

Step 4. Compensator input variables v1(k) and v

2(k)using control law (63) or (64).

r= [s0s1

s2

r0

r1 ]T (55) Step 5. Process input variables u1(k) and u2(k) according

to equation (34).â= [1 0 0 0 0]T (56)

For a reference signal with a step increase, equation This procedure is repeated in each sampling period.(43) can be written as A controller designed in this way ensures that, if the

reference signal w(k) changes course [ linearly increasingD1r1

=â1 (57)

signal changes to constant; see equation (46)], the outputwhere y(k) always tracks the reference signal accurately after

three sampling periods

D1= C 1 0 0 0

Õ1 1 0 bc10 Õ1 1 b

c20 0 Õ1 bc3

D (58)4 EXPERIMENTAL RESULTS

The controller is programmed in MATLAB–r1= [s0 s1 s2 r0 ]T (59)

SIMULINK and for industrial use the controller algor-â

1= [1 0 0 0]T (60) ithm was rewritten to language C++ .

Figure 6 shows the practical veri�cation results of theThe following expressions can be derived for the param-eters of polynomial R. real-time temperature control in the lower and upper





Fig. 6 Temperature control without a compensator and feedforward part of the control loop

furnace using two independent controllers without a e1(t) and e

2(t) respectively. A comparison between Figs6 and 7 and between Figs 8 and 9 shows that the compen-compensator and without a feedforward part of the con-

trol loop (R=Q ). Figure 7 shows the multivariable con- sator F and the feedforward part of controller, PÕ1R,substantially improve the quality of process control (thetrol veri�cation using the proposed decoupler and the

feedforward part of the controller in the control-loop. output process variable tracks the reference signal almostexactly). Only relatively small overshoots at the begin-Figures 8 and 9 show the time behaviour of the errors

Fig. 7 Temperature control with a compensator and feedforward part of the control loop

Fig. 8 Time behaviour of the error control without a compensator and feedforward part of the control loop





Fig. 9 Time behaviour of the error control with a compensator and feedforward part of the control loop

of decentralized PID controllers for TITO process.ning of control result from the adaptation of the param-Automatica, 1995, 31, 1011–1017.eter estimates. There are minor errors when the reference

4 MachacÊek, J. and Kotyk, J. Adaptive decoupling con-signal alters. When no compensator is used, the maxi-trol of distillation column. In Proceedings of the 3rdmum overshoots are, in the upper furnace, 4.26 ßC and,IEEE Conference on Control Applications, Strathclydein the lower furnace, 2.01 ßC. When the compensator isUniversity, Glasgow, August 1994, pp. 263–268 (IEEE

used, the maximum overshoots are, in the upper furnace, Control Systems Society, New York).2.34 ßC and, in the lower furnace, 1.05 ßC. 5 MachacÊek, J. and Kotyk, J. Multivariable control of distil-

lation column. In Proceedings of the 4th IFAC Symposiumon Dynamics and Control of Chemical Reactors, DistillationColumns and Batch Processes (DYCORD’95), Helsingor,5 CONCLUSIONSJune 1995, pp. 185–190 (International Federation ofAutomatic Control, New York).

In this contribution a multivariable self-tuning deadbeat 6 Luyben, W. L. Distillation decoupling. Am. Inst. Chem.controller with compensator to suppress undesirable Engrs J., 1970, 16, 198–202.

7 Waller, K. V. T. Decoupling in distillation. Am. Inst. Chem.interactions between input and output variables has beenEngrs J., 1974, 20, 592–594.proposed. This controller enables the nearly perfect

8 Tade, M. O., Bayoumi, M. M. and Bacon, D. W. Adaptivetracking of linear increases in the reference signal with-decoupling of a class of multivariabledynamic system usingout overshoots of the controlled output. The practicaloutput feedback. Instn Electl Engrs Proc., 1986, 133,implementation of the control algorithm con�rmed the265–275.

theoretical assumptions. The design method is relatively 9 Chien, I. L., Seborg, D. E. and Mellichamp, D. A. Self-simple, suYciently robust and suitable for the control of tuning control with decoupling. Am. Inst. Chem. Engrs J.,a large class of plants. The proposed multivariable self- 1987, 33, 1076–1088.tuning controller has been successfully used for tempera- 10 Kinnaert, M., Hanus, R. and Henrotte, J. I. A new decoup-ture control in both parts of a thermo-analyser furnace. ling precompensator for indirect adaptive control of multi-

variable linear systems. IEEE Trans. Autom. Control, 1987,AC-32, 455–459.

11 Wittenmark, B., Middleton, R. H. and Goodwin, G. C.ACKNOWLEDGEMENTS Adaptive decoupling of multivariable systems. Int. J.

Control, 1987, 46, 1993–2009.12 Peng, Y. A general decoupling precompensator for linearThis work was supported in part by the Grant Agency

multivariable systems with application to adaptive control.of the Czech Republic under Grants 102/99/1292 andIEEE Trans. Autom. Control, 1990, AC-35, 344–348.102/00/0526 and in part by the Ministry of Education of

13 Krishnaswamy, P. R., Shukla, N. V. and Deshpande, P. B.the Czech Republic under Grant MSM 2811 00001.Reference system decoupling for multivariable control. Ind.Engng Chem. Res., 1991, 30, 662–670.

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