Teaching Multivariable Control Using

the Quadruple-Tank Process1 2

Karl Henrik Johanssony, Alexander Horchz, Olle Wijkz, and Anders Hanssonz

yDepartment of Electrical Engineering and Computer Sciences

University of California, Berkeley, CA 94720-1770, U.S.A.

johans@eecs.berkeley.edu

zDepartment of Signals, Sensors and Systems

Royal Institute of Technology, S-100 44 Stockholm, Sweden

fhorch|olle|hanssong@e.kth.se

Abstract

A recent multivariable laboratory process is presentedtogether with its use in a graduate control course.The process is called the Quadruple-Tank Process anddemonstrates a multivariable level control problem.The multivariable zero dynamics of the system can bemade both minimumphase and nonminimumphase bysimply changing a valve. This makes the Quadruple-Tank Process suitable for illustrating many conceptsin linear and nonlinear multivariable control. In thispaper some of these are described together with thebasic setup of the process. Two computer exercises andone laboratory exercise have been developed as part ofa course in multivariable and nonlinear control. Theseare detailed and some experience from the course ispresented.

1 Introduction

Multivariable control is today taught in most controlcurricula. This includes both linear and nonlinear de-sign methods. Recently it has been an increased interestin emphasizing the limitations the process imposes oncontrol designs [5, 3, 19]. This is natural since new tech-nologies have opened possibilities for integrated solu-tions, where old control con�gurations may be changedto achieve an enhanced overall performance. A goodcontrol engineering course should be accompanied byhands-on experiments [2, 13, 1]. It is, however, nottrivial to �nd pedagogical examples to illustrate multi-variable performance limitations in feedback systems.

1The work by Karl Henrik Johansson was supported by the

Swedish Foundation for International Cooperation in Research

and Higher Education.2IEEE Conference on Decision and Control, Phoenix, AZ,

1999.

There exists only a handful of multivariable laboratoryprocesses, available frommanufactures such as QuanserConsulting, Educational Control Products, FeedbackInstruments, and TecQuipment. None of these illus-trate the e�ect of multivariable zeros on the closed-loopcontrol performance.

In this paper we describe a laboratory process, whichwas designed to illustrate the importance of multi-variable zeros and how the zeros may change due tovariations in the process. The process is called theQuadruple-Tank Process [8, 11, 10] and is a level con-trol problem based on four interconnected water tanksand two pumps. The inputs to the process are the volt-ages to the two pumps and the outputs are the wa-ter levels in the lower two tanks. The Quadruple-TankProcess can easily be built by using standard equip-ment available in many control laboratories. The setupis thus simple, but still the process can illustrate severalinteresting multivariable phenomena. One of its mainfeatures is that the zero dynamics can be made min-imum phase or nonminimum phase by simply chang-ing a valve. For the linearized model of the process,both the zero location and the zero direction havedirect physical interpretations. Also the relative gainarray has a straightforward meaning for the process.The Quadruple-Tank Process is thus suitable to use inteaching fundamentals of multivariable control.

The Quadruple-Tank Process was developed and built1996 at Lund Institute of Technology, Sweden, in or-der to illustrate the importance of multivariable zerolocation for control design [8]. Since then it has beenused extensively in the control curriculum in Lund. Anumber of student projects as part of courses in adap-tive control, system identi�cation, and real-time con-trol have been performed on the Quadruple-Tank Pro-cess. These projects consist of modeling from physi-cal or experimental data, control design, simulation,

p. 1

implementation, and evaluation together with a shortwritten and oral presentation. The Quadruple-TankProcess has also been the topic for master thesisprojects. Nunes [17] demonstrated decentralized PIDcontrol. Automatic tuning of multivariable PID con-trollers based on relay feedback was studied by Re-cica [18]. As a CalTech summer project, Grebeck [7]investigated what performance that was achievable forthe Quadruple-Tank Process using various optimal con-trol methods. He concluded that in the minimum-phase setting for the process there was no advantageof using centralized multivariable control, while in thenonminimum-phase setting H1 control gave better re-sponses. The Quadruple-Tank Process is also used inthe education at the Royal Institute of Technology inStockholm, as described in the paper, and at Univer-sity of Delaware. Other research institutes are currentlycopying the design and there is also ongoing commer-cial development.

The outline of the paper is as follows. A nonlinearmodel for the Quadruple-Tank Process based on phys-ical data is derived in Section 2. The location and thedirection of a multivariable zero of the linearized modelare derived in Section 3. It is shown that the valve po-sitions of the process uniquely determine if the systemis minimum phase or nonminimumphase. The relativegain array is discussed in Section 4. Section 5 givesan overview of our experiences in using the Quadruple-Tank Process in teaching. Some concluding remarks aregiven in Section 6. More details on the Quadruple-TankProcess are found in [10], see also [8, 11, 9].

2 The Quadruple-Tank Process

In this section we derive a physical model for theQuadruple-Tank Process. A schematic diagram of theprocess is shown in Figure 1. The target is to control thelevel in the lower two tanks with two pumps. The pro-cess inputs are v1 and v2 (input voltages to the pumps)and the outputs are y1 and y2 (voltages from level mea-surement devices). Mass balances and Bernoulli's lawyield

dh1dt

= �a1A1

p2gh1 +

a3A1

p2gh3 +

1k1A1

v1;

dh2dt

= �a2A2

p2gh2 +

a4A2

p2gh4 +

2k2A2

v2;

dh3dt

= �a3A3

p2gh3 +

(1� 2)k2A3

v2;

dh4dt

= �a4A4

p2gh4 +

(1� 1)k1A4

v1;

where Ai is the cross-section area of Tank i, ai thecross-section area of the outlet hole, and hi the waterlevel. The voltage applied to Pump i is vi and the cor-responding ow is kivi. The parameters 1; 2 2 [0; 1]

v1 v2

y1 y2Tank 1 Tank 2

Tank 3 Tank 4

Pump 1 Pump 2

Figure 1: Schematic diagram of the Quadruple-Tank Pro-cess. The water levels in Tank 1 and Tank 2 arecontrolled by two pumps. The positions of thevalves determine the location of a multivariablezero for the linearized model. The zero can beput in either the left or the right half-plane.

are determined from how the valves are set prior to anexperiment. The ow to Tank 1 is 1k1v1 and the owto Tank 4 is (1� 1)k1v1 and similarly for Tank 2 andTank 3. The acceleration of gravity is denoted g. Themeasured level signals are y1 = kch1 and y2 = kch2.

The transfer matrix of the linearized system is

G(s) =

2664 1c1

1 + sT1

(1� 2)c1(1 + sT3)(1 + sT1)

(1� 1)c2(1 + sT4)(1 + sT2)

2c21 + sT2

3775 ;(1)

where the time constants are

Ti =Ai

ai

s2h0

i

g; i = 1; : : : ; 4;

c1 = T1k1kc=A1, and c2 = T2k2kc=A2. Note the waythe parameters 1 and 2 enter the transfer matrix.Particularly, we see that if either 1 = 1 or 2 = 1,the transfer matrix is triangular and has no �nite ze-ros. This corresponds to that the ow through one ofthe valves are directed only to the corresponding lowertank. Physical data for the process are given in [10].

p. 2

3 Physical Interpretation of Zero

The zero locations and their directions of the trans-fer matrix (1) are considered in this section. It isshown that they have intuitive physical interpretationsin terms of how the valves 1 and 2 are set.

Zero location

The zeros of G are the zeros of the numerator polyno-mial of the rational function

detG(s) =c1c2

1 2Q

4

i=1(1 + sTi)

�

�(1 + sT3)(1 + sT4) �

(1� 1)(1� 2)

1 2

�:

It follows that the system is nonminimum phase for

0 < 1 + 2 < 1

and minimum phase for

1 < 1 + 2 < 2:

The multivariable zero being in the left or in right half-plane has a straightforward physical interpretation. Letqi denote the ow through Pump i and assume thatq1 = q2. Then the sum of the ows to the upper tanksis [2 � ( 1 + 2)]q1 and the sum of the ows to thelower tanks is ( 1 + 2)q1. Hence, the ow to the lowertanks is greater than the ow to the upper tanks ifthe system is minimum phase. The ow to the lowertanks is smaller than the ow to the upper tanks if thesystem is nonminimumphase. It is intuitively easier tocontrol y1 with v1 and y2 with v2, if most of the owsgoes directly to the lower tanks. There is thus an im-mediate connection between zero location and physicalintuition. The control problem is particularly hard ifthe total ow going to the left tanks (Tanks 1 and 3) isapproximately equal to the total ow going to the righttanks (Tanks 2 and 4). This corresponds to 1+ 2 � 1,i.e., a multivariable zero close to the origin.

Zero direction

An important di�erence between scalar systems andmultivariable systems is that not only the location ofa multivariable zero is important but also its direction.We de�ne the (output) direction of a zero z as a vector 2 R2 of unit length such as TG(z) = 0. If is equalto a unit vector, then the zero is only associated withone output. If this is not the case, then the e�ect of aright half-plane zero may be distributed between bothoutputs. In that sense, a multivariable right half-planezero must not deteriorate the performance as much asa corresponding scalar zero. For the transfer matrix Gin (1), the zero direction of a zero z > 0 is given by

= ( 1; 2)T such that

T

2664 1c1

1 + zT1

(1� 2)c1(1 + zT3)(1 + zT1)

(1� 1)c2(1 + zT4)(1 + zT2)

2c21 + zT2

3775 = 0:

Note that it follows from this equation that 1; 2 6= 0,so the zero is never associated with only one output.If we solve the equation for 2 and simplify, it followsthat

1 2

= �1� 1 1

�c2(1 + zT1)

c1(1 + zT4)(1 + zT2):

From this equation it is possible to conclude that if 1 issmall, then z is mostly associated with the �rst output.If 1 is close to one, then z is mostly associated withthe second output. Hence, for a given zero location, therelative size of 1 and 2 determines which output theright half-plane zero is related to.

The relation between 1 and 2 and the zero location zand the zero direction ( 1; 2)T can be described as amap ( 1; 2) 7! (z; 1= 2). Each valve position ( 1; 2)de�nes a unique zero con�guration (z; 1= 2). In thisway, any zero location and direction can be tested, orequivalently one valve knob can be used to choose zerolocation and the other zero direction.

4 Relative Gain Array

The relative gain array (RGA) was introduced by Bris-tol [4] as a measure of interaction in multivariable con-trol systems. The RGA � is de�ned as � = G(0) �G�T (0), where the asterisk denotes the Schur product(element-by-element matrixmultiplication) and �T in-verse transpose. It is possible to show that the elementsof each row and column of � sum up to one, so for a 2�2system the RGA is determined by the scalar � = �11.The RGA is used as a tool mainly in the process indus-try to decide on control structure issues such as input{output pairing for decentralized controllers [16, 12, 20].McAvoy [14] proposed that one should strive for a pair-ing with 0:67 < � < 1:50. The system is particularlyhard to control if � < 0.

The RGA of the Quadruple-Tank Process is easily de-rived to be given by

� = 1 2

1 + 2 � 1: (2)

Note that the RGA is only depending on the valve set-tings. Figure 2 shows a contour plot of � as a functionof 1 and 2. The edges of the box 1; 2 2 [0; 1] cor-responds to � = 0 and � = 1, respectively, as is shownin the �gure. The magnitude of � increases as 1 + 2

p. 3

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

� = 0

� = �1

� = 1� = 2

Figure 2: Contour plot of the RGA � as a function of thevalve parameters 1 and 2. The plot is drawnfor � = �10;�9; : : : ; 10. We have � � 0 belowthe straight line between (0; 1) and (1; 0) and� � 1 above the line. Close to the line, j�j islarge.

becomes close to one. There are no 1; 2 2 (0; 1) sothat � 2 (0; 1). From (2) we see for instance that � < 0if 1 + 2 < 1, which corresponds to the nonminimum-phase setting discussed previously.

If the valves of the quadruple-tank process are set suchthat 1 + 2 < 1, then the RGA analysis suggests thatthe input{output pairing v1{y2 and v2{y1 should bechosen for decentralized control. Let

eG =

�G21 G22

G11 G12

�be the linearized model with y1 and y2 permuted. TheRGA for eG is

e� = (1� 1)(1 � 2)

1� 1 � 2:

Hence, if 1 + 2 < 1 then e� > 0 so a decentralizedcontrol structure corresponding to eG is preferable ac-cording to the RGA. This is intuitive from physicalconsiderations, because 1 + 2 < 1 means that morewater is owing to the upper tanks than directly to thelower ones. Decentralized control is then easier usingv1 to control y2 and v2 to control y1, than vice versa.

5 Teaching Experiences

Recently a control course in multivariable and non-linear control was developed at the Royal Institute ofTechnology, where the Quadruple-Tank Process is used

extensively throughout the course. The prerequisites isa basic course in control. The recent textbook [6], whichcovers both linear and nonlinear control techniques, isused. Two computer exercises and a laboratory exer-cise based on the Quadruple-Tank Process are part ofthe course. They are described next.

Computer Exercise 1

In the �rst computer exercise the process dynam-ics is investigated by computing the poles and zerosof the system. The singular value frequency responseis investigated as well as the RGA. Step responsesare conducted in order to verify the coupling pre-dicted by the RGA. Comparisons are made between theminimum-phase case and the nonminimum-phase case.The conclusion is that the coupling is severe for thenonminimum-phase case. Then decentralized PI controlis investigated. A design is made by pairing the inputsand outputs of the process as suggested by the RGA.The PI controller is tuned such that for the minimum-phase case the cross-over frequency is 0.1 rad/sec witha phase margin of 80� and for the nonminimum-phasecase 0.006 rad/sec with 60�. The singular values of thesensitivity and complementary sensitivity functions areplotted in order to verify performance and robustness.The designs are further evaluated in simulations andthe coupling between the loops are investigated. It isseen that the interaction is large for the nonminimum-phase case.

Computer Exercise 2

In the second computer exercise PI controller designsbased on decoupling are investigated followed by themethod of Glover-McFarlane [15] to robustify a nomi-nal design. Both static and dynamic decoupling are in-vestigated. It is noticed that dynamic decoupling is su-perior for the minimum-phase case. The reason for thisis that the static decoupling is poor for frequencies closeto the desired cross-over frequency of the loop transferfunction. For the nonminimum-phase case static de-coupling is superior, because the static decoupling isgood enough for the lower bandwidth. At the end of thesecond computer exercise Glover-McFarlane's methodis used to robustify the design. Plotting the singularvalues of the sensitivity and complementary sensitiv-ity functions veri�es the improvement, see Figure 3.The �gure shows the sensitivity and the complementarysensitivity functions for both the minimum-phase andthe nonminimum-phase setting. Statically decoupledPI control as well as the robusti�ed Glover-McFarlanedesign are presented. Only the maximum singular val-ues are shown. The minimum-phase setting is almost amagnitude faster than the nonminimum-phase setting.(Note the di�erent frequency scales.) This illustratesthe limitation on closed-loop bandwidth that a multi-variable right half-plane zero imposes.

p. 4

10−3

10−2

10−1

100

10−2

10−1

100

Mag

nitu

de

Frequency (rad/sec)

10−4

10−3

10−2

10−1

10−2

10−1

100

Mag

nitu

de

Frequency (rad/sec)

Figure 3: Sensitivity functions and complementary sensi-tivity functions for the minimum-phase setting(upper diagram) and the nonminimum-phasesetting (lower diagram). PI control designswith static decoupling (dashed) and Glover-McFarlane designs (solid) are shown. Note thatthe bandwidth for the nonminimum-phase caseis almost a magnitude lower than for theminimum-phase case. This is due to the perfor-mance limitation that the multivariable righthalf-plane zero imposes.

Laboratory Exercise

In the laboratory exercise the students are asked toverify the theoretical results of this paper. Then exper-iments are performed in order to determine the physi-cal parameters of the process, such as the valve settingsand the proportional constants relating the pump volt-ages to the water ows. Then the designs from the com-puter exercises are redone using the identi�ed physicalmodel of the process. Manual control is investigated aswell as decentralized and statically decoupled PI con-trol. Both the minimum-phase and the nonminimum-phase cases are considered. As an example, Figure 4shows experiments for both these cases when staticallydecoupled PI control is applied. Note how much slowerthe responses for the nonminimum-phase process are

0 500 1000 15000.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

Nor

mal

ised

tank

leve

l

0 500 1000 1500 20000.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Time [s]

Nor

mal

ised

tank

leve

l

Figure 4: Experiments for the minimum-phase (upperdiagram) and nonminimum-phase (lower dia-gram) process controlled by statically decou-pled PI controllers. The levels y2 of Tank 2(upper curves) and y1 of Tank 1 (lower curves)are shown together with their setpoints. Tanklevels equal to one correspond to full tanks.

as compared to the minimum-phase process.

The experiments illustrate for the students that multi-variable control is important if process limitations suchas right half-plane zeros are present. They also showthat if there are no tight limitations, then scalar PIcontrol in each loop is su�cient. Added to these, theexercise clarify that there might be fundamental con-straints due to the process design that no linear feed-back controller can remove.

6 Conclusions

The Quadruple-Tank Process has been presented. It isa recent laboratory process that was designed in orderto illustrate various concepts in multivariable control.In particular, it is suited to demonstrate performancelimitations in multivariable control design caused by

p. 5

right half-plane zeros. This follows from that the pro-cess has a multivariable zero that in a direct way isconnected to the physical positions of two valves. Thepositions are given by the parameters 1; 2 2 (0; 1).It was shown in the paper that 1 + 2 determines thelocation of the zero, so that if 1 + 2 < 1 the systemis nonminimum phase and if 1 + 2 > 1 the systemis minimum phase. The quotient 1= 2 gives the zerodirection.

Some examples of how the Quadruple-Tank Processhas been used in teaching have also been presented.The conclusion of the teaching experiences is that theQuadruple-Tank Process is very well suited for demon-strating both the e�ects of coupling and performancelimitations in multivariable control systems. A particu-lar course on multivariable control was described. Com-puter exercises were shown to be a good preparation fora laboratory exercise. A remark on the laboratory ex-periments is that since the closed loop bandwidth forthe nonminimum-phase case is low, the experimentstake a relatively long time to conduct. It is there-fore crucial to implement the controllers such that thetransfer between manual and automatic mode is bump-less. We also found it important to inform the studentsabout the time the di�erent response experiments take.Prior to that some of the students thought that some-thing was wrong with their design and thus interruptedthe experiments to start to trouble shooting, althoughthey were only experiencing the large di�erence be-tween setpoint response times for minimum-phase andnonminimum-phase processes.

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