Robust Control of a Multivariable Experimental Four-Tank System

Rajanikanth Vadigepalli, Edward P. Gatzke, and Francis J. Doyle III*

Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716

A multivariable laboratory process of four interconnected water tanks is considered for modelingand robust control. The objective of the current study is to design and implement a variety ofcontrol algorithms and present corresponding robust control analysis for a multivariablelaboratory four-tank process. The control methodologies considered are (i) decentralized PIcontrol, (ii) “inner-outer” factorization-based multivariable internal model control (IMC), and(iii) µ-analysis-based H∞ control. The three control algorithms are comparatively analyzed usingstandard robustness measures for stability and peformance. The algorithms are implementedon the four-tank system, and the performance is compared in reference tracking and disturbancerejection cases. Using experimental data, the process dynamics and disturbance effects aremodeled. Input-output controllability analysis is performed to characterize achievable perfor-mance. A lumped unstructured uncertainty description is used to represent the variations inthe system. The three controllers designed are implemented on the four-tank experimentalapparatus using a Bailey control system with a MATLAB and Freelance software interface.The controllers are shown to provide stable and acceptable performance for reference trackingand disturbance rejection experiments. The multivariable IMC and H∞ controllers performedsimilarly and provided better performance than the decentralized PI control.

1. Introduction

Typical chemical plants are tightly integrated pro-cesses that exhibit nonlinear behavior and complexdynamic properties. For decades, the chemical processindustry has relied on single-loop linear controllers toregulate such systems. In many cases, the tuning ofthese controllers can be described as “heuristic”. Theperformance of such a control design in an uncertainprocess environment is difficult to predict. Although anonlinear process model of a system might be available,linear process models are usually developed for control-ler synthesis and system analysis. With a linear model,an uncertainty characterization can be used to math-ematically describe the model error and other variationsin the system. Desirable performance criteria can beestablished as well. Optimal controllers can be designedusing well-known tools such as H∞ synthesis andµ-analysis.1,2 These robust controller design techniqueshave been demonstrated in various experimental pro-cess applications.3-7

The four-tank system has attracted recent attentionbecause it exhibits characteristics of interest in bothcontrol research and education.8-10 A similar processwith three interconnected tanks has been used as abenchmark system for control research.7 The four-tanksystem exhibits elegantly complex dynamics that emergefrom a simple cascade of tanks. Such dynamic charac-teristics include interactions and a transmission zerolocation that are tunable in operation. With appropriate“tuning”, this system exhibits nonminimum-phase char-acteristics that arise completely from the multivariablenature of the problem. The four-tank system has beenused to illustrate both traditional and advanced mul-tivariable control strategies10-12 and as an educationaltool in teaching advanced multivariable control tech-

niques.9 The specific experiment presented in this studyis significantly larger in scale than all previouslypublished versions of the system by approximately 1-2orders of magnitude, and consequently, it introduces anumber of nonidealities including gravitational headsand piping friction losses, in addition to behaviorsspecific to large tanks such as vortices and splashingeffects on level measurement. These characteristics,combined with the complex interactive behavior, enablethe four-tank system to serve as a valuable educationaltool, as well as to provide interesting challenges incontrol design.10,12

The objective of the current study is to design andimplement a variety of control algorithms and presentcorresponding robust control analysis on a multivariablelaboratory four-tank process. The control methodologiesconsidered are (i) decentralized PI control, (ii) “inner-outer” factorization-based multivariable internal modelcontrol (IMC), and (iii) µ-analysis-based H∞ control. Thethree control algorithms are comparatively analyzedusing standard robustness measures for stability andpeformance. The algorithms are implemented on thefour-tank system and the performance is compared inreference tracking and disturbance rejection cases.

The particular design of the process considered in thispaper is described in ref 9. Two pumps are used toconvey water from a basin into four overhead tanks. Thetwo tanks at the upper level drain freely into the twotanks at the bottom level. The liquid levels in the bottomtwo tanks are measured. The piping system is such thateach pump affects the liquid levels of both measuredtanks. A portion of the flow from one pump is directedinto one of the lower-level tanks (where the level ismonitored). The rest of the flow from a single pump isdirected to the overhead tank that drains into the otherlower-level tank. By adjusting the bypass valves of thesystem, the amount of interaction between the inputsand the outputs can be varied. The process flowsheet isdisplayed in Figure 1. In the present study, additional

* Author to whom correspondence should be addressed.Phone: (302) 831-0760. Fax: (302) 831-1048. E-mail: fdoyle@Udel.edu.

1916 Ind. Eng. Chem. Res. 2001, 40, 1916-1927

10.1021/ie000381p CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 04/11/2001

flow disturbances are introduced into the upper-leveltanks. These external unmeasured disturbance flowscan either drain or fill the top tanks. The processvariations include uncertainties in the actuators, valvesettings, and head losses in the tanks. The objective ofthe present study is to design a robustly performingcontroller that provides stable and acceptable perfor-mance for flow disturbance rejection and setpoint track-ing.

2. Modeling and Parameter Estimation

A nonlinear mathematical model for a similar four-tank system is detailed in ref 11. The model used inthe present study includes the disturbance effect of flowsin and out of the upper-level tanks 3 and 4 as depictedin Figure 1. The differential equations representing themass balances in this four-tank system are

where hi is the liquid level in tank i; ai is the outletcross-sectional area of tank i; Ai is the cross-sectionalarea of tank i; νj is the speed setting of pump j, withthe corresponding gain kj; γj is the portion of the flowthat goes into the upper tank from pump j; and d1 andd2 are flow disturbances from tank 3 and tank 4,respectively, with corresponding gains kd1 and kd2. Theprocess manipulated inputs are ν1 and ν2 (speed settingsto the pumps), and the measured outputs are y1 and y2(voltages from level measurement devices). The mea-sured level signals are assumed to be proportional tothe true levels, i.e., y1 ) km1h1 and y2 ) km2h2. The levelsensors were calibrated so that km1 ) km2 ) 1.

This simple mass balance model adopts Bernoulli’slaw for flow out of an orifice. The tank areas Ai can bemeasured directly with adequate accuracy from theapparatus. Using tank drainage data, estimates of thecross-sectional outlet areas ai can also be determined.The steady-state operating conditions of ν1 ) 50% andν2 ) 50% are used for subsequent modeling and control-ler synthesis. Johansson and Nunes11 showed that theinverse response in the modeled outputs will occur whenγ1 + γ2 < 1. In the present system, the valves were setsuch that the nominal operating point exhibits inverseresponse.

For the purpose of parameter estimation, dynamicdata were collected from the laboratory system usinginput sequences as a series of steps in orthogonaldirections, i.e., (1, 1), (1, -1), (-1, 1), and (-1, -1). Thisapproach is intended to sufficiently excite the multi-variable process. To determine the model parameters,a nonlinear constrained optimization procedure thatminimizes the scaled 2-norm of the difference betweenthe nonlinear model and actual measurements is em-ployed. This optimization formulation is represented as

where at every time instant k ) 1, ..., N of the datarecord, each element of e(k) ∈ R2 is ej(k) ) [hj

nl(k) -hj

meas(k)]/hjnl(k), for j ) 1,2; the indices nl and meas

denote nonlinear model prediction and observed mea-surement, respectively. The optimal parameter valuesare computed with the constr function in MATLAB.Parameter values computed from the steady-state datawere used to initialize the optimization algorithm forerror minimization. Note that this optimization isnonlinear and based on the noisy data, and hence, it isnot guaranteed to result in globally optimal parametervalues. However, parameter constraints were imposedto restrict the optimization for γj to positive values lessthan 0.5 (as the nominal operating condition is in theinverse response regime for both levels) and pumpconstants kj to remain within (100% of the measuredsteady-state calibrations. The initialization data were“perturbed” across multiple optimization runs so thatthe estimates were close to optimal. The nominaloperating conditions and the estimated parameters ofthe nonlinear model are shown in Table 1. The esti-mated parameter values are found to be within theconstraints and are not on the constraint surface,indicating that the estimates are close to “true” values.The model fit and validation results are shown in Figure2. The residual error between the experimental data andthe nonlinear model are within (10%.

Figure 1. Schematic of the four-interconnected-tank system. Theliquid levels in tank 1 and tank 2 are measured.

dh1

dt) -

a1

A1x2gh1 +

a3

A1x2gh3 +

γ1k1

A1ν1

dh2

dt) -

a2

A2x2gh2 +

a4

A2x2gh4 +

γ2k2

A2ν2

(1)

dh3

dt) -

a3

A3x2gh3 +

(1 - γ2)k2

A3ν2 -

kd1d1

A3

dh4

dt) -

a4

A4x2gh4 +

(1 - γ1)k1

A4ν1 -

kd2d2

A4

Table 1. Nominal Operating Conditions and ParameterValues

symbol state/parameters value

h0 nominal levels 16.3, 13.7, 6.0, 8.1 cmν0 nominal pump settings 50, 50 %ai area of the drain in tank i 2.05, 2.26, 2.37, 2.07 cm2

Ai areas of the tanks 730 cm2

γ1 ratio of flow in tank 1 to flow in tank 4 0.3γ2 ratio of flow in tank 2 to flow in tank 3 0.3kj pump proportionality constants 7.45, 7.30 cm3/(s %)kdj disturbance gains 0.049, 0.049Ti time constants in the linearized model 65, 54.1, 34, 45.3 suj, ej, rj scaling factors 25%, 4 cm, 4 cmg gravitation constant 981 cm/s2

minγj,kj

∑k)1

N

||e(k)||22 j ) 1, 2 (2)

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1917

The linearized state-space model for these operatingconditions and parametric values is given as

where x ) [h1 h2 h3 h4]T, u ) [ν1 ν2]T, d ) [d1 d2]T, andy ) [h1 h2]T. For the linearized system, the timeconstants for the system are given as Ti ) (Ai/ai)

x(2hi0/g). The corresponding linear transfer function

matrix is

where cj ) (Tjkj/Aj), j ) 1,2. Note that individual transferfunctions in each of the input-output channels of G(s)do not have zeros. The RHP (right-half-plane) zero ofthe system is a multivariable characteristic and imposes

limitations on achievable performance as discussed inthe next section.

Figure 2 also shows the residual error betweennonlinear and linearized model with the estimatedparameters. This error is well within (5%, indicatingthat the linearized model is probably sufficient forcontroller design. From Figure 3, it can be seen thatboth the linear and the nonlinear models display aninverse response for an appropriate step change in theinputs (a positive change in one pump and a negativestep change in the other pump).

For disturbance model identification, submersiblepumps were used in the overhead tanks to pump waterout of the overhead tanks and into the lower basin. Withthe disturbance being activated at known times, processdata were collected. To determine the parameters forthe disturbance flow rates, the process data were usedin an optimization routine similar to the one used formodel parameter estimation. The identification andvalidation results are shown in Figure 4.

Prior to the control analysis and controller synthesis,the manipulated inputs were scaled by 25%. This scalingcorresponds to actual input levels in the range of 25-75%. Because of the nonlinearities introduced by thewater head in the piping and limited pump capacities,the pumps cannot operate satisfactorily below the 25%level. Hence, a larger input scaling factor could not bechosen even though the pumps could operate at more

Figure 2. Residuals between experimental data, nonlinear modeland linear approximations. (Top) Nonlinear model versus experi-mental data. The vertical line separates the data used forparameter estimation and model validation. (Bottom) Nonlinearversus linearized model.

x ) [- 1T1

0A3

A1T30

0 - 1T2

0A4

A2T4

0 0 - 1T3

0

0 0 0 - 1T4

]x +

[γ1k1

A10

0γ2k2

A3

0(1 - γ2)k2

A3

(1 - γ1)k1

A40

]u + [0 00 0

-kd1

A30

0 -kd2

A4

]d(3)

y ) [1 0 0 00 1 0 0 ]x

G(s) ) [ γ1c1

(T1s + 1)(1 - γ2)c2

(T1s + 1)(T3s + 1)(1 - γ1)c1

(T2s + 1)(T4s + 1)

γ2c2

(T2s + 1)] (4)

Figure 3. Validation of linearization and nonlinear models withexperimental step data.

Figure 4. Identification of linear disturbance model with experi-mental pulse data. The vertical line separates the identificationand validation data records.

1918 Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001

than 75%. The measured levels were scaled by a 4-cmdeviation corresponding to the maximum expectedchange in the liquid levels. The disturbance was scaledsuch that the maximum deviation obtained with thesubmersible pump corresponds to a disturbance of size1. These scaling factors are listed in Table 1.

3. Input-Output Controllability Analysis

Controllability analysis was performed on the systemto determine the limitations of feedback control. Thefollowing criteria were examined based on the control-lability analysis procedure detailed in ref 2.

Multivariable Interactions. The frequency-depend-ent relative gain array (RGA) elements for the linearsystem are displayed in Figure 5. At low frequency, theoff-diagonal elements of the RGA are on the order of 1,indicating that the decentralized control could be con-sidered with pairings [y1, u2] and [y2, u1]. The diagonalelements of the RGA matrix remain approximately 0.21at low frequency. This indicates some level of interac-tion, but adequate control should be possible.

Nonminimum-Phase (NMP) Characteristics. Thediagonal elements of the RGA change sign from steadystate to high frequency indicating the presence of a RHPzero in the system. Individual transfer function chan-nels of G(s) do not have RHP zeros. The multivariableRHP zero is found to be at 0.034 for the nominaloperating conditions and is associated with the inputdirection uz ) [-0.69, 0.73] and output direction νz )[-0.73, 0.69]. This is consistent with the inverse re-sponse observed in the real system when an increase isforced in one pump speed while the other is decreased.The RHP zero imposes an upper-bound constraint onthe achievable bandwidth as ωc e z/2. The RHP zerodirection indicates that the performance degradationcan be shifted entirely to either one of the outputs.13

The RHP zero arising from the multivariable nature ofthe system indicates that decentralized (or SISO) con-trollers have to be “retuned” for the multivariablesystem according to the inherent performance limita-tions resulting from their NMP characteristics.

Sensitivity to Uncertainty. The singular values forthe system as a function of frequency are plotted inFigure 5. The condition number, γ(G), is of low order ofmagnitude, implying that the plant is not ill-condi-

tioned. The condition number is higher at low frequency,indicating that the plant is more sensitive to unstruc-tured uncertainty at the steady state than at higherfrequencies.

Functional Controllability. The system is func-tionally controllable if G(s) has full row rank. For asquare MIMO system, this condition is equivalent todet[G(s)] * 0, ∀s, except for a finite number of zeros ofG(s). The four-tank system is functionally controllable,as this criterion is satisfied.

Input Saturation. Perfect control is achievable inthe presence of combined reference changes withoutsaturation of the inputs (in terms of the 2-norm) if thefollowing condition is satisfied2

where ωr is the frequency up to which reference trackingis required. Alternatively, this is a limitation on achiev-able bandwidth for reference tracking. A similar condi-tion for disturbance rejection is obtained by replacingR with Gd in eq 5. In the current problem, the errorscaling factor, ej, is chosen to be equal to referencescaling factor, rj. Therefore, the reference scaling matrix,R ) I. Hence, the perfect control conditions for referencetracking are directly obtained from the singular valuesof G. The low-frequency minimum singular value of theprocess, σ(jω), is greater than 1 (Figure 5). This indi-cates that adequate control should be possible; the inputmoves will be able to change the outputs by a sufficientamount while maintaining the scaled inputs and out-puts less than 1. The minimum singular value of theplant is greater than 1 up to a frequency of ω ) 0.015Hz. This is an upper bound on the controller bandwidth,ωc, due to input saturation considerations at highfrequency. This value depends on the scaling of theinputs and outputs. For the disturbance rejection prob-lem, one requires that σ(Gd

-1G) > 1, ∀ω. The currentprocess satisfies this criterion (Figure 5). As a result,perfect control for disturbance rejection can be achievedwith no limitation on the achievable bandwidth set bythe input saturation criterion. The RHP zero alsoimposes an upper-bound constraint on the achievablebandwidth as ωc e z/2 ) 0.017 Hz for the presentprocess. Based on the minimum singular value con-straint and RHP zero limitation, the achievable band-width for reference tracking is 0.015 Hz. Note that thetwo limitations almost overlap in the nominal case. Thisindicates that, if larger input space were available, i.e.,uj > 25%, the input saturation would occur at higherfrequency and the RHP zero limitation (ωc e 0.017 Hz)would be “active”. Uncertainty also plays a role in theactive limitation on achievable bandwidth as the inputsaturation and RHP zero limitations vary with param-eter changes. This aspect is discussed in the next sectionafter the details of uncertainty characterization of thefour-tank system are presented.

An interesting ramification of the adjustable bypassvalves (varying γj) is that one can adjust the degree ofinteraction in the system. If one assumes that thesystem is perfectly symmetric (including the bypass, γ),it is easy to show that the (1, 1) element of the RGA ofthe linearized model is given by

Figure 5. Input-output controllability analysis: |RGA(G)|, σ(G),and σ(Gd

-1G). The bandwidth for reference tracking is 0.015 Hz.Perfect control for disturbance rejection requires σ(Gd

-1G) g 1,∀ω.

σ(R-1G) g 1 ∀ω e ωr (5)

λ ) γ2

2γ - 1

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1919

which indicates that all values of λ less than zero andgreater than one are possible. Typically, controllerdesign for multivariable system involves extensiveanalysis of interactions and corresponding retuning ofthe controllers for implementation. However, in thisstudy, we present the case in which interactions areminimal while at the same time the multivariablenature of the system imposes considerable performancelimitations through a multivariable RHP zero. Thenominal operating point is appropriately chosen so thatthe retuning of the controllers is not affected by thecriteria based on multivariable interactions but isaffected by limitations based on RHP zero location.

4. Uncertainty Characterization

To rigorously analyze the robustness properties of thesystem, the plant-model error must be quantified.Sources of uncertainty in the four-tank system include(1) unmodeled and neglected dynamics; (2) valve posi-tion uncertainty, which affects the ratios of flows acrossthe top and bottom tanks; (3) uncertainty in the actua-tors; and (4) nonlinearities in the system. A lumpedunstructured input multiplicative uncertainty is con-sidered. The corresponding family of linear systemsconsidered is represented by

where G(ω) is the nominal model, Gp(ω) is the “actual”process, WI(ω) is the multiplicative frequency-dependentuncertainty bound, and ∆I is a linear time invariantoperator whose frequency response is an arbitrarycomplex operator with ||∆I(ω) ) ||∞ e 1, ∀ω. The closed-loop system with the real process Gp represented usingthis uncertainty structure is shown in Figure 6. Theuncertainty bound WI(ω) is chosen to satisfy the re-quirement that

where Gp is a perturbed plant and ΠI is the set of allperturbed plants. The parametric uncertainty can berigorously characterized through least-squares statisticsmethods.14 However, for this application, the perturbedplant Gp is obtained numerically by varying the param-eters γ1, γ2, k1, and k2 by (10% of the correspondingnominal value. These variations are found to be suf-ficient to capture the changes in steady-state operatinglevel for a range of approximately 6 cm. This “covers” abroad spectrum of operating levels in the nonminimum-phase regime. The experimental data and operationalexperience confirm this to be a reasonable level ofuncertainty. A diagonal uncertainty weight, WI, isconsidered assuming that both input channels have the

same uncertainty characteristics. This is a reasonableassumption as the system is nearly symmetric, and itis evident from the parameter estimates shown in Table1. The multiplicative uncertainty weight is found to beWI ) (s + 0.40 × 0.12)/[s/(0.22) + 0.12] and correspondsto the set of perturbed processes shown in Figure 7. Atsteady state, the system exhibits approximately 40%uncertainty, whereas at higher frequencies, the uncer-tainty drops to approximately 22%. This is in agreementwith operational experience, which indicates that thesystem exhibits similar dynamics when operated atdifferent steady-state levels in each of the minimum-phase and nonminimum-phase regimes. This higheruncertainty at steady state might impose limitations onthe steady-state performance of the system. An inte-grating action in the controller can overcome thislimitation.

The achievable bandwidth for closed-loop performanceis limited by input saturation and RHP zero location.Figure 8 shows the variation of the minimum singularvalue of G and the RHP zero location. For the uncer-tainty considered here, the input saturation upperbound is in the range 0.01-0.025 Hz, while the RHPzero location varies in the range 0.023-0.047 Hz, withcorresponding bandwidth limitation in the range 0.0115-0.0235 Hz. This overlap indicates that the input satura-tion and the RHP zero limitations are active separately

Figure 6. Block diagram of the closed-loop system with the realprocess Gp represented using a lumped unstructured multiplicativeinput uncertainty structure.

Gp(ω) ) G(ω)[I + WI(ω)∆I] (6)

||WI(ω)||∞ G supGp(ω)∈ΠI

||G-1(ω)(Gp(ω) - G(ω))||∞ ∀ω

(7)

Figure 7. Input multiplicative uncertainty weight, WI, as anupper bound on the process uncertainty due to (10% variation inparameters γ1, γ2, k1, and k2.

Figure 8. Variation in σ(G) and the RHP zero location due to(10% variation in parameters γ1, γ2, k1, and k2.

1920 Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001

on a per case basis within the domain of perturbedplants considered.

5. Controller Synthesis

The control algorithms considered in this work in-clude (i) decentralized PI control (PI), (ii) multivariableinternal model control (IMC), and (iii) µ-analysis-basedH∞ controller (H∞). For each of the controller designs, astable controller is desired that achieves reasonableperformance, given the process limitations. In thereference tracking problem, no disturbances were as-sumed to be present in the system. Additionally, noreference changes were assumed to occur for the dis-turbance rejection problem. A performance weight Wpof the following form is used2

where ωB/ is the desired controller bandwidth, A is the

steady-state offset, and M is a limit on the maximumvalue of the sensitivity function. For the H∞ controller,the controller bandwidth, ωB

/, and the peak sensitivitybound, M, are adjusted until the required performancecriteria are satisfied.

Typical robust performance criteria involve computa-tion of the sensitivity and complementary sensitivityfunctions. The sensitivity function S is the closed-loopoperator from d to y or from r to e. The complementarysensitivity function T is the closed-loop operator from rto y and is related to the sensitivity function as S + T) I, where I is the identity operator. For the classicalfeedback structure shown in Figure 6, S ) (I + GK)-1.For the IMC control structure, S ) (I - GK). For eachof the PI, IMC, and H∞ controllers, S and T arecomputed as functions of the frequency, and the nominalperformance (NP: ||WpS||∞ < 1), robust stability (RS:||WIT||∞ < 1), and robust performance (RP: structuredsingular value, µ < 1) criteria are evaluated.15 For eachcontrol algorithm, a robustly performing controller thatis also nominally stable is designed.

5.1. Robust PI Controller Design. A decentralizedPI controller as a combination of two SISO PI controllersis considered. Each PI controller is designed on the basisof the relay-tuning method.16 The “auto-tuning” methodusing a relay controller can be used to determine thestability limits of the system in terms of ultimate gainand ultimate period for small-amplitude experiments.The period of the closed-loop response is the ultimateperiod Pu, and the ultimate proportional controller gainis given by Kcu ) (4h/πA), where h is the amplitude ofthe input to the system and A is the output amplitude.Once the stability limits Kcu and Pu are known, theZiegler-Nichols stability margin design could be usedto calculate the parameters Kc and τI for the PI control-ler as Kc ) 0.45Kcu and τI ) Pu/1.2. The decentralizedPI controller thus designed is typically “detuned” toaccount for multivariable interactions and to providerobust stability and performance. In the case of the four-tank system, interactions are not as critical as the RHP-zero- and input-saturation-based performance limita-tions.

For the purpose of the decentralized PI controllerdesign, the RGA matrix indicates that the pairing y1 -u2 and y2 - u1 is appropriate. Note that the plant hasnegative gain in the nonminimum-phase setting, i.e.,

the determinant of G is negative. This indicates thatno controller K with positive gain (determinant) willstabilize the system under negative feedback. Thepairing y1 - u2 and y2 - u1 chosen after RGA analysisalways results in a K with negative gain as long as eachof the controllers have positive gain. Thus, the pairingis appropriate from the stability perspective.

Separate experiments were performed for each of theSISO pairings y1 - u2 and y2 - u1. In each case, theinput was cycled within (10% from the nominal operat-ing point of 50%. This was done to minimize the effectof nonlinearities on the controller design. The input andoutput time profiles used in the controller design areshown in Figure 9. The resulting PI controllers are [Kc1) 3.57, τI1 ) 46.8], and [Kc2 ) 3.78, τI2 ) 49.3]. Notethat each of these loops is theoretically second-order andhence does not have theoretical stability limits. Thismeans that the resulting PI controller is very aggres-sively tuned and does not provide stable nominalperformance when employed on the multivariable sys-tem. This is due to violation of the bandwidth limitationimposed by the RHP zero and the input saturationcriteria. The controllers need to be detuned to conformto these limitations. Also, the NP, RS, and RP criteriahave to be satisfied. The PI controller is detunedthrough an iterative ad hoc trial-and-error procedure.Figure 10 shows that the detuned decentralized PIcontroller satisfies the NS, RS, and RP criteria forrobust performance. This controller has a µ value of 0.99satisfying the robust performance criterion (RP: µ <1).15 The controller parameters are Kc1 ) 0.94, τI1 )187.3 and [Kc2 ) 0.99, τI2 ) 197.3]. A performancebandwidth (ωB

/) of 0.0025 Hz and a maximum allowablesensitivity peak (M) of 4 are both achieved. The imple-mentation results of this PI controller in referencetracking and disturbance rejection are discussed insection 6.

5.2. Robust Internal Model Controller Design.Internal model control (IMC) is a very effective methodof utilizing a process model for feedback control andrequires minimal on-line computation. For a full discus-sion of IMC, see the monograph by Morari and Zafiri-ou.17 Typically, IMC involves “inversion” of a portion ofthe model for use as a controller for the process.However, some portions of a linear process model cannotbe inverted. These noninvertible factors include timedelays and right-half-plane (RHP) zeros. In addition, aprocess model that is not semi-proper cannot be inverted

Wp(s) )s/M + ωB

/

s + AωB/

(8)Figure 9. Input-output data used for relay tuning of two SISOPI controllers. The pairing is (y1 - u2) and (y2 - u1).

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1921

directly. A linear filter could be added to make theprocess model invertible. The filter parameters are thenused in tuning the aggressiveness of the IMC controllerfor robust closed-loop performance.

The following inner-outer factorization for the stableprocess G(s) follows the procedure described in ref 17.The linear process transfer function can be written interms of the state-space matrices {A, B, C, D} as

where N(s) and M(s) are stable. Additionally, N(jω)H

N(jω) ) I. Given a factorization of D as D ) QIRI, thenoninvertible portion of the process, N(s), is given by

The invertible part of the process, M(s)-1, is

The operator Φ is given as

where X is the solution of the algebraic Riccati equation

The state-space description of N(s) is

For M-1(s), the state-space matrices are given as

In the current study, the orthogonal matrix QI isselected as identity. The true plant whose linearizedstate-space model is given in eq 3 is strictly proper. For

the purpose of the inner-outer factorization, the processhas to be semi-proper. Therefore, D is set to ∈I, wherethe scalar ∈ ) 0.1 here. Smaller values of ∈ give betterapproximations of the true plant but result in a control-ler that is sensitive to noise, whereas larger values of ∈result in increased plant-model mismatch. The value∈ ) 0.1 chosen in this study provided a good closed-loop performance while maintaining low sensitivity tomeasurement noise.

A schematic of the closed-loop system employing theinner-outer factorization-based IMC controller is shownin Figure 11. The controller for the IMC formulation isthe inverse of M(s), the invertible portion of the processmodel. For offset-free steady-state reference tracking inall channels, the product of the controller gain and theprocess model gain must be identity. The currentprocess model factorization does not guarantee this. Itcan conveniently be achieved by scaling the controllerby N(0)-1. Now, the noninvertible process model isN(s)N(0)-1, and the invertible process model isN(0)M-1(s). The IMC controller becomes M(s)N(0)-1.SISO IMC systems incorporate a scalar filter for strictlyproper process models so that the resulting controlleris semi-proper and realizable. In this multivariable case,the process model is already assumed to be semi-proper.Each of the error signals sent to the MIMO 2 × 2 IMCcontroller can be filtered by a first-order linear filter ofthe form Fi(s) ) (1/λis + 1). This filter allows for theaggressiveness of the controller to be tuned to achievedesired robust closed-loop performance. In this study,values of λi ) 50 s are chosen. The resulting controlleris stable, and all closed-loop transfer functions are foundto be stable. A performance bandwidth (ωB

/) of 0.005Hz and a maximum allowable sensitivity peak (M) of 3are both achieved. Figure 12 shows the NP, RS, and RPmeasures for the designed IMC controller. This control-ler has a µ value of 0.99, satisfying the robust perfor-mance criterion (RP: µ < 1).15

The process RHP zero is located at 0.034. The inputdirection corresponding to the RHP (as described in ref2) is [-0.69, 0.73], and the output direction is [-0.73,0.69]. In the nominal case, the process is identical tothe process model, and the filter time constants are setto 0. The complementary sensitivity function, T(s) , forthe nominal IMC system reduces to N(s)N(0)-1. Ideally,T(s) is identity at all frequencies. The presence of a RHPzero creates a performance limitation for the system.In ref 17, it is shown that T(s) for a nonminimum-phasesystem with a single zero can be arbitrarily selected sothat only a single row deviates from identity. Thisimplies that the performance degradation caused by theRHP zero can be driven into a single output channel.

Figure 10. Nominal performance (NP), robust stability (RS), androbust performance (RP) measures for the decentralized PIcontroller. The structured singular value, µ, is 0.99 as exhibitedby the magnitude of the RP measure. A controller bandwidth of0.0025 Hz and a peak sensitivity value of 4 are achieved.

G(s) ) C(sI - A)-1B + D ) N(s)M(s)-1

N(s) ) (C - QIΦ)[sI - (A - BRI-1Φ)]-1BRI

-1 + QI

M(s)-1 ) Φ(sI - A)-1B + RI

Φ ) QITC + (BRI

-1)TX

0 ) (A - BRI-1QI

TC)TX + X(A - BRI-1QI

TC) -

X(BRI-1)(BRI

-1)TX

AN ) A - BRI-1F BN ) BRI

-1

CN ) C - QIΦ DN ) QI

AM ) A BM ) B

CM ) Φ DM ) RI

Figure 11. Block diagram of the closed-loop system with theinternal model control (IMC) structure. Note the inner-outerfactorization of the plant G(s) into the invertible portion M(s)-1

and the noninvertible portion N(s). F(s) is the filter used in tuningthe aggresiveness of the controller to achieve desired robust closed-loop performance.

1922 Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001

The nominal complementary sensitivity function for the2 × 2 four-tank system with ideal performance in thefirst measurement is

and the complementary sensitivity function for thesystem with ideal performance in the second measure-ment is

where â1 and â2 are functions of the terms in the outputzero direction. Significant interaction can occur whenthe values of âi are large. For the given system, â1 is2.1 and â2 is 1.9. This implies that the choice of eitherinput for ideal tracking in the nominal case will haveessentially the same amount of interaction. This is anexpected result because of the symmetric nature of thesystem. For the developed controller formulation, a first-order realization of nominal T(s) is given as

This also demonstrates that the system and controllerformulation is symmetric. Both output channels shoulddemonstrate identical performance limitations and in-teractions. The implementation results of this IMCcontroller in reference tracking and disturbance rejec-tion cases are discussed in section 6.

5.3. Nominal H∞ Controller Design. A mixedsensitivity H∞-based controller is designed for thenominal four-tank system. This controller is obtainedby minimizing the quantity

where S is the sensitivity function, K is the controller,R () I) is the reference scaling matrix, and Wu is a filterthat penalizes control action in the neighborhood of thedesired bandwidth. In the current problem, Wu is chosento be equal to I as there is no specific requirement oninputs other than that they be maintained within (1.The controller thus designed will be referred to as the“nominal H∞ controller” for the rest of this discussion.

A sixth-order controller is designed through theγ-iteration procedure using hinfsyn function in µ-toolboxin MATLAB.18 A performance bandwidth (ωB

/) of 0.006Hz and a maximum allowable sensitivity peak (M) of 2were achieved through this controller design procedure.All closed-loop transfer functions with this controllerwere found to be stable, thus giving a nominal stableclosed-loop controller. To satisfy certain rank conditionsin H∞ controller synthesis, Gd must be semi-proper.18 Ifthe disturbance is assumed to affect the measured levelsdirectly, i.e., Gd ) I, then the controller syntheses forthe disturbance rejection and reference tracking areequivalent with R ) I in eq 9. This is a reasonableassumption, as the amplitude ratio of Gd is always lessthan 1 with a steady-state value of 0.9. However, theachieved bandwidth will be conservative for the distur-bance rejection case.

The sensitivity and input constraint plots are shownin Figure 13. It can be seen that the input constraintsare not violated for this controller (||WuKSR||∞ < 1). Therobust performance of this controller is characterizedusing µ-analysis. This approach involves the formulationof the performance problem as an appropriate robuststability problem with suitably defined uncertaintystructure.2,19 The resulting uncertainty structure isblock diagonal and can be represented in the M-∆structure used for µ-analysis.2,20 Figure 14 shows thatthis controller satisfies the nominal performance (NP:||WpS||∞ < 1) and robust stability (RS: ||WIT||∞ < 1)criteria. To guarantee robust performance, the struc-tured singular value (µ) has to be less than 1.15 Thenominal H∞ controller does not satisfy this robustperformance criterion. The µ value corresponding to theplot in Figure 14 is 1.25. This indicates that all of theuncertainty blocks considered in the RP problem shouldbe decreased in magnitude by a factor of 1.25 in orderto guarantee robust performance, i.e., both the plantuncertainty and the performance specification should

Figure 12. Nominal Performance (NP), Robust Stability (RS) andRobust Performance (RP) measures for the Internal Model Con-troller. The structured singular value, µ, is 0.99 as exhibited bythe magnitude of the RP measure. A controller bandwidth of 0.005Hz and a peak sensitivity value of 3 are achieved.

Figure 13. Sensitivity function and input constraints for thenominal system with the nominal H∞ controller. Input constraintsrequire that ||WuKSR||∞ < 1. A controller bandwidth of 0.006 Hzand a peak sensitivity value of 2 are achieved.

T1(s) ) [1 0â1s

s + ú-s + ús + ú ]

T2(s) ) [-s + ús + ú

â2ss + ú

0 1 ]

T(s) ) [5.82s + 132s + 1

42.74s42.74s + 1

42.74s42.74s + 1

-5.82s + 142.74s + 1

]

|| [WpSRWuKSR ] ||∞

(9)

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1923

be reduced by a scaling factor of 1.25. The skewed-µvalue is computed to determine the level of plantuncertainty for which the controller is robust withoutthe loss of achieved nominal performance. This analysisallows scaling of a particular source of uncertainty(plant uncertainty, in this case) so that the robustperformance criterion is satisfied.2 The skewed-µ valueof the system with the nominal H∞ controller is foundto be 2.1. This means that the designed H∞ controllerwould provide robust performance to approximately 48%of the level of plant input uncertainty considered in thisstudy. The design of a controller that is robust to thespecified level of plant uncertainty is presented in thenext section.

5.4. Robust H∞ Controller Design. A design of theµ-analysis-based H∞ controller that achieves robustperformance is presented in this section.1,15 Using thestandard D-K iteration procedure,19,15 a 28th-ordercontroller is obtained after three iterations. This con-troller has a µ value of 0.99. The computational toolsavailable in the µ-toolbox in MATLAB are used in thecontroller design.19 A performance bandwidth (ωB

/) of0.007 Hz and a maximum allowable sensitivity peak (M)of 2 are both achieved. The resulting controller is stable,and all closed-loop transfer functions are found to bestable. The nominal performance, robust stability, androbust performance indices for this controller are shownin Figure 15.

5.5. Input Constraints. In all of the controllerdesigns presented, input constraints are not explicitlyaccounted for, i.e., the controller design did not incor-porate the term WuKSR in the nominal performancemeasure. For all three designed controllers, σj(KS)increases to values greater than 1 within the bandwidthfrequency. This derivative action in the bandwidthfrequency could result in an input of size greater than1 for appropriate changes in the reference. On the otherhand, the performance bandwidth degraded consider-ably when the input size constraints were introducedinto the µ-analysis-based H∞ controller design. Similarresults apply to the PI and IMC controllers designedhere. As an alternative, a first-order prefilter could beimplemented on the setpoint changes to overcome thisissue. The time constant of the setpoint prefilter is basedon the corner frequency of σj(KSR) and is set to 50 s for

the current problem. Such a filter is not necessary forthe disturbance rejection problem, as Gd is found to bea sufficient prefilter for the disturbance. However, inpractice, the controllers provided better performancewhen the inputs are constrained to remain within 25-75% than when the reference filter is employed. This isexpected as the reference filter significantly detunes theclosed-loop response whereas “clipping” the inputs doesnot degrade closed-loop performance considerably aslong as the input is not saturated within the achievedbandwidth.

6. Results and Discussion

To validate the performance of the designed control-lers, closed-loop simulations were conducted prior toimplementation on the experimental system. The simu-lation and implementation results are discussed in thenext two subsections.

6.1. Simulation Results. For the purpose of simula-tion, the real process is simulated by the nonlinearstate-space model described in eq 1. A perturbed systemis simulated by increasing the parameters γ1, γ2, k1, andk2 by 10%. The setpoints were perturbed in the directionof the minimum singular value of the plant (increasein h1 and equal decrease in h2). Separate simulationswere performed for each of the robust-PI-, robust-IMC-,and µ-analysis-based H∞ controllers. Reference trackingresults for the perturbed system with the robust con-troller are shown in Figure 16. The robust IMC and H∞controllers provided better reference tracking perfor-mance than the decentralized PI controller. The pumplevels, as computed by the controllers, were not main-tained within the actuator constraints. However, clip-ping the pump inputs provided good performance with-out saturating the pumps at steady state. For the PIand H∞ controllers, the input constraints were activefor a brief period when the setpoint changed. However,the process contained sufficient input capacity to reachthe setpoint without saturating the pumps. Althoughthe designed H∞ and IMC controllers did not containintegral action (as defined theoretically), the perfor-mance weight Wp was designed such that only a verysmall offset (0.1%) is tolerated at steady state. This issufficiently close to integral action and could result inreset windup if the reference changes were introduced

Figure 14. Nominal performance (NP), robust stability (RS), androbust performance (RP) measures for the nominal H∞ controller.The structured singular value, µ, is 1.25 as exhibited by themagnitude of the RP measure. A controller bandwidth of 0.006Hz and a peak sensitivity value of 2 are achieved.

Figure 15. Nominal performance (NP), robust stability (RS), androbust performance (RP) measures for the µ-analysis-based H∞controller. The structured singular value, µ, is 0.99 as exhibitedby the magnitude of the RP measure.

1924 Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001

in such a dynamic manner as to continuously “excite”only the high-frequency components of the controllerresulting in input constraint violation. Such referencetrajectories were not explored in this study.

For the disturbance rejection case, a 10-min-long leakin tank 4 was simulated. All three controllers provideddesired performance with similar closed-loop peakchanges in the tank levels. The results for the distur-bance rejection case are shown in Figure 17. All of thecontollers attenuated the peak change in the measuredlevels by 60% compared to the open-loop case. In thesimulations, IMC and the H∞ controllers had similardisturbance rejection performance. The pump levelswere more oscillatory when the PI controller wasemployed. Pump inputs were maintained within theconstraints. An additional prefilter was not required inthis case as the filtering performed by Gd was found tobe sufficient.

6.2. Experimental Results. Controller implementa-tion was successfully carried out using the four-tankexperimental apparatus. To incorporate a controllermore advanced than P, PI, or PID, the Dynamic DataExchange interface was used to link MATLAB withFreelance, the software interface to the laboratory

experiment and control hardware. The state-spacecontroller was discretized using 1-s intervals. The levelsin the bottom tanks were sampled every second, andthe computed control move was implemented. Theclosed-loop time constant of the system was approxi-mately 2 orders of magnitude larger than the sampletime; hence, no discretization effects were observed.

Separate experiments were conducted for setpointtracking and disturbance rejection. The setpoint changeswere made in the direction of the minimum singularvalue of the plant to verify that the inputs weremaintained within the constraints. Figure 18 shows thesetpoint tracking results for the PI, IMC, and H∞controllers. As indicated by the simulations with theperturbed plant, the decentralized PI controller providedmore oscillatory changes in the tank levels and pumpinputs when implemented on the experimental system.However, the IMC and the H∞ controllers performedsimilarly, with the former being more sensitive to noise.As seen in Figure 18, the step change in the referenceproduced a derivative action on the inputs to the pumps.However, these input moves were constrained to (25%prior to communicating with the experimental hard-ware. This provides better performance than referenceprefiltering in this problem, as the designed prefilteradded considerable additional lag to system degradingthe closed-loop performance. The settling time of thetank levels was approximately 250-275 s.

For the disturbance rejection problem, a leak wasintroduced into the tank 4 by turning on the submers-ible pump. The leak was shut after 10 min by turningoff the submersible pump. The closed-loop performanceof the controllers for the disturbance rejection case isshown in Figure 19. All three controllers do providereasonable performance in rejecting the flow distur-bances. The open-loop change in the measured levels is3.5 cm, and the peak level change in the closed loop is2 cm for IMC and 1.75 cm for the H∞ and PI controllers.

The PI, IMC, and H∞ controllers provide robustperformance with different bandwidths and have dif-ferent robust performance characteristics, as is evidentfrom the corresponding RP measures in Figures 10, 12,and 15. The IMC and H∞ controllers provided similarperformance for both reference tracking and disturbance

Figure 16. Setpoint tracking simulation for the perturbed systemwith +10% change in the parameters γ1, γ2, k1, and k2. The levelchanges and pump inputs for each of the decentralized PI, IMC,and H∞ controllers are shown. The inputs are constrained toremain within 25-75%.

Figure 17. Disturbance rejection simulation for the perturbedsystem with +10% change in the parameters γ1, γ2, k1, and k2. Aleak is simulated in tank 4. The leak is on for 10 min and is thenturned off. The level changes and pump inputs for each of thedecentralized PI, IMC, and H∞ controllers are shown.

Figure 18. Setpoint tracking results for the experimental system.The level changes and pump inputs for each of the decentralizedPI, IMC, and H∞ controllers are shown. The inputs are constrainedto remain within 25-75%. Note the sensitivity of the IMC designto measurement noise as compared to the decentralized PI andH∞ controllers.

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1925

cases. This similarity is in tune with the similarity ofcorresponding measures for IMC and H∞ controllers(Figures 12 and 15). For disturbance rejection problem,all three controllers provide similar performance. Inputsto the closed-loop system, i.e., reference r and distur-bance d, need to sufficiently excite the system so thatvarious controllers can be differentiated from the closed-loop performance. It is plausible that the disturbancedynamic (Gd) is effectively “prefiltering” the leak viasubmersible pump. This could result in similar perfor-mance of all of the controllers in the disturbancerejection case by not exciting the system “fast enough”.However, the rapid setpoint changes excite the systemsufficiently to produce different closed-loop dynamicresponses for different controllers.

7. Conclusions

The design and implementation of robustly perform-ing PI, IMC, and H∞ controllers for an experimentalfour-tank process is presented. Flow disturbances areintroduced in the upper-level tanks to modify theoriginal process. Nonlinear optimization is used toidentify the model parameters from the dynamic ex-perimental data. The inverse response in the system isadequately captured using the nonlinear model param-eter identification. The achievable performance is char-acterized using acceptable control analysis. Lumpedunstructured input uncertainty is found to be sufficientto model various effects of the parametric and actuatoruncertainties in the system. A relay-tuning-based de-centralized PI controller, an inner-outer factorization-based IMC algorithm, and a µ-analysis-based H∞ opti-mal controller are designed. Implementation results onthe physical system show that all three controllersprovide stable and desired performance for both refer-ence tracking and disturbance rejection. The IMC andH∞ controllers performed similarly and provided betterperformance than the decentralized PI controller for thereference tracking problem. To improve the closed-loopperformance on the experimental system, the pumpsettings were constrained to remain within 25-75%rather than using a reference prefilter. The controllerdesign methods employed in this work are expected to

perform well for any of the linear time-invariant systemincluded in the uncertainty characterization. However,the experimental system is inherently nonlinear, andnot all of the nonlinear dynamics involving pipingoverheads and the stochastic nature of the apparatusare captured by the nonlinear state-space model em-ployed in this study (eq 1). Performance degradationfrom simulation to implementation could be due to theinherent stochastic behavior of the apparatus, whichvaries the steady states from one operation to another,potentially placing the system outside the uncertaintycharacterization area for which the controller is de-signed.

Acknowledgment

The authors gratefully acknowledge support from theOffice of Naval Research (Grant NOOO-14-96-1-0695)and University of Delaware Process Control and Moni-toring Consortium. The authors also acknowledge thehelpful comments of the reviewers of this manuscriptin shaping the focus of the paper.

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Figure 19. Disturbance rejection results for the experimentalsystem. A leak disturbance in tank 4 is introduced. The leak isturned on for 10 min and is then turned off. The level changesand pump inputs for each of the decentralized PI, IMC, and H∞controllers are shown. The inputs are maintained within theconstraints as sufficient prefiltering is provided by the disturbancedynamics.

1926 Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001

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Received for review April 5, 2000Revised manuscript received February 1, 2001

Accepted February 15, 2001

IE000381P

Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1927