On Control Structure Design for a Walking Beam Furnace ∗ †

H. Jafari1, M. Castano1, T. Gustafsson1 and G. Nikolakopoulos1,2

Abstract— The aim of this article is to introduce a novelsparse controller design for the temperature control of anexperimental walking beam furnace in steel industry. Adequatetracking of temperature references is essential for the qualityof the heated slabs. However, the design of the temperaturecontrol is hindered by the multivariable (non-square) dynamicbehavior of the furnace. These dynamics include significantloop interactions and time delays. Furthermore, a novel data-driven model, based on real life experimental data that relieson a subspace state representation in a closed loop approachis introduced. In the sequel, the derived model is utilized toinvestigate the controller’s structure. By applying the relativegain array approach a decentralized feedback controller isdesigned. However, in spite of the optimal and sparse designof the controller, there exists interaction between loops. Byanalyzing the interaction between the inputs-outputs with theΣ2 Gramian-based interaction methodology, a decoupled multi-variable controller is implied. The simulation result, basedon the experimental modeling of the furnace, shows that thecontroller can successfully decrease the interaction between theloops and track the reference temperature set-points.

I. INTRODUCTION

Walking Beam Furnaces (WBFs) are broadly utilizedin the modern steel industry to reheat the steel slabs toprescribed temperatures in order to change its mechanicalproperties. In a WBF, the slabs are walked in a cyclic move-ment, through heating zones of different temperatures. Froman energy efficiency perspective, this process plant is oneof the most energy consuming systems in the steel industry[1]. Therefore, an efficient controller is needed to minimizethe energy cost for the generated heat. Additionally, in theWBF, the temperature has an important effect on the energyexpenditure and the quality of the heated slabs and thus thecorresponding regulation of the temperature, for each specificzone of the furnace, is desired.

In order to control the temperature in an optimal approach,initially a reliable model of the furnace is needed and thisis the aim of the first contribution of this article. In theprevious literature, a variety of corresponding models havebeen investigated. More specifically, in [2] ComputationalFluid Dynamics (CFD) techniques were used for modeling,however the presented approach is computationally expen-sive for large industrial cases and it has more suitable results

*The work has received funding from the European Unions Horizon2020 Research and Innovation Programme under the Grant AgreementNo.636834, DISIRE

† Authors’ preprint of paper with the same title accepted for publicationin the IEEE Mediterranean Conference on Control and Automation.

1H. Jafari, M. Castano, T. Gustafsson and G. Nikolakopoulos are withControl Engineering group, Devision of Signals and Systems, Departmentof Computer Science, Electrical and Space Engineering, LuleaUniversity ofTechnology, Sweden. geonik@ltu.se

at steady state, while missing to represent the transientbehavior of the process.

From another approach, a considerable research in math-ematical modeling, based on the physical properties, hasbeen performed in the last decades. A simplified model, byconsidering first principles, was obtained in [3]. However,this model does not consider the radiative heat transfer andcomplex phenomena, such as turbulences. A model with lesscomputationally complexity was introduced in [4], which isonly valid for steady state operation. A novel mathematicalmodel, based on the zone method of radiation analysis andcombing this model with CFD has been performed in [5],while it was extended to a 3D approach in [6]. Althoughseveral studies have been performed for the physical model-ing, little attention has been paid for the black box modeling.Since unknown disturbances effect the WBF distinctly, blackbox modeling may be a better approach for this industrialplant, as in the works presented in [7] and [8] where theAuto-Regressive with an eXogenous term (ARX), modelingapproach for the reheating furnaces, have been adopted.However, studies on feedback loop data modeling are stilllacking.

In general, the concept of Process Control is always achallenging problem, since the development of the industrialplants lead to an increasing complexity of the utilized controlconfigurations. In the specific case of the WBF, distinctapproaches to design the temperature control scheme havebeen introduced. Until now, the most common strategy wasto design diagonal PID controllers, as the work in [9] and[10]. However, the performance of these controllers arenot always satisfactory, due to the existence of large timevarying delays and the high interaction between the controlloops [11]. Alternative popular control schemes, relies on theframework of the Model Predictive based Control (MPC) asin [12] and [13]. These approaches, in spite of a good MPC’sperformance in controlling the temperature and handling theconstraints, may not be practical in case of model mis-matches and high plant disturbances, while for a satisfactoryoperation, a high number of utilized tunning parameters isneeded. However, most of the previous studies do not takeinto account the coupling effects and the interactions ofinputs-outputs for the process plant to design the properoptimal control scheme.

The main novelty of this article, stems from the utilizationof structural decisions for the design of the WBF model,while additionally a novel sparse control scheme for thetemperature control of the walking beam furnace is intro-duced. First, a novel feedback based data-driven model, byconsidering disturbances to the furnace, is presented based

on real life experimental data. The obtained model is con-sidered as a multi-variable system with a non-square transferfunction. An additional novelty of this article is based on thesimplification of the previous derived plant model and thecorresponding study of the input-output interactions of thesystem by the utilization of the Relative Gain Array (RGA)theory and the Gramian based interaction measurement, Σ2.Finally, the reduced obtained model is used to design thecontroller configuration. The designed controller is testedwith real measurements of the furnace and the results werecompared with the decentralized PID controller.

The rest of this article is structured as it follows. Themodeling of the furnace is provided in Section II, whilethe input-output selection for reducing the model’s orderis described in the Section III. In the sequel, the controllerdesign, based on the obtained configuration is established inthe same Section, while simulation studies that prove theefficacy of the suggested scheme are depicted in SectionIV. Finally, the conclusions are drawn in Section V thatsummarizes the main outcome of this article, while definingthe future work directions.

II. WBF PLANT DESCRIPTION AND MODELING

Figure 1 depicts the schematic diagram of the experimentalWBF at Swerea MEFOS AB, Lulea, Sweden. This WBF,consists of three main temperature control zones. In the WBFoperation, initially the slabs enter the furnace through thehatch loading door and proceed to the first zone where arepreheated. The second zone heats the slabs to the maximumdesired temperature. Then, the heated slabs are entered thesoaking zone (Zone 3), where the temperature of the furnaceslightly increases and the temperature inside the slabs isnormally distributed. Finally the slabs are discharged throughthe unloading door.

Each control zone includes a pair of burners that fire thelight oil and are utilized as the corresponding actuators of theprocess. The supply rate of the oil and the air flow for thecombustion process are fed to the burners at each controlzone. The temperature of each zone is measured by twothermocouples, one at the wall of the furnace and the otherone at the ceiling. Additionally, the oxygen (O2) percentageinside the furnace and the internal pressure P of the furnaceare measured. The hot combustion gases are conveyed bytwo exhaust pipes located at zone 3. These exhaust pipesrecirculate the exhaust gases through the interior of the threezones in order to reuse the heat and later are fed to aheat exchanger in order to preheat the combustion air. Thisrecirculation of the combustion gases adds complexity to thedynamics of the process for representing a physical feedbackloop.

A mathematical model for the temperature of the furnaceby using heat and mass balance equations was driven by [3],while for clarity and consistency of this article it will be alsopresented as it follows:

dT1dt

=α

c1N1FO1 +

β

c1N1FA1(T1 − Tref1)

+ γ1c2P2

N1c1T2

√P 22 − P 2

1 (T1 − Tref1)

− γ1P2

N1T2

√P 22 − P 2

1 (T1 − Tref1)

− 1

N1(T1 − Tref )(FA1 + 0.76FO1)

− 1

c1N1Q1 −

1

c1N1W1

(1)

dT2dt

=α

c2N2FO2 +

β

c2N2FA2(T2 − Tref2)

+ γ2c3P3

N2c2T3

√P 23 − P 2

2 (T2 − Tref2)

− γ2P3

N2T3

√P 22 − P 2

2 (T2 − Tref2)

− 1

N2(T2 − Tref2)(FA2 + 0.76FO2)

− 1

c2N2Q2 −

1

c2N2W2

(2)

dT3dt

=α

c3N3FO3 +

β

c3N3FA3(T3 − Tref3)

− 1

N3(T3 − Tref3)(FA3 + 0.76FO3)

− 1

c3N3Q3 −

1

c3N3W3

(3)

where Ti i ∈ 1, 2, 3 is the temperature of each zone to becontrolled and Tref is the related reference temperature. FO,FA and P are the oil feed flow, air feed flow and pressurecorrespondingly inside of each heating zone respectively.The amount of energy, which is transfered to the slabsis represented by the variables Q and W that imply thetransfered energy to the wall of each zone of the furnace.N denotes the number of moles of the furnace gas and crepresents the heat capacity at each zone of the furnace. Theother parameters (α, β, γ) are the parameters of the systemto be identified. The above equations present a simplifiedanalysis of the modeling of the furnace, without consideringthe turbulence effect, heat transfer by radiation and the effectof spatial domain to the temperature of each zone of thefurnace. However, it can be used as conceptual illustrationof the structure of the system and determined the effectiveinputs and the outputs of the furnace. This investigation willbe later used in the data-driven modeling.

A. System identification

Subspace state-pace system identification method, is apowerful methodology to predict the dynamic of the systemfrom measured data [14]. By assuming that the temperatureat each zone of the furnace can be approximated by a linearfunction, the dynamics of the system can be described as:

xk+1 = Axk +Buk +Kek (4a)yk = Cxk +Duk + ek (4b)

where k ≥ 0 is discrete time, xk ∈ Rn is the state vector,uk ∈ Rr is the input vector, ek ∈ Rl is the zero mean

Fig. 1: The schematic block diagram of the Walking Beam Furnace under study.

white innovation process and yk ∈ Rm represents the outputvector, while A,B,C,D and K are constant matrices withproper dimensions. This form of representing the systemfor closed loop identification is a case of the so-calleddirect method. The direct method is preferred when operatingdata from the plant are available without knowledge of thecontrollers. The direct method ignores the closed loop in theformulation of the system, but requires that the noise models(K) are sufficiently complex to represent the influence of thecontroller [15].

Subspace system identification refers to the concept ofidentifying those constant matrices directly from measuredinput-output data. In this article, the Numerical algorithmsfor the Subspace State Space System IDentification (N4SID)with a Canonical Variable Algorithm (CVA) [16] are appliedto the WBF’s obtained experimental data. Initially, the vectorof stacked inputs, outputs and error are introduced in a futurehorizon as:

yfk = [yTk , yTk+1, ..., y

Tk+f−1]T (5a)

ufk = [uTk , uTk+1, ..., u

Tk+f−1]T (5b)

efk = [eTk , eTk+1, ..., e

Tk+f−1]T (5c)

where f ∈ Z+ is arbitrary chosen by the user and determinesthe number of the steps ahead prediction. By the subspacedefinition in (4a, 4b) the output variable can be redefined as:

yfk = Γxk + Φufk + Ψefk (6)

where Γ =

CCA

...CAf−1

is the observability matrix and

Φ =

D 0 ... 0CB D

CAB CB. . .

......

. . . 0CAf−2B CB D

,

Ψ =

I 0 ... 0CK I

CAK CK. . .

......

. . . 0CAf−2K CK I

.

The first step in this method is to estimate the state vectorsby the vectors of measured inputs and outputs. Thus, thestates matrix can be assumed as a function of j-steps-backinputs and outputs data as:

xk = Υpk (7)

where Υ is a matrix with unknown coefficient to be estimatedand

pk =

[yTk−1 yTk−2 ... yTk−juTk−1 uTk−2 ... uTk−j .

]T(8)

This estimation can be done by regressing yfk on pk and ufkfrom equation (6) and obtaining ΓK and Φ.

The next step is to estimate Υ and as a result the statesvector xk. In order to do that, first the effect of the futureinput on the future output should be removed by the linearregression problem defined as:

zk , yfk − Φufk ≈ ΓΥpk + Ψefk (9)

Then Υ can be estimated by solving the singular valuedecomposition on the correlation analysis on zk and pk as itfollows:

USV T = (Rzz)−12 (Rzp)(Rpp)−

12 (10)

where Rzp = 1N

N∑i=1

zkpTk is the correlation between zk and

pk, with U and V to be the orthonormal matrices of leftand right singular vectors and S is the diagonal matricesof singular values. Solving the above equation leads toestimating the CVA estimate of Υ. Finally, after estimatingthe states matrices by equation (7), the parameters of thesystem, defined by the matrices A,B,C,D and K can beestimated by the linear regression of state-space model in(4a and 4b) (see [16], [17]).

Unlike classical prediction methods, such as the ARXthat optimize effectively for one-step-ahead prediction, thepresented algorithm can optimize a k-steps-ahead prediction[18]. This property will reduce a bias in the prediction andwill conclude in a more accurate model of the system, whichwill be later used by the control scheme.

B. Model validation and simulation results

In order to identify the dynamics of the system bya data-driven methodology, a set of data from an on-line campaign for several days of the WBF’s operation

at MEFOS were utilized. After data cu-ration, consideringequations (1, 2 and 3) and the available measurementsthe variables of the inputs and outputs of the furnace areselected. Since only the pressure of the whole furnace ismeasured by a sensor, the effect of this input is neglectedin the identification process. Moreover, the exhaust flowof gases is consider as an another input applied to thefurnace. Therefore, the inputs for the system identificationare u = [FO1, FA1, FO2, FA2, FO3, FA3, Fexhaust]

T and thecorresponding outputs are y = [T1, T2, T3]T .

For avoiding an over-fitting of the data prediction, two setsof data are selected: the training and testing. The training setsare chosen in a way that capture all the important operatingcondition and consider a general case where both hatchdoors are open. For simplicity, it is assumed that there isno feed through and this results for the matrix D = 0. Theinitial estimated order of the system is determined by AkaikeInformation Criterion (AIC). Figure 2 depicts the 20 stepsahead prediction of temperature of each zone of the WBF onthe validation data set, with a sampling time of 10 seconds.

2000 4000 6000 8000 10000 12000

Data samples

700

800

900

1000

1100

1200

1300

Te

mp

rea

ture

[°

C]

T1

Model

T2

Model

T3

Model

T1

Meas

T2

Meas

T3

Meas

Fig. 2: Validation of model over testing data set with a 20steps ahead prediction. The dashed red line is the measure-ment data, while the solid blue line represents the model.

From the obtained results, it is straight forward that theresulted model has a very good performance in representingthe dynamics of the WBF, while at the same time has avery accurate matching. Furthermore, Figure 3 presents thechange of Predicted Root Mean Square Error (PRMSE) ofeach zone temperature for different k-steps-ahead prediction.

Obviously, by increasing the prediction horizon it resultsin a bigger prediction error and a less accuracy of the model.However, since the operating time for this furnace is in hours,this model can sufficiently predict the dynamic of the WBF.

III. CONTROL CONFIGURATION SELECTION

The design of a control system for a multivariable processusually involves the design of a control configuration priorto the synthesis of controller parameters [20]. In ControlConfiguration Selection (CCS) one approach is to simplifythe complete system model by selecting the most importantinput-output interconnections of the system and reduced thecomplexity of the model from the selected components [21].

20 40 60 80 100# Steps ahead prediction

0

10

20

45

40

50

mean P

RM

SE

of T

em

pera

ture

s [°C

]

Fig. 3: Illustration of ability of the prediction by changingthe prediction horizon. The mean of PRMSE for output tem-peratures is compared with changing the prediction horizon.

The derived simplified model can be later used to designthe controller. Moreover, CCS can be utilized to designa sparse and less complex controller that considers theinterconnections between all the loops.

In most of the industrial applications such as the WBF, thecontrol configuration is designed by using previous knowl-edge of the system, rules of thumb or even geographicalproximity between sensors and actuators [22]. Here, byapplying CCS methods such as classical Relative Gain Array(RGA) and the Σ2 Gramian-based interaction measure on theobtained model of the WBF, a systematic approach to designa sparse control configuration is desired.

A. Quantifying controllability and observability with grami-ans

The controllability (P ) and observability (Q) gramians forthe system in (4a) and (4b) can be calculated by solving thefollowing Lyapunov equations [23]:

AP + PAT +BBT = 0 (11)ATQ+QA+ CTC = 0 (12)

It is well-understood that the gramians can be used toquantify the system controllability and observability due totheir energy interpretations. The eigenvalues of P or Q areoften used for this quantification, since the states that aredifficult to reach are in the span of eigenvectors of P , whichcorrespond to small eigenvalues, and the states which aredifficult to observe are in the span of eigenvectors of Q whichcorrespond to small eigenvalues [24]. An issue is that thegramians depend on the realization of the process (selectionof states).

In the sequel of this section, gramians will be used toperform model reduction, input selection and control config-uration selection.

B. Model order reduction

Since numerical errors have been observed in the cal-culations of the gramians, a balanced truncation has beenperformed in order to reduce the number of the states of themodel [25].

The states that are related to the number of Hankel Singu-lar values 10−4 times smaller than the largest singular valueare removed, since an upper bound on the approximation

error can be calculated from the sum of the disregardedHankel Singular values [26].

C. Input Selection

The system presents 3 temperature measurements and 3actuators grouped in 3 oil flows, 3 air flows and exhaustflow. It is of convenience to perform input selection in orderto square down the plant for applying control techniques forsquared systems. The controllability gramian can be usedfor the selection of p inputs by evaluating the controllabilityof the reduced systems resulting from choosing any p-combination of inputs chosen from the r total number ofinputs [27]. If the maximum eigenvalue of the controllabilitygramian is considered, then the reduced B matrix (Bp),resulting from the optimal selection of p actuators is obtainedfrom the following integer optimization:

Bp = arg. maxBi=B(i,:),i∈(r

p)λ(P (A,Bi, C))

where λ(P (A,Bi, C)) is the maximum eigenvalue (λ ) of thecontrollability matrix P (A,Bi, C) of the minimal realizationof the reduced system formed by picking the columns of Bindicated by the vector i. The vector i belongs to the set

(rp

)of p-combinations of the first r integers.

D. Controller structure and controller design

After the selection of the reduced and simplified model,CCS can be performed to the identified interaction betweenthe loops, while RGA which was introduced by Bristol in[28] can be used for initial analysis of the system. Bydefinition, the RGA of a continuous process G(s) with equalnumber of inputs and outputs is defined as:

Λ = G(0)⊗G(0)−T (13)

where G(0)−T is the transpose of the inverse of G(0) and⊗ indicates element by element multiplication. Based onthe RGA pairing rules, it is preferred to select the pairingwith the related RGA value close to 1. Since, this can beinterpreted that there is no interaction effects in that pairing.Besides, pairings with negative values of the RGA shouldbe avoided, since it means that the gain of the subsystemchanges it sign when all the other loops are closed.

However, the RGA is only applicable in the designed ofdecentralized control configurations, where sensors and actu-ators are grouped in pairs and the loops are closed using onlySingle-Input-Single-Output controllers. This limitation leadsto improve the decision process for measurement/actuatorpairing with the gramian-based interaction measurementssuch as Σ2 [29].

By assuming the state space representation in (4a) and(4b). The Σ2 can be defined as:

Σ2 =

∥∥Gij

∥∥2

Σm,rh,l=1

∥∥Ghl

∥∥2

(14)

where∥∥Gij

∥∥2

is the H2 norm of the system transfer func-tion Gij . Since Σ2 is sensitive to input-output scaling, for

accuracy of the calculation it is required to scale the modelas [23]:

G(s) = D−1e G(s)Du

where De is the largest allowed control error and Du is thelargest allowed input change and G indicates the unscaledtransfer function.

The I/O pairing rules using this method is generally toselect the elements of Σ2m×m that have the value greaterthan the average 1

m2 . However, this would be a preliminaryconfiguration, which has to be tested, leading to a possibleredesign in favor to a configuration with larger contribution[30].

Additionally, Σ2 gives a measure of output controllability,since the H2-norm can be calculated from the gramians as:

||G(s)||2 =√trace(CPCT ) =

√trace(BTQB)

IV. SIMULATION RESULTS

The input selection for the WBF is performed byevaluating all the possible subsystems with p = 3 actuatorsand by choosing the subsystem that leads to the largervalue of controllability (λ(P )). After this analysis, thesubsystem formed by the 3 oil actuators: {FO1, FO2, FO3}and the 3 temperature measurements has been found tobe the combination, which achieves the largest maximumeigenvalue of the controllability gramian. The control actionon the air flows will be determined as a ratio from thecontrol action of the oil flows following the stoichiometricrelationship given by the combustion reaction.

The reduced subsystem model indicates that the mostinfluential inputs to control the WBF are the fuel feed flowat each zone of the furnace and the effect of air feed flowsand the exhaust of gases can be neglected to design thesparse controller. This statement can also be confirmed byobserving the data sets of the fuel feed flow and the airfeed flow. From Figure 4 it can be seen that from Pearson’slinear correlation coefficients [15] there is a high correlationbetween the Fuel feed flows and air feed flows (coefficientof correlation ≥ 0.5).

0

20

40

FO

1 [kg/h

]

100 200 300 400 500 600 700

20

40

FO

2 [kg/h

]

0 200 400 600FA [m

3/h]

0

20

40

FO

3 [kg/h

]

0.86

0.83

0.89

Fig. 4: Correlation between the fuel feed flow and air feedflow for each zone of the furnace. The value of Pearsoncorrelation coefficient is shown on each sub-figure.

The RGA is utilized to design the decentralized controlconfiguration initially. The RGA for the subsystem modelcan be derived as:

λ =

FO1 FO2 FO3 1.86 −1.77 0.91 T1

0.07 4.34 −3.41 T2

−0.93 −1.57 3.5 T3

Based on above calculation and the pairing rules mentionedin section III, the most appropriate pairing for design of thedecentralized controller are: FO1−T1, FO2−T2, FO3−T3.

However, this controller involved the selection of values ofRGA larger than 1, which are related to an ill-conditionedsystem. Also, RGA is unable to capture other loops per-turbations. Thus, it can be deduced that the WBF understudy in this work, will be hard to control with decentralizedcontroller.

In order to analyze the interaction between all the loops,Σ2 can be calculated from the scaled model as:

Σ2% =

FO1 FO2 FO3 9.69 3.92 15.72 T1

9.03 7.29 18.45 T2

7.36 5.93 22.62 T3

With the average contribution of 11.11% (1/9), the selectedelements to design multi-variable controller are: {FO1, FO3−T1} , {FO2, FO3 − T2} and {FO3 − T3}. This investiga-

tion results controller matrix C =

c11 0 c130 c22 c230 0 c33

. The

elements c13 and c23 can be designed by feed forwardactions. Figure 5 presents the comparison between the stepresponse of the designed decentralized controller and themulti-variable controller for the WBF in the feedback loop.It can be seen that the decoupled multi-variable system cansufficiently decreased the interaction between the other loops.The structure of the feed back controller is illustrated inFigure 6.

Fig. 6: The schematic of the designed multi-variable con-troller for WBF in the feedback loop.

To validate the designed multi-variable controller,a simulation on the experimental data of WBF wasperformed. As shown in Figure 7, after the change in theset points the controller is able to stabilized around the

set-points after 120[min].

150 300 450 600Time [min]

700

800

900

1000

1100

1200

1300

Te

mp

era

ture

[°C

]

T1

T2

T3

T1 ref

T2 ref

T3 ref

Fig. 7: Validation of the designed controller in the feedbackloop on the experimental set points. The dashed lines indicatethe reference temperature of each zones

V. CONCLUSIONS

The application of control structural design on the steelindustrial plant, walking beam furnace is demonstrated in thisarticle. Due to the high interaction between multiple inputs-outputs of this multi-variable plant, temperature control ofthis reheating furnace is a challenging task. A subspacestate space data-driven modeling was implemented that canestimate the dynamic behavior of the WBF. This model hasmultiple control actions and a non-square transfer function.Control configuration methods, derived the most influentialinputs to the outputs of the system and a subsystem of3 × 3 transfer function is derived. The reduced model isinitially analyzed by the RGA and the decentralized con-troller is designed. However, based on the produced results,the decentralized controller can not sufficiently control thetemperature of each zone of the furnace. A gramian basedinteraction measurement Σ2 recommends a sparse controllerstructure. The validation results of this suggested controller,successfully control the temperature of each zone of thefurnace and track the references set points.However, control design for this furnace is a trade offagainst the achieved performance and the cost of obtainingand maintaining the controllers, actuators and sensors. Theadvantage of the designed sparse controller is its ability tominimize this cost and the corresponding reduced complexityof the proposed control scheme, while achieving a sufficientperformance. As a future work, this performance can be com-pared with a centralized multi-variable controller. Moreover,in extension of this work, the control structure design canbe applied for the combustion process of this walking beamfurnace.

VI. ACKNOWLEDGEMENTS

This research was supported by the European UnionsHorizon 2020 Research and Innovation Programme underthe Grant Agreement No.636834, DISIRE.

0

0.5

1

To: T

1

From: FO1

0

0.5

1

To: T

2

0 20 40 60 80 100 120

0

0.5

1

To: T

3

From: FO2

0 20 40 60 80 100 120

From: FO3

Decentralized controllerSparse designed controller

0 20 40 60 80 100 120

time [min]

Am

plit

ude

Fig. 5: The comparison between the step response of decentralized controller and the sparse multi-variable controller in thefeedback loop. The decentralized controller results in perturbation between the loops.

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