Journal of Process Control 19 (2009) 216–230

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Journal of Process Control

journal homepage: www.elsevier .com/locate / jprocont

Decentralized robust PI controller design for an industrial boiler

Batool Labibi a,*, Horacio Jose Marquez b, Tongwen Chen b

a Advanced Process Automation and Control (APAC) Research Group, K.N. Toosi University of Technology, Tehran, Iranb Electrical and Computer Engineering Department, University of Alberta, Edmonton, Canada

a r t i c l e i n f o

Article history:Received 29 October 2007Received in revised form 23 April 2008Accepted 23 April 2008

Keywords:Industrial utility boilerInternal model controlNonlinear system modelingRobust decentralized control

0959-1524/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.jprocont.2008.04.013

* Corresponding author.E-mail addresses: labibi@kntu.ac.ir, blabibi@yahoo

a b s t r a c t

This paper presents a new scheme to design decentralized robust PI controllers for uncertain LTI multi-variable systems. Sufficient conditions for closed-loop stability and closed-loop diagonal dominance(almost decoupling) of a multivariable system are obtained. Satisfying these conditions and robust per-formance of the overall system are modeled as local robust performance problems. Then, by appropri-ately selecting the time constants of the closed-loop isolated subsystems in the IMC (Internal ModelControl) strategy, the defined local robust performance problems are solved. To design a decentralizedrobust PI controller for a real industrial utility boiler, a control oriented nonlinear model for the boileris identified. The nonlinearity of the system is modeled as uncertainty for a nominal LTI multivariable sys-tem. Using the new proposed method, a decentralized PI controller for the uncertain LTI model isdesigned. The designed controller is applied to the real system. The simulation results show the effective-ness of the proposed methodology.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Control of the interacting multivariable processes can be real-ized either by centralized MIMO controllers or by a set of SISO localcontrollers. For large-scale industrial processes, decentralized con-trol is preferred from the viewpoints of implementation, requiringfewer parameters to tune and loop failure tolerance of the resultingcontrol system [9].

A common design procedure in decentralized control is thesequential design strategy. Sequential design strategy involvesclosing and tuning one loop at a time [6]. However, the order ofdesigning local controllers in the individual loops may affect thecontrol quality. Besides, the closing of subsequent loops maychange the response of previously designed loops which leads tothe necessity of iteration [10]. The usual method for dealing withthe first problem in sequential design is to close the fast loops,which are relatively insensitive to the tuning of the lower loops,first. However, if the corresponding transfer function elementshas a right hand zero which is not a transmission zero of the plant,closing the fast loop first may not be possible [10]. To resolve thesecond problem, in [10], a strategy for sequential design of decen-tralized controllers for linear systems is presented. This method in-volves a simple estimate of the impact of closing previous loopsinto the design problem for the loop which is to be closed.Although, the proposed method is useful and easy to implement,but it does not give any solution for the first problem.

ll rights reserved.

.com (B. Labibi).

In control theory, the internal model control (IMC) method, firstproposed by Morari and Zafiriou [13], is a powerful and simplestrategy for designing controllers. The idea is to specify the desiredclosed-loop response and solve for the resulting controller [13,17].The decentralized type of the IMC strategy is also used in severalreferences such as [9,10]. In [10], a methodology for sequentialdesign of robust decentralized controllers is given. Each control-ler is parameterized to have only one IMC tuning parameter. There-by, initial values to tune parameters are selected and theparameters are updated by an iterative algorithm, such that the ro-bust performance is achieved. The design algorithm terminates ifthe closed-loop system is stabilized. In fact, in [10], a new propertycalled robust decentralized detunability (RDD) is introduced. If asystem has this property, any subset of the loops can indepen-dently be detuned to an arbitrary degree without influencing ro-bust stability. In the proposed method, the class of possibledecentralized IMC controllers are parameterized in terms of theIMC filter time constants. Then, the unknown time constants areconsidered as uncertainty and the possible bounds on time con-stants are obtained. Using l controller synthesis algorithms, thetime constants can be selected to ensure robust stability or robustperformance. The design procedure does not give the optimal filtertime constants, but it provides a range of values to guarantee ro-bust stability/performance, and robust decentralized detunability.The proposed design procedure can also be extended to includeother types of controllers such as PID controllers [10]. However,as it is stated by the authors in [10], the suggested algorithm is dif-ficult to apply for problems to achieve different bandwidths in thevarious loops.

B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 217

As a practical application of the decentralized IMC strategy,using the proposed method in [13], Ref. [7] has considered decen-tralized temperature control of a packed-bed chemical reactor. Infact, [7] models the off-diagonal elements of the MIMO plant as in-put multiplicative uncertainties which are used to design l inter-action measures. Then, employing the IMC tuning rules as abaseline, a decentralized PI controller is designed for the diagonalpart of the MIMO model by tuning a simple factor iteratively.

Designing decentralized robust PID controllers for power sys-tems is very important. These controllers have proven to be robustand extremely beneficial in the control of many important applica-tions. Extensive research has been done to tune PID controllers[1,2,8,11]. However, this problem still remains an active field of re-search in the control literature. In this area of research, [4] has sug-gested a decentralized robust control approach, by considering thediagonal part of the model as the design model, and the off-diagonaldynamics of the interconnected plant as uncertainty. Then, themaximum singular values or structured singular values of the mul-tiplicative uncertainty are used as interaction measures. In fact, themethod in [4] proposes an iterative two-step sub-optimal designprocedure to design a decentralized robust controller. This work fol-lows the basic framework of [13]. But, the interaction between sub-systems is measured by a revised passivity index. Due to use of boththe phase and gain information of the interactions, the offered algo-rithm may lead to a less conservative stability condition. To design adecentralized PID controller, the derived controller which is nor-mally a high order one, can be simplified to a lower order systemsuch as a PID controller. Compared to the previous methods, theproposed method may be less conservative. However, the offeredalgorithm in [4] is an iterative approach and requires complicatedcomputation where the dimension of the system is high.

To design robust PI controllers to regulate frequency and trackload in power systems, [5] formulates the load frequency control(LFC) problem as a multi-objective control problem via a mixedH2=H1 control technique. To design a robust PI controller, the con-trol problem is reduced to a static output feedback control synthe-sis, and then, it is solved using an iterative linear matrixinequalities (ILMI) algorithm. However, the proposed method isan iterative control design strategy. Generally, iterative algorithmssuffer from two major problems, convergence of the algorithm andfinding good initial values for designing parameters. So, presentinga non-iterative decentralized control strategy for industrial pro-cesses is of great importance.

The Control of industrial co-generation systems represents asignificant challenge for control engineers. The crucial part in apower plant is the boiler system. Due to the shrink and swelldynamics which causes a non-minimum phase behavior, the leveldrum control problem is very difficult [3]. In [21], robust controllerdesign for a linear two-input two-output model of a boiler/turbineunit which relates firing rate and turbine valve position inputs tothrottle pressure and megawatt outputs is given. The plant/modelmismatch is represented as output-multiplicative uncertainty.Then, the closed-loop performance and robustness of MIMO H1and l-synthesis control laws with those of MIMO control H2 laware compared. The comparative evaluation of applying the threedesigns to the model, shows that in power plant control problemsH1 and l-synthesis provide much better performance/robustnessthan H2 design.

Following the previously mentioned methods, in this paper, arobust decentralize PI controller for the utility boiler systems inthe Syncrude Canada Ltd. (SCL) is designed and applied to the realsystem. The SCL integrated energy facility located in Mildred Lake,Alberta, utilizes a complex header system for steam distribution.The normal plant operation requires tracking the steam demandwhile maintaining the steam pressure and the steam temperatureof the header at their respective set points, despite variations of the

steam load. Due to the physical characteristics, utility boilers areused to regulate the steam pressure [19]. To design a robust decen-tralized PI controller for the utility boilers in SCL, first, a non-iter-ative simple algorithm based on the IMC strategy is proposed.Sufficient conditions for closed-loop stability and diagonal domi-nance under a decentralized control are achieved. Based on the ob-tained sufficient conditions, a new interaction measure is defined,and the system is decomposed into the isolated subsystems, suchthat to minimize the obtained interaction measure. If the isolatedsubsystems are of high order, first order models are obtained.The error of approximation can be considered as multiplicativeuncertainty for each isolated subsystem. It will be shown thatachieving closed-loop diagonal dominance and robust stabilitycan be guaranteed by solving certain local robust performanceproblems to be defined. In the IMC strategy, by appropriatelyselecting the time constants of the closed-loop isolated subsys-tems, which are the IMC tuning parameters, these local problemscan be solved. The proposed approach has two major advantages:(1) the method is applicable to unstable systems as well. (2) Thetrue model uncertainties can be dealt within the same frameworkso that the decentralized controllers obtained can be robust. In or-der to control the drum boilers in the SCL, first identifying an accu-rate model for the system is necessary. Using the fundamentalphysical laws, the known physical constants of the system, andthe measured plant data, a fairly accurate nonlinear model forthe system is identified. The derived model captures all the essen-tial features of the actual boiler dynamics, including nonlinearities,non-minimum phase behavior, instabilities, and load disturbances.Then, the nonlinear model is linearized about its operating pointsand the nonlinearity is modeled as uncertainty for a nominal LTIsystem. Thereafter, based on the new proposed algorithm in thispaper, a decentralized PI controller for the system is designed.The resulting controller is applied to the real system using SYNSIM,to show the effectiveness of the proposed method. SYNSIM is asimulation package developed by the Syncrude Canada with thepurpose of simulating certain upset conditions and as a generaltool for stability analysis.

The rest of this paper is organized as follows: In Section 2, theproblem of finding suitable local dynamical controllers for subsys-tems of an LTI multivariable system is presented. In Section 3, byconsidering the interactions between the isolated subsystems asuncertainty, sufficient conditions for closed-loop stability, diagonaldominance and disturbance rejection are given. These conditionsare stated in the sensitivity functions of the closed-loop isolatedsubsystems. In Section 4, the new method for decentralized PI con-troller design is given. In Section 5, by considering the true modeluncertainty within the same framework of the interaction uncer-tainty, a method for robust decentralized PI controller design foran uncertain multivariable system is given. In Section 6, a nonlin-ear model of the drum boiler system is identified. Thereafter, inSection 7 the nonlinear model is linearized about its operatingpoints and the nonlinearity is modeled as uncertainty. Then, bysolving the appropriately defined local robust problems, a decen-tralized controller is designed. The designed controller is appliedto the real system and the simulation results are given. Finally,concluding results are given in Section 8.

2. Problem formulation

Consider an uncertain LTI multivariable system eGðsÞ with out-put multiplicative uncertainty as follows:

eGðsÞ ¼ ðI þ DðsÞ ��W3ðsÞÞGðsÞ; jDðjxÞj 6 1 8x; ð1Þwhere GðsÞ is the nominal plant, DðsÞ is any stable transfer matrixwhich at each frequency is less than or equal to one in norm, and��W3ðsÞ is the diagonal weighting matrix which contains the fre-

218 B. Labibi et al. / Journal of Process Control 19 (2009) 216–230

quency information for the uncertainties. Suppose the nominal sys-tem GðsÞ has the following state-space equations

_xðtÞ ¼ AxðtÞ þ BuðtÞ;yðtÞ ¼ CxðtÞ þ DuðtÞ;

ð2Þ

where x 2 Rn, u 2 Rm, y 2 Rm, A 2 Rn�n, B 2 Rn�m, C 2 Rm�n, andD 2 Rm�m.

Let

x ¼x1

..

.

xN

2664

3775; u ¼

u1

..

.

uN

2664

3775; y ¼

y1

..

.

yN

2664

3775; A ¼

A11 . . . A1N

..

.. . . ..

.

AN1 . . . ANN

2664

3775;ð3Þ

B ¼B11 . . . B1N

..

.. . . ..

.

BN1 . . . BNN

2664

3775; C ¼

C11 . . . C1N

..

.. . . ..

.

CN1 . . . CNN

2664

3775; and

D ¼D11 . . . D1N

..

.. . . ..

.

DN1 . . . DNN

2664

3775; ð4Þ

then, GðsÞ is composed of N linear time-invariant subsystems GiðsÞ,described by

_xiðtÞ ¼ Aiixi þ Biiui þXN

j¼1j–i

Aijxj þXN

j¼1j–i

Bijuj;

yiðtÞ ¼ Ciixi þ Diiui þXN

j¼1j–i

Cijxj þXN

j¼1j–i

Dijuj;

ð5Þ

with xi 2 Rni , ui 2 Rmi , yi 2 Rmi , Aii 2 Rni�ni , Bii 2 Rni�mi , Cii 2 Rmi�ni ,PNi¼1ni ¼ n, and

PNi¼1mi ¼ m. The terms

PNj¼1j–i

Aijxj,PN

j¼1j–iBijuj,PN

j¼1j–iCijxj, and

PNj¼1j–i

Dijuj are due to interactions of the other sub-

systems. The objective is to design a local PI controller given by

KiðsÞ ¼ Kci1þ TIis

TIis

� �; i ¼ 1; . . . ;N ð6Þ

for each isolated subsystem GiiðsÞ; described by

_xiðtÞ ¼ AiixiðtÞ þ BiiuiðtÞ;yiðtÞ ¼ CiixiðtÞ þ DiiuiðtÞ;

ð7Þ

such that the closed-loop subsystem is stabilized, and at the sametime effects of interactions of the other subsystems and uncertain-ties are minimized. By this, the decentralized controller

KðsÞ ¼ diagfKiðsÞg; ð8Þ

stabilizes the overall uncertain system given in (1), if some suffi-cient conditions are satisfied.

3. Closed-loop stability and diagonal dominance

In this section, sufficient conditions for closed-loop stability anddiagonal dominance of the nominal system are obtained. To thisend and in order to prove our theorems, the transformation pro-posed in [12] is used to transform the system given in (2) into anequivalent descriptor system representation. It should be notedthat this representation is only for proving the related theoremsand the control design will be done for conventional isolated sub-systems. Since designing a dynamic controller for a system can beconverted into designing a static controller for an augmented sys-tem, without loss of generality in this section we assume the de-signed controller is a static one.

Consider the system given by Eq. (2), to obtain an equivalentdescriptor representation form, all of the inputs and outputs ofthe system are defined as state variables. Then, the augmented sys-tem �GðsÞ has the following equations

�E _�xðtÞ ¼ �A�xðtÞ þ �BuðtÞ;yðtÞ ¼ �C�xðtÞ þ �DuðtÞ;

ð9Þ

where

�E ¼

0m1�m1 0m1�n1 0m1�m1. .

.0m1�mN 0m1�nN 0m1�mN

0n1�m1 In1�n1 0n1�m1. .

.0n1�mN 0n1�nN 0n1�mN

0m1�m1 0m1�n1 0m1�m1. .

.0m1�mN 0m1�nN 0m1�mN

..

. ... ..

.. . . ..

. ... ..

.

0mN�m1 0mN�n1 0mN�m1. .

.0mN�mN 0mN�nN 0mN�mN

0nN�m1 0nN�n1 0nN�m1. .

.0nN�mN InN�nN 0nN�mN

0mN�m1 0mN�n1 0mN�m1. .

.0mN�mN 0mN�mN 0mN�mN

266666666666666666664

377777777777777777775ð10Þ

�A¼

�Im1�m1 C11 D11. .

.0m1�mN C1N D1N

0n1�m1 A11 B11. .

.0n1�mN A1N B1N

0m1�m1 0m1�n1 �Im1�m1. .

.0m1�mN 0m1�nN 0m1�mN

..

. ... ..

.. . . ..

. ... ..

.

0mN�m1 CN1 DN1. .

.�ImN�mN CNN DNN

0nN�m1 AN1 BN1. .

.0nN�mN ANN BNN

0mN�m1 0mN�n1 0mN�m1. .

.0mN�mN 0mN�nN �ImN�mN

2666666666666666666664

3777777777777777777775

ð11Þ

�xT ¼ yT1 xT

1 uT1 . . . yT

N xTN uT

n

� �

u ¼u1

..

.

uN

2664

3775; y ¼

y1

..

.

yN

2664

3775 ð12Þ

�B ¼

0m1�m1. .

.0m1�mN

0n1�m1. .

.0n1�mN

Im1�m1. .

.0m1�mN

..

.. . . ..

.

0mN�m1. .

.0mN�mN

0nN�m1. .

.0nN�mN

0mN�m1. .

.ImN�mN

266666666666666666664

377777777777777777775

ð13Þ

�C ¼Im1�m1 0m1�n1 0m1�m1

. ..

0m1�mN 0m1�nN 0m1�mN

..

. ... ..

.. . . ..

. ... ..

.

0mN�m1 0mN�n1 0mN�m1. .

.ImN�mN 0mN�nN 0mN�mN

266664

377775ð14Þ

B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 219

and

�D ¼ 0m�m: ð15Þ

The transfer matrix of the closed-loop descriptor system in (9),�GclðsÞ is given by

�GclðsÞ ¼ �CðsE� �Aþ �BK �CÞ�1�BK þ �D

¼ ðI þ CðsI � AÞ�1BK þ DKÞ�1ðCðsI � AÞ�1Bþ DÞK; ð16Þ

which is equal to TðsÞ, the transfer matrix of system (2) under thedecentralized controller. Therefore, control of system (9) results incontrolling of system (2). Defining

�Ad ¼ diagf�Aiig; ð17Þ

with

�Aii ¼�Imi�mi

Cii Dii

0ni�miAii Bii

0mi�mi0mi�ni

�Imi�mi

264

375; ð18Þ

it is easy to show that the transfer matrices of the closed-loop sys-tems ðE; �Ad; �B; �C; �DÞ and ðAd; Bd;Cd;DdÞ under the decentralized con-troller K ¼ diagfKig are the same i.e.

TdðsÞ ¼ �C�P�BK ¼ ðIþ CdðsI�AdÞ�1BdK þDdKÞ�1ðCdðsI�AdÞ�1Bd þDdÞK;ð19Þ

where

�P ¼ ðsE� �Ad þ �BK �CÞ�1: ð20Þ

Defining

�H ¼ �A� �Ad; ð21Þ

the system ðE; �Ad; �B; �CÞ is a block-diagonal system and the matrix �Hcan be considered as uncertainty in the matrix �A.

3.1. Sufficient conditions for closed-loop stability

The next theorem provides sufficient conditions for closed-loopstability.

Theorem 1. Suppose the decentralized controller K stabilizes thediagonal system ðAd;Bd;Cd;DdÞ. Then the closed-loop original systemunder the decentralized controller is stable, if

k�P �Hk1 < 1; ð22Þ

where �P and �H are given in Eqs. (20) and (21), respectively andk � k1 is the maximum singular value of (�) [17].

Proof. The transfer matrix of the closed-loop system can be writ-ten as

TðsÞ ¼ �CðsE� �Ad þ �BK �C � �HÞ�1�BK ¼ �Cð�P�1 � �HÞ�1�BK

¼ �CðI � �P �HÞ�1�P�BK: ð23Þ

Since �P is stabilized by stabilizing the block-diagonal systemðAd; Bd;Cd;DdÞ, then the closed-loop system is stable if the transfermatrix ðI � �P �HÞ�1 is stable. The transfer matrix �P is stable and ifthe Nyquist plot of detðI � �P �HÞ does not encircle the origin or equiv-alently if the condition given in (22) is satisfied the closed-loop sys-tem is stable and the proof is complete. h

The matrix

�P ¼ diagf�Pig ð24Þ

with

�Pi ¼ ðsEi � �Aii þ �BiKi�CiiÞ�1 ð25Þ

�Ei ¼0mi�mi

0mi�ni0mi�mi

0ni�miIni�ni

0ni�mi

0mi�mi0mi�ni

0mi�mi

264

375 ð26Þ

�Bii ¼0mi�mi

0ni�mi

Imi�mi

264

375 ð27Þ

and�Cii ¼ Imi�mi

0mi�ni0mi�mi

� �is a block-diagonal matrix. Then, the following stability conditions

k�Pik1 < l�1ð�HÞ; i ¼ 1; . . . ;N; ð28Þ

where lð�Þ is the maximum structured singular value of (�) [17], givesufficient conditions for closed-loop stability at the subsystem level.

3.2. Sufficient conditions for diagonal dominance

Theorem 2. The closed-loop nominal system given in (2) under thedecentralized controller K is diagonal dominate, if

kSik1 <ai

j �w1ij; i ¼ 1; . . . ;N; ð29Þ

where ai is a positive scalar less than one and small enough, calledlocal diagonal dominance degree, �w1iðsÞ, the weighting function sat-isfies the following equation

j �w1iðsÞj >ffiffiffiffiNpjðCiiðsI � AiiÞ�1HABi

þ HCDij; ð30Þ

Si is the sensitivity function of the i-th closed-loop isolatedsubsystem

HABi¼ Ai1 Bi1 Ai2 Bi2 . . . Aii�1 Bii�1 0 0 Aiiþ1 Biiþ1 . . . AiN BiN½ �

ð31Þ

and

HCDi¼ Ci1 Di1 Ci2 Di2 . . . Cii�1 Dii�1 0 0 Ciiþ1 Diiþ1 . . . CiN DiN½ �:

ð32Þ

Proof. The closed-loop descriptor system under decentralizedcontrol has the following form

ð�C � �C�P �HÞ�X ¼ �C�P�BKR: ð33Þ

If

k�C�P �Hk1 < arminð�CÞ; ð34Þ

where rminð�Þ is the minimum singular value of (�), and a is a posi-tive scalar less than one, then [18]

�C � �C�P �H ffi �C ð35Þ

and TðsÞ, the transfer matrix of the closed-loop system is given bythe following equation

TðsÞ ffi eC �P�BK ¼ ðI þ CdðsI � AdÞ�1BdK þ DdKÞ�1

� ðCdðsI � AdÞ�1Bd þ DdÞK ¼ TdðsÞ; ð36Þ

which is a block-diagonal transfer matrix. Based on the definition of�C given in (14)

rminð�CÞ ¼ 1; ð37Þ

which means that by minimizing k�C�P �Hk1 such that

k�C�P �Hk1 < a; ð38Þ

220 B. Labibi et al. / Journal of Process Control 19 (2009) 216–230

the closed-loop system is diagonal dominant. We know [17]

k�C�P �Hk1 6ffiffiffiffiNpk�Cii

�Pi�Hik1; i ¼ 1; . . . ;N; ð39Þ

with

�Hi ¼0 Ci1 Di1 0 Ci2 Di2 . . . 0 Cii�1 Dii�1 0 0 0 0 Ciiþ1 Diiþ1 . . . 0 CiN DiN

0 Ai1 Bi1 0 Ai2 Bi2 . . . 0 Aii�1 Bii�1 0 0 0 0 Aiiþ1 Biiþ1 . . . 0 AiN BiN

0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 0 0 . . . 0 0 0

264

375: ð40Þ

Denoting

Si ¼ ðI þ CiiðsI � AiiÞ�1BiKi þ DiiKiÞ�1; ð41Þ

as the sensitivity function of the i-th closed-loop isolated subsys-tem, it can be shown that

�Cii�Pi

�Hi ¼ SiðCiiðsI � AiiÞ�1HABiþ HCDi

Þ: ð42Þ

Therefore, by defining �w1i as given in Eq. (30), by satisfying Eq.(29), the closed-loop system is diagonal dominant with the degreea ¼ maxfaig and the proof is complete. h

It follows that by designing local controllers such that

k �w1iSik1 < ai; i ¼ 1; . . . ;N; ð43Þ

with ai small enough, the closed-loop system is block-diagonaldominant. In fact, the transfer matrix of the closed-loop system isgiven as

TðsÞ ¼ �C�P �HðI � �P �HÞ�1�P�BK þ �C�P�BK: ð44Þ

In minimizing k�C�P �Hk1 < a, since k�Ck1 ¼ 1, we have

k�Ck1rminð�P �HÞ ¼ rminð�P �HÞ 6 k�C�P �Hk1 < a: ð45Þ

On the other hand, in [14] it is stated an m�m complex matrix Qdecomposed as

Q ¼ QD þ QC ; ð46Þ

with

Q D ¼ diagfQ 1;Q2; . . . ;QNg

Q C ¼

Q11 Q12 . . . Q 1N

Q21 Q22 . . . Q 2N

. . . . . . . . . . . .

QN1 QN2 . . . Q NN

26664

37775;

Qii may be zero or nonzero matrix, is said to be quasi-block-diago-nal dominance (QBDD) if there exists a matrix norm k � k such that

kQ CQ�1D k < 1: ð47Þ

By considering

Q D ¼ �C�P�BK ð48Þ

and

Q C ¼ �C�P �HðI � �P �HÞ�1�P�BK; ð49Þ

then

kQ CQ�1D k1 ¼ k�C�P �HðI � �P �HÞ�1�P�BKð�C�P�BKÞ�1k1 ð50Þ

and

kQ CQ�1D k1 6 k�C�P �Hk1kðI � �P �HÞ�1k1k�Cþk1k�C�P�BKð�C�P�BKÞ�1k1;

ð51Þ

where �Cþ is the pseudo-inverse of �C. Since

k�Cþk1 ¼1

rminð�CÞ¼ k�Ck1 ¼ 1; ð52Þ

then, we have

kQ CQ�1D k1 6

k�C�P �Hk11� rminð�P �HÞ

6a

1� a: ð53Þ

It is clear if

a1� a

6 1; ð54Þ

or equivalently

a 6 0:5; ð55Þ

the transfer matrix of overall closed-loop system is a quasi-block-diagonal matrix in sense of the concept introduced in [14]. Then,the stability and performance of the system can be inferred directlyfrom the stability and performance of the block- diagonal transfermatrix TdðsÞ ¼ �C�P�BK [16].

3.3. Disturbance rejection

In order to have good disturbance rejection characteristics forthe closed-loop system, an appropriately defined weighting matrixcan be chosen such that

k ��W1ðsÞSk1 6 1; ð56Þ

where S is the sensitivity matrix of the overall closed-loop system.Since the closed-loop system is diagonal dominant, if the diagonaldominance degree is small and ��W1ðsÞ is chosen as a block-diagonalmatrix

��W1ðsÞ ¼ diagf ��w1iðsÞ; ð57Þ

then, with a good approximation we can show by designing adecentralized controller to minimize the norm of weighted sensitiv-ity matrix of the closed-loop block-diagonal system such that

k ��W1ðsÞSdk1 6 1; ð58Þ

where

Sd ¼ diagfSig ð59Þ

is the sensitivity matrix of the closed-loop block-diagonal system,or equivalently designing local controllers such that

k��w1iðsÞSik1 6 1; i ¼ 1; . . . ;N; ð60Þ

the condition given in (56) is satisfied.

4. Decentralized PI design

In this section, a new method for decentralized PI controller de-sign is given. Before doing so, however, we revisit the SISO PI de-sign problem.

4.1. SISO PI Design using Internal Model Control Method

In designing a PI controller for a SISO system, approximation ofhigh order processes by first order models is a common practice[11]. Once an approximated model is obtained, a PI controllerbased on IMC method can be designed. Even though many indus-trial processes meet the assumptions sufficiently to be modeledwith a first order model, there do exist many plants which cannot

B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 221

be well approximated by first order systems. To avoid instabilitydue to approximation, in this paper the error of approximation isconsidered as multiplicative uncertainty for isolated subsystems.Then, designing a local PI can be converted into solving an appro-priately defined local robust performance problem. The strategy isdescribed in the following subsection.

4.2. Decentralized PI controller design

In this subsection, a new tuning criterion for MIMO PI control-lers is proposed. The next theorem gives a methodology for strictlyproper subsystems. Proper subsystems will be considered later.

Theorem 3. Consider the i-th isolated subsystem which is approxi-mated with a first order model and the approximation error isconsidered as multiplicative uncertainty weight �w3iðsÞ. Then, theclosed-loop MIMO system can be stabilized if sci for the i-th isolatedclosed-loop subsystem is selected such that

kj 1ai

�w1iðsÞsci

ssci

sþ 1j þ j �w3iðsÞ

1sci

sþ 1jk1 < 1 8x; i ¼ 1; . . . ;N:

ð61Þ

Proof. By designing a local PI controller for the i-th isolated sub-system by an IMC based method, the i-th closed-loop subsystemhas the following sensitivity and complementary sensitivity func-tions, respectively [17]

Si ¼sci

ssci

sþ 1ð62Þ

and

Ti ¼1

scisþ 1

: ð63Þ

To attenuate the interactions between subsystems by local control-lers, the sensitivity functions of each isolated subsystem should sat-isfy condition (29). This condition can be considered as a nominalperformance problem. If the isolated subsystem can not be approx-imated sufficiently well by a low order model, the modeling errormay be considered as multiplicative uncertainty given by weightingfunction �w3iðsÞ. Then, the i-th approximated closed-loop isolatedsubsystem is stable if and only if [17]

kTiðsÞ �w3iðsÞk1 < 1: ð64Þ

The i-th sensitivity and complementary sensitivity functions shouldsatisfy the conditions given in (29) and (64), respectively, to haveclosed-loop diagonal dominance (nominal performance) for theoverall system and robust stability for the subsystems. This is a ro-bust performance problem for the isolated subsystems. By consider-ing definitions given in (62) and (63), the closed-loop system isdiagonal dominant if the local robust performance problems givenin (61) are solved and the proof is complete. h

According to the theorem, by selecting appropriate values forsci

’s and ai < 0:5 to solve local problems (61), the closed-loop sta-bility and diagonal dominance are guaranteed.

For proper subsystems the proposed method should be modi-fied as we now explain. Suppose that the reduced order model ofthe i-th subsystem is given by a first order system

Gi ¼aisþ bi

sþ ci; ð65Þ

where ai, bi and ci are known constants of the transfer function. Thelocal PI controller can be selected as

KiðsÞ ¼ kisþ ci

s; ð66Þ

which results in

TiðsÞ ¼ðaisþ biÞki

ðaiki þ 1Þsþ bikið67Þ

SiðsÞ ¼s

ðaiki þ 1Þsþ bikið68Þ

and the constant ki can be selected such that

kj 1ai

�w1iðsÞSiðsÞj þ j �w3iðsÞTiðsÞk1 < 1; 8x i ¼ 1; . . . ;N: ð69Þ

5. Robust decentralized PI control

Consider the uncertain system given in (1). The uncertainty ismodeled as diagonal multiplicative output uncertainty (In this sec-tion, without loss of generality we consider output multiplicativeuncertainty. It is clear the same result can be obtained for inputmultiplicative uncertainty as well.). The closed-loop uncertain sys-tem is robust stable if and only if for multiplicative output uncer-tainty [17]

k ��W3Tk1 < 1; ð70Þ

where ��W3ðsÞ ¼ diagf ��w3iðsÞg is a diagonal matrix representing mul-tiplicative uncertainty of the system. If the closed-loop system isdiagonal dominant with a good degree of diagonal dominance, thenwith a good approximation TðsÞ ffi TdðsÞ and the performance of theclosed-loop system can be inferred from the block-diagonal part ofthe transfer matrix. We can show by solving the robust stabilityproblem

k ��W3Tdk1 < 1; ð71Þ

or equivalently by solving local robust stability problems

k��w3iTik1 < 1; ð72Þ

the condition given in (70) can be satisfied.Now, suppose the objective is to design a robust decentralized

PI controller for an uncertain plant with multiplicative uncertaintyto solve the robust stability problem given in Eq. (70). In addition,the closed-loop system should have the desirable disturbancerejection characteristics. It was shown that for closed-loop diago-nal dominance, the local robust performance problems given in(61), for treating uncertainty in the system, the conditions givenin (72), and for good disturbance rejection, the conditions givenin (60) should be satisfied. Combining these conditions for design-ing a robust decentralized PI controller for an uncertain LTI multi-variable, the problem can be converted into solving the followingmodified local robust performance problems

w1iðsÞsci

ssci

sþ 1

��������þ w3iðsÞ

scisþ 1

��������

��������1< 1 8x; i ¼ 1; . . . ;N; ð73Þ

with

max�w1iðsÞai

��������; j ��w1iðsÞj

6 jw1iðsÞj; 8x; i ¼ 1; . . . ;N ð74Þ

and

maxfj�w3iðsÞj; j ��w3iðsÞjg 6 jw3iðsÞj; 8x; i ¼ 1; . . . ;N: ð75Þ

So far, we have shown designing a robust decentralized PI controllerfor an uncertain MIMO system can be reduced to solving the localrobust performance problems given in (73)–(75).

Remark 1. Decomposing the matrices A, B, C and D into block-diagonal and non-block-diagonal matrices is arbitrary. It mayhappen that the open-loop isolated subsystems have some purelyimaginary poles. Then, at these frequencies the norm

222 B. Labibi et al. / Journal of Process Control 19 (2009) 216–230

ci ¼ kCiiðsI � AiiÞ�1HABiþ HCDi

k1; ð76Þ

approaches infinity and conditions (30) can not be satisfied for sta-ble weighting functions. To avoid this problem, we can decomposethe matrices A, B, C and D, such that the isolated subsystems arecontrollable, observable and stable and the interactions betweensubsystems are minimized. This approach introduces some flexibil-ity into the design procedure and makes the proposed methodapplicable to unstable systems as well. In addition, as it is knownin the IMC method, the poles of the isolated subsystems are zerosof the local PI controllers. If the pole of an isolated subsystem isfar from the imaginary axis, then, the integration time of the de-signed local PI controller decreases and closed-loop response is fas-ter but, is also more oscillatory. If the pole is close to the imaginaryaxis, then, the integration time increases and the closed-loop re-sponse creeps slowly toward the set point [2]. Then, by decompos-ing the matrix A appropriately, it is possible to control the shape ofthe closed-loop response as well. To do an appropriate decomposi-tion, appropriate local LMI optimization problems are introduced.From Eq. (42) we can observe by minimizing the norm given in(76), the interactions between closed-loop isolated subsystems willbe decreased. In fact, this norm can be considered as an interactionmeasure for the overall closed-loop system. Suppose the matrices A,B, C and D are decomposed such that bAii, bBii, bCii and bDii are the statematrices of the i-th isolated subsystem, and

bHABi¼ Ai1 Bi1 . . . Aii�1 Bii�1 Aii � bAii Bii � bBii . . . AiN BiN

h ið77Þ

and

bHCDi¼ Ci1 Di1 . . . Cii�1 Dii�1 Cii � bCii Dii � bDii CiN DiN

h i:

ð78Þ

By virtue of the bounded real lemma, minimizing the normci ¼ kbCiiðsI � bAiiÞ�1 bHABi

þ bHCDik1 is equivalent to finding the local

Lyapunov matrices Pi > 0 such that [17]

: ð79Þ

In the above equations, the off-diagonal-blocks Aij; i–j, Bij; i–j,Cij; i–j and Dij; i–j are fixed. Because, Ad ¼ diagfbAiig,Bd ¼ diagfbBiig, Cd ¼ diagfbCiig and Dd ¼ diagfbDiig should be block-diagonal. But, the diagonal blocks bAii, bBii, bCii and bDii can be selectedto minimize the norm. The local LMI optimization problem given in(79) is not affine in the variables bAii and bBii. To solve the problemwe introduce new LMI variables

Ri ¼ PibAii ð80Þ

Si ¼ PibBii ð81Þ

and the LMI constraint given in (79) is converted into

; ð82Þ

with

Q i ¼ PiAi1 PiBi1 . . . PiAii�1 PiBii�1 PiAii�Ri PiBii� Si . . . PiAiN PiBiN½ �ð83Þ

Yi ¼ bHCDi: ð84Þ

Now, the LMI optimization problem given in (82) is affine in LMIvariables Pi, Ri, and Si. After solving the local LMI optimization prob-

lems, the matrices bCii and bDii are obtained and the matrices bAii andbBii can be calculated as

bAii ¼ P�1i Ri ð85Þ

and

bBii ¼ P�1i Si: ð86Þ

To avoid fast modes in the matrix bAii, new LMI constraints can beadded to optimization problem (82), to limit the eigenvalues of bAii

or assign them in the desired region [17].

Remark 2. Based on the discussion given in Theorem 2, the closed-loop system is diagonal dominant, if

jSij <1ffiffiffiffi

NpjCiiðsI � AiiÞ�1HABi

þ HCDij; 8x; i ¼ 1; . . . ;N:

In the above relation, at high frequencies, the left and right handsides of the relation approach to one and 1ffiffiffi

NpjHCDi

j, respectively. Then,

to satisfy these conditions at high frequencies, kHCDik1 should be

less than or equal to 1ffiffiffiNp . This is, however not always the case. For

solving this problem, it is possible to use similarity transformations.Since similarity transformations do not affect output feedback andthe overall system is observable, it is possible by using theobservability matrix of the overall system to find an appropriatetransformation to transform the original system into the output-decentralized form, where the matrix C is block diagonal [12]. Then,Cij ¼ 0; i–j, and for strictly proper systems (D ¼ 0), at highfrequencies these conditions will always be satisfied. But, for propersystems, the proposed methodology is applicable only whenkD� diagfDiigk1 is less than 1ffiffiffi

Np .

Since, our objective is to design a decentralized PI controller fora nonlinear multivariable system, based on the discussion given sofar, an algorithm to design a robust decentralized PI controller fornonlinear multivariable systems is given as follows:

Algorithm 1

(a) Linearize the system about its operating points. Select aplant as the nominal plant and model the nonlinearity asoutput multiplicative uncertainty. Define the diagonalweighting matrix ��W3ðsÞ.

(b) By solving the local LMI optimization problems given in (82),decompose the nominal plant into isolated subsystems suchthat they are stable, controllable and observable and theinteractions between subsystems are minimized. IfkC � diagfbCiigk1 is not less than 1ffiffiffi

Np , apply the similarity

transformation to the system to convert the matrix C intoa block-diagonal one and do the decomposition again. Then,check kD� diagfbDiigk1 if this norm is greater than 1ffiffiffi

Np , stop,

the method is not applicable to the system. Otherwise go tostep (c).

(c) Based on the decomposition, define the weighting functions�w1iðsÞ; i ¼ 1; . . . ;N; such that the conditions given in (30) aresatisfied.

(d) If the isolated subsystems are not first order, use an appro-priate methodology to approximate the transfer function ofeach isolated subsystem with a first-order model. Definethe error of approximation as multiplicative uncertainty.Based on the local approximation models, define appropriateweighting functions �w3iðsÞ; i ¼ 1; . . . ;N.

B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 223

(e) Derive appropriate weighting functions w1iðsÞ and w3iðsÞusing Eqs. (74) and (75). Select sci

for the i-th isolated sub-system, such that the local robust performance problemsgiven in (73) are solved.

(f) Based on the IMC method, design local PI’s using theobtained time constants in step (e).

6. Utility boiler

The utility boilers in Syncrude Canada are water tube drumboilers. Since, steam is used for generating electricity and processapplications, demand for steam is variable. The control objectiveof the co-generation system is to track steam demand while main-taining steam pressure and steam temperature of the header attheir respective set-points. The objective of this paper is to designa controller, so that the utility boiler system keeps stability andreaches the desired performance. In the system, the principal inputvariables are u1, feedwater flow rate (kg/s); u2, fuel flow rate (kg/s);and u3, attemperator spray flow rate (kg/s). The principal outputvariables are y1, drum level (m); y2, drum pressure ðkPaÞ; and y3,steam temperature �C [19].

To design a decentralized PI controller for the utility boiler sys-tem, first it is necessary to identify a fairly accurate model for thesystem. The identified model is a control oriented model. Since, thedesigned controller will be a decentralized controller, we try toidentify the system such that the matrices, A, B, C and D are as closeas possible to block-diagonal matrices. Indeed, because in the pro-posed method, the subsystems should be approximated by first or-der models, we try to derive a model with the lowest possibleorder. According to the field experience, the utility boilers in theSCL work mainly in three typical operating points, which are lowload, normal load, and high load operating points [19]. The param-eters of the different operating points are listed in the following ta-ble (see Table 1).

6.1. Steam flow dynamics

Steam flow plays an important role in the drum-boiler dynam-ics. Steam flow from the drum to the header, through the superheaters, is assumed to be a function of the pressure drop fromthe drum to the header. We use a modified form of the Bernoulli’slaw to represent flow versus pressure, with friction [15]. Thisexpression is written as

qs ¼ Kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � P2header

q; ð87Þ

where qs is the steam mass flow rate, K is a constant, and x2 andPheader are the upstream and downstream pressures, respectively.The constant K is chosen to produce agreement between measuredflow and pressure drop at a reference condition. Because, for thereal system Pheader ¼ 6306ðkPaÞ, by measuring the steam flow anddrum pressure in the real system, the value of K is identified andthe steam flow in the system can be modeled as

Table 1Parameters of the different operating points

Low load Normal load High load

Steam flow rate 50.12 kg/s 68.94 kg/s 83.58 kg/s

Steady state values u10 ¼ 50:12 kg=s u10 ¼ 68:36 kg=s u10 ¼ 81:74 kg=su20 ¼ 2:62 kg=s u20 ¼ 3:67 kg=s u20 ¼ 4:48 kg=su30 ¼ 0 kg=s u30 ¼ 0:58 kg=s u30 ¼ 1:84 kg=sy10 ¼ 1 m y10 ¼ 1 m y10 ¼ 1 my20 ¼ 6523:6 kpa y20 ¼ 6711:7 kpa y20 ¼ 6894 kpay30 ¼ 483:19 �C y30 ¼ 500 �C y30 ¼ 500 �C

qs ¼ 0:03ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

: ð88Þ

6.2. Drum pressure dynamics

To model the pressure dynamics, first step identification is doneto observe the behavior of the system. By applying step inputs tothe three different inputs at different operating points, we observefor a step increase in the feedwater and fuel flow, the system be-haves like a first order system with the same time constant. Byapplying a step to spray flow input, the system behaves like a firstorder system with different time constant. It means, using twostate variables can describe the pressure dynamics fairly accu-rately. But, because of minimizing the interactions between sub-systems due to the matrix B in the model, we neglect the effectof the spray flow input on the drum pressure and based on the fun-damental physical laws given in [3], and the observation form thestep identification, the dynamics for the drum pressure is chosenas follows:

_x2 ¼ ðc1x2 þ c2Þqs þ c3u1 þ c4u2; ð89Þy2 ¼ x2: ð90Þ

To obtain the constants in the pressure equation given in (89), stepinputs are applied to water flow and fuel flow inputs separately atdifferent operating points. The information gathered from the initialstep responses gives an initial guess for the time constant and theconstants in Eq. (89). Then, a random input sequence (PRBS) with6 sec. sampling interval is applied to each of two input variables.The PRBS data provides a much richer input signal to obtain a moreconsistent model through identification. To obtain the parametervalues, we equate the transfer function coefficients from the sym-bolic linearization of Eq. (89) with the identified transfer function.At the low load

X2ðsÞU1ðsÞ

¼ c3

s� A1¼ �0:0404ðsþ 0:0144Þ ; ð91Þ

X2ðsÞU2ðsÞ

¼ c4

s� A1¼ 3:025ðsþ 0:0144Þ ; ð92Þ

at the normal load

X2ðsÞU1ðsÞ

¼ c3

s� A2¼ �0:0404ðsþ 0:0111Þ ; ð93Þ

X2ðsÞU2ðsÞ

¼ c4

s� A2¼ 3:025ðsþ 0:0111Þ ; ð94Þ

and at the high load we have

X2ðsÞU1ðsÞ

¼ c3

s� A3¼ �0:0404ðsþ 0:0096Þ ; ð95Þ

X2ðsÞU2ðsÞ

¼ c4

s� A3¼ 3:025ðsþ 0:0096Þ ; ð96Þ

with

Ai ¼ c1i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2i � ð6306Þ2q

þ ðc1ix2i þ c2iÞx2iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2i � ð6306Þ2q ; i ¼ 1;2;3; ð97Þ

where c1i, c2i and x2i i ¼ 1;2;3 are the constants and drum pressureat low load, normal load and high load, respectively. From Eqs.(91)–(97) the constants given in (89) can be calculated as

c1 ¼ �6:1687� 10�6;

c2 ¼ �0:08;c3 ¼ �0:0404;c4 ¼ 3:025:

224 B. Labibi et al. / Journal of Process Control 19 (2009) 216–230

Finally, the dynamics of the drum pressure can be modeled as

_x2 ¼ ð�1:8506� 10�7x2 � 0:0024Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

� 0:0404u1 þ 3:025u2; ð98Þ

y2ðtÞ ¼ x2ðtÞ þ p0; ð99Þ

where p0 ¼ 8:0715, p0 ¼ �0:6449 and p0 ¼ �6:8555 for low, normaland high load, respectively. At the three operating points, the initialconditions are x20 ¼ 6523:6, x20 ¼ 6711:5 and x20 ¼ 6887:9 for low,normal and high load, respectively.

6.3. Drum level dynamics

Identification of the water level dynamics is a difficult task.Applying step inputs to the inputs separately, show that the leveldynamics is unstable. By increasing the water flow rate, the levelincreases and by increasing the fuel flow, the level decreases. Threeinputs, water flow, fuel flow and steam flow affect on the drumwater level. Simulation results show that there should exist anintegrator in the model. Let x1, and VT denote the fluid densityand total volume of the system, then we have

_x1 ¼u1 � qs

VT; ð100Þ

where VT ¼ 155:1411. By doing several experiments, it was ob-served that the dynamics of the drum level can be given by

y1 ¼ c5x1 þ c6qs þ c7u1 þ c8u2 þ c9: ð101Þ

The constants ci; i ¼ 5; . . . ;9 should be identified from the plantdata. By applying PRBS signal with sampling time equal to 6 sec.and using the plant data, the constants given in (101) are calculatedas c5 ¼ 0:0101571, c6 ¼ 0:00612875, c7 ¼ 0:019814, c8 ¼ 0:001, andc9 ¼ �6:1982. The initial values of x1 at the three operating pointsare given by x10 ¼ 678:15, x10 ¼ 667:1, and x10 ¼ 654:628 for low,normal and high load, respectively.

6.4. Steam temperature

In the utility boiler, the steam temperature must be kept at acertain level to avoid overheating of the super-heaters. To iden-tify a model for steam temperature, first step identification isused. By applying a step to the water flow input, steam temper-ature increases and the steam temperature dynamics behaveslike a fist order system. Applying a step to the fuel flow input,the steam temperature increases and the system behaves like asecond order system. Applying a step to the spray flow input,steam temperature decreases and the system behaves like a firstorder system. Then, a third order system is selected for thesteam temperature model. This step identification gives an initialguess for local time constants and gains. By considering steamflow as input and applying input PRBS at the three operatingpoints, local linear models for the steam temperature dynamicsare defined. Combining the local linear models, the followingnonlinear model is identified for all three operating points witha good fitness.

_x3ðtÞ ¼ �0:0211ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

þ x4 � 0:0010967u1

þ 0:0475u2 þ 3:1846u3; ð102Þ

_x4ðtÞ ¼ 0:0015ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

þ x5 þ 0:001u1 þ 0:32u2

� 2:9461u3; ð103Þ

_x5ðtÞ ¼ �1:278� 10�3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

� 0:00025831x3

� 0:029747x4 � 0:8787621548x5 � 0:00082u1

� 0:2652778u2 þ 2:491u3; ð104Þ

y3 ¼ x3 þ T0; ð105Þ

where T0 ¼ 443:3579, T0 ¼ 446:4321, and T0 ¼ 441:9055 for lowload, normal load and high load, respectively. At three operatingpoints, we have x30 ¼ 42:2529, x40 ¼ 3:454, x50 ¼ �3:45082, forlow load, x30 ¼ 49:0917, x40 ¼ 2:9012, x50 ¼ �2:9862, for normalload, and x30 ¼ 43:3588, x40 ¼ �0:1347 and x50 ¼ �0:2509 for highload.

6.5. The model

Combining the achieved results so far, the identified model forthe utility boiler is given as follows:

_x1ðtÞ ¼u1 � 0:03

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

155:1411;

_x2ðtÞ ¼ ð�1:8506� 10�7x2 � 0:0024Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

� 0:0404u1 þ 3:025u2;

_x3ðtÞ ¼ �0:0211ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

þ x4 � 0:0010967u1

þ 0:0475u2 þ 3:1846u3;

_x4ðtÞ ¼ 0:0015ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

þ x5 þ 0:001u1 þ 0:32u2 � 2:9461u3;

_x5ðtÞ ¼ �1:278� 10�3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

� 0:00025831x3

� 0:029747x4 � 0:8787621548x5 � 0:00082u1

� 0:2652778u2 þ 2:491u3;

y1ðtÞ ¼ 0:010157116x1 þ 1:8386� 10�4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

� 0:001u1 þ 0:019814u2 � 6:1982;

y2ðtÞ ¼ x2;

y3ðtÞ ¼ x3;

qsðtÞ ¼ 0:03ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � ð6306Þ2q

:

The parameters of the different operating points for the modelare listed in the following table (See Table 2).

In addition, the following limit constraints exist for the threecontrol variables:

0 6 u1 6 120; ð106Þ0 6 u2 6 7; ð107Þ0 6 u3 6 10; ð108Þ� 0:017 6 _u2 6 0:017: ð109Þ

To compare the derived model and the real system, PRBS-like sig-nals with T = 6 sec. are used to stimulate the real system and theidentified model. The experiments were performed at low, normaland high load. Figs. 1–3 show responses in drum pressure, drum

Table 2Parameters of different operating points for the identified nonlinear model

Low load Normal load High load

Steam flow rate 49.18 kg/s 68.9256 kg/s 83.1251 kg/s

Steady state values u10 ¼ 50:12 kg=s u10 ¼ 68:9256 kg=s u10 ¼ 83:1251 kg=su20 ¼ 2:6618 kg=s u20 ¼ 3:6867 kg=s u20 ¼ 4:4761 kg=su30 ¼ 0 kg=s u30 ¼ 0:58 kg=s u30 ¼ 1:84 kg=sx10 ¼ 678:15 x10 ¼ 677:1 x10 ¼ 654:628x20 ¼ 6523:6 x20 ¼ 6711:5 x20 ¼ 6887:9x30 ¼ 42:2529 x30 ¼ 49:0917 x30 ¼ 43:3388x40 ¼ 3:454 x40 ¼ 2:9012 x40 ¼ �0:1347x50 ¼ �3:4082 x50 ¼ �2:9862 x50 ¼ �0:2509

0 10 20 30 40 50 60 70 80 90 100

6.528

6.53

6.532

Pr.

(M

Pa.

)

Low load

0 10 20 30 40 50 60 70 80 90 1000.99

1

1.01

Leve

l (m

.)

0 10 20 30 40 50 60 70 80 90 100486

486.5

487

Tem

p. (

C)

0 10 20 30 40 50 60 70 80 90 10050.5

51

Time (min.)

SF

(kg

/s)

Fig. 1. Outputs of the utility boiler (solid) and model (dashed) at low load.

0 10 20 30 40 50 60 70 80 90 1006.68

6.7

6.72

6.74

Pr.

(M

Pa.

)

Normal load

0 10 20 30 40 50 60 70 80 90 1000.98

1

1.02

Leve

l (m

.)

0 10 20 30 40 50 60 70 80 90 100498500502504506508

Tem

p. (

C)

0 10 20 30 40 50 60 70 80 90 10065

70

75

Time (min.)

SF

(kg

/s)

Fig. 2. Outputs of the utility boiler (solid) and model (dashed) at normal load.

B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 225

level, steam temperature, and steam flow for perturbations in theinputs of the system at low, normal, and high load, respectively.As it can be observed, there is a good agreement between the modeldata and the experiment data.

7. Controller design

The identified nonlinear model is linearized about its operatingpoints. So, there are three linearized model at low load, normal

0 10 20 30 40 50 60 70 80 90 1006.85

6.9

Pr.

(M

Pa.

)

High load

0 10 20 30 40 50 60 70 80 90 100

0.98

1

1.02Le

vel (

m.)

0 10 20 30 40 50 60 70 80 90 100495

500

505

Tem

p. (

C)

0 10 20 30 40 50 60 70 80 90 10080

85

Time (min.)

SF

(kg

/s)

Fig. 3. Outputs of the utility boiler (solid) and model (dashed) at high load.

226 B. Labibi et al. / Journal of Process Control 19 (2009) 216–230

load, and high load. The transfer matrix of the normal load modelwhich is in the middle of the uncertainty interval, is selected as thenominal plant to minimize the uncertainty profile. The state spacematrices of the normal load model are given as follows:

A ¼

0 �0:0002 0 0 0

0 �0:0111 0 0 0

0 �0:0616 0 1 0

0 0:0044 0 0 1

0 �0:0037 �0:0003 �0:0297 �0:8788

266666664

377777775;

B ¼

0:0064 0 0

�0:0404 3:0250 0

�0:0011 0:0475 3:1846

0:0010 0:32 �2:9461

�0:0008 �2:6528 2:4910

266666664

377777775; ð110Þ

C ¼0:0102 0:0005 0 0 0

0 1 0 0 00 0 1 0 0

264

375; D ¼

�0:001 0:0198 00 0 00 0 0

264

375:

ð111Þ

The uncertainty of the change of variables in the three different lin-ear models is modeled as output multiplicative uncertainty by thefollowing weighting matrix:

��W3ðsÞ ¼0 0 00 0:05

sþ0:02 00 0 0:2

264

375: ð112Þ

The first subsystem associated with the drum level is an unsta-ble system. By solving local LMI optimization problem (82) for thefirst subsystem, the nominal system is decomposed into the fol-lowing isolated subsystems:

A11 ¼ �0:0156; B11 ¼ 0:0064; C11 ¼ 0:0102; D11 ¼ 0:001;

ð113Þ

A22 ¼ �0:0288; B22 ¼ 3:025; C22 ¼ 1; D22 ¼ 0; ð114Þ

A33 ¼0 1 0

0 0 1

�0:0003 �0:0297 �0:8788

264

375;

B33 ¼3:1846

�2:9461

2:4910

264

375; C33 ¼ 1 0½ �; D33 ¼ 0: ð115Þ

According to this decomposition, based on the interaction be-tween subsystems, the following appropriately defined weightingmatrix is obtained to satisfy the equations given in (30)

�W1ðsÞ ¼

0:0198sþ0:000532sþ0:02 0 0

0 0:333sþ0:09425sþ0:031 0

0 0 0:0014sþ12:16sþ7

264

375: ð116Þ

The first two subsystems are first order systems with the followingtransfer functions

g1ðsÞ ¼�0:001ðs� 0:04968Þðsþ 0:0156Þ ; g2ðsÞ ¼

3:025ðsþ 0:0288Þ : ð117Þ

The third subsystem is a third order system which can be approxi-mated by

g3ðsÞ ¼�3:075ðsþ 0:25Þ ; ð118Þ

with the following multiplicative uncertainty weighting function

�w33ðsÞ ¼ 0:8: ð119Þ

The selected weighting function in (119) seems to be conservative.However, this selection simplifies the controller design procedure.Now, according to the proposed algorithm in the paper, designingthe local PI controllers is reduced to solving the local robust controlproblems given in (73) for the following uncertain isolatedsubsystems

g1ðsÞ ¼�0:001ðs� 0:04968Þðsþ 0:0156Þ ;

w11ðsÞ ¼0:0198sþ 0:000532

sþ 0:02; w31ðsÞ ¼ 0; ð120Þ

0 5 10 15 20 25 30480

500

520

Time (min.)

Tem

p. (

C)

0 5 10 15 20 25 306700

6800

6900P

re. (

KP

a.)

0 5 10 15 20 25 30

0.95

1

1.05

Leve

l (m

.)

Fig. 4. Switching from normal load to high load (output signals).

0 5 10 15 20 25 300.95

1

1.05

Leve

l (m

.)

0 5 10 15 20 25 306700

6800

6900

Pre

. (K

Pa.

)

0 5 10 15 20 25 30480

500

520

Time (min.)

Tem

p. (

C)

Fig. 5. Switching from high load to normal load (output signals).

0 2 4 6 8 10 12 14 16 18 200.9

1

1.1

Leve

l (m

.)

0 2 4 6 8 10 12 14 16 18 206500

6600

6700

Pre

. (K

Pa.

)

0 2 4 6 8 10 12 14 16 18 20

480

500

Time (min.)

Tem

p. (

C)

Fig. 6. Switching from normal load to low load (output signals).

B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 227

228 B. Labibi et al. / Journal of Process Control 19 (2009) 216–230

g2ðsÞ ¼3:025

ðsþ 0:0284Þ ; w12ðsÞ ¼0:333sþ 0:09425

sþ 0:031;

w32ðsÞ ¼0:005ðsþ 0:02Þ ; ð121Þ

g3ðsÞ ¼�3:075ðsþ 0:25Þ ; w13ðsÞ ¼

0:0014sþ 12:16sþ 7

; w33ðsÞ ¼ 0:8:

ð122Þ

By selecting sc1 ¼ 75, a1 ¼ 10, sc2 ¼ 33:0579, a1 ¼ 10, sc3 ¼ 21:7,and a3 ¼ 2, the local robust problems in (73) are solved. Usingthe obtained time constants for the isolated subsystems, the lo-cal PI controllers are designed based on the IMC method asfollows:

KðsÞ ¼212 1þ 1

62:2424s

� �0 0

0 0:01 1þ 135:23s

� �0

0 0 �0:015 1þ 14s

� �264

375:ð123Þ

0 5 10 10

50

100

FW

flow

(kg

/s)

0 5 10 12

4

6

FF

(kg

/s)

0 5 10 10

5

10

Time

SF

(kg

/s)

Fig. 7. Switching from normal load

0 5 10 160

80

FW

flow

(kg

/s)

0 5 10 1

3

4

5

FF

(kg

/s)

0 5 10 10

1

2

Time

SF

(kg

/s)

Fig. 8. Switching from high load to

The designed controller is applied to the real nonlinear system.In order to compensate the constraints given in (106)–(109) oncontrol signals, as explained in [20], these constraints can be ig-nored at the design stage, and the effects of the constraints arecompensated after the controller design, using anti-windupbump-less transfer (AWBT) techniques [20]. Then, for each isolatedsubsystem an AWBT compensator is designed. Applying the de-signed controller with the decentralized AWBT compensator tothe nonlinear system, Figs. 4–6 show the responses of the closed-loop system in switching from normal load to high load, from highload to normal load and from normal load to low load, respectively.These figures show good set point tracking of the closed-loop sys-tem. Figs. 7–9 show the related control signals and that the con-straints given on control signals are satisfied. Fig. 10 comparesthe step responses for the designed decentralized PI controllerand the existing multiloop controller in switching from normalload to high load. It can be observed that the closed-loop systemunder the multiloop controller has a large overshoot for drum pres-sure and steam temperature change. The controller outputs forboth controllers are compared in Fig. 11 which shows the controlsignals for the new controller are less aggressive.

5 20 25 30

5 20 25 30

5 20 25 30 (min.)

to high load (control signals).

5 20 25 30

5 20 25 30

5 20 25 30 (min.)

normal load (control signals).

0 2 4 6 8 10 12 14 16 18 200

50

100

FW

flow

(kg

/s)

0 2 4 6 8 10 12 14 16 18 200

2

4

FF

(kg

/s)

0 2 4 6 8 10 12 14 16 18 200

1

2

Time (min)

SF

(kg

/s)

Fig. 9. Switching from normal load to low load (control signals).

0 2 4 6 8 10 12 14 16 18 20

1

1.1

Leve

l (m

.)

0 2 4 6 8 10 12 14 16 18 20460480500520540

Time (min)

Tem

p. (

C)

0 2 4 6 8 10 12 14 16 18 206700

6800

6900

Pre

. (K

Pa.

)

Fig. 10. Switching from normal load to high load (output signals) (solid: the new PI controller. Dashed: the original multiloop controller).

0 2 4 6 8 10 12 14 16 18 2040

60

80

100

FW

flow

(kg

/s)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

Time (min)

SF

(kg

/s)

0 2 4 6 8 10 12 14 16 18 20

4

5

6

FF

(kg

/s)

Fig. 11. Switching from normal load to high load (control signals) (solid: the new PI controller. Dashed: the original multiloop controller).

B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 229

230 B. Labibi et al. / Journal of Process Control 19 (2009) 216–230

8. Conclusion

In this paper, a method for robust decentralized PI controllerdesign for an industrial utility boiler is proposed. Sufficient condi-tions for robust stability and diagonal dominance of the overallclosed-loop system are derived. These conditions are based onthe sensitivity functions of the closed-loop isolated subsystemsand are formulated as local robust performance problems. It isshown, by appropriately selecting the time constants of theclosed-loop isolated subsystems, these sufficient conditions aresatisfied. The method is applicable to systems with any order, sta-ble/unstable and minimum-/non-minimum-phase. Then, a fairlyaccurate nonlinear model for the boiler system in the SyncrudeCanada Ltd. is identified. For the identified model, a decentralizedPI controller is designed. Applying the designed controller to thereal industrial utility system shows the effectiveness of the pro-posed method.

Acknowledgement

This research is supported by the Natural Sciences and Engi-neering Research Council of Canada, and Syncrude Canada Ltd.

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