Decentralized robust PI controller design for an industrial boiler

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 b

Advanced Process Automation and Control (APAC) Research Group, K.N. Toosi University of Technology, Tehran, Iran 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 2007 Received in revised form 23 April 2008 Accepted 23 April 2008

Keywords: Industrial utility boiler Internal model control Nonlinear system modeling Robust decentralized control

a b s t r a c t This paper presents a new scheme to design decentralized robust PI controllers for uncertain LTI multivariable 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 performance of the overall system are modeled as local robust performance problems. Then, by appropriately selecting the time constants of the closed-loop isolated subsystems in the IMC (Internal Model Control) strategy, the defined local robust performance problems are solved. To design a decentralized robust PI controller for a real industrial utility boiler, a control oriented nonlinear model for the boiler is identified. The nonlinearity of the system is modeled as uncertainty for a nominal LTI multivariable system. Using the new proposed method, a decentralized PI controller for the uncertain LTI model is designed. The designed controller is applied to the real system. The simulation results show the effectiveness of the proposed methodology. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Control of the interacting multivariable processes can be realized either by centralized MIMO controllers or by a set of SISO local controllers. For large-scale industrial processes, decentralized control is preferred from the viewpoints of implementation, requiring fewer parameters to tune and loop failure tolerance of the resulting control system [9]. A common design procedure in decentralized control is the sequential design strategy. Sequential design strategy involves closing and tuning one loop at a time [6]. However, the order of designing local controllers in the individual loops may affect the control quality. Besides, the closing of subsequent loops may change the response of previously designed loops which leads to the necessity of iteration [10]. The usual method for dealing with the 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 elements has 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 the second problem, in [10], a strategy for sequential design of decentralized controllers for linear systems is presented. This method involves a simple estimate of the impact of closing previous loops into 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. * Corresponding author. E-mail addresses: [email protected], [email protected] (B. Labibi). 0959-1524/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2008.04.013

In control theory, the internal model control (IMC) method, first proposed by Morari and Zafiriou [13], is a powerful and simple strategy for designing controllers. The idea is to specify the desired closed-loop response and solve for the resulting controller [13,17]. The decentralized type of the IMC strategy is also used in several references such as [9,10]. In [10], a methodology for sequential design of robust decentralized controllers is given. Each controller is parameterized to have only one IMC tuning parameter. Thereby, initial values to tune parameters are selected and the parameters are updated by an iterative algorithm, such that the robust performance is achieved. The design algorithm terminates if the closed-loop system is stabilized. In fact, in [10], a new property called robust decentralized detunability (RDD) is introduced. If a system has this property, any subset of the loops can independently be detuned to an arbitrary degree without influencing robust stability. In the proposed method, the class of possible decentralized IMC controllers are parameterized in terms of the IMC filter time constants. Then, the unknown time constants are considered as uncertainty and the possible bounds on time constants are obtained. Using l controller synthesis algorithms, the time constants can be selected to ensure robust stability or robust performance. The design procedure does not give the optimal filter time constants, but it provides a range of values to guarantee robust stability/performance, and robust decentralized detunability. The proposed design procedure can also be extended to include other types of controllers such as PID controllers [10]. However, as it is stated by the authors in [10], the suggested algorithm is difficult to apply for problems to achieve different bandwidths in the various loops.

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As a practical application of the decentralized IMC strategy, using the proposed method in [13], Ref. [7] has considered decentralized temperature control of a packed-bed chemical reactor. In fact, [7] models the off-diagonal elements of the MIMO plant as input multiplicative uncertainties which are used to design l interaction measures. Then, employing the IMC tuning rules as a baseline, a decentralized PI controller is designed for the diagonal part of the MIMO model by tuning a simple factor iteratively. Designing decentralized robust PID controllers for power systems is very important. These controllers have proven to be robust and extremely beneficial in the control of many important applications. Extensive research has been done to tune PID controllers [1,2,8,11]. However, this problem still remains an active field of research in the control literature. In this area of research, [4] has suggested a decentralized robust control approach, by considering the diagonal part of the model as the design model, and the off-diagonal dynamics of the interconnected plant as uncertainty. Then, the maximum singular values or structured singular values of the multiplicative uncertainty are used as interaction measures. In fact, the method in [4] proposes an iterative two-step sub-optimal design procedure to design a decentralized robust controller. This work follows the basic framework of [13]. But, the interaction between subsystems is measured by a revised passivity index. Due to use of both the phase and gain information of the interactions, the offered algorithm may lead to a less conservative stability condition. To design a decentralized PID controller, the derived controller which is normally a high order one, can be simplified to a lower order system such as a PID controller. Compared to the previous methods, the proposed method may be less conservative. However, the offered algorithm in [4] is an iterative approach and requires complicated computation where the dimension of the system is high. To design robust PI controllers to regulate frequency and track load in power systems, [5] formulates the load frequency control (LFC) problem as a multi-objective control problem via a mixed H2 =H1 control technique. To design a robust PI controller, the control problem is reduced to a static output feedback control synthesis, and then, it is solved using an iterative linear matrix inequalities (ILMI) algorithm. However, the proposed method is an iterative control design strategy. Generally, iterative algorithms suffer from two major problems, convergence of the algorithm and finding good initial values for designing parameters. So, presenting a non-iterative decentralized control strategy for industrial processes is of great importance. The Control of industrial co-generation systems represents a significant challenge for control engineers. The crucial part in a power plant is the boiler system. Due to the shrink and swell dynamics which causes a non-minimum phase behavior, the level drum control problem is very difficult [3]. In [21], robust controller design for a linear two-input two-output model of a boiler/turbine unit which relates firing rate and turbine valve position inputs to throttle pressure and megawatt outputs is given. The plant/model mismatch is represented as output-multiplicative uncertainty. Then, the closed-loop performance and robustness of MIMO H1 and l-synthesis control laws with those of MIMO control H2 law are compared. The comparative evaluation of applying the three designs to the model, shows that in power plant control problems H1 and l-synthesis provide much better performance/robustness than H2 design. Following the previously mentioned methods, in this paper, a robust decentralize PI controller for the utility boiler systems in the Syncrude Canada Ltd. (SCL) is designed and applied to the real system. 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 demand while maintaining the steam pressure and the steam temperature of the header at their respective set points, despite variations of the

steam load. Due to the physical characteristics, utility boilers are used to regulate the steam pressure [19]. To design a robust decentralized PI controller for the utility boilers in SCL, first, a non-iterative simple algorithm based on the IMC strategy is proposed. Sufficient conditions for closed-loop stability and diagonal dominance under a decentralized control are achieved. Based on the obtained sufficient conditions, a new interaction measure is defined, and the system is decomposed into the isolated subsystems, such that to minimize the obtained interaction measure. If the isolated subsystems are of high order, first order models are obtained. The error of approximation can be considered as multiplicative uncertainty for each isolated subsystem. It will be shown that achieving closed-loop diagonal dominance and robust stability can be guaranteed by solving certain local robust performance problems to be defined. In the IMC strategy, by appropriately selecting the time constants of the closed-loop isolated subsystems, which are the IMC tuning parameters, these local problems can be solved. The proposed approach has two major advantages: (1) the method is applicable to unstable systems as well. (2) The true model uncertainties can be dealt within the same framework so that the decentralized controllers obtained can be robust. In order to control the drum boilers in the SCL, first identifying an accurate model for the system is necessary. Using the fundamental physical laws, the known physical constants of the system, and the measured plant data, a fairly accurate nonlinear model for the system is identified. The derived model captures all the essential 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 points and the nonlinearity is modeled as uncertainty for a nominal LTI system. Thereafter, based on the new proposed algorithm in this paper, 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 a simulation package developed by the Syncrude Canada with the purpose of simulating certain upset conditions and as a general tool for stability analysis. The rest of this paper is organized as follows: In Section 2, the problem of finding suitable local dynamical controllers for subsystems of an LTI multivariable system is presented. In Section 3, by considering the interactions between the isolated subsystems as uncertainty, sufficient conditions for closed-loop stability, diagonal dominance and disturbance rejection are given. These conditions are stated in the sensitivity functions of the closed-loop isolated subsystems. In Section 4, the new method for decentralized PI controller design is given. In Section 5, by considering the true model uncertainty within the same framework of the interaction uncertainty, a method for robust decentralized PI controller design for an uncertain multivariable system is given. In Section 6, a nonlinear model of the drum boiler system is identified. Thereafter, in Section 7 the nonlinear model is linearized about its operating points and the nonlinearity is modeled as uncertainty. Then, by solving the appropriately defined local robust problems, a decentralized controller is designed. The designed controller is applied to the real system and the simulation results are given. Finally, concluding results are given in Section 8. 2. Problem formulation e Consider an uncertain LTI multivariable system GðsÞ with output multiplicative uncertainty as follows:

 ðsÞÞGðsÞ; e GðsÞ ¼ ðI þ DðsÞW 3

jDðjxÞj 6 1 8x;

ð1Þ

where GðsÞ is the nominal plant, DðsÞ is any stable transfer matrix which at each frequency is less than or equal to one in norm, and  ðsÞ is the diagonal weighting matrix which contains the freW 3

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

_ xðtÞ ¼ AxðtÞ þ BuðtÞ;

ð2Þ

yðtÞ ¼ CxðtÞ þ DuðtÞ;

where x 2 Rn , u 2 Rm , y 2 Rm , A 2 Rnn , B 2 Rnm , C 2 Rmn , and D 2 Rmm . Let

2

3 x1 6 . 7 7 x¼6 4 .. 5; xN

2

3 u1 6 . 7 7 u¼6 4 .. 5; uN

2

3 y1 6 . 7 7 y¼6 4 .. 5; yN

2

A11 6 . 6 A ¼ 4 ..

... ...

3 A1N 7 ... 7 5; ð3Þ

2

3

2

B11 . . . B1N C 11 6 . 6 . . 7 6 7 6 . . B¼4 . ... . 5; C ¼ 4 .. BN1 . . . BNN C N1 2 3 D11 . . . D1N 6 . .. 7 7 D¼6 ... . 5; 4 .. DN1 . . . DNN

3

C 1N 7 ... 7 ... 5; . . . C NN ...

and

ð4Þ

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

x_i ðtÞ ¼ Aii xi þ Bii ui þ

N X

Aij xj þ

j¼1j–i

j¼1j–i

N X

N X

yi ðtÞ ¼ C ii xi þ Dii ui þ

C ij xj þ

j¼1j–i ni

N X

mi

Bij uj ; ð5Þ Dij uj ;

j¼1j–i

with xi 2 R , ui 2 R , yi 2 R , Aii 2 Rni ni , Bii 2 Rni mi , C ii 2 Rmi ni , PN PN PN PN i¼1 ni ¼ n, and i¼1 mi ¼ m. The terms j¼1j–i Aij xj , j¼1j–i Bij uj , PN PN j¼1j–i C ij xj , and j¼1j–i Dij uj are due to interactions of the other sub-

mi

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

  1 þ T Ii s ; K i ðsÞ ¼ K ci T Ii s

i ¼ 1; . . . ; N

yi ðtÞ ¼ C ii xi ðtÞ þ Dii ui ðtÞ;

where

6 0m1 m1 6 6 6 0n1 m1 6 6 6 6 0m1 m1 6 6  .. E¼6 . 6 6 6 6 0mN m1 6 6 60 6 nN m1 4 0mN m1

0m1 n1

0m1 m1

In1 n1

0n1 m1

0m1 n1 .. .

0m1 m1 .. .

0mN n1

0mN m1

0nN n1

0nN m1

0mN n1

0mN m1

3

..

.

0m1 mN

0m1 nN

..

.

0n1 mN

0n1 nN

..

.

0m1 mN .. .

0m1 nN .. .

... .. . 0mN mN .. . 0nN mN .. . 0mN mN

0mN nN InN nN 0mN mN

0m1 mN 7 7 7 0n1 mN 7 7 7 7 0m1 mN 7 7 7 .. 7 . 7 7 7 0mN mN 7 7 7 0nN mN 7 7 5 0mN mN ð10Þ

2 C 11 D11 6 Im1 m1 6 6 6 0n m A11 B11 6 1 1 6 6 6 0m1 m1 0m1 n1 Im1 m1 6 6  .. .. .. A¼6 6 . . . 6 6 60 C N1 DN1 6 mN m1 6 6 6 0n m AN1 BN1 6 N 1 4 0mN m1 0mN n1 0mN m1

3

..

.

0m1 mN

C 1N

D1N

..

.

0n1 mN

A1N

B1N

..

.

0m1 mN

0m1 nN

0m1 mN

.. .

... .. . ImN mN

.. .

.. .

C NN

DNN

..

.

0nN mN

ANN

BNN

..

.

0mN mN

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

0mN nN ImN mN ð11Þ

 xT ¼ yT1 ð7Þ

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

KðsÞ ¼ diagfK i ðsÞg;

ð9Þ

ð6Þ

for each isolated subsystem Gii ðsÞ; described by

x_ i ðtÞ ¼ Aii xi ðtÞ þ Bii ui ðtÞ;

 xðtÞ þ BuðtÞ; x_ ðtÞ ¼ A  E  xðtÞ þ DuðtÞ;  yðtÞ ¼ C 2

. . . ANN

AN1

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

ð8Þ

stabilizes the overall uncertain system given in (1), if some sufficient conditions are satisfied. 3. Closed-loop stability and diagonal dominance In this section, sufficient conditions for closed-loop stability and diagonal dominance of the nominal system are obtained. To this end and in order to prove our theorems, the transformation proposed in [12] is used to transform the system given in (2) into an equivalent descriptor system representation. It should be noted that this representation is only for proving the related theorems and the control design will be done for conventional isolated subsystems. Since designing a dynamic controller for a system can be converted into designing a static controller for an augmented system, without loss of generality in this section we assume the designed controller is a static one.

xT1

2

3 u1 6 . 7 7 u¼6 4 .. 5; uN 2 6 0m1 m1 6 6 6 0n1 m1 6 6 6 6 Im1 m1 6 ¼6 .. B 6 . 6 6 6 6 0mN m1 6 6 60 6 nN m1 4 0mN m1

uT1

. . . yTN

xTN



2

3 y1 6 . 7 7 y¼6 4 .. 5 yN ..

ð12Þ

3

0m1 mN 7 7 .. 7 . 0n1 mN 7 7 7 .. 7 . 0m1 mN 7 7 7 .. 7 ... . 7 7 .. 7 . 0mN mN 7 7 7 .. . 0nN mN 7 7 5 .. . ImN mN .

2 6 6 ¼6 C 6 4

uTn

Im1 m1 .. .

0m1 n1 .. .

0m1 m1 .. .

0mN m1

0mN n1

0mN m1

ð13Þ

..

3

0m1 mN .. ... . .. . ImN mN .

0m1 nN .. .

0m1 mN .. .

0mN nN

0mN mN

7 7 7 7 5 ð14Þ

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

and

with

 ¼ 0mm : D

ð15Þ

The transfer matrix of the closed-loop descriptor system in (9),  cl ðsÞ is given by G

 þ BK   BK þ D  cl ðsÞ ¼ CðsE   CÞ A G 1 

¼ ðI þ CðsI  AÞ1 BK þ DKÞ1 ðCðsI  AÞ1 B þ DÞK;

ð16Þ

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

 ii g; Ad ¼ diagfA

ð17Þ

with

2

Imi mi

 ii ¼ 6 A 4 0ni mi 0mi mi

C ii

Dii

Aii

Bii

0mi ni

Imi mi

i

i

i

ð18Þ

i

0m m 0m n 2 i i 3 i i 0mi mi 7  ii ¼ 6 B 4 0ni mi 5 Imi mi

i

i

  ii ¼ Im m C i i

0mi ni

ð25Þ ð26Þ

0mi mi ð27Þ

and

0mi mi



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

 kPi k1 < l1 ðHÞ;

3 7 5;

 ii þ B  i K i C ii Þ1 Pi ¼ ðsEi  A 2 3 0mi mi 0mi ni 0mi mi 6 7 i ¼ 4 0n m In n 0n m 5 E

i ¼ 1; . . . ; N;

ð28Þ

where lðÞ is the maximum structured singular value of () [17], give sufficient conditions for closed-loop stability at the subsystem level. 3.2. Sufficient conditions for diagonal dominance

it is easy to show that the transfer matrices of the closed-loop sys d ; B;  DÞ  C;  and ðAd ; Bd ; C d ; Dd Þ under the decentralized contems ðE; A troller K ¼ diagfK i g are the same i.e.

P  BK  ¼ ðI þ C d ðsI  Ad Þ1 Bd K þ Dd KÞ1 ðC d ðsI  Ad Þ1 Bd þ Dd ÞK; T d ðsÞ ¼ C

ð19Þ where

 d þ BK  1 :  ¼ ðsE  A  CÞ P

ð20Þ

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

kSi k1 <

ai  1i j jw

i ¼ 1; . . . ; N;

ð29Þ

where ai is a positive scalar less than one and small enough, called  1i ðsÞ, the weighting function satlocal diagonal dominance degree, w isfies the following equation

 1i ðsÞj > jw

Defining

;

pffiffiffiffi NjðC ii ðsI  Aii Þ1 HABi þ HCDi j;

ð30Þ

ð21Þ

Si is the sensitivity function of the i-th closed-loop isolated subsystem

 d ; B;  is a block-diagonal system and the matrix H  CÞ  the system ðE; A  can be considered as uncertainty in the matrix A.

HABi ¼ ½ Ai1 Bi1 Ai2 Bi2 . . . Aii1 Bii1 0 0 Aiiþ1 Biiþ1 . . . AiN BiN 

3.1. Sufficient conditions for closed-loop stability

and

A d;  ¼A H

The next theorem provides sufficient conditions for closed-loop stability. Theorem 1. Suppose the decentralized controller K stabilizes the diagonal system ðAd ; Bd ; C d ; Dd Þ. Then the closed-loop original system under the decentralized controller is stable, if

 kP Hk 1 < 1;

ð22Þ

 and H  are given in Eqs. (20) and (21), respectively and where P k  k1 is the maximum singular value of () [17]. Proof. The transfer matrix of the closed-loop system can be written as

ð23Þ

 is stabilized by stabilizing the block-diagonal system Since P ðAd ; Bd ; C d ; Dd Þ, then the closed-loop system is stable if the transfer  is stable and if  HÞ  1 is stable. The transfer matrix P matrix ðI  P  HÞ  does not encircle the origin or equivthe Nyquist plot of detðI  P alently if the condition given in (22) is satisfied the closed-loop system is stable and the proof is complete. h

ð32Þ

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

P  PBKR:  HÞ  X ¼C  ðC  C

ð33Þ

If

P   Hk  kC 1 < armin ðCÞ;

ð34Þ

where rmin ðÞ is the minimum singular value of (), and a is a positive scalar less than one, then [18]

ð35Þ

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

eP  BK  ¼ ðI þ C d ðsI  Ad Þ1 Bd K þ Dd KÞ1 TðsÞ ffi C  ðC d ðsI  Ad Þ1 Bd þ Dd ÞK ¼ T d ðsÞ;

ð36Þ

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

 ¼ 1; rmin ðCÞ

ð37Þ

P  Hk  which means that by minimizing kC 1 such that

The matrix

 ¼ diagfP i g P

HCDi ¼ ½ C i1 Di1 C i2 Di2 . . . C ii1 Dii1 0 0 C iiþ1 Diiþ1 . . . C iN DiN :

C P  H  ffiC C

 d þ BK  P    HÞ  ¼ Cð  1  HÞ   C  1 BK  1 BK TðsÞ ¼ CðsE A  P   HÞ  1 P BK: ¼ CðI

ð31Þ

ð24Þ

P  Hk  kC 1 < a;

ð38Þ

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

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

 kC PHk 1 6

pffiffiffiffi  ii Pi H  ik ; N kC 1

i ¼ 1; . . . ; N;

then, we have

ð39Þ

kQ C Q 1 D k1 6

with

2

0 C i1 i ¼ 6 H 4 0 Ai1 0 0

Di1

0 C i2

Di2

. . . 0 C ii1

Dii1

0 0 0 0 C iiþ1

Diiþ1

. . . 0 C iN

Bi1

0

Ai2

Bi2

... 0

Aii1

Bii1

0 0 0 0

Aiiþ1

Biiþ1

... 0

AiN

0

0

0

0

... 0

0

0

0 0 0 0

0

0

... 0

0

Denoting

DiN

ð53Þ

3

7 BiN 5: 0

ð40Þ

It is clear if 1

1

Si ¼ ðI þ C ii ðsI  Aii Þ Bi K i þ Dii K i Þ ;

ð41Þ

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

 ii Pi H  i ¼ Si ðC ii ðsI  Aii Þ1 HAB þ HCD Þ: C i i

ð42Þ

 1i as given in Eq. (30), by satisfying Eq. Therefore, by defining w (29), the closed-loop system is diagonal dominant with the degree a ¼ maxfai g and the proof is complete. h It follows that by designing local controllers such that

 1i Si k1 < ai ; kw

i ¼ 1; . . . ; N;

ð43Þ

with ai small enough, the closed-loop system is block-diagonal dominant. In fact, the transfer matrix of the closed-loop system is given as

P P  BK  þC  BK:   HðI  P  HÞ  1 P TðsÞ ¼ C

ð44Þ

P   Hk  In minimizing kC 1 < a, since kCk1 ¼ 1, we have

ð45Þ

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

Q ¼ QD þ QC;

ð46Þ

with

Q N2

6 1;

ð54Þ

or equivalently

a 6 0:5;

ð55Þ

the transfer matrix of overall closed-loop system is a quasi-blockdiagonal matrix in sense of the concept introduced in [14]. Then, the stability and performance of the system can be inferred directly from the stability and performance of the block- diagonal transfer P  BK  [16]. matrix T d ðsÞ ¼ C 3.3. Disturbance rejection In order to have good disturbance rejection characteristics for the closed-loop system, an appropriately defined weighting matrix can be chosen such that

ð56Þ

where S is the sensitivity matrix of the overall closed-loop system. Since the closed-loop system is diagonal dominant, if the diagonal  ðsÞ is chosen as a block-diagonal dominance degree is small and W 1 matrix

 ðsÞ ¼ diagfw  1i ðsÞ; W 1

ð57Þ

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

Q D ¼ diagfQ 1 ; Q 2 ; . . . ; Q N g 2 3 Q 11 Q 12 . . . Q 1N 6Q 7 6 21 Q 22 . . . Q 2N 7 QC ¼ 6 7; 4 ... ... ... ... 5 Q N1

a 1a

 ðsÞSk 6 1; kW 1 1

 rmin ðPHÞ P  ¼ rmin ðP  HÞ  6 kC  Hk  kCk 1 1 < a:

 ðsÞS k 6 1; kW 1 d 1

ð58Þ

where

. . . Q NN

Q ii may be zero or nonzero matrix, is said to be quasi-block-diagonal dominance (QBDD) if there exists a matrix norm k  k such that

kQ C Q 1 D k < 1:

ð47Þ

Sd ¼ diagfSi g

P  BK  QD ¼ C

ð48Þ

and

ð59Þ

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

 1i ðsÞSi k 6 1; kw 1

By considering

i ¼ 1; . . . ; N;

ð60Þ

the condition given in (56) is satisfied. 4. Decentralized PI design

P  BK;   HðI  P  HÞ  1 P QC ¼ C

ð49Þ

then

    1      1 kQ C Q 1 D k1 ¼ kC P HðI  P HÞ P BKðC P BKÞ k1

ð50Þ

In this section, a new method for decentralized PI controller design is given. Before doing so, however, we revisit the SISO PI design problem. 4.1. SISO PI Design using Internal Model Control Method

and

kQ C Q 1 D k1

P  Hk  kðI  P  HÞ  k kC k kC P  BKð  C  BKÞ  6 kC P k1 ; 1 1 1 1

þ

1

ð51Þ  þ is the pseudo-inverse of C.  Since where C

kC þ k1 ¼

 P Hk  kC a 1  6 1  a: 1  rmin ðPHÞ

1  rmin ðCÞ

 ¼ kCk 1 ¼ 1;

ð52Þ

In designing a PI controller for a SISO system, approximation of high order processes by first order models is a common practice [11]. Once an approximated model is obtained, a PI controller based on IMC method can be designed. Even though many industrial processes meet the assumptions sufficiently to be modeled with a first order model, there do exist many plants which cannot

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be well approximated by first order systems. To avoid instability due to approximation, in this paper the error of approximation is considered as multiplicative uncertainty for isolated subsystems. Then, designing a local PI can be converted into solving an appropriately defined local robust performance problem. The strategy is described in the following subsection.

which results in

ðai s þ bi Þki ðai ki þ 1Þs þ bi ki s Si ðsÞ ¼ ðai ki þ 1Þs þ bi ki

T i ðsÞ ¼

ð67Þ ð68Þ

and the constant ki can be selected such that 4.2. Decentralized PI controller design

kj In this subsection, a new tuning criterion for MIMO PI controllers is proposed. The next theorem gives a methodology for strictly proper subsystems. Proper subsystems will be considered later. Theorem 3. Consider the i-th isolated subsystem which is approximated with a first order model and the approximation error is  3i ðsÞ. Then, the considered as multiplicative uncertainty weight w closed-loop MIMO system can be stabilized if sci for the i-th isolated closed-loop subsystem is selected such that

kj

1

ai

 1i ðsÞ w

sc i s 1  3i ðsÞ j þ jw jk < 1 8x; i ¼ 1; . . . ; N: sc i s þ 1 sc i s þ 1 1 ð61Þ

Proof. By designing a local PI controller for the i-th isolated subsystem by an IMC based method, the i-th closed-loop subsystem has the following sensitivity and complementary sensitivity functions, respectively [17]

Si ¼

sc i s sci s þ 1

ð62Þ

and

1 : Ti ¼ sc i s þ 1

8x i ¼ 1; . . . ; N:

 3i ðsÞT i ðsÞk1 < 1;  1i ðsÞSi ðsÞj þ jw w

ð63Þ

ð64Þ

The i-th sensitivity and complementary sensitivity functions should satisfy the conditions given in (29) and (64), respectively, to have closed-loop diagonal dominance (nominal performance) for the overall system and robust stability for the subsystems. This is a robust performance problem for the isolated subsystems. By considering definitions given in (62) and (63), the closed-loop system is diagonal dominant if the local robust performance problems given in (61) are solved and the proof is complete. h

ð69Þ

5. Robust decentralized PI control Consider the uncertain system given in (1). The uncertainty is modeled as diagonal multiplicative output uncertainty (In this section, without loss of generality we consider output multiplicative uncertainty. It is clear the same result can be obtained for input multiplicative uncertainty as well.). The closed-loop uncertain system is robust stable if and only if for multiplicative output uncertainty [17]

 Tk < 1; kW 3 1

ð70Þ

 ðsÞ ¼ diagfw  3i ðsÞg is a diagonal matrix representing mulwhere W 3 tiplicative uncertainty of the system. If the closed-loop system is diagonal dominant with a good degree of diagonal dominance, then with a good approximation TðsÞ ffi T d ðsÞ and the performance of the closed-loop system can be inferred from the block-diagonal part of the transfer matrix. We can show by solving the robust stability problem

 T k < 1; kW 3 d 1

To attenuate the interactions between subsystems by local controllers, the sensitivity functions of each isolated subsystem should satisfy condition (29). This condition can be considered as a nominal performance problem. If the isolated subsystem can not be approximated sufficiently well by a low order model, the modeling error may be considered as multiplicative uncertainty given by weighting  3i ðsÞ. Then, the i-th approximated closed-loop isolated function w subsystem is stable if and only if [17]

 3i ðsÞk1 < 1: kT i ðsÞw

1

ai

ð71Þ

or equivalently by solving local robust stability problems

 3i T i k < 1; kw 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 uncertainty to solve the robust stability problem given in Eq. (70). In addition, the closed-loop system should have the desirable disturbance rejection characteristics. It was shown that for closed-loop diagonal dominance, the local robust performance problems given in (61), for treating uncertainty in the system, the conditions given in (72), and for good disturbance rejection, the conditions given in (60) should be satisfied. Combining these conditions for designing a robust decentralized PI controller for an uncertain LTI multivariable, the problem can be converted into solving the following modified local robust performance problems

  w1i ðsÞ 

   sci s   w3i ðsÞ  þ < 1 8x; i ¼ 1; . . . ; N;  sci s þ 1 sci s þ 11

ð73Þ

with According to the theorem, by selecting appropriate values for sci ’s and ai < 0:5 to solve local problems (61), the closed-loop stability and diagonal dominance are guaranteed. For proper subsystems the proposed method should be modified as we now explain. Suppose that the reduced order model of the i-th subsystem is given by a first order system

ai s þ bi Gi ¼ ; s þ ci

ð65Þ

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

K i ðsÞ ¼ ki

s þ ci ; s

ð66Þ

 

w  1i ðsÞ  ; jw  1i ðsÞj 6 jw1i ðsÞj; max  a 

8x ;

i ¼ 1; . . . ; N

ð74Þ

i

and

 3i ðsÞjg 6 jw3i ðsÞj;  3i ðsÞj; jw maxfjw

8x;

i ¼ 1; . . . ; N:

ð75Þ

So far, we have shown designing a robust decentralized PI controller for an uncertain MIMO system can be reduced to solving the local robust performance problems given in (73)–(75). Remark 1. Decomposing the matrices A, B, C and D into blockdiagonal and non-block-diagonal matrices is arbitrary. It may happen that the open-loop isolated subsystems have some purely imaginary poles. Then, at these frequencies the norm

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ci ¼ kC ii ðsI  Aii Þ1 HABi þ HCDi k1 ;

ð76Þ

approaches infinity and conditions (30) can not be satisfied for stable weighting functions. To avoid this problem, we can decompose the matrices A, B, C and D, such that the isolated subsystems are controllable, observable and stable and the interactions between subsystems are minimized. This approach introduces some flexibility into the design procedure and makes the proposed method applicable to unstable systems as well. In addition, as it is known in the IMC method, the poles of the isolated subsystems are zeros of the local PI controllers. If the pole of an isolated subsystem is far from the imaginary axis, then, the integration time of the designed local PI controller decreases and closed-loop response is faster but, is also more oscillatory. If the pole is close to the imaginary axis, then, the integration time increases and the closed-loop response creeps slowly toward the set point [2]. Then, by decomposing the matrix A appropriately, it is possible to control the shape of the closed-loop response as well. To do an appropriate decomposition, 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 will be decreased. In fact, this norm can be considered as an interaction measure for the overall closed-loop system. Suppose the matrices A, b ii , B b ii and D b ii , C b ii are the state B, C and D are decomposed such that A matrices of the i-th isolated subsystem, and

h i b AB ¼ A B . . . A b b H i i1 i1 ii1 Bii1 Aii  A ii Bii  B ii . . . AiN BiN ð77Þ and

h b CD ¼ C H i1 i

Di1

. . . C ii1

Dii1

b ii C ii  C

b ii Dii  D

C iN

i DiN : ð78Þ

By virtue of the bounded real lemma, minimizing the norm b CD k is equivalent to finding the local b AB þ H ci ¼ k Cb ii ðsI  Ab ii Þ1 H i i 1 Lyapunov matrices Pi > 0 such that [17]

:

ð79Þ

b CD : Yi ¼ H i

ð84Þ

Now, the LMI optimization problem given in (82) is affine in LMI variables Pi , Ri , and Si . After solving the local LMI optimization probb ii and b ii and D b ii are obtained and the matrices A lems, the matrices C b ii can be calculated as B

b ii ¼ P1 Ri A i

ð85Þ

and

b ii ¼ P1 Si : B i

ð86Þ

b ii , new LMI constraints can be To avoid fast modes in the matrix A b ii added to optimization problem (82), to limit the eigenvalues of A or assign them in the desired region [17]. Remark 2. Based on the discussion given in Theorem 2, the closedloop system is diagonal dominant, if

1 jSi j < pffiffiffiffi ; NjC ii ðsI  Aii Þ1 HABi þ HCDi j

8x;

i ¼ 1; . . . ; N:

In the above relation, at high frequencies, the left and right hand sides of the relation approach to one and pffiffiNffijH1 j, respectively. Then, CDi

to satisfy these conditions at high frequencies, kHCDi k1 should be less than or equal to p1ffiffiNffi. 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 and the overall system is observable, it is possible by using the observability matrix of the overall system to find an appropriate transformation to transform the original system into the outputdecentralized form, where the matrix C is block diagonal [12]. Then, C ij ¼ 0; i–j, and for strictly proper systems (D ¼ 0), at high frequencies these conditions will always be satisfied. But, for proper systems, the proposed methodology is applicable only when kD  diagfDii gk1 is less than p1ffiffiNffi. Since, our objective is to design a decentralized PI controller for a nonlinear multivariable system, based on the discussion given so far, an algorithm to design a robust decentralized PI controller for nonlinear multivariable systems is given as follows: Algorithm 1

In the above equations, the off-diagonal-blocks Aij ; i–j, Bij ; i–j, b ii g, C ij ; i–j and Dij ; i–j are fixed. Because, Ad ¼ diagf A b b b Bd ¼ diagf B ii g, C d ¼ diagf C ii g and Dd ¼ diagf D ii g should be blockb ii , B b ii and D b ii , C b ii can be selected diagonal. But, the diagonal blocks A to minimize the norm. The local LMI optimization problem given in b ii and B b ii . To solve the problem (79) is not affine in the variables A we introduce new LMI variables

b ii Ri ¼ P i A b ii Si ¼ P i B

ð80Þ ð81Þ

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

;

ð82Þ

with Q i ¼ ½ Pi Ai1 P i Bi1 . . . P i Aii1 P i Bii1 P i Aii  Ri P i Bii  Si . . . P i AiN P i BiN 

ð83Þ

(a) Linearize the system about its operating points. Select a plant as the nominal plant and model the nonlinearity as output multiplicative uncertainty. Define the diagonal  ðsÞ. weighting matrix W 3 (b) By solving the local LMI optimization problems given in (82), decompose the nominal plant into isolated subsystems such that they are stable, controllable and observable and the interactions between subsystems are minimized. If b ii gk is not less than p1ffiffiffi, apply the similarity kC  diagf C 1 N transformation to the system to convert the matrix C into a block-diagonal one and do the decomposition again. Then, b ii gk if this norm is greater than p1ffiffiffi, stop, check kD  diagf D 1 N the method is not applicable to the system. Otherwise go to step (c). (c) Based on the decomposition, define the weighting functions  1i ðsÞ; i ¼ 1; . . . ; N; such that the conditions given in (30) are w satisfied. (d) If the isolated subsystems are not first order, use an appropriate methodology to approximate the transfer function of each isolated subsystem with a first-order model. Define the error of approximation as multiplicative uncertainty. Based on the local approximation models, define appropriate  3i ðsÞ; i ¼ 1; . . . ; N. weighting functions w

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(e) Derive appropriate weighting functions w1i ðsÞ and w3i ðsÞ using Eqs. (74) and (75). Select sci for the i-th isolated subsystem, such that the local robust performance problems given in (73) are solved. (f) Based on the IMC method, design local PI’s using the obtained time constants in step (e).

6. Utility boiler The utility boilers in Syncrude Canada are water tube drum boilers. Since, steam is used for generating electricity and process applications, demand for steam is variable. The control objective of the co-generation system is to track steam demand while maintaining steam pressure and steam temperature of the header at their respective set-points. The objective of this paper is to design a controller, so that the utility boiler system keeps stability and reaches the desired performance. In the system, the principal input variables are u1 , feedwater flow rate (kg/s); u2 , fuel flow rate (kg/s); and u3 , attemperator spray flow rate (kg/s). The principal output variables 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 system, first it is necessary to identify a fairly accurate model for the system. The identified model is a control oriented model. Since, the designed controller will be a decentralized controller, we try to identify the system such that the matrices, A, B, C and D are as close as possible to block-diagonal matrices. Indeed, because in the proposed method, the subsystems should be approximated by first order models, we try to derive a model with the lowest possible order. According to the field experience, the utility boilers in the SCL work mainly in three typical operating points, which are low load, normal load, and high load operating points [19]. The parameters of the different operating points are listed in the following table (see Table 1). 6.1. Steam flow dynamics Steam flow plays an important role in the drum-boiler dynamics. Steam flow from the drum to the header, through the super heaters, is assumed to be a function of the pressure drop from the drum to the header. We use a modified form of the Bernoulli’s law to represent flow versus pressure, with friction [15]. This expression is written as

qs ¼ K

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x22  P 2header ;

ð87Þ

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

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qs ¼ 0:03 x22  ð6306Þ2 : 6.2. Drum pressure dynamics

To model the pressure dynamics, first step identification is done to observe the behavior of the system. By applying step inputs to the three different inputs at different operating points, we observe for a step increase in the feedwater and fuel flow, the system behaves like a first order system with the same time constant. By applying a step to spray flow input, the system behaves like a first order system with different time constant. It means, using two state variables can describe the pressure dynamics fairly accurately. But, because of minimizing the interactions between subsystems due to the matrix B in the model, we neglect the effect of the spray flow input on the drum pressure and based on the fundamental physical laws given in [3], and the observation form the step identification, the dynamics for the drum pressure is chosen as follows:

x_ 2 ¼ ðc1 x2 þ c2 Þqs þ c3 u1 þ c4 u2 ;

ð89Þ

y2 ¼ x2 :

ð90Þ

To obtain the constants in the pressure equation given in (89), step inputs are applied to water flow and fuel flow inputs separately at different operating points. The information gathered from the initial step responses gives an initial guess for the time constant and the constants in Eq. (89). Then, a random input sequence (PRBS) with 6 sec. sampling interval is applied to each of two input variables. The PRBS data provides a much richer input signal to obtain a more consistent model through identification. To obtain the parameter values, we equate the transfer function coefficients from the symbolic linearization of Eq. (89) with the identified transfer function. At the low load

X 2 ðsÞ c3 0:0404 ¼ ; ¼ U 1 ðsÞ s  A1 ðs þ 0:0144Þ X 2 ðsÞ c4 3:025 ¼ ; ¼ U 2 ðsÞ s  A1 ðs þ 0:0144Þ

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 u20 u30 y10 y20 y30

u10 u20 u30 y10 y20 y30

u10 u20 u30 y10 y20 y30

¼ 50:12 kg=s ¼ 2:62 kg=s ¼ 0 kg=s ¼1m ¼ 6523:6 kpa ¼ 483:19  C

¼ 68:36 kg=s ¼ 3:67 kg=s ¼ 0:58 kg=s ¼1m ¼ 6711:7 kpa ¼ 500  C

¼ 81:74 kg=s ¼ 4:48 kg=s ¼ 1:84 kg=s ¼1m ¼ 6894 kpa ¼ 500  C

ð91Þ ð92Þ

at the normal load

X 2 ðsÞ c3 0:0404 ¼ ; ¼ U 1 ðsÞ s  A2 ðs þ 0:0111Þ X 2 ðsÞ c4 3:025 ¼ ; ¼ U 2 ðsÞ s  A2 ðs þ 0:0111Þ

ð93Þ ð94Þ

and at the high load we have

X 2 ðsÞ c3 0:0404 ; ¼ ¼ U 1 ðsÞ s  A3 ðs þ 0:0096Þ X 2 ðsÞ c4 3:025 ¼ ; ¼ U 2 ðsÞ s  A3 ðs þ 0:0096Þ

ð95Þ ð96Þ

with

Ai ¼ c1i Table 1 Parameters of the different operating points

ð88Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc1i x2i þ c2i Þx2i x22i  ð6306Þ2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; x22i  ð6306Þ2

i ¼ 1; 2; 3;

ð97Þ

where c1i , c2i and x2i i ¼ 1; 2; 3 are the constants and drum pressure at 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  106 ; c2 ¼ 0:08; c3 ¼ 0:0404; c4 ¼ 3:025:

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

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

x_ 5 ðtÞ ¼ 1:278  103

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 2 ¼ ð1:8506  107 x2  0:0024Þ x22  ð6306Þ2  0:0404u1 þ 3:025u2 ; y2 ðtÞ ¼ x2 ðtÞ þ p0 ;

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

ð99Þ

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 level dynamics is unstable. By increasing the water flow rate, the level increases and by increasing the fuel flow, the level decreases. Three inputs, water flow, fuel flow and steam flow affect on the drum water level. Simulation results show that there should exist an integrator in the model. Let x1 , and V T denote the fluid density and total volume of the system, then we have

u1  qs ; VT

ð100Þ

where V T ¼ 155:1411. By doing several experiments, it was observed that the dynamics of the drum level can be given by

y1 ¼ c5 x1 þ c6 qs þ c7 u1 þ c8 u2 þ c9 :

ð101Þ

The constants ci ; i ¼ 5; . . . ; 9 should be identified from the plant data. By applying PRBS signal with sampling time equal to 6 sec. and using the plant data, the constants given in (101) are calculated as c5 ¼ 0:0101571, c6 ¼ 0:00612875, c7 ¼ 0:019814, c8 ¼ 0:001, and c9 ¼ 6:1982. The initial values of x1 at the three operating points are 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

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 3 ðtÞ ¼ 0:0211 x22  ð6306Þ2 þ x4  0:0010967u1 ð102Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 4 ðtÞ ¼ 0:0015 x22  ð6306Þ2 þ x5 þ 0:001u1 þ 0:32u2  2:9461u3 ;

ð103Þ

ð104Þ

y3 ¼ x3 þ T 0 ;

ð105Þ

where T 0 ¼ 443:3579, T 0 ¼ 446:4321, and T 0 ¼ 441:9055 for low load, normal load and high load, respectively. At three operating points, we have x30 ¼ 42:2529, x40 ¼ 3:454, x50 ¼ 3:45082, for low load, x30 ¼ 49:0917, x40 ¼ 2:9012, x50 ¼ 2:9862, for normal load, and x30 ¼ 43:3588, x40 ¼ 0:1347 and x50 ¼ 0:2509 for high load. 6.5. The model Combining the achieved results so far, the identified model for the utility boiler is given as follows:

x_1 ðtÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1  0:03 x22  ð6306Þ2 155:1411

;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 2 ðtÞ ¼ ð1:8506  107 x2  0:0024Þ x22  ð6306Þ2  0:0404u1 þ 3:025u2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 3 ðtÞ ¼ 0:0211 x22  ð6306Þ2 þ x4  0:0010967u1 þ 0:0475u2 þ 3:1846u3 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 4 ðtÞ ¼ 0:0015 x22  ð6306Þ2 þ x5 þ 0:001u1 þ 0:32u2  2:9461u3 ; x_ 5 ðtÞ ¼ 1:278  103

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x22  ð6306Þ2  0:00025831x3

 0:029747x4  0:8787621548x5  0:00082u1  0:2652778u2 þ 2:491u3 ; y1 ðtÞ ¼ 0:010157116x1 þ 1:8386  104

In the utility boiler, the steam temperature must be kept at a certain level to avoid overheating of the super-heaters. To identify a model for steam temperature, first step identification is used. By applying a step to the water flow input, steam temperature increases and the steam temperature dynamics behaves like a fist order system. Applying a step to the fuel flow input, the steam temperature increases and the system behaves like a second order system. Applying a step to the spray flow input, steam temperature decreases and the system behaves like a first order system. Then, a third order system is selected for the steam temperature model. This step identification gives an initial guess for local time constants and gains. By considering steam flow as input and applying input PRBS at the three operating points, local linear models for the steam temperature dynamics are defined. Combining the local linear models, the following nonlinear model is identified for all three operating points with a good fitness.

þ 0:0475u2 þ 3:1846u3 ;

 0:2652778u2 þ 2:491u3 ;

ð98Þ

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

x_1 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x22  ð6306Þ2  0:00025831x3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x22  ð6306Þ2

 0:001u1 þ 0:019814u2  6:1982;

y2 ðtÞ ¼ x2 ; y3 ðtÞ ¼ x3 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qs ðtÞ ¼ 0:03 x22  ð6306Þ2 : The parameters of the different operating points for the model are listed in the following table (See Table 2). In addition, the following limit constraints exist for the three control variables:

0 6 u1 6 120;

ð106Þ

0 6 u2 6 7;

ð107Þ

0 6 u3 6 10;  0:017 6 u_ 2 6 0:017:

ð108Þ ð109Þ

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

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B. Labibi et al. / Journal of Process Control 19 (2009) 216–230 Table 2 Parameters 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 u20 ¼ 2:6618 kg=s u30 ¼ 0 kg=s x10 ¼ 678:15 x20 ¼ 6523:6 x30 ¼ 42:2529 x40 ¼ 3:454 x50 ¼ 3:4082

u10 ¼ 68:9256 kg=s u20 ¼ 3:6867 kg=s u30 ¼ 0:58 kg=s x10 ¼ 677:1 x20 ¼ 6711:5 x30 ¼ 49:0917 x40 ¼ 2:9012 x50 ¼ 2:9862

u10 ¼ 83:1251 kg=s u20 ¼ 4:4761 kg=s u30 ¼ 1:84 kg=s x10 ¼ 654:628 x20 ¼ 6887:9 x30 ¼ 43:3388 x40 ¼ 0:1347 x50 ¼ 0:2509

Level (m.)

487

SF (kg/s)

1.01

Temp. (C)

Pr. (MPa.)

Low load 6.532 6.53 6.528 0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

1 0.99

486.5 486 51 50.5

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

Pr. (MPa.) Level (m.)

1.02

Temp. (C)

508 506 504 502 500 498

SF (kg/s)

Normal load 6.74 6.72 6.7 6.68

75

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

1 0.98

70 65

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

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

7. Controller design The identified nonlinear model is linearized about its operating points. So, there are three linearized model at low load, normal

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6.9 6.85

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

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20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

1.02 1 0.98

SF (kg/s)

Temp. (C)

Level (m.)

Pr. (MPa.)

High load

505 500 495 85 80

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

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

2

0 0:0002 6 6 0 0:0111 6 6 A ¼ 6 0 0:0616 6 6 0 0:0044 4

0

0

0

0

0

0

0

1

0

0

0

1

2

0

1

0

3

6 A33 ¼ 4

0

0

1

7 5;

3 B33

7 7 7 7 7; 7 7 5

0:0003 0:0297 0:8788 3 3:1846 6 7 ¼ 4 2:9461 5; C 33 ¼ ½ 1 0 ; D33 ¼ 0: 2

2:4910 According to this decomposition, based on the interaction between subsystems, the following appropriately defined weighting matrix is obtained to satisfy the equations given in (30)

0 0:0037 0:0003 0:0297 0:8788 3 0:0064 0 0 7 6 0 7 6 0:0404 3:0250 7 6 7 6 3:1846 7; B ¼ 6 0:0011 0:0475 7 6 6 0:0010 0:32 2:9461 7 5 4 2

0:0008 2 6 C¼4

2:6528

2 0:0198sþ0:000532 ð110Þ

3

2 6 D¼4

0

1

7 0 0 0 5;

0

0

1 0 0

0:001 0:0198 0

3

0

0

7 0 5:

0

0

0 ð111Þ

The uncertainty of the change of variables in the three different linear models is modeled as output multiplicative uncertainty by the following weighting matrix:

2

0

 ðsÞ ¼ 6 0 W 4 3

0

0

0:05 sþ0:02

0

0

3

7 0 5: 0:2

B11 ¼ 0:0064;

ð112Þ

C 11 ¼ 0:0102;

D11 ¼ 0:001; ð113Þ

A22 ¼ 0:0288;

B22 ¼ 3:025;

0

0

3

0

0:333sþ0:09425 sþ0:031

0

7 5:

0

0

0:0014sþ12:16 sþ7

ð116Þ

g 1 ðsÞ ¼

0:001ðs  0:04968Þ ; ðs þ 0:0156Þ

g 2 ðsÞ ¼

3:025 : ðs þ 0:0288Þ

ð117Þ

The third subsystem is a third order system which can be approximated by

g 3 ðsÞ ¼

3:075 ; ðs þ 0:25Þ

ð118Þ

with the following multiplicative uncertainty weighting function

The first subsystem associated with the drum level is an unstable system. By solving local LMI optimization problem (82) for the first subsystem, the nominal system is decomposed into the following isolated subsystems:

A11 ¼ 0:0156;

 1 ðsÞ ¼ 6 W 4

sþ0:02

The first two subsystems are first order systems with the following transfer functions

2:4910

0:0102 0:0005 0 0 0

ð115Þ

C 22 ¼ 1;

D22 ¼ 0;

ð114Þ

 33 ðsÞ ¼ 0:8: w

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, designing the local PI controllers is reduced to solving the local robust control problems given in (73) for the following uncertain isolated subsystems

0:001ðs  0:04968Þ ; ðs þ 0:0156Þ 0:0198s þ 0:000532 ; w11 ðsÞ ¼ s þ 0:02

g 1 ðsÞ ¼

w31 ðsÞ ¼ 0;

ð120Þ

227

Level (m.)

1.05

Pre. (KPa.)

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

6900

1 0.95 0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

6800 6700

Temp. (C)

520 500 480

Time (min.)

1.05 1 0.95 0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

6900 6800 6700 520

Temp. (C)

Pre. (KPa.)

Level (m.)

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

500 480

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

Level (m.)

1.1 1

Pre. (KPa.)

0.9

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

6700 6600 6500

Temp. (C)

0

500 480

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

228

g 2 ðsÞ ¼

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

3:025 ; ðs þ 0:0284Þ

w32 ðsÞ ¼

g 3 ðsÞ ¼

w12 ðsÞ ¼

0:333s þ 0:09425 ; s þ 0:031

0:005 ; ðs þ 0:02Þ

3:075 ; ðs þ 0:25Þ

ð121Þ

w13 ðsÞ ¼

0:0014s þ 12:16 ; sþ7

w33 ðsÞ 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. Using the obtained time constants for the isolated subsystems, the local PI controllers are designed based on the IMC method as follows:



0

1 0:01 1 þ 35:23s

0 0

0



3

0

7 0 5:  1 0:015 1 þ 4s

FW flow (kg/s)

ð123Þ

100 50 0

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

FF (kg/s)

6 4 2

SF (kg/s)

10 5 0

Time (min.)

FW flow (kg/s)

Fig. 7. Switching from normal load to high load (control signals).

80 60

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

5

FF (kg/s)

6 KðsÞ ¼ 4

 1 212 1 þ 62:2424s

SF (kg/s)

2

The designed controller is applied to the real nonlinear system. In order to compensate the constraints given in (106)–(109) on control signals, as explained in [20], these constraints can be ignored at the design stage, and the effects of the constraints are compensated after the controller design, using anti-windup bump-less transfer (AWBT) techniques [20]. Then, for each isolated subsystem an AWBT compensator is designed. Applying the designed controller with the decentralized AWBT compensator to the nonlinear system, Figs. 4–6 show the responses of the closedloop system in switching from normal load to high load, from high load to normal load and from normal load to low load, respectively. These figures show good set point tracking of the closed-loop system. Figs. 7–9 show the related control signals and that the constraints given on control signals are satisfied. Fig. 10 compares the step responses for the designed decentralized PI controller and the existing multiloop controller in switching from normal load to high load. It can be observed that the closed-loop system under the multiloop controller has a large overshoot for drum pressure and steam temperature change. The controller outputs for both controllers are compared in Fig. 11 which shows the control signals for the new controller are less aggressive.

4 3

2 1 0

Time (min.) Fig. 8. Switching from high load to normal load (control signals).

229

FW flow (kg/s)

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

100 50 0

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

FF (kg/s)

4 2 0

SF (kg/s)

2 1 0

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

Pre. (KPa.)

Level (m.)

1.1 1

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

6900 6800

Temp. (C)

6700 540 520 500 480 460

Time (min)

FW flow (kg/s)

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

100 80 60 40

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

FF (kg/s)

6 5 4

SF (kg/s)

6 4 2 0

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

230

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

8. Conclusion In this paper, a method for robust decentralized PI controller design for an industrial utility boiler is proposed. Sufficient conditions for robust stability and diagonal dominance of the overall closed-loop system are derived. These conditions are based on the sensitivity functions of the closed-loop isolated subsystems and are formulated as local robust performance problems. It is shown, by appropriately selecting the time constants of the closed-loop isolated subsystems, these sufficient conditions are satisfied. The method is applicable to systems with any order, stable/unstable and minimum-/non-minimum-phase. Then, a fairly accurate nonlinear model for the boiler system in the Syncrude Canada Ltd. is identified. For the identified model, a decentralized PI controller is designed. Applying the designed controller to the real industrial utility system shows the effectiveness of the proposed method. Acknowledgement This research is supported by the Natural Sciences and Engineering Research Council of Canada, and Syncrude Canada Ltd. References [1] K.J. Astrom, T.H. Hagglund, New tuning methods for PID controllers, in: Proceedings of the 3rd European Control Conference, 1995, pp. 2456–2462. [2] K.J. Astrom, T. Hagglund, PID Controllers: Theory, Design and Tuning, second ed., Instrument Society of America, Research Triangle Park, 1995. [3] K.J. Astrom, R.D. Bell, Drum-boiler dynamics, Automatica 36 (2000) 363–378. [4] J. Bao, P.L. Lee, F. Wangc, W. Zhou, Y. Samyudia, A new approach to decentralised process control using passivity and sector stability conditions, Chemical Engineering Communications 182 (2000) 213–237.

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