A design framework for overlapping controllers and its industrial application

ARTICLE IN PRESS Control Engineering Practice 17 (2009) 97– 111

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

A design framework for overlapping controllers and its industrial application Adarsha Swarnakar , Horacio Jose Marquez, Tongwen Chen Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G 2V4

a r t i c l e in fo

abstract

Article history: Received 11 March 2007 Accepted 26 May 2008 Available online 7 July 2008

This paper presents a new practical framework for output feedback control design with overlapping information structure constraints. In comparison to the earlier work, the proposed method removes some restrictions in the control design algorithm by utilizing congruence transformations, simplifications, and the reciprocal variant of the projection lemma. This leads to a less conservative solution than previous design methods because the choice of some design parameters by trial and error is eliminated. Moreover, in some cases the structural constraint of having a diagonal Lyapunov function in linear matrix inequalities (LMIs) is removed. The results are extended to capture a more general scenario of output feedback control design for nonlinear interconnected systems. The validity of the proposed approach is demonstrated through applications to an industrial utility boiler and to a multi-area power system. Simulation experiments using a nonlinear simulation package of utility boilers called SYNSIM reveal that the proposed design strategy overcomes control problems in the present plant and maintains stability in the presence of sudden load variations. Furthermore, the performance of the overlapping controllers is found to be better than existing PI controllers in the plant. & 2008 Elsevier Ltd. All rights reserved.

Keywords: LMI applications Overlapping control Nonlinear systems Industrial utility boiler Power systems Control applications

1. Introduction In many practical systems, specific structures of controllers are used instead of a centralized architecture. Motivations for this preference include less modeling effort, connective stability, less communication overhead, and wide acceptance by operators in the industry, to mention just a few. Moreover, stability of the closed loop system in the presence of uncertainties (both in subsystems and their interconnections) leads to an additional robustness property (Siljak & Zecevic, 2005). For these reasons the last few decades have seen increasing research interest in the design of multi-loop control systems (Siljak, 1991; Siljak, Stipanovic, & Zecevic, 2002; Siljak & Zecevic, 2005; Swarnakar, Marquez, & Chen, 2007). However, the design algorithms of decentralized structures are relatively complex due to nonconvexity (Rotkowitz & Lall, 2006). It is well known that if some state information is shared among subsystems, the concept of overlapping control arises (Siljak, 1991; Siljak & Zecevic, 2005; Zecevic & Siljak, 2005). Local controllers use this extra information to improve the stability and performance of the overall closed loop system. Practical scenarios where this kind of situation arises include platooning vehicles, power systems, web handling systems, and traffic light

control systems (Benlatreche, Knittel, & Ostertag, 2008; Siljak & Zecevic, 2005; Stankovic, Chen, Matausek, & Siljak, 1999; Stankovic, Stanojevic, & Siljak, 2000; Zecevic & Siljak, 2005). However, in many cases, finding an explicit solution to overlapping control design is still a problem. In the past, expansion and inclusion principles were generally used for designing overlapping control laws (Ikeda, Siljak, & White, 1984; Siljak, 1991; Stankovic & Siljak, 2001). In the expansion principle, systems are stretched out into a space where they appear to be decoupled. The control design can then be viewed as a decentralized control problem in the expanded region. Using the inclusion principle, the control laws are then converted into the original space for the application purpose. One important problem, which may crop up while using this approach is that the method is not appropriate if some subsystem ðA22 Þ is unstable (Zecevic & Siljak, 2005). Eigenvalues of this subsystem represent fixed modes of the expansion space and limit the practical appeal of this algorithm. In Zecevic and Siljak (2005) and Siljak and Zecevic (2005), an approach towards the elimination of this weakness was shown. Their method solves static state feedback problems for both Type I and Type II overlapping (Fig. 1), where the input matrix and the control law have the following forms: 2

 Corresponding author. Tel.: +1780 492 3368; fax: +1780 492 1811.

E-mail addresses: [email protected] (A. Swarnakar), [email protected] (H.J. Marquez), [email protected] (T. Chen). 0967-0661/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2008.05.009

B11 6B B ¼ 4 21 0

0

3

B22 7 5; B32

" K¼

K11 0

K12 K23

# 0 , K24

(1)

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A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

Fig. 1. Type I and Type II overlapping (Zecevic & Siljak, 2005).

where Type II corresponds to B21 ¼ B22 ¼ 0. The method blends linear matrix inequalities (LMIs) (Boyd, Ghaoui, Feron, & Balakrishnan, 1994) and system expansion to overcome the aforementioned problem, which is elegant and meritorious. However, in the Type II overlapping case, the selection of some parameters in the optimization problem remains an open question. The parameters are generally selected on a trial and error basis to convert the nonlinear optimization problem into LMIs. Moreover, for both Type I and Type II overlapping, the algorithm requires a Lyapunov function with some special structural constraint and, in some cases, a diagonal version of Lyapunov functions. This is restrictive and could lead to infeasibility of the optimization problem; hence, it may not be user friendly to control engineers. In this paper the authors propose two different techniques to solve the foregoing problems. Different congruence transformations, some simplifications, change of controller variables (Chilali & Gahinet, 1996), and the reciprocal variant of the projection lemma (Apkarian, Tuan, & Bernussou, 2001) are used to obtain less conservative LMI solutions. This is possible because the use of diagonal Lyapunov functions and choice of parameters by trial and error are not required in this approach. The method is extended to capture a general scenario of output feedback control design and the results are generalized to include large-scale nonlinear interconnected systems. Some interesting observations of the algorithm, which are a source of attraction to both theorists and practitioners, are as follows: (1) Method I: A general algorithm which deals with both Type I and Type II overlapping has been developed for linear as well as nonlinear systems. The method can handle static state feedback, static output feedback, full order, and reduced order dynamic output feedback control designs. There is no need to select parameters by trial and error or to impose structural constraints on the Lyapunov function. The overlapping control design problem is converted into an optimization problem that involves LMIs and only one equality constraint of the form: Q ¼ MT M. This constraint is then relaxed as: " # Q MT X0, (2) % I and an iterative algorithm is used to compute controller parameters. The objective function value strictly decreases in each step, proving the convergence of this algorithm. It should be noted that Q ¼ MT M corresponds to the boundary of the convex sets in (2). As the optimization involves only one equality constraint, few iterations are required for the control

design. Moreover, each step entails solving LMIs (feasibility problem, eigenvalue problem) and there is no requirement for an initial guess. Generalization of the results to N nonlinear interconnected systems is straightforward. The algorithm can also accommodate many other structures of the controllers, namely, decentralized design, or control design when overlapping states are shared by multiple subsystems and bordered block diagonal (BBD) structure (Siljak & Zecevic, 2005). This makes the results general and increases the applicability of the algorithm to a number of practical systems. (2) Method II: The control design problem for Type II overlapping is converted into a sequential two-part optimization problem using different congruence transformations, simplifications, and change of controller variables. A two-step method, similar to that employed by Zhu and Pagilla (2007), is used for computing controller parameters. The advantage of this approach is that no iteration is required and the control law can be obtained in two steps. However, this method requires block diagonal Lyapunov functions and cannot handle static output feedback control designs (additional non-convex rank constraints are required).

To show that the approach is practical, two engineering problems are considered. In the first case, an overlapping load frequency control law for a two-area power system (a benchmark example) is designed. The areas are represented by the subsystems and the tie lines are the overlapping parts. It is shown that the scenario corresponds to a Type II overlapping case and the designed controller keeps the system frequency and the inter-area tie line power to desired values in the presence of load variations. The stability is studied in the presence of a generating rate constraint (GRC) and the results are compared with results from the decentralized design. In the second case the control problem of the Syncrude Canada integrated energy facility is considered. The utility part (among other divisions, namely, mining, extraction, and upgrading) of this plant consists of a boiler system (utility boilers, CO boilers, and once through steam generators), an electricity generating system (steam and gas turbines), and a header system (6.306, 4.24, 1.068, and 0.372 MPa). In most cases, decentralized PI controllers of the present plant work well; however, the 6.306 MPa header pressure shows oscillatory behavior under load fluctuations that the controllers are unable to damp out quickly. This happens due to nonlinearities and other interactions that arise from avoiding the multi-variable characteristics of the plant. To overcome this problem and bring overall stability under load variations, the authors identified a nonlinear interconnected model of utility boilers and the 6.306 MPa header. The pressure equation is obtained by data fitting and the drum water level is obtained on the basis of physical laws. Inputs to the model are feedwater flow rate, firing rate (to control air and fuel flow), and attemperator spray flow rate; the outputs are drum level, header pressure, and steam temperature. Different overlapping controllers are designed and their performance under load variations are compared with the existing PI controllers in a Syncrude nonlinear simulation package called SYNSIM. The controllers are linear, so they can be easily implemented. The rest of this paper is organized as follows. The instrumental tools used throughout are introduced in Section 2. A solution to Type I and Type II overlapping control is given in Section 3, which involves the use of a Lyapunov function without any structural constraint. Different cases are considered under which the extension to output feedback control design

ARTICLE IN PRESS A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

and to overlapping control laws for nonlinear interconnected systems are noteworthy. In Section 4, a sequential two-part optimization procedure to solve control problems in the Type II framework is introduced. Section 5 deals with the application of the proposed design strategy to a two-area power system and to an industrial utility boiler. Finally, Section 6 concludes the paper.

99

controller of the form u ¼ Ky (K in (1)), and a dynamic output feedback overlapping control law x_ k1 ¼ Ak1 xk1 þ Bk11 C1 x1 þ Bk12 C2 x2 , u1 ¼ Ck1 xk1 þ Dk11 C1 x1 þ Dk12 C2 x2 , x_ k2 ¼ Ak2 xk2 þ Bk21 C2 x2 þ Bk22 C3 x3 , u2 ¼ Ck2 xk2 þ Dk21 C2 x2 þ Dk22 C3 x3 .

(4)

In both cases, the closed loop system is given by

2. Instrumental tools

^ D x þ hr ðx Þ, ¯ cl þ hr ðxcl Þ ¼ A ¯ d CÞx x_ cl ¼ ðA0 þ BK cl cl Throughout this paper, the following instrumental tools are used. Lemma 2.1 (Reciprocal projection lemma: Apkarian et al., 2001). Let X be any given positive-definite matrix. The following statements are equivalent:

ST þ WT

%

X

A11 6 6 A21 6 6 ^ D ¼ 6 A31 A 6 6 6 0 4 0

# o0

is feasible with respect to W. Here, S is a square matrix of size compatible with W (a symmetric matrix), which appears in the control design algorithm. The matrices W and S can contain elements that are affine/ non-affine in the controller parameters. The slack variable W provides additional flexibility and degree of freedom in a variety of problems. In Section 3, it will be shown that this additional variable forms a cornerstone in the design of overlapping controller. Lemma 2.2 (Schur’s complement method: Boyd et al., 1994). For a negative definite matrix Uo0, the following two statements are equivalent: h

i 12 (1) U ¼ U%11 U U22 o0. T (2) U22 o0; U11  U12 U1 22 U12 o0. 3. A solution to overlapping control design Consider a nonlinear process of the form (Siljak et al., 2002; Siljak & Zecevic, 2005; Swarnakar et al., 2007; Zecevic & Siljak, 2005)

n

T

hr ðxcl Þhr ðxcl Þpa2 xTcl HT Hxcl ¼ a2 xTcl HTl Hl xcl .

2

W þ X  ðW þ WT Þ

x_ ¼ Ax þ Bu þ hðxÞ;

where for static output feedback: A0 ¼ A, B¯ ¼ B, Kd ¼ K, C¯ ¼ C, xcl ¼ x, and hr ðxcl Þ ¼ hðxÞ is bounded by (6)

However, for the case of dynamic output feedback with xTcl ¼ ½xT1 xT2 xT3 xTk1 xTk2 ,

(1) W þ S þ ST o0. (2) The LMI problem "

(5)

y ¼ Cx,

(3) m

p

where x 2 R is the state, u 2 R is the control input, y 2 R is the output, C ¼ diag ðC1 ; C2 ; C3 Þ, and 2 3 2 3 B11 0 A11 A12 A13 6A 7 6 7 A ¼ 4 21 A22 A23 5; B ¼ 4 B21 B22 5. A31 A32 A33 0 B32 Here, B21 ¼ B22 ¼ 0 for Type II overlapping and C ¼ I for full state feedback. The function hðxÞ is assumed to be uncertain, but bounded by (Siljak et al., 2002; Siljak & Zecevic, 2005; Swarnakar et al., 2007; Zecevic & Siljak, 2005) T

h ðxÞhðxÞpa2 xT HT Hx, where H is a constant matrix and a is a scalar parameter that reflects the degree of robustness. This kind of nonlinearity and Lipschitz-continuous averaged nonlinearity (Iannelli, Johansson, Joensson, & Vasca, 2008) can be found in power systems, power converters, etc. Consider a static output feedback overlapping

2

A12

A13

0

A22

A23

0

A32

A33

0

0

0

0

0

0

0

Ak1

6 6 0 6 6 6 Ck1 4 0

0

Bk11

3

2 0 7 6 7 6 07 60 7 6 6 07 7 þ 60 7 6 7 05 6 4I 0 0 0

Bk12

0

Bk21

0

Dk11

Dk12

Ck2

0

Dk21

B11

0

B21

0

0

0

0

3

0

Ak2

0

I 2

7 B22 7 7 7 B32 7 7 7 0 7 5

0

0

0

6 0 76 7 Bk22 6 76 6 C 7 1 0 76 56 6 0 Dk22 4 0

3

0

0

0

I

0

0

0

0

0

0

C2

0

0

0

C3

0

¯ ¯ d C, ¼ A0 þ BK

0

3

7 I7 7 7 07 7 7 07 5 0 (7)

and 2 6 T hr ðxcl Þhr ðxcl Þpa2 xTcl 4

3

HT H

0

0

0

7 T 0 5xcl ¼ a2 xTcl Hl Hl xcl .

0

0

0

0

(8)

In the following theorem, sufficient conditions are provided for the existence of a stabilizing overlapping control law for the nonlinear system in (5). It is assumed that ðA; BÞ is stabilizable, ðC; AÞ is detectable and the system has no unstable fixed modes for the control structure in (1). Theorem 3.1. If there exists a controller Kd such that the following optimization is feasible min g subject to XP 40, 2 T T Q AT0 þ C¯ KTd B¯ þ MT 6 6 % I 6 6 6 % % 6 6 6 % % 4 %

%

(9) XP

HTl

XP  M

0

0

0

I

0

0

%

gI

0

%

%

I

T

3 7 7 7 7 7o0, 7 7 7 5

(10)

Q ¼ MT M, then the system in (5) is asymptotically stable for nonlinearities satisfying the quadratic constraint in (6) or (8). Proof. Please see Appendix A. Corroborated by many simulations, it has been found that by replacing X with rI (where r40 is a tuning parameter) instead of I gives faster convergence speed. This is because, the inequality in

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A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

Fig. 2. Physical meaning of parameters r and z.

and with dynamic output feedback control law

(28) is then equivalent to

2

 Xo0, 1 I þ YHTl Hl Y þ rI  ðW þ WT Þo0, g Iþ

(11)

1 1 ^T T ^ YHTl Hl Y þ rI  ðW þ WT Þ þ ðYA D þ W ÞðAD Y þ WÞo0. g r

In Theorem 3.1, the positive definite matrix XP (which has no structural constraint) is decoupled from the controller Kd. This is an important advantage, because other structures of Kd (decentralized design, control design when x2 in Fig. 1 is reachable from u1 or u2 only, etc.) can be assigned. Moreover, it is possible to design reduced order dynamic controllers, because different orders of Ak1 and Ak2 in (7) can be imposed. In some cases, Theorem 3.1 yields a controller with fast dynamics. Therefore, additional pole placement constraints (Chilali & Gahinet, 1996) should be added. Remark 3.1. Consider an interconnected system of the form x_ ¼ Ax þ Bu þ hint ðxÞ; T

y ¼ Cx, T

(13) T

where hint ðxÞ ¼ ½h1 ðxÞ h2 ðxÞ h3 ðxÞT . The functions hi ðxÞ (for i ¼ 1; 2; 3) contain the nonlinearities in subsystems and in the interconnections. They are assumed to be bounded by a quadratic inequality (Siljak et al., 2002; Swarnakar et al., 2007) T

hi ðxÞhi ðxÞpa2i xT HTi Hi x;

i ¼ 1; 2; 3,

(14)

where ai ’s are interconnection parameters. Here, with static output feedback control law ! ! 3 3 X X T hint ðxcl Þhint ðxcl ÞpxTcl a2i HTi Hi xcl ¼ xTcl a2i HTil Hil xcl , i¼1

i¼1

0

0

0

3

7 0 5 xcl .

HTi Hi

(12)

I

%

0

0 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

^ T þ WT ÞðA ^ D Y þ WÞ It is clear that (11) is negative definite and 1r ðYA D is positive definite (or positive semi-definite). Therefore, in most cases, (12) can be easily satisfied by increasing r, since it decreases ^ T þ WT ÞðA ^ D Y þ WÞ (Fig. 2), and the positive contribution of ½1r ðYA D this decrease is much more pronounced compared to the decrease of negative definiteness in I þ 1g YHTl Hl Y þ rI  ðW þ WT Þ. This makes the optimization problem feasible and leads to faster convergence speed. The equality constraint can also be reduced to the form: Q ¼ ðXP þ MÞT ðXP þ MÞ, which is useful in some applications due to extra degree of freedom provided by XP . A linear system is a special case of (3) with hðxÞ ¼ 0, hence, the optimization problem becomes 2 3 T T Q AT0 þ C¯ KTd B¯ þ MT XP  MT 6 7 T XP 40; Q ¼ M M; 4 % 5o0. I 0 %

6 T hi ðxcl Þhi ðxcl Þpa2i xTcl 4

HTi Hi

l

l

For large-scale interconnected systems, where the nonlinearities in the ith subsystem satisfy the constraint (Siljak et al., 2002; Swarnakar et al., 2007) T

hi ðt; xÞhi ðt; xÞpa2i xT HTi Hi x;

i ¼ 1; 2; 3; . . . ; N

(15)

the following optimization problem should be solved for the controller parameters min

g1 þ g2 þ    þ gN

subject to XP 40, 2 T T Q AT0 þ C¯ KTd B¯ þ MT 6 6 6 % I 6 6 6 % % 6 6 6 % % 6 6 6 6 % % 6 6 6 % % 4 %

%

XP

HT1l



HTNl

X P  MT

0

0



0

0

I

0



0

0

%

g1 I



0

0

%

%

..

.. .

.. .

%

%

%

gN I

0

%

%

%

%

I

.

3 7 7 7 7 7 7 7 7 7o0, 7 7 7 7 7 7 7 5

Q ¼ MT M. Hence, the generalization of the result here is very straightforward. Since, (14) is a special case of (15), the optimization follows along the same lines. Remark 3.2. For solving the optimization problem using the available numerical software, a key idea is to relax the equality constraint as "

Q

MT

%

I

# X0,

(16)

and then apply a cone complementary linearization (CCL) algorithm (Ghaoui, Oustry, & Rami, 1997) for computing the controller parameters. The following algorithm shows a modified version of the CCL method. Computational method: As the equality constraint Q ¼ MT M corresponds to the boundary of the convex set in (16), let Hb 9fXP ; Kd ; Q ; M; (9), (10) and (16) are satisfiedg. Here, Hb is a closed and convex set.

ARTICLE IN PRESS A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

Algorithm OC (overlapping control): (1) Find the feasible set ðX0P ; K0d ; Q 0 ; M0 Þ 2 Hb . Let k :¼ 0. (2) Solve the following convex optimization problem for the variables ðXP ; Kd ; Q ; MÞ 2 Hb : trace½Q  ðMk ÞT M  MT Mk 

min Hb

subject to (9), (10) and (16). (3) Substitute the values of ðXP ; Kd ; MÞ in (29). If the condition is satisfied then output the feasible solutions ðXP ; Kd ; Q ; MÞ. EXIT. (4) Set k ¼ k þ 1, ðXkP ; Kkd ; Q k ; Mk Þ ¼ ðXP ; Kd ; Q ; MÞ, and go to step 2. This remark shows that each step in the algorithm entails solving LMIs and an arbitrary initial guess is not required. Remark 3.3. It is important to note that the optimization %

problem

in

the

kth

step,

~J k ¼ min ~J k ¼ min trace½Q k þ Q 

ðMk ÞT M  MT Mk , subject to (9), (10), (16) and the step 2 in algorithm OC are equivalent. This is because Q k is a constant matrix; therefore, both optimization problems have the same solution. Using ideas from literatures (Theorem 3.2 of Tao & Zhao, %

k 2007), it can be easily derived that ~J X0 and the sequence 1 2 fJ~ ; J~ ; . . .g is monotonically decreasing and convergent. Moreover, another way to solve the algorithm OC is to expand the set to include the equality constraint by substituting Q with Q þ zI (z40, say), and to stop the iterative algorithm to output the feasible solution if ½Q k  ðMk ÞT Mk ozI. This is due to the fact that the equality constraint is obtained at the boundary of (16), while the LMI solver always tries to achieve solutions in the interior of this set (strict inequalities). Consequently, the parameter z is introduced to expand the set such that the boundary appears inside and (9) as well as (10) are satisfied (Fig. 2) under this situation. %

%

Remark 3.4. It is apparent that the conditions in Theorem 3.1 are not convex owing to a matrix equality Q ¼ MT M. In this regard, one question may be fascinating to many control engineers: when these conditions are reduced to convex ones? To answer this question, if the condition (10) in Theorem 3.1 is replaced by 2 6 6 6 6 6 6 4

T

T

%

AT0 þ C¯ KTd B¯ þ MT I

XP 0

%

%

I

0

%

%

%

gI

%

%

%

ðM þ MT Þ þ I

%

HTl 0

3 X P  MT 7 7 0 7 7o0, 0 7 7 5 0 I

(which yields MT Mp based on ðM  IÞ ðM  IÞX0 ðM þ MT Þ þ I), then the resulting controller synthesis problem can be reduced to a convex one. T

Remark 3.5. It is interesting to note that in power systems the overlapping states are shared by a number of subsystems. Under this situation, the algorithms presented in this paper can be easily ^ D which is used, because the optimization problem involves A ^ D ¼ A0 þ BK ¯ Therefore, differ¯ d C. affine in controller parameters A ent structures for Kd and B¯ can be assigned (different control laws and different overlapping). For a three-area power system, Kd has the structure 2 3 K11 K12 0 K14 0 0 6 0 K24 0 7 K22 K23 Kd ¼ 4 0 5. 0 0 0 K34 K35 K36

101

Example 3.1. Consider the system in Zecevic and Siljak (2005): 2 3 2 3 1 4 0 1 0 6 7 6 7 x_ ¼ 4 1 2 2 5x þ 4 0 0 5u. 0 2 3 0 1 Here, the open loop system has eigenvalues at l ¼ 0:166, 3:08 þ j1:58 and 3:08  j1:58, and the goal is to stabilize the system with an overlapping control law. In Zecevic and Siljak (2005), system expansion and LMIs   4:06 12:8 0 K¼ . 0 3:27 11:6 Using the Algorithm OC, the stabilizing static controller in and the dynamic controller in (7) are given by " # 5:65 5:99 0 , Kstatic ¼ 0 1:79 7:34 2 9:4929 0 0:0069 0:1184 0 6 6 0 9:4873 0 0:1756 0:0676 6 Kdynamic ¼ 6 6 1 0 9:9991 7:3232 0 4 0

1

0

4:8233

(1)

3 7 7 7 7. 7 5

10:8633

The number of iterations is 5 for the static controller and 7 for the dynamic controller. The optimum value of the objective function  is ~J ¼ 7.8435e-005 for static control, and the Q and M matrices are given by 2

27:09 6 Q ¼ 4 8:35 4:05

8:35 20:13 16:70

3 4:05 16:70 7 5; 20:69

2

5:1865 6 M ¼ 4 0:1339 0:4208

1:8787 1:5256 3:7788

3 0:4225 0:2728 7 5, 4:5218

which satisfy Q ¼ MT M. 4. Two-step optimization method for overlapping control design The advantage of this approach is that no iteration is required and the overlapping control law for Type II can be obtained in two steps. The idea behind this method is straightforward. For linear as well as nonlinear systems, if the asymptotic stability condition is given by " # F11 F12 o0, (17) F¼ % F22 where F22 and F11 are affine in controller parameters and bilinear terms appear in F12 , then (1) Solve the feasibility problem F22 o0 to calculate some of the controller parameters. (2) Define n ¼ diagðx1 I; x2 I; . . .Þ40. Substitute the variables from step 1 and solve the optimization (Zhu & Pagilla, 2007) " # F11 F12 o0, min x1 þ x2 þ    ; subject to Fx ¼ % nF22 where x can be considered as a tuning variable to guarantee a feasible solution in the second step. In the following, a dynamic output feedback overlapping control design problem for a nonlinear interconnected system is converted into the form of (17), using different transformations, simplifications, and new variable definitions. It helps to utilize the two-step approach.

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4.1. Dynamic output feedback overlapping control design Consider the nonlinear interconnected system in (13), where a dynamic output feedback overlapping controller has to be designed. With the control law in (4), the closed loop system is given by 2 _ 3 x1 2 3 0 A11 þ B11 Dk11 C1 B11 Ck1 A12r þ B11 Dk12r C2r 6 7 6 x_ k1 7 6 7 6 7 6 7 Bk11 C1 Ak1 Bk12r C2r 0 6 7 7 6 x_ 2 7 ¼ x_ cl ¼ 6 6 7 xcl 6 7 6 A21r 0 A22r þ B32r Dk2r Cr B32r Ck2 7 6 7 4 5 6 x_ 3 7 4 5 0 0 Bk2r Cr Ak2 x_ k2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2

I

6 60 6 6 þ6 60 6 60 4

0

0

^D A

3

Here, h i   A12r ¼ A12 A13 ; Bk12r ¼ Bk12 0 ; C2r ¼ diagðC2 ; 0Þ, h i Dk2r ¼ Dk21 Dk22 ; Cr ¼ diagðC2 ; C3 Þ; Dk12r ¼ ½Dk12 0, " # " # A22 A23 0 Bk2r ¼ ½Bk21 Bk22 ; A22r ¼ ; B32r ¼ , B32 A32 A33 " # A21 . A21r ¼ A31 After using a quadratic Lyapunov function, different congruence transformations, and change of controller variables (Chilali & Gahinet, 1996), the asymptotic stability conditions for the closed loop system are given by the following optimization problem:

diag

"

X1

6 6 6 6 6 6 6 6 6 6 6 6 6 4

F11

F12

%

F22 % % % %

where

#

# " ;

X2

I

#!

40, Y1 % Y2 " # 2 1 3 2 1 3 2 1 33 T2l T3l T1l G1 0 4 5 4 5 4 57 7 0 G2 T21l T22l T23l 7 7 7 7 I 0 0 0 7 7o0, 7 % g1 I 0 0 7 7 7 % % g2 I 0 7 5 % % % g3 I %

2"

I

11

2r

2r

^ 2 ¼ Dk . D 2r (19)

G

subject to

^ 1 ¼ Y1 A11 X1 þ B ^ 1 C1 X1 þ Y1 B11 Ck MT þ N1 Ak MT , A 1 1 1 1 ^ ^ 2 Cr X2 þ Y2 B32r Ck MT þ N2 Ak MT , A2 ¼ Y2 A22r X2 þ B 2 2 2 2 ^ 2 ¼ Y2 B32r Dk þ N2 Bk , ^ 1 ¼ Y1 B11 Dk þ N1 Bk ; B B C^ 1 ¼ Dk11 C1 X1 þ Ck1 MT1 ; C^ 2 ¼ Dk2r Cr X2 þ Ck2 MT2 , ^ 1 ¼ Dk ; D ^ k ¼ Dk ; ^ k ¼ Y1 B11 Dk þ N1 Bk ; D B 12 12r 12r 11 12 12r

0

g1 þ g2 þ g3

In (18),

11

72 3 07 7 h1 ðxÞ 7 76 6 7 I 07 7 4 h2 ðxÞ 5 . 7 7 0 I 5 h3 ðxÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} 0 0 0 hint ðxÞ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

min

The terms h111 , h12r1 , h22r1 are the elements of the matrix HT1l . Similar expressions exist for T12l , T22l , T13l and T23l in terms of the elements in HT2l and HT3l , respectively, where

According to the two-step method, the following steps should be used for computing the controller parameters. Step 1: Maximize the interconnection bounds a1 ; . . . ; a3 ða2i ¼ 1=gi Þ by solving the optimization problem min

g1 þ g2 þ g3 " # X2 I 40, subject to % Y2 2 F22 ½0 G2  T21l 6 6 % I 0 6 6 6 % g1 I F22 ¼ 6 % 6 6 % % 6 % 4 %

%

%

T22l 0 0 g2 I %

T23l

3

7 0 7 7 7 0 7 7o0. 7 7 0 7 5 g3 I

^ 2, B ^ 2 , C^ 2 , and D ^ 2 . Now, compute N2 , M2 square, This gives X2 , Y2 , A T and invertible from N2 M2 ¼ U2 S2 VT2 ¼ svdðI  Y2 X2 Þ, which pro1=2 1=2 vides N2 ¼ U2 S2 and M2 ¼ V2 S2 . Next, the controller parameters are calculated from ^ Ck2 ¼ ðC^ 2  Dk2r Cr X2 ÞðMT2 Þ1 ; Bk2r ¼ N1 2 ðB2  Y 2 B32r Dk2r Þ, 1 ^ ^ A ¼ N ðA2  Y2 A22r X2  B2 Cr X2  Y2 B32r C MT ÞðMT Þ1 , k2

k2

2

2

2

^ 2. Dk2r ¼ D Step 2: Define the tuning parameter n ¼ diagðdiagðx2 I; x3 IÞ; x4 I; x1 I; x2 I; x3 IÞ. Using X2 , Y2 and other parameters from step 1 solve min

x1 þ x2 þ x3 þ x4 " # X1 I subject to 40; % Y1

"

F11

F12

%

nF22

n40;

# o0,

(18)

ARTICLE IN PRESS A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

^ 1, where F12 ¼ ½F12 ½G1 0 T11l T12l T13l . This gives X1 , Y1 , A ^ 1, B ^ k and D ^ k . Hence, the rest of the controller ^ 1 , C^ 1 , D B 12 12 parameters in (19) can be computed in a similar fashion using N1 MT1 ¼ svdðI  Y1 X1 Þ.

5. Applications to a two-area power system and an industrial utility boiler The algorithms developed in Sections 3 and 4 are applied to a power system and utility boilers. 5.1. Two-area power system In the two-area power system shown in Fig. 3, the numerical values of the parameters are obtained from (Yang, Ding, & Yu, 2002). The areas represent the subsystems and the tie lines are the overlapping parts. Therefore, the controllers can be designed to share the overlapping state ðDPtie Þ to improve the performance of the overall system. The controllers minimize the system frequency deviations Df 1 in area 1 as well as Df 2 in area 2 under the influence of load disturbances P D1 and P D2 in the two areas. The overall system is of ninth order and the output measurements (frequency deviations) as well as the system input matrix B are given by

103

(static and dynamic) decentralized and overlapping controllers are designed. They have the following form: " # " # 0:4638 0 0:4098 0:2778 0 ; , 0 0:4638 0 0:2778 0:4098 2 3 5:1443 0 1:9535 2:8866 0 6 7 6 0 5:1443 0 2:8866 1:9535 7 6 7 6 7. 6 7 1 0 0:7943 0:3099 0 4 5 0 1 0 0:3099 0:7943 From Figs. 5 and 6, it can be seen that the controllers are capable of attenuating most of the oscillations. The performance of the first order dynamic controller is better than the static overlapping control law, which in turn shows a better response than static decentralized control. For performance assessment, an integral time absolute error (ITAE) criteria of the following form is

S1

u1

Area 1

ΔPtie which reveals Type II overlapping (Fig. 4). In Figs. 5 and 6, the frequency deviations in the two areas due to the disturbance of DP D1 ¼ 0:01 pu in area 1 are shown (by dotted lines, without controller). This and the Nyquist array with the column Gershgorin circles (Fig. 7, first row) on the diagonal element show that the system is highly interacting. Gershgorin circles for the first subsystem (g 11 and g 12 ) only are drawn because the transfer functions of both subsystems are the same. The responses in Figs. 5 and 6 also show that local controllers should be designed to minimize the oscillations. By using the algorithm OC for a linear system in Section 3, output feedback

+

K1 ΔPc1 + s Integral part

+

+

ΔPG1

Governor

ΔPT1 1 1 sTT1

+

Turbine

ΔP12

ΔPtie

KP1 1 sTP1

Δf1

Generator T12 s

+ -

a12 u2 K2 s

ΔPc2

+

+

-

Integral part B2

1 1 sTG1

u1

a12

+

ΔPD1 -

1 1 sTG2 Governor

ΔPG2

Area 2

Fig. 4. Overlapping scenario of two-area power system.

1 R1

B1 +

S3

u2

1 1 sTT2 Turbine

ΔPT2 +

ΔP21 ΔPD2

1 R2

Fig. 3. Two-area power system (Yang et al., 2002).

KP2 1 sTP2 Generator

Δf2

ARTICLE IN PRESS A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

10

0.005

0

IMAG

0.01

−10

0

0

10

−5 −10

20

2

−0.01

−2

−2 −4

15

20

Fig. 5. Frequency deviation of the first area with output feedback controllers.

0

2

−1

0

1

0.5

0 −2

0 −0.5 −1 −2

−4 −2

0.01

−2

1

2

0.015

0

2

4

Fig. 7. Nyquist array with column Gershgorin circles of the first area: without controller (first row), with static output feedback controller (second row), and with dynamic output feedback controller (third row).

0.005 Δ f2 [Hertz]

0

IMAG

10 Time [sec]

IMAG

5

5

0

−6

−0.025 0

0

2

IMAG

−2 −4

−0.02

−5

4

0 without controller with static output feedback overlapping controller with static output feedback decentralized controller first order dynamic output feedback overlapping controller

−0.015

g12

5

−20

−0.005

IMAG

Δ f1 [Hertz]

0

10 g11 IMAG

104

0 Table 1 Jfre values with static and first order dynamic overlapping controller

−0.005

No control

Static decentralized

Static overlapping

Dynamic overlapping

28.3728

6.2166

4.3148

3.0808

−0.01 without controller with static output feedback overlapping controller with static output feedback decentralized controller first order dynamic output feedback overlapping controller

−0.015 −0.02 0

5

10 Time [sec]

15

x 10−3

20

Control signals

10

overlapping control decentralized control

Fig. 6. Frequency deviation of the second area with output feedback controllers.

u1

5

used (Yang et al., 2002):

0

20 0

tjDf 1 ðtÞjdt.

Table 1 shows the values of J fre for different controllers, which verifies that a dynamic overlapping controller is better. The control signals in Fig. 8 give some idea of economic issues. It is clear that it takes more effort to control the system and there are more transients with a static output feedback decentralized controller than with an overlapping controller. Hence, the overlapping control law may lead to less wear and tear of the control valve and requires less steam, which in turn reduces fuel consumption. The second row of Fig. 7 shows the Gershgorin circles of the closed loop system with a static output feedback decentralized controller. It is noticeable that some circles enclose the origin only at low frequencies, whereas at medium and high frequencies the system is diagonal dominant. With the first order dynamic output feedback decentralized controller in the third row of Fig. 7, the radii of circles at medium frequencies are very small compared to those in the second row. Hence, this controller is capable of minimizing the effects of interactions between different loops and has better performance. To overcome the oscillations completely,

−5 0

5

10

15

20

10

15

20

x 10−3 6 4 u2

J fre ¼

Z

2 0 −2 0

5

Time [sec] Fig. 8. Control signals with static output feedback controllers.

the authors then designed state feedback controllers for which all the local states are available for measurement. Responses with the static state feedback overlapping controller, decentralized controller, and the controller designed based on the two-step

ARTICLE IN PRESS A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

approach are shown in Figs. 9 and 10. The controllers are now capable of attenuating the oscillations completely. The response with the decentralized controller has some transients, but with the overlapping controller the response is very smooth. It should be noted that using the two-step approach, the frequency deviations have less undershoot, but the response is slow. Extension of these results to multi-area power systems is straightforward.

0.005 0

Δ f1 [Hertz]

−0.005 −0.01

5.2. Industrial utility boiler

−0.015 state feedback overlapping controller state feedback decentralized controller overlapping controller using two step approach

−0.02

The effectiveness of the design strategies in Sections 3 and 4 can be demonstrated through application to an industrial utility boiler. As shown in Fig. 11, a part of the interconnected system at Syncrude consists of utility boilers (UB 201–UB 203), CO-type boilers (C.O.1 and C.O.2), and once through steam generators (OTSG1 and OTSG2) developed by Innovative Technologies, Ontario. Steam from different boilers is collected in the 900# (900 pounds or 6.306 MPa) header which is then used in: (1) cokers for extracting bitumen from oil sands, (2) other low pressure headers (600#, 150#, and 50#) for extraction, upgrading (conversion of bitumen heavy oil into lighter components like naphtha and diesel oil), building heating, etc., and (3) turbines (G1, G2, G4, and G6) for generating electricity. The difference between utility boiler and CO boiler lies in the type of fuels they use; UBs utilize natural gases such as methane and ethane and CO boilers exploit coker-off gases. OTSGs are heat exchangers (without any drum) between the incoming feedwater and the outgoing hot gases from G3 and G5. Due to characteristics of different boilers (time constants and usage of different fuels), utility boilers maintain the 900# header pressure, CO boilers have self loops for the steam flow, and OTSGs maintain their own steam temperature. The function of the let down stations is to reduce the steam pressure and to act as an interface between different headers. Recently, a new UE-1 system consisting of two CO boilers (CO3 and CO4) was introduced in the plant to take care of additional load demands. Here, the work is concentrated on redesigning the utility boiler control system whose main task is to regulate the steam pressure of the 900# header and to maintain the drum level and the steam temperature at their set points (1 m and 500  C, respectively).

−0.025 0

5

10 Time [sec]

15

20

Fig. 9. Frequency deviation of the first area with state feedback controllers.

0 −0.002 −0.004 −0.006 Δ f2 [Hertz]

105

−0.008 −0.01 state feedback overlapping controller state feedback decentralized controller overlapping controller using two step approach

−0.012 −0.014 −0.016 −0.018 0

5

10 Time [sec]

15

20

Fig. 10. Frequency deviation of the second area with state feedback controllers.

Natural gas

Feedwater

spray

Coker-off gas spray

UE-1 system Plt. 25-1 CO3-C04

OTSG 1-2

UB 201-203

9/6 900 #

800 kpph

C.O.1-C.O.2

190 kpph

750 kpph

600 #

900 # G2

G3

50 MW

50 MW

25 MW

6/150

G1

G4 50 MW

G5 25 MW

G6 65 MW

Hot gases

150 #

Condenser 150/50

6/50

150/50

50 # Tumblers

Trim heaters

Deaerator Fig. 11. A part of interconnected system at Syncrude.

ARTICLE IN PRESS 106

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To this end, a nonlinear model of the utility boiler and the 900# header is first developed. Inputs to this system are feedwater flow rate, firing rate (output of which is then fanned out into fuel demand and air demand), and attemperator spray flow rate; the outputs are drum water level, header pressure, and steam temperature. It is assumed that steam flow ðqs Þ is a function of the pressure drop from the drum boiler to the 900# header (Bernoulli’s law), as shown in Fig. 12. This can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (20) qs ¼ ks y2drum  y2header ¼ 30 y2drum  y2header , where yheader and ydrum are header and drum pressures (in kPa), respectively. The master control block in Fig. 12 contains a complex logic (high/low select, rate limit, and PI loops), therefore, at first, a linear model at normal operating point (using the MATLAB identification toolbox) was developed from the firing rate ðu2 Þ to the fuel flow ðx5 Þ, which is governed by x_ 5 ¼ x6 þ 0:1758u2 ;

x_ 6 ¼ 0:001833x5  0:1731x6  0:0177u2 .

The air flow usually bears a constant relationship with the fuel flow, i.e., air flow ¼ 18:8x5 ; hence, it is not separately identified. Similarly, the differential equation for the steam flow rate ðx2 Þ is given by x_ 2 ¼ x3 þ 0:009151u1 þ 0:02988x5 þ 0:2239u3 , x_ 3 ¼ 0:001864x2  0:1533x3  0:001987u1 þ 0:03634x5  0:03288u3 ,

where u1 represents the feedwater flow rate and u3 is the attemperator spray flow rate (in kg/s). The intermediate variables x3 and x6 affect the dynamics of the steam flow rate and the fuel flow rate, respectively. To model the drum water level, the physical relations developed in Bell and Astrom (1979, 1987a,b) and Pellegrinetti and Bentsman (1996) were used: u1  x 2 u1  x 2 , ¼ VT 155:1411 1 K ½k e þ ru1  þ x2 , qe ¼ 1þK b f 1þK 1 ½vw V T x7 þ k1 ar þ T s qe . ylevel ¼ Ad x_ 7 ¼

Dylevel ¼ 0:01028x7 þ 0:0044963x2 þ 0:035154x5  5:71107uw1  7:2741,

where an additional feedthrough term uw1 ¼ 105 u1 is added to obtain a good match between the model and the process data. The numerical value 7:2741 accounts for slight disagreement in the operating point of the drum level equation. Following the models of the pressure equation in Bell and Astrom (1979, 1987a,b) and incorporating some heuristic knowledge of boiler behavior (to accommodate the data from SYNSIM), the effect of the fuel flow rate ðx5 Þ and the feedwater flow rate ðu1 Þ on the drum pressure is formulated as x_ 1 ¼ r 1 x1 þ ðr 2 x5 þ r 5 Þcv þ r 3 x5  r 4 u1 ,

Feedwater flow

that (21) does not consider the effects of attemperator spray flow rate ðu3 Þ on drum pressure. Therefore, an identification test was carried out that gives x_ 4 ¼ 0:08x4 þ 0:0006u3 . It is clear that ydrum ¼ x1 þ x4 and the steam flow out of the boiler drum is related to the drum pressure by qs ¼ x2 ¼ 30cv ydrum (from (20)), which yields cv ¼ x2 =30ðx1 þ x4 Þ. The parameters r 1 –r 5 are obtained using a graphical technique through a curve fit to the data from SYNSIM (dynamical experiments) and plant specifications. Finally, the nonlinear differential equation for the drum pressure is governed by " ( ) # 104 x2 _x1 ¼  0:0157x1 þ 1:866 þ 0:00157 x5  0:0000395u1 ðx1 þ x4 Þ ( ) 104 x2  3:545 þ 0:099333, x1 þ x4 ydrum ¼ x1 þ x4 .

The overall nonlinear model and the fourth order linear model for steam temperature dynamics have shown good fit with the data from SYNSIM (both in short term and long term characteristics). This is shown in Figs. 13–16 and the fitness in other operating regions is also good. The model for the steam

Master Control Air flow

Drum level Drum-boiler and Superheaters

(22)

The constant 0:099333 takes care of the disagreement in the operating point of the nonlinear pressure equation and is chosen to minimize the offset between the simulated data and the measurement. To obtain the parameters r 1 –r 5 , symbolic linearization or nonlinear regression techniques can also be used. From (20) and (22), the 900# header pressure can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 yheader ¼ x21 þ x24 þ 2x1 x4  2 . 900

Steam flow

Fuel flow

(21)

where cv is referred to as an imaginary control valve position from qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  drum to header cv ¼ 1  ðy2header =y2drum Þ . It should be noted

x_ 4 ¼  0:08x4 þ 0:0006u3 ;

Here, x7 is the fluid density (mixture of steam and water in the system) and V T is the total volume of the drum, the downcomers and the risers ð¼ 155:1411 m3 Þ. The constant K is a measure of the change in mass of steam generated in the boiler per unit mass lost from the steam space, ef describes the energy flow rate (which depends linearly on the fuel flow x5 and can be obtained from measured data), Ad is drum area, vw is the specific volume of water, T s is the increase in water volume per unit increase in evaporation rate, kb ¼ 1=hfg and r ¼ ðhw  hf Þ=hfg . Here, hfg is the latent heat of evaporation, hf is the enthalpy of saturated water, and hw is the enthalpy of feedwater. After some calculations and substituting values from the steam table at a saturation pressure, the value of qe is obtained. The expression for the quality of the steam ðar Þ

Firing rate

in the system, based on volume, is determined by curve fitting with the data from SYNSIM (which gives a relation in terms of x2 , x5 , and u1 ). Finally, after substituting the construction parameters and steam table data and making some adjustments, the model for the drum water level (deviation about mean) is given by

Header node Drum pressure Steam temperature

Attemperator spray flow Fig. 12. Modeling of the utility boiler and the header.

900 # header pressure

ARTICLE IN PRESS A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

temperature dynamics is governed by

x_ 11 ¼  2:35  106 x8  0:000531x9  0:0346x10  0:8159x11 þ 0:001391u1  0:05352x5  1:108u3 ,

Model and process output

1

Fuel flow [kg/sec]

x_ 8 ¼ x9  0:002324u1 þ 0:5772x5 þ 2:194u3 , x_ 9 ¼ x10 þ 0:002323u1  0:08838x5  1:859u3 , _x10 ¼ x11  0:001799u1 þ 0:06898x5 þ 1:436u3 ,

107

process model

0.5 0 −0.5

ysteam ¼ x8 .

1.2

Measured Output and Simulated Model Output Measured Output model Fit: 98.37% 10

1.5

1.6

1.7

1.8

Firing rate

0.05 0 −0.05 −0.1 1.2

1.3

1.4

1.5

1.6

1.7

1.8 x 104

Fig. 14. Validation of the firing rate.

Steam temperature [C]

15

1.4

0.1

Drum level [m]

In the first case (full overlapping), the feedwater controller uses the extra measurement of header pressure (in addition to drum level) and the firing rate controller utilizes the extra measurement of steam temperature (in addition to header pressure) to control the header pressure. In the second case (partial overlapping), only the firing rate controller shares the measurement of steam

1.3

x 104

Drum pressure [kPa]

Linearization of the overall model at the normal operating point has one pole at the origin (associated with water dynamics) and one RHP zero at 0:0619, which reveals non-minimum phase characteristics. Next, based on the algorithm of Section 3, the following stabilizing overlapping controllers were designed: 2 3 11:98s  12:92 5:318s þ 3:154 0 6 7 s þ 1:122 s þ 1:122 6 7 6 7 6 27:61s þ 33:76 113:4s  139:2 7 6 7, 0 Kfull ¼ 6 7 s þ 1:227 s þ 1:227 6 7 6 7 3:797s  6:5 5 4 0 0 s þ 1:908 2 3 11:97s  13:05 0 0 6 7 s þ 1:134 6 7 6 7 6 61:08s þ 74:81 113:5s  139:3 7 6 7. 0 Kpartial ¼ 6 7 s þ 1:227 s þ 1:227 6 7 6 7 3:797s  6:506 5 4 0 0 s þ 1:91

Model validation

0.8 0.6

process nonlinear model

0.4 0

500

1000

1500

2000

0

500

1000

1500

2000

0

500

1000

1500

2000

6500 6450 6400

480 470 460 450 Time [sec]

Steam flow [kg/sec]

5

Fig. 15. Validation of the model.

0

−5

−10

−15 1.2

1.3

1.4

1.5

1.6

1.7

1.8 x 104

Fig. 13. Validation of the steam flow.

temperature. In order to test the design, simulations were done under several perturbed conditions and the designed controllers were implemented in SYNSIM. These simulations took into account interactions from other subsystems, namely, CO boilers, OTSGs, tie lines, and turbine-generator units G1–G6 (Fig. 11). Fig. 17 shows the stabilizing effect of the overlapping controllers revealing good regulation under high load conditions. Figs. 18 and 19 show responses caused by a load change in the 50# (50 lb) header and Figs. 20 and 21 show responses of different process variables due to a load change of 30 kpph on the 900# header. In both cases, plots of total steam flow rate, 900# steam temperature, and 50# header pressure show non-oscillatory behavior. Responses with overlapping controllers are smoother than those of the existing PI controllers and there is also suppression in amplitude. The improvement in the 50# header steam pressure shown in Fig. 20 could lead to an enhancement in power production, because the 50# header pressure is the back pressure of the turbines (G1, G2, and G4). There are slight deviations in

ARTICLE IN PRESS 108

A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

Two different approaches to solving the overlapping control design problem are introduced. In the first case, an iterative algorithm is used to obtain the controller parameters. This method is applicable to a vast array of overlapping control problems including static state feedback, static output feedback, full order, and reduced order dynamic output feedback control designs. The method eliminates the necessity to choose parameters by trial and error and removes the structural

20% load change in 50 pound header Steam flow [kg/sec]

40 35 500

1000

1500

2000

2.5 Fuel flow [kg/sec]

6. Conclusions

Process inputs 45

0

2 1.5 0

500

1000

1500

50# header pressure

0 −0.05 500

1000 Time [sec]

1500

280 278 276 0

500

1000

1500

2000

2500

3000

0

500

1000

1500

2000

2500

3000

500 499 498 497

51.95 full overlapping partial overlapping PI controller

51.9 51.85 51.8

2000

0

Drum level [m]

500

1000

1500

2000

Time [sec]

Fig. 16. Inputs to the process.

Fig. 18. Sudden load change in the 50# header.

1.001 1 full overlapping

0.999

partial overlapping

0.998 Header pressure [kPa]

0

282

2000

0.05

Steam temperature [°C]

Spray flow [kg/sec]

plant. Since this controller is linear and of only third order, it is simple to implement and test in practice.

900# steam temperature [°C]

Feedwater flow [kg/sec]

drum level (1%) and header pressure (0.09% for partial overlapping) from steady state values due to lack of integrators in the overlapping controllers. However, response of the partial overlapping controller is within range (very less offset) and is acceptable in the present plant. Moreover, this limitation is compensated by improvement in the 900# header steam pressure, which shows no oscillations, the main concern in practice. Fig. 17 reveals the robustness of controllers, since they are designed at normal load conditions and are operated under high load conditions. It is clear that the partial overlapping controller Kpartial, where only the firing rate controller is using the extra measurement of steam temperature, provides a better result than the full overlapping controller and the decentralized PI controllers of the

0

500

1000

1500

2000

2500

0

500

1000

1500

2000

2500

0

500

1000

1500

2000

2500

6307 6306 6305 6304 500.1 500.05 500 499.95 Time [sec] Fig. 17. Performance of the overlapping controllers under high load conditions.

2500

3000

ARTICLE IN PRESS

20% load change in 50 pound header

1 0.98

Steam temperature [°C]

Header pressure [kPa]

0

500

1000

1500

2000

2500

3000

6310 6300 6290 500

1000

1500

2000

2500

3000

3500

500.2 500 499.8 0

500

1000

1500

2000

2500

3000

1.05

3500

0

500

1000

1500

2000

2500

0

500

1000

1500

2000

2500

0

500

1000

1500

2000

2500

6320 6300 6280 6260

500.5 500 499.5 Time [sec]

Fig. 19. Responses during load change in the 50# header.

Fig. 21. Header pressure response during sudden load change.

Steam flow [kg/sec]

Load change of 30 kpph

900# steam

temperature [°C]

50# header pressure

Acknowledgments

290 285 280 275 500

1000

1500

2000

505 partial overlapping

full overlapping

PI controller

0.95

Time [sec]

0

full overlapping

partial overlapping

1

3500

6320

0

109

Load change of 30 kpph

PI controller

partial overlapping

Drum level [m]

full overlapping

Header pressure [kPa]

1.02

Steam temperature [°C]

Drum level [m]

A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

This research is supported by Syncrude Canada Inc. and the Natural Sciences and Engineering Research Council of Canada (NSERC), through a Collaborative Research and Development program. The authors would like to thank the anonymous reviewers for their constructive criticism and suggestions to improve the technical content and readability of this paper.

PI controller

500

Appendix A 495 0

500

1000

1500

2000

Proof of Theorem 3.1. Consider a quadratic Lyapunov function v ¼ xTcl Pxcl . The sufficient conditions for stability of the closed loop system can be expressed as P40, and

51.96 51.94

^ Dx ^ T Px þ hT ðx ÞPx þ xT PA v_ ¼ x_ Tcl Pxcl þ xTcl Px_ cl ¼ xTcl A cl cl cl cl r D cl

51.92 0

500

1000 Time [sec]

1500

2000

Fig. 20. Stabilizing effect of the overlapping controllers.

constraint on the Lyapunov function. In the second case, a twostep approach is employed that requires no iteration. However, the first approach is found to be superior to the second in several aspects. Simulation results in SYNSIM show that the stabilizing effect of the designed controllers is good under normal and perturbed conditions. Moreover, when only the firing rate controller (which controls header pressure) is utilizing the extra measurement of steam temperature, the performance of the closed loop system is better (no header oscillations, minimum offset) than in the case of full overlapping. The presented algorithm has the capability to capture other overlapping cases in addition to Type I and Type II. Future work will concentrate on obtaining a low order nonlinear model and controlling the firing rate of CO boilers along with the utility boilers. Special attention will be paid to reducing the fuel consumption during load variations, leading to a more economic system.

þ xTcl Phr ðxcl Þo0. The above inequality can be written as # " T #" xcl ^D P ^ P þ PA A T T D ½xcl hr ðxcl Þ o0, hr ðxcl Þ % 0

(23)

and the nonlinear quadratic bound in (6) or (8) is equivalent to # " #" xcl a2 HTl Hl 0 T ½xTcl hr ðxcl Þ X0. (24) hr ðxcl Þ 0 I Combination of (23) and (24) according to the S-procedure (Boyd et al., 1994) gives P40 and # " T #" xcl ^ D þ ta2 HT H ^ P þ PA P A T l l ½xTcl hr ðxcl Þ D o0. (25) hr ðxcl Þ % tI The parameter t allows control engineers to combine several quadratic inequalities into a single inequality. Since, tIo0, from (25), the conditions for stability are P40 and " T # ^ D þ ta2 HT H ^ P þ PA P A l l D o0. (26) % tI Pre- and post-multiplying P40 by tP1 and tP1 , respectively, and (26) by diagðtP1 ; IÞ and diagðtP1 ; IÞ, respectively, the new

ARTICLE IN PRESS 110

A. Swarnakar et al. / Control Engineering Practice 17 (2009) 97–111

where M ¼ WY1 . written as

relations are " tP1 :P:tP1 40;

^TP tP1 ðA D

^ D þ ta2 HT H ÞtP1 þ PA l l

#

tI tI

%

o0.

Since, t is a positive scalar, defining Y ¼ tP1 , the conditions are Y40 and (27)

2 6 6 6 6 6 6 6 6 6 6 4

This LMI cannot be used to compute the controller parameters because it is not affine in Kd . Therefore, using the Schur’s complement, (27) can be written as ^ T þ 1 YHT H Y þ Io0, ^ D Y þ YA A l l D g

Iþ

T 1 g YHl Hl Y

T

þ X  ðW þ W Þ

#

^ T þ WT YA D

o0,

(28)

X

%

Y1 Y1 þ 1g HTl Hl þ Y1 ðI  W  WT ÞY1

^ T þ Y1 WT A D

%

I

# o0.

This can be expanded as 2

1 T H H 4g l l

3

^ T þ Y1 WT A D

þ Y1 ðI  W  WT ÞY1

"

5

Y1

I

%

½Y1

# ðIÞ1

0

0o0.

T Since, the inequality is in the form of U11  U12 U1 22 U12 o0, application of the Schur’s complement method gives 2 3 1 T ^ T þ Y1 WT Y1 H H þ Y1 ðI  W  WT ÞY1 A D 6g l l 7 6 7 6 7o0. % I 0 4 5 % % I

Again, expanding 2 1 T 1 6 Y ðI  W  W ÞY 6 6 % 4

^ T þ Y1 WT A D I

%

½Hl

0

%

3 2 3 HTl Y1 7 6 7 7 6 1  0 7 0 7 5ðgIÞ 5 4 0 I

0o0,

and applying the Schur’s complement method 2 6 6 6 6 6 6 4

Y1 Y1  Y1 M  MT Y1

^ T þ MT A D

Y1

%

I

0

%

%

I

%

2 6 6 6 ¼6 6 6 4

ðY

1

%

T

 M ÞðY

%

HTl

3

7 7 0 7 7 7 0 7 5 gI

^ T þ MT A D

Y1

%

I

0

%

%

I

%

%

%

1

^ T þ MT A D

Y1

%

I

0

%

%

I

%

%

%

h  Y1  M

T

 MÞ  M M

2

T 6 M M 6 % 6 6 6 % 6 6 % 4

0

^ T þ MT A D

%

where X can be any given positive definite matrix and W is a decision variable. Hence, selecting X ¼ I and pre- and postmultiplying by diagðY1 ; IÞ and diagðY1 ; IÞ, respectively, "

MT M

0

HTl

the

above

inequality

can

be

3

3 7 2 1 Y  MT 7 0 7 6 7 7 6 7 7 0 7 0 76 6 7ðIÞ1 7 6 7 7 4 0 5 7 7 5 0 gI

i 0 o0,

which gives

where g ¼ 1=a2 . It should be noted that the inequality is in the form of W þ S þ ST o0, with W ¼ I þ 1g YHTl Hl Y. Therefore, application of the reciprocal projection lemma gives "

Finally,

HTl

3

7 7 0 7 7o0, 7 0 7 5 gI

Y1

HTl

I

0

0

%

I

%

%

0 gI

%

%

%

3 Y1  MT 7 7 0 7 7 7o0. 0 7 7 0 5 I T

(29)

T

T

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