ARTICLE IN PRESS. Control Engineering Practice

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1 Control Engineering Practice (9) 9 Contents lists available at ScienceDirect Control Engineering Practice journal homepage: 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 TG V article info Article history: Received March Accepted May 8 Available online July 8 Keywords: LMI applications Overlapping control Nonlinear systems Industrial utility boiler Power systems Control applications abstract 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. & 8 Elsevier Ltd. All rights reserved.. 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, ). For these reasons the last few decades have seen increasing research interest in the design of multi-loop control systems (Siljak, 99; Siljak, Stipanovic, & Zecevic, ; Siljak & Zecevic, ; Swarnakar, Marquez, & Chen, ). However, the design algorithms of decentralized structures are relatively complex due to nonconvexity (Rotkowitz & Lall, ). It is well known that if some state information is shared among subsystems, the concept of overlapping control arises (Siljak, 99; Siljak & Zecevic, ; Zecevic & Siljak, ). 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 Corresponding author. Tel.: ; fax: addresses: adarsha@ece.ualberta.ca (A. Swarnakar), marquez@ece.ualberta.ca (H.J. Marquez), tchen@ece.ualberta.ca (T. Chen). control systems (Benlatreche, Knittel, & Ostertag, 8; Siljak & Zecevic, ; Stankovic, Chen, Matausek, & Siljak, 999; Stankovic, Stanojevic, & Siljak, ; Zecevic & Siljak, ). 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, 98; Siljak, 99; Stankovic & Siljak, ). 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 ða Þ is unstable (Zecevic & Siljak, ). Eigenvalues of this subsystem represent fixed modes of the expansion space and limit the practical appeal of this algorithm. In Zecevic and Siljak () and Siljak and Zecevic (), 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. ), where the input matrix and the control law have the following forms: B " B ¼ B B ; K ¼ K # K, () K K B 9-/$ - see front matter & 8 Elsevier Ltd. All rights reserved. doi:./j.conengprac.8..9

2 98 A. Swarnakar et al. / Control Engineering Practice (9) 9 Fig.. Type I and Type II overlapping (Zecevic & Siljak, ). where Type II corresponds to B ¼ B ¼. The method blends linear matrix inequalities (LMIs) (Boyd, Ghaoui, Feron, & Balakrishnan, 99) 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, 99), and the reciprocal variant of the projection lemma (Apkarian, Tuan, & Bernussou, ) 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: () 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 M T M. This constraint is then relaxed as: Q M T X, () % 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 M T M corresponds to the boundary of the convex sets in (). 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, ). This makes the results general and increases the applicability of the algorithm to a number of practical systems. () 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 (), 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 (.,.,.8, and. MPa). In most cases, decentralized PI controllers of the present plant work well; however, the. 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. 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. A solution to Type I and Type II overlapping control is given in Section, 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

3 A. Swarnakar et al. / Control Engineering Practice (9) 9 99 and to overlapping control laws for nonlinear interconnected systems are noteworthy. In Section, a sequential two-part optimization procedure to solve control problems in the Type II framework is introduced. Section deals with the application of the proposed design strategy to a two-area power system and to an industrial utility boiler. Finally, Section concludes the paper.. Instrumental tools Throughout this paper, the following instrumental tools are used. Lemma. (Reciprocal projection lemma: Apkarian et al., ). Let X be any given positive-definite matrix. The following statements are equivalent: () W þ S þ S T o. () The LMI problem W þ X ðwþw T Þ S T þ W T o % X 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, it will be shown that this additional variable forms a cornerstone in the design of overlapping controller. Lemma. (Schur s complement method: Boyd et al., 99). For a negative definite matrix Uo, the following two statements are equivalent: h i () U ¼ U % U U o. () U o; U U U UT o.. A solution to overlapping control design Consider a nonlinear process of the form (Siljak et al., ; Siljak & Zecevic, ; Swarnakar et al., ; Zecevic & Siljak, ) _x ¼ Ax þ Bu þ hðxþ; y ¼ Cx, () where x R n is the state, u R m is the control input, y R p is the output, C ¼ diag ðc ; C ; C Þ, and A A A B A ¼ A A A ; B ¼ B B. A A A B Here, B ¼ B ¼ 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., ; Siljak & Zecevic, ; Swarnakar et al., ; Zecevic & Siljak, ) h T ðxþhðxþpa x T H T 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, 8) can be found in power systems, power converters, etc. Consider a static output feedback overlapping controller of the form u ¼ Ky (K in ()), and a dynamic output feedback overlapping control law _x k ¼ A k x k þ B k C x þ B k C x, u ¼ C k x k þ D k C x þ D k C x, _x k ¼ A k x k þ B k C x þ B k C x, u ¼ C k x k þ D k C x þ D k C x. () In both cases, the closed loop system is given by _x cl ¼ðA þ BK d CÞx cl þ h r ðx cl Þ¼ ^A D x cl þ h r ðx cl Þ, () where for static output feedback: A ¼ A, B ¼ B, K d ¼ K, C ¼ C, x cl ¼ x, and h r ðx cl Þ¼hðxÞ is bounded by h T r ðx clþh r ðx cl Þpa x T cl HT Hx cl ¼ a x T cl HT l H lx cl. () However, for the case of dynamic output feedback with x T cl ¼½xT xt xt xt k x T k Š, A A A B A A A B B ^A D ¼ A A A þ B I I I A k B k B k I A k B k B k C C k D k D k C C k D k D k C and ¼ A þ BK d C, () H T H h T r ðx clþh r ðx cl Þpa x T cl x cl ¼ a x T cl HT l H lx cl. (8) In the following theorem, sufficient conditions are provided for the existence of a stabilizing overlapping control law for the nonlinear system in (). 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 (). Theorem.. If there exists a controller K d such that the following optimization is feasible min g subject to X P, (9) Q A T þ C T K T B T d þ M T X P H T l X P M T % I % % I o, () % % % gi % % % % I Q ¼ M T M, then the system in () is asymptotically stable for nonlinearities satisfying the quadratic constraint in () or (8). Proof. Please see Appendix A. Corroborated by many simulations, it has been found that by replacing X with ri (where r is a tuning parameter) instead of I gives faster convergence speed. This is because, the inequality in

4 A. Swarnakar et al. / Control Engineering Practice (9) 9 Fig.. Physical meaning of parameters r and z. (8) is then equivalent to Xo, I þ g YHT l H ly þ ri ðw þ W T Þo, () I þ g YHT l H ly þ ri ðw þ W T Þþ r ðy ^A T D þ WT Þð ^A D Y þ WÞo. () It is clear that () is negative definite and r ðy ^A T D þ WT Þð ^A D Y þ WÞ is positive definite (or positive semi-definite). Therefore, in most cases, () can be easily satisfied by increasing r, since it decreases the positive contribution of ½ r ðy ^A T D þ WT Þð ^A D Y þ WÞŠ (Fig. ), and this decrease is much more pronounced compared to the decrease of negative definiteness in I þ g YHT l H ly þ ri ðwþw T Þ. This makes the optimization problem feasible and leads to faster convergence speed. The equality constraint can also be reduced to the form: Q ¼ðX P þ MÞ T ðx P þ MÞ, which is useful in some applications due to extra degree of freedom provided by X P. A linear system is a special case of () with hðxþ ¼, hence, the optimization problem becomes X P ; Q ¼ M T M; Q A T þ C T K T d B T þ M T X P M T % I % % I o. In Theorem., the positive definite matrix X P (which has no structural constraint) is decoupled from the controller K d. This is an important advantage, because other structures of K d (decentralized design, control design when x in Fig. is reachable from u or u only, etc.) can be assigned. Moreover, it is possible to design reduced order dynamic controllers, because different orders of A k and A k in () can be imposed. In some cases, Theorem. yields a controller with fast dynamics. Therefore, additional pole placement constraints (Chilali & Gahinet, 99) should be added. Remark.. Consider an interconnected system of the form _x ¼ Ax þ Bu þ h int ðxþ; y ¼ Cx, () where h int ðxþ ¼½h T ðxþ ht ðxþ ht ðxþšt. The functions h i ðxþ (for i ¼ ; ; ) contain the nonlinearities in subsystems and in the interconnections. They are assumed to be bounded by a quadratic inequality (Siljak et al., ; Swarnakar et al., ) h T i ðxþh iðxþpa i xt H T i H ix; i ¼ ; ;, () where a i s are interconnection parameters. Here, with static output feedback control law! h T int ðx X clþh int ðx cl Þpx T cl a i HT i H X i x cl ¼ x T cl a i HT i l H il!x cl, i¼ i¼ and with dynamic output feedback control law H T h T i H i i ðx clþh i ðx cl Þpa i xt cl x cl. fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} H T i l H il For large-scale interconnected systems, where the nonlinearities in the ith subsystem satisfy the constraint (Siljak et al., ; Swarnakar et al., ) h T i ðt; xþh iðt; xþpa i xt H T i H ix; i ¼ ; ; ;...; N () the following optimization problem should be solved for the controller parameters min g þ g þþg N subject to X P, Q A T þ C T K T B T d þ M T X P H T l H T N l X P M T % I % % I % % % g I o,.. % % % %... % % % % % g N I % % % % % % I Q ¼ M T M. Hence, the generalization of the result here is very straightforward. Since, () is a special case of (), the optimization follows along the same lines. Remark.. For solving the optimization problem using the available numerical software, a key idea is to relax the equality constraint as Q M T X, () % I and then apply a cone complementary linearization (CCL) algorithm (Ghaoui, Oustry, & Rami, 99) for computing the controller parameters. The following algorithm shows a modified version of the CCL method. Computational method: As the equality constraint Q ¼ M T M corresponds to the boundary of the convex set in (), let H b 9fX P ; K d ; Q ; M; (9), () and () are satisfiedg. Here, H b is a closed and convex set.

5 A. Swarnakar et al. / Control Engineering Practice (9) 9 Algorithm OC (overlapping control): () Find the feasible set ðx P ; K d ; Q ; M ÞH b. Let k :¼. () Solve the following convex optimization problem for the variables ðx P ; K d ; Q; MÞ H b : min trace½q ðm k Þ T M M T M k Š Hb subject to (9), () and (). () Substitute the values of ðx P ; K d ; MÞ in (9). If the condition is satisfied then output the feasible solutions ðx P ; K d ; Q; MÞ. EXIT. () Set k ¼ k þ, ðx k P ; Kk d ; Q k ; M k Þ¼ðX P ; K d ; Q ; MÞ, and go to step. This remark shows that each step in the algorithm entails solving LMIs and an arbitrary initial guess is not required. Remark.. It is important to note that the optimization k problem in the kth step, ~ J % ¼ min ~ J k ¼ min trace½q k þ Q ðm k Þ T M M T M k Š, subject to (9), (), () and the step 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. of Tao & Zhao, ), it can be easily derived that ~ J k% X and the sequence f ~ J % ; ~ 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 (z, say), and to stop the iterative algorithm to output the feasible solution if ½Q k ðm k Þ T M k ŠozI. This is due to the fact that the equality constraint is obtained at the boundary of (), 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 () are satisfied (Fig. ) under this situation. Remark.. It is apparent that the conditions in Theorem. are not convex owing to a matrix equality Q ¼ M T 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 () in Theorem. is replaced by ðm þ M T ÞþI A T þ C T K T B T d þ M T X P H T l X P M T % I % % I o, % % % gi % % % % I based on ðm IÞ T ðm IÞX (which yields M T Mp ðm þ M T ÞþI), then the resulting controller synthesis problem can be reduced to a convex one. Remark.. 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 used, because the optimization problem involves ^A D which is affine in controller parameters ^A D ¼ A þ BK d C. Therefore, different structures for K d and B can be assigned (different control laws and different overlapping). For a three-area power system, K d has the structure K K K K d ¼ K K K. K K K Example.. Consider the system in Zecevic and Siljak (): _x ¼ x þ u. Here, the open loop system has eigenvalues at l ¼ :, :8 þ j:8 and :8 j:8, and the goal is to stabilize the system with an overlapping control law. In Zecevic and Siljak (), system expansion and LMIs : :8 K ¼. : : Using the Algorithm OC, the stabilizing static controller in () and the dynamic controller in () are given by : :99 K static ¼, :9 : 9:99 :9 :8 9:8 : : K dynamic ¼. 9:999 : :8 :8 The number of iterations is for the static controller and for the dynamic controller. The optimum value of the objective function is ~ J ¼.8e- for static control, and the Q and M matrices are given by :9 8: : :8 :88 : Q ¼ 8: : : ; M ¼ :9 : :8, : : :9 :8 :88 :8 which satisfy Q ¼ M T M.. 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 " F ¼ F # F o, () % F where F and F are affine in controller parameters and bilinear terms appear in F, then () Solve the feasibility problem F o to calculate some of the controller parameters. () Define n ¼ diagðx I; x I;...Þ. Substitute the variables from step and solve the optimization (Zhu & Pagilla, ) " min x þ x þ; subject to F x ¼ F # F o, % nf 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 (), using different transformations, simplifications, and new variable definitions. It helps to utilize the two-step approach.

6 A. Swarnakar et al. / Control Engineering Practice (9) 9.. Dynamic output feedback overlapping control design Consider the nonlinear interconnected system in (), where a dynamic output feedback overlapping controller has to be designed. With the control law in (), the closed loop system is given by _x A þ B D k C B C k A r þ B D kr C r _x k B k C A k B kr C r _x ¼ _x cl ¼ x A r A r þ B r D kr C r B r C k cl _x B kr C r A k fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} _x k I h ðxþ þ I h ðxþ. I h ðxþ fflfflfflfflffl{zfflfflfflfflffl} h intðxþ fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} G Here, h i A r ¼ A A ; Bkr ¼ B k ; C r ¼ diagðc ; Þ, h i D kr ¼ D k D k ; C r ¼ diagðc ; C Þ; D kr ¼½D k Š, A A B kr ¼½B k B k Š; A r ¼ ; B r ¼, A A B A A r ¼. A After using a quadratic Lyapunov function, different congruence transformations, and change of controller variables (Chilali & Gahinet, 99), the asymptotic stability conditions for the closed loop system are given by the following optimization problem: min g þ g þ g! X I X I subject to diag ;, % Y % Y F F G T l T l T l % F G T l T l T l % I o, % % g I % % % g I % % % % g I where ^AD The terms h, h r, h r are the elements of the matrix H T l. Similar expressions exist for T l, T l, T l and T l in terms of the elements in H T l and H T l, respectively, where In (8), ^A ¼ Y A X þ ^B C X þ Y B C k M T þ N A k M T, ^A ¼ Y A r X þ ^B C r X þ Y B r C k M T þ N A k M T, ^B ¼ Y B D k þ N B k ; ^B ¼ Y B r D kr þ N B kr, ^C ¼ D k C X þ C k M T ; ^C ¼ D kr C r X þ C k M T, ^B k ¼ Y B D kr þ N B kr ; ^D ¼ D k ; ^Dk ¼ D kr ; ^D ¼ D kr. (9) According to the two-step method, the following steps should be used for computing the controller parameters. Step : Maximize the interconnection bounds a ;...; a ða i ¼ =g i Þ by solving the optimization problem min g þ g þ g X I subject to, % Y F ½ G Š T l T l T l % I F ¼ % % g I o. % % % g I % % % % g I This gives X, Y, ^A, ^B, ^C, and ^D. Now, compute N, M square, and invertible from N M T ¼ U S V T ¼ svdði Y X Þ, which provides N ¼ U S = and M ¼ V S =. Next, the controller parameters are calculated from C k ¼ð^C D kr C r X ÞðM T Þ ; B kr ¼ N ð ^B Y B r D kr Þ, A k ¼ N ð ^A Y A r X ^B C r X Y B r C k M T ÞðMT Þ, D kr ¼ ^D. Step : Define the tuning parameter n ¼ diagðdiagðx I; x IÞ; x I; x I; x I; x IÞ. Using X, Y and other parameters from step solve min x þ x þ x þ x X I subject to ; n; % Y F F o, % nf (8)

7 A. Swarnakar et al. / Control Engineering Practice (9) 9 where F ¼½F ½G Š T l T l T l Š. This gives X, Y, ^A, ^B, ^C, ^D, ^Bk and ^D k. Hence, the rest of the controller parameters in (9) can be computed in a similar fashion using N M T ¼ svdði Y X Þ.. Applications to a two-area power system and an industrial utility boiler The algorithms developed in Sections and are applied to a power system and utility boilers... Two-area power system In the two-area power system shown in Fig., the numerical values of the parameters are obtained from (Yang, Ding, & Yu, ). The areas represent the subsystems and the tie lines are the overlapping parts. Therefore, the controllers can be designed to share the overlapping state ðdp tie Þ to improve the performance of the overall system. The controllers minimize the system frequency deviations Df in area as well as Df in area under the influence of load disturbances P D and P D 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 (static and dynamic) decentralized and overlapping controllers are designed. They have the following form: :8 :98 :8 ;, :8 :8 :98 : :9 :88 : :88 :9. :9 :99 :99 :9 From Figs. and, 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 u S Area which reveals Type II overlapping (Fig. ). In Figs. and, the frequency deviations in the two areas due to the disturbance of DP D ¼ : pu in area are shown (by dotted lines, without controller). This and the Nyquist array with the column Gershgorin circles (Fig., first row) on the diagonal element show that the system is highly interacting. Gershgorin circles for the first subsystem (g and g ) only are drawn because the transfer functions of both subsystems are the same. The responses in Figs. and also show that local controllers should be designed to minimize the oscillations. By using the algorithm OC for a linear system in Section, output feedback ΔP tie u S Area Fig.. Overlapping scenario of two-area power system. B R ΔP D + + ΔP - ΔP T - K ΔP G c K P s + st G st T + st + - P ΔP Integral part u Governor Turbine Generator ΔP tie T s + - Δf a u a ΔP - + ΔP + K c ΔP G ΔP T K P s + st G st T st P Integral part Governor Turbine Generator ΔP D B R Δf Fig.. Two-area power system (Yang et al., ).

8 A. Swarnakar et al. / Control Engineering Practice (9) 9. g g. IMAG IMAG Δ f [Hertz] Δ f [Hertz] without controller with static output feedback overlapping controller with static output feedback decentralized controller first order dynamic output feedback overlapping controller. Fig.. Frequency deviation of the first area with output feedback controllers.. without controller. with static output feedback overlapping controller with static output feedback decentralized controller first order dynamic output feedback overlapping controller. Fig.. Frequency deviation of the second area with output feedback controllers. IMAG IMAG IMAG IMAG.. Fig.. 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). Table J fre values with static and first order dynamic overlapping controller No control Static decentralized Static overlapping Dynamic overlapping x Control signals overlapping control decentralized control used (Yang et al., ): u J fre ¼ Z tjdf ðtþjdt. Table 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. 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., 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, u x 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

9 A. Swarnakar et al. / Control Engineering Practice (9) 9 Δ f [Hertz] Δ f [Hertz]..... state feedback overlapping controller state feedback decentralized controller overlapping controller using two step approach. Fig. 9. Frequency deviation of the first area with state feedback controllers state feedback overlapping controller state feedback decentralized controller overlapping controller using two step approach.8 Fig.. Frequency deviation of the second area with state feedback controllers. approach are shown in Figs. 9 and. 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... Industrial utility boiler The effectiveness of the design strategies in Sections and can be demonstrated through application to an industrial utility boiler. As shown in Fig., a part of the interconnected system at Syncrude consists of utility boilers (UB UB ), CO-type boilers (C.O. and C.O.), and once through steam generators (OTSG and OTSG) developed by Innovative Technologies, Ontario. Steam from different boilers is collected in the 9# (9 pounds or. MPa) header which is then used in: () cokers for extracting bitumen from oil sands, () other low pressure headers (#, #, and #) for extraction, upgrading (conversion of bitumen heavy oil into lighter components like naphtha and diesel oil), building heating, etc., and () turbines (G, G, G, and G) 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 G and G. Due to characteristics of different boilers (time constants and usage of different fuels), utility boilers maintain the 9# 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- system consisting of two CO boilers (CO and CO) 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 9# header and to maintain the drum level and the steam temperature at their set points ( m and C, respectively). UE- system Coker-off Natural gas spray Feedwater gas spray Plt. - CO-C 9/ UB - OTSG - C.O.-C.O. # 9 # 8 kpph 9 kpph kpph 9 # G G G G G G MW MW MW MW MW MW / Hot gases # Condenser # / / / Tumblers Trim heaters Deaerator Fig.. A part of interconnected system at Syncrude.

10 A. Swarnakar et al. / Control Engineering Practice (9) 9 To this end, a nonlinear model of the utility boiler and the 9# 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 ðq s Þ is a function of the pressure drop from the drum boiler to the 9# header (Bernoulli s law), as shown in Fig.. This can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q s ¼ k s y drum y ¼ y header drum y, () header where y header and y drum are header and drum pressures (in kpa), respectively. The master control block in Fig. 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 ðu Þ to the fuel flow ðx Þ, which is governed by _x ¼ x þ :8u ; _x ¼ :8x :x :u. The air flow usually bears a constant relationship with the fuel flow, i.e., air flow ¼ 8:8x ; hence, it is not separately identified. Similarly, the differential equation for the steam flow rate ðx Þ is given by _x ¼ x þ :9u þ :988x þ :9u, _x ¼ :8x :x :98u þ :x :88u, where u represents the feedwater flow rate and u is the attemperator spray flow rate (in kg/s). The intermediate variables x and x 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 (99, 98a,b) and Pellegrinetti and Bentsman (99) were used: _x ¼ u x ¼ u x V T :, q e ¼ þ K ½k be f þ ru Šþ K þ K x, y level ¼ ½v w V T x þ k a r þ T s q A e Š. d Here, x 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 : m 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, e f describes the energy flow rate (which depends linearly on the fuel flow x and can be obtained from measured data), A d is drum area, v w is the specific volume of water, T s is the increase in water volume per unit increase in evaporation rate, k b ¼ =h fg and r ¼ðh w h f Þ=h fg. Here, h fg is the latent heat of evaporation, h f is the enthalpy of saturated water, and h w is the enthalpy of feedwater. After some calculations and substituting values from the steam table at a saturation pressure, the value of q e is obtained. The expression for the quality of the steam ða r Þ in the system, based on volume, is determined by curve fitting with the data from SYNSIM (which gives a relation in terms of x, x, and u ). 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 Dy level ¼ :8x þ :9x þ :x :u w :, where an additional feedthrough term u w ¼ u is added to obtain a good match between the model and the process data. The numerical value : accounts for slight disagreement in the operating point of the drum level equation. Following the models of the pressure equation in Bell and Astrom (99, 98a,b) and incorporating some heuristic knowledge of boiler behavior (to accommodate the data from SYNSIM), the effect of the fuel flow rate ðx Þ and the feedwater flow rate ðu Þ on the drum pressure is formulated as _x ¼ r x þðr x þ r Þc v þ r x r u, () where c v is referred to as an imaginary control valve position from qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi drum to header c v ¼ ðy header =y drum Þ. It should be noted that () does not consider the effects of attemperator spray flow rate ðu Þ on drum pressure. Therefore, an identification test was carried out that gives _x ¼ :8x þ :u. It is clear that y drum ¼ x þ x and the steam flow out of the boiler drum is related to the drum pressure by q s ¼ x ¼ c v y drum (from ()), which yields c v ¼ x =ðx þ x Þ. The parameters r r 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 " ( ) # _x ¼ :x þ :8 x þ : x :9u ðx þ x Þ ( ) : x þ :99, x þ x _x ¼ :8x þ :u ; y drum ¼ x þ x. () The constant :99 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 r, symbolic linearization or nonlinear regression techniques can also be used. From () and (), the 9# header pressure can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y header ¼ x þ x þ x x x. 9 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. and the fitness in other operating regions is also good. The model for the steam Feedwater flow Steam flow Firing rate Master Control Fuel flow Air flow Drum-boiler and Superheaters Drum level Drum pressure Header node 9 # header pressure Attemperator spray flow Steam temperature Fig.. Modeling of the utility boiler and the header.

11 A. Swarnakar et al. / Control Engineering Practice (9) 9 temperature dynamics is governed by _x 8 ¼ x 9 :u þ :x þ :9u, _x 9 ¼ x þ :u :888x :89u, _x ¼ x :99u þ :898x þ :u, _x ¼ : x 8 :x 9 :x :89x þ :9u :x :8u, y steam ¼ x 8. 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 :9, which reveals non-minimum phase characteristics. Next, based on the algorithm of Section, the following stabilizing overlapping controllers were designed: :98s :9 s þ : K full ¼ K partial ¼ :8s þ : s þ : :s þ : :s 9: s þ : s þ :, :9s : s þ :98 :8s þ :8 :s 9: s þ : s þ :. :9s : s þ :9 :9s : s þ : 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 Steam flow [kg/sec] Measured Output and Simulated Model Output Measured Output model Fit: 98.% x Fig.. Validation of the steam flow. Fuel flow [kg/sec] Firing rate Drum level [m] Drum pressure [kpa]. Model and process output process model x... Steam temperature [C] x.8 Fig.. Validation of the firing rate. Model validation. process nonlinear model. 8 Fig.. Validation of the model. 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 G G (Fig. ). Fig. shows the stabilizing effect of the overlapping controllers revealing good regulation under high load conditions. Figs. 8 and 9 show responses caused by a load change in the # ( lb) header and Figs. and show responses of different process variables due to a load change of kpph on the 9# header. In both cases, plots of total steam flow rate, 9# steam temperature, and # 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 # header steam pressure shown in Fig. could lead to an enhancement in power production, because the # header pressure is the back pressure of the turbines (G, G, and G). There are slight deviations in

12 8 A. Swarnakar et al. / Control Engineering Practice (9) 9 drum level (%) and header pressure (.9% 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 9# header steam pressure, which shows no oscillations, the main concern in practice. Fig. 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 K partial, 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 plant. Since this controller is linear and of only third order, it is simple to implement and test in practice.. Conclusions 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 Feedwater flow [kg/sec] Process inputs Steam flow [kg/sec] % load change in pound header Fuel flow [kg/sec].. 9# steam temperature [ C] Spray flow [kg/sec].. # header pressure full overlapping partial overlapping PI controller Fig.. Inputs to the process. Fig. 8. Sudden load change in the # header. Drum level [m] Header pressure [kpa]. full overlapping.999 partial overlapping.998 Steam temperature [ C] Fig.. Performance of the overlapping controllers under high load conditions.

13 A. Swarnakar et al. / Control Engineering Practice (9) 9 9 Drum level [m] Header pressure [kpa] % load change in pound header. full overlapping partial overlapping PI controller.98 9 Drum level [m] Header pressure [kpa]. Load change of kpph partial overlapping full overlapping PI controller.9 8 Steam temperature [ C] Steam temperature [ C]. 99. Fig. 9. Responses during load change in the # header. Fig.. Header pressure response during sudden load change. Steam flow [kg/sec] 9# steam temperature [ C] # header pressure Load change of kpph partial overlapping full overlapping PI controller.9 Fig.. 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. Acknowledgments 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. Appendix A Proof of Theorem.. Consider a quadratic Lyapunov function v ¼ x T cl Px cl. The sufficient conditions for stability of the closed loop system can be expressed as P, and _v ¼ _x T cl Px cl þ x T cl P_x cl ¼ x T cl ^A T D Px cl þ h T r ðx clþpx cl þ x T cl P ^A D x cl þ x T cl Ph rðx cl Þo. The above inequality can be written as " ½x T cl h T r ðx ^A T D clþš P þ P ^A # D P x cl o, () h r ðx cl Þ % and the nonlinear quadratic bound in () or (8) is equivalent to ½x T cl h T r ðx a H T l clþš l x cl X. I h r ðx cl Þ () Combination of () and () according to the S-procedure (Boyd et al., 99) gives P and ½x T cl " h T r ðx ^A T D clþš # D þ ta H T l H l P x cl o. % ti h r ðx cl Þ () The parameter t allows control engineers to combine several quadratic inequalities into a single inequality. Since, tio, from (), the conditions for stability are P and ^A T D P þ P ^A D þ ta H T l H l P o. () % ti Pre- and post-multiplying P by tp and tp, respectively, and () by diagðtp ; IÞ and diagðtp ; IÞ, respectively, the new

14 A. Swarnakar et al. / Control Engineering Practice (9) 9 relations are tp :P:tP ; tp ð ^A T D P þ P ^A D þ ta H T l H lþtp ti o. % ti Since, t is a positive scalar, defining Y ¼ tp, the conditions are Y and () This LMI cannot be used to compute the controller parameters because it is not affine in K d. Therefore, using the Schur s complement, () can be written as ^A D Y þ Y ^A T D þ g YHT l H ly þ Io, where g ¼ =a. It should be noted that the inequality is in the form of W þ S þ S T o, with W ¼ I þ g YHT l H ly. Therefore, application of the reciprocal projection lemma gives " I þ g YHT l H ly þ X ðwþw T Þ Y ^A T # D þ WT o, (8) % X 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 diagðy ; IÞ and diagðy ; IÞ, respectively, Y Y þ g HT l H l þ Y ði W W T ÞY ^AT D þ Y W T o. % I This can be expanded as g HT l H l þ Y ði W W T ÞY ^AT D þ Y W T Y ð IÞ % I ½Y Šo. Since, the inequality is in the form of U U U UT o, application of the Schur s complement method gives g HT l H l þ Y ði W W T ÞY ^AT D þ Y W T Y % I o. % % I Again, expanding Y ði W W T ÞY ^AT D þ Y W T Y H T l % I ð giþ ½H l Šo, % % I and applying the Schur s complement method Y Y Y M M T Y ^AT D þ MT Y H T l % I % % I % % % gi ðy M T ÞðY MÞ M T M ^A T D þ MT Y H T l % I ¼ o, % % I % % % gi where M ¼ WY. Finally, the above inequality can be written as M T M ^A T D þ MT Y H T l Y % I M T % % I ð IÞ % % % gi h i Y M o, which gives M T M ^A T D þ MT Y H T l Y M T % I % % I o. (9) % % % gi % % % % I Substituting Q ¼ M T M, X P ¼ Y and ^A T D ¼ AT þ C T K T B T d, the result follows. & References Apkarian, P., Tuan, H. D., & Bernussou, J. (). Continuous-time analysis, eigenstructure assignment, and H synthesis with enhanced linear matrix inequalities (LMI) characterizations. IEEE Transactions on Automatic Control,, 9 9. Bell, R. D., & Astrom, K. J. (99). A low-order nonlinear dynamic model for drum boiler-turbine-alternator units. Technical Report TFRT-, Lund Institute of Technology (pp. 9). Bell, R. D., & Astrom, K. J. (98a). Dynamic models for boiler turbine alternator units: Data logs and parameter estimation for a MW unit. Technical Report TFRT-9, Lund Institute of Technology (pp. ). Bell, R. D., & Astrom, K. J. (98b). Simplified models for boiler turbine units. Technical Report TFRT-9, Lund Institute of Technology (pp. ). Benlatreche, A., Knittel, D., & Ostertag, E. (8). Robust decenteralized control strategies for large-scale web handling systems. Control Engineering Practice, (),. Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (99). LMIs in system and control theory. Philadelphia: SIAM. Chilali, M., & Gahinet, P. (99). H design with pole placement constraints: An LMI approach. IEEE Transactions on Automatic Control, (), 8. Ghaoui, L. E., Oustry, F., & Rami, M. A. (99). A cone complementary linearization algorithm for static output feedback and related problems. IEEE Transactions on Automatic Control,,. Iannelli, L., Johansson, K. H., Joensson, U. T., & Vasca, F. (8). Subtleties in the averaging of a class of hybrid systems with applications to power converters. Control Engineering Practice,, 9 9. Ikeda, M., Siljak, D. D., & White, D. E. (98). An inclusion principle for dynamic systems. IEEE Transactions on Automatic Control,,. Pellegrinetti, G., & Bentsman, J. (99). Nonlinear control oriented modeling a benchmark problem for controller design. IEEE Transactions on Control Systems Technology, (),. Rotkowitz, M., & Lall, S. (). A characterization of convex problems in decentralized control. IEEE Transactions on Automatic Control,, 8. Siljak, D. D. (99). Decentralized control of complex systems. Boston: Academic Publisher. Siljak, D. D., Stipanovic, D. M., & Zecevic, A. I. (). Robust decentralized turbine/ governor control using linear matrix inequalities. IEEE Transactions on Power Systems, (),. Siljak, D. D., & Zecevic, A. I. (). Control of large-scale systems: Beyond decentralized feedback. Annual Reviews in Control, 9, 9 9. Stankovic, S. S., Chen, X. B., Matausek, M. R., & Siljak, D. D. (999). Stochastic inclusion principle applied to decentralized automatic generation control. International Journal of Control,, 88. Stankovic, S. S., & Siljak, D. D. (). Contractibility of overlapping decentralized control. Systems and Control Letters,, Stankovic, S. S., Stanojevic, M. J., & Siljak, D. D. (). Decentralized overlapping control of a platoon of vehicles. IEEE Transactions on Control System Technology, 8, 8 8.