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1 This article was downloaded by: [EBSCOHost EJS Content Distribution] On: 0 June 009 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, -41 Mortimer Street, London W1T JH, UK International Journal of Control Publication details, including instructions for authors and subscription information: A double homotopy method for decentralised control design Charudatta S. Mehendale a ; Karolos M. Grigoriadis a a Department of Mechanical Engineering, University of Houston, Houston, TX 04 First Published:October008 To cite this Article Mehendale, Charudatta S. and Grigoriadis, Karolos M.(008)'A double homotopy method for decentralised control design',international Journal of Control,81:10, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 International Journal of Control Vol. 81, No. 10, October 008, A double homotopy method for decentralised control design Charudatta S. Mehendale* and Karolos M. Grigoriadis Department of Mechanical Engineering, University of Houston, Houston, TX 04 (Received July 00; in final form 10 December 00) An iterative algorithm that is based on the idea of two homotopy paths is proposed for output feedback decentralised H 1 control design. The approach follows a bilinear matrix inequality (BMI) formulation of the decentralised control problem. Along one homotopy path the BMI problem, which is non-convex, is locally linearised and solved. Along the second homotopy path the controller is deformed at each step so that in the end it attains a decentralised structure. The proposed computational algorithm can also be applied to obtain reducedorder decentralised controllers. Numerical examples are used to demonstrate the efficacy of the proposed algorithm. Keywords: decentralised control; convex optimisation; homotopy; reduced order control 1. Introduction The analysis and design of decentralised control systems has been an actively researched area for the last 5 years and still attracts considerable attention (Sandell et al. 198; Siljak 1991). Decentralised control structures arise naturally when dealing with large scale systems such as, power systems, communication networks and space structures (Siljak 1991; Michel and Miller 19). A complex large scale system can be represented as the interconnection of smaller interacting subsystems. The goal of decentralised control is to design for each subsystem a local controller which uses local measurements, to generate a control action for that subsystem. The local objective is subsystem stability. The local controllers however need to satisfy global objectives such as the internal stability and performance of the interconnected system. Decentralised control design constraints also arise in applications that have controller implementation (hardware and software) constraints that limit the amount of information that can be handled by the control structure. Also, many plants have legacy control systems that do not support the implementation of complex multivariable controllers. A need to update the existing legacy control systems by employing powerful modern control design and analysis techniques leads to decentralised control problems. It should be noted that, one cannot always characterise the plant as an interconnection of subsystems. A survey of applications of decentralised control in power systems, communication networks, urban traffic networks, manufacturing networks, economic systems and space systems can be found in Sandell et al. (198). There exists a wide variety of design techniques for decentralised control and we will briefly review some relevant methods here. Earlier approaches for decentralised design were based on the idea of vector Lyapunov functions (Siljak 1991). Each subsystem is assigned a Lyapunov function chosen based on subsystem properties without taking into account interconnection terms. These are grouped into a vector function, which is then used to analyse the stability and performance of the overall system. The results are conservative and useful only when interconnection terms are small. In recent years, linear matrix inequalities (LMIs) have emerged as a powerful tool for solving a variety of control problems (Skelton et al. 1998). LMI formulations of control problems are also derived from a Lyapunov based approach. The advantage of posing control synthesis problems in terms of LMIs is that these are convex optimisation problems and can be solved efficiently using interior point methods (Boyd et al. 1994). Naturally, there has been an effort towards using LMI methods to solve the general decentralised control problem. However, the general output feedback decentralised control problem is a bilinear matrix inequality (BMI) non-convex problem (Shiau and Chow 1996; Tan and Grigoriadis 000). Further, BMI problems are known to be NP-hard (Toker and Ozbay 1995). *Corresponding author. mehendalec@yahoo.com ISSN print/issn online ß 008 Taylor & Francis DOI: /

3 International Journal of Control 1601 One important research direction is to use techniques from numerical optimisation to solve the decentralised control design problem. However, so far, there are no methods with guaranteed convergence results available in the literature. Geromel et al. (1994) proposed to solve the BMI associated with the decentralised control problem by over-constraining the problem to obtain a convex formulation. Tan and Grigoriadis (000) apply the alternating projection method for the resolution of the non-convex BMI problem. Scorletti and Duc (001) propose an algorithm for decentralized control design based on a result on stability of interconnection of operators (Moylan and Hill 198; Scorletti and El Ghaoui 1998). Design of dynamic output feedback decentralised controllers is investigated via a separation principle and a numerical cross decomposition algorithm by Oliveira et al. (000). However, as noted by the authors, working with a level constraint on the H 1 norm, the proposed method could fail for large values of, even though a decentralised controller which achieves smaller may exist. Another approach proposed to solve the BMI problem is by iteratively solving LMI problems. Shiau and Chow (1996) propose a heuristic solution of the BMI problem for the decentralised state feedback H 1 control problem, wherein a decentralised controller parametrisation is used. The idea is to first find a centralised controller (centralised refers to a controller with no structure) and then determine a parameter that decentralises the centralised controller. This idea is used, coupled with a homotopy approach by Zhai et al. (001), for the decentralised output feedback H 1 control problem. Here, the controller matrices are deformed from full matrices defined by a centralised H 1 controller, to block diagonal matrices which describe a decentralised control structure. First, the BMI is slightly perturbed in a desired fixed manner such that the inequality still holds. This is possible for small enough perturbations by continuity arguments. Next, the perturbed BMI is solved by fixing one of the matrix variables and solving the resulting projected LMI for the other matrix variable. The process is repeated, then, by fixing the second variable and solving the resulting LMI for the first variable; and so on. Hence, locally, the procedure acts like a coordinate descent method. The convergence of the algorithm is not guaranteed and depends on the initial centralised controller. A path-following (homotopy) method for locally solving BMI problems is proposed by Hassibi et al. (1999). The BMI is linearised using a first-order perturbation approximation. The resulting LMI is then solved to compute a perturbation that slightly improves performance. The perturbations have to be small enough so that the perturbed variables satisfy the original BMI. However, an initial solution is needed to start the iteration. It is stated that... given a reduced order, decentralised, or fixed architecture controller we could iteratively design for lower values of induced L norm (Hassibi et al. 1999). This method is used by Bao et al. (1999) for the design of robust multi-loop PID controllers for a distillation column. It is assumed that an initial decentralised controller is available and is then improved upon. Hence, for the output feedback decentralised control problem the path-following method cannot be applied directly. In this paper, an algorithm for output feedback decentralised H 1 control design using the idea of two homotopy paths is proposed. Along one homotopy path the controller matrices are deformed from a centralised structure to a block diagonal decentralised structure. Along the other path, designs are slightly improved (in the sense of making off diagonal blocks zero) by solving the linear approximation to the BMI problem. The proposed method is different from the approach followed by Zhai et al. (001), in that, the BMI is resolved by locally linearising, instead of treating the BMI as a double LMI by holding in turn the controller parameters and the Lyapunov matrix fixed. In general, the order of the decentralised controller is the same as the order of the starting centralised controller. In order to design reduced order decentralised controllers, one needs to either start with a reduced order centralised controller or choose a slightly different form for the deformation of the centralised controller matrices. The paper is organised as follows: x describes the output feedback decentralised H 1 control problem; x details the idea behind the proposed two homotopy paths and states the computational algorithm; this section also discusses computational aspects, compares the proposed approach with previous methods and provides comments about extensions to reduced order decentralised controller designs. The validity of the proposed algorithm is illustrated with design examples in x 4. Section 5 concludes the paper.. Problem statement Consider a nth order linear time-invariant system with the following state space representation 9 _x p ¼ A p x p þ B p u þ D p w >= y ¼ C p x p þ B y u þ D y w ð1þ >; z ¼ M p x p þ D z w; where x p (t) R n is the state vector, y(t) R e is the output of interest, z(t) R p is the measured signal,

4 160 C.S. Mehendale and K.M. Grigoriadis w(t) R q is the disturbance vector containing both process and measurement noise, and u(t) R r is the control input. The output feedback H 1 control problem is to design a feedback controller of order ^n with state space representation ) _x c ¼ A c x c þ B c z ðþ u ¼ C c x c þ D c z such that the closed loop system ) _x ¼ðAþBGM Þx þðdþbge Þw y ¼ðCþHGM Þx þðfþhge Þw is internally stable and the energy to energy gain kyk ee ¼ sup L wl f0gkwk L is minimised. Here, x ¼ 4 x T p x T T c denotes the augmented closed-loop state vector, and the augmented matrices are defined by A p 0 D p B p 0 A D B I ^n 4 C F H 5¼ 4 C p 0 D y B y 0 : M E G T 6 M p 0 D z D T c B T 4 c 5 0 I ^n 0 C T c A T c The output feedback decentralised H 1 control problem, is to design for the plant in (1) a family of dynamic controllers _x ci ) ¼ A ci x ci þ B ci z i u i ¼ C ci x ci þ D ci z i i ¼ 1,..., N, where x ci R ^n i is the state of the ith local controller, z i R p i is the partial measurement available to each local controller and u i R r i is the local control P input. We define z ¼½z T 1 ;...; zt N ŠT, u ¼½u T 1 ;...; ut N ŠT ; N i¼1 p i ¼ p; and P N i¼1 r i ¼ r: Then, the total order of all local controllers is P N i¼1 ^n i ¼ ^n: We collect the controller state and the coefficient matrices as x c ¼½x T c 1 ; x T c ;...; x T c N Š T A cd ¼ diag A c1 ; A c ;...; A cn B cd ¼ diag B c1 ; B c ;...; B cn C cd ¼ diag C c1 ; C c ;...; C cn D cd ¼ diag D c1 ; D c ;...; D cn ðþ ð4þ ð5þ ð6þ The closed-loop system with the decentralised controller is given by Equation () by simply replacing G by G d, where G d ¼ 4 D cd C cd : ðþ B cd A cd It is well known that the closed loop system () is internally stable and satisfies a bound on the energy to energy gain ee, if there exists a symmetric matrix P such that P > 0 ð8þ BRLðG; PÞ PðA þ BGMÞþðAþBGMÞ T P ¼ 4 6 ðd þ BGEÞ T 4 P I 5 < 0; ðc þ HGMÞ ðf þ HGEÞ I where () denotes the terms induced by symmetry. Clearly the matrix inequality in (9) is bilinear due to terms involving products of the unknown matrix variables P and G. When G has no specific structure the problem can be solved by eliminating G from (9) using the so-called projection lemma, and solving the resulting set of matrix inequalities (Boyd et al. 1994; Skelton et al. 1998). If the desired controller order equals the plant order, that is if ^n ¼ n; then a full order controller is to be designed and the resulting set of matrix inequalities define a convex LMI problem which can be solved very efficiently. However, if a reduced order controller is desired, we have in addition to LMI constraints, a non-convex rank constraint. Methods for obtaining reduced order controllers can be found in Skelton et al. (1998), Grigoriadis and Skelton (1996), El Ghaoui et al. (199) and references therein. Further, the application of the projection lemma is not possible if G is constrained to have a decentralised structure. In this paper, we propose a double homotopy algorithm which can be used to obtain a decentralised controller, given a starting centralised controller. The starting centralised controller may be full order or reduced order, and the order of the resulting decentralised controller can be chosen to be equal to or less than the starting controller. The next section proposes a method to locally solve the above BMI problem (9) and suggests a homotopy path that gradually deforms a centralised controller G to a decentralised one G d.. Double homotopy path method The idea is to solve the BMI (9) by linearising it around a nominal solution. Suppose P 0 and G 0 denote a solution of (9). Now let P ¼ P 0 þ P, with P ð9þ

5 International Journal of Control 160 symmetric, and G ¼ G 0 þ G. We illustrate the linearisation procedure for the product terms P(A þ BGM) and P(D þ BGE) in Equation (9). The lineariation of the first term results in PðA þ BGMÞ ¼ðP 0 þ PÞ½A þ BðG 0 þ GÞM Š ¼ P 0 ða þ BG 0 MÞþP 0 BðGÞM þ PðA þ BG 0 MÞþðPÞBðGÞM P 0 ða þ BG 0 MÞþP 0 BðGÞM þ PðA þ BG 0 MÞ: Similarly PðD þ BGEÞ ¼ðP 0 þ PÞ½D þ BðG 0 þ GÞEŠ P 0 ðd þ BG 0 EÞþP 0 BðGÞE þ PðD þ BG 0 EÞ: The approximations are valid if the perturbations P and G are small and hence, their product is negligible to the first-order. Now, the linearised approximation of (8) and (9) obtained by neglecting all second order terms is P 0 þ P > 0 ð10þ 4 where and F 11 ðp 0 ; G 0 Þ F T 1 ðp 0; G 0 Þ I C þ HðGÞM D þ HðGÞE I 5< 0 ð11þ F 11 ðp 0 ; G 0 Þ¼P 0 A þ A T P 0 þ P A þ A T P þ P 0 BðGÞM þ M T ðgþ T B T P T 0 F 1 ðp 0 ; G 0 Þ¼P 0 B þ P B þ P 0 BðGÞE 9 A ¼ A þ BG 0 M; B ¼ D þ BG 0 E; >= C ¼ C þ HG 0 M; >; D ¼ F þ HG 0 E: ð1þ The inequality (11) is linear in the unknown perturbations P and G and can be solved using efficient interior point algorithms (Boyd et al. 1994). Once (10) and (11) have been solved for P and G, one needs to make sure that P ¼ P 0 þ P and G ¼ G 0 þ G satisfy (9). The idea for a decentralised H 1 control design using homotopy is to deform the centralised H 1 controller at each step with a perturbation G which leads to decentralisation and if desired, controller order reduction. The perturbation G can be determined by solving the set of linear inequalities (10) and (11). A computational algorithm is proposed next that specifies the exact manner in which G is deformed and the structure of G..1 Computational algorithm Step 0: Given the plant in (1) design a centralised suboptimal H 1 controller G 0 of a desired order with a performance level 0. Compute the corresponding Lyapunov matrix P 0 by solving the LMI obtained by fixing G 0 and 0 in (9). Let G have the same structure as the desired decentralised control structure G d. Let G 0,d denote the block diagonal part of G 0 that has the same decentralised structure as G d. Similarly let G 0,off denote the complement of G 0,d such that G 0 ¼ G 0,d þ G 0,off. Step 1: Choose " min 1, " min < " 0 < 1 and ¼ 0. Set k ¼ 1, " ¼ " 0. Step : Set G k ¼ G k 1 ":G 0;off. Let A ¼ A þ B G k M; B ¼ D þ B G k E; C ¼ C þ H G k M and D ¼ F þ H G k E. Solve the following convex optimisation problem min k P;G subject to P ¼ P T, P k 1 þ P > 0 ð1þ F 11 ðp k 1 ; G k Þ 4 F T 1 ðp k 1; G k Þ k I 5< 0; ð14þ C þ HðGÞM D þ HðGÞE k I 9 kpk <" kp k 1 k; >= kgk <" kg 0 k; >; k k 1 : ð15þ Step : Set G k ¼ G k þ G and P k ¼ P k 1 þ P. Check that the triple (G k, P k, k ) satisfies the bounded real lemma (9). If (G k, P k, k ) satisfy (9), go to Step 4. If not, set " ¼ "/ until " < " min and repeat Step. If " < " min, we conclude that the algorithm does not converge and a different starting centralised controller is required. Step 4: Recalculate P k by fixing G k and k in (9) such that the condition number of P k is minimised, or in other words, log det P 1 k is minimised. This is a convex problem. This step improves the conditioning of the overall algorithm. Step 5: Set ¼ þ ". If ¼ 1 go to Step 6, else set k ¼ k þ 1, " ¼ min(" 0,1 ) and go to Step. Step 6: The desired decentralised H 1 controller is given by G k.

6 1604 C.S. Mehendale and K.M. Grigoriadis. Remarks on the proposed algorithm The proposed algorithm can be explained as follows: At the kth iteration, in Step of the proposed algorithm, the BMI (9) is slightly perturbed by perturbing the controller G k 1 using the term "G 0,off. This perturbation is applied sufficient number of times until it reduces the structure of the controller to that of a decentralised one. Next, we find small perturbations G and P using the linearised BRL (11) such that performance is improved. It should be noted that the perturbation G has a structure similar to the desired decentralised controller and hence only affects the block diagonal part of the controller. The off-diagonal part is slowly removed by the term "G 0,off at each iteration. A homotopy path is followed between the off-diagonal part of the initial centralised controller and the desired final off-diagonal component, which has all zero entries. The algorithm starts in Step 0 with the design of a centralised suboptimal H 1 controller G 0 of a desired order with a performance level 0. The order of the decentralised controller obtained using the proposed algorithm may be chosen to be less than the order of the starting centralised controller. For the case of a full order controller design, that is ^n ¼ n; this centralised controller design can be carried out using an LMI approach (Gahinet and Apkarian 1994; Gahinet 1996; Skelton et al. 1998) which involves finding the Lyapunov matrix P 0 or a Ricatti based state space approach (Zhou and Doyle 199). In cases where a reduced order decentralised controller is desired, two approaches may be followed. One approach is to design a reduced order centralised controller G 0 to start the iterations. The design of a reduced order controller G 0 is a non-convex problem and iterative methods, such as those proposed by or Grigoriadis and Skelton (1996) or El Ghaoui et al. (199) may be used. Another approach as suggested by the authors in Zhai et al. (001) is to start with a full order centralised controller G 0 and augment the desired decentralised controller structure G d of order ^n; by an unobservable or uncontrollable stable system of order l ¼ n ^n. The perturbation G for the proposed homotopy algorithm is then selected such that it updates only the controllable and observable parts of the desired decentralised controller leading to a reduced order controller. The conditioning of the proposed algorithm improves if the optional Step 4 is executed during each iteration. This involves minimising in each iteration k, the objective function log det P 1 k ; which is equivalent to the minimising the volume of the ellipsoid E k ¼ 4 fzjz T P k z 1g: This is a convex problem and can be solved directly using an ellipsoid algorithm (Boyd et al. 1994) or by posing it as an eigenvalue problem (Nesterov and Nemirovskii 1994). The final decentralised controller obtained using the proposed algorithm also depends on the initial step size " used to initiate the iterations. Also, the convergence is observed to be faster when the initial Lyapunov matrix P 0 is well conditioned. Further, the off-diagonal entries of the controller could be perturbed via a different structure until all off-diagonal entries become zero. Each step of the proposed algorithm involves the solution of a structured LMI problem where the fundamental complexity is low and polynomial-time algorithms are available for solution (Nesterov and Nemirovskii 1994). However, it is known that general decentralised output feedback stabilisation using a norm bounded controller is an NP-hard problem (Toker and Ozbay 1995) indicating that it is rather unlikely to find a polynomial-time solution procedure. Hence, we expect that only low size and medium size problems will be able to be approached with the proposed algorithm.. Relevance to past approaches A related algorithm for decentralised design using homotopy has been proposed in Zhai et al. (001) and it is summarised below. Consider a matrix function HðG d ; P;Þ¼BRLðð1 ÞG 0 þ G d ; PÞ; where G d is the desired decentralized controller and BRL(G,P) is defined by (9). The term (1 )G 0 þ G d defines the homotopy interpolation. As changes from 0 to 1, a path is followed from the centralised controller to the desired decentralised controller. The BMI is slightly perturbed by slowly increasing at each iteration. This is possible for small enough increase in by continuity arguments. Next, the perturbed BMI H(G d, P, ) < 0 for P > 0 is solved by fixing one of the matrix variables (either P or G d ) and solving the resulting projected LMI for the other matrix variable. The process is repeated by then fixing the second variable and solving the resulting LMI for the first variable. The algorithm continues until a decentralised controller is found ( ¼ 1) or till can not be increased further. Next, our proposed homotopy method is compared to the above described approach. The proposed algorithm (x.1) finds a linear perturbation to both P and G in the same step, also allowing to increase at the same time. There is an extra degree of freedom in letting k k 1 in Step. Clearly if (G k 1, P k 1, k 1 ) satisfy (9) then so does (G k 1, P k 1, k ) for k k 1, and the linearisation

7 International Journal of Control 1605 around (G k 1, P k 1, k 1 ) is still valid with k 1 replaced by k. This allows the search to be extended to a larger ball around (G k 1, P k 1 ). The conditions in Step on the norms of P and G (15) guarantee that the perturbations obtained at each step are small. There are no bounds on the norms of P and G in Zhai et al. (001). This could result in controllers with large gains. It should be noted that convergence of the proposed algorithm is not guaranteed. Non-convergence of the proposed algorithm does not imply that a decentralised H 1 controller does not exist for the given performance level 0. However, it will be seen in the design example that the proposed algorithm converges to a decentralised controller, whereas, the algorithm proposed in Zhai et al. (001) does not converge. Remark 1: It is not known how it can be guaranteed a priori by the appropriate selection of the initial centralised controller, that every point on the homotopy path corresponds to a stabilising controller. Hence, the convergence of the proposed method, as well as, the method in Zhai et al. (001) depends on the starting centralised controller. 4. Numerical examples This section demonstrates the application of the proposed algorithm to various decentralised control design examples. The examples provide a comparison with past approaches and illustrate the convergence properties of the algorithm. A reduced order decentralised control design case is also examined. 4.1 Example 1: Inverted pendulums The example is borrowed from Geromel et al. (1994) and Scorletti and Duc (001). The problem is the control of two inverted pendulums in cascade, which is a highly unstable system and is difficult to control. The plant is given by (1) with :8 0 9:8 0 A p ¼ ; B 1 p ¼ ; 9:8 0 : C p ¼ ; B 0 0 y ¼ ; M p ¼ ; D p ¼ I 44 : The plant has two control inputs. It is assumed that only the first state variable of each subsystem is measured for feedback. A suboptimal full order centralised controller with guaranteed H 1 performance level 0 ¼ 15 is designed to act as the starting point for the proposed algorithm. The centralised H 1 controller obtained is where G 0 ¼ D c C c ; B c A c 1:46 1:0 :85 46:5 1:9 10:10 11:96 16: A c ¼ 6 4 4:01 80:49 10: 1191:9 5 ; :4 19:6 18:45 4: 1: :4 5:6 :14 B c ¼ 6 4 0:84 1:8 5 ; 5:09 6:1 0:56 19:4 1:5 90:0 C c ¼ ; 0:49 41:1 4:6 615: D c ¼ 0 : The actual closed loop H 1 performance level achieved by G 0 is ¼.. Starting with this centralised controller and " ¼ 0.001, the proposed algorithm gives a decentralised controller where G d ¼ D cd C cd ; B cd A cd 51:6 4: :15 :5 0 0 A cd ¼ : :4 5 ; 0 0 8:0 459:4 49: :44 0 B cd ¼ : :94 99: 6: C cd ¼ ; 0 0 6:89 616:4 D cd ¼ 11: 0 ; 0 4:01

8 1606 C.S. Mehendale and K.M. Grigoriadis with upper bound on the H 1 performance k ¼ The actual H 1 performance level achieved by G d is ¼ The proposed algorithm converged in 1000 iterations (with steps of " ¼ 0.001). If a suboptimal full order centralised controller with guaranteed H 1 performance level 0 ¼ 5 is designed to act as the starting point for the proposed algorithm, convergence is obtained after more than iterations. Using the algorithm in Scorletti and Duc (001) for output feedback decentralised design, the guaranteed upper bound for H 1 performance level is ¼ 14.4, while the achieved performance level is ¼. Hence, it can be observed that the performance is improved by more than 50% using the double homotopy algorithm proposed in the paper. Also, the method proposed in Scorletti and Duc (001) resulted in controllers with large gains as opposed to the controllers obtained using the proposed double homotopy method where the controller gains are relatively small. For comparison purposes, the algorithm in Zhai et al. (001) was also used for the same design problem with the same starting centralised controller. However, this algorithm failed to converge to a decentralised controller. 4. Example The second example is borrowed from Veillette et al. (199) where the design of reliable control systems is investigated. This numerical example is reproduced in Oliveira et al. (000) to test separation based numerical algorithms for decentralised control design. The open loop system matrices are A p ¼ ; B 1 0 p ¼ ; D p ¼ C 6 p ¼ ; B y ¼ ; D y ¼ M p ¼ ; D z ¼ : The objective is to design two controllers of order two such that the first control input is based on the first output and the second control input is based on the second output. The proposed algorithm is initialised with a sub-optimal centralised controller design with guaranteed H 1 performance level 0 ¼.54 and an initial step size of " ¼ The proposed algorithm converged in 1 iterations giving a decentralised controller with guaranteed H 1 performance level ¼.66 and an actual performance level of ¼.8. The decentralised controller gains are 5:5 : :16 5: 0 0 A cd ¼ :5 96:1 5 ; 0 0 9:461 50:94 0: :1 0 B cd ¼ : :19 1:98 6: C cd ¼ ; :48 :115 D cd ¼ 0:95 0 : 0 0:49 4. Example : Reduced order decentralised control design The last example is borrowed from Zhai et al. (001) wherein a two channel system is considered, with each channel utilising one measurement and generating one control input. The disturbance vector w and the error vector y both lie in R 4. The order of the system is n ¼ 8, and two reduced order controllers of orders ^n 1 ¼ and ^n ¼ are to be designed for the two channels, respectively. The total dimension of the two local controllers is ^n 1 þ ^n ¼ 5 which is less than the order of the controlled system. The plant matrices are not reproduced here due to space considerations. Following the remarks in x., we append the desired reduced order decentralised controller with a stable uncontrollable system of order three and use the proposed algorithm. For this system, the minimum H 1 disturbance attenuation level achieved by a centralised controller is., and a suboptimal centralised controller with a guaranteed H 1 performance level of 0 ¼.4 is chosen as a starting controller. The centralised controller is balanced using grammian based balancing. This procedure transforms the states and sorts them by their controllability and observability indices. With an initial step size of " ¼ 0.01 the algorithm converged in 100 iterations with a guaranteed H 1 disturbance attenuation level of ¼.466. The actual H 1 disturbance attenuation

9 International Journal of Control 160 level was computed to be ¼.. The reduced order decentralised controller matrices are 0:14 : :984 : A cd ¼ 0 0 5:46 0:66 0:0 ; :00 1:191 : :458 :149 4:009 0:4 0 :05 0 B cd ¼ 0 0: : :04 0:419 : C cd ¼ ; 0 0 0:4 0:44 0:8 D cd ¼ 0:10 0 : 0 0: Concluding remarks An iterative algorithm is proposed to solve the BMI problem that appears in the decentralised output feedback H 1 control. The method is based on two homotopy path-followings and provides an alternative to past approaches. Along one of the paths, the centralised controller is slowly deformed to a decentralized controller. Along the second path the solution to the BMI (9) is improved upon by locally linearising along a nominal point. Reduced order decentralised controllers can be obtained by either starting with an appropriate reduced order centralised controller, or by appending the decentralised controller structure with a stable uncontrollable or unobservable system. The design method is applied to the control of two inverted pendulums in cascade, which is a highly unstable system and generally considered difficult to control. It is seen that, with the proposed algorithm, improved H 1 performance levels can be achieved. Additional examples demonstrate the convergence properties of the algorithm and the ability to solve reduced order decentralised control problems. The proposed algorithm can be extended to decentralised stabilisation problems, H control design or multi-objective control synthesis problems. The convergence of the proposed algorithm depends on the starting centralised controller choice. It would be worthwhile investigating non-linear homotopy methods to solve decentralised control problems, as they may alleviate the dependence on the initialising controller. References Bao, J., Forbes, J.F., and McLellan, P.J. (1999), Robust Multi-Loop PID Controller Design: A Successive Semi- Definite Programming Approach, Industrial Engineering Chemistry Research, 8, Boyd, S.P., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994), Linear Matrix Inequalities in Systems and Control Theory, Philadelphia, PA: SIAM. El Ghaoui, L., Oustry, F., and AitRami, M. (199), A Cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems, IEEE Transactions on Automatic Control, 4, Gahinet, P. (1996), Explicit Controller Formulas for LMI-Based H 1 Synthesis, Automatica,, Gahinet, P., and Apkarian, P. (1994), A Linear Matrix Inequality Approach to H 1 Control, International Journal of Robust and Nonlinear Control, 4, Geromel, J.C., Bernussou, J., and Peres, P.L.D. (1994), Decentralized Control Through Parameter Space Optimization, Automatica, 0, Grigoriadis, K.M., and Skelton, R.E. (1996), Low-Order Control Design for LMI Problems Using Alternating Projections, Automatica,, Hassibi, A., How, J., and Boyd, S. (1999), A Path-Following Method for Solving BMI Problems in Control, in Proceedings of the American Control Conference, San Diego, California. Michel, A.N., and Miller, R.K. (19), Qualitative Analysis of Large Scale Dynamical Systems, San Diego, CA: Academic Press. Moylan, P.J., and Hill, D.J. (198), Stability Criterion for Large Scale Systems, IEEE Transactions on Automatic Control,, Nesterov, Y., and Nemirovskii, A. (1994), Interior-Point Polynomial Algorithms in Convex Programming, 1 of Studies in Applied Mathematics, Philadelphia, PA: SIAM. Oliveira, M.C.D., Geromel, J.C., and Bernussou, J. (000), Design of Dynamic Output Feedback Decentralized Controllers Via a Separation Procedure, International Journal of Control,, Sandell, N.S., Varaiya, P., Athans, M., and Safonov, M.G. (198), Survey of Decentralized Control Methods for Large Scale Systems, IEEE Transactions on Automatic Control,, Scorletti, G., and Duc, G. (001), An LMI Approach to Decentralized H 1 Control, International Journal of Control, 4, Scorletti, G., and El Ghaoui, L. (1998), Improved LMI Conditions for Gain Scheduling and Related Control Problems, International Journal of Robust and Nonlinear Control, 8, Shiau, J.K., and Chow, J.H. (1996), Robust Decentralized State Feedback Control Design Using an Iterative Linear Matrix Inequality Algorithm, in Proceedings of the 1th Triennial World Congress, San Francisco, CA. Siljak, D.D. (1991), Decentralized Control of Complex Systems, 184 of Mathematics in Science and Engineering, Boston: Academic Press.

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