DSpace at VNU: Combining Convex-Concave Decompositions and Linearization Approaches for Solving BMIs, With Application t...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 1377 Combining Convex–Concave Decompositions and Linearization Approaches for Solving BMIs, With Application to Static Output Feedback Quoc Tran Dinh, Suat Gumussoy, Wim Michiels, Member, IEEE, and Moritz Diehl, Member, IEEE Abstract—A novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality (BMI) constraints The key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings At each iteration of the algorithm the concave part is linearized, leading to a convex subproblem Applications to various output feedback controller synthesis problems are presented In these applications, the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality (LMI) constraints The performance of the algorithm has been benchlibrary marked on the data from the e COMPl ib Index Terms—Bilinear matrix inequality (BMI), convex–concave decomposition, linear time-invariant system, semidefinite programming, static feedback controller design I INTRODUCTION PTIMIZATION involving matrix constraints has a broad interest and applications in static state/output feedback controller design, robust stability of systems, topology optimization, see, e.g., [3], [5], [16], and [18] Many problems in these fields can be reformulated as an optimization problem with linear matrix inequality (LMI) constraints [5], [18] which can be solved efficiently and reliably by means of interior O Manuscript received February 16, 2011; revised July 28, 2011; accepted October 27, 2011 Date of publication December 05, 2011; date of current version May 23, 2012 This work was supported by Research Council KUL: CoE EF/05/006 Optimization in Engineering (OPTEC), IOF-SCORES4CHEM, GOA/10/009 (MaNet), GOA/10/11, ST1/09/33, several PhD/postdoc and fellow grants; Flemish Government: FWO: Ph.D./postdoc grants, projects G.0452.04, G.0499.04, G.0211.05, G.0226.06, G.0321.06, G.0302.07, G.0320.08, G.0558.08, G.0557.08, G.0588.09, G.0377.09, G.0712.11, research communities (ICCoS, ANMMM, MLDM); IWT: Ph.D Grants, Belgian Federal Science Policy Office: IUAP P6/04; EU: ERNSI; FP7-HDMPC, FP7-EMBOCON, ERC-HIGHWIND, Contract Research: AMINAL Other: Helmholtz-viCERP, COMET-ACCM Recommended by Associate Editor F Dabbene Q Tran Dinh was with the Faculty of Mathematics-Mechanics-Informatics, Hanoi University of Science, Hanoi, Vietnam He is now with the Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center (OPTEC), Katholieke Universiteit Leuven, B-3001 Leuven-Heverlee, Belgium (e-mail: quoc.trandinh@esat.kuleuven.be) S Gumussoy was with the Department of Computer Science and Optimization in Engineering Center (OPTEC), Katholieke Universiteit B-3001 Leuven, Belgium He is currently with MathWorks, Natick MA, 01760 USA (e-mail: suat.gumussoy@mathworks.com) W Michiels is with the Department of Computer Science and Optimization in Engineering Center (OPTEC), Katholieke Universiteit Leuven, B-3001 Leuven, Belgium (e-mail: wim.michiels@cs.kuleuven.be) M Diehl is with the Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center (OPTEC), Katholieke Universiteit Leuven, B-3001 Leuven, Belgium (e-mail: moritz.diehl@esat.kuleuven.be) Digital Object Identifier 10.1109/TAC.2011.2176154 point methods for semidefinite programming (SDP) [3], [21] and efficient open-source software tools such as Sedumi [27] and SDPT3 [29] However, solving optimization problems involving nonlinear matrix inequality constraints is still a big challenge in practice The methods and algorithms for nonlinear matrix constrained optimization problems are still limited [8], [10], [16] In control theory, many problems related to the design of a reduced-order controller can be conveniently reformulated as a feasibility problem or an optimization problem with bilinear matrix inequality (BMI) constraints by means of, for instance, Lyapunov’s theory The BMI constraints make the problems much more difficult than the LMI ones due to their nonconvexity and possible nonsmoothness It has been shown in [4] that the optimization problems involving BMI are NP-hard Several approaches to solve optimization problems with BMI constraints have been proposed For instance, Goh et al [11] considered problems in robust control by means of BMI optimization using global optimization methods Hol et al in [15] proposed to used a sum-of-squares approach to fixed order -infinity synthesis Apkarian and Tuan [2] proposed local and global methods for solving BMIs also based on techniques of global optimization These authors further considered this problem by proposing parametric formulations and difference of two convex functions (DC) programming approaches A similar approach can be found in [1] However, finding a global optimum in optimization with BMI constraints is in general impractical and global optimization methods are usually recommended only for low dimensional problems Our method developed in this paper is classified as a local optimization method which aims to find a local optimum based on solving a sequence of convex semidefinite programming problems The approach in this paper is to generalize the idea of DC programming to optimization with convex-concave matrix inequality constraints However, this is not only a technical extension since many characterizations of standard nonlinear programming are no longer preserved in nonlinear semidefinite programming, see, e.g., [25], [28] Moreover, converting a nonlinear semidefinite programming problem into a standard nonlinear programming one usually requires some spectral functions which are related to the eigenvalues of matrix mappings The resulting problem is in general nonconvex and nonsmooth, see, e.g., [7] Sequential semidefinite programming method for nonlinear SDP and its application to robust control was considered by Fares et al in [9] Thevenet et al [30] studied spectral SDP 0018-9286/$26.00 © 2011 IEEE 1378 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 methods for solving problems involving BMI arising in controller design Another approach is based on the fact that problems with BMI constraints can be reformulated as problems with LMI constraints and additional rank constraints In [22], Orsi et al developed a Newton-like method for solving problems of this type In this paper, we are interested in optimization problems arising in static output feedback controller design for a linear, time-invariant system of the form (1) B Outline of the paper The remainder of the paper is organized as follows Section II provides some preliminary results which will be used in what follows Section III presents the formulation of optimization problems involving convex-concave matrix inequality constraints and a fundamental assumption, Assumption A1 The algorithm and its convergence results are presented in Section IV Applications to optimization problems in static feedback controller design and numerical benchmarking are given in Section V The last section contains some concluding remarks II PRELIMINARIES is the state vector, is the performance where is the input vector, is the performance input, is the physical output vector, is output, state matrix, is input matrix, and is the output matrix Using a static feedback controller of the form with , we can write the closed-loop system as follows: (2) The stabilization, , optimization and other control problems for this closed-loop system will be considered A Contribution Many control problems can be expressed as optimization problems with BMI constraints and these optimization problems can conveniently be reformulated as optimization problems with difference of two positive semidefinite convex (psd-convex) mappings (or convex-concave decomposition) constraints (see Definition 2.1 below) In this paper, we propose to use this reformulation leading to a new local optimization method for solving some classes of optimization problems involving BMI constraints We provide a practical algorithm and prove the convergence of the algorithm under certain standard assumptions The algorithm proposed in this paper is very simple to implement by using available SDP software tools Moreover, it does not require any globalization strategy such as line-search procedures to guarantee global convergence to a local minimum The method still works in practice for nonsmooth optimization problems, where the objective function and the concave parts are only subdifferentiable, but not necessarily differentiable Note that our method is different from the standard DCA approach in [24], [26] since we work directly with positive semidefinite matrix inequality constraints instead of transforming into DC representations as in [1], [2] We show that our method is applicable to many control problems in static state/output feedback controller design The numerical results are benchmarked using the data from the library Note, however, that this method is also applicable to other nonconvex optimization problems with matrix inequality constraints which can be written as convex-concave decompositions be the set of symmetric matrices of size , , and Let be the set of symmetric positive semidefinite, resp., resp., positive definite matrices For given matrices and in , the relation (resp., ) means that (resp., ) and (resp., ) is (resp., ) The quantity is an inner product of two matrices and defined on , where is the trace of matrix Definition 2.1: [25] A matrix-valued mapping is said to be positive semidefinite convex (psd-convex) on a convex subset if for all and , one has (3) then is said If (3) holds true for instead of for to be strictly psd-convex on Alternatively, if we replace in (3) by then is said to be psd-concave on It is obvious that any convex function is psd-convex with is said to be strongly convex with A function if is convex parameter The derivative of a matrix-valued mapping at is a linear mapping from to which is defined by For a given convex set , the matrix-valued mapping is said to be differentiable on a subset if its derivative exists at every The definitions of the second order derivatives of matrix-valued mappings can be found, e.g., be a linear mapping defined as in [25] Let , where for The ad, is defined as joint operator of , for any Lemma 2.2: if and a) A matrix-valued mapping is psd-convex on the function is only if for any convex on b) A mapping is psd-convex on if and only if for all and in , one has (4) Proof: The proof of the statement a) can be found in [25] for any If is psdWe prove b) Let TRAN DINH et al.: COMBINING CONVEX–CONCAVE DECOMPOSITIONS AND LINEARIZATION APPROACHES FOR SOLVING BMIS convex then Now, is convex We have Hence, for all We conclude that (4) holds Conversely, if (4) holds then, for any , we have , which is Thus is convex equivalent to By virtue of a), the mapping is psd-convex For simplicity of discussion, throughout this paper, we assume that all the functions and matrix-valued mappings are twice differentiable on their domain [25], [30] However, this assumption can be reduced to the subdifferentiability of the objective function and the concave parts of the convex-concave decompositions of the matrix-valued mappings as in Definition 2.3 below Definition 2.3: A matrix-valued mapping is said to be a psd-convex-concave mapping if can be represented as a difference of two psd-convex mappings, i.e., , where and are psd-convex is called a psd-DC (or psd-convex-concave) The pair decomposition of Note that each given psd-convex-concave mapping possesses many psd-convex-concave decompositions 1379 The following lemma shows that the bilinear matrix form (5) can be decomposed as a difference of two psd-convex mappings Lemma 3.1: and are a) The mappings The mapping is psd-convex on psd-convex on can b) The bilinear matrix form be represented as a psd-convex-concave mapping of at least three forms: (6) The statement b) provides at least three different explicit psd convex-concave decompositions of the bilinear form Intuitively, we can see that the first decomposition has a “strong curvature” on the second term, while the second and the third decompositions have “less curvature” on the second term due to the compensation between and The following result will be used to transform Schur psdconvex constraints to LMI constraints Lemma 3.2: Then the matrix inequality a) Suppose that is equivalent to (7) III OPTIMIZATION OF CONVEX-CONCAVE MATRIX INEQUALITY CONSTRAINTS b) Suppose that , , then we have: A Psd-Convex-Concave Decomposition of BMIs Instead of using the vector as a decision variable, we use Note from now on the matrix as a matrix variable in -column vector that any matrix can be considered as an by vectorizing with respect to its columns, i.e., The inverse mapping of is called Since and are linear operators, the psd-convexity is still preserved under these operators given by A mapping , where , is called a Schur psd-convex1 mapping Let us consider a bilinear matrix form (5) By using the Kronecker product, we can write where , are appropriate identity matrices, Kronecker product Hence, the vectorization of deed a bilinear form of two vectors 1Due to Schur’s complement form (8) The proof of this lemma immediately follows by applying Schur’s complement and Lemma 2.2[6] We omit the proof here B Optimization Involving Convex-Concave Matrix Inequality Constraints Let us consider the following optimization problem: s.t (9) as denotes the is inand where is convex, is a nonempty, closed and ( ) are psd-convex convex set, and Problem (9) is referred to as a convex optimization problem with psd-convex-concave matrix inequality constraints Let be a polyhedral in Then, if is nonlinear or one of or ( is nonlinear then (9) is a the mappings ( ) are linear nonlinear semidefinite program If then (9) is a convex nonlinear SDP problem Otherwise, it is a nonconvex nonlinear SDP problem Let us define as the Lagrange function of (9), where ( ) 1380 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 considered as Lagrange multipliers The generalized KKT condition of (9) is presented as (10) Here, is the normal cone of at defined as if , otherwise A pair satisfying (10) is called a KKT point, is called a stationary point and is the corresponding multiplier of (9) The generalized optimality condition for nonlinear semidefinite programming can be found in the literature, e.g., [25], [28] Let us denote by (11) the feasible set of (9) and by which is defined by the relative interior of where is the set of classical relative interiors of [6] The following condition is a fundamental assumption in this paper is nonempty Assumption A1: Note that this assumption is crucial for our method, because, as we shall see, it requires a strictly feasible starting point Finding such a point is in principle not an easy task However, in many problems, this assumption is always satisfied In Section V, we will propose techniques to determine a starting point for the control problems under consideration IV THE ALGORITHM AND ITS CONVERGENCE In this section, a local optimization method for finding a stationary point of problem (9) is proposed Motivated from the DC programming algorithm developed in [24] and the convexconcave procedure in [26] for scalar functions, we develop an iterative procedure for finding a stationary point of (9) The main idea is to linearize the nonconvex part of the psd-convexconcave matrix inequality constraints and then transform the linearized subproblem into a quadratic semidefinite programming problem The subproblem can be either directly solved by means of interior point methods or transformed into a quadratic problem with LMI constraints In the latter case, the resulting problem can be solved by available software tools such as Sedumi [27] and SDPT3 [29] A The Algorithm Suppose that is a given point, the linearized problem of (9) around is written as Here, we add a regularization term into the objective function of is a given matrix that projects the original problem, where in a certain subspace of and is a regularization parameter Since ( ) are psd-convex and the objective function is convex, problem (12) is convex The linearized convex-concave SDP algorithm for solving (9) is described as follows Algorithm 1: Initialization: Choose a positive number and a matrix Find an initial point Set Perform the following steps: Iteration : For Step 1) Solve the convex semidefinite program (12) to obtain a solution for a given tolerance Step 2) If then terminate Otherwise, update and (if necessary), set and go back to Step The following main property of the method makes an implebelongs to the relmentation very easy If the initial point , then Alative interiors of the feasible set , i.e., gorithm generates a sequence which still belongs to In particular, no line-search procedure is needed to ensure global convergence This property follows from the fact that the linearization of is its an overestimate of this mapping (in the concave part the sense of the positive semidefinite cone), i.e., which is equivalent to Hence, if the subproblem (12) has a solution then it is feasible to (9) Geometrically, Algorithm can be seen as an inner approximation method The main tasks of an implementation of Algorithm consist of: ; • determining an initial point • solving the convex semidefinite program (12) repeatedly As mentioned before, since is nonconvex, finding an initial in is, in principle, not an easy task However, point in some practical problems, this can be done by exploiting the special structure of the problem (see the examples in Section V) To solve the convex subproblem (12), we can either implement an interior point method and exploit the structure of the problem or transform it into a standard SDP problem and then make use of available software tools for SDP The regularizaand the projection matrix can be fixed at tion parameter appropriate choices for all iterations, or adaptively updated is a solution of (12) linearized at then Lemma 4.1: If it is a stationary point of (9) is a multiplier associated with Proof: Suppose that , substituting into the generalized KKT condition (39) of is a stationary point of (9) (12) we obtain (10) Thus, B Convergence Analysis s.t (12) In this subsection, we restrict our discussion to the following special case TRAN DINH et al.: COMBINING CONVEX–CONCAVE DECOMPOSITIONS AND LINEARIZATION APPROACHES FOR SOLVING BMIS Assumption A2: The mappings ( ) are Schur psd-convex and is formed by a finite number of LMIs In adwith a convexity parameter dition, is convex quadratic on This assumption is only technical for our implementation If is Schur psd-convex then the linearized conthe mapping straints of problem (12) can directly be transformed into LMI ( ) can constraints (see Lemma 3.2) In practice, be general psd-convex mappings and can be a general convex function Under Assumption A2, the convex subproblem (12) can be transformed equivalently into a quadratic semidefinite program of the form s.t (13) where is a linear mapping from to , , and is a symmetric matrix, by means of Lemma 3.2 A vector is said to satisfy the Slater condition of (13) if Suppose that the triple satisfies the KKT condition of (13) (see [10]), where is a primal stationary point, is a Lagrange multiplier and is a slack variable associated with and Then, problem (13) is said to satisfy the if strict complementarity condition at Let be a stationary point of (13) We say that is a feasible direction to (13) if is a feasible point of (13) for all sufficiently small As in [10], we assume that the second order sufficient condition holds for (13) at with modulus if for all feasible directions at with , one has We say that the convex problem (13) is solvable and satisfies the strong second of order sufficient condition if there exists a KKT point the KKT system of (13) that satisfies the second order sufficient condition and the strict complementary condition Assumption A3: The convex subproblem (12) is solvable and satisfies the strong second order sufficient condition Assumption A3 is standard in optimization and is usually used to investigate the convergence of the algorithms [9], [10], [25] is a The following lemma shows that descent direction of problem (9) whose proof can be found in the Appendix is a sequence genLemma 4.2: Suppose that erated by Algorithm Then: : a) The following inequality holds for (14) is the convexity parameter of where , b) If there exists at least one constraint , , to be strictly feasible at , i.e., then provided that and is full-row-rank then is a suffic) If cient descent direction of (9), i.e., for all The following theorem shows the convergence of Algorithm in a particular case 1381 Theorem 4.3: Under Assumptions A1, A2, and A3, suppose that is bounded from below on , where is assumed to be be a sequence generated by Albounded in Let gorithm starting from Then if either is strongly and is full-row-rank for all convex or then every accumulation point of is a KKT point of (9) Moreover, if the set of the KKT points of converges to a (9) is finite then the whole sequence KKT point of (9) be the sequence of Proof: Let sample points generated by Algorithm starting from For a given , let us define the following mapping: (15) Then, is a multivalued mapping and it can be considered as the solution mapping of the convex subproblem (12) Note generated by Algorithm satisfies that the sequence for all We first prove that is a closed mapping Indeed, since the convex subproblem (12) satisfies Slater’s condition and has a solution that satisfies the strict complementarity and the second order sufficient condition, by applying Theorem in [10] we conclude that the mapping is differentiable in a neighborhood of the solution In particular, it is closed due to the compactness of On the other hand, since is either strongly convex or for all and is full-row-rank, it follows from Lemma 4.2 that the objective function is strictly , i.e., for all monotone on Since and is compact, is also compact Applying Theorem in [20] we conclude that every belongs to the set of stalimit points of the sequence tionary points Moreover, since is bounded from below and is full-row rank, and either is strongly convex or it follows from (14) that Thereis connected and if is finite then the whole sequence fore, converges to in Remark 4.4: The condition that is quadratic in Assumption can be relaxed to being twice continuously differentiable However, in this case, we need a direct proof for Theorem 4.3 instead of applying Theorem in [10] V APPLICATIONS TO ROBUST CONTROLLER DESIGN In this section, we apply the method developed in the previous sections to the following static state/output feedback controller design problems: 1) Sparse linear static output feedback controller design; 2) Spectral abscissa and pseudospectral abscissa optimization; optimization; 3) optimization; 4) synthesis 5) and mixed We used the system data from [13], [23] and the library [17] All the implementations are done in Matlab 7.11.0 (R2010b) running on a PC Desktop Intel(R) Core(TM)2 Quad CPU Q6600 with 2.4 GHz and GB RAM We use the YALMIP package [19] as a modeling language and SeDuMi 1.1 as a SDP 1382 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 solver [27] to solve the LMI optimization problems arising in Algorithm at the initial phase (Phase 1) and subproblem (12) We also benchmarked our method with various examples and compared our results with HIFOO [12] and PENBMI [14] for all control problems HIFOO is an open-source Matlab package for fixed-order controller design It computes a fixed-order controller using a hybrid algorithm for nonsmooth, nonconvex optimization based on quasi-Newton updating and gradient sampling PENBMI [14] is a commercial software for solving optimization problems with quadratic objective and BMI constraints, which is freely licensed for academic purposes We initialized the initial controller for HIFOO and the BMI parameters for PENBMI to the initial values of our method As shown in [22], we can reformulate the spectral abscissa feasibility problem as a rank constrained LMI feasibility problem Therefore, we also compared our results with LMIRank [22] (a MATLAB toolbox for solving rank constrained LMI feasibility problems) by implementing a simple procedure for solving the spectral abscissa optimization Note that all problems addressed here lead to at least one BMI constraint To apply the method developed in the previous sections, we propose a unified scheme to treat these problems 1) Scheme A.1: Step 1) Find a convex-concave decomposition of the BMI constraints as Step 2) Find a starting point Step 3) For a given , linearize the concave part to obtain the convex constraint , where is the linearization of at Step 4) Reformulate the convex constraint as an LMI constraint by means of Lemma 3.2 Step 5) Apply Algorithm with an SDP solver to solve the given problem If we denote by A Sparse Linear Constant Output-Feedback Design Let us consider a BMI optimization problem of sparse linear constant output-feedback design given as s.t (16) Here, matrices , , are given with appropriate dimensions, and are referred to as variables and is a weighting parameter The objective function consists of two terms: the first is to stabilize the system (or to maximize the decay rate) term and the second one is to ensure the sparsity of the gain matrix This problem is a modification of the first example in [13] Let us illustrate Scheme A.1 for solving this problem , where is the identity 1) Step 1: Let matrix Then, applying Lemma 3.1 we can write and then the BMI constraint in (16) can be written equivalently as a psd-convex-concave matrix inequality constraint (of a variable formed from as ) as follows: (20) Note that the objective function of (16) is convex but nonsmooth which is not directly suitable for the sequential SDP approach in [8], but, the nonconvex problem (16) can be reformulated in the form of (9) by using slack variables 2) Steps 2–5: The implementation is carried out as follows: ) Set Phase (Determine a starting point , where is the maximum real part of the eigenvalues of the matrix, and as the solution of the LMI feasibility compute problem (21) originates from the property The above choice for renders the left-hand size of (21) negative that semidefinite (but not negative definite) Phase Perform Algorithm with a starting point found at Phase Let us now illustrate Step of Scheme A.1 After linearizing the concave part of the convex-concave reformulation of the we obtain the last BMI constraint in (16) at linearization (22) is a linear mapping of , , and Now, where by applying Lemma 3.2, (22) can be transformed into an LMI constraint: With the above approach we solved problem (16) for the same system data as in [13] Here, matrices , and are given, respectively as (17) (18) In our implementation, we use the decomposition (18) (19) and TRAN DINH et al.: COMBINING CONVEX–CONCAVE DECOMPOSITIONS AND LINEARIZATION APPROACHES FOR SOLVING BMIS The weighting parameter is chosen by Algorithm is terminated if one of the following conditions is satisfied: • subproblem (12) encounters a numerical problem; ; • , reaches; • the maximum number of iterations, • or the objective function is not significantly improved after two successive iterations, i.e., for some and , where In this example, Algorithm is terminated after 15 iterations, whereas the objective function is not significantly improved iteration, matrix only has three However, after the nonzero elements, while the decay rate is 1.17316 This value after is much higher than the one reported in [13], six iterations We obtain the gain matrix as With this matrix, the maximum real part of the eigenvalues , is of the closed-loop matrix in (2), Simultaneously, and due to the inactiveness of the BMI Note that constraint in (16) at the second iteration B Spectral Abscissa and Pseudo-Spectral Abscissa Optimization One popular problem in control theory is to optimize the spectral abscissa of the closed-loop system Briefly, this problem is presented as an unconstrained optimization problem of the form (23) where is , denotes the real part the spectral abscissa of and is the spectrum of of Problem (23) has many drawbacks in terms of numerical solution due to the nonsmoothness and non-Lipschitz continuity of [7] the objective function In order to apply the method developed in this paper, problem (23) is reformulated as an optimization problem with BMI constraints of the form, see, e.g., [7], [18] s.t (24) Here, matrices , , and are and and the scalar given Matrices are considered as variables If the optimal value of (24) is strictly positive then the closed-loop feedback controller stabilizes the linear system Problem (24) is very similar to (16) Therefore, using the same trick as in (16), we can reformulate (24) in the form of then (9) More precisely, if we define the bilinear matrix mapping can be represented TABLE I COMPUTATIONAL RESULTS FOR (24) IN COMPl 1383 ib as a psd-convex-concave decomposition of the form (18) and problem (24) can be rewritten in the form of (9) We implement Algorithm for solving this resulting problem using the same parameters and the stopping criterions as in Section V-B In addition, we regularize the objective function by adding the term , with The maximum number of iterations is set to 150 We test for several problems in and compare our results with the ones reported by HIFOO, PENBMI, and LMIRank For LMIRank, we implement the algorithm proposed in at and [22] We initialize the value of the decay rate perform an iterative loop to increase as The algorithm is terminated if either the problems [22, (12) or (21)] with a correspondence can not be solved or the maximum number of iterations is reached The numerical results of four algorithms are reported in Table I Here, we initialize the algorithm in HIFOO with the same initial guess Since PENBMI and our methods solve the same BMI problems, they are initialized by the same initial values for , , and The notation in Table I consists of: Name is the name of prob, are the maximum real part of the eigenlems, values of the open-loop and closed-loop matrices , , respectively; iter is the number of iterations, time[s] is the CPU time in seconds The columns titled HIFOO, LMIRank, and PENBMI give the maximum real part of the eigenvalues of the closed-loop system for a static output feedback controller computed by available software HIFOO [12], LMIRank [22], and PENBMI [14], respectively Our results can be found in the sixth column The entries with a dash sign indicate that there is no feasible solution found Algorithm fails or makes only slow progress towards a local solution with six problems: AC18, Problems AC5 DIS5, PAS, NN6, NN7, NN12 in to avoid nuand NN5 are initialized with a different matrix merical problems 1384 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 Note that Algorithm as well as the algorithms implemented in HIFOO, LMIRank, and PENBMI are local optimization methods, which only report a local minimizer and these solutions may not be the same Because the LMIRank package can only handle feasibility problems, it cannot directly be used to solve problem (24) Therefore, we have used a direct search procedure for finding The computational time of the overall procedure is much higher than the other methods for the majority of the test problems To conclude this subsection, we show that our method can also be applied to solve the problem of optimizing the pseudo-spectral abscissa in static feedback controller designs This problem is described as follows (see [7], [18]): s.t (25) and , then this problem is formulated as that the following optimization problem with BMI constraints [17]: s.t (26) is positive definite Otherwise, Here, we also assume that instead of with in (26) we use In order to apply Algorithm for solving problem (26), a is required This task can be done starting point by performing some extra steps called Phase The algorithm is now split in two phases as follows 1) Phase 1: (Determine a starting point ) Step 1) If then we set Otherwise, go to Step Step 2) Solve the following optimization problem with LMI constraints: where as before and as in (24) and Using the same notation applying the statement b) of Lemma 3.2, the BMI constraint in this problem can be transformed into a psd-convex-concave one If we denote the linearization of , i.e., iteration by at the s.t (27) where If this problem has and then terminate Phase and a solution together with as a starting point using for Phase Otherwise, go to Step Step 3) Solve the following feasibility problem with LMI constraints: Find and such that then the linearized constraint in the subproblem (12) can be represented as an LMI thanks to Lemma 3.2: Hence, Algorithm can be applied to solve problem (25) Remark 5.1: If we define then the bilinear matrix can be rewritten as mapping to obtain and , where is a given regulariza, where tion factor Compute is a pseudo-inverse of , and resolve problem If problem (27) has a solution (27) with and then set and terminate Phase Otherwise, perform Step Step 4) Apply the method in Section V-C to solve the following BMI feasibility problem: Find and such that: (28) Using this decomposition, one can avoid the contribution of maon the bilinear term Consequently, Algorithm may trix work better in some specific problems C Optimization: BMI Formulation In this subsection, we consider an optimization problem arising in synthesis of the linear system (1) Let us assume then go back to If this problem has a solution Step Otherwise, declare that no strictly feasible point is found 2) Phase 2: (Solve problem (26)) Perform Algorithm with found at Phase the starting point Note that Step of Phase corresponds to determining a full state feedback controller and approximating it subsequently with an output feedback controller Step of Phase is usually TRAN DINH et al.: COMBINING CONVEX–CONCAVE DECOMPOSITIONS AND LINEARIZATION APPROACHES FOR SOLVING BMIS time consuming Therefore, in our numerical implementation, we terminate Step after finding a point such that Remark 5.2: The algorithm described in Phase is finite It is terminated either at Step if no feasible point is found or at Step if a feasible point is found Indeed, if a feasible matrix is found at Step then the first BMI constraint of (27) is Thus, we can find an appropriate feasible with some , which implies the second matrix such that LMI constraint of (27) is satisfied Consequently, problem (27) has a solution The method used in Phase is closely heuristic It can be improved when we apply to a certain problem However, as we can see in the numerical results, it performs quite acceptable for the majority of the test problems In the following numerical examples, we implement Phase and Phase of the algorithm using the decomposition for the BMI form at the left-top corner of the first constraint in (26) The regularization parameters and the stopping criterion for Algorithm are chosen as in Section V-B and and the We test the algorithm for many problems in computational results are reported in Table II For the comparison purpose, we also carry out the test with HIFOO [12] and PENBMI [14], and the results are put in the columns marked by HIFOO and PENBMI in Table II, respectively The initial and the BMI parameters for controller for HIFOO is set to Here, PENBMI are initialized with are the dimensions of problems, the columns norm of the closed-loop titled HIFOO and PENBMI give the system for the static output feedback controller computed by HIFOO and PENBMI; iter and time[s] are the number of iterations and CPU time in second of Algorithm , respectively, included Phase and Phase Problems marked by “b” mean that Step in Phase is performed In Table II, we only report the problems that were solved by Algorithm The numerical results allow us to conclude that Algorithm 1, PENBMI and HIFOO report similar values for the majority of the test prob lems in then the second LMI constraint of (26) becomes If a BMI constraint H TABLE II SYNTHESIS BENCHMARKS ON COMPl 1385 ib PLANTS be written as an LMI constraint Therefore, Algorithm can be applied to solve problem (29) in the case D Optimization: BMI Formulation Alternatively, we can also apply Algorithm to solve the optimization with BMI constraints arising in optimization of , then this the linear system (1) Let us assume that problem is reformulated as the following optimization problem with BMI constraints [17]: s.t (29) (31) which is equivalent to , where Since is convex on [see Lemma 3.1 a)], this BMI constraint can be reformulated as a convexconcave matrix inequality constraint of the form Here, as before, and at the top-corner of the The bilinear matrix term first constraint can be decomposed as (17) or (18) Therefore, we can use these decompositions to transform problem (31) into (9) After linearization, the resulting subproblem is also rewritten as a standard SDP problem by applying Lemma 3.2 We omit this specification here To determine a starting point, we perform Phase which is similar to the one carried out in the -optimization subsection (30) at as By linearizing the concave term (see [6]), the resulting constraint can 1386 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 1) Phase 1: (Determine a starting point ) Step 1) If then set Otherwise, go to Step Step 2) Solve the following optimization with LMI constraints: H TABLE III SYNTHESIS BENCHMARKS ON COMPl ib PLANTS s.t (32) and If this problem has a solution and then terminate Phase and using together with as a starting point for Phase Otherwise, go to Step Step 3) Solve the following feasibility problem of LMI constraints: where and Find such that: to obtain , and Compute , where is a pseudo-inverse of , and resolve problem (32) with and then set If problem (32) has a solution and terminate Phase Otherwise, perform Step Step 4) Apply the method in Section V-C to solve the following BMI feasibility problem: Find and such that E Optimization: BMI Formulation and optimization problems, Motivated from the in this subsection we consider the mixed synthesis , and the problem Let us assume that performance output is divided in two components, and Then the linear system (1) becomes (33) (34) then go back to If this problem has a solution Step Otherwise, declare that no strictly feasible point for (31) is found As in the problem, Phase of the is also terminated after finite iterations In this subsection, we also test this algorithm for several problems in using the same parameters and the stopping criterion as in the previous subsection The computational results are shown in Table III The numerical results computed by HIFOO and PENBMI are also included in Table III Here, the notation is as same as in Table II, except that denotes the -norm of the closed-loop system for the static output feedback controller We can see from Table III that the optimal values reported by Algorithm and HIFOO are almost similar for many problems whereas in general PENBMI has difficulties in finding a feasible solution The mixed control problem is to find a static output , the -norm of the feedback gain such that, for closed loop from to is minimized, while the -norm from to is less than some imposed level [5], [18], [23] This problem leads to the following optimization problem with BMI constraints [23]: s.t (35) TRAN DINH et al.: COMBINING CONVEX–CONCAVE DECOMPOSITIONS AND LINEARIZATION APPROACHES FOR SOLVING BMIS where , and Note that if , the identity matrix, then of static state feedback this problem becomes a mixed design problem considered in [23] In this subsection, we test Algorithm for the static state feedback and output feedback cases 1) Case 1: The static state feedback case ( ) First, we apply the method in [23] to find an initial point via solving two optimization problems with LMI constraints Then, we use the same approach as in the previous subsections to transform problem (35) into an optimization problem with psd-convexconcave matrix inequality constraints Finally, Algorithm is implemented to solve the resulting problem For convenience of implementation, we introduce a slack variable and then with an replace the objective function in (31) by additional constraint In the first case, we test Algorithm with three problems The first problem was also considered in [13] with 1387 The results obtained by Algorithm for solving problems DIS4 and AC16 in this paper confirm the results reported in [23] 2) Case 2: The static output feedback case As before, we first propose a technique to determine a starting point for Algorithm We described this phase algorithmically as follows 3) Phase 1: (Determine a starting point ) then set Otherwise, go Step 1) If to Step Step 2) Solve the following linear SDP problem: s.t (36) where , , and If this problem has and then terminate an optimal solution for a starting Phase Set point of Algorithm in Phase Otherwise go to Step Step 3) Solve the following LMI feasibility problem: and Find If the tolerance is chosen then Algorithm converges after 17 iterations and reports the value with This result is similar to the one shown in [23] If we regularize the subproblem (12) with and then the number of iterations is reduced to ten iterations [17] In this The second problem is DIS4 in and as in [23] Algoproblem, we set rithm converges after 24 iterations with the same tolerance It reports and with If we regularize the subproblem (12) with and then the number of iterations is 18 [17] In this exThe third problem is AC16 in ample we also choose and as in the previous problem As mentioned in [23], if we choose a starting , then the LMI problem can not be solved by the value SDP solvers (e.g., Sedumi, SDPT3) due to numerical problems Thus, we rescale the LMI constraints using the same trick as in [23] After doing this, Algorithm converges after 298 itera The value of reported tions with the same tolerance and with in this case is and such that: (37) to obtain a solution , and Set , where is the pseudo-inverse of Solve again problem (36) with If problem (36) has solution then terminate Phase Otherwise, perform Step Step 4) Solve the following optimization with BMI constraints: s.t (38) to obtain an optimal solution corresponding to then set the optimal value If and go back to Step to determine , and Otherwise, declare that no strictly feasible point of problem (35) is found 1388 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 H =H TABLE IV SYNTHESIS BENCHMARKS ON COMPl ib PLANTS Since at Step of Phase 1, it requires to solve an optimization problem with two BMI constrains This task is usually expensive In our implementation, we only terminate this step after find a strictly feasible point with a feasible gap 0.1 If matrix is invertible then matrix at Step is Hence, we can ignore Step of Phase To avoid the numerical problem in Step 3, we can reformulate problem (37) equivalently to the following one: Find and have applied our method to design static feedback controllers for various problems in robust controller design The algorithm is easy to implement using the current SDP software tools The numerical results are also reported for the benchmark collec The algorithm requires a strictly feasible tion in starting point which is determined by Phase This phase is implemented based on some heuristic techniques which may need to solve a feasibility problem with BMI constraints In the preup to vious numerical examples, Phase costs from of the total time depending on each problem Note, however, that our method depends crucially on the psdconvex-concave decomposition of the BMI constraints In practice, it is important to look at the specific structure of the problems and find an appropriate psd-convex-concave decomposition for Algorithm The method proposed can be extended to general nonlinear semidefinite programming, where the psdconvex-concave decomposition of the nonconvex mappings are available From a control design point of view, the application to more general reduced order controller synthesis problems and the extension towards linear parameter varying or time-varying systems are future research directions APPENDIX Proof of Lemma 4.2: For any matrices , we have From Step is a solution of the convex sub1 of Algorithm 1, we have is the corresponding multiplier, under problem (12) and Assumption 3, they must satisfy the following generalized Kuhn–Tucker condition: such that: (39) We test the algorithm described above for several problems in with the level values and In these Thus, examples, we assume that the output signals and The pawe have rameters and the stopping criterion of the algorithm are chosen as in Section V-D The computational results are reported in and Here, are the Table IV with and norms of the closed-loop systems for the static , the comoutput feedback controller, respectively With putational results show that Algorithm satisfies the condition for all the test problems While, with , there are problems reported infeasible, which are de-constraint of three problems AC3, AC11, noted by “-” The and NN8 is active with respect to Noting that for convexity of , it follows from the first line of (39) and the that VI CONCLUDING REMARKS We have proposed a new algorithm for solving many classes of optimization problems involving BMI constraints arising in static feedback controller design The convergence of the algorithm has been proved under standard assumptions Then, we (40) TRAN DINH et al.: COMBINING CONVEX–CONCAVE DECOMPOSITIONS AND LINEARIZATION APPROACHES FOR SOLVING BMIS On the other hand, we have (41) Since and are psd-convex, applying Lemma 2.2 we have and Summing up these inequalities we obtain Using the fact that , this inequality implies that (42) into (40) and then combining the conseSubstituting quence, (41), (42) and the last line of (39) to obtain (43) Now, since is the solution of the convex subproblem (12) One has Moreover, linearized at since , we have Substituting this inequality into (43), we obtain This inequality is indeed (14) which proves the item such a) If there exists at least one and then that Substituting this inwhich equality into (43) we conclude that proves the item b) The last statement c) follows directly from the inequality (14) REFERENCES [1] T Alamo, J M Bravo, M J Redondo, and E F Camacho, “A setmembership state estimation algorithm based on DC programming,” Automatica, vol 44, no 1, pp 216–224, 2008 [2] P Apkarian and H D Tuan, “Robust control via concave optimization: Local and 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current research focuses on methods for nonlinear optimization, especially sequential convex programming approaches, structured large-scale convex optimization, and distributed optimization Suat Gumussoy received the B.S degrees in electrical and electronics engineering and mathematics from Middle East Technical University, Ankara, Turkey, in 1999 and the M.S and Ph.D degrees in electrical and computer engineering from The Ohio State University, Columbus, in 2001 and 2004, respectively He was a System Engineer in electronic self-protection system design for F-16 aircraft at Mikes Inc., New York (2005–2007), and a Software Quality Engineer in MATLAB control toolboxes at MathWorks, Natick, MA (2007–2008) He was a Postdoctoral Researcher in the Computer Science Department, Katholieke Universiteit Leuven, Leuven, Belgium (2008–2011) He is currently a Senior Software Developer of Robust Control Toolbox at MathWorks His general research interests are control, optimization, and scientific computing His academic study has focused on optimization based control methods on the fixed-order robust controller design for finite-dimensional and time-delay systems and their numerical implementations IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 6, JUNE 2012 Wim Michiels (M’02) received the M.Sc degree in electrical engineering and the Ph.D degree in computer science from the Katholieke Universiteit Leuven, Leuven, Belgium, in 1997 and 2002, respectively He was a fellow of the Research Foundation—Flanders (2002–2008) and a Postdoctoral Research Associate at the Eindhoven University of Technology, Eindhoven, The Netherlands (2007) In October 2008, he was appointed Associate Professor at Katholieke Universiteit Leuven, where he leads a research team within the Numerical Analysis and Applied Mathematics Division He has authored the monograph Stability and Stabilization of Time-Delay Systems An Eigenvalue Based Approach (SIAM, 2007, with S.-I Niculescu), more than 50 articles in scientific journals in the area of control and numerical mathematics, and he has been coeditor of three books His research interests include control and optimization, dynamical systems, numerical linear algebra, and scientific computing His work has focused on the analysis and control of systems described by functional differential equations and on large-scale linear algebra problems, with applications in engineering and the bio-sciences Dr Michiels has been co-organizer of several workshops and conferences in the area of numerical analysis, control, and optimization, including the 5th IFAC Workshop on Time-Delay Systems (Leuven, 2004) and the 14th Belgian–French–German Conference on Optimization (Leuven, 2009) He is member of the IFAC Technical Committee on Linear Control Systems and associate editor of the journal Systems and Control Letters Moritz Diehl (M’09) received the Ph.D degree from the Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany, in 2001 Since 2006, he has been a Professor with the University of Leuven (K.U Leuven), Belgium, and Principal Investigator of K.U Leuven’s Optimization in Engineering Center OPTEC His research is centered around embedded optimization algorithms for use in model predictive control, real-time optimization, and moving horizon estimation His general interests are in structure exploitation for optimization in engineering, convex optimization, dynamic optimization He works on real-world applications of optimization and control in mechatronics, robotics, sustainable energy, and chemical engineering ... state vector, is the performance where is the input vector, is the performance input, is the physical output vector, is output, state matrix, is input matrix, and is the output matrix Using a static. .. transformed into a psd -convex-concave one If we denote the linearization of , i.e., iteration by at the s.t (27) where If this problem has and then terminate Phase and a solution together with. .. feedback controller and approximating it subsequently with an output feedback controller Step of Phase is usually TRAN DINH et al.: COMBINING CONVEX–CONCAVE DECOMPOSITIONS AND LINEARIZATION APPROACHES