P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the Riccati differential inequality on [0; 1] The importance of this inequality is well known in the LMI approach to H1 control, and the above inequality can be written AQ f Q f A Q f C C 1 Q f Gammafl I 0 5 where NR denotes a basis for the null space of . See [26] for more details on this approach. Here we have given a different intuition to the LMI solution for linear time invariant problems, and extended this approach to cover linear time varying systems also. 6.4 Terminal constraints We therefore might expect that one possible method for constructing ....

P. Gahinet and P. Apkarian. Linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 4(4):421--448, 1994.

....predictably) on some measurable parameter. Important developments toward being able to deal with the latter class of systems came through the restatement of the conditions for the existence of suboptimal controllers for LTI plants in terms of the existence of solutions to two Riccati inequalities ([33], 77] 78] Using affine matrix inequalities these can be reformulated as convex constraints on the space of real symmetric positive definite matrices, and allow some of the assumptions made in [24] to be relaxed [33] The inequality approach was adapted by Packard [67] to synthesise linear ....

.... LTI plants in terms of the existence of solutions to two Riccati inequalities ( 33] 77] 78] Using affine matrix inequalities these can be reformulated as convex constraints on the space of real symmetric positive definite matrices, and allow some of the assumptions made in [24] to be relaxed [33]. The inequality approach was adapted by Packard [67] to synthesise linear parameter dependent controllers for parameter dependent plants. By expressing both the plant and the controller as an LFT in the unknown parameter(s) he made use of the optimally scaled small gain theorem to pose the ....

P. Gabinet and P. Apkarian, A Linear Matrix Inequality Approach to H Control, Inter- national Journal of Robust and Nonlinear Control, Vol.4, pp 421-448, 1994.

.... set up will lead to a solution for the state space case, analogous to those on the standard H# problem obtained in [1] This double Riccati equation solution and its variations have been the subject of very intensive research, see e.g. 5, 14] and generalizations in [10, 11] 6, 7] 3, 4] and [2]. Assume the plant P full is given in input state output representation by x = Ax Bu Gd y = Cx Dd f = Hx Ju (2) The state variable x is assumed to take its values in R . The following three regularity conditions are assumed to hold: A.1: D is surjective and J is injective, ....

P. Gahinet and P. Apkarian, A linear matrix inequality approach to H# -control, International Journal of Robust and Nonlinear Control, volume 4, pages 421-448, 1994.

.... for generating u, output feedback control laws of the form u = t) can be envisioned; a simple example of such a control law is constant output feedback u = Ky(t) If in addition to the measured output, the uncertainty # in a polytopic system (or # in an LFR system) is measurable in real time [8, 9, 10, 11, 12, 13], a control law u = #, t) or u = #, t) that explicitly depends on the uncertainty can be implemented. This is the so called gain scheduled controller. The problem of synthesizing robustly stabilizing constant state feedback for both polytopic and LFR systems, using quadratic Lyapunov ....

P. Gahinet and P. Apkarian, "A linear matrix inequality approach to H# control," Int. J. Robust and Nonlinear Contr., vol. 4, pp. 421--448, 1994.

....by Doyle et al. 23] It is based on the solution of two coupled Riccati equations, with a constraint expressed in terms of a spectral radius. The class of all controllers are built around the socalled central controller and parametrized by a free parameter Q . However, as pointed out in [30] and [53] a major de ciency of the controller design approach in [23] is that Q is dicult to choose in practice. Moreover, the results in [23] can only be applicable to regular systems. The main feature of solving the Riccati equations requires the tuning of a symmetric positive de nite matrix, ....

....approach of [23] dicult to use. To get around these problems, an LMI approach has been proposed to solve the H 1 control problem. The solvability conditions are given in terms of LMI and the parametrization of all H 1 suboptimal controllers including reduced order controllers are proposed in [30, 53]. These results are applicable to both regular and singular plants. Solving the LMI can be done eciently by using interior point methods [74, 11] To show the main features of the LMI approach to solving the H 1 control problem, we consider the following linear continuous time invariant ....

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P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. Int. J. Robust Nonlinear Control, 4:421-448, 1994.

....representation, and the weighting func2 tional is a constant two variable polynomial matrix. In this case, our general results of part I lead to solvability conditions in terms of algebraic Riccati equations and inequalities, as studied before in [1] 4] 7] 12] 13] 14] 15] [3], 18] 16] 17] 5] 6] 9] 10] In section 6 of the paper, we address the important issue of feedback implementability. We nd conditions under which a controller that renders a system dissipative, can implemented as a feedback controller The proofs of the results can be found in section ....

.... lead to a solution for the state space case, analogous to those on the standard H1 problem obtained in [1] This double Riccati equation solution and its variations have been the subject of very intensive research, see e.g. 7, 18] and generalizations in [12, 13] 16, 17] 9, 10] 5, 6] and [3]. Whereas most of the existing literature deals with the problem of nding an internally stabilizing controller such that the H1 norm of the closed loop transfer function is strictly less than 1, we deal with the problem of making the H1 norm of this transfer function less than or equal to 1, ....

P. Gahinet and P. Apkarian, A linear matrix inequality approach to H1 -control, International Journal of Robust and Nonlinear Control, volume 4, pages 421-448, 1994.

....dependent nature of the designed controllers. Many extensions and generalizations have been done since the appearance of these two conference papers [34] 27] 31] 21] More recently, the LMI treatment has been extended to cope with robust performance synthesis for continuous time systems [17], 2] 22] A fairly complete review of LMIs and their roles in control theory is given in [8] also a software toolbox for the numerical solutions to LMIs is available [18] This paper is presented in an axiomatic fashion in the order: Section II.1, Section III.1, Section IV, and Section V. The ....

P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H1 Control, Int. J. of Robust and Nonlinear Control, Vol.4, pp.421--448, 1994.

....efficient. A typical example of this is the computation of solutions of Riccati inequalities, appearing in H1 control. For these problems, under appropriate regularity hypotheses, the feasibility of the Riccati matrix inequality implies the solvability of the algebraic Riccati equation [34]. In this case, it is not necessary to solve LMIs, but instead just solve Riccati equations, at a lower computational cost. Similarly, the results in this chapter show that for a certain class of LMIs, the optimal solution can be computed by alternative, faster methods than general purpose LMI ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 4(4):421--448, 1994.

....always assumed to be nonempty. 4 2.1 The matrix inequalities (2.9) can be written in terms of a rank constrained LMI. This in turn, will lead to a new characterization of the Lyapunov certi cates for the static output feedback stabilizability. First, we recall the following result. Lemma 2. 1 ([15]) Let M 2 S n , B 2 R n m , and C 2 R p n . Then there exists a matrix K 2 R m p such that M BKC C T K T B T 0; 2.14) if and only if there exists (M) 0 such that for all (M) M BB T ; and M C T C: 2.15) The proof of Lemma 2.1 involves the application ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 43:421-448, 1994.

....(quadratic) Lyapunov function to enforce the specifications. If we restrict attention to a subset of these specifications, then our conditions are either necessary or sufficient, or are equivalent to conditions obtained recently in the literature. Our approach follows closely that developed in [10,5,3,8,7] for single objective, output feedback problems. Note that the paper by Scherer [11] published shortly after submission of this paper, uses a similar approach, and includes many results given here. Notation. For a matrix P , P 0 (resp. P 0) means P is symmetric positive definite (resp. ....

....that satisfy constraints (15) 16) 17) 18) and (21) we can reconstruct a controller as follows. Any of the inequalities (15) 16) 17) 18) implies Q 0, P Q Gamma1 . Without loss of generality, we may assume P Q Gamma1 (this restricts our search to full order controllers only [5]) We can choose an arbitrary invertible matrix M 2 R n Thetan , and set P as in (5) Find B; C from (14) Then, find A such that (19) holds. With P fixed, this is simply an LMI problem in A, which is guaranteed to be feasible. The choice of M has no effect on the controller s ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. Int. J. Robust and Nonlinear Contr., October 1992.

....cl k fl. Although there is a growing interest in H 1 type problems for stochastic systems there are, as yet, relatively few publications in the area. Most of them deal with continuous time systems. In the deterministic context the discrete time H 1 control problem has been analyzed e.g. in [2]. In the stochastic field [1] studies a state feedback version of the H 1 control problem for infinite dimensional discrete time stochastic bilinear systems in which the underlying model corresponds to the special case where the matrix B 0 in (1) is zero. In the present paper we develop a ....

.... ker [B 2 D 12 ] and GammaS A SA Gamma C 1 C 1 A SB 1 Gamma C 1 D 11 B 1 SA Gamma D 11 C 1 fl 2 I B 1 SB 1 Gamma D 11 D 11 0 on ker [C 2 D 21 ] These are equivalent to the LMI solvability conditions for the discrete time H 1 control problem, see [2]. c) In the deterministic case the two linear matrix inequalities (respectively, the corresponding Riccati inequalities) have the same structure and they are decoupled. In the stochastic case the first matrix inequality is coupled to the second. Moreover the two inequalities are of a very ....

P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H 1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

....is a growing interest in stability radii and H 1 type problems for stochastic systems there are as yet relatively few publications in the area. Most of them deal with continuous time systems. In the deterministic context the discrete time H 1 control problem has been analyzed e.g. in [24] [8] and [26] In the stochastic field [2] deals with a state feedback version of the H 1 control problem for discrete time stochastic bilinear systems. The underlying model corresponds to the special case where all the matrices B i 0 in (1) are zero; however, the basic state space is an infinite ....

....a rational Riccati inequality. The proof of these results proceeds via the analysis of an associated finite time optimal control problem. In Section 3 we solve the disturbance attenuation problem for systems of the form (1) Here we follow the LMI approach developed for deterministic systems in [8], 18] and for continuous time stochastic systems in [14] Necessary and sufficient conditions are derived for the existence of a stabilizing controller ensuring that the norm of the closed loop perturbation operator is below a prespecified bound. These conditions take the form of coupled linear ....

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P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H 1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

....types of disturbances in (1) is essential to obtain a full generalisation of the H 1 control problem to the stochastic context. As special cases it contains on the one hand (A 0 = 0; B 0 = 0) the general deterministic H 1 control problem (without regularity assumptions) as stated e.g. in [8], 9] 17] 25] 26] On the other hand it also includes the purely stochastic case where B 1 = 0, see [16] It may seem odd that we use the same disturbance vector v in both the deterministic and the stochastic disturbance terms. But this is in fact more general since distinct disturbance ....

....Section 2 deals with a problem of system analysis (under which conditions has the input output operator L of a stable stochastic system a norm kLk fl ) the synthesis problem of H 1 control will be treated in Section 3. Here we follow the LMI approach developed for deterministic systems, see [8], 17] The idea is to apply the Stochastic Bounded Real Lemma to the compensated system Sigma cl in such a way that the matrices of compensator parameters which achieve stability and kL cl k fl are characterized by a linear matrix inequality. Then applying the projection lemma [8] we are able ....

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P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H 1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

....corresponding finite energy output signals z of the closed loop system. The size of this linear operator is measured by the induced norm and the problem is to determine, for any fl 0 whether or not there exists a stabilizing controller K achieving kL cl k fl. The LMI approach to H 1 control [3], 6] has shown that the mathematical centrepiece of the state space theory of deterministic H 1 control is the Bounded Real Lemma. This lemma characterizes the norm of the perturbation operator of a given system in terms of a parametrized algebraic Riccati Inequality or the corresponding ....

.... the stochastic system (1) can be resolved, see [5] The stabilizing controllers K achieving kL cl k fl are characterized by LMI s which in the special case that A 0 = 0; B 0 = 0 reduce to the LMI s characterizing the fl Gammasuboptimal controllers in the general deterministic case, see [3], 6] If B 0 = 0 and D = 0 the rational Riccati equation (13) reduces to the algebraic Riccati equation: PA A P A 0 PA 0 Gamma C C Gamma fl Gamma1 PBB P = 0: In [8] Morozan has used the corresponding differential Riccati equation to characterize the norm of the perturbation ....

P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

....in the literature. The proof of the existence result proceeds via an analysis of the corresponding finite time optimal control problem. In Section 3 we solve the disturbance attenuation problem for systems of the form (1) Here we follow the LMI approach developed for deterministic systems in [5], 12] In Section 4 we discuss the simplification of the results under regularity assumptions and derive explicit formulae for full order stabilizing compensators achieving kL cl k fl. Finally in section 5 we apply the previous results to obtain estimates for the stability radius of a ....

....or not there is a compensator (45) which stabilizes system (42) internally and achieves kL cl k fl. Such controllers will be called suboptimal of level fl. To solve the disturbance attenuation problem we follow the LMI approach developed by Gahinet and coworkers for the deterministic case [5]. A proof of the following lemma can be found in [5] Lemma 3.1 (Projection Lemma) Suppose N 2 K Thetam ; M 2 K n Thetam and H 2 Hm (K ) Then the linear matrix inequality H N X M M XN 0 has a solution X 2 K n Theta if and only if H is positive definite on ker N and on ker ....

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P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H 1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

....gain bounded systems, positive real systems, and sector bounded systems, follow simply by substituting their respective power functions in the general results. The bounded realness lemma for characterizing unity gain bounded systems, or bounded real systems[9, 10] and the LMI form of this condition[14, 15] follow by substituting the power function for bounded real systems, that is, setting R = I; Q = GammaI ; and N = 0; in the results of Theorem 3.1. Corollary 3.1 Consider a stable LTI system, Sigma; with a minimal realization, A; B;C;D) The following statements are equivalent. 1. The LTI ....

....with respect to the given quadratic power function. Note that the problem of maximization of a linear objective function subject to LMI constraints is a convex programming problem. Thus, a global maximum for the parameter t exists, and may be computed using efficient numerical technqiues [14, 34]. Often, it is desirable to obtain the coefficient matrices, Q = Q T ; R = R T ; and N of the power function, such that a given LTI system is dissipative with respect to that power function. Note that the matrix inequality of the dissipativity lemma is linear with respect to the matrix P = P ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to h1 control. International Journal of Robust and Nonlinear Control, 4:421--448, 1994.

....are equivalent: 1. AX XA 0 Q 0. 2. For all a 0, aI Gamma A)X(aI Gamma A 0 ) Gamma (aI A)X(aI A 0 ) Gamma 2aQ 0: Proof: The proposition can be verified by simply expanding the left hand side of the second inequality above and dividing both sides by a 0. 2 Lemma 2. 2 ([12]) Let M 2 SR n Thetan , P 2 R n Thetap , and Q 2 R q Thetan . Then the following statements are equivalent: 1. There exists a matrix K 2 R p Thetaq such that, M PKQ Q 0 K 0 P 0 0: 2. There exists a positive number fl such that for all fl fl, M flP P 0 ; and M flQ ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 43:421--448, 1994.

....if kT 1 k1 fl, where kT 1 k1 : sup w2L 2 ;kwk 2 =1 kT 1 wk 2 defines the operator norm of T 1 induced by signals in L 2 . In the LTI case, there is a wellknown test on the state space matrices which characterizes stability of A and kT 1 k1 fl, the so called Bounded Real Lemma (BRL) [57, 22, 26]. It is not difficult to prove the following generalization to LTV systems. Recall that time functions are bounded, and that a symmetric valued time function X is strictly positive (X AE 0) if there exists an ffl 0 with X(t) fflI for all t 0. Theorem 1 (LTV Strict Bounded Real Lemma) The time ....

....following observations: The BRL inequality in (8) is, for a fixed X , linear in the controller parameters. Hence one can eliminate these parameters using the so called Projection Lemma (Lemma 7) what leads to a suboptimality test in terms of linear inequalities in (parts) of X and X Gamma1 [22, 26, 49]. In the mixed problem (8) we have to fulfill three inequalities what makes it impossible to eliminate all controller parameters from the final characterization. Instead, we intend to keep as many (transformed) controller parameters as possible such that, still, matrix inequalities result which ....

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Gahinet, P., Apkarian, P.: A linear matrix inequality approach to H1 control. Int. J. of Robust and Nonlinear Control 4 (1994) 421--448

...., V V are both invertible, and that U U = 0 and V V = 0. Then inf K2C p Thetam oe (Q UKV ) 21) if and only if V Gamma Q Q Gamma 2 I Delta V 0 (22) and U Gamma QQ Gamma 2 I Delta U 0: 23) Proof: See [1] 2 A related lemma [4, 5], goes as follows. Lemma 3.2 Given a hermitian matrix Q 2 C n Thetan and U , V , U and V as above. Then Q UKV (UKV ) 0 (24) is solvable for K 2 C p Thetam if and only if ae U QU 0 V QV 0: 25) Proof: Necessity of (25) is clear: for instance, U U = 0 implies ....

....) 0 (24) is solvable for K 2 C p Thetam if and only if ae U QU 0 V QV 0: 25) Proof: Necessity of (25) is clear: for instance, U U = 0 implies U QU 0 when pre and post multiplying (24) by U and U respectively. For details on the sufficiency part, see [4, 5]. 2 In [5] the set of K solving (24) is parametrized. 3.5 Complex synthesis If we restrict the problem to only have complex uncertainties (real uncertainties can be conservatively treated us complex ones) the following lemma from [10] applies. Lemma 3.3 Let Q, U , V , U and V be given as ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 1:656--661, October 1992.

....V 2 C m Thetan have full column rank and row rank respectively. Then inf K2C p Thetam oe (Q UKV ) 22) if and only if V Gamma Q Q Gamma 2 I Delta V 0 (23) and U Gamma QQ Gamma 2 I Delta U 0: 24) Proof: See [2] 2 A related lemma [5, 6], goes as follows. Lemma 4.3 Given a hermitian matrix Q 2 C n Thetan and U , and V as above. Then Q UKV (UKV ) 0 (25) is solvable for K 2 C p Thetam if and only if ae U QU 0 V QV 0: 26) Proof: Necessity of (26) is clear: for instance, U U = 0 ....

....(25) is solvable for K 2 C p Thetam if and only if ae U QU 0 V QV 0: 26) Proof: Necessity of (26) is clear: for instance, U U = 0 implies U QU 0 when pre and post multiplying (25) by U and U respectively. For details on the sufficiency part, see [5, 6]. 2 In [6] the set of K solving (25) is parametrized. 4.3 Complex synthesis If we restrict the problem to only have complex uncertainties, the following lemma from [12] applies. Lemma 4.4 Let Q, U , and V be given as above. Then inf K2C p Thetam D2D oe Gamma D(Q UKV )D ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 1:656--661, October 1992.

....a coupling condition and a number of additional linear matrix inequalities. Moreover we give explicit formulae for suboptimal controllers. These results are obtained by using an inequality approach to de2 terministic H 1 control theory developed by Gahinet and his co workers, see [10] [11]. Whereas in the deterministic case the suboptimal controllers can be characterized by a pair of Riccati equations and a coupling condition, it is not possible in the stochastic case to replace both the Riccati and the Liapunov inequalities by equalities. This will be illustrated by an example. We ....

....of dynamic compensators of the form (53) that stabilize the system and achieve a stability radius r w K (A; D i ; E i ) i2N ) oe, for a given oe 0. We follow an approach based on inequalities similar to the one that Gahinet developed in his approach to the H 1 control problem, see [10] [11]. We proceed in two steps. First we derive some necessary conditions and then we show that these conditions are also sufficient for building a stabilizing compensator of dimension n which achieves the required stability radius. We will make use of the following criterion for the positive ....

[Article contains additional citation context not shown here]

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. Int. J. Robust and Nonlinear Control, 4: 421-448, 1994.

....types of disturbances in (1) is essential to obtain a full generalisation of the H 1 control problem to the stochastic context. As special cases it contains on the one hand (A 0 = 0; B 0 = 0) the general deterministic H 1 control problem (without regularity assumptions) as stated e.g. in [7], 8] 14] 20] 21] On the other hand it also includes the purely stochastic case where B 1 = 0, see [13] It may seem odd that we use the same disturbance vector v in both the deterministic and the stochastic disturbance terms. But this is in fact more general since distinct disturbance ....

....Section 2 deals with a problem of system analysis (under which conditions has the input output operator L of a stable stochastic system a norm kLk fl ) the synthesis problem of H 1 control will be treated in Section 3. Here we follow the LMI approach developed for deterministic systems, see [7], 14] The idea is to apply the Bounded Real Lemma to the compensated system Sigma cl in such a way that the matrices of compensator parameters which achieve stability and kL cl k fl are characterized by a linear matrix inequality. Then applying the projection lemma [7] we are able to obtain ....

[Article contains additional citation context not shown here]

P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

....on the performance criteria variable. Linear Matrix Inequality(LMI) has become more and more important in control engineering . Not only because of the way it represents the problem, but also because of the state of art interior point method which can solve the problem efficiently. In paper [2], LMI conditions for the H1 control of linear systems are given based on the bounded real lemma. While in [5] a complete LMI design procedure for linear H1 controller is presented. In this paper we also want to convert the problem into LMI conditions. Then by finding the feasible solution of ....

P. Gahinet and P. Apkarian, A linear matrix inequality approach to H1 Control, Int. Journal of Robust and Nonlinear Control, Vol. 4, No. 4, pp. 421-428, 1994

....I) f0g for all 2 oe(A) C 0 . If we drop the structural constraint X = X and consider instead A X X A Q 0, things become considerably easier. The even more general inequality A XB B X A Q 0 has recently proved important in the H1 problem and variations thereof [3, 1]. This inequality is solvable (and one can explicitly construct solutions) iff Q ker(A) 0 and Q ker(B) 0. In this paper we need a generalization to the LMI k X j=1 (A j X j B j B j X j A j ) Q 0 (1) for which we can indeed give an algebraic solvability test and a procedure to ....

P. Gahinet, P. Apkarian, A linear matrix inequality approach to H1 control, Int. J. of Robust and Nonlinear Control 4 (1994) 421-448.

....[0; 1) including 1. 3. There is a P (Hermitian or not) that satisfies (15) iff A is nonsingular, and G(s) G(s) H 0 at s = 0 and s = 1. Pi Proof: Item 1 is the KYP lemma in a strict inequality version (see e.g. Willems [11] Item 3 is a consequence of Lemma 3. 1 of Gahinet and Apkarian [6], which says that if F 0 is Hermitian, then F 0 V H PU U H P H V 0 for some P iff U H F 0 U 0 and V H F 0 V 0, where U and V are matrices that span the kernel of U and V respectively. Items 1 and 3 are included for comparison only. We prove Item 2. Define E ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to h 1 control. Int. J. of Robust and Nonlinear Control, 4:421--448, 1994.

....(11) is solvable for Theta if and only if ae U T PsiU 0 V T PsiV 0: 12) Proof. Necessity of (12) is clear: for instance, U T U = 0 implies U T PsiU 0 when pre and post multiplying (11) by U T and U respectively. For details on the sufficiency part, see [3]. 2 Using this lemma, step (ii) in Algorithm 3.1 can be simplified by removing C and D. To achieve this we start by rewriting (9) as 2 6 6 4 A T P P A P B C T 0 B T P Gammafl I 0 Theta C 0 0 Gammafl I 3 7 7 5 U Theta C D V T V C T D ....

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 1:656--661, October 1992 (to appear in).

.... system (4) and (7) with the loop w Theta = Theta(t)z Theta open, that is, 8 : x xK = A c x c B c w w z z = C c x c D c w w : 10) The Bounded Real Lemma conditions (8) are further simplied by means of the Projection Lemma [14, 21], and we obtain the following characterization amenable to numerical computations. Theorem 2.1 Consider the LFT plant (4) where Theta is ranging over the polytope P dened in (5) Let NX and N Y denote any bases of the null spaces of (C 2 ; D 2 Theta ; D 21 ; 0) and (B T 2 ; D T Theta2 ; D ....

P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H1 Control, Int. J. Robust and Nonlinear Control, 4 (1994), pp. 421448.

....and or global optimization techniques. 1 Introduction A number of challenging problems in robust control theory fall within the class of rank minimization problems subject to LMI (convex) constraints. An important example is provided by the reduced order H1 control problem. It has been shown in [20, 4, 13] that there exists a k th order controller solving the H1 control problem of a plant with n th order if and only if one can nd a pair of symmetric matrices (X; Y ) with dimension n Theta n such that for some H1 performance level fl the following holds. X; Y; fl) 2 L; 1) Rank X I I Y n k ....

....of the closed loop system x = A BKC)x; x(0) x 0 (15) converges asymptotically to zero as time increases. An equivalent formulation [15] is the existence of a symmetric matrix X such that the matrix inequalities (A BKC) T X X(A BKC) 0 (16) X 0 (17) hold. Using the Projection Lemma [4] in (16) The problem is then easily reformulated as the matrix inequalities of (17) together with N T C (A T X XA)NC 0; 18) N T B T (Y A T AY )N B T 0 (19) with Y = X Gamma1 . Thus, as in Section 2, this problem is equivalent to that 0 be the optimal value of the following ....

P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H1 Control, Int. J. Robust and Nonlinear Control, 4 (1994), pp. 421448.

....and randomized numerical experiments. 1 Introduction A number of challenging problems in robust control theory fall within the class of rank minimization problems subject to LMI (convex) constraints. An important example is provided by the reduced order H1 control problem. It has been shown in [28, 10, 20] that there exists a k th order controller solving the H1 control problem ioe one can nd a pair of symmetric matrices (X; Y ) such that for some H1 performance level fl the following holds. L(X;Y; fl) 0 ; 1) Rank X I I Y n k ; 2) where n designates the plant s order and (1) is an LMI in ....

P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H1 Control, Int. J. Robust and Nonlinear Control, 4 (1994), pp. 421448.

.... (v) nally, compute AK , BK and CK with the help of AK = N Gamma1 (X Y N M T b AK Gamma X(A Gamma B 2 DKC 2 )Y Gamma b BKC 2 Y Gamma XB 2 b CK )M GammaT (54) BK = N Gamma1 ( b BK Gamma XB 2 DK ) 55) CK = b CK Gamma DKC 2 Y )M GammaT : 56) The reader might consult [17, 16, 22, 37, 1] for details on this construction. 6. Numerical examples . In this section, the concepts and tools developed above are illustrated by some numerical examples. All LMI related computations were performed on a ULTRA 1 SUN station using the LMI Control Toolbox [19] 6.1. Stability analysis. We ....

....statement (i) of the theorem. In turn, the latter conditions are equivalent to the existence of continuously parameter dependent symmetric matrices X( and Y ( such that LMI conditions (38) 40) hold for all ( on H Theta H d . This assertion is a slight extension of the main result in [44, 17, 22]. Assume a (closed loop) Lyapunov function establishing L 2 gain performance is V (x; xK ; x xK T P ( x xK ; where P : X N N T ; P Gamma1 : Y M M T : It trivially holds that I Gamma XY = NM T : Then, dening Pi Y : Y I M T 0 ; Pi X ....

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P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H1 Control, Int. J. Robust and Nonlinear Control, 4 (1994), pp. 421448.

....for all parameter trajectories Delta(t) dened by (6) It is now well known that such problems can be handled via a suitable generalization of the Bounded Real Lemma involving adequately dened class of scalings. The Bounded Real Lemma conditions are then simplied by means of the projection Lemma [13, 25]. Proposition 3.1 Consider the LFT plant governed by (5) where Delta is ranging over a polytopic set P dened in (6) Let KX and KY denote any bases of the null spaces of [C2 ; D 2 Delta ; D21 ] and [B T 2 ; D T Delta2 ; D T 12 ] respectively. Then, there exists a controller such that the ....

P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H1 Control, Int. J. Robust and Nonlinear Control, 4 (1994), pp. 421448.

....problems such as robustness analysis, or Linear Parameter Varying (LPV) control are then thoroughly discussed and illustrated by examples. 1 Introduction Linear matrix inequalities (LMIs) have emerged as a very powerful tool in the analysis and synthesis for robust control problems (see e.g. [6, 8, 12] and references therein) From a computational point of view, LMIs are very attractive because they consist of convex constraints and therefore can be solved by very efficient convex optimization interior points methods with worst case polynomial complexity [14, 22] From the viewpoint of robust ....

P. Gahinet, P. Apkarian, A linear matrix inequality approach to H1 control, Inter. J. of Robust and Nonlinear Control 4(1994), 421-448.

....Frank and Wolfe algorithms. 1 Introduction A number of challenging problems in robust control theory fall within the class of rank minimization problems subject to LMI (convex) constraints. An important example is provided by the reducedorder H1 control problem. It has been shown in [24, 8, 18] that there exists a k th order controller solving the H1 control problem if and only if one can nd a pair of symmetric matrices (X; Y ) satisfying LMIs constraint with Rank X I I Y n k ; 1) ONERA CERT, Control System Dept. 2 av. Edouard Belin, 31055 Toulouse, FRANCE Email : ....

P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H1 Control, Int. J. Robust and Nonlinear Control, 4 (1994), pp. 421448.

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P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H1 control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

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P. Gahinet and P. Apkarian. linear matrix inequality approach to H1 control. International J. Robust and Nonlinear Control 4: 421-448, 1994.

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P. Gahinet and P. Apkarian. A Linear Matrix Inequality approach to H control. Int. J. Robust and Nonlinear Control 4: 421-448, 1994.

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P. Gahinet and P. Apkarian. A Linear Matrix Inequality Approach to H# Control. Int. J. of Robust and Nonlinear Control, 4:421--448, 1994.

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P. Gahinet and P. Apkarian, "A linear matrix inequality approach to H -control," Int. J. Robust Nonlinear Control, vol. 4, pp. 421--448, 1994.

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P. GAHINET and P. APKARIAN. A linear matrix inequality approach to H # control. Int. J. Robust and Nonlinear Control, 4:421--448, 1994.

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P. Gahinet and P.Apkarian. A linear matrix inequality approachtoH1 control. Int. J. Robust Non. Contr.: Special Issue on H1 Control, 4:421-438, 1994.

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P. Gahinet and P. Apkarian. linear matrix inequality approach to H1 control. International J. Robust and Nonlinear Control 4: 421-448, 1994.

No context found.

P. Gahinet and P. Apkarian, A linear matrix inequality approach to H1 Control, Int. Journal of Robust and Nonlinear Control 4(4):421-428 (1994).

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P. Gahinet and P. Apkarian, A linear matrix inequality approach to H1 Control, Int. Journal of Robust and Nonlinear Control, Vol. 4, No. 4, pp. 421-428, 1994

No context found.

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 4(4):421--448, 1994.

No context found.

P. Gahinet and P. Apkarian. A linear matrix inequality approach to H1 control. International Journal of Robust and Nonlinear Control, 4(4):421--448, 1994.

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