S. Boyd, Q. Young, Structured and simultaneous Lyapunov functions for system stability problems, Int. J. Control, vol. 49, 1989, pp. 2215--2240.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... we say that the system is Q stable if there exists a real differentiable positive definite matrix function P(p) P(p) 0 such that tP(P(t) A(p(t) P(p(t) P(p(t) A(p(t) 0 If we restrict the matrix P to be constant then we recover the familiar definition of quadratic stability ( 7] [11], 12] see also the review article in [17] For presentation purposes we now drop p dependence in the equations although it will be implicitly assumed that the matrices are functions of p throughout. Proposition 2.4.7 The notion of Q stability is a system property, independent of any particular ....

S.Boyd and Q. Yang, Structured and Simultaneous Lyapunov Functions for System Stability Problems, Int. J. Control, Vol.49, No.6, pp 2215-2240, 1989. 232

....general theory. Examples of results that use such additional structure include the so called Switching Theorem which plays a role in the supervisory control of uncertain linear systems [4] and conditions for existence of a common Lyapunov function which exploit positive realness [78, Chapter 33] [79], 80] We 15 believe that to make significant further progress, one must stay in close contact with particular applications that motivate the study of switched systems. Acknowledgment It is a pleasure to thank Joo Hespanha for illuminating discussions on many topics related to the material of ....

S. Boyd and Q. Young, "Structured and simultaneous Lyapunov functions for system stability problems," Int. J. Control, vol. 49, pp. 2215-2240, 1989.

....to Lyapunov s theorem, also enables the extension of Proposition 2 to more general settings. Consider the problem of the existence of P satisfying P 0; A 1 P PA 1 0; A 2 P PA 2 0: 11) The matrix P can be interpreted as defining a common or simultaneous quadratic Lyapunov function [BY89, BEFB94, SN98, SN99, SN00] that proves the stability of the time varying system x = A(t)x; A(t) l(t)A 1 (1 l(t) A 2 ; l(t) 2 [0; 1] for all t: An application of Theorem ALT 1 immediately yields a necessary and sufficient condition for (11) to be feasible: There do not exist Z 1 ; Z 2 2 S n such that diag(Z 1 ; Z 2 ....

S. Boyd and Q. Yang. Structured and simultaneous Lyapunov functions for system stability problems. Int. J. Control, 49(6):2215--2240, 1989.

....theory. Examples of results that use such additional structure include the so called Switching Theorem which plays a role in the supervisory control of uncertain linear systems [4] and conditions for 15 existence of a common Lyapunov function which exploit positive realness [78, Chapter 33] [79], 80] We believe that to make significant further progress, one must stay in close contact with particular applications that motivate the study of switched systems. Acknowledgment It is a pleasure to thank Jo ao Hespanha for illuminating discussions on many topics related to the material of ....

S. Boyd and Q. Young, "Structured and simultaneous Lyapunov functions for system stability problems," Int. J. Control, vol. 49, pp. 2215--2240, 1989.

....of query based adaptive control systems concerns the selection of P . Prior work [Lemmon 1993] assumed that P was known and an example was used in which P was the identity matrix. Unfortunately, the set of plants for which P = I is extremely small. A better way to determine P is suggested in [Boyd 1989]. In this approach, it is assumed that the system matrix, A, lies in a matrix interval, A 1 ; A 2 ] Letting M i (i = 1; N) be a finite set of square symmetric matrices, where N = n(n 1) 2 and n here indicates the number of gains to be determined. we define a basis for a matrix G, G i ....

....where N = n(n 1) 2 and n here indicates the number of gains to be determined. we define a basis for a matrix G, G i = GammaM i Phi (A T 1 M i M i A 1 ) Phi (A T 2 M i M i A 2 ) 13) where A Phi B denotes the block diagonal matrix with A and B on its diagonals. With these definitions Boyd [Boyd 1989] has shown that there exists a Lyapunov matrix P for which all matrices in [A 1 ; A 2 ] are P stable if and only if there exist positive coefficients a i (i = 1; N) such that G = N X i=1 a i G i 0 (14) Note that since equation 14 is affine in the coefficients a i , linear ....

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S. Boyd, Q. Yang. Structured and Simultaneous Lyapunov Functions for System Stability Problems. Int. J. Control, 1989, vol. 49, No. 6, 22152240. 13

.... C 2 C q Thetaq such that (A Gamma j I) H C H C (A Gamma j I) 0; 8A 2 A (49) Taking C = C = C H 0, the above reduces to quadratic stability: A H C CA 0; 8A 2 A (50) Example of using the multiplier approach to quadratic stability analysis can be found in Boyd and Yang [3] and Boyd et al. 2] A follow up paper [13] offers several other interesting properties of the multiplier approach. Namely, an equivalence among several multiplier schemes is established. The computation of the new upper bounds is formally formulated as an generalized eigenvalue problem which ....

S. Boyd and Q. Yang, "Structured and Simultaneous Lyapunov Functions for System Stability Problems," Int. J. Contr., vol. 49, pp. 2215-2240, 1989.

....the values of resistors and capacitors in electrical circuits, etc. Even though this problem is NP hard in general, a number of more or less conservative tests are available to estimate stability regions. These include Kharitonov s theorem and related results [26, 5] quadratic stability tests [22, 6, 29, 25], and the real or Km stability margins [14, 15, 33, 34] y The first and third authors are with INRIA Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France. E mail: gahinet colorado.inria.fr z The second author is with CERT ONERA, 2 av. Edouard Belin, 31055 Toulouse ....

....Lyapunov function V (x) x T Px proves stability of (1. 1) for all parameter trajectories (t) When each entry of A( is a ratio of multilinear functions of , it is shown in [20] that finding an adequate P amounts to solving a system of Lyapunov inequalities, which is a convex program [6]. Quadratic stability guarantees stability against arbitrarily fast parameter variations. As a result, this test can be very conservative for constant or slowly varying parameters, even in its refined form discussed in [40] To reduce conservatism in the case of constant parameters, Barmish ....

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Boyd, S., and Q. Yang, "Structured and Simultaneous Lyapunov Functions for System Stability Problems," Int. J. Contr., 49 (1989), pp. 2215-2240.

.... be taken linear time invariant, the H1 theory and its ramifications offer powerful synthesis tools for achieving robust performance (see, e.g. 10, 12, 24] and references thererin) Even though parametric uncertainty remains delicate to handle, the synthesis [9, 5] and Lyapunov based techniques [7, 27, 36, 17] give satisfactory results in many applications. In comparison, available design techniques for uncertain linear time varying (LTV) plants are relatively immature. Recall that a LTV plant is any linear system governed by state equations of the form: x(t) A(t) x(t) B(t) u(t) y(t) C(t) x(t) ....

Boyd, S., and Q. Yang, "Structured and Simultaneous Lyapunov Functions for System Stability Problems," Int. J. Contr., 49 (1989), pp. 2215-2240.

....quadratic stability of X is a reasonable choice for a sufficient condition for robust Stability. In Section II we discuss the significance of resorting to this stringent type of stability. The problem of robust quadratic stability has received a considerable amount of attention in literature e.g. [1], 6] 8] 10] 12] 13] 16] 18] 19] 24] Formally, if we denote by P the set of symmetric positive definite matrices, then the system (1 2) is said to be robustly quadratically stable if there exists a single P 2 P which simultaneously satisfies all the algebraic Lyapunov inclusions ....

S. Boyd, Q. Yang, "Structured and Simultaneous Lyapunov Functions for System Stability Problems", Int. J. Contr., Vol. 49, No. 6, pp. 2215-2240, 1989.

....Engineering Department, Stanford University, Stanford CA 94305 We consider nonlinear systems dz dt = f(x(t) where Df(x(t) is known to lie in the convex hull of L matrices A1, AL C R X. For such systems, quadratic Lyapunov functions can be determined using convex programming techniques [1]. This paper de scribes an algorithm that either finds a quadratic Lyapunov function or terminates with a proof that no quadratic Lyapunov function exists. The algorithm is an interior point method based on the theory developed by Nesterov and Nemirovsky [2] 1. AN EQUIVALENT OPTIMIZATION ....

S. Boyd and Q. Yang. Structured and simultaneous Lyapunov functions for system stability problems. InL d. Conrol, 49(6):2215-2240, 1989.

.... work includes Cullum et al. [CDW75] Craven and Mond [CM81] Polak and Wardi [PW82] Fletcher [Fle85] Shapiro [Sha85] Friedland et al. FNO87] Goh and Teo [GT88] Panier [Pan89] Allwright [All89] Overton [Ove88, Ove92, OW93, OW92] Ringertz [Rin91] Fan and Nekooie [FN92] and Fan [Fan92] In [BY89], Boyd and Yang use the cutting plane algorithm and Shor s subgradient method [Sho85] to solve eigenvalue minimization problems that arise in control theory. They also describe a saddle point method for eigenvalue mimimization due to Pyatnitski and Skorodinsky [PS83] Interior point methods for ....

....of computing the Newton direction. 7 An example 7.1 A Lyapunov function search problem We consider a simple example of determining a Lyapunov function that optimizes a decay rate estimate for a linear differential inclusion. More detail on this and similar problems can be found in [BGFB93] or [BY89]. We consider the differential equation i (t)G i where G i 2 R N ThetaN (and do not depend on t) and the i (t) satisfy i (t) 1, i (t) 0, but are otherwise arbitrary. Given any P = P 0, let V (z) z P z. For y(t) satisfying (84) we have dt V (y(t) ....

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S. Boyd and Q. Yang. Structured and simultaneous Lyapunov functions for system stability problems. Int. J. Control, 49(6):2215--2240, 1989.

....is defined as sup u6=0 k Delta(u)k 2 kuk 2 : Research supported in part by AFOSR under contract F49620 92 J 0013. kuk 2 stands for the usual L 2 norm of u. The matrices P in (4) are input output scalings that leave the block structure of the system (1,2) invariant. For more details, see [1]. 2. Computation of the Robustness Measure The strict bounded real lemma states the following (see e.g. 2] For a given transfer matrix H(s) C(sI Gamma A) B, with (A; B; C) minimal, the following statements are equivalent: 1. H(s) is stable and kHk1 fl. 2. There exists X = X 0 ....

S. Boyd and Q. Yang, Structured and simultaneous Lyapunov functions for system stability problems, Int. J. Control, 49(6):2215--2240, 1989.

....F49620 92 J 0013. 1= H) where the L 2 gain of an operator Delta is defined as sup u6=0 k Delta(u)k 2 kuk 2 : kuk 2 stands for the usual L 2 norm of u. The matrices P in (4) are input output scalings that leave the block structure of the system (1,2) invariant. For more details, see [1]. 2. Computation of the Robustness Measure The strict bounded real lemma states the following (see e.g. 2] For a given transfer matrix H(s) C(sI Gamma A) Gamma1 B, with (A; B; C) minimal, the following statements are equivalent: 1. H(s) is stable and kHk1 fl. 2. There exists X = X ....

S. Boyd and Q. Yang, Structured and simultaneous Lyapunov functions for system stability problems, Int. J. Control, 49(6):2215--2240, 1989.

....existence of P 0 such that A(t) T P PA(t) 0; A(t) 2 Co fA 1 ; ALg : 7) If there exists such a P , we say the DI (6) is quadratically stable. Condition (7) is equivalent to P 0; A T i P PA i 0; i = 1; L; which is a linear matrix inequality in P (see for example [7, 8, 9, 10]) Thus, determining quadratic stability is an LMIP. V is sometimes called a simultaneous quadratic Lyapunov function since it proves stability of each of A 1 ; AL . 3.3. Lyapunov functions and state feedback Consider the system (6) with state feedback: dx dt = A(t)x(t) B(t)u(t) ....

S. Boyd and Q. Yang. Structured and simultaneous Lyapunov functions for system stability problems. Int. J. Control, 49(6):2215--2240, 1989.

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S. Boyd, Q. Young, Structured and simultaneous Lyapunov functions for system stability problems, Int. J. Control, vol. 49, 1989, pp. 2215--2240.

No context found.

S. BOYD and Q. YANG. Structured and simultaneous Lyapunov functions for system stability problems. Int. J. Control, 49(6):2215--2240, 1989.

No context found.

S. Boyd, and Q. Yang, "Structured and simultaneous Lyapunov functions for system stability problems," International Journal of Control, vol. 49, 2215--2240, 1989.

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