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Method of centers for minimizing generalized eigenvalues
 Linear Algebra Appl
, 1993
"... We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fr ..."
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Cited by 60 (12 self)
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We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a selfconcordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.
Structured and Simultaneous Lyapunov Functions for System Stability Problems
, 2001
"... It is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic Lyapunov function of a certain structure which establishes stability of x = Ax for some appropriate A. The existence of such a Lyapunov function can be determined by solving a c ..."
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Cited by 26 (4 self)
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It is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic Lyapunov function of a certain structure which establishes stability of x = Ax for some appropriate A. The existence of such a Lyapunov function can be determined by solving a convex program. We present several numerical methods for these optimization problems. A simple numerical example is given.
Control system analysis and synthesis via linear matrix inequalities
 Control Conference, American
, 1982
"... A wide variety of problems in systems and control theory can be cast or recast as convex problems that involve linear matrix inequalities (LMIs). For a few very special cases there are \analytical solutions " to these problems, but in general they can be solved numerically very e ciently. In many ca ..."
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Cited by 13 (1 self)
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A wide variety of problems in systems and control theory can be cast or recast as convex problems that involve linear matrix inequalities (LMIs). For a few very special cases there are \analytical solutions " to these problems, but in general they can be solved numerically very e ciently. In many cases the inequalities have the form of simultaneous Lyapunov or algebraic Riccati inequalities; such problems can be solved in a time that is comparable to the time required to solve the same number of Lyapunov or Algebraic Riccati equations. Therefore the computational cost of extending current control theory that is based on the solution of algebraic Riccati equations to a theory based on the solution of (multiple, simultaneous) Lyapunov or Riccati inequalities is modest. Examples include: multicriterion LQG, synthesis of linear state feedback for multiple or nonlinear plants (\multimodel control"), optimal transfer matrix realization, norm scaling, synthesis of multipliers for Popovlike analysis of systems with unknown gains, and many others. Full details can be found in the references cited. 1.
Computation of Lyapunov Functions for Smooth Nonlinear Systems using Convex Optimization
 AUTOMATICA
, 1999
"... It is shown that for smooth nonlinear systems conditions for the existence of a Lyapunov function that guarantees uniform exponential stability can be formulated as linear inequalities defined pointwise in the statespace when assuming a general linearly parameterized class of smooth nonquadratic L ..."
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Cited by 12 (1 self)
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It is shown that for smooth nonlinear systems conditions for the existence of a Lyapunov function that guarantees uniform exponential stability can be formulated as linear inequalities defined pointwise in the statespace when assuming a general linearly parameterized class of smooth nonquadratic Lyapunovfunction candidates. Hence, computation of the Lyapunov function involves the solution of a convex largescale optimization problem using linear or quadratic programming. The optimization criterion can for example be selected to find a Lyapunov function which predicts fast decay rate or large region of attraction. Analysis of the tradeoff between accuracy and computational complexity as well as possible conservativeness of the procedure is given particular attention. The procedure is illustrated using numerical examples.
Piecewise Lyapunov functions for robust stability of linear timevarying systems
 Systems & Control Letters
, 1997
"... In this paper, we investigate the use of twoterm piecewise quadratic Lyapunov functions for robust stability of linear timevarying systems. By using the socalled Sprocedure and a special variable reduction method, we provide numerically efficient conditions for the robust asymptotic stability of ..."
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Cited by 7 (0 self)
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In this paper, we investigate the use of twoterm piecewise quadratic Lyapunov functions for robust stability of linear timevarying systems. By using the socalled Sprocedure and a special variable reduction method, we provide numerically efficient conditions for the robust asymptotic stability of the linear timevarying systems involving the convex combinations of two matrices. An example is included to demonstrate the usefulness of our results. Key words: Robust stability, Piecewise Lyapunov function, Sprocedure, Linear matrix inequality. 1 Introduction The quadratic stability approach is popularly used for robust stability analysis of timevarying uncertain systems. This approach, however, may lead to very conservative results. Alternatively, nonquadratic Lyapunov functions have been used to improve the estimate of robust stability (see [1, 2, 6, 7, 9, 10]). The difficulty with nonquadratic Lyapunov functions is that the resulting optimization problem is typically nonconvex. I...