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 self-concordant 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 non-heuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several non-heuristic 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.

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.

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 state-space when assuming a general linearly parameterized class of smooth non-quadratic Lyapunov-function candidates. Hence, computation of the Lyapunov function involves the solution of a convex large-scale 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. 1 1 Introduction This work describes a procedure for computing a Lyapunov function (if one exists) for the equilibrium point x = 0 for the class of non-autonomous ...

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 efficiently. 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 ("multi-model control"), optimal transfer matrix realization, norm scaling, synthesis of multipliers for Popov-like...

In this paper, we investigate the use of two-term piecewise quadratic Lyapunov functions for robust stability of linear time-varying systems. By using the so-called S-procedure and a special variable reduction method, we provide numerically efficient conditions for the robust asymptotic stability of the linear time-varying 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, S-procedure, Linear matrix inequality. 1 Introduction The quadratic stability approach is popularly used for robust stability analysis of time-varying uncertain systems. This approach, however, may lead to very conservative results. Alternatively, non-quadratic Lyapunov functions have been used to improve the estimate of robust stability (see [1, 2, 6, 7, 9, 10]). The difficulty with non-quadratic Lyapunov functions is that the resulting optimization problem is typically non-convex. I...