The continuous- and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H∞-suboptimal controllers, including reduced-order controllers. The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMI's. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H1 controllers and bear important connections with the controller order and the closed-loop Lyapunov functions. Thanks to such connections, the LMI-based characterization of H∞ controllers opens new perspectives for the refinement of H∞ design. Applications to cancellation-free design and controller order reduction are discuss...

This paper is concerned with the design of gain-scheduled controllers for Linear Parameter-Varying systems. Two alternative LMI characterizations are investigated. Both characterizations are amenable to a finite number of LMI conditions either via a gridding of the parameter range or via grid-free techniques which rely on multi-convexity concepts. Practicality and implementation issues are discussed and examples are provided. 1 Introduction The gain-scheduling problem has been the subject of a great deal of research over recent years, both from theoretical and practical viewpoints. This renewed interest probably stems from the development of new techniques and software which allow for a more rigorous and systematic treatment of the gain-scheduling problem. The classical approach to this problem essentially consists of repeated design syntheses associated with some scheduling strategy connecting locally designed controllers. Such schemes, however, lack supporting theories that guarante...

An important class of linear time-varying systems consists of plants where the state-space matrices are fixed functions of some time-varying physical parameters `. Small Gain techniques can be applied to such systems to derive robust time-invariant controllers. Yet, this approach is often overly conservative when the parameters ` undergo large variations during system operation. In general, much higher performance can be achieved by control laws that incorporate available measurements of ` and therefore "adjust" to the current plant dynamics. This paper discusses extensions of H∞ synthesis techniques to allow for controller dependence on time-varying but measured parameters. When this dependence is linear fractional, the existence of such gain-scheduled H1 controllers is fully characterized in terms of linear matrix inequalities (LMIs). The underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available. The formalism and...

This paper addresses the problem of evaluating the maximum norm vector within the intersection of several ellipsoids. This difficult non-convex optimization problem frequently arises in robust control synthesis. Linear matrix inequality relaxations of the problem are enumerated. Two randomized algorithms and several ellipsoidal approximations are described. Guaranteed approximation bounds are derived in order to evaluate the quality of these relaxations. 1 Introduction 1.1 Problem Statement In this paper we consider the optimization problem p opt = max x x 0 x s.t. x 2 F (1) where x is a vector in R n and the set F is the intersection of m ellipsoids F = E 1 " E 2 " \Delta \Delta \Delta " Em (2) Corresponding Author. E-mail: henrion@laas.fr defined as E i = fx : x 0 P i x 1g (3) for P i a given symmetric positive definite matrix in R n\Thetan . Feasible set F is the intersection of m centered ellipsoids in R n , hence F is convex and centered about the origin. It i...

We consider the following quasiconvex optimization problem: minimize the largest eigenvalue of a symmetric definite matrix pencil depending on parameters. A new form of optimality conditions is given, emphasizing a complementarity condition on primal and dual matrices. Newton's method is then applied to these conditions to give a new quadratically convergent interior-point method which works well in practice. The algorithm is closely related to primaldual interior-point methods for semidefinite programming. 1. Introduction Many matrix inequality problems in control can be cast in the form: minimize the maximum eigenvalue of the Hermitian definite pencil (A(x); B(x)), w.r.t. a parameter vector x, subject to positive definite constraints on B(x) and sometimes also on other Hermitian matrix functions of x. The maximum eigenvalue is a quasiconvex function of the pencil elements and therefore of the parameter vector x if A, B depend affinely on x. This quasiconvexity reduces to convexity i...

An important class of linear time-varying systems consists of plants where the state-space matrices are fixed functions of some time-varying physical parameters `. Small Gain techniques can be applied to such systems to derive robust time-invariant controllers. Yet, this approach is often overly conservative when the parameters ` undergo large variations during system operation. In general, much higher performance can be achieved by control laws that incorporate available measurements of ` and therefore "adjust" to the current plant dynamics. This paper discusses extensions of H1 synthesis techniques to allow for controller dependence on time-varying but measured parameters. When this dependence is linear fractional, the existence of such gain-scheduled H1 controllers is fully characterized in terms of linear matrix inequalities (LMIs). The underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available. The formalism and derivatio...

We present an interior-point method for convex fractional programming. The proposed algorithm converges in polynomial time, just as in the case of a convex problem, even though convex fractional programs are only pseudo-convex. More precisely, the rate of convergence is measured in terms of the area of two-dimensional convex sets C k containing the optimal points, and the area of C k is reduced by a constant factor c ! 1 at each iteration. The factor c depends only on the self-concordance parameter of a barrier function associated with the feasible set. We present an outline of a practical implementation of the proposed method, and we report results of a few numerical experiments. 1. Introduction Interior-point methods for the solution of nonlinear programming problems were already introduced in the 1950s and 1960s; see [6] and the references given there. In the 1970s, new and seemingly superior approaches, such as sequential quadratic programming techniques, were developed, and as a ...

Sufficient conditions for the robust stability of a class of uncertain systems, with several different assumptions on the structure and nature of the uncertainties, can be derived in a unified manner in the framework of integral quadratic constraints. These sufficient conditions, in turn, can be used to derive lower bounds on the robust stability margin for such systems. The lower bound is typically computed with a bisection scheme, with each iteration requiring the solution of a linear matrix inequality feasibility problem. We show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem, which can be solved very efficiently using standard algorithms. We illustrate this with several important, commonly-encountered special cases: Diagonal, nonlinear uncertainties; diagonal, memoryless, time-invariant sector-bounded ("Popov") uncertainties; structured dynamic uncertainties; and structured pa...

this article is to provide an overview of the state of the art of