This paper presents synthesis procedures for the design of both robust and gain-scheduled H ∞ static output feedback controllers for discrete-time linear systems with timevarying parameters. The parameters are assumed to vary inside a polytope and have known bounds on their rate of variation. The geometric properties of the polytopic domain are exploited to derive a finite set of linear matrix inequalities that consider the bounds on the rate of variation of the parameters. A numerical example illustrates the proposed approach.

SUMMARYThis paper presents synthesis conditions for the design of gain‐scheduled dynamic output feedback controllers for discrete‐time linear parameter‐varying systems. The state‐space matrix representation of the plant and of the controller can have a homogeneous polynomial dependency of arbitrary degree on the scheduling parameter. As an immediate extension, conditions for the synthesis of a multiobjective ℋ︁∞ and ℋ︁2 gain‐scheduled dynamic feedback controller are also provided. The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real‐time. If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results. The geometric properties of the uncertainty domain are exploited to derive finite sets of linear matrix inequalities based on the existence of a homogeneous polynomially parameter‐dependent Lyapunov function. An application of the control design to a realistic engineering problem illustrates the benefits of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.

This paper presents a theory of noncommutative functions which results in an algorithm for determining where they are "matrix convex". Of independent interest is a theory of noncommutative quadratic functions and the resulting algorithm which calculates the region where they are "matrix positive". This is accomplished via a theorem (a type of Positivstellensatz) on writing noncommutative quadratic functions with noncommutative rational coefficients as a weighted sum of squares. Furthermore the paper gives an LDU algorithm for matrices with noncommutative entries and conditions guaranteeing that the decomposition is successful.The motivation for the paper comes from systems engineering. Inequalities, involving polynomials in matrices and their inverses, and associated optimization problems have become very important in engineering. When these polynomials are "matrix convex" interior point methods apply directly. A difficulty is that often an engineering problem presents a matrix polynomial whose convexity takes considerable skill, time, and luck to determine. Typically this is done by looking at a formula and recognizing "complicated patterns involving Schur complements"; a tricky hit or miss procedure. Certainly computer assistance in determining convexity would be valuable. This paper, in addition to theory, describes a symbolic algorithm and software which represent a beginning along these lines.The algorithms described here have been implemented under Mathematica and the noncommutative algebra package NCAlgebra. Examples presented in this article illustrate its use. Mathematics Subject Classification (2000). Primary 47L99, 93A99; Secondary 15A39, 14P99.

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