S.R. Souza

Convex analysis of output feedback control problems: robust stability and performance

This paper addresses the problem of optimal H/sub 2/ control by output feedback. Necessary and sufficient conditions on the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function. The results can be easily extended to deal with linear uncertain systems, where uncertainties are supposed to belong to convex bounded domains providing an H/sub 2/-guaranteed cost output feedback control. Thanks to the properties of the above-mentioned function, we show that under certain conditions, convex programming tools can be used for numerical purposes. Examples illustrate the theoretical results.

H ∞ guaranteed cost control for uncertain continuous-time linear systems

Abstract This paper deals with the H ∞ guaranteed cost control problems for continuous-time uncertain systems. It consist of the determination of a stabilizing state feedback gain which imposes on all possible closed-loop models an H ∞ -norm upper bound γ > 0. Assuming that the uncertain domain is convex-bounded and the uncertain system is quadratic-stabilizable with γ disturbance attenuation, it is shown how to determine, by means of a convex programming problem, the global minimum of γ. As a particular and important case, for precisely known linear systems, the last problem reduces to the classical H ∞ optimal control problem. The results follow from the definition of a special parameter space on which the above-mentioned problems are convex.

H/sub /spl infin// control of discrete-time uncertain systems

This paper considers H/sub /spl infin// control problems involving discrete-time uncertain linear systems. The uncertainty is supposed to belong to convex-bounded domains, and no additional assumptions are made (as, for instance, matching conditions). Two H/sub /spl infin// guaranteed cost control problems are solved. The first one concerns the determination of a state feedback gain (if one exists) in such way the H/sub /spl infin// norm of a certain transfer function remains bounded by a prespecified H/sub /spl infin// level for all possible models. The second one includes this bound as an additional variable to be minimized to achieve the smallest feasible limiting bound. The results follow from the simple geometry of those problems which are shown to be convex in the particular parametric space under consideration. An example illustrates the theory. >

Output Feedback Stabilization of Uncertain Systems through a Min/Max Problem*

Abstract This paper addresses the problem of output feedback control design for linear uncertain continuous-time systems, where uncertainties are supposed to belong to convex-bounded domains. Necessary and sufficient conditions are provided concerning the existence of a linear static stabilizing output feedback. No extra assumptions such as matching conditions are considered. Based on these conditions, an auxiliary min/max problem can be formulated, allowing us to achieve robust output feedback control using convex programming. These results are illustrated by some examples.

Optimal H/sub 2/ control by output feedback

This paper addresses the problem of optimal H/sub 2/ control by output feedback. Necessary and sufficient conditions concerning the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function. The results can be easily extended to deal with linear uncertain systems, where uncertainties are supposed to belong to convex bounded domains, providing an H/sup 2/ guaranteed cost output feedback control. Due to the geometrical properties of the above mentioned function, convex programming tools can be used for numerical purposes.<<ETX>>

Convex analysis of output feedback structural constraints

Any output feedback control problem can be viewed as a state feedback control problem where the feedback gain is subjected to a structural constraint. In this note, we discuss the geometry of this constraint in a convex framework. The main purpose of the study is to obtain an equivalent mathematical description which exhibits important geometrical properties.<<ETX>>

Optimal H ∞ -state feedback control for continuous-time linear systems

This paper proposes a convex programming method to achieve optimal ℋ∞-state feedback control for continuous-time linear systems. State space conditions, formulated in an appropriate parameter space, define a convex set containing all the stabilizing control gains that guarantee an upper bound on the ℋ∞-norm of the closed-loop transfer function. An optimization problem is then proposed, in order to minimize this upper bound over the previous convex set, furnishing the optimal ℋ∞-control gain as its optimal solution. A limiting bound for the optimum ℋ∞-norm can easily be calculated, and the proposed method will achieve minimum attenuation whenever a feasible state feedback controller exists. Generalizations to decentralized and output feedback control are also investigated. Numerical examples illustrate the theory.This paper proposes a convex programming method to achieve optimal ℋ∞-state feedback control for continuous-time linear systems. State space conditions, formulated in an appropriate parameter space, define a convex set containing all the stabilizing control gains that guarantee an upper bound on the ℋ∞-norm of the closed-loop transfer function. An optimization problem is then proposed, in order to minimize this upper bound over the previous convex set, furnishing the optimal ℋ∞-control gain as its optimal solution. A limiting bound for the optimum ℋ∞-norm can easily be calculated, and the proposed method will achieve minimum attenuation whenever a feasible state feedback controller exists. Generalizations to decentralized and output feedback control are also investigated. Numerical examples illustrate the theory.

Quadratic Stabilizability of Linear Uncertain Systems with Prescribed H ∞ Norm Bounds 1

Abstract This paper presents a method to synthesize a linear state feedback control for dynamic continuous-time linear systems. The major improvement is that the method takes into account, as a design requirement, an upper bound to the H ∞ norm of a specified closed-loop transfer function. The condition that guarantees the H ∞ norm bound, giving the linear state feedback gain, has convexity properties that allow the controller design to be applied to uncertain systems with convex-bounded uncertainties. Therefore, using the state space models and Lyapunov conditions, the method solves jointly both quadratic stabilizability and H ∞ control problems.

Convex analysis of discrete-time uncertain H/sub infinity / control problems

Two classical problems involving discrete-time systems are analyzed. The first one concerns the quadratic stabilizability with uncertainties in convex bounded domains, which naturally covers the important class of interval matrices. In that problem, there is no need to introduce any kind of matching conditions, which is an important improvement compared with other results available in the literature. The second problem is defined by simply adding to the first problem some prespecified closed-loop transfer function H/sub infinity / norm bound. Assuming the state is available for feedback, the geometry of both problems is thoroughly analyzed. They turn out to be convex on the parameter space.<<ETX>>

Optimal H 2 Control for Uncertain Linear Systems

This paper proposes a method based on convex programming to calculate a guaranteed cost stabilizing state feedback control, for both continuous-time and discrete-time uncertain linear systems. In the uncertain case, it provides a guaranteed cost, i.e., an upper bound for the H2 norm of the closed-loop transfer function. In the absence of uncertainties, the numerical algorithm furnishes, under certain conditions, exactly the same optimal control gain obtained by the classical Linear Quadratic Problem. Thanks to the convexity of the proposed conditions, additional constraints can be easily taken into account as, for instance, robustness against actuators failure. Examples illustrate the theoretical results.