José Claudio Geromel

A new discrete-time robust stability condition☆

A new robust stability condition for uncertain discrete-time systems with convex polytopic uncertainty is given. It enables to check stability using parameter-dependent Lyapunov functions which are derived from LMI conditions. It is shown that this new condition provides better results than the classical quadratic stability. Besides the use of a parameter-dependent Lyapunov function, this condition exhibits a kind of decoupling between the Lyapunov and the system matrices which may be explored for control synthesis purposes. A numerical example illustrates the results.

Extended H 2 and H norm characterizations and controller parametrizations for discrete-time systems

This paper presents new synthesis procedures for discrete-time linear systems. It is based on a recently developed stability condition which contains as particular cases both the celebrated Lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters. These new synthesis conditions have some nice properties: (a) they can be expressed in terms of LMI (linear matrix inequalities) and (b) the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic Lyapunov function used to test stability. This second feature is important for several reasons. First, structural constraints, as those appearing in the decentralized and static output-feedback control design, can be addressed less conservatively. Second, parameter dependent Lyapunov function can be considered with a very positive impact on the design of robust H 2 and H X control problems. Third, the design of controller with mixed objectives (also gain-scheduled controllers) can be addressed without employing a unique Lyapunov matrix to test all objectives (scheduled operation points). The theory is illustrated by several numerical examples.

Output feedback control of Markov jump linear systems in continuous-time

This paper addresses the dynamic output feedback control problem of continuous-time Markovian jump linear systems. The fundamental point in the analysis is an LMI characterization, comprising all dynamical compensators that stabilize the closed-loop system in the mean square sense. The H/sub 2/ and H/sub /spl infin//-norm control problems are studied, and the H/sub 2/ and H/sub /spl infin// filtering problems are solved as a by product.

Stability and Stabilization of Continuous-Time Switched Linear Systems

This paper addresses two strategies for the stabilization of continuous-time, switched linear systems. The first one is of open loop nature (trajectory independent) and is based on the determination of a minimum dwell time by means of a family of quadratic Lyapunov functions. The relevant point on dwell time calculation is that the proposed stability condition does not require the Lyapunov function to be uniformly decreasing at every switching time. The second one is of closed loop nature (trajectory dependent) and is designed from the solution of what we call Lyapunov-Metzler inequalities from which the stability condition (including chattering) is expressed. Being nonconvex, a more conservative but simpler-to-solve version of the Lyapunov-Metzler inequalities is provided. The theoretical results are illustrated by means of examples.

Brief An improved approach for constrained robust model predictive control

In this paper, we present a new technique to address constrained robust model predictive control. The main advantage of this new approach with respect to other well-known techniques is the reduced conservativeness. Specifically, the technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several Lyapunov functions each one corresponding to a different vertex of the uncertaintys polytope.

On a convex parameter space method for linear control design of uncertain systems

This paper presents a new procedure for continuous and discrete-time linear control systems design. It consists of the definition of a convex programming problem in the parameter space that, when solved, provides the feedback gain. One of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation, yielding solutions to problems that have raised a great deal of interest within the last few years. This is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices. In this last case, necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and, to the authors’ knowledge, this is one of the first numerical procedures able to handle and solve this interesting design problem for high-order, continuous-time or discrete-time linear models. The theory is illustrated by examples.

a linear programming oriented procedure for quadratic stabilization of uncertain systems

Abstract This paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty. A constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain. Some experiments are presented.

Optimal linear filtering under parameter uncertainty

This paper addresses the problem of designing a guaranteed minimum error variance robust filter for convex bounded parameter uncertainty in the state, output, and input matrices. The design procedure is valid for linear filters that are obtained from the minimization of an upper bound of the error variance holding for all admissible parameter uncertainty. The results provided generalize the ones available in the literature to date in several directions. First, all system matrices can be corrupted by parameter uncertainty, and the admissible uncertainty may be structured. Assuming the order of the uncertain system is known, the optimal robust linear filter is proved to be of the same order as the order of the system. In the present case of convex bounded parameter uncertainty, the basic numerical design tools are linear matrix inequality (LMI) solvers instead of the Riccati equation solvers used for the design of robust filters available in the literature to date. The paper that contains the most important and very recent results on robust filtering is used for comparison purposes. In particular, it is shown that under the same assumptions, our results are generally better as far as the minimization of a guaranteed error variance is considered. Some numerical examples illustrate the theoretical results.

Dynamic Output Feedback Control of Switched Linear Systems

This paper is devoted to stability analysis and control design of switched linear systems in both continuous and discrete-time domains. A particular class of matrix inequalities, the so-called Lyapunov--Metzler inequalities, provides conditions for open-loop stability analysis and closed-loop switching control using state and output feedback. Switched linear systems are analyzed in a general framework by introducing a quadratic in the state cost determined from a series of impulse perturbations. Lower bounds on the cost associated with the optimal switching control strategy are derived from the determination of a feasible solution to the Hamilton--Jacobi--Bellman inequality. An upper bound on the optimal cost associated with a closed-loop stabilizing switching strategy is provided as well. The solution of the output feedback problem is based on the construction of a full-order linear switched filter whose state variable is used by the mechanism for the determination of the switching rule. Throughout, the theoretical results are illustrated by means of academic examples. A realistic practical application related to the optimal control of semiactive suspensions in road vehicles is reported.

Robust Filtering of Discrete-Time Linear Systems with Parameter Dependent Lyapunov Functions

Robust filtering of linear time-invariant discrete-time uncertain systems is investigated through a new parameter dependent Lyapunov matrix procedure. Its main interest relies on the fact that the Lyapunov matrix used in stability checking does not appear in any multiplicative term with the uncertain matrices of the dynamic model. We show how to use such an approach to determine high performance H2 robust filters by solving a linear problem constrained by linear matrix inequalities (LMIs). The results encompass the previous works in the quadratic Lyapunov setting. Numerical examples illustrate the theoretical results.