Quantised and sampled output feedback for nonlinear systems

Abstract

In this paper we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantised. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimisation problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented in order to illustrate the applicability of the proposed design method.

Advances in attractive ellipsoid method for robust control design

“…We now explain (qualitatively) the main advances in AEM we developed in this paper. The nonstationary character of the dynamic systems (1) and (5) makes it impossible a direct application of the conventional LMI-based robust control design that involves the AEM techniques (see other works 3,6,7,9,12,31,32 ). Recall that the classic AEM was developed under the strict assumptions of stationarity for the given dynamic system.…”

“…The problem of powerful computer-oriented algorithms for the robust control of nonlinear dynamic systems has attracted a lot of attention, thus both theoretical results and real-world applications were developed (see, eg, the work of Khalil 1 ). The newly elaborated attractive ellipsoid method (AEM) (see other works [2][3][4][5][6][7][8][9][10][11][12][13] ) constitutes a useful engineering oriented development of the theory of invariant sets and proposes a self-closed effective computational approach to the robust control design of systems with uncertainties. Recall that a set in the state space of a system is said to be positively invariant if any trajectory initiated in this set remains inside the set at all future time instants.…”

“…A "small-size" ellipsoidal attractive (or invariant) set is evidently a geometrically motivated region that can also be defined as a "practical stability" region for the system under consideration. We refer to related works 3,[6][7][8][9][10][11][12][13] for the necessary analytic background, some existing modifications of the AEM, and numerous applications of this methodology.…”

Robust feedback design for stabilization of nonlinear systems with sampled-data and quantized output: Atractive ellipsoid method

“…More precisely, let C 0 (R, R 2n ) be the space of all continuous functions of R into R 2n , differentiable almost everywhere; let R > 0 and P > 0 be (2n × 2n)-dimensional matrices and let α > 0 be a scalar. Following [23], propose the functional V : R × C 0 (R, R 2n ) → R + , defined as…”

“…In this regard, the present work can be seen as an extension of the work presented in [22] to the case when quantization phenomena are present. Notice that the same problem has been considered in [23] where the feedback has built using the state estimates generated by the Luenberger-like filter. Here the designing of robust full-order dynamic feedback is considered.…”