Derivative Of Lyapunov Research Articles

ON THE LYAPUNOV APPROACH TO ROBUST STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS

In the Lyapunov approach employed in this paper, known in the literature as Lyapunov control, or min-max control, robust, global uniform asymptotic stability is achieved by a discontinuous control law which ensures that the Lyapunov derivative is negative despite bounded uncertainty. For that, it is assumed that the uncertainties satisfy certain matching conditions, and that a Lyapunov function for the nominal plant is available. To obtain lower control magnitudes, this paper develops control laws which counter the uncertainties on a component-wise basis, rather than the usual normic one. Both the basic discontinuous control law, which is proved to provide robust global uniform asymptotic stability, and a continuous app roximation, which is proved to ensure global uniform ultimate boundedness, are derived. Application to model following is given. We adapt recent results on robust quadratic stabilization of nominally linear time-invariant plants subjected to nonlinear, bo unded and unmatched uncertain perturbations, to extend our results to this important class of systems; this is illustrated by two examples.

A riccati equation approach to the stabilization of uncertain linear systems

This paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems. The systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set. Furthermore, these uncertain parameters may be time-varying. The method used to establish asymptotic stability of the closed loop system (obtained when the feedback control is applied) involves the use of a quadratic Lyapunov function. The main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic Lyapunov function. Once the Lyapunov function has been obtained, it is used to construct the stabilizing feedback control law. The fundamental idea behind the algorithm presented involves constructing an upper bound for the Lyapunov derivative corresponding to the closed loop system. This upper bound is a quadratic form. By using this upper bounding procedure, a suitable Lyapunov function can be found by solving a certain matrix Riccati equation. Another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic Lyapunov function.