Lyapunov Functions and Feedback in Nonlinear Control

Abstract

Summary. The method of Lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. For nonlinear systems, it turns out to be essential to consider nonsmooth Lyapunov functions, even if the underlying control dynamics are themselves smooth. We synthesize in this article a number of recent developments bearing upon the regularity properties of Lyapunov functions. A novel feature of our approach is that the guidability and stability issues are decoupled. For each of these issues, we identify various regularity classes of Lyapunov functions and the system properties to which they correspond. We show how such regularity properties are relevant to the construction of stabilizing feedbacks. Such feedbacks, which must be discontinuous in general, are implemented in the sample-and-hold sense. We discuss the equivalence between open-loop controllability, feedback stabilizability, and the existence of Lyapunov functions with appropriate regularity properties. The extent of the equivalence confirms the cogency of the new approach summarized here.

Pseudospectral Feedback Control: Foundations, Examples and Experimental Results

“…In extending this concept for general nonlinear control systems (i.e. not necessarily aerospace) we follow Clarke 16 and note that there are essentially two principal objectives in all of control theory: 1) Positional : the state trajectory, t → x, must remain in or approach a given set of R N x ;…”

“…stability) objective can also be addressed by way of optimal control theory by linking the Bellman value function to a Lyapunov function as facilitated by nonsmooth calculus. 2,16 For the type of feedback control promulgated in this paper, we have provided such a link in a complementary paper, 6 but under a stronger assumption of smoothness of the Lyapunov function. In this paper, we outline the issues, principles and certain implementation details.…”

“…In the next section, we consider the more general time-varying feedback law, k : R × R Nx → R Nu , as this is central to our new approach. In following Clarke, 16 we let π = {t i } i≥0 be a partition of the time interval, [0, ∞),…”

“…Given an initial condition, x 0 , a π-trajectory, x(·), corresponding to k(x) is defined as follows: 16 From t 0 to t 1 , we generate a classical solution to the differential equation,…”

“…These theorems directly connect the "epsilons and deltas" of abstract mathematics to practical digital computer implementations of optimal control. 16,26 An implementation of the sample-and-hold feedback concept is illustrated in Fig. 2.…”