Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction

Abstract

Abstract-We present a novel numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. Our proposed approach constructs a continuous piecewise affine (CPA) function given a suitable partition of the state space, called a triangulation, and values at the vertices of the triangulation. The vertex values are obtained from a Lyapunov function in a classical converse Lyapunov theorem and verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Furthermore, by refining the triangulation, we show that it is always possible to construct a CPA Lyapunov function. Numerical examples are presented demonstrating the effectiveness of the proposed method.

Computing continuous and piecewise affine lyapunov functions for nonlinear systems

“…The second diculty is that Sontag's lemma on KL-estimates is not constructive and, to the best of the authors' knowledge, given an arbitrary β ∈ KL, there are currently no constructive techniques for nding In practice, this causes no diculty since whether or not a computed CPA [T ] function is a CPA[T ] Lyapunov function relies only on the verication of the linear inequalities (7). Similarly, approximation errors caused by the use of low-order integration methods, inaccurate stability estimates, or incorrect time horizons may result in a poor approximation of the Yoshizawa function but may nonetheless lead to a CPA [T ] function that satises inequalities (7) and is hence a CPA[T ] Lyapunov function.…”

“…In the linear programming approach used in [1,5,6,14], the linear inequalities are used as constraints in a linear program and, hence, a solution necessarily satises (7). By contrast, in this paper, we propose xing the vertex values by a computational procedure described in the next section followed by verifying the inequalities (7 …”

“…This signicantly simplies the initial setup of the computational problem as choosing a computational domain larger than the basin of attraction does not lead to an infeasible problem necessitating a renement of the computational domain. First and second order examples can be found in [7]. 6 …”

“…Similarly, approximation errors caused by the use of low-order integration methods, inaccurate stability estimates, or incorrect time horizons may result in a poor approximation of the Yoshizawa function but may nonetheless lead to a CPA [T ] function that satises inequalities (7) and is hence a CPA[T ] Lyapunov function.…”

“…The usefulness of Theorem 2 is that it reduces the verication that a given function V ∈ CPA[T ] is a Lyapunov function for (1) to the verication of a nite number of inequalities (7). In the linear programming approach used in [1,5,6,14], the linear inequalities are used as constraints in a linear program and, hence, a solution necessarily satises (7). By contrast, in this paper, we propose xing the vertex values by a computational procedure described in the next section followed by verifying the inequalities (7 …”