Revised CPA method to compute Lyapunov functions for nonlinear systems

Abstract

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium's basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C 2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction

“…The proofs of Theorem 2 and Corollary 1 can be proved in a manner similar to [5,Theorem 1]. We omit the details.…”

“…Finding a candidate CPA Lyapunov function can be done as in [1], [5], [6], [14], via linear programming. Alternatively, as in this paper, one can define V 2 CPA[T ] by computing suitable values at the vertices of the simplices of T and then verify the inequalities (3).…”

“…In order to do this, we clearly need a solution to (1) from each such x. As a closed form solution is generally not available, we will resort to numerical integration in order to calculate V (x) given by (5). For this approach to be numerically tractable, we require that the time horizon T (x) given by (11) not be too large.…”

“…However, in order to make use of Theorem 3 we would clearly prefer that the Yoshizawa-Lyapunov function be twice continuously differentiable. Standard smoothing techniques can be applied to the Yoshizawa-Lyapunov function to obtain a smooth function, however these techniques do not generally provide us with an easily calculable function such as (5). Rather, we would prefer to demonstrate directly that the YoshizawaLyapunov function inherits the regularity of the vector field of (1).…”

“…An approach using linear programming to compute feasible values for the CPA function at the simplex vertices was proposed in [14] with refinements in [6], [1], [5]. However, since linear programming is not very efficient, the question remains if the values at the vertices cannot be fixed by more efficient means with a subsequent speedy test of the validity of the linear inequalities.…”