Computation of Lyapunov functions for nonlinear discrete systems by linear programming

Given an autonomous discrete time system with an equilibrium at the origin and a hypercube D containing the origin, we state a linear programming problem, of which any feasible solution parameterizes a continuous and piecewise affine (CPA) Lyapunov function V : D -> R for the system. The linear programming problem depends on a triangulation of the hypercube. We prove that if the equilibrium at the origin is exponentially stable, the hypercube is a subset of its basin of attraction, and the triangulation fulfils certain properties, then such a linear programming problem possesses a feasible solution. We present an algorithm that generates such linear programming problems for a system, using more and more refined triangulations of the hypercube. In each step the algorithm checks the feasibility of the linear programming problem. This results in an algorithm that is always able to compute a Lyapunov function for a discrete time system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibrium's basin of attraction. The system is assumed to have a right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type such as linear, piecewise affine and so on. Our approach is a non-trivial adaptation of the CPA method to compute Lyapunov functions for continuous time systems to discrete time systems

Linear Programming based Lower Bounds on Average Dwell-Time via Multiple Lyapunov Functions

With the objective of developing computational methods for stability analysis of switched systems, we consider the problem of finding the minimal lower bounds on average dwell-time that guarantee global asymptotic stability of the origin. Analytical results in the literature quantifying such lower bounds assume existence of multiple Lyapunov functions that satisfy some inequalities. For our purposes, we formulate an optimization problem that searches for the optimal value of the parameters in those inequalities and includes the computation of the associated Lyapunov functions. In its generality, the problem is nonconvex and difficult to solve numerically, so we fix some parameters which results in a linear program (LP). For linear vector fields described by Hurwitz matrices, we prove that such programs are feasible and the resulting solution provides a lower bound on the average dwell-time for exponential stability. Through some experiments, we compare our results with the bounds obtained from other methods in the literature and we report some improvements in the results obtained using our method.Comment: Accepted for publication in Proceedings of European Control Conference 202

Computation and verification of Lyapunov functions

Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determining arbitrary compact subsets of the basin of attraction. The method is applied to two examples

Wendland functions a C++ code to compute them

In this paper we present a code in C++ to compute Wendland functions for arbitrary smoothness parameters. Wendland functions are compactly supported Radial Basis Functions that are used for interpolation of data or solving Partial Differential Equations with mesh-free collocation. For the computations of Lyapunov functions using Wendland functions their derivatives are also needed so we include this in the code. Wendland functions with a few fixed smoothness parameters are included in some C++ libraries, but for the general case the only code freely available was implemented in MAPLE taking advantage of the computer algebra system. The aim of this contribution is to allow scientists to use Wendland functions in their C++ code without having to implement them themselves. The computed Wendland functions are polynomials and their coefficients are computed and stored in a vector, which allows for efficient computation of their values using the Horner scheme

Analysing dynamical systems towards computing complete Lyapunov functions

Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits

Automatic determination of connected sublevel sets of CPA Lyapunov functions

Lyapunov functions are an important tool to determine the basin of attraction of equilibria. In particular, the connected component of a sublevel set, which contains the equilibrium, is a forward invariant subset of the basin of attraction. One method to compute a Lyapunov function for a general nonlinear autonomous differential equation constructs a Lyapunov function, which is continuous and piecewise affine (CPA) on each simplex of a fixed triangulation. In this paper we propose an algorithm to determine the largest connected sublevel set of such a CPA Lyapunov function and prove that it determines the largest subset of the basin of attraction that can be obtained by this Lyapunov function

Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions

We study stability for dynamical systems specifed by autonomous stochastic diferential equations of the form dX(t) = f(X(t))dt + g(X(t))dW(t), with (X(t))t≥0 an Rd -valued Ito process and ˆ (W(t))t≥0 an RQ-valued Wiener process, and the functions f : Rd → Rd and g : Rd → Rd×Q are Lipschitz and vanish at the origin, making it an equilibrium for the system. Te concept of asymptotic stability in probability of the null solution is well known and implies that solutions started arbitrarily close to the origin remain close and converge to it. Te concept therefore pertains exclusively to system properties local to the origin. We wish to address the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can they then be started? To this end we defne a probabilistic version of the basin of attraction, the y-BOA, with the property that any solution started within it stays close and converges to the origin with probability at least y. We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on y-BOA.This work was supported by The Icelandic Research Fund, Grant no. 152429-051.Peer Reviewe