This paper deals with the problem of the estimation of regions of asymptotic stability for continuous, autonomous, nonlinear systems. The first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields. In the second part certain topological considerations are first developed and the "trajectory reversing method" is then presented together with a theorem on which it is based. In the final part, several examples of application are reported, showing the efficiency of the proposed technique for low-order (second and third) systems.

On the estimation of asymptotic stability regions: State of the art and new proposals

We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description. The approach is demonstrated on several numerical examples.

https://homepages.laas.fr/henrion/Papers/roa.pdf

Convex Computation of the Region of Attraction of Polynomial Control Systems

A constructive methodology for estimating stability regions of general nonlinear dynamical systems is developed. The constructive methodology starts with a given Lyapunov function (either a global or a local Lyapunov function) and yields a sequence of Lyapunov functions which are then used to estimate the stability region. The resulting sequence of estimated stability regions is shown to be a strictly monotonic increasing sequence, and yet each of them remains inside the entire stability region. The significance of this methodology includes: its ability to reduce significantly the conservativeness in estimating the stability regions; its computational efficiency; its adaptability; and its sound theoretical basis. >

Stability regions of nonlinear dynamical systems: a constructive methodology

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them.
Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ different methods such as series expan- sion, linear programming, linear matrix inequalities, collocation methods, al- gebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function.

https://www.aimsciences.org/article/exportPdf?id=ee7e4111-9a13-4e44-95c8-14c5557278fc

Review on computational methods for Lyapunov functions

A computational method for determining quadratic lyapunov functions for non-linear systems

The determination of a stability domain of a control system, the motion of which is described by nonlinear differential equations, is often the object of intensive experimental and theoretical attack. This paper, partly tutorial and partly a presentation of new results, describes a method for obtaining a solution to this problem proposed recently by the Russian mathematician, V. I. Zubov. The tutorial part outlines the fundamental principles of V. I. Zubov's procedure for constructing Lyapunov functions for non-linear systems. If the construction problem can be solved, it leads to a Lyapunov function which uniquely defines the exact boundary of the stability region. For the application of the method, several simple examples are treated in which the exact stability region is found in analytic closed form. Since the construction procedure requires the solution of a linear partial differential equation, there are many cases for which an exact analytic solution is not possible. In some of these cases, however, it is possible to construct an approximate series solution which is always at least as good as the usual quadratic form Lyapunov function. The series construction procedure has been programmed (in IBM 7070 FORTRAN language) for a broad class of differential equations of the second order. A simple example solved by the digital computer program is described.

M. Patel

Papers

Control engineering applications of V. I. Zubov's construction procedure for Lyapunov functions

Genetic Algorithm Based Sliding Mode Controller for Underactuated 2-DoF Gyroscope