Review on computational methods for Lyapunov functions

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 di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic 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

A Numerical Technique For Stability Analysis Of Linear Switched Systems

In this report the ray-gridding approach, a new numerical technique for the stability analysis of linear switched systems is presented. It is based on uniform partitions of the state-space in terms of ray directions which allow refinable families of polytopes of adjustable complexity to be examined for invariance. In this framework the existence of a polyhedral Lyapunov function that is common to a family of asymptotically stable subsystems can be checked efficiently via simple iterative algorithms. The technique can be used to prove the stability of switched linear systems, classes of linear time-varying systems and Linear Differential Inclusions. We als

Robust Controller Synthesis for Hybrid Systems Using Modal Logic

. In this paper, we formulate and robustly solve a quite general class of hybrid controller synthesis problems. The type of controller we investigate is the switching control mechanism of a hybrid automaton (via guard and mode invariant sets), and the robustness result is with respect to variations in the right hand sides of the dierential equations that depend continuously on a parameter. We present a novel methodology for controller design and synthesis which uses modal logic as a formalism for reasoning about sets of plant states, and various operators on sets arising from the dierential equations and from metric tolerance relations on the state space. 1 Introduction In general terms, a hybrid system H can be said to satisfy a performance speci cation robustly if every system H 0 in some nominated variation class around H also satis es that speci cation. Likewise, a synthesis procedure for a class of control problems can be called robust if the nominal closed-loop hyb..

Proving the correctness of the implementation of a control-command algorithm

Abstract. In this article, we study the interactions between a controlcommand program and its physical environment via sensors and actuators. We are interested in finding invariants on the continuous trajectories of the physical values that the program is supposed to control. The invariants we are looking for are periodic sequences of intervals that are abstractions of the values read by the program. To compute them, we first build octrees that abstract the impact of the program on its environment. Then, we compute a period of the abstract periodic sequence and we finally define the values of this sequence as the fixpoint of a monotone map. We present a prototype analyzer that computes such invariants for C programs using a simple specification language for describing the continuous environment. It shows good results on classical benchmarks for hybrid systems verification. 1 Introduction. The behavior of an embedded, control-command program depends on both