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

On invariant asymptotic observers

paper number: CDC02-INV0903. For dynamics ˙x = f(x) with output y = h(x) invariant with respect to a transformation group G, we define invariant asymptotic observer of the form ˙ ˆx = ˆ f(ˆx, y) wherey = h(x) is the measured output and ˆx an estimation of the unmeasured state x. Such a definition is motivated by a class of chemicalreactors, treated in details, when the group of transformations corresponds to unit changes and the output y to ratio of concentrations. We propose a constructive method that guaranties automatically the observer invariance ˙ ˆx = ˆ f(ˆx, y): it is based on invariant vector fields and scalar functions, called invariant estimation errors, that can be computed via Darboux-Cartan moving frame methods. The observer convergence remains, in the general case, an open problem. But for the class of chemicalreactors considered here, the invariant observer convergence is proved by showing that, in a Killing metric associated to the action of G, the symmetric part of the Jacobian matrix ∂ ˆ f/∂ˆx is definite negative (contraction). Key words: asymptotic observers, moving-frame method, invariant, symmetries, contraction, chemicalreactors.