In Robust control design of nonlinear uncertain systems, stability theory plays an important role in engineering systems. For any given control system, it is crucial to have a stable system since an unstable control system is useless. Lyapunov . stability criterion is a general and useful procedure for studying the stability of nonlinear systems. The Lyapunov stability theory include two method...

In this paper we make use of the alternative converse Lyapunov theorem presented in [1] for specific classes of systems. We show that the developed converse Lyapunov theorem can be used to establish non–conservatism of a particular type of Lyapunov functions. Most notably, a proof that the existence of conewise linear Lyapunov functions are non– conservative for globally exponentially stable (G...

We study new formulae based on Lyapunov exponents for entropy, mutual information, and capacity of finite state discrete time Markov channels. We also develop a method for directly computing mutual information and entropy using continuous state space Markov chains. Our methods allow for arbitrary input processes and channel dynamics, provided both have finite memory. We show that the entropy ra...

This paper presents a constructive method for showing exponential stability of autonomous nonlinear systems consisting of state-dependent weighted linear systems. This kind of system representation is common in for instance fuzzy systems or is the result of an exact or approximative description of an arbitrary nonlinear vector field. Stability is shown by joining multiple local Lyapunov functio...

Lyapunov exponents, important invariants of a complicated dynamical process, can be difficult to determine from experimental data. In particular, when using embedding theory to build chaotic attractors in a reconstruction space, extra “spurious” Lyapunov exponents can arise that are not Lyapunov exponents of the original system. By studying the local linearization matrices that are key to a pop...

In this paper we introduce a generalized class of filtered Lyapunov functions, which are Lyapunov functions with time-varying parameters satisfying certain differential equations. Filtered Lyapunov functions have the same stability properties as Lyapunov functions. We give tools for designing composite filtered Lyapunov functions for cascaded systems. These functions are used to design globally...

In this article, we consider chaotic behavior happened in nonsmooth dynamical systems. To quantify such a behavior, a computation of Lyapunov exponents for chaotic orbits of a given nonsmooth dynamical system is focused. The Lyapunov exponent is a very important concept in chaotic theory, because this quantity measures the sensitive dependence on initial conditions in dynamical systems. Therefo...

Introduction 1 1. Lyapunov exponents of dynamical systems 3 2. Examples of systems with nonzero exponents 6 3. Lyapunov exponents associated with sequences of matrices 18 4. Cocycles and Lyapunov exponents 24 5. Regularity and Multiplicative Ergodic Theorem 31 6. Cocycles over smooth dynamical systems 46 7. Methods for estimating exponents 54 8. Local manifold theory 62 9. Global manifold theor...

In this paper, finite-time boundedness and stabilization problems for a class of switched linear systems with time-varying exogenous disturbances are studied. Firstly, the concepts of finite-time stability and finite-time boundedness are extended to switched linear systems. Then, based on matrix inequalities, some sufficient conditions under which the switched linear systems are finite-time bou...