An alternative converse Lyapunov theorem for discrete-time systems

Abstract

This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete-time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi-global practical Lyapunov functions for globally asymptotically stable systems. Furthermore, for specific classes of systems, the developed converse theorem can be used to establish non-conservatism of a particular type of Lyapunov functions. Most notably, a proof that conewise linear Lyapunov functions are non-conservative for GES conewise linear systems is given and, as a by-product, tractable construction of polyhedral Lyapunov functions for linear systems is attained.

A delta-sampling verification theorem for discrete-time, possibly discontinuous systems

“…The values of σi(δ) are, in order, the following: 0.5750, 0.0750, 0.1000, 0.1313, 0.1781, for i ∈ Z [1,5] . Observe again that σi(δ) < γi, for all i ∈ Z [1,8] , which validates the correctness of the obtained result in the context of Theorem 3.3.…”

“…The classical notions are recovered for a unitary step. These relaxed notions were recently studied in a number of papers, including [9], [17], [3], [8] and [4]. The most attractive feature of these tools is that, under the assumption of exponential stability, any proper candidate set and candidate function is a finite-step invariant set and FSLF, respectively, for some step value.…”

On Path-Complete Lyapunov Functions: Geometry and Comparison

“…We provide an algorithmic sufficient condition in Theorem IV.4, inspired by existing ad-hoc proofs for particular cases such as the ones presented in [1, Section 4 and 5], [33,Theorem 3.5], [16,Theorem 20]. Finally, in Section V, we comment on our results and conclude our work.…”

On the computation of Lyapunov functions for discrete-time nonlinear systems

“…This advantage was already exploited in the above mentioned papers for developing scalable and non-conservative stability tests for linear and switched linear discrete-time systems. Moreover, recently, [18] introduced an alternative converse theorem that gives an explicit construction of a standard LF from a FSLF.…”

“…The results of [18] consider a global setting, i.e., the properties hold for all states in R n , and therefore, they require an analytical solution, which depends on the vector field of the particular system. In this paper, we restrict the space to a compact set, which allows for a formulation of the converse result in [18] as a nonlinear optimization problem.…”