We propose the notion of nondecreasing Lyapunov functions which can be used to prove stability or other properties of the system in question. This notion is in particular useful in studying switched or hybrid systems. We illustrate the concept by a general construction of such a nondecreasing Lyapunov function for a class of planar hybrid systems. It is noted that this class encompasses switche...

This paper presents a new approach for the stability analysis of TakagiSugeno (T-S) fuzzy systems. An idea is investigated to use piecewise quadratic Lyapunov function with low amount of computations. This class of Lyapunov function candidates is much richer than the common quadratic Lyapunov function. By exploiting the piecewise continuous Lyapunov function, we derive stability conditions that...

Dynamical behaviors of the competitive Lotka-Volterra system even for 3 species are not fully understood. In this paper, we study this problem from the perspective of the Lyapunov function. We construct explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-species case, (2) a 3-species model, and (3) the model...

We show that for any positive integer d, there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cann...

This paper proposes a variable structure controller (VS) with sliding sector for a hybrid system. At first, the extremum seeking control algorithm is used to find a Lyapunov function for the hybrid system. In general, a sliding sector is defined as a subset of the state space inside which some norm of state decreases. In the paper, then a sliding sector is designed for each subsystem of the hyb...

In this paper the existence of a quadratic control Lyapunov function for bilinear systems is considered. The existence of a control Lyapunov function ensures the existence of a control law which ensures the global asymptotic stability of the closed loop control system. In this paper we will derive conditions for the existence of a control Lyapunov function for bilinear systems. These conditions...

We investigate linear programming relaxations to synthesize Lyapunov functions that establish the stability of a given system over a bounded region. In particular, we attempt to discover functions that are more readily useful inside symbolic verification tools for proving the correctness of control systems. Our approach searches for a Lyapunov function, given a parametric form with unknown coef...

Zusammenfassung Der Beitrag beschreibt ein LyapunovFunktionen-basiertes Führungsfilter für lineare Strecken mit Stellgrößenbegrenzung. Ziel des Filters ist es, die Abhängigkeit zwischen der zu stabilisierenden Ruhelage und der Größe des mit Lyapunov-Funktionen nachweisbaren Einzugsbereichs zu überwinden. Hierdurch kann erreicht werden, dass für alle Ruhelagen im System der gleiche Einzugsbereic...

It is shown that for smooth nonlinear systems conditions for the existence of a Lyapunov function that guarantees uniform exponential stability can be formulated as linear inequalities deened pointwise in the state-space when assuming a general linearly parameterized class of smooth non-quadratic Lyapunov-function candidates. Hence, computation of the Lyapunov function involves the solution of ...