Vector Lyapunov Functions Research Articles

Vector Lyapunov functions and conditional stability for systems of impulsive differential-difference equations

AbstractBy means of piecewise continuous vector functions, which are analogues of the classical Lyapunov functions and via the comparison method, sufficient conditions are found for conditional, stability of the zero solution of a system of impulsive differential-difference equations.

Stability analysis of discrete systems

The present review includes the following sections: formulation of the stability problem for discrete systems, parameter-variation formula and comparison method, development of Lyapunov's direct method for discrete systems, sufficient conditions for the practical stability of a discrete system, conditions for the stability of a discrete system on a finite interval, application of Lyapunov's vector functions, and closing remarks.

Nonuniqueness Theorem for a Singular Cauchy Problem

Abstract A general nonuniqueness theorem is given for ordinary differential equations with singularities. The criterion uses vector Lyapunov functions and extends the previously known criteria ones. The applicability is illustrated by several examples.

Stability analysis of discontinuous dynamical systems using vector Lyapunov functions

In this paper, we establish several stability results for discontinuous dynamical systems defined onR+=[0, ∞) which make use of vector Lyapunov functions. For the scalar case, these results yield in particular theprincipal Lyapunov stability results for discontinuous dynamical systems reported earlier. We demonstrate the applicability of our results by studying a class of interconnected discontinuous dynamical systems and several specific examples.

Decentralized techniques for the analysis and control of Takagi-Sugeno fuzzy systems

This paper discusses decentralized parallel distributed compensator design for Takagi-Sugeno fuzzy systems. The fuzzy system is viewed as an interconnection of subsystems some of which are strongly connected, while others being weakly connected. The necessary theory is developed so that one can associate this fuzzy system with another one in a higher dimensional space, the so-called expanded space, design decentralized parallel distributed compensators in the expanded space, then contract the solution for implementation on the original fuzzy system. In this respect, connective stability of the open loop and closed loop of the interconnected system is analyzed via the concepts of vector Lyapunov functions and M-matrices. Different Lyapunov functions generate different results for the discrete-time fuzzy system, quadratic Lyapunov generating the superior of the two. Following a similar approach, stabilization of the closed-loop fuzzy system using local parallel distributed compensators is investigated.

A comparison theory for the stability analysis of general hybrid systems

In some of our recent work, a general model for hybrid dynamical systems was proposed whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time . By transforming this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but defined on real time, we established the Principal Lyapunov Stability Results as well as Lagrange Stability Results. Making use of stability preserving mappings, results which comprise a general comparison theory for hybrid dynamical systems can be developed and these results can be specialized by employing vector Lyapunov functions. It then can be shown that the latter results can be used to yield the Principal Lyapunov Stability Results discussed above as special cases. Due to space limitations, only a summary of the paper is presented.

Stability and attractivity of solutions of differential equations with impulses at fixed times

In this paper we consider the dynamics of solutions of impulsive differential equations with fixed time moments of impulsive effects on the basis of comparison methods and vector Lyapunov functions. We propose sufficient conditions on the following dynamic properties: stability, attractivity, and some combinations of them.

Diagonal stability in the large-scale system approach

Diagonal type Lyapunov functions are ubiquitous in interconnected systems analysis and design, either by the so called vector or scalar Lyapunov function approaches. Clear justification for this phenomenon has not been given in the extensive literature on the field. This paper explains the reasons for this ubiquity, actually justifying the necessity of diagonal stability in this context.

Stabilization of Nonlinear Systems Using Vector Lyapunov Functions

This paper deals with the stabilization problem of nonlinear systems using vector Lyapunov methods. The stabilization of general nonlinear systems is posed in terms of the existence of some vector Lyapunov function which together with comparison principles allows us to compute the stabilizing controller. The design procedure consists of giving a comparison system satisfying the desired stability requirements and proposing some vector Lyapunov function then, the stabilizing control law is computed from certain differential inequality. For nonlinear systems affine in the control input the computation of a closedform expression for the control law is indeed guaranteed by the existence of some control Lyapunov function.

Vector Lyapunov functions in impulsive variable-time differential systems

Vector Lyapunov functions in impulsive variable-time differential systems

Research of dynamics of strategic weapons of the ”multipole world” by a method of Vector Lyapunov functions

Research of dynamics of strategic weapons of the ”multipole world” by a method of Vector Lyapunov functions

Stability analysis of fuzzy multivariable systems: Vector Lyapunov function approach

A design methodology for the stabilisation of a class of nonlinear systems which can be identified in the form of a fuzzy dynamic model is presented. The identified model is a fuzzy aggregation of a set of local linear models. The stability condition of the fuzzy model is devised by using the vector Lyapunov function approach and some important issues of stability analysis are discussed. A parameter optimisation algorithm is proposed to construct a set of local linear control laws for the fuzzy system. The performance of the fuzzy model based control technique is illustrated by a simulation example.

Vector Lyapunov Function Method: Theory and Application to Complex Industrial Systems

Subject of this lecture is Vector Lyapunov functions (VLF) method in nonlinear theory for large-scale systems and there industrial applications. Abstract control theory like axiomatic theory of system of processes with controls and perturbations: concepts, dynamic properties, discrete and continuous time: axiomatic VLF method. Algorithms for inference of theorems on dynamical properties for nonlinear systems. Software packages for theorem inference, for stability analysis of nonlinear systems. Application to industrial problems. Nonlinear dynamics of orbital telescopes and geostationary spacecraft. Nonlinear methods, results and software for dynamic investigations of fail-safe gyromoment attitude control systems of communication spacecrafts. Method oriented software DYNAMICS for CAD-systems.

Estimation of the Asymptotic Stability Domain and Asymptotic Behaviour for Nonlinear Neutral Systems

Abstract This paper deals with the estimation of both stability domain and asymptotic behaviours of a class of nonlinear neutral functional differential equations (FDE). The approach for solving such a complex problem is based on comparison methods: the solutions of the original system are compared (by upper-bounding) to the behaviours of some more simple ordinary differential equation (ODE). It is combined with the use of regular vector Lyapunov's functions, that appear to yield less conservative results than usual scalar norms or Lyapunov's functions.

Sign and stability of equilibria in quasi-monotone positive nonlinear systems

The paper deals with the problem of finding the extension of a well-established connection between sign and stability of nontrivial equilibria in positive linear systems to the class of quasi-monotone positive nonlinear systems. The mathematical framework to provide the sufficient conditions guaranteeing such a relationship is the stability theory for nonlinear systems, especially the comparison principles and the method of vector Lyapunov functions. As an illustration of our results we present an example of a double-effect evaporator to study the sign-stability property of its only equilibrium point.

An algorithm to find the smallest possible values in a matrix to satisfy the M-matrix condition

M-matrix is an important tool for the construction of vector Lyapunov functions for linear and nonlinear large-scale systems. It is also widely used in the conditions related to decentralized control of large-scale systems. An algorithm and its intended application to large-scale system decentralized controller designs are presented in this paper.

Control Design of Nonlinear Systems Using Fuzzy Model and Genetic Algorithm

A control design methodology for stabilization of a class of nonlinear systems is presented. The systems are identified in the form of a fuzzy model and its structure is a fuzzy aggregation of a set of local linear models. The stability condition of the fuzzy dynamic model is determined by using the vector Lyapunov function approach. A parameter optimization algorithm is proposed to construct a set of local linear control laws for the fuzzy systems by implementing the modified Genetic Algorithm. The performance of the fuzzy model based control technique is illustrated by a simulation example.

Stability analysis of hereditary iterative processes

By employing the theory of difference inequalities, several comparison theorems in the context of vector-Lyapunov functions are developed. These comparison theorems are applied to study various stability properties of the equilibrium state of the hereditary iterative processes. Examples are given to demonstrate the obtained results. The method provides a tool to investigate the fundamental problem of stability versus hereditary effect.

New Directions in the Method of Vector Lyapunov Functions: Applications to Large-Scale Dynamic Systems

New Directions in the Method of Vector Lyapunov Functions: Applications to Large-Scale Dynamic Systems

Global attractivity and stability of a scalar nonlinear difference equation

Lyapunov functions are constructed and used to establish that an equilibrium of a scalar nonlinear difference equation is globally stable. General conditions for the local stability of an equilibrium of a difference equation which is described by a nonsmooth function are established. Vector Lyapunov functions are constructed and used to establish conditions for an unstable equilibrium to be globally attractive. These results are useful for applications which use scalar nonlinear difference equations as models. They provide insight into conditions for the global convergence of a scalar iterative process in computation.