Continuous Lyapunov Research Articles

Efficient computation of Lyapunov functions for Morse decompositions

We present an efficient algorithm for constructing piecewise constant Lyapunov functions for dynamics generated by a continuous nonlinear map defined on a compact metric space.We provide a memory efficient data structure for storing nonuniform grids on which the Lyapunov function is defined and give bounds on the complexity of the algorithm for both time and memory.We prove that if the diameters of the grid elements go to zero, then the sequence of piecewise constant Lyapunov functions generated by our algorithm converge to a continuous Lyapunov function for the dynamics generated the nonlinear map.We conclude by applying these techniques to two problems from population biology.

Stability of sets of stochastic functional differential equations with impulse effect

In this paper, we study the stability of sets for a class of impulsive stochastic functional differential equations. By employing piecewise continuous Lyapunov functions with Razumikhin methods, some sufficient conditions are established to guarantee the stability of sets of impulsive stochastic functional differential equations and we also show that the impulses play an important role in the stability of stochastic functional differential equations. Three examples are presented to illustrate the effectiveness of the results obtained.

H∞ controller design for affine fuzzy systems based on piecewise Lyapunov functions in finite-frequency domain

This paper studies the problem of state feedback controller design for a class of nonlinear systems described by continuous-time affine fuzzy models. Based on a piecewise continuous Lyapunov function combined with S-procedure and some matrix decoupling techniques, a novel control law is derived in the formulation of linear matrix inequalities (LMIs) in finite-frequency domain. First, a so-called finite-frequency H∞ performance index is defined, which extends the standard H∞ performance. Then, a sufficient condition is derived such that the fuzzy closed-loop system satisfies a finite-frequency H∞ performance. By introducing the S-procedure and adding slack variables, piecewise controllers are designed to deal with disturbances in the low-, middle-, and high-frequency domain, respectively. The proposed piecewise controller design method can get a better disturbance-attenuation performance when the frequency ranges of disturbances are known beforehand. Finally, two examples are given to illustrate the effectiveness and superiority of the new results.

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise affine function can be parameterized by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterized function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.

A Lyapunov view on positive Harris recurrence of multiclass queueing networks

In this paper we prove the positive Harris recurrence of the underlying Markov process of a multiclass queueing network. Using the fact that under certain conditions the stability of a fluid network is equivalent to the existence of a continuous Lyapunov function, we propose a new method to conclude that the underlying Markov process is positive Harris recurrent if the associated fluid network model is stable by using explicitly the Lyapunov function of the fluid network.

Stabilizing Controller Design for a Special Class of PWA Systems using Discontinuous Piecewise Quadratic Lyapunov Functions

this paper a new controller is proposed to stabilize an especial class of hybrid piecewise affine systems. In this study, for the first time, the stabilizing controller is designed based on discontinuous piecewise quadratic Lyapunov functions which decrease the conservation and propose a wider class of applicable Lyapunov functions as it omits the continuity condition in boundary points compared to continuous piecewise quadratic Lyapunov functions. In addition the stability conditions are formulated in the form of Bilinear Matrix Inequalities (BMI) problem. To solve the proposed problem, BMI is defined in the form of a multi-objective nonlinear optimization problem which has been solved through using genetic Algorithm (GA).

Synchronisation control of dynamical networks subject to variable sampling and actuators saturation

This study investigates aperiodic sampled-data synchronisation of complex dynamical network (CDN) incorporate dynamics of actuators saturation. The sampling intervals considered here are time-varying which allowed to be variable within the lower and upper bounds. In order to fully captures the sawtooth structure characteristic of the sampled-data systems, a novel time-dependent continuous Lyapunov functional is employed to obtain less conservative criteria. Furthermore, the generalised sector bound condition is utilised for the estimate about the basin of the attraction. Based on the local stability condition, the sampled-data controller is constructed to guarantee the synchronisation of CDN in the presence of actuators saturation. The obtained sufficient conditions can be cast in two optimisation cases to maximise the admissible upper bound of sampling instants or enlarge estimate about the domain of attraction for the closed-loop systems. Subsequently, the validity and applicability of the criteria are verified through numerical examples.

Computing continuous and piecewise affine lyapunov functions for nonlinear systems

We present a numerical technique for the computation of a Lyapunov functionfor nonlinear systems with an asymptotically stable equilibrium point. The proposed approachconstructs a partition of the state space, called a triangulation, and then computes values atthe vertices of the triangulation using a Lyapunov function from a classical converse Lyapunovtheorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuousand Piecewise Affine (CPA) function.Verification that the obtained CPA function is a Lyapunov function is shownto be equivalent to verification of several simple inequalities.Numerical examples are presented demonstrating different aspects of the proposed method.

Class library in C++ to compute Lyapunov functions for nonlinear systems

Lyapunov functions are of fundamental importance in the stability analysis of dynamical systems. Unfortunately, the construction of Lyapunov functions for nonlinear systems is in general a very dificult problem. We present software written in C++ that computes Lyapunov functions for n-dimensional, nonlinear, time-continuous dynamical systems defined through systems of autonomous differential equations. The software implements the computation of continuous and piecewise afine (CPA) Lyapunov functions through linear programming (LP) and the computation of Lyapunov functions using radial basis functions (RBF) and collocation. In the former case a common Lyapunov function for a finite set of nonlinear systems can be computed and it is guaranteed to be a true Lyapunov function, i.e. to fulfill the defining properties of a Lyapunov function exactly and rigorously (as opposed to approximately, which is so often the case with numerically contrived results). In the latter case the computed function is smooth and is guaranteed to be a Lyapunov function outside of an arbitrary small neighbourhood of the equilibrium, if the collocation points are close enough. However, it is not obvious how to determine what is close enough. A Lyapunov property test for the CPA interpolation of the computed RBF solution is therefore also part of the software.

Lyapunov Function Verification: MATLAB Implementation

Lyapunov functions are a mathematical generalization of the dissipative energy concept of physics. Lyapunov functions are the centerpiece of the Lyapunov-stability theory for dynamical systems in general. Here we present a simple method for checking the validity of a quadratic Lyapunov function, which is constructed for the linearization of a nonlinear system and does not in general satisfy the condition of having a negative orbital derivative on the origins basin of attraction. The current work also extends previous work done on continuous and piecewise afine (CPA) Lyapunov functions by permitting more general triangulations than have been used in that context, namely Delaunay triangulations. Delaunay triangulations have been studied intensively in the literature and allow local refinements of the triangulation. The third contribution of this paper is a freely available MATLAB implementation of the methods proposed.

Adaptive Synchronization of Networked Euler-Lagrange Systems with Directed Switching Topology

In this paper, the cooperative control problem of networked Euler-Lagrange systems with parametric uncertainties and unidirectional interaction is addressed under dynamically changing topology. As the communication graph evolves over time, a distributed control law via local effective interactions is designed. Adaptive techniques are used to deal with parametric uncertainties in the dynamics. With a continuous Lyapunov function, it is obtained that synchronization can still be achieved asymptotically as long as the union graph of the switching topologies has a directed spanning tree frequently enough. Extensions to disturbance rejection problems are also addressed using simple disturbance-observer or sliding mode control scheme. Illustrative examples with comparing simulation in the context of attitude synchronization of five non-identical spacecraft are further presented to show the effectiveness of the proposed cooperative control strategy.

A Converse Lyapunov Theorem and Robustness for Asymptotic Stability in Probability

A converse Lyapunov theorem is established for discrete-time stochastic systems with non-unique solutions. In particular, it is shown that global asymptotic stability in probability implies the existence of a continuous Lyapunov function, smooth outside of the attractor, that decreases in expected value along solutions. The keys to this result are mild regularity conditions imposed on the set-valued mapping that characterizes the update of the system state, and the ensuing robustness of global asymptotic stability in probability to sufficiently small state-dependent perturbations.

Forward and Converse Lyapunov Theorems for Discrete Dynamical Systems

This technical note addresses the stability analysis of nonlinear dynamic systems. Three main contributions are made. First, we show that the standard assumption of a continuous Lyapunov function can be (and in some cases must be) relaxed. We introduce the concept of the `weak' Lyapunov function, which requires that an annulus condition be satisfied. We believe that this annulus condition is a more natural construct, because it is precisely what is needed to make the forward Lyapunov theorem true. Second, we provide an example of a nonlinear system with stable equilibrium point that cannot be shown to be stable with a continuous Lyapunov function. Finally, we demonstrate a simpler and less restrictive proof of the converse Lyapunov theorem.

Integral manifolds for uncertain impulsive differential–difference equations with variable impulsive perturbations

In the present paper sufficient conditions for the existence of integral manifolds of uncertain impulsive differential–difference equations with variable impulsive perturbations are obtained. The investigations are carried out by means the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s functions.

Finite-time state-feedback control for a class of stochastic high-order nonlinear systems

This paper discusses the problem of global finite-time stabilization in probability for a class of stochastic high-order nonlinear systems whose drift and diffusion terms satisfy lower-triangular growth conditions. By adopting adding one power integrator technique and constructing twice continuous differential Lyapunov functions, a continuous state-feedback controller is recursively designed. Based on stochastic finite-time stability theorem, it is proved that the solution of the closed-loop system is finite-time stable in probability. Several simulation examples are given to illustrate the effectiveness of the proposed design procedure.

Integral manifolds of impulsive fractional functional differential systems

In this paper, a class of impulsive Caputo fractional functional differential systems with variable impulsive perturbations is investigated. Sufficient conditions for the existence of integral manifolds are obtained. The main results are proved by means of piecewise continuous Lyapunov functions and the fractional comparison principle. The demonstrated techniques can be applied in studying properties of many applied problems of diverse interest.

Lowest-rank solutions of continuous and discrete Lyapunov equations over symmetric cone

The low-rank solutions of continuous and discrete Lyapunov equations are of great importance but generally difficult to compute in control system analysis and design. Fortunately, Mesbahi and Papavassilopoulos (1997) [30] showed that with the semidefinite cone constraint, the lowest-rank solutions of the discrete Lyapunov inequality can be efficiently solved by a linear semidefinite programming. In this paper, we further show that the lowest-rank solutions of both the continuous and discrete Lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations, the symmetric cone linear programming problems. Therefore, they are polynomial-time solvable. Since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case, our results also answer an open question proposed by Recht, Fazel and Parrilo (2010) [36].

Computation of Lyapunov functions for nonlinear discrete time systems by linear programming

Given an autonomous discrete time system with an equilibrium at the origin and a hypercube containing the origin, we state a linear programming problem, of which any feasible solution parameterizes a continuous and piecewise affine (CPA) Lyapunov function for the system. The linear programming problem depends on a triangulation of the hypercube. We prove that if the equilibrium at the origin is exponentially stable, the hypercube is a subset of its basin of attraction, and the triangulation fulfils certain properties, then such a linear programming problem possesses a feasible solution. We present an algorithm that generates such linear programming problems for a system, using more and more refined triangulations of the hypercube. In each step the algorithm checks the feasibility of the linear programming problem. This results in an algorithm that is always able to compute a Lyapunov function for a discrete time system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibrium's basin of attraction. The system is assumed to have a right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type such as linear, piecewise affine and so on. Our approach is a non-trivial adaptation of the CPA method to compute Lyapunov functions for continuous time systems to discrete time systems.

Integralφ0-Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

This paper establishes a criterion on integralφ0-stability in terms of two measures for impulsive differential equations with “supremum” by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method. Meantime, an example is given to illustrate our result.

On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays

This paper is concerned withpth moment input-to-state stability (p-ISS) and stochastic input-to-state stability (SISS) of impulsive stochastic systems with time delays. Razumikhin-type theorems ensuringp-ISS/SISS are established for the mentioned systems with external input affecting both the continuous and the discrete dynamics. It is shown that when the impulse-free delayed stochastic dynamics arep-ISS/SISS but the impulses are destabilizing, thep-ISS/SISS property of the impulsive stochastic systems can be preserved if the length of the impulsive interval is large enough. In particular, if the impulse-free delayed stochastic dynamics arep-ISS/SISS and the discrete dynamics are marginally stable for the zero input, the impulsive stochastic system isp-ISS/SISS regardless of how often or how seldom the impulses occur. To overcome the difficulties caused by the coexistence of time delays, impulses, and stochastic effects, Razumikhin techniques and piecewise continuous Lyapunov functions as well as stochastic analysis techniques are involved together. An example is provided to illustrate the effectiveness and advantages of our results.