Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem

The classical Matrosov theorem concludes uniform asymptotic stability of time-varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi-definite derivative along solutions) and another auxiliary function with derivative that is strictly nonzero where the derivative of the Lyapunov function is zero (Mastrosov in J Appl Math Mech 26:1337–1353, 1962). Recently, several generalizations of the classical Matrosov theorem have been reported in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005). None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) which is a very useful analysis and controller design tool for nonlinear systems. Inspired by generalized Matrosov conditions in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005), we provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function-based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a simple Euler-Lagrange system controlled by an adaptive controller and use this result to determine an ISS controller.     

Two conditions concerning common quadratic Lyapunov functions forlinear systems

Concerning a pair of linear systems, some triangle conditions for the existence of a common quadratic Lyapunov function are presented in this paper. These conditions are derived from a criterion for judging the semipositiveness of a linear map defined on symmetric matrices     

Further constructions of control-Lyapunov functions and stabilizing feedbacks for systems satisfying the Jurdjevic-Quinn conditions

For a broad class of nonlinear systems, we construct smooth control-Lyapunov functions. We assume our systems satisfy appropriate generalizations of the Jurdjevic-Quinn conditions. We also design state feedbacks of arbitrarily small norm that render our systems integral-input-to-state stable to actuator errors.     

Strict Lyapunov functions for time-varying systems

Uniformly asymptotically stable periodic time-varying systems for which is known a Lyapunov function with a derivative along the trajectories non-positive and negative definite in the state variable on non-empty open intervals of the time are considered. For these systems, strict Lyapunov functions are constructed.     

Lyapunov functions for diagonally dominant systems

This paper deals with the construction of Lyapunov functions for the finite dimensional linear system?= Ax when the entries of the generating matrixA satisfy various conditions requiring dominance of its diagonal elements and nonnegativity of its off-diagonal elements. The particular case in which the system defines a Markov chain is given special attention and it is shown that the results then imply certain inequalities which have an intuitively appealing information theoretic significance.     

Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions

In this article, the fuzzy Lyapunov function approach is considered for stabilising continuous-time Takagi-Sugeno fuzzy systems. Previous linear matrix inequality (LMI) stability conditions are relaxed by exploring further the properties of the time derivatives of premise membership functions and by introducing slack LMI variables into the problem formulation. The relaxation conditions given can also be used with a class of fuzzy Lyapunov functions which also depends on the membership function first-order time-derivative. The stability results are thus extended to systems with large number of rules under membership function order relations and used to design parallel-distributed compensation (PDC) fuzzy controllers which are also solved in terms of LMIs. Numerical examples illustrate the efficiency of the new stabilising conditions presented.     

Time-varying Lyapunov functions for linear time-varying systems

Adopting special time-varying Lyapunov-function candidates, it is shown that the Popov criterion ensures large-scale asymptotic stability for linear time-varying feedback systems with relaxed conditions on the time-varying gain fc(t). The main result is that li(t)/k(t) need not be bounded for all finite t and need be bounded only when ibecomes arbitrarily large. Again, in the derivations we make use of the Lefschetz version of the Kalman-Yakubovich lemma.     

Conjugate Lyapunov functions for saturated linear systems

Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper's methods.     

Quadratic-type Lyapunov functions for singularly perturbed systems

Asymptotic and exponential stability of nonlinear singularly perturbed systems are investigated via Lyapunov stability techniques. A quadratic-type Lyapunov function for a singularly perturbed system is obtained as a weighted sum of quadratic-type Lyapunov functions of two lower order systems. Estimates of domain of attraction, of upper bound on perturbation parameter, and of degree of exponential stability are obtained. The method is illustrated by studying the stability of a synchronous generator connected to an infinite bus.     

Homogeneous Lyapunov functions and necessary conditions for stabilization

We provide necessary conditions for the stabilization of nonlinear control systems with the additional requirement that a time-invarianthomogeneous Lyapunov function exists for the closed-loop system.The authors gratefully acknowledge research support from the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology, and Culture, and from the EC-Science Project SC1-0433-C(A). The first author is Charge de recherches F.N.R.S, on leave from CESAME, Université Catholique de Louvain, Belgium. He acknowledges partial support from the following organizations: National Science Foundation under Grant ECS-9203491, Air Force Office of Scientific Research under Grant F-49620-92-J-0495, Belgian American Educational Foundation, and North Atlantic Treaty Organization. The scientific responsibility rests with the authors.     

On vector Lyapunov functions for stochastic dynamical systems

Vector Lyapunov functions are used in the stability analysis of large-scale stochastic systems described by Itô differential equations (with stochastic disturbances in the subsystems and in the interconnecting structure). Sufficient conditions for asymptotic stability and exponential stability with probability 1 and in probability are established, in all cases the objective is the same: to analyze large-scale systems in terms of their lower order (and simpler) subsystems and in terms of their interconnecting structure. Use of the method presented makes it often possible to circumvent difficulties usually encountered when the Lyapunov method is applied to high-dimensional systems and to systems with complicated interconnecting structure. In order to demonstrate the usefulness of the present approach, a specific example is considered.     

Lyapunov conditions for input-to-state stability of impulsive systems

This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS, but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS, but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS, and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case, we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network.     

三阶系统族的共同二次Lyapunov函数

研究了寻找三阶系统族的共同二次Lyapunov函数问题.针对给定的稳定的三阶系统提出了寻找二次Lyapunov函数集的方法,然后获得了三阶系统族具有共同二次Lyapunov函数的充分条件.该充分条件易于构造,从而具有较强的工程实用性.文中实例验证了所得结果的有效性.     

Energy-based Lyapunov functions for forced Hamiltonian systems withdissipation

In this paper, we propose a constructive procedure to modify the Hamiltonian function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for nonzero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions can be easily constructed. Interestingly enough, for linear systems the resulting Lyapunov function is the incremental energy; thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are given     

Composite quadratic Lyapunov functions for constrained control systems

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.     

Homogeneous Lyapunov functions for systems with structured uncertainties

The problem addressed in this paper is the construction of homogeneous polynomial Lyapunov functions (HPLFs) for linear systems with time-varying structured uncertainties. A sufficient condition for the existence of an HPLF of given degree is formulated in terms of a linear matrix inequalities (LMI) feasibility problem. This condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the Lyapunov function. The maximum ?_∞ norm of the parametric uncertainty for which there exists a homogeneous polynomial Lyapunov function is computed by solving a generalized eigenvalue problem. The construction of such Lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in LMI form. Comparisons with other classes of Lyapunov functions through numerical examples taken from the literature show that HPLFs are a powerful tool for robustness analysis.     

Motion planning for control-affine systems satisfying low-order controllability conditions

This paper is devoted to the motion planning problem for control-affine systems by using trigonometric polynomials as control functions. The class of systems under consideration satisfies the controllability rank condition with the Lie brackets up to the second order. The approach proposed here allows to reduce a point-to-point control problem to solving a system of algebraic equations. The local solvability of that system is proved, and formulas for the parameters of control functions are presented. Our local and global control design schemes are illustrated by several examples.     

切换系统的二类共同Lyapunov 函数

研究切换系统的共同Lyapunov函数存在问题.对于一类正切换系统,给出了共同Lyapunov函数存在的充分条件.当系统矩阵集为二阶矩阵紧集时,给出了判断共同Lyapunov函数存在的方法,并给出了计算共同Lyapunov函数的算法.最后通过算例验证了所提出算法的有效性.     

Families of Lyapunov functions for nonlinear systems in criticalcases

Lyapunov functions are constructed for nonlinear systems of ordinary differential equations whose linearized system at an equalized point possesses either a simple zero eigenvalue or a complex conjugate pair of simple, pure imaginary eigenvalues. The construction is explicit, and yields parameterized families of Lyapunov functions for such systems. In the case of a zero eigenvalue, the Lyapunov functions contain quadratic and cubic terms in the state. Quartic terms appear as well for the case of a pair of pure imaginary eigenvalues. Predictions of local asymptotic stability using these Lyapunov functions are shown to coincide with those of pertinent bifurcation-theoretic calculations. The development of the paper is carried out using elementary properties of multilinear functions. The Lyapunov function families thus obtained are amenable to symbolic computer coding