Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions

Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is mandatory not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the domain of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.     

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

This paper investigates linear systems with polynomial dependence on time-invariant uncertainties constrained in the simplex via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs). It is shown that a sufficient condition for establishing whether the system is either stable or unstable can be obtained by solving a generalized eigenvalue problem. Moreover, this condition is also necessary by using a sufficiently large degree of the HPD-QLF.     

LMI conditions for quadratic and robust D-stability of interval systems

Sufficient conditions for the quadratic D-stability and further robust D-stability of interval systems are presented in this paper. This robust D-stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities （LMIs） defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust D-stability of interval systems.     

Integral quadratic constraint approach vs. multiplier approach

Integral quadratic constraints (IQC) arise in many optimal and/or robust control problems. The IQC approach can be viewed as a generalization of the classical multiplier approach in the absolute stability theory. In this paper, we study the relationship between the two approaches for robust stability analysis. Using a slightly modified multiplier approach, we show that the existence of an IQC is equivalent to the existence of a multiplier in most known cases. It is hoped that this result provides some new insight into both approaches and makes them more useful in robust control applications.     

Stability of an analog optimization circuit for quadratic programming

We study the stability of an analog optimization circuit that solves quadratic programming (QP) problems. The circuit dynamics are modeled as a switched affine system. A piece-wise quadratic Lyapunov function and the KYP lemma are used to derive the stability criterion. The stability criterion characterizes the range of critical circuit parameters for which the QP circuit is globally exponentially stable.     

Robust control of state delayed systems with polytopic type uncertainties via parameter-dependent Lyapunov functionals

This paper considers the problem of robust stability and robust stabilization for linear systems with a constant time-delay in the state and subject to real convex polytopic uncertainty. First of all, for robust stability problem, we exploit new matrix inequalities characterization of delay-dependent quadratic stability results, demonstrate that it allows the use of parameter-dependent Lyapunov functionals, and develop control design methods based on linear matrix inequalities (LMIs) for solving the robust control problem. Next, the problem of determining the maximum time-delay under which the system remains stable is cast into a generalized eigenvalue problem and thus solved by LMI techniques. Finally, illustrative examples are given to demonstrate the advantage of these new representations.     

LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions

The robust stability of uncertain linear systems in polytopic domains is investigated in this paper. The main contribution is to provide a systematic procedure for generating sufficient robust stability linear matrix inequality conditions based on homogeneous polynomially parameter-dependent Lyapunov matrix functions of arbitrary degree on the uncertain parameters. The conditions exploit the positivity of the uncertain parameters, being constructed in such a way that: as the degree of the polynomial increases, the number of linear matrix inequalities and free variables increases and the test becomes less conservative; if a feasible solution exists for a certain degree, the conditions will also be verified for larger degrees. For any given degree, the feasibility of a set of linear matrix inequalities defined at the vertices of the polytope assures the robust stability. Both continuous and discrete-time uncertain systems are addressed, as illustrated by numerical examples.     

Necessary and sufficient conditions for quadratic controllability of a class of uncertain systems

We introduce a controllability notion for uncertain systems, namely, quadratic controllability. For a specific class of norm bounded uncertain systems, we show that this notion is equivalent to controllability of a nominal system and the satisfaction of a certain structural condition by the uncertainty. For systems in which the uncertain parameters affect all control inputs, this structural condition is simply the familiar matching condition commonly encountered in the literature on robust and adaptive control of uncertain systems.     

Asymptotic feedback stabilization: A sufficient condition for the existence of control Lyapunov functions

In this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control. We offer sufficient conditions for the existence of ‘Control Lyapunov functions’ that according to [3,23] and [28–30] guarantee stabilization at a specified equilibrium by means of a feedback law, which is smooth except possibly at the equilibrium. We note that the results of the paper present a local nature.     

Comments and improvement on “Robust control of state delayed systems with polytopic type uncertainties via parameter-dependent Lyapunov functionals”

The errors are pointed out in the above paper. And then, a new delay-dependent condition is presented for the robust stability of uncertain time-delay systems, which is an improvement on the corresponding results in the above paper.     

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.     

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.     

参数不确定性广义周期时变系统的鲁棒稳定性分析

基于广义周期时变系统允许的充分必要条件,提出了参数不确定性广义周期时交系统鲁棒稳定的概念,并得到了该类系统鲁棒稳定的充分必要条件．研究在状态反馈控制下保证闭环系统鲁棒稳定的条件,给出了一族状态反馈鲁棒稳定器的设计方法．引入广义周期时变系统二次稳定的概念,讨论了二次稳定性与鲁棒稳定性的关系．最后通过数值算例说明了所得的主要结果,     

On the region of attraction of nonlinear quadratic systems

Quadratic systems play an important role in the modelling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems, it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the region of attraction of the equilibrium. The proposed algorithm requires the solution of a suitable feasibility problem involving linear matrix inequalities constraints. An example illustrates the effectiveness of the proposed procedure by exploiting a population interaction model of three species.     

Robust filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions

This paper addresses the design of robust filters for linear continuous-time systems subject to parameter uncertainty in the state-space model. The uncertain parameters are supposed to belong to a given convex bounded polyhedral domain. Two methods based on parameter-dependent Lyapunov functions are proposed for designing linear stationary asymptotically stable filters that assure asymptotic stability and a guaranteed performance, irrespective of the uncertain parameters. The proposed filter designs are given in terms of linear matrix inequalities which depend on a scalar parameter that should be searched for in order to optimize the filter performance.     

On the Lyapunov stability of discrete-time processes modeled by difference inclusions

The subject of this note is the stability in the sense of Lyapunov of difference inclusions. Connection with the convergence of algorithms is also discussed.     

Discrete-time stochastic control systems: A continuous Lyapunov function implies robustness to strictly causal perturbations

Discrete-time stochastic systems employing possibly discontinuous state-feedback control laws are addressed. Allowing discontinuous feedbacks is fundamental for stochastic systems regulated, for instance, by optimization-based control laws. We introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations. Under basic regularity conditions, the existence of a continuous stochastic Lyapunov function is sufficient to establish that asymptotic stability in probability for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. Robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state-dependent perturbations, and in a semiglobal practical sense in the case of persistent perturbations. An example shows that a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case disturbances that are not strictly causal. Our positive results are also illustrated by examples.     

A method for construction of stability regions by Lyapunov functions

This article proposes a new method of constructing a sequence of Lyapunov functions which provides more adequate approximation for the exact stability region via the regions which are bounded by a level surfaces of the functions.     

Stability analysis of interconnected systems with possibly unstable subsystems

A direct method is presented for stability analysis of nonlinear interconnected dynamical systems. A new scalar Lyapunov function is considered as weighted sum of individual Lyapunov functions for each free subsystem and individual scalar functions related separately to each connection. Sufficient conditions are obtained for asymptotic stability of the equilibrium state by testing the definity of two constant square matrices whose dimension is equal to the number of subsystems. This method can assure stability of systems with possible unstable subsystems. A simple numerical example is included to illustrate this theory.     

Robust stability analysis of time-varying systems using time-varying quadratic forms

We consider the robust stability of time-varying linear systems subject to structured time-varying uncertainty. We provide an alternate proof of the result that robust stability holds if and only if a scaled small-gain condition holds asymptotically. The new proof employs an extension of the so-called S-procedure losslessness theorem to time-varying quadratic forms which is of independent interest.