Common Quadratic Lyapunov Function Research Articles

Development of Stability Research on Takagi-Sugeno Fuzzy Control Systems and Approximation of the Necessary and Sufficient Conditions

The controller's design and analysis based on Takagi-Sugeno (T-S) models play important roles in the research of fuzzy control. When fuzzy control is being widely applied, the stability research on T-S fuzzy control supports the applications on the theoretical standpoint. The development of the stability research on T-S fuzzy control is surveyed in this paper. For the Lyapunov functions, this paper considers common quadratic Lyapunov functions, piecewise quadratic Lyapunov functions, fuzzy Lyapunov functions, nonquadratic Lyapunov functions and homogenously polynomially parameterized Lyapunov functions. For the control laws, this paper considers parallelly distributed compensation, non-parallel distributed compensation laws and homogenously polynomially parameterized control laws. By extensively applying the Pólya's theorem and the techniques for homogenous polynomials, the stability conditions are gradually developed towards necessary and sufficient. It is very important to recognize and master this development in order to further study fuzzy control and the related control theories.

Quadratic Stability and Singular SISO Switching Systems

In this note, we consider the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices are of the form <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ghT</i> , and where one of the matrices is singular. A necessary and sufficient condition for the existence of such a function is given in terms of the spectrum of the product <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ghT</i> ) . The technical note also contains a spectral characterization of strictly positive real transfer functions which are strictly proper. Examples are presented to illustrate our results.

An extension of LaSalle’s Invariance Principle for a class of switched linear systems

In this paper LaSalle’s Invariance Principle for switched linear systems is studied. Unlike most existing results in which each switching mode in the system needs to be asymptotically stable, in this paper the switching modes are allowed to be only Lyapunov stable. Under certain ergodicity assumptions, an extension of LaSalle’s Invariance Principle for global asymptotic stability of switched linear systems is proposed provided that the kernels of derivatives of a common quadratic Lyapunov function with respect to the switching modes are disjoint (except the origin).

Stability analysis and H ∞ control for uncertain stochastic piecewise-linear systems

The problem of the stochastic stability and H∞ control for uncertain stochastic piecewise-linear systems is studied. The stability analysis is based on Lyapunov functions that are continuous and piecewise quadratic. It is shown that the stability in the mean square for uncertain stochastic piecewise-linear systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. It is also demonstrated via a numerical example that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions. The H∞ controllers can also be designed by solving a set of bilinear matrix inequalities (BMIs) based on the powerful piecewise quadratic Lyapunov function.

Quadratic stabilization of switched nonlinear systems

In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated. When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law. The results of this paper are also applied to switched linear systems.

Exponential stability of simultaneously triangularizable switched systems with explicit calculation of a common Lyapunov function

In this note, a common quadratic Lyapunov function is explicitly calculated for a linear hybrid system described by a family of simultaneously triangularizable matrices. The explicit construction of such a function allows not only obtaining an estimate of the convergence rate of the exponential stability of the switched system under arbitrary switching but also calculating an upper bound for the output during its transient response. Furthermore, the presented result is then extended to the case where the system is affected by parametric uncertainty, providing the corresponding results in terms of the nominal matrices and uncertainty bounds.

Stability of equilibria of some switched non-linear systems with applications to control synthesis for electrohydraulic servomechanisms

Stability of the zero solution is analyzed for a family of switched systems indexed by a parameter, each system having = 0 in the spectrum of the Jacobian matrix calculated in zero. It is proved that existence of a common quadratic Lyapunov function for some lower dimensional linear systems is sufficient to ensure local uniform stability of the zero solution of the switched non-linear system and a regular asymptotic behaviour. An application to control synthesis for stabilizing equilibria in a switched non-linear control system modelling an electrohydraulic servomechanism is given.

Common Lyapunov solutions for two matrices whose difference has rank one

Real stable matrices A and B with rank of A - B equal to one have a common Lyapunov solution if and only if their product AB has no real negative eigenvalue. This was proved by Shorten and Narendra [R.N. Shorten, K.S. Narendra, On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form, IEEE Trans. Automat. Control 48 (4) (2003) 618–621], whose proof is based on the fundamental results of Kalman on Luré’s problem. In this paper we give an alternative proof of this result and its generalization to the general regular inertia case, and to the case when the matrices A and B are complex.

Padé Approximations of and preservation of quadratic Lyapunov functions

AbstractWe investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Robust pole placement controller design in LMI region for uncertain and disturbed switched systems

This paper concerns the state feedback control for continuous-time, disturbed and uncertain linear switched systems with arbitrary switching rules. The main result of this work consists in getting a LMI (Linear Matrix Inequalities) condition guaranteeing a robust pole placement according to some desired specifications. Then, external disturbance attenuation with a fixed rate according to the H ∞ criterion is ensured. This is obtained thanks to the existence of a common quadratic Lyapunov function for all sub-systems. Finally, an academic example illustrates the efficiency of the developed approach.

Stability Analysis and $H_{\infty}$ Controller Design of Discrete-Time Fuzzy Large-Scale Systems Based on Piecewise Lyapunov Functions

This paper is concerned with stability analysis and H(infinity) decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of J interconnected discrete-time Takagi-Sugeno (T-S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The H(infinity) controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.

Stability and robust stabilization of discrete-time switched systems with time-delays: LMI approach

We present sufficient linear matrix inequality conditions for asymptotic stability and stabilizability of switched discrete-time linear systems subject to time-delays and norm bounded uncertainties. Namely, if these LMIs are solvable then, the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of switched time-delay systems satisfying these conditions possesses a quadratic common Lyapunov function. We also discuss the implication of this result on the stabilizability of this class of systems by switching controllers that use common Lyapunov functions. We show that even the analysis is carried out through a common Lyapunov function, the applied controller is mode dependent.

Stabilization of Networked Stochastic Time-Delay Fuzzy Systems With Data Dropout

This paper deals with the problem of stabilization for networked stochastic systems with transmitted data dropout. The plant in the networked control system (NCS) under consideration is a discrete stochastic time-delay nonlinear system represented by a Takagi-Sugeno fuzzy model. Exponential stability criteria of the NCS are developed by using a common quadratic Lyapunov function and a fuzzy Lyapunov function, respectively. A stabilization controller with convergence rate constraint can be designed by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. Three numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Approximation of the joint spectral radius using sum of squares

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.

A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control

In this paper we consider the asymptotic stability of a class of discrete-time switching linear systems, where each of the constituent subsystems is Schur stable. We first present an example to motivate our study, which illustrates that the bilinear transform does not preserve the stability of a class of switched linear systems. Consequently, continuous time stability results cannot be transformed to discrete time analogs using this transformation. We then present a subclass of discrete-time switching systems that arise frequently in practical applications. We prove that global attractivity for this subclass can be obtained without requiring the existence of a common quadratic Lyapunov function (CQLF). Using this result, we present a synthesis procedure to construct switching stabilizing controllers for an automotive control problem, which is related to the stabilization of a vehicle’s roll dynamics subject to switches in the center of gravitys (CG) height.

Extended Lie Algebraic Stability Analysis for Switched Systems with Continuous-Time and Discrete-Time Subsystems

Extended Lie Algebraic Stability Analysis for Switched Systems with Continuous-Time and Discrete-Time SubsystemsWe analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.

On convergence properties of piecewise affine systems

In this paper convergence properties of piecewise affine (PWA) systems are studied. In general, a system is called convergent if all its solutions converge to some bounded globally asymptotically stable steady-state solution. The notions of exponential, uniform and quadratic convergence are introduced and studied. It is shown that for non-linear systems with discontinuous right-hand sides, quadratic convergence, i.e., convergence with a quadratic Lyapunov function, implies exponential convergence. For PWA systems with continuous right-hand sides it is shown that quadratic convergence is equivalent to the existence of a common quadratic Lyapunov function for the linear parts of the system dynamics in every mode. For discontinuous bimodal PWA systems it is proved that quadratic convergence is equivalent to the requirements that the system has some special structure and that certain passivity-like condition is satisfied. For a general multimodal PWA system these conditions become sufficient for quadratic convergence. An example illustrating the application of the obtained results to a mechanical system with a one-sided restoring characteristic, which is equivalent to an electric circuit with a switching capacitor, is provided. The obtained results facilitate bifurcation analysis of PWA systems excited by periodic inputs, substantiate numerical methods for computing the corresponding periodic responses and help in controller design for PWA systems.

Constructing Common Quadratic Lyapunov Functions for a Class of Stable Matrices

Narendra and Balakrishnan proposed a way to construct a common quadratic Lyapunov function (CQLF) [1], whena set of stable matrices are commutative. The purpose of this paper is to generalize the method to non-commutative and non-solvable case. A modified constructing algorithm is proposed and certain conditions are provided to assure the resulting matrix being a CQLF. Next, the problem discussed is when a stable matrix can be added to a set of matrices with CQLF to construct a new CQLF for the enlarged set.

Lyapunov Stability of Complementarity and Extended Systems

A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear time-invariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finite-dimensional linear complementarity problem (LCP), whose constant vector is a linear function of the differential variable. Continuing the authors’ recent investigation of the LCS from the combined point of view of system theory and mathematical programming, this paper addresses the important system-theoretic properties of exponential and asymptotic stability for an LCS with a C$^1$ state trajectory. The novelty of our approach lies in our employment of a quadratic Lyapunov function that involves the auxiliary algebraic variable of the LCS; when expressed in the state variable alone, the Lyapunov function is piecewise quadratic, and thus nonsmooth. The nonsmoothness feature invalidates standard stability analysis that is based on smooth Lyapunov functions. In addition to providing sufficient conditions for exponential stability, we establish a generalization of the well-known LaSalle invariance theorem for the asymptotic stability of a smooth dynamical system to the LCS, which is intrinsically a nonsmooth system. Sufficient matrix-theoretic copositivity conditions are introduced to facilitate the verification of the stability properties. Properly specialized, the latter conditions are satisfied by a passive-like LCS and certain hybrid linear systems having common quadratic Lyapunov functions. We provide numerical examples to illustrate the stability results. We also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities, based on a new converse theorem for ODEs with B-differentiable right-hand sides. The latter theorem asserts that the existence of a “B-differentiable Lyapunov function” is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system.

Stability Criteria for Switched and Hybrid Systems

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.