Showing papers on "Lyapunov equation published in 2006"

Stability of a class of linear switching systems with time delay

Abstract: We consider a switching system composed of a finite number of linear delay differential equations (DDEs). It has been shown that the stability of a switching system composed of a finite number of linear ordinary differential equations (ODEs) may be achieved by using a common Lyapunov function method switching rule. We modify this switching rule for ODE systems to a common Lyapunov functional method switching rule for DDE systems and show that it stabilizes our model. Our result uses a Riccati-type Lyapunov functional under a condition on the time delay.

Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions

Abstract: In this paper, we develop a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration. The only assumptions we make about the system are well-posedness of the algebraic loop and local stability. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of bounded energy disturbances and the nonlinear L2 gain. The regional analysis is established through an effective treatment of the algebraic loop and the saturation/deadzone function. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Adjustable parameters are incorporated into the differential inclusions to reflect the regional property. The main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain Lyapunov function where the PDI or the NDI is valid. With quadratic Lyapunov functions, conditions for stability and performances are derived as linear matrix inequalities (LMIs). To obtain less conservative conditions, we use a pair of conjugate non-quadratic Lyapunov functions, the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the corresponding LMI conditions as special cases, hence the BMI results are guaranteed to be as good as the LMI results. In most examples, the BMI results are significantly better than the LMI results

On the stability and control of nonlinear dynamical systems via vector Lyapunov functions

Abstract: Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we extend the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. Furthermore, we present a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii-LaSalle invariant set theorem. In addition, we introduce the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, and show that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. Moreover, using control vector Lyapunov functions, we construct a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. Furthermore, we establish connections between the recently developed notion of vector dissipativity and optimality of the proposed decentralized feedback control law. Finally, the proposed control framework is used to construct decentralized controllers for large-scale nonlinear systems with robustness guarantees against full modeling uncertainty.

Gain‐scheduled ℋ︁2 controller synthesis for linear parameter varying systems via parameter‐dependent Lyapunov functions

Abstract: This paper deals with the problem of gain-scheduled ℋ2 control for linear parameter-varying systems. The system state–space model matrices are affinely parameterized and the admissible values of the parameters and their rate of variation are supposed to belong to a given convex bounded polyhedral domain. Based on a parameter-dependent Lyapunov function, a linear matrix inequality methodology is proposed for designing a gain-scheduled state feedback ℋ2 controller, where the feedback gain is a matrix fraction of polynomial matrices with quadratic dependence on the scheduling parameters. Copyright © 2005 John Wiley & Sons, Ltd.

Fuzzy lyapunov method for stability conditions of nonlinear systems

Abstract: This paper proposes a fuzzy Lyapunov method for stability analysis of nonlinear systems represented by Tagagi-Sugeno (T-S) fuzzy model. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Based on fuzzy Lyapunov functions, some stability conditions are derived to ensure nonlinear systems are asymptotic stable. By using parallel distributed compensation (PDC) scheme, we design a nonlinear fuzzy controller for the nonlinear system. This control problem will be reformulated into linear matrix inequalities (LMI) problem.

Analysis and Synthesis of Robust Control Systems Using Linear Parameter Dependent Lyapunov Functions

Abstract: This note provides sufficient robust stability conditions for continuous time polytopic systems. They are obtained from the Frobenius-Perron Theorem applied to the time derivative of a linear parameter dependent Lyapunov function and are expressed in terms of linear matrix inequalities (LMI). They contain as special cases, various sufficient stability conditions available in the literature to date. As a natural generalization, the determination of a guaranteed H2 cost is addressed. A new gain parametrization is introduced in order to make possible the state feedback robust control synthesis using parameter dependent Lyapunov functions through linear matrix inequalities. Numerical examples are included for illustration

Discrete Chaos-I: Theory

Abstract: We propose a definition of the discrete Lyapunov exponent for an arbitrary permutation of a finite lattice. For discrete-time dynamical systems, it measures the local (between neighboring points) average spreading of the system. We justify our definition by proving that, for large classes of chaotic maps, the corresponding discrete Lyapunov exponent approaches the largest Lyapunov exponent of a chaotic map when Mrarrinfin, where M is the cardinality of the discrete phase space. In analogy with continuous systems, we say the system has discrete chaos if its discrete Lyapunov exponent tends to a positive number, when Mrarrinfin. We present several examples to illustrate the concepts being introduced

Conjugate convex Lyapunov functions for dual linear differential inclusions

Abstract: Tools from convex analysis are used to show how stability properties and Lyapunov inequalities translate when passing from a linear differential inclusion (LDI) to its dual. In particular, it is proved that a convex, positive definite function is a Lyapunov function for an LDI if and only if its convex conjugate is a Lyapunov function for the LDIs dual. Examples show how such duality effectively doubles the number of tools available for assessing stability of LDIs.

Lyapunov Stability of Complementarity and Extended Systems

Abstract: 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.

A Parametric Lyapunov Equation Approach to the Design of Low Gain Feedback

Abstract: Low gain feedback has found several applications in constrained control systems, robust control and nonlinear control Low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero Such feedback gains can be constructed either by an eigenstructure assignment algorithm or through the solution of a parametric algebraic Riccati equation (ARE) The eigenstructure assignment approach leads to feedback gains in the form of a matrix polynomial in the parameter, while the ARE approach requires the solution of an ARE for each value of the parameter This paper proposes an alternative approach to low gain feedback design based on the solution of a parametric Lyapunov equation Such an approach possesses the advantages of both the eigenstructure assignment approach and the ARE based approach It also avoids the possible numerical stiffness in solving a parametric ARE and the structural decomposition of the open loop system that is required by the eigenstructure assignment approach

Shorter horizons for model predictive control

Abstract: We provide a less conservative (compared with our previous results) lower bound on the horizon length for a stabilizing model predictive control algorithm where the terminal cost is not assumed to be a local control Lyapunov function. Our main additional assumption is that the value function is bounded by a linear function of a measure of the state uniformly in the horizon length.

Stability of switched systems with time-varying delays: delay-dependent common Lyapunov functional approach

Abstract: By using a descriptor system approach and the linear matrix inequality (LMI) technique, we investigate the problem of delay-dependent common Lyapunov functions (DCLFs) for switched linear systems with mode-dependent delays. The main contribution of this paper is that we establish the relationship between the delay-dependent common Lyapunov functions for switched delay systems and the well-known common Lyapunov functions (CLFs) for the corresponding systems without delays.

Factorized Solution of Lyapunov Equations Based on Hierarchical Matrix Arithmetic

Abstract: We investigate the numerical solution of large-scale Lyapunov equations with the sign function method. Replacing the usual matrix inversion, addition, and multiplication by formatted arithmetic for hierarchical matrices, we obtain an implementation that has linear-polylogarithmic complexity and memory requirements. The method is well suited for Lyapunov operators arising from FEM and BEM approximations to elliptic differential operators. With the sign function method it is possible to obtain a low-rank approximation to a full-rank factor of the solution directly. The task of computing such a factored solution arises, e.g., in model reduction based on balanced truncation. The basis of our method is a partitioned Newton iteration for computing the sign function of a suitable matrix, where one part of the iteration uses formatted arithmetic while the other part directly yields approximations to the full-rank factor of the solution. We discuss some variations of our method and its application to generalized Lyapunov equations. Numerical experiments show that the method can be applied to problems of order up to ** (105) on workstations.

A numerical analysis of the Nash strategy for weakly coupled large-scale systems

Abstract: This note discusses the feedback Nash equilibrium of linear quadratic N-player Nash games for infinite-horizon large-scale interconnected systems. The asymptotic structure along with the uniqueness and positive semidefiniteness of the solutions of the cross-coupled algebraic Riccati equations (CAREs) is newly established via the Newton-Kantorovich theorem. The main contribution of this study is the proposal of a new algorithm for solving the CAREs. In order to improve the convergence rate of the algorithm, Newton's method is combined with a new decoupling algorithm; it is shown that the proposed algorithm attains quadratic convergence. Moreover, it is shown for the first time that solutions to the CAREs can be obtained by solving the independent algebraic Lyapunov equation (ALE) by using the reduced-order calculation

LMI-based criteria for stability of high-order neural networks with time-varying delay☆

Abstract: This paper investigates the stability of a class of high-order neural networks with time-varying delay, which can be considered as an expansion of Hopfield neural networks and is seldom considered in the literature. Based on the Lyapunov stability theory and linear matrix inequality (LMI) technique, sufficient conditions guaranteeing the global exponential stability of the equilibrium point are presented. Two examples are given to show the effectiveness of the proposed conditions. The obtained results are also shown to be different from and more general than existing ones.

A computational approach to conley's decomposition theorem

Abstract: Background. The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. Method of approach. We describe explicit algorithms for computing dynamical structures for the combinatorial multivalued maps. Results. We provide computational complexity bounds and numerical examples. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. Conclusions. The computed discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map.

Numerical Linear Algebra for Model Reduction in Control and Simulation

Abstract: e approximation, balanced truncation, Lyapunov equation, Krylov subspace method. MSC (2000) 65F30,93B11,41A20,65F50 Model reduction is an ubiquitous tool in analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In the pas t decades many approaches have been developed for reducing the order of a given model. Often these methods have been derived in parallel in different disciplines with particular applic ations in mind. We will discuss some of the most prominent methods used for linear systems, compare their properties and highlight similarities. In particular, we will emphasize the role of recent developments in numerical linear algebra in the different approaches. Effic iently using these new techniques, the range of applicability of some of the methods has considerably widened. Copyright line will be provided by the publisher

Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies

Abstract: We study the hyperbolicity of a class of horseshoes exhibiting an internal tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly not uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.

Constant gain observer for continuous-discrete time uniformly observable systems

Abstract: For many physical processes, and in particular, chemical engineering processes, output measurements are discrete in time. Continuous-discrete observers combine a continuous time dynamical model with discrete time output measurements to estimate the unknown state or/and parameters. In this paper, we synthesized a new high gain continuous-discrete time observer based on the observability canonical form. Contrarily to Extended Kalman techniques, the gain of this observer is a constant one and its calculation is based on a discrete stationary Lyapunov equation. Simulation results related to an academic biotechnological process are given in order to illustrate the performances of the proposed estimators.

Positive Forms and Stability of Linear Time-Delay Systems

Abstract: We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that stability implies that there exists a quadratic Lyapunov function on the state space, although this is in general infinite dimensional. We give an explicit parametrization of a finite-dimensional subset of the cone of Lyapunov functions. Positivity of this class of functions is enforced using sum-of-squares polynomial matrices. This allows the computation to be formulated as a semidefinite program.

Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions

Abstract: Counterexamples are given which show that a linear switched system (with controlled switching) that can be stabilized by means of a suitable switching law does not necessarily admit a convex Lyapunov function. Both continuous and discrete-time cases are considered. This fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched systems. In particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a Lyapunov function.

Control of cascade systems with time delay -- the integral cross-term approach

Abstract: Presence of a time delay introduces a limitation to achievable performance and robustness of industrial control systems. This paper presents a Lyapunov based stabilization and control method for nonlinear cascade systems with time delay. The approach uses a construction of a composite Lyapunov functional for a stable cascade obtained in an intermediate step of the procedure. The functional consists of Lyapunov functions or functionals for the subsystems and a cross-term defined by an infinite integral. A strong robustness guarantee for the closed loop system is available and the stabilization can be achieved even through distributed-delay control channels.