Showing papers on "Lyapunov equation published in 2002"
TL;DR: The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration and shows that the second condition is, in this case, less conservative.
Abstract: This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.
1,398 citations
TL;DR: The aim is to present a framework for understanding questions of incremental stability fully compatible with the well-known input-to-state stability approach.
Abstract: Deals with several notions of incremental stability. In other words, the focus is on stability of trajectories with respect to one another, rather than with respect to some attractor. The aim is to present a framework for understanding such questions fully compatible with the well-known input-to-state stability approach. Applications of the newly introduced stability notions are also discussed.
731 citations
TL;DR: In this paper, the authors present new synthesis procedures for discrete-time linear systems based on a recently developed stability condition which contains as particular cases both the celebrated Lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters.
Abstract: This paper presents new synthesis procedures for discrete-time linear systems. It is based on a recently developed stability condition which contains as particular cases both the celebrated Lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters. These new synthesis conditions have some nice properties: (a) they can be expressed in terms of LMI (linear matrix inequalities) and (b) the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic Lyapunov function used to test stability. This second feature is important for several reasons. First, structural constraints, as those appearing in the decentralized and static output-feedback control design, can be addressed less conservatively. Second, parameter dependent Lyapunov function can be considered with a very positive impact on the design of robust H 2 and H X control problems. Third, the design of controller with mixed ...
675 citations
10 Dec 2002
TL;DR: In this article, a relaxation of Lyapunov's direct method has been proposed to include systems with equality, inequality, and integral constraints, which allows robustness analysis to be performed.
Abstract: A relaxation of Lyapunov's direct method has been proposed elsewhere that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In the paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain non-polynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain non-polynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.
498 citations
Book•
01 Jan 2002TL;DR: In this paper, Lyapunov stability theory of differential equations and nonuniform hyperbolic theory of local manifold theory have been studied in the context of smooth hyperbolc measures.
Abstract: Introduction Lyapunov stability theory of differential equations Elements of nonuniform hyperbolic theory Examples of nonuniformly hyperbolic systems Local manifold theory Ergodic properties of smooth hyperbolc measures Bibliography Index
346 citations
TL;DR: It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.
Abstract: This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX+XAT=-BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A.
This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.
302 citations
TL;DR: It is shown that a smooth Lyapunov function exists for a family of time-varying discrete systems if these systems are robustly globally asymptotically stable.
290 citations
TL;DR: In this article, the authors studied the regularity properties of the Lyapunov exponent of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency.
Abstract: We study regularity properties of the Lyapunov exponent L of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove joint continuity of L, in frequency and energy, at every irrational frequency.
216 citations
TL;DR: It is shown how to use a new parameter dependent Lyapunov matrix procedure to determine high performance H2 robust filters by solving a linear problem constrained by linear matrix inequalities (LMIs).
Abstract: Robust filtering of linear time-invariant discrete-time uncertain systems is investigated through a new parameter dependent Lyapunov matrix procedure. Its main interest relies on the fact that the Lyapunov matrix used in stability checking does not appear in any multiplicative term with the uncertain matrices of the dynamic model. We show how to use such an approach to determine high performance H2 robust filters by solving a linear problem constrained by linear matrix inequalities (LMIs). The results encompass the previous works in the quadratic Lyapunov setting. Numerical examples illustrate the theoretical results.
214 citations
TL;DR: It is shown that the Lyapunov equation of the work by Lewis can lead to incorrect conclusion about stability, and an attempt to correct that theorem with a generalized Lyap unov equation similar to the original one leads naturally to the generalized equation of Takaba et al.
Abstract: In this paper, we revisit the Lyapunov theory for singular systems. There are basically two well-known generalized Lyapunov equations used to characterize stability for singular systems. We start with the Lyapunov theorem of the work by Lewis. We show that the Lyapunov equation of that theorem can lead to incorrect conclusion about stability. Some cases where that equation can be used are clarified. We also show that an attempt to correct that theorem with a generalized Lyapunov equation similar to the original one leads naturally to the generalized equation of Takaba et al.
207 citations
TL;DR: A new sufficient condition for the robust stability of continuous-time uncertain linear systems with convex bounded uncertainties is proposed in this note based on linear matrix inequalities formulated at the vertices of the uncertainty polytope, which provide a parameter dependent Lyapunov function that assures the stability of any matrix inside the uncertainty domain.
Abstract: A new sufficient condition for the robust stability of continuous-time uncertain linear systems with convex bounded uncertainties is proposed in this note. The results are based on linear matrix inequalities (LMIs) formulated at the vertices of the uncertainty polytope, which provide a parameter dependent Lyapunov function that assures the stability of any matrix inside the uncertainty domain. With the aid of numerical procedures based on unidimensional search and the LMIs feasibility tests, a simple and constructive way to compute robust stability domains can be established.
TL;DR: This paper investigates the decay rate of the Hankel singular values of linear dynamical systems by relating the solution to a numerically low-rank Cauchy matrix determined by the poles of the system.
TL;DR: In this article, necessary and sufficient conditions are derived for the existence of a common quadra-tic Lyapunov function for a finite number of stable second order linear time-invariant systems.
Abstract: In this paper, necessary and sufficient conditions are derived for the existence of a common quadra-tic Lyapunov function for a finite number of stable second order linear time-invariant systems. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: This paper discusses the application of Lyapunov theory in chaotic systems to the dynamics of tracer gradients in two-dimensional flows and the spatial distribution of finite time LyAPunov exponents is discussed in relation to stirring properties.
Abstract: This paper discusses the application of Lyapunov theory in chaotic systems to the dynamics of tracer gradients in two-dimensional flows. The Lyapunov theory indicates that more attention should be given to the Lyapunov vector orientation. Moreover, the properties of Lyapunov vectors and exponents are explained in light of recent results on tracer gradients dynamics. Differences between the different Lyapunov vectors can be interpreted in terms of competition between the effects of effective rotation and strain. Also, the differences between backward and forward vectors give information on the local reversibility of the tracer gradient dynamics. A numerical simulation of two-dimensional turbulence serves to highlight these points and the spatial distribution of finite time Lyapunov exponents is also discussed in relation to stirring properties.
TL;DR: In this paper, the search for the Lyapunov functions is formulated as a linear matrix inequality (LMI) problem for hybrid systems with affine as well as non-linear vector fields.
Abstract: This paper addresses issues concerning exponential stability and robustness of hybrid systems. Stability conditions using Lyapunov techniques are given. The search for the Lyapunov functions is formulated as a linear matrix inequality (LMI) problem for hybrid systems with affine as well as non-linear vector fields. It is shown how the Lyapunov approach most advantageously also can be used to guarantee stability despite the presence of model uncertainties. Several examples are given to illustrate the theory.
TL;DR: It is shown that the strong delay-independent stability is equivalent to the feasibility of certain linear matrix inequality (LMI), that is to the existence of a quadratic Lyapunov-Krasovskii functional, independent of the (nonnegative) value of the delay.
Abstract: In this note, the delay-independent stability of delay systems is studied. It is shown that the strong delay-independent stability is equivalent to the feasibility of certain linear matrix inequality (LMI), that is to the existence of a quadratic Lyapunov-Krasovskii functional, independent of the (nonnegative) value of the delay. This constitutes the analogue of some well-known properties of finite-dimensional systems. This result is then applied to study delay-independent stability of systems with polytopic uncertainties.
TL;DR: In this article, it was shown that the Poincare return time of a typical cylinder is at least its length, and that the Lyapunov exponent and dimension can be expressed via return times.
Abstract: We show that the Poincare return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.
TL;DR: In this article, a study of downhole vibrations of drill-strings bottom-hole assemblies is undertaken, where a method of statistical linearization is used, and expressions for computing the equivalent linear system of the bottomhole assembly are presented.
Abstract: ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC2000-251AbstractA study of downhole vibrations of drill-strings bottom-hole assemblies is undertaken. The lateral behaviorof the system is of interest. The nonlinear nature of the problem is addressed by considering a lateral clear-ance between the drill-string and the borehole that induces a stiffening of the system when exceeded. Thestochastic input force is defined by its power spectral density and it is applied laterally to the bit. The methodof statistical linearization is used, nd expressions for computing the equivalent linear system of the bottom-hole assembly are presented. The adopted procedure involves a prefiltering of the bit excitation to derive adynamic system under white-noise and colored white-noise excitations. Then, the Lyapunov equation for thecovariance of the linearized system is solved. Further, a Monte-Carlo simulation is conducted by means ofan auto-regressive moving-average digital filter, and the equations of motion are integrated by the Newmarkmethod. Numerical results pertaining to data obtained by measurement-while-drilling tools are presented.The study facilitates the assessment of the appropriateness of the method of statistical linearization for “realworld” problems encountered even in conservative industrial applications such as drilling.
TL;DR: A new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class /spl Kscr//spl Lscr/ function for the decay estimate and for describing the influence of the perturbation is presented.
Abstract: We present a new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class /spl Kscr//spl Lscr/ function for the decay estimate and for describing the influence of the perturbation. We show the relation to the original ISS formulation and describe characterizations by means of suitable Lyapunov functions. As applications, we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems.
TL;DR: In this paper, the authors present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents.
Abstract: We present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. The main tool is local ergodicity theory for non-uniformly hyperbolic systems.
05 Aug 2002
TL;DR: In this paper, the authors provide a construction method of a smooth homogeneous Lyapunov function associated with a discontinuous homogeneous system, which is locally and asymptotically stable.
Abstract: This paper provides a construction method of a smooth homogeneous Lyapunov function associated with a discontinuous homogeneous system, which is locally and asymptotically stable First, we analyze two similar converse Lyapunov theorems for differential inclusions and unify them into a simple theorem Next, we propose a new definition of homogeneous differential inclusion Then, we construct a smooth, homogeneous Lyapunov function associated with the homogeneous differential inclusion Finally, we show that the order of homogeneity of a homogeneous system indicates the speed of convergence
10 Dec 2002
TL;DR: In this paper, the stability of discrete-time linear systems subject to actuator saturation is analyzed using a saturation-dependent Lyapunov function, which leads to less conservative estimate of the domain of attraction, which is based on the solution of an LMI optimization problem.
Abstract: In this paper, the stability of discrete-time linear systems subject to actuator saturation is analyzed using a saturation-dependent Lyapunov function. This saturation-dependent Lyapunov function captures the real-time information on the severity of actuator saturation and leads to less conservative estimate of the domain of attraction, which is based on the solution of an LMI optimization problem. Numerical examples are presented to show the effectiveness of the proposed method.
TL;DR: In this article, the authors consider the problem of robust stability for linear systems with a constant time-delay in the state and subject to real convex polytopic uncertainty, and propose a generalized eigenvalue problem to determine the maximum of time delay within which the system remains stable.
Abstract: This paper considers the problem of robust stability for linear systems with a constant time-delay in the state and subject to real convex polytopic uncertainty. Both delay-independent and delay-dependent stability conditions are characterized by linear matrix inequalities which allow the use of parameter-dependent Lyapunov functionals to analyse the system stability against parameter perturbations, and therefore improve the conservativeness resulting from the single Lyapunov functional. In order to determine the maximum of time-delay within which the system remains stable, the problem can be cast into a generalized eigenvalue problem and solved by standard LMI solvers. Two examples are included to illustrate the proposed method.
TL;DR: In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions, and the authors identify classes of switching sequences that result in stable trajectories.
Abstract: In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. In particular, we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov functions, we compose 'global' Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space. The approach is applied to both discrete-time and continuous-time switched linear systems.
TL;DR: In this paper, generalized Lyapunov equations with special right-hand sides have been applied in stability theory and control problems for descriptor systems, including matrix pencils, and generalized stability theorems.
TL;DR: In this article, a linear program for an ordinary differential equation is presented, of which a feasible solution defines a continuous piecewise affine linear Lyapunov function for the differential equation.
Abstract: An algorithm that derives a linear program for an ordinary differential equation is presented, of which a feasible solution defines a continuous piecewise affine linear Lyapunov function for the differential equation. The linear program can be generated for an arbitrary region containing an equilibrium of the differential equation. The domain of the Lyapunov function is the region used in the generation of the linear program. The Lyapunov function secures the asymptotic stability of the equilibrium and gives a lower bound on its region of attraction.
TL;DR: In this article, an elementary formula about the average expansion of certain products of 2 by 2 matrices is given. But it is not shown that this formula holds for all products of spectral radius.
Abstract: We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. Indeed, we show that equality holds in Herman’s result. Finally, we give a result about the growth of the spectral radius of products.
TL;DR: Another necessary and sufficient stability criterion is developed via this generalized Lyapunov equation, which reduces the stability test into that of an approximate FDLCP model whose transition matrix can be determined explicitly.
Abstract: Asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems is studied by harmonic analysis. It is first shown that stability can be examined with what we call the harmonic Lyapunov equation. Another necessary and sufficient stability criterion is developed via this generalized Lyapunov equation, which reduces the stability test into that of an approximate FDLCP model whose transition matrix can be determined explicitly. By extending the Gerschgorin theorem to linear operators on the linear space l/sub 2/, yet another disc-group criterion is derived, which is only sufficient. Stability of the lossy Mathieu equation is analyzed as a numerical example to illustrate the results.
TL;DR: The general method of Lyapunov functionals construction for stability investigation of stochastic hereditary systems is considered and some features of this method for difference systems which allow one to use the method more effectively are discussed.