Showing papers on "Lyapunov equation published in 2001"
TL;DR: The input-to-state stability property and small-gain theorems are introduced as the cornerstone of new stability criteria for discrete-time nonlinear systems.
1,179 citations
TL;DR: It is shown that extending the new discrete time stability condition proposed by de Oliveira et al. to the case of time varying uncertainty leads to a necessary and sufficient condition for the computation of such a Lyapunov function.
657 citations
TL;DR: In this article, the authors introduce a weaker notion of stability, which can be viewed as a dual to Lyapunov's theorem, which is used for stability analysis of ordinary differential equations and has a convexity property related to control synthesis.
350 citations
TL;DR: Based on the Lyapunov stability theorem as well as some facts about the positive definiteness and inequality of matrices, a new sufficient condition is presented for the existence of a unique equilibrium point and its global asymptotic stability for delayed CNNs as mentioned in this paper.
Abstract: Based on the Lyapunov stability theorem as well as some facts about the positive definiteness and inequality of matrices, a new sufficient condition is presented for the existence of a unique equilibrium point and its global asymptotic stability for delayed CNNs. This condition imposes constraints on the feedback matrices independent of the delay parameter. This condition is less restrictive than that given in earlier references.
330 citations
TL;DR: In this paper, the discretized Lyapunov functional method for the stability problem of time-delay systems is further refined using a combination of integral inequality and variable elimination technique.
Abstract: The discretized Lyapunov functional method for the stability problem of time-delay systems is further refined using a combination of integral inequality and variable elimination technique. As a result, the computational requirement is further reduced for the same discretization mesh. For systems without uncertainty, the convergence to the analytical solution is greatly accelerated. For uncertain systems, the new method is much less conservative. Numerical examples are presented to illustrate the effectiveness of the method.
245 citations
TL;DR: Modifications to the representation of the bounded real lemma (BRL) in the linear continuous time-invariant case are introduced, reducing the overdesign that occurs in the analysis and the design of systems with polytopic-type uncertainties.
Abstract: Simple modifications to the representation of the bounded real lemma (BRL) in the linear continuous time-invariant case are introduced. These modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic-type uncertainties. A different Lyapunov function accompanies each of the vertices of the uncertainty polytope, thus eliminating the need for quadratic stability or stabilizability. The advantages of these new representations are demonstrated by way of two examples.
233 citations
25 Jun 2001
TL;DR: The stability of Takagi-Sugeno fuzzy models via the so-called fuzzy Lyapunov function which is a multiple Lyap unov function is discussed, which gives the stability conditions for open-loop fuzzy systems.
Abstract: This paper discusses the stability of Takagi-Sugeno fuzzy models via the so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. Based on a fuzzy Lyapunov approach, we gives the stability conditions for open-loop fuzzy systems. All the conditions derived here are represented in terms of linear matrix inequalities (LMIs) and contain upper bounds of the time derivative of premise membership functions as LMI variables. Hence, the treatment of the upper bounds play an important and effective role in system analysis and design. In addition, relaxed stability conditions are also derived by considering the property of the time derivative of premise membership functions. Several analysis and design examples illustrate the utility of the fuzzy Lyapunov approach.
202 citations
02 Dec 2001
TL;DR: It is shown that the global stability of the closed loop fuzzy control systems can be established, and the control laws can be obtained by solving a set of linear matrix inequalities (LMI).
Abstract: This paper presents a controller synthesis method for fuzzy dynamic systems based on a piecewise smooth Lyapunov function. It is shown that the global stability of the closed loop fuzzy control systems can be established, and the control laws can be obtained by solving a set of linear matrix inequalities (LMI).
141 citations
TL;DR: In this article, the stability of a singular point for planar discontinuous differential equations with a line of discontinuities was studied, for the most generic cases, by computing some kind of Lyapunov constants.
122 citations
TL;DR: In this paper, the authors propose a fortified boundary control law and an adaptation law for Burgers' equation with unknown viscosity, where no a priori knowledge of a lower bound on visosity is needed, and prove that the closed-loop system, including the parameter estimator as a dynamic component, is globally H1 stable and well posed.
Abstract: In this paper, we propose a fortified boundary control law and an adaptation law for Burgers' equation with unknown viscosity, where no a priori knowledge of a lower bound on viscosity is needed. This control law is decentralized, i.e., implementable without the need for central computer and wiring. Using the Lyapunov method, we prove that the closed-loop system, including the parameter estimator as a dynamic component, is globally H1 stable and well posed. Furthermore, we show that the state of the system is regulated to zero by developing an alternative to Barbalat's Lemma which cannot be used in the present situation. Copyright © 2001 John Wiley & Sons, Ltd.
114 citations
TL;DR: In this paper, sufficient conditions ensuring that a nonlinear system with disturbances having a delay is globally asymptotically stable independent of delay are given, relying extensively on a characterization of the stability property in terms of Lyapunov function.
TL;DR: In this article, the stability problem for systems with distributed delay is considered using discretized Lyapunov functional, where coefficients associated with the distributed delay are assumed to be piecewise constant, and the discretization mesh may be non-uniform.
Abstract: The stability problem for systems with distributed delay is considered using discretized Lyapunov functional. The coefficients associated with the distributed delay are assumed to be piecewise constant, and the discretization mesh may be non-uniform. The resulting stability criteria are written in the form of linear matrix inequality. Numerical examples are also provided to illustrate the effectiveness of the method. The basic idea can be extended to a more general setting with more involved formulation.
TL;DR: This note proposes a new approach to robust state observer construction derived by including an extra term and adopting the Lyapunov stability theorem, and requires the solution of an algebraic Riccati equation.
Abstract: This note proposes a new approach to robust state observer construction. The scheme is derived by including an extra term and adopting the Lyapunov stability theorem, and requires the solution of an algebraic Riccati equation. The scheme is implemented in a gas-fired furnace system example. Satisfactory results have been obtained, though in general special attention need be paid to some numerical issues in implementing the scheme.
01 Nov 2001
TL;DR: In this paper, the asymptotic stability of a class of neutral systems with multiple discrete and distributed time delays is considered. And the Lyapunov stability theorem and a linear matrix inequality approach are applied to solve the stability problem for such systems.
Abstract: The asymptotic stability of a class of neutral systems with multiple discrete and distributed time delays is considered. The Lyapunov stability theorem and a linear matrix inequality (LMI) approach are applied to solve the stability problem for such systems. Discrete-delay-independent and discrete-delay-dependent criteria are proposed to guarantee stability for the systems considered. Some numerical examples are given to illustrate that the results obtained are not conservative.
TL;DR: In this paper, the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory is studied, where the authors propose to use a feedback controller such that the target elliptic orbit becomes a locally asymptotically stable periodic orbit in the closed-loop dynamics.
Abstract: We present a study of the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory specific to this problem. The construction of Lyapunov functions is based on the fact that a non-degenerate Keplerian orbit is uniquely described by its angular momentum and Laplace (- Runge-Lenz) vectors. We suggest a Lyapunov function, which gives a feedback controller such that the target elliptic orbit becomes a locally asymptotically stable periodic orbit in the closed-loop dynamics. We show how to perform a global transfer between two arbitrary elliptic orbits based on the local transfer result. Finally, a second Lyapunov function is presented that works only for circular target orbits.
01 Jan 2001
TL;DR: This work presents a MIMO analog to the well known Lyapunov-based MRAC SISO design by making use of a new control parametrization derived from a factorization of the high-frequency gain matrix K/sub p/=SDU.
Abstract: The design of model-reference adaptive control (MRAC) for MIMO linear systems has not yet been achieved, in spite of significant efforts, the completeness and simplicity of its SISO counterpart. One of the main obstacles has been the generalization of the SISO assumption that the sign of the high-frequency gain is known. Here we overcome this obstacle and present a MIMO analog to the well known Lyapunov-based MRAC SISO design. Our algorithm makes use of a new control parametrization derived from a factorization of the high-frequency gain matrix K/sub p/=SDU, where S is symmetric positive definite, D is diagonal, and U is unity upper triangular. Only the signs of the entries of D or, equivalently, the signs of the leading principal minors of K/sub p/, are assumed to be known.
TL;DR: In this article, the authors extend Lyapunov's second method for random dynamical systems and random sets, together with matching notions of attraction and stability, and prove that a random set is asymptotically stable if and only if it has a LyAPunov function.
TL;DR: It is shown that strong delay-independent stability of delay system is equivalent to the feasibility of certain linear matrix inequality (LMI) related to quadratic Lyapunov–Krasovskii functionals.
TL;DR: Some new sufficient conditions for the robust stability of a class of hybrid nonlinear systems with polytopic uncertainties are derived that do not require the Lyapunov functions to be non-increasing along each subsystem nor the whole sequence of the switching.
Abstract: We analyze the discrete behavior to identify all kinds of cycles of hybrid nonlinear systems and then study the continuous behavior along each kind of cycle. Based on these analysis, we construct some continuous functions to bound Lyapunov functions along all subsystems and identify a subsequence of time points where the Lyapunov functions are non-increasing. We use these results to derive some new sufficient conditions for the robust stability of a class of hybrid nonlinear systems with polytopic uncertainties. These conditions do not require the Lyapunov functions to be non-increasing along each subsystem nor the whole sequence of the switching. Furthermore, they do not require the knowledge of continuous trajectory.
02 Dec 2001
TL;DR: A dual design concept that minimizes/maximizes upper bounds of the time derivative of premise membership functions in a Takagi-Sugeno fuzzy system and a stability margin concept for switching speed among if-then rules are introduced.
Abstract: Presents a dual design problem via multiple Lyapunov functions. Based on a multiple Lyapunov approach, we introduce two important concepts. One is a dual design concept that minimizes/maximizes upper bounds of the time derivative of premise membership functions in a Takagi-Sugeno fuzzy system. The other is a stability margin concept for switching speed among if-then rules. It is related to avoiding conservatism of stability analysis and guaranteeing system response performance. A design example illustrates the utility of the dual design via the multiple Lyapunov function approach.
TL;DR: In this paper, equivalent conditions for exact observability of diagonal systems with a finite-dimensional output operator are given, in terms of the eigenvalues and the Lyapunov solutions of finite dimensional subsystems.
TL;DR: In this article, the authors studied the relationship between the Fast Lyapunov Indicator values and the order of periodic orbits and provided a simple model to explain the relationship that they have found between the values of the Fast Lipschitz Indicator, the ordering of the periodic orbits, and also the minimum number of iterations needed to obtain the Fast lphindicator values.
Abstract: The computation on a relatively short time of a quantity, related to the largest Lyapunov Characteristic Exponent, called Fast Lyapunov Indicator allows to discriminate between ordered and weak chaotic motion and also, under certain conditions, between resonant and non resonant regular orbits. The aim of this paper is to study numerically the relationship between the Fast Lyapunov Indicator values and the order of periodic orbits. Using the two-dimensional standard map as a model problem we have found that the Fast Lyapunov Indicator increases as the logarithm of the order of periodic orbits up to a given order. For higher order the Fast Lyapunov Indicator grows linearly with the order of the periodic orbits. We provide a simple model to explain the relationship that we have found between the values of the Fast Lyapunov Indicator, the order of the periodic orbits and also the minimum number of iterations needed to obtain the Fast Lyapunov Indicator values.
01 Apr 2001
TL;DR: This approach significantly reduces the computational load associated with determining closed loop stability as the input dimension increases, and includes a consideration of the input membership functions.
Abstract: In this paper we address the stability of a class of non-linear fuzzy systems that can be decomposed into a set of local models characterized as Takagi-Sugeno models This new approach includes a consideration of the input membership functions Via this approach, a reduction in the number of candidate Lyapunov functions and associated linear matrix inequalities (LMIs) is produced This approach significantly reduces the computational load associated with determining closed loop stability as the input dimension increases
TL;DR: An overview of the subject of regularity in nonlinear control theory is given, in contexts as various as necessary conditions for optimal control, the existence of Lyapunov functions, and the construction of stabilizing feedbacks.
TL;DR: The solution to the algebraic Riccati equation (ARE) with indefinite sign quadratic term related to the H/sub /spl infin// control problem for singularly perturbed systems is presented by means of a Kleinman type algorithm.
Abstract: We present the solution to the algebraic Riccati equation (ARE) with indefinite sign quadratic term related to the H/sub /spl infin// control problem for singularly perturbed systems by means of a Kleinman type algorithm. The resulting algorithm is very efficient from the numerical point of view because the ARE is solvable even if the quadratic term has an indefinite sign. Moreover, the resulting iterative algorithm is quadratically convergent. We also present an algorithm for solving the generalized algebraic Lyapunov equation on the basis of the fixed point algorithm.
TL;DR: In this paper, the authors used multiple Lyapunov functions to establish sufficient criteria for locating the limit sets of solutions of stochastic differential equations, which can be used to construct the Lyapinov functions much more easily in applications.
TL;DR: Differential constraints relating the finite-time Lyapunov exponents to the characteristic directions are derived and are realized with exponential accuracy in time, which has implications for the efficiency of chaotic mixing in the advection-diffusion equation.
Abstract: Constraints are found on the spatial variation of finite-time Lyapunov exponents of two- and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of separation, along characteristic directions, of neighboring trajectories. The solution of the equations is a coordinate transformation that takes initial conditions (the Lagrangian coordinates) to the state of the system at a later time (the Eulerian coordinates). This coordinate transformation naturally defines a metric tensor, from which the Lyapunov exponents and characteristic directions are obtained. By requiring that the Riemann curvature tensor vanish for the metric tensor (a basic result of differential geometry in a flat space), differential constraints relating the finite-time Lyapunov exponents to the characteristic directions are derived. These constraints are realized with exponential accuracy in time. A consequence of the relations is that the finite-time Lyapunov exponents are locally small in regions where the curvature of the stable manifold is large, which has implications for the efficiency of chaotic mixing in the advection–diffusion equation. The constraints also modify previous estimates of the asymptotic growth rates of quantities in the dynamo problem, such as the magnitude of the induced current.
TL;DR: In this paper, a team algorithm based on piecewise quadratic simultaneous Lyapunov functions for robust stability analysis and control design of uncertain time-varying linear systems is introduced.
Abstract: A team algorithm based on piecewise quadratic simultaneous Lyapunov functions for robust stability analysis and control design of uncertain time-varying linear systems is introduced. The objective is to use robust stability criteria that are less conservative than the usual quadratic stability criterion. The use of piecewise quadratic Lyapunov functions leads to a non-convex optimization problem, which is decomposed into a convex subproblem in a selected subset of decision variables, and a lower-dimensional non-convex subproblem in the remaining decision variables. A team algorithm that combines genetic algorithms (GA) for the non-convex subproblem and interior-point methods for the solution of linear matrix inequalities (LMI), which form the convex subproblem, is proposed. Numerical examples are given, showing the advantages of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: The moment Lyapunov exponents of a two-dimensional system under real-noise excitation, an Ornstein-Uhlenbeck process, are studied in this article.
04 Dec 2001
TL;DR: In this paper, a Lyapunov-Krasovskii functional is constructed for linear coupled systems of delay differential and functional equations, and sufficient conditions for delay-dependent stability are given in terms of linear matrix inequalities.
Abstract: The Lyapunov second method is developed for linear coupled systems of delay differential and functional equations. By means of conventional approaches, such equations may be reduced to neutral systems, and the known results for the latter may be exploited. In this paper, we introduce a new approach by constructing a Lyapunov-Krasovskii functional that corresponds directly to the descriptor form of the system. Moreover, by representing a neutral system in the descriptor form, we obtain new stability criteria for neutral systems which lead to results that are less conservative than the existing results. Sufficient conditions for delay-dependent stability are given in terms of linear matrix inequalities. Illustrative examples show the effectiveness of the method.