Stability of equilibrium points definitions in this section we consider nth order nonlinear time varying continuous time ct systems of the form x ft, x, xto xo l. Global stability of discretetime competitive population. The following statements regarding system 14 are equivalent. Control systemsintroduction to digital controls wikibooks. The interest in this class of systems has been motivated traditionally by sampleddata systems in which a process is sampled periodically and then controlled via a computer.
It is known from the literature that letting the lyapunov function depend on the timevarying. Lyapunov stability analysis of the implicit discretetime twisting control algorithm article pdf available in ieee transactions on automatic control pp99. In this paper, we consider the generalized lyapunov stability analysis for a linear discretetime timeinvariant. These extended versions can be necessary in evaluating. The interest in this class of systems has been motivated traditionally by sampleddata systems in which a process is sampled periodically and then controlled via a. N is a matrix of order n, is asymptotically discrete stable if the spectrum of a lies inside the open unit disk. Kellett and teel, on the robustness of klstability for difference inclusions. On stability analysis of linear discretetime switched. Stability of multistable nonlinear discretetime systems. This page was last edited on 14 november 2019, at 00.
The purpose of this paper is to prove the generalized lyapunov stability theorem for discretetime systems. Discontinuous discrete time systems kellett and andrew r. We introduce a linear transfer operatorbased lyapunov measure as a new tool for stability veri. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local. Ru er z yschool of electrical engineering and computer science.
Stability analysis of discretetime lure systems archive ouverte hal. Converse lyapunov theorems for discretetime systems. This paper presents a lyapunovkrasovskii methodology for asymptotic stability of discrete time delay systems. Lyapunov stability theorem digital systems a digital system with the system matrix a is asymptotically stable if and only if there exists a unique matrix m that satisfies the lyapunov equation for every positive definite matrix n. Pdf lyapunov stability analysis of the implicit discrete.
Consider the discretetime dt linear timeinvariant lti system. The idea of lyapunov stability can be extended to infinitedimensional manifolds, where it is known as structural stability, which concerns the behavior of different but nearby solutions to differential equations. Lyapunov function and relate it to these various stability notions. Summary discretetime model reference adaptive control mrac is considered with both least squares and projection algorithm parameter identi. Exploiting this fact, we derive much more general versions of the lyapunov foster criteria, which allow for time varying test functions. In 41, parrilo and jadbabaie prove that stable switched linear systems in discrete. A lyapunovkrasovskii methodology for asymptotic stability of discrete time delay systems sretenb. A note on the lyapunov stability of periodic discrete time systems. Lyapunov stability is named after aleksandr mikhailovich lyapunov, a russian mathematician who defended the thesis the general problem of stability of motion at kharkov university in 1892. Stability analysis and control of discretetime systems. It is known from the literature that letting the lyapunov function depend on the timevarying dynamic improves the. Lyapunovs second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a.
Lyapunov and nonlyapunov stability of linear discrete. Computation of lyapunov functions for nonlinear discrete. Request pdf lyapunov and nonlyapunov stability of linear discrete time delay systems this paper extends some of the basic results in the area of lyapunov asymptotic and finite time and. Kellettz abstractin this paper, we present a new approach for computing lyapunov functions for nonlinear discretetime systems with an asymptotically stable equilibrium at the origin. The most classical and viable way of studying the uniform asymptotic stability of such a system is to check for the existence of a quadratic lyapunov function. We can use this version of the lyapunov equation to define a condition for stability in discrete time systems. It is known from the literature that letting the lyapunov function depend on the time varying dynamic improves the. Nonmonotonic lyapunov functions for stability of discrete time nonlinear and switched systems amir ali ahmadi and pablo a. Tu berlin discretetime control systems 12 nyquist criterion wellknown stability test for continuoustime systems. The discussion is illustrated with numerical examples. Several numerical tests illustrate the behavior of the proposed method.
Stochastic stability analysis of discrete time system. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. Discrete time system stability concepts 465 one of the aims of the present note is to strengthen and unify some of the theorems appearing in 7 and 3, in particular showing that a stochastic version of lyapunov stability holds under less demanding hypotheses than. Lyapunov and nonlyapunov stability of linear discrete time delay systems 1149 analysis of nonlinear singular and implicit dynamic systems in terms of the generic qualitative and quantitative concepts, which contain technical and practical stability types as special cases, have been introduced and studied in bajic 1988, 1992. Equivalences of stability properties for discretetime nonlinear systems and extensions author. Moreover, using the method of the lyapunov functionals, some efficient criteria for stochastic stability are obtained. In part ii of this series of papers 25, we developed various such forms of stability for markov processes. Lyapunov functions for nonlinear discretetime systems. Pdf in this paper, we consider the generalized lyapunov stability analysis for a discretetime system described by a high order differencealgebraic.
Computation of lyapunov functions for nonlinear discrete time. Equivalences of stability properties for discrete time nonlinear systems and extensions author. In the discretetime system, a generalized lyapunov stability analysis based on a qdf has never been studied so far. This and related equations are named after the russian mathematician aleksandr lyapunov. Lyapunov function and relate it to these v arious stabilit y notions. To determine the stability of the closedloop system when the openloop system is given. Parrilo abstract we relax the monotonicity requirement of lyapunov s theorem to enlarge the class of functions that can provide certicates of stability. Lyapunov and nonlyapunov stability of linear discrete time.
The research presented in this thesis considers the stability analysis and control of discretetime systems with delay. D r satisfying 3 and 4 is called a lya punov function. Kellett and teel, on the robustness of kl stability for difference inclusions. A be a discretetime stable lti system, and p pt 0 be a solution of the discretetime lyapunov equation atpa. The model 1 is assumed to have a unique interior xed point. Computation of lyapunov functions for discretetime. The paper deals with the stability properties of linear discretetime switched systems with polytopic sets of dynamics. A lyapunov krasovskii methodology for asymptotic stability of discrete time delay systems. Exploiting this fact, we derive much more general versions of the lyapunovfoster criteria, which allow for timevarying test functions. Lyapunov stability analysis of the implicit discrete time twisting control algorithm article pdf available in ieee transactions on automatic control pp99.
Asymptotic stability of positive 2d linear systems, proceedings of the th scientific conference on computer applications in electrical engineering, poznan, poland. M are, respectively, replaced by their discreteanalogs x a t x a m and x a x a t m. The lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. Solve discretetime lyapunov equations matlab dlyap. The following stability condition proposed by sugeno and tanaka is well known based on a common lyapunov function vxk xktpxk. Computation of lyapunov functions for discretetime systems. A class of lyapunov functions is proposed for discretetime linear systems interconnected with a cone bounded nonlinearity. Equivalences of stability properties for discretetime. Global stability of discretetime competitive population models. Lmis in controldiscretetime systemsdiscretetime lyapunov stability. A linear complementarity system lcs is a piecewise linear dynamical system consisting of a linear timeinvariant ordinary di. Our method constitutes the extension to maps of the split lyapunov method developed for di. Stability analysis of ts fuzzy systems has been pursued mainly based on lyapunov stability theory but with. The following lemma relates the discretetime lyapunov equation 1 to the continuoustime lyapunov equation 6, and introduces a notation that we shall use to state our results.
Lyapunovstable discretetime model reference adaptive control. First of all, several definitions on stability are introduced, such as stability, asymptotical stability, and p th moment exponential stability. Let v be a given positive vector and vx q n i1 x v i i. On common quadratic lyapunov functions for stable discrete. Stability and l gain analysis for discretetime lti. Numerical methods to compute lyapunov functions for nonlinear discrete time. Two models of positive 2d discrete time linear lyapunov systems are introduced. M are, respectively, replaced by their discrete analogs x a t x a m and x a x a t m. A lyapunovkrasovskii methodology for asymptotic stability of. November 15, 2009 1 1 lyapunov theory of stability introduction. Nonmonotonic lyapunov functions for stability of discrete.
The origin of discrete time ts fuzzy system described by 2 with. Stochastic stability analysis of discrete time system using. In this paper, an alternative approach to the construction of lyapunov functions for discretetime systems is proposed. The general problem of the stability of motion, math. A lyapunovkrasovskii methodology for asymptotic stability of discrete time delay systems. Stability condition of an lti discretetime system example consider a causal lti discretetime system with an impulse response for this system therefore s discrete time lyapunov stability of large matrices is proposed. Pdf a generalized lyapunov stability theorem for discretetime. Stability analysis and control of discretetime systems with. The research presented in this thesis considers the stability analysis and control of discrete time systems with delay. Lmis in controldiscretetime systemsdiscretetime lyapunov. We give ecologicallymotivated sucient conditions for global stability of an interior. Discrete lyapunov equation an overview sciencedirect topics. Lyapunovstable discretetime model reference adaptive control s.
Similar to the proof for strong asymptotic stability s. Tu berlin discrete time control systems 12 nyquist criterion wellknown stability test for continuous time systems. Global stability of discretetime competitive population models 3 the framework that we use is as follows. These are analogous to and based on stability concepts in discrete. This paper studies the stability for nonlinear stochastic discretetime systems. Positive 2d discretetime linear lyapunov systems in. A lyapunovkrasovskii methodology for asymptotic stability. We can use this version of the lyapunov equation to define a condition for stability in discretetime systems. Positive discretetime linear lyapunov systems, proceedings of the 15th mediterranean conference of control and automation, med, athens, greece.
Converse lyapunov theorem is presented in paper by lin et al. Inputtostate stability iss applies lyapunov notions to systems with inputs. Stochastic stability analysis of discrete time system using lyapunov measure umesh vaidya, senior member, ieee, abstractin this paper, we study the stability problem of a stochastic, nonlinear, discretetime system. Stability condition of an lti discretetime system example consider a causal lti discrete time system with an impulse response for this system therefore s stability analysis and control of discretetime systems with delay.
Lyapunov and non lyapunov stability of linear discrete time delay systems 1149 analysis of nonlinear singular and implicit dynamic systems in terms of the generic qualitative and quantitative concepts, which contain technical and practical stability types as special cases, have been introduced and studied in bajic 1988, 1992. Pdf a note on the lyapunov stability of periodic discrete. A note on the lyapunov stability of periodic discretetime systems. Computation of lyapunov functions for discretetime systems using the yoshizawa construction huijuan li, sigur. The method combines lyapunov theory with krylov subspace techniques. Can be reformulated to handle discretetime systems.
Discrete lyapunov equation an overview sciencedirect. Stability of nonlinear stochastic discretetime systems. Lyapunovs second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a dynamical system or systems of differential equations. Motivated by the importance and application of discrete dynamical systems, this paper presents a new lyapunov characterization which is an extension of conventional lyapunov characterization for multistable discrete time nonlinear systems. Can be reformulated to handle discrete time systems. The paper deals with the stability properties of linear discrete time switched systems with polytopic sets of dynamics. Here it is shown that various versions of probabilistic stability criteria, 36 are in fact special cases of a discrete form of dynkins formula. A note on the lyapunov stability of periodic discretetime. A generalized lyapunov stability theorem for discretetime. Lyapunov stability stability of equilibrium points 1. Paper by jiang and wang presents the property of input to state stability for discretetime systems. Ru er z yschool of electrical engineering and computer science z school of mathematical and physical sciences university of.
In the discretetime case, the equation corresponding to the c. This was established under the assumption that the system dynamics is continuous and locally lipschitz. Such matrices are said to be discrete stable in the sense of lyapunov, because of the connection with lyapunov s theory see. Lyapunov stability for nonautonomous timevarying systems for asymptotic stability of timevarying system timederivative of lyapunov function must be negativede. In 9, ahmadi and parrilo show that in the continuous time case, existence of the lyapunov function of mason et al. Global asymptotic stability if a linear system has uniform asymptotic stability, it also is globally stable x tfxt system is asymptotically stable for any. Lecture 12 basic lyapunov theory stanford university.
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