Robust stability of large-scale systems which contain time-invariant lur'e-type uncertainties in subsystems and norm-bounded uncertainties in interconnections

In this paper a new frequency-domain condition for the robust stability of a class of large-scale systems is derived. It is assumed that each subsystem contains one time-invariant nonlinearity satisfying a sector condition, and that the interconnecting functions are linearly bounded by the norms of the scalar outputs of subsystems. In deriving the above such Lur'e-type Lyapunov functions of subsystems are constructed so that their weighted sum is a Lyapunov function of the overall system. This condition is potential to give a sharper result than the circle-condition-type result which was obtained previously. Furthermore, a method to estimate the domain of attraction using the above Lyapunov function is also given when the sector condition is satisfied only locally.     

Stability criteria for large-scale systems

Recent research into large-scale system stability has proceeded via two apparently unrelated approaches. For Lyapunov stability, it is assumed that the system can be broken down into a number of subsystems, and that for each subsystem one can find a Lyapunov function (or something akin to a Lyapunov function). The alternative approach is an input-output approach; stability criteria are derived by assuming that each subsystem has finite gain. The input-output method has also been applied to interconnections of passive and of conic subsystems. This paper attempts to unify many of the previous results, by studying linear interconnections of so-called "dissipative" subsystems. A single matrix condition is given which ensures both input-output stability and Lyapunov stability. The result is then specialized to cover interconnections of some special types of dissipative systems, namely finite gain systems, passive systems, and conic systems.     

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.     

Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable

This paper is concerned with the stability and robust stability of switched positive linear systems (SPLSs) whose subsystems are all unstable. By means of the mode-dependent dwell time approach and a class of discretized co-positive Lyapunov functions, some stability conditions of switched positive linear systems with all modes unstable are derived in both the continuous-time and the discrete-time cases, respectively. The copositive Lyapunov functions constructed in this paper are timevarying during the dwell time and time-invariant afterwards. In addition, the above approach is extended to the switched interval positive systems. A numerical example is proposed to illustrate our approach.      

Stability analysis of interconnected systems with possibly unstable subsystems

A direct method is presented for stability analysis of nonlinear interconnected dynamical systems. A new scalar Lyapunov function is considered as weighted sum of individual Lyapunov functions for each free subsystem and individual scalar functions related separately to each connection. Sufficient conditions are obtained for asymptotic stability of the equilibrium state by testing the definity of two constant square matrices whose dimension is equal to the number of subsystems. This method can assure stability of systems with possible unstable subsystems. A simple numerical example is included to illustrate this theory.     

Exponential Stabilization of Switched Discrete‐Time Systems with All Unstable Modes

This paper studies the exponential stabilization of switched discrete‐time systems whose subsystems are unstable. A new sufficient condition for the exponential stability of the class of systems is proposed. The result obtained is based on the determination of a lower bound of the maximum dwell time by virtue of the multiple Lyapunov functions method. The key feature is that the given stability condition does not need the value of the Lyapunov function to uniformly decrease at every switching instant. An example is provided to illustrate the effectiveness of the proposed result.     

PIECEWISE LYAPUNOV FUNCTIONS FOR SWITCHED SYSTEMS WITH AVERAGE DWELL TIME

The stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems are investigated by using piecewise Lyapunov functions incorporated with an average dwell time approach. It is shown that if the average dwell time is chosen sufficiently large and the total activation time ratio between Hurwitz stable and unstable subsystems is not smaller than a specified constant, then exponential stability of a desired degree is guaranteed. The above result is also extended to the case where nonlinear norm‐bounded perturbations exist.     

Quasi-ISS/ISDS observers for interconnected systems and applications

This paper considers interconnected nonlinear dynamical systems and studies observers for such systems. For single systems the notion of quasi-input-to-state dynamical stability (quasi-ISDS) for reduced-order observers is introduced and observers are investigated using error Lyapunov functions. It combines the main advantage of ISDS over input-to-state stability (ISS), namely the memory fading effect, with reduced-order observers to obtain quantitative information about the state estimate error. Considering interconnections quasi-ISS/ISDS reduced-order observers for each subsystem are derived, where suitable error Lyapunov functions for the subsystems are used. Furthermore, a quasi-ISS/ISDS reduced-order observer for the whole system is designed under a small-gain condition, where the observers for the subsystems are used. As an application, we prove that quantized output feedback stabilization for each subsystem and the overall system is achievable, when the systems possess a quasi-ISS/ISDS reduced-order observer and a state feedback law that yields ISS/ISDS for each subsystem and therefor the overall system with respect to measurement errors. Using dynamic quantizers it is shown that under the mentioned conditions asymptotic stability can be achieved for each subsystem and for the whole system.     

Stability analysis and H/sub /spl infin// controller design of fuzzy large-scale systems based on piecewise Lyapunov functions

This paper presents a novel approach to stability analysis of a fuzzy large-scale system in which the system is composed of a number of Takagi-Sugeno (T-S) fuzzy subsystems with interconnections. The stability analysis is based on Lyapunov functions that are continuous and piecewise quadratic. It is shown that the stability of the fuzzy large-scale systems can be established if a piecewise 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/sub /spl infin// controllers can also be designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions.     

Stability analysis and H infinity controller design of fuzzy large-scale systems based on piecewise Lyapunov functions.

This paper presents a novel approach to stability analysis of a fuzzy large-scale system in which the system is composed of a number of Takagi-Sugeno (T-S) fuzzy subsystems with interconnections. The stability analysis is based on Lyapunov functions that are continuous and piecewise quadratic. It is shown that the stability of the fuzzy large-scale systems can be established if a piecewise 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 infinity controllers can also be designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions.     

Stability of linear switched differential algebraic equations with stable and unstable subsystems

This article studies linear switched differential algebraic equations (DAEs), which contains stable and unstable subsystems. We prove sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The result shows that stability is preserved under switching with an average dwell time and an additional condition involving consistency projectors holds. Furthermore, we also give an example to illustrate the result.     

Global stabilization via local stabilizing actions

Stabilization of a linear, time-invariant system via stabilization of its main diagonal subsystems is the underlying problem in all diagonal dominance techniques for decentralized control. In these techniques as well as all Nyquist-based techniques, sufficient conditions are obtained under the assumption that the collection of the unstable poles of all diagonal subsystems is the same as the unstable poles of the overall system. We show that this assumption is by itself enough to construct a solution to the problem at least in cases where the diagonal subsystems have disjoint poles.     

时滞半Markov切换随机系统输入–状态稳定性分析的时变驻留时间条件方法

本文针对时滞半Markov切换随机系统,建立了一种基于时变驻留时间条件的不定多重Lyapunov-Razumikhin函数方法,给出了系统输入–状态稳定性和积分输入–状态稳定性判别条件.一方面,构造了一种不定多重Lyapunov函数,不要求每个子系统对应的Lyapunov-Razumikhin函数导数总是保持负定,从而放宽了对于子系统稳定性要求,甚至允许了不稳定子系统的存在;另一方面,提出了一种时变驻留时间条件来表示半Markov切换信号的切换次数与子系统驻留时间之间的关系,利用时变函数对切换次数进行估计,间接放松了不定多重Lyapunov-Razumikhin函数选取的限制.最后,数值算例验证了所提方法的有效性.     

Sufficient conditions for almost sure stability of jump linearsystems

This paper derives less conservative conditions for almost-sure stability of jump linear systems by using the stochastic version of Lyapunov's second method. To ensure the required stability, the Lyapunov function is only required to be nonincreasing along each of the specially organized subsequences of “jumgings”. This allows one to handle the case where some or even all subsystems are unstable and provides more choices of Lyapunov functions     

Stability of a class of interconnected evolution systems

Stability conditions for a class of interconnected systems modeled by linear abstract evolution equations and a memoryless nonlinearity are derived. These conditions are stated in terms of the passivity of each of the subsystems and can be considered as a partial generalization of the hyperstability theorem. A Lyapunov function approach is used in the proof without requiring the positive definiteness of the Lyapunov function. Application to the robustness analysis of the infinite-dimensional linear quadratic regulator is also discussed     

Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems

Exponential stability and robust exponential stability relating to switched systems consisting of stable and unstable nonlinear subsystems are considered in this study. At each switching time instant, the impulsive increments which are nonlinear functions of the states are extended from switched linear systems to switched nonlinear systems. Using the average dwell time method and piecewise Lyapunov function approach, when the total active time of unstable subsystems compared to the total active time of stable subsystems is less than a certain proportion, the exponential stability of the switched system is guaranteed. The switching law is designed which includes the average dwell time of the switched system. Switched systems with uncertainties are also studied. Sufficient conditions of the exponential stability and robust exponential stability are provided for switched nonlinear systems. Finally, simulations show the effectiveness of the result.     

Stability analysis for a special class of dynamical networks

In this article, stability analysis and decentralised control problems are studied for a special class of linear dynamical networks. Necessary and sufficient conditions for stability and stabilisability under a decentralised control strategy are given for this type of linear networks. Especially, two types of linear regular networks, star-shaped networks and globally coupled networks, are studied in detail, respectively. A dynamical network can be viewed as a large-scale system composed of some subsystems with some coupling structures, based on this, the relationship between the stability of a network and the stability of its corresponding subsystems is studied. Different from the discussions that the subsystems in networks vary with different coupling structures (Duan, Z.S., Wang, J.Z., Chen, G.R., and Huang, L. (2008), ‘Stability Analysis and Decentralised Control of a Class of Complex Dynamical Networks’, Automatica, 44, 1028–1035), the subsystems in network discussed in this article remain unchanged with different interconnections which is the same as in general large-scale system. It is also pointed out that some subsystems must be made unstable for the whole network to be stable in some special cases. Moreover, the controller design method based on parameter-dependent Lyapunov function is provided.     

Robust non-fragile guaranteed cost control of uncertain large-scale systems with time-delays in subsystem interconnections

In this paper, the robust non-fragile guaranteed cost-control problem is studied for a class of uncertain linear large-scale systems with time-delays in subsystem interconnections and given quadratic cost functions. The uncertainty in the system is assumed to be norm-bounded and time-varying. Also, the state-feedback gains for subsystems of the large-scale system are assumed to have norm-bounded controller gain variations. The problem is to design a state feedback control law such that the closed-loop system is asymptotically stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. Sufficient conditions for the existence of such controllers are derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. A parameterized characterization of the robust non-fragile guaranteed cost controllers is given in terms of the feasible solutions to a certain LMI. A numerical example is given to illustrate the proposed method.     

Explicit construction of quadratic Lyapunov functions for the small gain,positivity, circle,and popov theorems and their application to robust stability. Part I: Continuous-time theory

The purpose of this paper is to construct Lyapunov functions to prove the key fundamental results of linear system theory, namely, the small gain (bounded real), positivity (positive real), circle, and Popov theorems. For each result a suitable Riccati-like matrix equation is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Lyapunov functions for the small gain and positivity results are also constructed for the interconnection of two transfer functions. A multivariable version of the circle criterion, which yields the bounded real and positive real results as limiting cases, is also derived. For a multivariable extension of the Popov criterion, a Lure-Postnikov Lyapunov function involving both a quadratic term and an integral of the nonlinearity, is constructed. Each result is specialized to the case of linear uncertainty for the problem of robust stability. In the case of the Popov criterion, the Lyapunov function is a parameter-dependent quadratic Lyapunov function.     

Stability and asynchronous stabilization for a class of discrete-time switched nonlinear systems with stable and unstable subsystems

The stability analysis and asynchronous stabilization problems for a class of discrete-time switched nonlinear systems with stable and unstable subsystems are investigated in this paper. The Takagi-Sugeno (T-S) fuzzy model is used to represent each nonlinear subsystem. Through using the T-S fuzzy model, the studied systems are modeled into the switched T-S fuzzy systems. By using the switching fuzzy-basis-dependent Lyapunov functions (FLFs) approach and mode-dependent average dwell time (MDADT) technique, the stability conditions for the open-loop switched T-S fuzzy systems with unstable subsystems and asynchronous stabilization conditions for the closed-loop switched T-S fuzzy systems with unstable subsystems are obtained. Both the stability results and asynchronous stabilization results are derived in terms of linear matrix inequalities (LMIs). Finally two numerical examples are provided to illustrate the effectiveness of the results obtained.