切换系统的二类共同Lyapunov 函数

研究切换系统的共同Lyapunov函数存在问题.对于一类正切换系统,给出了共同Lyapunov函数存在的充分条件.当系统矩阵集为二阶矩阵紧集时,给出了判断共同Lyapunov函数存在的方法,并给出了计算共同Lyapunov函数的算法.最后通过算例验证了所提出算法的有效性.     

Two conditions concerning common quadratic Lyapunov functions forlinear systems

Concerning a pair of linear systems, some triangle conditions for the existence of a common quadratic Lyapunov function are presented in this paper. These conditions are derived from a criterion for judging the semipositiveness of a linear map defined on symmetric matrices     

On the stability of fuzzy systems

Studies the global asymptotic stability of a class of fuzzy systems. It demonstrates the equivalence of stability properties of fuzzy systems and linear time invariant (LTI) switching systems. A necessary and sufficient condition for the stability of such systems are given, and it is shown that under the sufficient condition, a common Lyapunov function exists for the LTI subsystems. A particular case when the system matrices can be simultaneously transformed to normal matrices is shown to correspond to the existence of a common quadratic Lyapunov function. A constructive procedure to check the possibility of simultaneous transformation to normal matrices is provided     

On the asymptotic stability of switched homogeneous systems

The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L>0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.     

Globally asymptotical stability of discrete-time analog neuralnetworks

Some globally asymptotical stability criteria for the equilibrium states of a general class of discrete-time dynamic neural networks with continuous states are presented using a diagonal Lyapunov function approach. The neural networks are assumed to have the asymmetrical weight matrices throughout the paper. The resulting criteria are described by the diagonal stability of some matrices associated with the network parameters. Some novel stability conditions represented by either the existence of the positive diagonal solutions of the Lyapunov equations or some inequalities are given. Using the equivalence between the diagonal stability and the Schur stability for a nonnegative matrix, some simplified global stability conditions are also presented. Finally, some examples are provided for demonstrating the effectiveness of the global stability conditions presented.     

A relaxed stability criterion for T-S fuzzy discrete systems

Stability conditions for Tanaka-Sugeno (T-S) fuzzy discrete systems based on a single quadratic Lyapunov function or a weighting dependent Lyapunov function have been mentioned in lots of literature. The existence of a common matrix P is required in the former, and r (rules' number) positive matrices satisfying r2 Lyapunov inequalities are necessary in the latter. In this paper, the weighting dependent Lyapunov function is used again. Moreover according to the idea of the firing rule group and the distance estimation between two successive states of the system, the relaxed stability criterion of T-S fuzzy discrete system is proposed.     

切换线性奇异系统输出反馈容许控制

The admissibility analysis and robust admissible control problem of the uncertain discrete- time switched linear singular (SLS) systems for arbitrary switching laws are investigated. Based on linear matrix inequalities, some sufficient conditions are given for: A) the existence of generalized common Lyapunov solution and the admissibility of the SLS systems for arbitrary switching laws, B) the existence of static output feedback control laws ensuring the admissibility of the closed-loop SLS systems for arbitrary switching laws and norm-bounded uncertainties.     

Gain-scheduled L-one control for linear parameter-varying systems with parameter-dependent delays

This paper deals with the problem of gain-scheduled L-one control for linear parameter-varying (LPV) systems with parameter-dependent delays. The attention is focused on the design of a gain-scheduled L-one controller that guarantees being an asymptotically stable closed-loop system and satisfying peak-to-peak performance constraints for LPV systems with respect to all amplitude-bounded input signals. In particular, concentrating on the delay-dependent case, we utilize parameter-dependent Lyapunov functions (PDLF) to establish peak-to-peak performance criteria for the first time where there exists a coupling between a Lyapunov function matrix and system matrices. By introducing a slack matrix, the decoupling for the parameter-dependent time-delay LPV system is realized. In this way, the sufficient conditions for the existence of a gain-scheduled L-one controller are proposed in terms of the Lyapunov stability theory and the linear matrix inequality (LMI) method. Based on approximate basis function and the gridding technique, the corresponding controller design is cast into a feasible solution problem of the finite parameter linear matrix inequalities. A numerical example is given to show the effectiveness of the proposed approach.     

Robustness of stability through necessary and sufficient Lyapunov-like conditions for systems with a continuum of equilibria

The equivalence between robustness to perturbations and the existence of a continuous Lyapunov-like mapping is established in a setting of multivalued discrete-time dynamics for a property sometimes called semistability. This property involves a set consisting of Lyapunov stable equilibria and surrounded by points from which every solution converges to one of these equilibria. As a consequence of the main results, this property turns out to always be robust for continuous nonlinear dynamics and a compact set of equilibria. Preliminary results on reachable sets, limits of solutions, and set-valued Lyapunov mappings are included.     

Observer-based residual generation for fault diagnosis for non-affine non-linear polynomial systems

This paper proposes an observer-based residual generator (OBRG) for diagnosing faults in a continuous non-affine system with polynomial non-linearities up to any finite degree where the fault and an unknown input affect both the system and part of the output. Firstly, given certain assumptions and the use of defined extended vectors, a parameterized polynomial system is considerred for which a compact set of sufficient conditions is given for the existence of a candidate OBRG. Conditions for error stability (by a Lyapunov method) and detectability are given. The calculation steps in the design of the OBRG are shown to involve the solution of three linear equations (with parameterizations) and the calculation of a set of constant matrices (for detectability of faults). A result is then given establishing that the design holds for a much wider class of systems. The residual design is applied to a real three-tank system.     

LPV重复过程的H_∞滤波新方法

将时滞线性参数变化(LPV)思想应用于重复过程中,研究了其H∞滤波问题。基于参数依赖Lyapunov函数方法推出了该重复过程的稳定性和滤波器设计的充分条件。通过投影定理引入两个附加矩阵,解除了重复过程矩阵和依赖于参数的Lyapunov函数矩阵之间的耦合,使得到的条件便于求解。仿真实例证实了该设计方法的有效性。     

时滞LPV重复过程的H∞控制

将时滞线性参数变化(LPV)思想应用于重复过程,研究其H∞控制问题.基于参数依赖Lyapunov函数方法,给出了该重复过程的稳定性和控制器设计的充分条件;同时通过投影定理引入两个附加矩阵,解除了重复过程矩阵和依赖于参数的Lyapunov函数矩阵之间的耦合,使得到的条件便于求解.仿真实例表明了该设计方法的有效性.     

On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form

In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices A/sub 1/ and A/sub 2/ are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product A/sub 1/A/sub 2/ does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result.     

On absolute stability of Lur''''e control systems with multiple non-linearities using linear matrix inequalities

Necessary and suffcient conditions for the existence of a Lyapunov function in the Lur'e form to guarantee the absolute stability of Lur'e control systems with multiple non-linearities are discussed in this paper. It simplifies the existence problem to one of solving a set of linear matrix inequalities (LMIs), If those LMIs are feasible, free parameters in the Lyapunov function,such as the positive definite matrix and the coefficients of the integral terms, are given by the solution of the LMIs. Otherwise, this Lyapunov function does not exist. Some sufficient conditions are also obtained for the robust absolute stability of uncertain systems.A numerical example is provided to demonstrate the effectiveness of the proposed method.     

On absolute stability of Lur＇e control systems with multiple non-linearities using linear matrix inequalities

Necessary and suffcient conditions for the existence of a Lyapunov function in the Lur‘e form to guarantee the absolute stability of Lur‘e control systems with multiple non-linearities are discussed in this paper. It simplifies the existence problem to one of solving a set of linear matrix inequalities (LMIs). If those LMIs are feasible, free parameters in the Lyapunov function, such as the positive definite matrix and the coefficients of the integral terms, are given by the solution of the LMIs. Otherwise, this Lyaptmov function does not exist. Some sufficient conditions are also obtained for the robust absolute stability of uncertain systems. A numerical example is provided to demonstrate the effectiveness of the proposed method.     

On the simultaneous diagonal stability of linear discrete-time systems

This paper studies the set of time-invariant linear discrete-time systems in which each system has a diagonal quadratic Lyapunov function. First, it is shown that there is generally no common diagonal quadratic Lyapunov function for such a set of systems even if the set is assumed to be commutative. It is also shown that the commutativity assures the existence of a common diagonal quadratic Lyapunov function inside the set of 2×2 systems or the set of nonnegative systems. Then, two simple topological results are presented concerning the simultaneous diagonal stability on the set of nonnegative systems. The first is a measure of the difference of matrices that assures the simultaneous diagonal stability. The second is a measure of the commutativity of matrices.     

On linear co-positive Lyapunov functions for sets of linear positive systems

In this paper we derive necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for a finite set of linear positive systems. Both the state dependent and arbitrary switching cases are considered. Our results reveal an interesting characterisation of “linear” stability for the arbitrary switching case; namely, the existence of such a linear Lyapunov function can be related to the requirement that a number of extreme systems are Metzler and Hurwitz stable. Examples are given to illustrate the implications of our results.     

Some common fixed point theorems for generalized nonlinear contractive mappings

Sufficient conditions for the existence of a common fixed point for generalized nonlinear contractive mappings are derived. As applications, some results on the set of best approximation for this class of mappings are also obtained. The proved results generalize and extend various known results in the literature.     

Stability analysis of interval matrices: improved bounds

New sufficient conditions for the stability of interval matrices are presented, based on the Lyapunov stability approach as well as on the frequency-domain approach. The conditions help to overcome the conservatism of the existing criteria for the stability of interval matrices based on a matrix approach. Several numerical examples are given to demonstrate the new conditions and to compare them with results previously reported.     

一类概率依赖的随机网络系统的随机容错设计

针对网络控制系统的传感器、执行器和网络时延具有随机性这一现象,引入相互独立的Bernoulli随机变量序列作为传感器失效故障、执行器失效故障的开关矩阵,将时延的变化区间任意分成两个小区间,引进Bernoulli随机变量并将其概率分布引入到系统矩阵中,建立随机的网络化控制系统模型,研究了该系统存在传感器失效故障、执行器失效故障及二者同时失效故障情况下的随机容错控制问题。基于Lyapunov稳定性理论,结合线性矩阵不等式技术,给出了确保系统均方指数稳定的条件,该条件不仅依赖于时延的上下界,还依赖于时延的概率分布。数值实例说明结论是有效的。