具有时变状态滞后的非线性2-D 离散系统的稳定性与控制

考虑一类由局部状态空间Fornasini-Marchesini(FM LSS)第二模型描述的,具有时变状态滞后非线性二维(2-D)离散系统的稳定性分析和控制问题.时变状态滞后项的上、下界为正整数,非线性项满足Lipschitz条件.首先,通过引入一个含有时滞上、下界的新Lyapunov函数,给出了系统的稳定性准则;然后设计了状态反馈控制器以保证系统的稳定性,进而,状态反馈控制律可由线性矩阵不等式求得;最后通过数值算例表明了所得结果的有效性.     

Delay-Independent Stability Conditions for Time-Varying Nonlinear Uncertain Systems

A new stability criterion for time-varying systems consisting of linear and norm bounded nonlinear terms with uncertain time-varying delays is formulated. An explicit delay-independent sufficient stability condition is formulated in the terms of the transition matrix of the given linear part without delay and the bounds for the uncertain terms. The obtained condition turns out to be also necessary if the matrix of the linear part is time-invariant and symmetric; it is shown that these systems satisfy the well-known Aizerman's conjecture. The obtained criterion is contrasted by some of stability estimates available in the literature for these kinds of systems; in all cases the proposed criterion provides less conservative stability bounds.     

时滞乘积型Lyapunov泛函的时变时滞系统稳定性新判据

应用新的Lyapunov泛函研究时变时滞系统的渐近稳定性问题.首先,基于Bessel-Legendre积分不等式和改进的自由矩阵积分不等式,提出两个新的时滞乘积型Lyapunov泛函,其充分利用时滞$d(t)$与二次型函数的乘积信息以及时滞$h-d(t)$与二次型函数的乘积信息;然后,采用Bessel-Legendre积分不等式和改进的逆凸组合方法估计时滞乘积型Lyapunov泛函导数的积分项,得到两个低保守性的时滞相关稳定性判据;最后,通过两个数值算例进行验证,结果表明,与最近相关方法相比,所提出方法可获得更大的允许时滞上界,进一步表明了所提出方法的可行性和低保守性.     

New stability conditions for uncertain T-S fuzzy systems with interval time-varying delay

This paper focuses on the stability problem for uncertain T-S fuzzy systems with interval time-varying delay. The system uncertainties are assumed to be time-varying and norm-bounded. The time-varying delay is considered as either being differentiable uniformly bounded with delay-derivative bounded by constant interval, or being fast-varying case with no restrictions on the delay derivative. Since we employ a novel Lyapunov-Krasovskii functional (LKF) which contains the information on the time-varying delay, and estimate the upper bound of its derivative less conservatively and adopt the convex optimization approach, some less conservative delay-derivative-dependent stability conditions are obtained in terms of linear matrix inequalities (LMIs), without using any free weighting matrix. These conditions are derived that depends on both the upper and lower bounds of the delay derivatives. Finally, some numerical examples are given to demonstrate the effectiveness and reduced conservatism of the proposed method.     

Guardian map approach to robust stability of linear systems withconstant real parameter uncertainty

New sufficient conditions are derived for stability robustness of linear time-invariant state-space systems with constant real parameter uncertainty. These bounds are obtained by applying a guardian map to the uncertain system matrices. Since this approach is only valid for constant real parameter uncertainty, these bounds do not imply quadratic stability, which guarantees robust stability with respect to time-varying uncertainty but is often conservative with respect to constant real parameter uncertainty. Numerical results are given to compare the new bounds with bounds obtained previously by means of Lyapunov methods     

带有干扰的线性时变系统的非线性鲁棒控制

研究了含有未知时变参数和有界干扰的单输入单输出线性时变系统的鲁棒控制问题.系统时变参数只要求光滑有界而不限制为慢时变或参数上界已知.利用时变的状态变换得到新的动态系统,基于Backstepping方法,设计出一种非线性鲁棒控制器.通过适当选择控制器参数,可以保证闭环系统是全局渐近稳定的.仿真例子表明了算法的有效性.     

On designing memory state feedback controller for linear systems with interval time-varying delay

This paper is concerned with the design of a memory state feedback controller for linear systems with interval time-varying delays. The time delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds of time-varying delay are available. First, a less conservative delayrange-dependent stability criteria is proposed by using a new interval fraction method. In the process of controller synthesis, the history information of system is considered in the controller design by introducing the lower delay state. Moreover, the usual memoryless state feedback controller for the underlying systems could be considered as a special case of the memory case. Finally, two numerical examples are given to show the effectiveness of the proposed method.     

Stability of non-linear feedback systems with time-varying linear sub-system (plant) †

A new approach is used to examine the stability of feedback systems comprising a time-varying linear subsystem and a time-varying non-linearity. The paper examines three stability aspects of the systems dynamic response: boundedness, unboundedness, and asymptotic decay. In addition to qualitative criteria, the results provide explicit quantitative bounds on the system response. The distinct time-domain method of analysis presented stipulates rather weak a priori restrictions on the nature of the linear and non-linear parts of the system, thus admitting a relatively broad class of systems. In the general time-varying non-linear feedback system considered, the non-linearity may exhibit hysteresis, and the linear subsystem may be non-causal, and may comprise any number of feed-forward differentiators of any order. The criteria obtained are given a simple graphical interpretation. A transformation which trades time-varying gains between plant and non-linearity is introduced. A number of examples demonstrate that the present results can predict stability information not obtainable from other existing criteria. In one example of a system with stationary plant, the present criteria prove superior to the Circle Theorem.     

Exponential stability and solution bounds for systems with bounded nonlinearities

Estimation of Lyapunov exponents of systems with bounded nonlinearities plays an essential part in their robust control. Known results in this field are based on the Gronwall inequality yielding relatively conservative bounds for Lyapunov exponents. In this note, we obtained more accurate upper bounds for the general Lyapunov exponent for systems consisting of a known linear time-varying part and an unknown nonlinear component with a bounded Lipschitz constant at zero. Consequently, a sufficient condition for exponential stability of this system is formulated. The systems are indicated for which the obtained bound is precise, i.e., cannot be improved without additional information on the nonlinear term. In the presence of a persisting perturbation, an upper bound for the solution norm is derived and expressed in the norm of the solution of the corresponding linear system. Using the obtained results, a condition for exponential stability of a linear time-varying control system with a nonlinear feedback is derived. Numerical results are obtained for a second-order time-varying system and for the Lienard equation; in the latter case they are favorably compared with stability conditions previously obtained using the Lyapunov function method.     

Robust stability analysis of linear continuous/discrete-timesystems with output feedback controllers

The robust stability of linear systems with output feedback controllers and time-varying uncertain parameters is considered. The robust stability bounds for time-varying uncertain parameters are given using the Lyapunov method. When there are no uncertain parameters in the input and/or output matrices, it is shown that the result for continuous-time systems is the same as that presented by K.M. Zhou and P.P. Khargonekar (1987), and the result for discrete-time systems is better than that of S.R. Kolla (1989) for the same example     

Improved results on stability and stabilization criteria for uncertain linear systems with time-varying delays

In this paper, the problems of stability and stabilization for linear systems with time-varying delays and norm-bounded parameter uncertainties are considered. By constructing augmented Lyapunov functionals and utilizing auxiliary function-based integral inequalities, improved delay-dependent stability and stabilization criteria for guaranteeing the asymptotic stability of the system are proposed with the framework of linear matrix inequalities. Four numerical examples are included to show that the proposed results can reduce the conservatism of stability and stabilization criteria by comparing maximum delay bounds.     

New approach for the stability analysis of linear time-varying state-space models

A new approach is proposed to establish stability bounds on the parameters of linear time-varying systems. In particular, stability of the Mathieu equation is investigated and stability boundaries are established. The proposed method is computationally simple to apply and can be calculated easily by computer. Examples are given to illustrate the improvement of the proposed method over that reported by Yedavalli and Kolla (1988).     

Improved bounds for linear discrete-time systems with structuredperturbations

A time-domain analysis of the stability robustness of linear discrete-time systems subject to time-varying structured perturbations is considered. The Lyapunov stability theory is used to obtain bounds on the perturbation such that the systems remain stable. It is shown that these bounds are less conservative than the existing ones. This is illustrated via two numerical examples     

Quantitative measures of robustness for systems including delayedperturbations

Asymptotically stable linear systems subject to delayed time-varying and nonlinear perturbations are considered. Razumikhin-type theorems are used to obtain easy-to-compute bounds on the perturbations so that the systems remain stable. Results indicate that if delayed perturbations are included, then the bound is reduced as compared to the one for nondelayed perturbations. However, in certain cases previously obtained bounds for the nondelayed perturbations guarantee stability even when delayed perturbations are in effect     

一类线性时滞系统的鲁棒稳定性分析

针对一类具有范数有界不确定性和2个继发时变时滞的线性时滞不确定系统,研究了其时滞依赖鲁棒稳定性问题.通过定义充分利用时变时滞上下界信息的新型Lyapunov-Krasovskii泛函,并结合时滞系统相关处理方法和线性矩阵不等式方法,得到了时滞线性不确定系统鲁棒渐近稳定所满足的条件.为了降低结论的保守性,对某些项进行了较紧致的估计.此外,并未引入自由权矩阵.最后并通过2个数值仿真证实了方法的有效性和优越性.     

Balanced truncation of linear time-varying systems

In this paper, balanced truncation of linear time-varying systems is studied in discrete and continuous time. Based on relatively basic calculations with time-varying Lyapunov equations/inequalities we are able to derive both upper and lower error bounds for the truncated models. These results generalize well-known time-invariant formulas. The case of time-varying state dimension is considered. Input-output stability of all truncated balanced realizations is also proven. The method is finally successfully applied to a high-order model.     

Time-varying discrete-time linear systems with bounded rates of variation: Stability analysis and control design

This paper investigates the problems of robust stability analysis and state feedback control design for discrete-time linear systems with time-varying parameters. It is assumed that the time-varying parameters lie inside a polytopic domain and have known bounds on their rate of variation. A convex model is proposed to represent the parameters and their variations as a polytope and linear matrix inequality relaxations that take into account the bounds on the rates of parameter variations are proposed. A feasible solution provides a parameter-dependent Lyapunov function with polynomial dependence on the parameters assuring the robust stability of this class of systems. Extensions to deal with robust control design as well as gain-scheduling by state feedback are also provided in terms of linear matrix inequalities. Numerical examples illustrate the results.     

具有区间时变时滞2-D 离散系统的时滞相关稳定与控制

针对具有区间时变时滞2-D离散系统,利用时滞相关方法,研究其稳定性与控制问题.首先选取含有时滞项上、下界的一个新的Lyapunov函数,对其差分时考虑所有项,得到了基于线性矩阵不等式(LMI)的时滞相关稳定性准则;然后给定时变时滞项的下界,再由一个凸优化问题最大化其上界,进而通过状态反馈实现系统的时滞相关控制,且求解LMI可得到增益矩阵;最后,利用数值算例说明了所得结果有效且优于已有成果.     

LPV system analysis via quadratic separator for uncertain implicitsystems

This paper considers a class of linear systems containing time-varying parameters whose behavior is not known exactly. We assume that the parameters vary within known intervals and there are known bounds on their rates of variation. Our objective is to give a computationally verifiable condition that guarantees stability of the system for all possible parameter variations. We first point out that the information on the rate bounds can be exploited by considering an augmented system described by an implicit model. We then develop a tool, called the quadratic separator, for stability analysis of uncertain implicit systems. Using this tool, a sufficient stability condition is obtained for the original linear parameter-varying system. Moreover, we show that the stability condition obtained is equivalent to the existence of a Lyapunov function that depends on the parameters in a linear fractional manner. Finally, the computational aspects of the proposed stability conditions are addressed     

Average consensus seeking of high-order continuous-time multi-agent systems with multiple time-varying communication delays

The average consensus problem of high-order multi-agent systems with multiple time-varying communication delays is investigated in this paper. By using the idea of state decomposition, the condition for guaranteeing average consensus is converted into verifying the stability of zero equilibrium of disagreement system. Considering multiple time-varying communication delays, Lyapunov-Krasovskii approach in time-domain is employed to analyze the stability of zero equilibrium. With the help of Free-weighting Matrices (FWM) approach, the tolerant upper bounds on communication delays can be obtained through solving feasible linear matrix inequalities (LMIs). Delay-dependent stability criteria for both strongly-connected fixed and switching topologies are provided in the main results. Further, the conclusion is extended to the case of jointly-connected switching topologies. Numerical examples and simulation results are given to demonstrate the effectiveness and the benefit on reducing conservativeness of the proposed method.