Simultaneous controller design for time-varying linear systems

In this paper, we establish the parametrization of all simultaneously stabilizing controllers for two time-varying linear systems. A new approach to the simultaneous stabilization of three time-varying linear systems is also provided.     

A parametric frequency response method for non-linear time-varying systems

A new parametric frequency response algorithm is introduced to investigate linear and non-linear dynamic systems with time-varying parameters. In the new algorithm the time-varying parameters are regarded as additional inputs of the systems and the non-linear generalised frequency response functions for multi-input-single-output systems are then employed to obtain Zadeh's system functions from a differential equation representation. The parametric frequency response method reveals how the time-varying parameters affect the behaviour of the systems through a time-varying term. The new method can be applied to both linear and non-linear time-varying systems.     

H∞-output feedback controller design for linear systems withtime-varying delayed state

This paper considers the H∞-controller design problem for linear systems with time-varying delays in states. The authors obtain sufficient conditions for the existence of H∞ controllers in terms of three linear matrix inequalities (LMIs). These sufficient conditions are dependent on the maximum value of the time derivative of time-varying delay. Furthermore, they briefly explain how to construct such controllers from the positive-definite solutions of their LMIs and give an example     

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.     

线性时变广义系统的能控性与能观性问题

讨论了线性时变系统和线性时变广义系统的两个基本问题, 得到了两种判定时变系统能控性与能观性的必要条件, 该判定条件只依赖于系统矩阵A(t)和输入矩阵B(t), 不必计算系统的系统状态转移矩阵, 使得判别时变系统能控性与能观性易于实现. 说明了本文结论是线性定常系统相应结论的自然扩展. 对进一步深入研究时变系统和时变广义系统具有实际启发作用.     

Decoupling and tracking control using eigenstructure assignment for linear time-varying systems

This work is concerned with the assignment of a desired PD-eigenstructure for linear time-varying systems. Despite its well-known limitations, gain scheduling control appeared to be a focus of the research efforts. Scheduling of frozen-time, frozen-state controllers for fast time-varying dynamics is known to be mathematically fallacious, and practically hazardous. Therefore, recent research efforts are being directed towards applying time-varying controllers. In this paper, (a) we introduce a differential algebraic eigenvalue theory for linear time-varying systems, and then (b) a novel decoupling and tracking control scheme is proposed by using the PD-eigenstructure assignment scheme via a differential Sylvester equation and a Command Generator Tracker for linear time-varying systems. The PD-eigenstructure assignment is utilized as a regulator. A feedforward gain for tracking control is computed by using the command generator tracker. The whole design procedure of the proposed PD-eigenstructure assignment scheme is systematic in nature. The scheme could be used to determine stability of linear time-varying systems easily as well as to provide a new horizon of designing controllers for the linear time-varying systems. The presented method is illustrated by numerical examples.     

Robust exponential stability of time-varying linear systems

This paper investigates the robustness of time-varying linear systems under a large class of complex time-varying perturbations. Previous results⁸ which were restricted to bounded linear perturbations of output feedback type are generalized to unbounded and nonlinear perturbations of multi-output feedback type. We establish a lower bound for the stability radius of these systems and show how it may be possible to improve the bound using time-varying scalar transformations of the state, input and output variables. The results are applied to derive Gershgorin type stability criteria for time-varying linear systems.     

一个新型充要指数稳定性定理及其初步应用

针对一般滞后型动态系统,本文建立了一种新型的充分必要指数稳定性定理以及新的稳定性分析方法,进一步扩展了李雅普诺夫函数的概念.初步地探讨了将新方法用于多时变时滞线性系统和线性定常系统的稳定性分析.和文献中已有的结果不同,所建立的稳定性判据不依赖于时变时滞的变化率.因此,所提方法适合具有非常快变的时变时滞的系统.文中也指出了进一步改进应用结果的研究方向.     

Switching strategy in tracking constant references for linear time-varying delay systems with actuator failures

Tracking of command constant references is a tracking control problem of fundamental importance. This problem for linear time-varying delay systems in the presence of actuator failures is solved in this article. The linear time-varying delay system with actuator failures is modelled as a switched linear time-varying delay system by utilising switching systems theory and a suitable control scheme. The polytopic technique is employed in order to cope with the time-varying, delay-derivative term in case its absolute value might exceed one. On the grounds of the average dwell-time approach, a switching tracking control law that is immune to actuator failures is designed and the conditions are derived such that the output of the linear time-varying delay system tracks given constant command references. Two illustrative examples along with the respective numerical and simulation results are presented to demonstrate the effectiveness of the proposed method.     

一类线性时变系统的输出反馈控制

基于线性时不变系统能控能观标准型变换及非线性系统高增益观测器方法,本文研究了一类线性时变系统 的输出反馈控制问题. 通过引入时变的状态变量坐标变换,分别设计了线性时变系统的状态反馈控制器、状态观测器以及基于 状态观测器的输出反馈控制器. 进一步地,本文分别证明了观测器动态误差是渐近收敛于零的,而状态反馈控制器以及输出反馈控制器可以 保证闭环系统的渐近稳定性.     

学习辨识:最小二乘算法及其重复一致性

针对重复时变系统, 提出学习辨识方法用于估计系统的时变参数. 讨论了有限时间作业区间上重复运行的时变系统以及周期时变系统两种情形. 文中给出最小二乘学习算法的推导过程, 并分析了所提算法的收敛性. 结果表明, 当重复持续激励条件成立时, 提出的学习算法具有重复一致性, 能够给出时变参数的完全估计. 通过数值算例进一步验证了学习算法的有效性.     

连续时间时变参数系统的指数稳定性和Lp指数稳定性

本文给出具有慢时变参数的连续时间线性齐次系统指数稳定的一个一般性条件，并给出具有随机时变参数的连续时间线性齐次系统Ｌｐ指数稳定的一个充分条件。这些结果可用来分析带有未建模动态，确定性噪声或随机噪声及慢时变参数的自适应控制系统的稳定性问题。     

线性时变周期系统的鲁棒稳定及H∞控制

给出带有结构性不确定的线性时变周期系统二次稳定的充要条件,并指出为该系统设计 鲁棒动态补偿器和具有H∞指标r的鲁棒动态补偿器问题都可以转化成相应的确定性辅助线 性时变周期系统的H∞控制问题.     

基于H表示的时变随机Markov 跳跃系统的能观性

研究时变连续和离散随机Markov 跳跃系统(SMJSs) 的能观性问题. 基于H-表示方法将时变SMJSs 转化为等价的时变线性系统, 根据线性系统理论得到时变连续和离SMJSs 的能观性Gramian 矩阵判据. 数值仿真表明了所得结论的正确性.     

On robustness and stabilization of linear systems with delayednonlinear perturbations

In this paper, by employing a Razumikhin-type theorem, a robust stability criterion for a class of linear systems subject to delayed time-varying nonlinear perturbations is given. Then, stabilization of a class of linear systems subject to mismatched delayed time-varying nonlinear perturbations by linear controller is considered. For such systems, conditions which ensure closed-loop asymptotic stability are given. Numerical examples are given to illustrate the results     

Orthonormal function parametrisation of model-predictive control for linear time-varying systems

It is well known that some practical difficulties are involved in the implementation of stabilising model predictive control for time-varying systems. In order to address the difficulty of computational load, this paper extends the orthonormal function method for model predictive control to linear time-varying systems. We provide sufficient conditions for a sub-optimal model predictive controller to be stabilising for a time-varying system. It is also shown that the orthonormal parametrisation method enables us to reduce the number of decision variables significantly and with a satisfactory performance. In addition, it is shown that orthonormality and, the called for, long prediction horizons are not necessary for stability. Examples are provided, illustrating the effectiveness of the method for linear time-varying systems.     

Frequency-domain $L_2$-stability conditions for time-varying linear and nonlinear MIMO systems

The paper deals with the g2-stability analysis of multi-input-multi-output （MIMO） systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monotone and quasi-monotone nonlin- earities. For nonlinear MIMO systems that are described by differential equations, most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates. In contrast, a non-Lyapunov framework is employed here to derive new and more general g2-stability conditions in the frequency domain. These conditions have the following features： i） They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block, ii） For certain cases of the periodic time-varying gain, they contain, depending on the multiplier function chosen, no restrictions on the normalized rate of variation of the time-varying gain, but, for other periodic/aperiodic time-varying gains, they do. Overall, even when specialized to periodic-coefficient linear and nonlinear MIMO systems, the stability conditions are distinct from and less restrictive than recent results in the literature. No comparable results exist in the literature for aperiodic time-varying gains. Furthermore, some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented. Examples are given to illustrate a few of the stability theorems.     

The graph topology for linear plants with applications to nonlinearrobust stabilization

The graph topology is defined for the set of finite-dimensional linear time-varying plants that are internally stabilizable by output feedback. This follows some earlier results of M. Vidyasagar et al. (1982) on time-invariant systems. Feedback connections of linear time-varying plants and general nonlinear time-varying controllers are considered, and it is shown that the graph topology is the weakest topology in which feedback stabilization is a robust property. Using this result, necessary and sufficient conditions for robust stabilizability of families of linear time-varying plants with parametric uncertainty are derived     

迭代学习控制在线性时变系统终端控制问题中的应用研究

An iterative learning control algorithm based on shifted Legendre orthogonal polynomials is proposed to address the terminal control problem of linear time-varying systems. First, the method parameterizes a linear time-varying system by using shifted Legendre polynomials approximation. Then, an approximated model for the linear time-varying system is deduced by employing the orthogonality relations and boundary values of shifted Legendre polynomials. Based on the model, the shifted Legendre polynomials coefficients of control function are iteratively adjusted by an optimal iterative learning law derived. The algorithm presented can avoid solving the state transfer matrix of linear time-varying systems. Simulation results illustrate the effectiveness of the proposed method.     

Stability analysis of linear time-varying systems

This paper addresses the issue of the stability of linear time-varying systems. Recently developed elemental perturbation bound analysis is extended to obtain stability bounds on the parameters of linear time-varying systems. In particular, systems governed by the Mathieu equation are considered. The bounds obtained with the proposed method are compared with those of the circle criterion and matrix decomposition methods. The proposed method is computationally simple and the bounds obtained fare well with the other methods considered.