线性系统稳定性与最优性的关系(Ⅱ)

我们在[1]中提出并讨论了与R.E.Kalman的最优控制反问题不同的另一类线性最优控制的反问题:任给一渐近稳定的定常线性系统和一个非负二次型性能指标,问是否可以从该稳定系统中分解出一个状态反馈,使得这个状态反馈就是给定指标下的最优控制。本文将上述问题加以推广,对线性离散系统和线性时变系统得出了相应的结论,进而得出[1]中提出的渐近稳定系统和最优系统之间的对应关系是一切线性系统的内在特征。     

Asymptotic behavior of Timoshenko beam with dissipative boundary feedback

This paper is concerned with the stabilization problem of Timoshenko beam in the presence of linear dissipative boundary feedback controls. Using C₀-semigroups theory we establish the existence and the uniqueness of solution of the proposed closed loop system. In order to consider the asymptotic behavior of the closed loop system, we first discuss the existence of nonzero solution of a closely related boundary value problem. Then we derive various necessary and sufficient conditions for the system to be asymptotically stable. Finally, we prove the equivalence between the exponential stability and the asymptotic stability for the closed loop system.     

Commuting and stable feedback design for switched linear systems

In the paper, commuting and stable feedback design for switched linear systems is investigated. This problem is formulated as to build up suitable state feedback controller for each subsystem such that the closed-loop systems are not only asymptotically stable but also commuting each other. A new concept, common admissible eigenvector set (CAES), is introduced to establish necessary/sufficient conditions for commuting and stable feedback controllers. For second-order systems, a necessary and sufficient condition is established. Moreover, a parametrization of the CAES is also obtained. The motivation comes from stabilization of switched linear systems which consist of a family of LTI systems and a switching law specifying the switching between them, where if all the subsystems are stable and commuting each other, then the total system is stable under arbitrary switching.     

Global stabilization of switched control systems with time delay

In this paper, the stabilization problem of switched control systems with time delay is investigated for both linear and nonlinear cases. First, a new global stabilizability concept with respect to state feedback and switching law is given. Then, based on multiple Lyapunov functions and delay inequalities, the state feedback controller and the switching law are devised to make sure that the resulting closed-loop switched control systems with time delay are globally asymptotically stable and exponentially stable.     

Periodical stabilization of switched linear systems

Periodical stabilization problems for switched linear systems are investigated in this paper. For autonomous switched systems, if there exists a stable convex combination of the subsystems, then a periodically switching signal can be constructed such that the overall system is asymptotically stable. Based on this fact, for switched control systems, corresponding sufficient conditions are presented under which constant/switching direct/observer-based state feedback controller can be designed such that the corresponding closed-loop systems are asymptotically stable under some periodically switching signal. Some numerical examples are given to illustrate our results.     

An application of maximal dissipative sets in control theory

A model of a distributed-boundary control system is considered. Assume the uncontrolled system possesses an exponential asymptotically stable zero solution. We then construct suboptimal feedback controls for the distributed and boundary control problems via the direct method of Liapunov. Furthermore, existence-uniqueness of the synthesized control systems is proven by applying the theory of nonlinear semigroups and maximal dissipative sets. Applications to diffusion equations are given.     

Memory feedback controller design for stochastic Markov jump distributed delay systems with input saturation and partially known transition rates

This paper considers the problem of stabilization for a class of stochastic Markov jump distributed delay systems with partially known transition rates subject to saturating actuators. By employing local sector conditions and an appropriate Lyapunov function, a state memory feedback controller is designed to guarantee that the resulted closed-loop constrained systems are mean-square stochastic asymptotically stable. Some sufficient conditions for the solution to this problem are derived in terms of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.     

Robust variance-constrained control for a class of continuous time-delay systems with parameter uncertainties

In this paper, we consider the robust variance-constrained control problem for uncertain linear continuous time-delay systems subjected to parameter uncertainties. The purpose of this multi-objective control problem is to design a static state feedback controller that does not depend on the parameter uncertainties such that the resulting closed-loop system is asymptotically stable and the steady-state variance of each state is not more than the individual pre-specified value simultaneously. Using the linear matrix inequality approach, the existence conditions of such controllers are derived. A parameterized representation of the desired controllers is presented in terms of the feasible solutions to a certain linear matrix inequality system. An illustrative numerical example is provided to demonstrate the effectiveness of the proposed results.     

具有动态边界的非均匀Euler-Bernoulli梁的边界反馈镇定

本讨论一端带有重物的Euler-Bernoulli梁的边界反馈镇定问题。在生物的质量忽略不计而只考虑重物的转动惯量的情况下，证明了同时在梁的自由端施加力和力矩反馈，闭环系统的能量可被指数镇定。进而对于系统只有力反馈或只有力矩反馈的情况，得到了闭环系统（指数）稳定的充分必要条件。     

具有边界控制的线性Timoshenko型系统的指数稳定性

研究多孔弹性材料在实际应用中的稳定性问题.多孔物体的动力学行为由线性Timoshenko型方程描述,这样的系统一般只是渐近稳定但不指数稳定,假定系统在一端简单支撑,另一端自由,在自由端对系统施加边界反馈控制,讨论闭环系统的适定性和指数稳定性.首先,证明了由闭环系统决定的算子A是预解紧的耗散算子、生成C0压缩半群,从而得到了系统的适定性.进一步通过对系统算子A的本征值的渐近值估计,得到算子谱分布在一个带域,相互分离的,模充分大的本征值都是A的简单本征值.通过引入一个辅助算子A0,利用算子A0的谱性质以及算子A与A0之间的关系,得到了A的广义本征向量的完整性以及Riesz基性质.最后利用Riesz基性质和谱分布得到闭环系统的指数稳定性.     

Guaranteed Cost Control for Uncertain Neutral Stochastic Systems via Dynamic Output Feedback Controllers

This paper deals with the problem of guaranteed cost control for uncertain neutral stochastic systems. The parameter uncertainties are assumed to be time-varying but norm-bounded. Dynamic output feedback controllers are designed such that, for all admissible uncertainties, the resulting closed-loop system is mean-square asymptotically stable and an upper bound on the closed-loop value of the cost function is guaranteed. By employing a linear matrix inequality (LMI) approach, a sufficient condition for the solvability of the underlying problem is obtained. A numerical example is provided to demonstrate the potential of the proposed techniques.     

具有终端不等式约束的线性二次控制问题

本文研究具有终端不等式约束的线性二次控制问题,得到了最优控制的反馈形式.所得到的反馈形式与相应的无约束问题的Riccati微分方程和一个函数矩阵的线性方程的解有关.     

Guaranteed cost control of time-delay chaotic systems

This article studies a guaranteed cost control problem for a class of time-delay chaotic systems. Attention is focused on the design of memory state feedback controllers such that the resulting closed-loop system is asymptotically stable and an adequate level of performance is also guaranteed. Using the Lyapunov method and LMI (linear matrix inequality) framework, two criteria for the existence of the controller are derived in terms of LMIs. A numerical example is given to illustrate the proposed method.     

慢变线性切换系统的镇定(英文)

The stabilization problem of systems that switch among a finite set of slowly varying linear systems with arbitrary switching frequency is discussed.It is shown that if the entries of the pointwise stabilizing feedback gain matrix are continuously differentiable functions of the entries of the system coefficient matrices,then the closed-loop system is uniformly asymptotically stable if the rate of time variation of the system coefficient matrices is sufficiently small.     

Decentralized Reliable Control of Interconnected Time-Delay Systems Against Sensor Failures

In this paper, we study the problem of designing decentralized reliable feedback control methods under a class of control failures for a class of linear interconnected continuous-time systems having internal subsystem time-delays and additional time-delay couplings. These failures are described by a model that takes into consideration possible outages or partial failures in every single actuator of each decentralized controller. The decentralized control design is performed through two steps. First, a decentralized stabilizing reliable feedback control set is derived at the subsystem level through the construction of appropriate Lyapunov-Krasovskii functional and, second, a feasible linear matrix inequalities procedure is then established for the effective construction of the control set under different feedback schemes. Two schemes are considered: the first is based on state-measurement and the second utilizes static output-feedback. The decentralized feedback gains in both schemes are determined by convex optimization over linear matrix inequalities. We characterize decentralized linear matrix inequality-based feasibility conditions such that every local closed-loop subsystem of the linear interconnected delay system is delay-dependent robustly asymptotically stable with an γ-level ℒ₂-gain. The developed results are tested on a representative example.     

GLOBAL ATTRACTIVITY IN A PERIODIC COMPETITION SYSTEM WITH FEEDBACK CONTROLS

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具有两个加性输入时滞系统的状态反馈控制

主要考虑了具有两个加性输入时滞的网络化控制系统的状态反馈控制问题.首先考虑具有两个加性时滞的系统稳定性,构造适当的Lyapunov泛函,不需要所有的矩阵正定,只需其整体正定.其次在估计Lyapunov泛函沿系统的导数时,针对时滞范围的不同,引入了不同的松弛矩阵,得出系统保守性较小的稳定条件.在此基础上,研究了系统状态反馈控制,最后得出闭环系统渐近稳定的充分条件,并给出控制器的设计方法.     

Positive real control for 2-D discrete delayed systems via output feedback controllers

This paper considers the problem of positive real control for two-dimensional (2-D) discrete delayed systems in the Fornasini–Marchesini second local state-space model. Attention is focused on the design of dynamic output feedback controllers, which guarantee that the closed-loop system is asymptotically stable and the closed-loop transfer function is extended strictly positive real. We first present a sufficient condition for extended strictly positive realness of 2-D discrete delayed systems. Based on this, a sufficient condition for the solvability of the positive real control problem is obtained in terms of a linear matrix inequality (LMI). When the LMI is feasible, an explicit parametrization of a desired output feedback controller is presented. Finally, we provide a numerical example to demonstrate the application of the proposed method.     

Some results for impulsive problems via Morse theory

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects.     

Finite spectrum assignment and complete damping of a differential system of the neutral type by a single controller

For a linear autonomous system of the neutral type with commensurable delays, we construct a dynamic feedback that ensures a complete damping of the original system and an asymptotically stable finite spectrum of the closed-loop system. The results are illustrated by examples.