Parametric state feedback design of linear distributed-parameter systems

The parametric approach for the design of state feedback controllers has been formulated so far only for linear lumped-parameter systems. It yields an explicit parametric expression for the state feedback gain given the closed-loop eigenvalues and the set of corresponding parameter vectors. This contribution presents a parameterisation of state feedback controllers for linear distributed-parameter systems with scalar state and distributed control. By introducing the closed-loop eigenvalues and the parameter vectors as design parameters, an explicit expression for the state feedback is obtained. In contrast to the pure eigenvalue assignment, the parameterisation allows the assignment not only of the closed-loop eigenvalues but also of the closed-loop eigenfunctions. The usefulness of the proposed parametric approach is demonstrated by decoupling the transfer behaviour of a MIMO diffusion system with respect to its dominant modes.     

ESA in high‐order linear systems via output feedback

This paper considers eigenstructure assignment in high‐order linear systems via output feedback. Parametric expressions for the left and right closed‐loop eigenvectors associated with the finite closed‐loop eigenvalues and two simple and complete parametric solutions for the feedback gain matrices are obtained on the basis of the parametric solutions of the generalized high‐order Sylvester matrix equations. This approach does not impose any restrictions on the closed‐loop eigenvalues. An illustrative example shows the effect of the proposed approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

鲁棒比例与部分状态导数反馈控制系统设计

基于广义线性系统比例与部分状态导数反馈参数化特征结构配置结果和矩阵对广义特征值灵敏度结果, 得到了关于开环系统矩阵中摄动元素的闭环特征值灵敏度的参数表达式, 并在此基础上提出了广义线性系统比例与部分状态导数反馈最小灵敏度特征值配置的有效算法. 该算法不含有"返回"过程, 允许闭环特征值在希望的区域内方便地参与优化. 一个算例说明了算法的简单性和有效性.     

Insensitive and robust control design via output feedback eigenstructure assignment

Insensitive and robust control design using output‐feedback eigenstructure assignment for linear multivariable systems is considered in this paper. A parametric expression of closed‐loop eigenvectors and generalized eigenvectors is developed. It can cope with the case where the closed‐loop eigenvalues are multiple and/or the same as the open‐loop ones so that the system to be designed can be uncontrollable and/or unobservable. The controller designed via output‐feedback eigenstructure assignment is expressed by proposed parameter vectors. The freedom provided by output‐feedback eigenstructure assignment is used to optimize some performance functions which are used to measure the sensitivity of the closed‐loop matrix and the robustness of the closed‐loop system. Copyright © 2000 John Wiley & Sons, Ltd.     

Minimum norm output feedback design under specified eigenvalue areas

Starting from a new parametric expression for the state feedback controller of a linear, time-invariant system a procedure is derived which reduces the design of output feedback controllers to an unconstrained optimization problem where all free parameters, both the closed-loop eigenvalues and the invariant parameter vectors, are used for its numerical solution by a gradient-based search technique. Thus minimal norm output feedback controllers can be designed that place the closed-loop eigenvalues within specified regions of the eigenvalue plane.     

Some necessary and sufficient conditions for eigenstructure assignment

Some necessary and sufficient conditions for closed-loop eigenstructure assignment by output feedback in time-invariant linear multivariable control systems are presented. A simple condition on a square matrix necessary and sufficient for it to be the closed-loop plant matrix of a given system with some output feedback is the basis of the paper. Some known results on entire eigenstructure assignment are deduced from this. The concept of an inner inverse of a matrix is employed to obtain a condition concerning the assignment of an eigenstructure consisting of the eigenvalues and a mixture of left and right eigenvectors.     

Closed-loop eigenstructure assignment by state feedback in multivariable linear systems with slow and fast modes

It is shown that the recent results of Moore (1976) and Porter and D'Azzo (1977) can be readily extended so as to provide a computationally attractive method of closed-loop eigenstructure assignment by state feedback in multivariable systems with ‘ slow ’ and ‘ fast ’ modes. The general results are illustrated by designing a state-feedback controller for a third-order system with ‘ slow ’ and ‘ fast ’ modes which assigns asymptotically the entire closed-loop eigenstructure as characterized by both ‘ slow ’ and ‘ fast ’ eigenvalues and eigenvectors.     

Parametric eigenstructure assignment by output-feedback control: the case of multiple eigenvalues

This paper deals with closed-loop eigenstructure assignment by output-feedback control in linear multivariable systems for the general case of multiple eigenvalues. The approach is algebraic in nature and operates directly on the closed-loop characteristic equation using fundamental properties of determinants and their derivatives. A compact form for the controller gain matrix is derived and expressed in terms of a set of parameter vectors that specify the non-uniqueness of the solution of the eigenvalue-assignment problem. Some of these parameter vectors assume restricted forms while others are free. A computational scheme is given in which the free parameter vectors are determined in such a way as to assign the corresponding eigenvectors and generalized eigenvectors in a weighted least-square-error sense. An illustrative numerical example is included.     

A parametric approach to finite-dimensional control of linear distributed-parameter systems

This article considers the design of finite-dimensional compensators for distributed-parameter systems using eigenvalue assignment. The proposed compensator consists of an observer estimating additional outputs and a static feedback of the measurable and the estimated outputs. Since the additional outputs can be asymptotically reconstructed, the compensator can be designed using the separation principle, i.e. the closed-loop eigenvalues are given by the observer eigenvalues and the eigenvalues resulting from the static output feedback control. In order to solve the corresponding eigenvalue assignment problem, the parametric approach for the design of static output feedback controllers in finite-dimensions is extended to distributed-parameter systems. By using a parameter optimisation it is possible to assign all closed-loop eigenvalues within specified regions of the complex plane in order to stabilise the system and to assure a desired control performance. A heat conductor is used to demonstrate the proposed design procedure.     

基于特征结构配置的最小灵敏度控制器设计

本文描述了一种多变量控制系统的最小灵敏度控制器的设计方法.该方法除将闭环系统 的特征值配到给定点外,还利用状态反馈所剩余的自由度,对闭环特征向量实行部分配置,以 满足其它设计指标,如低参数灵敏度和良好的响应特性等.设计举例和仿真表明,采用这种最 小灵敏度控制器,被控系统既能获得满意的瞬态响应,又能降低状态轨迹对参数扰动的灵敏 度.本设计采用的算法是以解矩阵Sylvester方程为主.     

Arbitrary pole placement with the extended Kautsky–Nichols–van Dooren parametric form

We consider the classic problem of pole placement by state feedback. The well-known eigenstructure assignment algorithm of Kautsky, Nichols, and van Dooren is extended to obtain a parametric formula for the pole-placing feedback matrix that can deliver any desired closed-loop eigenvalues, with any desired multiplicities.     

Gramian assignment based on the Lyapunov equation

A theorem on the assignability of the controllability Gramian using state feedback is presented. The approach is based on the Lyapunov equation. Specifically, it shows that the set of Gramian matrices which are assignable using state feedback lies in a vector space and is precisely the intersection of that vector space with the set of positive definite matrices. The results show that it may be possible to reshape the structure of the system indirectly by specifying a desired closed-loop Lyapunov function, rather than by directly assigning eigenvalues or eigenvectors     

Boundary feedback exponential stabilization of a Timoshenko beam with both ends free

In the present paper we consider the boundary feedback stabilization of a Timoshenko beam with both ends free. We propose boundary feedback control law that makes the closed loop system dissipative. Using asymptotic analysis techniques, we give explicit asymptotic formula of eigenvalues of the closed loop system, and prove the Riesz basis property of eigenvectors and generalized eigenvectors. By a detailed analysis of spectrum of the closed loop system, we show that the closed system is exponentially stable.     

基于特征结构配置的二阶线性系统鲁棒容错控制设计

研究基于特征结构配置的二阶线性系统鲁棒容错控制设计问题,目的是重新设计状态反馈控制律,使得故障闭环系统和正常闭环系统具有相同的特征值.两闭环系统的特征向量依最小二乘法接近,而且能通过极小化灵敏度指标提高系统的鲁棒性.基于状态反馈特征结构配置的参数化结果,将系统灵敏度指标优化问题转化为含有约束条件的优化问题,并提出了鲁棒容错控制设计方法.数值算例及其仿真结果验证了所提出设计方法的有效性.