Stability of a class of coupled non-linear time-varying systemsf†

The paper describes a two-stage method for the design of proportional-derivative output feedback for pole placement in multivariable systems. In the first stage, a number of poles are assigned by means of proportional output feedback. In the second stage, the assigned poles are preserved and a number of additional poles are positioned using proportional-derivative output feedback. The method is extended to the design of proportional-multiple-derivative output feedback and is illustrated by a numerical example.     

Pole assignment using dynamic output feedback compensators

The design of dynamic output feedback compensators for the pole assignment of linear time-invariant multivariable systems is considered. A systematic procedure is developed for the synthesis of the transfer function matrix of the compensator. A certain number of poles can be arbitrarily assigned by the proposed compensator. Moreover, in order to assign more poles using a compensator of the same order as before, a new approximate assignment approach based on the least-square-error algorithm is proposed. A numerical example is given to demonstrate the effectiveness of the proposed approaches.     

A parametric solution to the pole assignment problem using dynamicoutput-feedback

A technique is presented for pole placement of linear time-invariant systems using dynamic feedback. A previously developed method for partial pole assignment using constant feedback is generalized to the dynamic output-feedback case. Subject to a mild assumption on the number of complex conjugate poles to be assigned, it is almost always possible to arbitrarily assign all the closed-loop system poles using a compensator of order [(n-φ)/max(m,l)] using this new method. Here, n, m, and l are the order of the system, and the number of inputs and outputs, respectively, and φ Δ/=max(m,l)+[max(m,l)/2]+…+[max(m,l)/min(m,l)] where [x] denotes the nearest integer lower than or equal to x (i.e., floor (x)), and [x] denotes the nearest integer greater than or equal to x (i.e., ceiling (x)). An equivalent result is that using a compensator of order q, it is almost always possible to arbitrarily assign min(n+q,(max(m,l)+1)q+φ) closed-loop system poles. Only the normal procedures of linear algebra are required to implement the technique. Note that φ⩾l+m-1 and, therefore, the result is stronger than previous exact pole assignment results. Since it does not involve iteration or any other numerical techniques, it is possible to implement the method symbolically and, therefore, to obtain general parametric solutions to the pole assignment problem. The freedom in this design approach can also often be used to guarantee the internal stability and/or robustness of the resulting closed-loop system     

Pole assignment using dynamic compensators with prespecified poles

The paper describes a simple method for the design of physically realizable dynamic output feedback compensators with prespecified poles to achieve pole assignment in single-input, single-output and multivariable systems. In this method, the numerator parameters of the compensator transfer function are used to position the poles of the augmented closed-loop system consisting of the plant and the compensator at specified locations in the complex plane. It is shown that some or all poles of the closed-loop system can be positioned arbitrarily depending on the order of the compensator. The method is first developed for single-input systems. It is then shown to apply directly to single-output systems and is extended to multivariable systems by restricting the compensators to have unity rank. A number of illustrative numerical examples are given     

输出反馈极点配置的一种新方法

有关文献已证明,完全可控可观系统(ABC)通过定常输出反馈K,至少可配置max(m,r) 以及几乎总可以配置min(n,m+r-1)个极点任意接近事先指定的对称分布值,并给出了 配置的方法--并矢法.本文给出了输出反馈系统配置max(m,r)和min(n,m+r-1)个 互异不属于σ(A)的极点的非并矢方法,并矢法是其特殊情况.此方法也适用于状态反馈系统 及带动态补偿器输出反馈系统的极点配置.     

输出反馈极点配置的直接方法

本文研究了线性时不变能控能观系统x=Ax+Bu,y=Cx应用输出比例反馈和动态 补偿器任意配置闭路极点问题.文中借助于[sI-A]-1B矩阵的右既约分解矩阵,将闭路系统 特征多项式表示成p×p维矩阵行列式表示式,基于这一表示式建立了计算反馈矩阵和设计 动态补偿器的简单、实用的新方法.证明了应用输出比例反馈和动态补偿器可任意配置闭路 极点数分别为η≤min{max {m+(p-1) [m/p], p+(m-1) [p/m],n}和η0≤min {v+max {vm+m+(p-1)[m/p],vp+p+(m-1)[p/m]},n+v} (其中n和v分别为控制对 象和动态补偿器的阶数,p=rankB,m=rankC),文章最后举例说明了这种方法的应用.     

Guaranteed dominant pole placement with PID controllers

Pole placement is a well-established design method for linear control systems. Note however that with an output feedback controller of low-order such as the PID controller one cannot achieve arbitrary pole placement for a high-order or delay system, and then partially or hopefully, dominant pole placement becomes the only choice. To the best of the authors’ knowledge, no method is available in the literature to guarantee dominance of the assigned poles in the above case. This paper proposes two simple and easy methods which can guarantee the dominance of the two assigned poles for PID control systems. They are based on root locus and Nyquist plot respectively. If a solution exists, the parametrization of all the solutions is explicitly given. Examples are provided for illustration.     

Some new bounds related to output feedback pole placement

A number of new results regarding linear output feedback compensation are presented. In particular, it is shown that the rank of an appropriately defined real matrix Ω represents an upper bound on the number of closed-loop poles which can be completely and arbitrarily assigned via constant gain output feedback. A new bound on the minimum number of dynamical elements required for complete and arbitrary closed-loop pole placement is also defined in terms of the observability index of a certain single-input system.     

Minimal-order compensators for decoupling and arbitrary pole placement in linear multivariable systems

Linear multivariable systems that can be decoupled by state-variable feedback are considered. The problem of arbitrary pole placement and/or decoupling by means of output feedback via a dynamic compensator is first obtained through algebraic decoupling theory, Then an optimization problem in the free compensator parameters is derived by defining a performance index that measures both the amount of coupling in the closed-loop system and, when pole placement is also desired, the error in the closed-loop pole locations. The theory necessary for solving this optimization problem by means of standard gradient algorithms is developed. Minimal-order solutions are obtained because the compensator structure considered is quite general. Two illustrative numerical examples are presented.     

带有超大型挠性网状天线航天器姿控系统的参数化多目标设计

本文研究了带有超大型挠性附件的卫星的姿态控制问题.关于挠性航天器振动抑制与姿态控制,绝大多数已有的控制方法都是针对某一单一指标提出的.然而工程上要同时兼顾精度、快速性、平稳性、挠性部件的振动抑制以及各种鲁棒性,因此挠性航天器的姿态控制系统设计实际上是一个典型的多目标设计问题.本文针对具有超大挠性网状天线卫星的俯仰通道姿态系统,提出了一种基于输出反馈的鲁棒极点配置的参数化多目标设计方法.首先给出了系统能控与能观的充分必要条件,然后给出了动态补偿器以及特征向量矩阵的参数化表达,并在此基础上进一步对自由参向量进行了多目标优化,使得控制系统具有:1)配置到期望区域的极点; 2)较低的特征值灵敏度;3)较强的高阶未建模动态抑制能力; 4)尽可能小的控制增益.最后,本文根据卫星工程参数进行了控制器设计与仿真验证,仿真结果表明本文提出的方法可以在动态响应、高阶未建模动态抑制能力、控制量峰值等方面优于传统的PID控制器.     

Decoupling and arbitrary pole placement in linear systems using output feedback

Linear multivariable systems are considered which can be decoupled by state-variable feedback, but arbitrary pole placement and decoupling cannot be achieved simultaneously. These systems cannot be decoupled using output feedback. A compensator is presented which allows decoupling and arbitrary pole placement in the augmented system using output feedback.     

极点配置问题线性可解的充要条件

本文研究线性系统在输出反馈作用下的极点配置问题,得到了系统可由"线性化"方法任 意配置n个(等于系统的维数)闭环极点的充要条件.并给出了一种新的极点配置方法.     

Pole assignment for linear periodic systems by memoryless outputfeedback

We consider linear periodic discrete-time systems. We are interested in the problem of placing the poles of the monodromy map by means of periodic output feedback of the same or multiple period. It is well known that, in general, the poles of time-invariant systems cannot be assigned by constant output feedback. This is in contrast with what can be obtained in the context of time-variant systems. The main contribution of this paper is that periodic output feedback suffices for pole placement of periodic systems     

输出反馈配置极点的新方法

本文定义了系统的最大输出均匀分布指数t_m。给出了利用输出反馈配制闭环极点的新方法,对几乎所有的系统,能任意近似配置的极点个数为min[n,(m-1)t_m+r]。     

Pole assignment with output feedback

For pole assignment with output feedback, appropriate partitioning of the coefficient matrix of the system numerator transfer function matrix leads to simplified design equations to solve for the vectors of the unity rank output feedback matrix. This concept is applied to derive the pole placement equations for the proportional-derivative output feedback dynamic compensator. The simplicity of the design procedure is illustrated with a numerical example.     

Stabilization of Single‐Input LTI Systems by Proportional‐Derivative Feedback

This article presents an efficient solution to the stabilization pole placement problem for single‐input linear time‐invariant (LTI) systems by proportional‐derivative (PD) feedback. For a controllable system, any arbitrary closed‐loop poles can be placed in order to achieve the desired closed‐loop system performance. Its derivation is based on the transformation of linear system into Hessenberg form by a special coordinate transformation before solving the pole placement problem. The available degrees of freedom offered by PD feedback are utilized to obtain closed‐loop systems with small gains. So, the minimization problem for a suitably chosen cost function is formulated. Simulation results are included to show the effectiveness of the proposed approach.     

Pole assignment for linear time-invariant systems by periodic memoryless output feedback

It is well known that the poles of a linear time-invariant controllable and observable system can be assigned arbitrarily by state feedback. When only the output is available, pole assignment is still possible by means of dynamic output feedback. In this paper the potential of time-varying memoryless output feedback is considered. It is shown that, up to some technical conditions, it is indeed possible to allocate the poles of a linear time-invariant discrete-time system by memoryless output feedback with periodic gains. The period of the gains is (n + 1) with n the order of the system. The power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach.     

Robust pole assignment for linear control systems in a circular region using novel global harmony search algorithm

This paper presents a new approach to the problem of robust pole assignment in a circular region using novel global harmony search (NGHS) algorithm. Based on geometric principles, the position information of poles in the circular region is depicted and the rules of dynamic selection of poles from the circular region are determined. This ensures the algorithm select poles dynamically from the circular region. In order to get a set of poles and the state feedback controller which allow the system to have a maximum allowable perturbation or uncertainty, the upper bound of perturbation or uncertainty is optimized by the NGHS algorithm for the poles in the circular region. In contrast to most existing methods, an optimization method using NGHS algorithm for the dynamic selection of poles, makes the closed-loop system show better robustness. Finally, the simulation results demonstrate the effectiveness of the proposed approach.     

Structural properties and poles assignability of LTI singular systems under output feedback

This paper studies the structural properties of both finite poles and the infinite pole of linear time-invariant singular systems under output feedback. Three main problems are studied, namely, (1) the algebraic structures of both finite poles and the infinite pole; (2) the assignability of finite poles and the elimination of the infinite pole by output feedback; and (3) the controllability and observability of the system with minimal number of inputs and outputs. New generic solutions to these problems are presented in terms of some new concepts defined in this paper including the geometric multiplicity of the infinite pole, the finite and impulsive output feedback cycle indices of the system. Determination of these multiplicities and indices are discussed. An assignability equivalence is established between the variable finite poles and the poles of a controllable and observable non-singular system. The number of the independent infinite poles that can be reduced is given in terms of the system matrices. The minimal number of inputs and outputs that guarantee controllability and observability are shown to be the output feedback cycle indices.     

On constraints in pole assignment by static feedback

A design method based on pole assignment considerations for an nth order linear time-invariant single-input p-output system by means of static (constant gain) feedback is presented for the case in which the rank p of an output matrix C is less than the number of system states n. The constraints in pole assignment are described in the form of linear relationships between the coefficients of the closed-loop characteristic polynomial and some open-loop system parameters. It has been shown that the problem of pole assignment is equivalent to the problem of solving a set of n linear equations with p unknowns in such a way that n – p equations are made linearly dependent by proper choice of coefficients of the closed-loop characteristic polynomial. As a result, all n system poles can be shifted to selected locations which satisfy the constraints. Of particular interest are the design methods for the cases when the number of arbitrarily assigned poles are equal to p and p – I.