Pole assignment for uncertain systems in a specified disk by statefeedback

This paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system, the uncertainty being norm bounded. For this the “quadratic d stabilizability” concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived. This condition expressed as a parameter dependent discrete Riccati equation enables one to design the control gain matrix by solving iteratively a discrete Riccati equation     

Pole assignment for uncertain systems in a specified disk by output feedback

In this paper the problem of pole assignment in a disk by output feedback for continuous-or discrete-time uncertain systems is addressed. A necessary and sufficient condition for quadratic d stabilizability by output feedback is presented. This condition is expressed in terms of two parameter-dependent Riccati equations whose solutions satisfy two extra conditions. An output d stabilization algorithm is derived and a controller formula given.     

Pole assignment in a specified circular region using a bilinear transformation onto the unit circle

An algorithm for pole assignment in a specified circular region using a bilinear transformation onto the unit circle is considered, A Lyapunov type test is developed to check that all the eigenvalues of a discrete system are within a circle in the complex plane. Pole placement in a circular region by LQ state feedback design is investigated for discrete linear systems. Perturbation bounds are derived, using bilinear transformation, for all closed-loop poles of a perturbed discrete system to remain inside a specified circular region. Illustrative examples are included     

Pole assignment problem: a structural investigation

Based on the structure of a closed-loop system under a specified feedback pattern, a qualitative analysis of the problem of pole assignability is considered. The problem is first formulated algebraically, in terms of the relation p = g(f) between the vector p of the closed-loop characteristic polynomial coefficients and the vector f of the non-zero elements of the feedback matrix. Then, translation to the structural framework is achieved by means of two theorems which give graph-theoretical sufficient conditions for solvability. These structural conditions also guarantee genericity of pole assignability.     

Pole assignment in real Schur form

In this paper an algorithm for pole assignment in real Schur form by gain state feedback is given. The proposed algorithm shifts, at each iteration, 1 or 2 poles in the last diagonal block of the closed-loop system. Only the last few columns in real Schur form are altered and therefore the Schur form is preserved at each iteration. After the completion of each iteration, the orthogonal similarity transformation is applied so that a new block is moved to the last diagonal block and the iterations continue. The desired feedback matrix K at the end of the algorithm is found to be the sum of the component matrices, each of which assign one or two poles.     

Pole assignment with robust stability

This paper uses convex analysis for the pole assignment design of discrete-time SISO systems incorporating robust stability against norm bounded parametric perturbations in the plant transfer function. The method involves designing an overparameterized pole assignment controller for the nominal plant with the overparameterization chosen to reduce the size of changes in the closed-loop characteristic equation which result from plant perturbations. Sufficiency bounds on the l_p norm of the perturbation guaranteeing stability are obtained     

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.     

Robustness of pole assignment in a specified region for perturbed systems

This paper deals with the pole location of a perturbed matrix A + E. Two perturbation classes are discussed: one with structured perturbations |E| ≤ U (a given non-negative matrix) and one where only the spectral norm ||E|| is given. Sufficient conditions are derived for the eigenvalues of A + E to be located within a circle C(α, r). The case of a diagonalizable matrix A is also considered and a sufficient condition for robust stability of a class of perturbed discrete systems is derived. Numerical examples are presented for illustration.     

区域极点配置问题的研究方法

对区域极点配置的现有研究成果进行综述.将区域极点配置的基本方法归结为代数Riccati方程(ARE)方法和线性矩阵不等式(LMI)方法.列出了基于区域极点配置方法的主要研究成果,包括最优控制、鲁棒性、H2性能、H∞性能等方面.最后给出了几点研究展望.     

Pole assignment using matrix generalized inverses

The problem of pole assignment in a completely controllable linear time-invariant system dx/t = Ax + Bu, y = Cx is considered. A method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement. The method does not require solution of sets of non-linear equations or manipulation of polynomial matrices, and no knowledge of eigenvalues and/or eigenvectors of A is necessary. If the computed matrix K and the given matrix C satisfy a consistency condition, a matrix Kν such that KνC = K can be directly obtained from K and the desired pole-placement can be realized by an output feedback law u = r + Kνy.     

Pole assignment by state transition graph

The pole assignment problem is considered, using the graph representation of a matrix. The parametrization of controllers which have a specified characteristic polynomial is given. A simple algorithm based on graphs is presented and two examples are given     

Pole assignment with optimal spectral conditioning

This paper presents a novel method for pole assignment with robustness measured in terms of the spectral condition number of the closed-loop eigenvector matrix. It is established that the spectral condition number can be minimized asymptotically via a sequence of unconstrained minimizations on some auxiliary objective functions. Moreover, the sequence of minimizers converges to a minimizer of the spectral condition number. A numerical algorithm with analytical formulas of the gradient of the auxiliary objective functions is provided. The efficiency and effectiveness of the approach is demonstrated via an example.     

Pole assignment for systems over rings

A very simple proof of the pole assignment theorem for systems over a principal ideal domain (and other rings) is given. Furthermore, an algorithm is presented. Extensions are also indicated.     

Pole assignment using full-rank output-feedback compensators

An algorithm is presented for the systematic design of full-rank output-feedback compensators which allow arbitrary pole assignment in the resulting closed-loop system. The desired compensator of least degree is constructed from a minimal sequence of dyads. The underlying theory provides a unification of previous results.     

极点配置固定滞后稳态Kalman平滑器

基于稳态Kakman预报器和白噪声估计理论, 应用控制理论中的极点配置原理, 提出了极点配置固定滞后稳态Kalman平滑器. 它们不仅是全局渐近稳定的, 而且通过配置平滑器的极点可使初始平滑估值的影响按指数衰减迅速消失. 它们避免了计算最优初始平滑估值, 可减小计算负担. 一个雷达跟踪系统的仿真例子说明了其有效性     

Pole assignment via the schur form

We propose an algorithm for the state-feedback pole assignment problem. The algorithm is the first of its kind, making direct use of the Schur form, and minimizing the departure from normality of the closed-loop poles for a given first Schur vector x₁. The robust pole assignment problem can then be solved via choosing x₁ optimally. Several numerical examples were presented to illustrate the feasibility of the algorithm.