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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 796143, 13 pages
http://dx.doi.org/10.1155/2010/796143
Research Article

The Robust Pole Assignment Problem for Second-Order Systems

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 13 April 2010; Accepted 12 October 2010

Academic Editor: J. Rodellar

Copyright © 2010 Hao Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Pole assignment problems are special algebraic inverse eigenvalue problems. In this paper, we research numerical methods of the robust pole assignment problem for second-order systems. The problem is formulated as an optimization problem. Depending upon whether the prescribed eigenvalues are real or complex, we separate the discussion into two cases and propose two algorithms for solving this problem. Numerical examples show that the problem of the robust eigenvalue assignment for the quadratic pencil can be solved effectively.

1. Introduction

Pole assignment problems are special algebraic inverse eigenvalue problems [1, 2]. The nature model for the vibrating systems, arising in a wide range of applications, especially in the design and analysis of vibration structures, such as bridges, buildings, and airplanes, can be described by the second-order differential equation. The properties of systems of second-order differential equation are governed by its associated quadratic eigenvalue problem (QEP). To avoid unwanted oscillations of the vibratory system, pole assignment can change the poles of the system by choosing a control force and improve its stability.

Consider the following second-order matrix differential equation: where the dots denote differentiation with respect to time, and and are real symmetric matrices; is positive definite (denoted by ). Separation of variables where , is a constant vector, in (1.1), leads to the following quadratic eigenvalue problem: where is quadratic pencil. In general, the eigenvalues of are named poles of system (1.1). In engineering, the dynamics of equation (1.1) can be modified by applying a control force , where . The relation (1.1) now becomes The special choice where and are real matrices, is called state feedback control, and (1.1) becomes The associated quadratic eigenvalue problem becomes where is a closed-loop pencil.

The quadratic eigenvalue assignment problem is stated explicitly as follows.

Problem QEA
Given real matrices and , and a set of complex numbers , closed under complex conjugation, find real matrices such that the eigenvalues of are equal to .

Conditions for the existence of solutions to problem QEA are known as in the following theorem.

Theorem 1.1 (see [3]). Solutions to problem QEA exist for every set of self-conjugate complex numbers if and only if the system (1.1) is completely controllable, that is

In a realistic situation, it is desirable to choose the feedback to ensure that the eigenstructure of closed-loop system is as robust, or insensitive to perturbation in the system matrices and , as possible to the following inverse eigenvalue problem, known as the robust quadratic eigenvalue assignment problem.

Problem RQEA
Given real matrices and a set of as in problem QEA, find real matrices and matrix such that is nonsingular, satisfying and the eigenstructure of the closed-loop system (1.6) is as robust as possible.

We remark that the requirement that the matrix is diagonal, together with the invertibility of . These require the prescribed eigenvalues to be distinct. In the next section, we derive conditions for the solution of problem RQEA.

In the majority of methods that have been proposed for solving problem RQEA, the second-order control system (1.1) is rewritten as a first-order system. There are two difficulties in using this approach. The first is the linear system which has a double dimension of the original quadratic system and, hence, the computational work used to solve the problem is greater than necessary. The second difficulty arises because all the exploitable properties such as definiteness, and sparsity, of the coefficient matrices and , usually offered by a practice problem, will be completely destroyed. So it is natural to wonder if solutions of the robust pole assignment problem can be obtained without resorting to a first-order reformulation.

Over the past years, many techniques for pole assignment without linearization have been proposed. Chu and Datta [4] gave a method to solve Problem RQEA. Nichols and Kautsky [5] recently have proposed a numerical method to solve this problem without linearization; the measures of robustness are subject to structured perturbations.

Datta and Sarkissian [6] proposed a direct partial modal approach to solve the partial eigenvalue assignment problem for second-order systems. It is “direct,” because the solutions are obtained directly in the second-order system without any types of reformulations; It is “partial modal,” because only a part of spectral data is needed for the solution. This method do not, however, ensure the robustness of the closed-loop system. Qian [7], Qian and Xu [8] recently have proposed a direct method to solve the robust partial pole assignment problem for second-order systems which seems more efficient and reliable. Bai et al. ([9]) gave a new optimization approach for solving this problem.

According to new measures ([10, 11]), we suggest in this paper two numerical methods for solving the problem RQEA. Numerical results show that this problem can be solved effectively.

We begin by presenting the solutions to problem RQEA without linearization. In Section 3, we describe a new measure of robustness and formulate the problem as an optimization problem. Two numerical methods are developed in Section 4, and numerical results are given in Section 5.

2. Solution to Problem RQEA

Without loss of generality, We assume in this paper that the system (1.1) is completely controllable, and is of full column rank. The next theorem provides necessary and sufficient conditions for the existence of solutions to problem RQEA.

Theorem 2.1 (see [5]). Let be such that is nonsingular, where , then there exist real matrices , satisfying condition (1.9) of problem RQEA if and only if where with orthogonal and nonsingular. The matrices are given by

An immediate consequence of Theorem 2.1 is the following.

Corollary 2.2. Let be the right eigenvector of corresponding to the prescribed eigenvalue , then where denotes right nullspace.

For every , find an orthogonal basis, comprised by columns of matrix for the space . Observe that the matrix can be obtained by the QR decomposition of .

Theorem 2.3. If the system (1.1) is completely controllable, then where denotes the dimension of space .

Proof. From Theorem 1.1, we can get then So we can have , from which (2.5) easily follows.

According to the previous theory, we can write the following: where , and if , then . Because in the case where is complex the associated eigenvector is a complex vector, in order to ensure that the computed feedback matrices are real, the eigenvector corresponding to the conjugate eigenvalue must be taken to be the conjugate vector .

3. The Measures of Robustness

In this section, we present the measures of robustness for the second-order system. The eigenvectors of the closed-loop system can be selected to minimize the measure of robustness.

Consider a matrix , if is nondefective, namely, then there exists a nonsingular matrix , such that where , let then the sensitivity of the eigenvalues of of to perturbations in the components of depends upon the magnitude of the condition number , where Hence, every reasonable measure of the magnitude of the vector is a reflection of the robustness of the system.

If is a multiple eigenvalue of and then the sensitivity of eigenvalue depends upon the magnitude of the number Therefore, it is natural to take as global measure of the sensitivity of eigenvalue.

In essence, the aim of the robust eigenvalue assignment problem is to select eigenvectors , such that and the vectors are as orthogonal as possible to each other. Therefore, in this paper we first consider the following measure : where

Observe that if is nonsingular, then the QEP (1.7) can be formulated as the following standard eigenvalue problem: Let be the matrix comprised by the right eigenvectors of (3.9), then By (2.8), we have From (3.7) it follows that where Thus, we must solve an unconstrained optimization problem where and are defined by (3.12) and (3.13), and .

4. Numerical Methods

Unfortunately, it is difficult to solve the optimization problem (3.15). Depending upon whether the prescribed eigenvalues are real or complex, we separate the discussion into two cases.

Case 1. Assume that the prescribed eigenvalues are real.
In this case, where the eigenvectors are real vectors, (3.12) becomes Let where , then minimizing the measure may be reduced to solving the following nonlinear programming problem with constrains:

Let we can prove easily that the programming problem (4.3) is equivalent to the following programming problem with inequality constrains:

Let We consider a multiparameter eigenvalue problem Using the Kuhn-Tucker optimality ([12, 13]), we can prove that every solution to problem (4.5) is necessarily a solution to problem (4.7).

In this section, we develop an algorithm, called Horst algorithm ([14]), to solve (4.7), for finding a global optimal solution of problem (4.3).

Algorithm 1. Initialization Step
Let be the termination scalar. Given choose an initial vector .

Main Step
    Find the decomposition (2.2) of and an orthonormal basis, comprised by the columns of the matrix , for the subspaces , defined by (2.4). Let .
For , we compute
Set . If then is an approximation optimal solution, go to if , then replace by and repeat step
Let , where is defined by (2.8), and construct feedback matrices by solving

Case 2. Assume that prescribed eigenvalues are complex.
In this case, it is almost impossible to solve optimization problem (3.15) without costing too much. Based on this ideal, we consider , where is defined by (3.10), and use some similar techniques developed by Kautsky et al. in 1985 ([15]).
Obviously, achieves the minimum if and only if is unitary.

Let then . Let the columns of matrix comprise an orthonormal basis, for the subspace , and and let be a normalized vector orthogonal to . The objective here is to choose vectors such that each vector is as orthogonal as possible to the space . Observe that orthogonal to is equivalent to choose such that the angle between and is minimized.

Obviously, must satisfy The coefficient matrix of equation (4.12) is a matrix, so we can easily solve from equation (4.12), and let be the normalized projection of onto the space , then This method is then to sweep through the columns of replacing the th column in turn with the normalized projection . Because the prescribed eigenvalue is complex, two columns need to be altered simultaneously. Repeat the previous procedure until satisfying the stopping criterion or reaching the maximal amount of iteration. We can choose as the stopping criterion.

It is worthwhile to point out that the procedure is not guaranteed to converge, that is, the criterion (4.14) may not be satisfied. To ensure the end of iteration, we need to set a maximal amount of iteration . Now we develop an algorithm, called orthogonal vector method, for solving the problem RQEA.

Algorithm 2.
Initialization Step
Let be the termination scalar, is the maximal amount of iteration. Given and .

Main Step
Find the decomposition (2.2) of and an orthonormal basis, comprised by the columns of the matrix , for the subspaces , defined by (4.10).
Select an initial matrix , defined by (3.10), such that and is nonsingular. Let .
If , do for  compute the solution of equation (4.12) and normalize ;  compute ;  compute ;  update the th column of with ; compute .  If , then go to else continue;
let ; repeat until convergence.
let the first rows of be construct feedback matrices by solving

Although this method does not guarantee convergence, it is simple to implement, and numerical results show that it often leads to better conditioned closed-loop systems.

5. Numerical Results

To illustrate the performance of the present algorithms, in this section we give some numerical examples, which were carried out using MATLAB 6.1.

Example 5.1. In the first example we examine a case from [5], where the matrix is very ill conditioned. The system matrices and are defined by The system is undamped and the open-loop eigenvalues are The desired closed-loop eigenvalues are given by

With Algorithm 1, we choose an initial vector with and , and take . The corresponding matrix has condition number , and after three iterations, it is reduced to . The computed feedback matrices are given by

Example 5.2. In the second example, we examine a case from [16]. The system matrices and are defined by The open-loop eigenvalues are The desired closed-loop eigenvalues are given by

With Algorithm 2, we choose , and the initial matrix is generated by a random selection of vectors from each subspace. The corresponding matrix has condition number , and after eight iterations, it is reduced to . The computed feedback matrices are given by

Acknowledgment

This research was supported by the National Natural Science Foundation of China under Grant no. 10926130.

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