# Parametric Eigenstructure Assignment Submission

^{1}School of Mechanical Engineering, Wuhan Polytechnic University, Wuhan 430023, China^{2}Institute of Electrical and Mechanical Engineering, Beijing Information Science and Technology University, Beijing 100101, China

Copyright © 2015 Zhigang Hu et al. 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.

A new approach for the partial eigenvalue and eigenstructure assignment of undamped vibrating systems is developed. This approach deals with the constant output feedback control with the collocated actuator and sensor configuration, and the output matrix is also considered as a design parameter. It only needs those few eigenpairs to be assigned as well as mass and stiffness matrices of the open-loop vibration system and is easy to implement. In addition, this approach preserves symmetry of the systems. Numerical example demonstrates the effectiveness and accuracy of the proposed approach.

#### 1. Introduction

Eigenstructure assignment techniques based on active feedback control have attracted much attention and have been widely used for vibration suppression during the past three decades. The extensive research results can be found in review articles [1, 2] and references therein. As the dynamics of a structural system is naturally described by a set of second-order differential equations and control theory and estimation techniques are established for first-order realization of the systems, the majority of previous researches are made via transferring second-order equations to first-order configuration (see, e.g., [3]). In the past ten years, in order not to increase the dimension of the equations and to preserve the symmetry and sparsity of the structural matrices, many authors proposed their works based directly on the second-order equation models [4–12]. On the other hand, it is needed to change only few undesirable eigenvalues and/or corresponding eigenvectors which are purposefully assigned to desired values, and it is desirable to keep all other eigenpairs unchanged. So some methods have been proposed to implement the partial eigenvalue or eigenstructure assignment [13–18]. For these methods, the process of applying a control, whether state feedback or output feedback, usually produces a closed-loop matrix which is no longer symmetric. However, it is sometimes necessary for the closed-loop system to satisfy the reciprocity principle of the structure. For systems with this requirement one restriction is that the available controls may need to be symmetric, as indicated in [19].

In this respect, Elhay [19] used the symmetric rank-one matrix modification and derived an explicit solution for a symmetry preserving partial eigenvalue assignment method for the generalised eigenvalue problem. But the method is difficult to implement in feedback control. Ram [20] solved the eigenvalue assignment problem for the vibrating rod. More recently, Liu and Li [21] suggested a method for the symmetry preserving partial eigenvalue assignment of undamped structural systems. They adapted the method proposed in [22] to the requirement of the symmetry preserving. Their results involve complex mathematical calculation. In this paper we propose a new approach for the symmetry preserving eigenstructure assignment of undamped vibrating systems, which is also applicable to the partial eigenvalue assignment. This approach is concerned with the constant output feedback control with the collocated actuator and sensor configuration and uses a partial eigenstructure modification formulation for the incremental mass and stiffness matrices to be satisfied. This formulation was recently obtained by Zhang et al. [18]. Our method proposed in this paper is easy to implement and the calculation procedure is relatively simple and clear.

The rest of the paper is organized as follows. Section 2 presents some preliminaries and the problem description. Section 3 gives our approach. Section 4 provides a numerical example to show the accuracy and effectiveness of the proposed approach. Finally, conclusions are drawn in Section 5.

#### 2. Preliminaries and Problem Statement

##### 2.1. Preliminaries

Consider an -degree-of-freedom undamped vibration system that is modelled by the following set of second-order ordinary differential equations:where is displacement vector, is the vector of external forces, and are constant mass and stiffness matrices, respectively. In general, is symmetric and positive definite, and is symmetric and positive semidefinite; that is, , .

It is well known that if is a fundamental solution of (1), then the natural frequency and the mode shape vector must satisfy the following generalised eigenvalue equation:where is the square of the th natural frequency , called the th eigenvalue, and is the corresponding th eigenvector. Equation (2) can be written in a compact representation as follows:where and make up the complete eigenstructure of system (1) and satisfies the mass-normalised condition .

Suppose that the system described by (1) is modified by the incremental matrices and . Then the motion of the modified system is governed byand it satisfies the following eigenmatrix equation:where and are the complete eigenstructure of modified system (4).

In [18] a necessary and sufficient condition was proposed for the incremental mass and stiffness matrices that modify some eigenvalues or eigenpairs while keeping other eigenpairs unchanged, which is shown in the following:where and are submatrices of and and are composed of eigenvalues and eigenvectors to be modified in system (1), respectively. It implies that if and satisfy (6), the following equation then holds:where and are submatrices of and and are composed of unchanged eigenvalues and eigenvectors of system (1). Here and . Equation (7) means that and are also eigenpairs of the modified system. When only the stiffness modification is concerned, (6) reduces towhich is crucial to address the partial eigenvalue or eigenstructure assignment problem by static output feedback control in this paper.

##### 2.2. Problem Statement

When considering the feedback control, we set in (1), where is full rank constant control matrix and the control vector. If the real constant static output feedbackis applied to system (1), where is an output feedback gain matrix to be determined and is the output or measurement vector, where is a real constant output matrix, then the closed-loop system becomeswhich is supposed to have desired eigenvalues or eigenpairs.

In this paper, we consider and as design variables as well. Moreover, a special situation, , is concerned. Namely, it means the use of collocated actuator and sensor pairs, and the number of output measurements is equal to the number of inputs . The problem of partial eigenvalue or eigenstructure assignment is then formulated as follows: assuming that system (1) with and is controllable and observable, given matrices and , the subset

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