A modified optimization method for robust partial quadratic eigenvalue assignment using receptances and system matrices
Introduction
We consider the following second-order feedback control system where M, C and K are all n-by-n matrices, t means time, is an n-vector, and the n-by-m real matrix B is a control matrix (), and is the control m-vector with the unknown n-by-m real feedback matrices F and G.
Such second-order control system arises in the active vibration control for many vibrating structures in structural engineering such as vibration control in structural dynamics [15], [24], [45], large flexible space structure control theory [6], [7], [28], [29], control of mechanical descriptor systems [34], damped-gyroscopic system control [25], earthquake engineering control [14], etc. In many engineering applications, M, C and K denote the mass, damping and stiffness matrices, which are obtained by using the finite element technique and are all real symmetric with M being positive definite and K being positive semi-definite, and denote the displacement, velocity and acceleration vectors, respectively.
Many vibrating structures may suffer dangerous vibrations (e.g., resonance) when they encounter some external forces (e.g., storm, turbulence and earthquake). These dangerous vibrations are generally generated when the external frequencies are close to the natural frequencies of vibrating structures. The dynamics of the system (1.1) are governed by the natural frequencies and mode shapes. Mathematically, this is related to the following quadratic eigenvalue problem [49]: where is called the open-loop quadratic matrix pencil, λ is called an eigenvalue of with associated eigenvector . If M is nonsingular, then has 2n finite eigenpairs . By applying the separated variable to (1.1), we obtain the closed-loop system where is called the closed-loop quadratic matrix pencil.
Let and stand for the set of all complex matrices and the set of all real matrices, respectively. The partial quadratic eigenvalue assignment problem (PQEAP) for the second-order control system (1.1) aims to find the feedback matrices such that the closed-loop pencil possesses the desired eigenvalues instead of a few unwanted eigenvalues () of the open-loop pencil , which may cause dangerous vibrations, while keeps the remaining large number of eigenpairs of the open-loop pencil unchanged (i.e., the no spill-over property is preserved). The robust and minimum norm PQEAP aims to find a solution to the PQEAP such both the sensitivity of the closed-loop eigenvalues and the feedback norms are minimized.
The PQEAP has been studied in second-order setting itself for practical engineering applications. Datta, Elhay and Ram [16] gave a solution to the PQEAP for the single-input case (). Since then, there are much work for solving the PQEAP with single-input and multi-input controls [12], [17], [18], [20], [40], [43]. For practical application, the robust and minimum norm PQEAP has been widely discussed [2], [8], [9], [11], [12], [13], [19], [20], [30], [31], [32], [39], [40], [51].
Recently, the measured receptances have been used for pole assignment problems and the PQEAP [4], [35], [36], [37], [41], [42], [44], [46], [47], [50]. In particular, the measured receptances and system matrices have been used for solving the robust and minimum norm PQEAP [1], [3], [5], [48], [50].
In this paper, we reformulate the robust and minimum norm PQEAP as an minimization problem. This is motivated by the recent papers [3], [4], [5]. In [5], an optimization method was presented for the robust and minimum norm PQEAP, where a new cost function was provided for measuring the closed-loop eigenvalue sensitivity. However, in this optimization problem, there exist additional equality constraints involving several determinants of linear functions of the parameter matrix. In [4], a parameterized solution to the PQEAP was provided, where only a small linear system was needed. In [3], a gradient-based optimization method was proposed for solving the minimum norm PQEAP. In this paper, by combining the idea in [5] and [4], an minimization problem is reformulated for solving the robust and minimum norm PQEAP, where the cost function is used to measure both the sensitivity of the closed-loop eigenvalues and the feedback norms. This is modified version of the optimization problem proposed in [5]. For the proposed minimization problem, there is no additional equality constraint. We propose a modified gradient-based optimization method for solving the minimization problem, where the explicit expression of the cost function is derived based on a small linear system-based parametric solution to the PQEAP in [4]. In our method, the measured receptances, the system matrices and a few unwanted open-loop eigenvalues and associated eigenvectors are all employed as in [3], [4], [5]. To implement our method in real operations, the real form of our optimization method is also presented.
The main contributions of this paper are as follows: A gradient-based optimization method and its real form are proposed for solving the robust and minimum norm PQEAP.
- (i)
The measured receptances, the system matrices, and a few unwanted open-loop eigenvalues and associated eigenvectors are employed. In particular, the required eigendata may be computed using existing computational methods or by measurement.
- (ii)
A cost function is provided to measure both the sensitivity of the closed-loop eigenvalues and the feedback norms.
- (iii)
The explicit gradient expression of the cost function is derived.
- (iv)
The real form of our method is also provided.
We also give some numerical examples to illustrate the effectiveness of our method by comparing the proposed method with the methods in [3], [4]. We can see from the later numerical results that the proposed method works better than the methods in [3], [4] in reducing the sensitivity of the closed-loop eigenvalues.
Section snippets
Notation
In what follows, we use the following notation. Let I be the identity matrix of appropriate dimension. Let , and denote the transpose, the conjugate and the conjugate transpose of a matrix accordingly. Let be a vector obtained by stacking the columns of A on top of one another. Let , , and be the largest singular value, the smallest singular value, and the condition number of a nonsingular matrix accordingly, where means the matrix
Robust and minimum norm PQEAP using receptances and system matrices
In this section, we focus on the robust and minimum norm PQEAP based on the measured receptances and system matrices. We reformulate the robust and minimum norm PQEAP as a minimization problem. Then we propose a gradient-based optimization method for solving the minimization problem. We first need the following result on the parameterized solution to the PQEAP for the multiple-input control system (1.1) [4, Theorem 2.5].
Lemma 3.1 Given a nontrivial matrix such that whenever
Real form of the robust and minimum norm PQEAP by receptances and system matrices
In this section, we propose the real form of Algorithm 3.1 for solving the robust and minimum norm PQEAP by using receptances and system matrices. As noted in [3], for the real form of Algorithm 3.1, the cost function J defined in problem (3.3) is a nonlinear nonconvex function of the real reformulation of Γ so that problem (3.3) becomes an unconstrained optimization problem.
By hypothesis, the unwanted open-loop eigenvalues and the assigned closed-loop eigenvalues appear in
Illustrative numerical examples
In this section, we present some numerical examples to show the effectiveness of Algorithms 3.1 and 4.1 for solving the robust and minimum norm PQEAP. Our numerical tests were implemented in MATLAB 9.0. In our tests, we set , and denote by “CT.” the total computing time in seconds. Also, the symbols “tol1”, “tol2”, and “tol3” stand for the upper bounds of the errors of the closed-loop eigenvalues and eigenvectors accordingly, i.e.,
Conclusions
In this paper, we focus on the robust partial quadratic eigenvalue assignment problem for multiple-input vibration control systems. We give a cost function for measuring the sensitivity of the closed-loop eigenvalues and present a minimization problem for reducing the feedback norms and the sensitivity of the closed-loop eigenvalues simultaneously. By using the measured receptances and the system matrices, we present an explicit formula for the gradient of the objective function where only a
Acknowledgements
The authors would like to thank the editor and the referees for their valuable comments.
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- 1
The research of this author is partially supported by the National Natural Science Foundation of China (No. 11632015 and No. 11871430).
- 2
The research of this author was partially supported by the National Natural Science Foundation of China (No. 11671337) and the Fundamental Research Funds for the Central Universities (No. 20720180008).