Robust eigenvalue assignment with maximum tolerance to system uncertainties (original) (raw)
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Optimisation approach to robust eigenstructure assignment
IEE Proceedings - Control Theory and Applications, 1999
A systematic optimisation approach to robust eigenstructure assignment for control systems with output feedback is presented. The proposed scheme assigns the maximum allowable number of closed-loop eigenvalues to desired locations, and determines the corresponding closedloop eigenvectors as close to desired ones as possible. Additionally, the stability of the remaining closed-loop eigenvalues is guaranteed by the satisfaction of an appropriate Lyapunov equation. The overall design is robust with respect to time-varying parameter perturbations. The approach is applied to a literature example, where it is shown to capture the shape of the desired transient response.
Journal of Sound and Vibration, 2016
We propose an optimization approach to the solution of the partial quadratic eigenvalue assignment problem (PQEVAP) for active vibration control design with robustness (RPQEVAP). The proposed cost function is based on the concept of sensitivities over the sum and the product of the closed-loop eigenvalues, introduced recently in our paper. Explicit gradient formula for the solutions using state feedback and derivative feedback are derived as functions of a free parameter. These formulas are then used to build algorithms to solve RPQEVAP in a numerically efficient way, with no need to compute new eigenvectors, for both state feedback and state-derivative feedback designs. Numerical experiments are carried out in order to demonstrate the effectiveness of the algorithms and to compare the proposed method with other methods in the literature, thus showing its effectiveness.
Robust eigenvalue assignment for generalized systems
Automatica, 1992
In this paper we examine the problem of robust pole placement using state-feedback in generalized systems. We develop a robustness theory for the finite generalized spectrum of the system as a partial problem, and for the "infinite" pole placement problem as a second partial problem where perfect conditioning is always achievable. We also give bounds between the complete closed-loop robust eigenvalue problem and the two partial ones. The basic tool that is exploited in the presented theory, is the concept of chordal metric. In this paper we take advantage of this notion and present a compact theory for the robust eigenvalue assignment problem in generalized systems. The proposed theory is easy to implement, retrieves the results for the state-variable case as a special case, and takes advantage of well-known computational results.
The Partial Quadratic Eigenvalue Assignment Problem is the problem of reassigning a small number of undesirable eigenvalues of a quadratic matrix pencil using feedback. The problem arises in controlling resonance in vibrating structures and also in stabilizing control systems. The solution involves the computation of a pair of feedback matrices. For practical effectiveness, the magnitudes of the feedback norms need to be reduced and the conditioning of the closed-loop eigenvalues needs to be improved. In this paper, we propose new optimization methods for solving these problems. An important practical aspect of these methods is that the gradient formulas needed to solve the underlying unconstrained optimization problems are computed using only a small number of eigenvalues and eigenvectors of the quadratic pencil, which are all that can be computed using the state-of-the-art computational techniques.
Journal of Low Frequency Noise, Vibration and Active Control
Based on the notions of spectrum sensitivities, proposed by us earlier, we develop a novel optimization approach to deal with robustness in the closed-loop eigenvalues for partial quadratic eigenvalue assignment problem arising in active vibration control. A distinguished feature of this new approach is that the objective function is composed of only the system and the closed-loop feedback matrices. It does not need an explicit knowledge of the eigenvalues and eigenvectors. Furthermore, the approach is applicable to both the state-feedback and derivative feedback designs. These features make the approach viable to design an active vibration controller for practical applications to large real-life structures. A comparative study with existing algorithms and a study on the transient response of a real-life system demonstrate the effectiveness, superiority, and competitiveness of the proposed approach.
Robust Partial Quadratic Eigenvalue Assignment Problem: Spectrum Sensitivity Approach
2016
We propose an optimization approach to the solution of the partial quadratic eigenvalue assignment problem (PQEVAP) for active vibration control design with robustness (RPQEVAP). The proposed cost function is based on the concept of sensitivities over the sum and the product of the closed-loop eigenvalues, introduced recently in our paper. Explicit gradient formula for the solutions using state feedback and derivative feedback are derived as functions of a free parameter. These formulas are then used to build algorithms to solve RPQEVAP in a numerically efficient way, with no need to compute new eigenvectors, for both state feedback and state-derivative feedback designs. Numerical experiments are carried out in order to demonstrate the effectiveness of the algorithms and to compare the proposed method with other methods in the literature, thus showing its effectiveness.
Robust eigenstructure assignment with a control design package
1989
Eigenstructure assignment for linear systems is basically an inverse eigenvalue problem. Two possible solutions to this problem are presented here that have been implemented on the Ctrl-C@ Commercial Design Package. Both algorithms use singular-value decomposition and produce robust (well-conditioned) answers to the statefeedback pole-assignment problem. An cxample is included to assess the relative strengths of the two techniques.
Robust Eigenvalue Assignment in Descriptor Systems via Output Feedback
Asian Journal of Control, 2008
Based on a recently proposed parametric approach for eigenstructure assignment in descriptor linear systems via output feedback, the robust eigenvalue assignment problem in descriptor linear systems via output feedback is solved. The problem aims to assign a set of finite closed-loop eigenvalues which have minimum sensitivities with respect to perturbations in the closed-loop coefficient matrices, while at the same time, guarantee the closed-loop regularity. The approach optimizes the design parameters existing in the closed-loop eigenvectors to achieve the minimum eigenvalue sensitivities, and use the extra degree of freedom existing in the solution of the gain matrix to further minimize the magnitude of the gain matrix and enhance the robustness of the closed-loop regularity. The approach allows the finite closed-loop eigenvalues to be optimized within desired regions, and is demonstrated to be simple and effective.
New method of parametric eigenvalue assignment in state feedback control
IEE Proceedings - Control Theory and Applications, 1994
A new method is described for the assignment of eigenvalues of closed-loop plants in linear time-invariant multivariable systems. The parameterisation of controllers is based on the derivation of zero eigenvalue assignment by implementation of vector companion forms. This method is computationally very attractive and can be used for optimisation of the feedback matrix which assigns the closed-loop eigenvalues (from the set of real, complex conjugates or even those of the open-loop system) to the desired locations. A numerical example is presented to illustrate some advantages of this new explicit parameterisation of the controller gain matrix. Paper 1157D (C8), first received Zlst lune 1993 and in revised form 8th