Single-input eigenvalue assignment algorithms: A close look (original) (raw)
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Unification and Improvement of Certain Methods for Eigenvalue Assignment
Dirasat Engineering Sciences, 2011
Certain methods of eigenvalue assignment namely: the two stage method, the entire eigenstructure method, and the recursive method are shown to have similar structure and form. Exposing the similarities between the methods has lead to the development of a new method of assignment. The study resulted in a modification of the conceptually simple method of entire eigenstructure ending with a new modified method requiring fewer number of eigenvectors. Finally, the three methods are contrasted against each other, guiding the user when to use any appropriate one.
Simplified Methods for Eigenvalue Assignment
Advances in Pure Mathematics, 2015
A state feedback method of reduced order for eigenvalue assignment is developed in this paper. It offers immediate assignment of m eigenvalues, with freedom to assign the remaining n m − eigenvalues. The method also enjoys a systematic one-step application in the case where the system has a square submatrix. Further simplification is also possible in certain cases. The method is shown to be applicable to uncontrollable systems, offering the simplest control law when having maximum uncontrollable eigenvalues.
Robust eigenvalue assignment with maximum tolerance to system uncertainties
Journal of Guidance, Control, and Dynamics, 1991
For a linear time-invariant system with a feedback controller, the closed-loop eigenvalues perturb due to system uncertainties. Given an allowable tolerance for the closed-loop eigenvalue perturbation, an algorithm is developed to obtain a state feedback controller that maximizes the uncertainty tolerance of the open-loop system matrix. The design procedure is based on an existing eigenvalue assignment technique using Sylvester's equation. A robustness condition is derived to guarantee satisfaction of a specified closed-loop perturbation tolerance. Finally, an iterative algorithm is presented for easy numerical implementation to compute the robust controller, and a numerical example is given for illustration.
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Proceedings of 36th Midwest Symposium on Circuits and Systems
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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
A simple solution to the optimal eigenvalue assignment problem
IEEE Transactions on Automatic Control, 1999
The problem of the optimal eigenvalue assignment for multiinput linear systems is considered. It is proven that for an n-order system with m independent inputs, the problem is split into the following two sequential stages. Initially, the n 0 m eigenvalues are assigned using an n 0 m-order system. This assignment is not constrained to satisfy optimality criteria. Next, an m-order system is used to assign the remaining m eigenvalues in such a way that linear quadratic optimal criteria are simultaneously satisfied. Therefore, the original n-order optimal eigenvalue assignment problem is reduced to an m-order optimal assignment problem. Moreover, the structure of the equivalent m-order system permits further simplifications which lead to solutions obtained by inspection.
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.