Numerical algorithms for eigenvalue assignment by constant and dynamic output feedback (original) (raw)

Single-input eigenvalue assignment algorithms: A close look

SIAM journal on matrix analysis and applications, 1998

A close-look is given at the single-input eigenvalue assignment methods. Several previously known backward stable QR algorithms are tied together in a common framework of which each is a special case, and their connection to an explicit expression for the feedback vector is exposed. A simple new algorithm is presented and its backward stability is established by round-o-error analysis. The di erences between this new algorithm and the other QR algorithms are discussed. Also, the round-o error analysis of a simple recursive algorithm for the problem (Datta (1987)) is presented. The analysis shows that the latter is reliable, and the reliability can be determined during the execution of the algorithm rather cheaply. Finally, some numerical experiments comparing some of the methods are reported.

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.

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

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.

Efficient eigenvalue assignments for general linear MIMO systems

Automatica, 1995

Abstrad-This paper deals with the transformation of linear, multi-input multi-output (MIMO) systems into Frobenius canonical form, with the ultimate objective of developing a new, computationally efficient methodology for a poleplacement procedure. Both time-invariant and time-varying systems are considered. The conventional pole placement steps for time-invariant SISO (single-input single-output) systems are generalized for both classes. This is a unique study of the expansion of the pole placement capability, in particular for time-varying MIMO systems. This depth of generalization has been neglected in the past due to its complex formulation.

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.

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.

PARTIAL EIGENSTRUCTURE ASSIGNMENT BY STATE FEEDBACK: AN LMI APPROACH

A new methodology of the partial eigen-structure assignment by state feedback via Linear matrix inequality (LMI) is given. It enables to adopt the Sylvester matrix equa-tion ¢ ¡ ¤ £ ¦ © ¡ and to check sta-bility using parameter dependent Lyapunov functions which are derived from LMI con-ditions. We show in this work that it is possible to avoid some limiting assumptions needed for the resolution of the Sylvester equation by using a reduced-order system obtained by the projection of the trajectories of original system onto a subspace associ-ated with the undesirable open-loop eigen-values.