Abstract
The discrete time Lyapunov equation is used in many applications and there is interest in its inverse and direct solutions. New methods are proposed to obtain solutions for cases where the system matrix is in controllable canonical form. The approach is based on the relationship between the discrete Lyapunov equation and the entries of one of the stability tables presented by Jury. It is shown that the inverse solution, which is based on this stability table, can be obtained using LDLt decomposition. Also the direct solution of the discrete Lyapunov equation can be obtained directly from the entries of this stability table. The proposed algorithms are illustrated by numerical examples.
Highlights
The inverse solution of the Discrete time Lyapunov matrix equation has been considered in applications, such state-covariance assignment [39], the generation of q-Markov covers for discrete systems [40] and the design of Fuzzy controllers [7]
In this paper our aim is to propose a new method to obtain the inverse solution of the Discrete time Lyapunov matrix equation for the case where the system matrix is in controllable canonical form
Given a single-input discrete system {Ac, bc} in controllable canonical form, the symmetric matrix Xc which is the solution of the Lyapunov equation in (4) can be obtained as follows: 1) Find the quantities ∆j, j = 1, 2, · · ·, n using the stability table in [15], [1]
Summary
Arises in many applications, for instance in the study of discrete time system stability [15], in covariance calculation [36], [37], in the iterative solution of the matrix Riccati equation [19] and in the optimal constant output feedback problem [11], [27]. In this paper our aim is to propose a new method to obtain the inverse solution of the Discrete time Lyapunov matrix equation for the case where the system matrix is in controllable canonical form. New methods for obtaining the inverse and the direct solutions of the Discrete time Lyapunov matrix equation for linear discrete systems with the system matrix in controllable canonical form are presented. They are based on the properties of the Mansour matrix [1] and lead to solutions which are closely related to the ∆j’s, the entries of the stability table presented in [16], [1].
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