Regularization of Linear Discrete-Time Periodic Descriptor Systems by Derivative and Proportional State Feedback

Abstract

In this paper, we consider the regularization problem for the linear time-varying discrete-time periodic descriptor systems by derivative and proportional state feedback controls. Sufficient conditions are given under which derivative and proportional state feedback controls can be constructed so that the periodic closed-loop systems are regular and of index at most one. The construction procedures used to establish the theory are based on orthogonal and elementary matrix transformations and can, therefore, be developed to a numerically efficient algorithm. The problem of finite pole assignment of periodic descriptor systems is also studied.

Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations

“…This type of equations arises in the context of periodic state feedback problems and in model reduction of periodic descriptor systems when the solutions of the noncausal matrix equations associated with the systems (Kuo et al (2004);Benner et al (2011a);Chu et al (2007)) are sought. Note that in that case Q l (k) = I n −P l (k) and Q r (k) = I n − P r (k), where P l (k), P r (k) are the spectral projectors onto the k-th left and right deflating subspaces of the periodic matrix pairs {(E k , A k )} K−1 k=0 corresponding to the finite eigenvalues.…”

Projected Generalized Discrete-Time Periodic Lyapunov Equations and Balanced Realization of Periodic Descriptor Systems

“…In a similar fashion to the Kronecker canonical form for a regular matrix pair, we can transform a regular set of periodic matrix pairs into a periodic Kronecker canonical form [23] (see also [34] for the history of the canonical form).…”

“…. , K − 1, then these K values are defined as the indices [23] of a regular set of periodic matrix pairs {(E k , A k )} K−1 k=0 . Hence we define the index of the periodic descriptor system (1.1) as ν ≡ max{ν 0 , ν 1 , · · · , ν K−1 }.…”

“…where the matrices E k , A k ∈ R n×n , B k ∈ R n×m , C k ∈ R p×n are periodic with period K ≥ 1, i.e., E k = E k+K , A k = A k+K , B k = B k+K , C k = C k+K , and the matrices E k are allowed to be singular for all k. Recently, this class of periodic descriptor systems (1.1) is discussed and studied extensively in the problem of solvability and conditionability [36], the computation of H ∞ -norm and system zeros [35,50], and the compensating and regularization problems for periodic descriptor systems [9,23].…”

Robust admissibility of uncertain switched singular systems

“…Notice that the proportional and derivative feedback controller has been studied and applied in various applications (Jing 1994;Bunse-Gerstner, Byers, Mehrmann, and Nichols 1999;Fridman and Shaked 2002;Duan and Zhang 2003;Kuo, Lin, and Xu 2004;Mao and Chu 2010). For example this controller is used for stabilisation and regularization of linear descriptor systems (Jing 1994;Duan and Zhang 2003;Kuo et al 2004) and for decentralised singular system by decentralised output feedback (Wang and Soh 1999), for exact feedback linearization of nonlinear system with application to induction motor speed control (Boukas and Habetler 2004) and also for H 1 -control of systems with state delay (Fridman and Shaked 2002).…”