Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection

Abstract

In analogy to the Kučera-Youla parametrization we construct and parametrize all stabilizing controllers of a stabilizable linear periodic discrete-time input/output system, the plant. We establish a necessary and sufficient algebraic condition for the existence of controllers among these for which the output of the plant tracks a given reference signal in spite of disturbance signals on the input and the output of the plant. With a minor additional assumption the tracking stabilizing controllers are robust. As in the linear time-invariant (LTI) case the reference and disturbance signals are assumed to be generated by an autonomous system. Our results are the analogues for periodic behaviors of the corresponding LTI results of Vidyasagar. A completely different approach to stabilization and control of discrete periodic systems was developed by Bittanti and Colaneri. We derive a categorical duality between periodic behaviors over the time-axis of natural numbers and finitely generated modules over a suitable noncommutative ring of difference operators and use this for the proof of the main stabilization and control results. Morita's theory of equivalences between module categories is employed as an essential algebraic tool. All results of the paper are constructive.

A robust reduced-order observers design approach for linear discrete periodic systems

“…To design a periodic observer for system (1), a natural thought is to design a periodic observer for system (16) firstly, then by linear inverse transformation of state x(t), to obtain the estimation of original state x(t). From (16), we can see that for the transformed state x t , its substate x 1t is the output y t of the original system (1). So it can be used directly without reconstruction.…”

“…Theorem 3.3. Consider the completely observable periodic discrete-time linear system (1). If matrices Q t , A t , B t , C t are determined by equations (11)- (12), and matrices L t are generated from Algorithm 1, then a periodic reduced-order observer for system (1) with dimension n − m can be given by…”

“…That is to say, for any x 0 , z 0 , and u t the composite system composed of system (1) and system (27) satisfies the following relationship…”

“…where 0 m,n denotes the zero element of R m×n , and I n denotes the identity matrix of dimension n. The notions and results concerning the reachability and observability of the periodic system (1) have been introduced and analyzed in [3]. We provide them in the following.…”