Decomposition of Second-Order Discrete-Time Linear Time-Varying Systems Into First-Order Commutative Pairs

Abstract

In this study, necessary and sufficient conditions for the decomposition of any second-order discrete-time linear time-varying system as a commutative pair of two first-order systems are presented. Commutativity conditions with zero initial conditions are first expressed by Theorem 1. Then, the conditions under nonzero initial conditions are studied and presented by Theorem 2. The results including decomposition formulas are well verified by examples worked by using MATLAB Simulink tool. The importance of the paper subject is emphasized in view of some engineering applications for getting better performance characteristics such as sensitivity, robustness and stability. In fact, the results of this paper can be used when the synthesis of a second-order system is made by cascaded and commutative pairs of simple first-order subsystems; and by using the explicit decomposition formulas derived in the paper this can be achieved very easily. Since different decompositions lead to different performance characteristics, one should choose the appropriate ones among them.