Decomposition of Third Order Linear Time-Varying System into its Second and First Order Commutative Pairs

TL;DR

This paper establishes necessary and sufficient conditions for decomposing third order linear time-varying systems into commutative first and second order pairs, providing explicit formulas for their realization.

Contribution

It introduces a method to decompose third order systems into commutative first and second order subsystems with explicit parameter formulas, including conditions for non-zero initial states.

Findings

Derived explicit formulas for system realization.

Established conditions for commutative decomposition.

Addressed decomposition with non-zero initial conditions.

Abstract

Decomposition is a common tool for synthesis of many physical systems. It is also used for analyzing large scale systems which then known as tearing and reconstruction. On the other hand, commutativity of cascade connected systems have gained a grade deal of interest and its possible benefits have been pointed out on the literature. In this paper, the necessary and sufficient conditions for decomposition of any third order linear time-varying system as a commutative pair of first and second order systems of which parameters are also explicitly expressed, are investigated. Further, additional requirements in case of non-zero initial conditions are derived. This paper highlights the direct formulas for realization of any third order linear time-varying system as a series (cascade) connection of first and second order subsystems. This series connection is commutative so that it is independent from the sequence of subsystems in the connection. Hence, the convenient sequence can be decided by considering the overall performance of the system when the sensitivity, disturbance and robustness effects are considered. Realization covers transient responses as well as steady state responses.