Extension Principle and Controller Design for Systems With Distributed Time-Delay

Abstract

Extension principle is defined for systems with distributed time-delay and the necessary and sufficient conditions for one system being an extension of the other are presented. Preservation of stability properties between two systems, one of which is an extension of the other is also discussed and it is shown that when the expanded system is an extension of the original system, (i) the original system is bounded-input bounded-output stable if and only if the expanded system is bounded-input bounded-output stable and (ii) the original system is exponentially stable if the expanded system is exponentially stable. Controller design using the extension principle is then considered. It is shown that, if the expanded system is an extension of the original system, then any controller designed for the expanded system can be contracted for implementation on the original system. Furthermore, if the controller designed for the expanded system stabilizes the expanded system and satisfies certain performance requirements, then the contracted controller stabilizes the original system and satisfies corresponding performance requirements for the original system. Finally, overlapping decompositions and controller design using overlapping decompositions are demonstrated. A highway traffic congestion control problem is then considered to demonstrate a possible application of the presented controller design approach.