New state transformations of time-delay systems with multiple delays and their applications to state observer design

This paper addresses the problem on how to systematically compute state transformations of a general class of time-delay systems with multiple time delays in the state and output vectors, and applications of these state transformations to state observer design. First, a forward state transformation problem is studied. For this, we present a two-stage coordinate transformation method to transform any given time-delay system into an observable canonical form where time delays appear in the input and output vectors, but not in the state vector. The significance of such a coordinate transformation is that a Luenberger-type state observer can be easily designed in the new coordinate system. Then, a backward state transformation problem is studied which allows us to reconstruct the original state vector of the system. Therefore, by using both the forward and the backward state transformations, state observers for time-delay systems can be systematically designed. Our approach based on the new state transformations enables the design of state observers of a more general class of time-delay systems than existing works in the literature. Conditions for the existence of the state transformations and an algorithm for computing them are provided in this paper. Numerical examples and simulation results are given to illustrate the effectiveness of our proposed design method.