The equivalence of model-following systems designed in the time domain and frequency domain

There are two approaches to the design of optimal model-following systems. One is in the time domain (modern synthesis), and the other is in the frequency domain (classical synthesis). In this paper, an equality is derived from the expanded Riccati equation. This equality represents the relation among the input signal, reference signal, and reference model. Together with the return difference equality, the equivalence of these two approaches is discussed. Based on this equivalence, an optimal model-following system can either be designed by the modern synthesis approach or the classical synthesis approach. For clarity and simplicity, only single-input/single-output systems are considered. An example is used to illustrate the results