Approximated modeling and minimal realization of transfer function matrices with multiple time delays

The modeling and minimal realization techniques for a specific multiple time-delay continuous-time transfer function matrix with a delay-free denominator and a multiple (integer/fractional) time-delay numerator matrix have been developed in the literature. However, this is not the case for a general multiple time-delay continuous-time transfer function matrix with multiple (integer/fractional) time delays in both the denominator and the numerator matrix. This paper presents a new approximated modeling and minimal realization technique for the general multiple time-delay transfer function matrices. According to the proposed technique, an approximated discrete-time state-space model and its corresponding discrete-time transfer function matrix are first determined, by utilizing the balanced realization and model reduction methods with the sampled unit-step response data of the afore-mentioned multiple time-delay (known/unknown) continuous-time systems. Then, the modified Z-transform method is applied to the obtained discrete-time transfer function matrix to find an equivalent specific multiple time-delay continuous-time transfer function matrix with multiple time delays in only the inputs and outputs, for which the existing control and design methodologies and minimal realization techniques can be effectively applied. Illustrative examples are given to demonstrate the effectiveness of the proposed method.