Discretisation of continuous-time dynamic multi-input multi-output systems with non-uniform delays

Input and output time delays in continuous-time (CT) dynamic systems impact such systems differently as their effects are encountered before and after the state dynamics. Given a fixed sampling time, input and output signals in multiple-input multiple-output (MIMO) systems may exhibit any combination of the following four cases: no delays, integer-multiple delays, fractional delays and integer-multiple plus fractional delays. A common pitfall in the digital control of delayed systems literature is to only consider the system timing diagram to derive the discrete-time (DT) equivalent model; hence, effectively `lump` the delays across the system as one total delay. DT equivalent models for systems with input delays are radically different than those with output delays. Existing discretisation techniques for delayed systems usually consider the delays to be integer-multiples of the sampling time. This study is intended to serve as a reference for systematically deriving DT equivalent models of MIMO systems exhibiting any combination of the four delay cases. This algorithm is applied towards discretising an MIMO heat exchanger process with non-uniform input and output delays. A significant improvement towards the CT response was noted when applying this algorithm as opposed to rounding the delays to the closest integer-multiple of the sampling time.