Relay Feedback Based Time Domain Modeling of Linear 3-by-3 MIMO System

Relay feedback tests are carried out on multi input multi output systems. Analytical expressions for relay responses in time domain for the interactive transfer functions (off-diagonal elements) of the systems are derived and are validated. Generalized expressions are formulated based on the observation that a relay feedback test consists of a series of step inputs and a stable limit cycle implies a convergent infinite series. MIMO systems with lower-order transfer functions with different time delays are used to illustrate the derivation. These analytical expressions, in a closed form solution are useful to identify unknown system parameters, to understand interactions, to diagnosis the fault at the earlier stage, and subsequently for auto tuning. This closed-loop equation not only enables us to assess the effect of interactions between inputs and outputs but is also useful to back calculate/estimate the exact process model parameters, appearing in the analytical expressions for linear 3-by-3 MIMO systems, from experimental response.