A Multiple-Input Multiple-Output Cepstrum

This letter extends the concept of scalar cepstrum coefficients from single-input single-output linear time invariant dynamical systems to multiple-input-multiple-output (MIMO) models, making use of the Smith-McMillan form of the transfer function. These coefficients are interpreted in terms of poles and transmission zeros of the underlying dynamical system. We present a method to compute the MIMO cepstrum based on input/output signal data for systems with square transfer function matrices (i.e., systems with as many inputs as outputs). This allows us to do a model-free analysis. Two examples to illustrate these results are included: a simple MIMO system with three inputs and three outputs, of which the poles and zeros are known exactly, that allows us to directly verify the equivalences derived in this letter, and a case study on realistic data. This case study analysis data coming from a (model of) a non-isothermal continuous stirred tank reactor, which experiences linear fouling. We analyze normal and faulty operating behavior, both with and without a controller present. We show that the cepstrum detects faulty behavior, even when hidden by controller compensation. The code for the numerical analysis is available online.