Closed-loop identication of MIMO systems: a new look at identifiability and experiment design

This paper addresses a question that has been posed to us:“Is it necessary to excite all reference signals for the identification of a multivariable system operating in closed loop with a linear time-invariant controller?” In this paper we follow a careful re-examination of the notions of identifiability and informative experiments for multi-input multi-output (MIMO) closed-loop systems, which provides a negative answer to this question. Our analysis also allows to establish conditions on the controller complexity that guarantee existence of a unique global minimum of the identification criterion in the absence of external excitation; these conditions extend to the MIMO case conditions that were known for the SISO case. We illustrate our results for 2-input 2-output systems by presenting various possible experiment designs that produce a unique global minimum for the identification criterion.