Subsystem identification of feedback and feedforward systems with time delay

We present an algorithm for identifying discrete-time feedback-and-feedforward subsystems with time delay that are interconnected in closed loop with a known subsystem. This frequency-domain algorithm uses only measured input and output data from a closed-loop discrete-time system, which is single input and single output. No internal signals are assumed to be measured. The orders of the unknown feedback and feedforward transfer functions are assumed to be known. We use a two-candidate-pool multi-convex-optimization approach to identify not only the feedback and feedforward transfer functions but also the feedback and feedforward time delay. The algorithm guarantees asymptotic stability of the identified closed-loop transfer function. The main analytic result shows that if the data noise is sufficiently small and the cardinality of the feedback-candidate-pool set is sufficiently large, then the identified feedforward and feedback delays are equal to the true delays, and the parameters of the identified feedforward and feedback transfer functions are arbitrarily close to the true parameters. This subsystem identification algorithm has application to modeling human-in-the-loop behavior. To demonstrate this application, we apply the new subsystem identification algorithm to data obtained from a human-in-the-loop control experiment in order to model the humans’ feedback and feedforward (with delay) control behavior.