Model reference adaptive control using the delta operator

Reference is made to a recent paper by G.C. Goodwin, R. Lozano Leal, D.Q. Mayne, and R.H. Middleton (Automatica, vol.22, p.199-207, 1986), where it is shown that a cancellation of nonminimum-phase zeros appearing in discrete-time model reference adaptive control (MRAC) at a fast sampling rate can be easily avoided if the model to be identified is expressed in terms of the delta operator. A new proof of stability of the MRAC is given using the delta operator which, in contrast to the proof given in the above paper, does not require an assumption that a system consisting of a model determined by estimated parameters and a control law based on certainty equivalence principle is exponentially stable. A simple parameterization is proposed for the discrete-time MRAC using the delta operator which allows application of the usual (i.e. without dead zone) estimation algorithms