Is Monopoli ’ s Model Reference Adaptive Controller Correct ? ∗

In 1974 R. V. Monopoli published a paper [1] in which he posed the now classical model reference adaptive control problem, proposed a solution and presented arguments intended to establish the solution's correctness. Subsequent research [2] revealed a flaw in his proof which placed in doubt the correctness of the solution he proposed. Although provably correct solutions to the model reference adaptive control problem now exist {see [3] and the references therein}, the problem of deciding whether or not Monopoli's original proposed solution is in fact correct remains unsolved. The aim of this note is to review the formulation of the classical model reference adaptive control problem, to describe Monopoli's proposed solution, and to outline what's known at present about its correctness. The classical model reference adaptive control problem is to develop a dynamical controller capable of causing the output y of an imprecisely modelled siso process P to approach and track the output y ref of a pre-specified reference model M ref with input r. The underlying assumption is that the process model is known only to the extent that it is one of the members of a pre-specified class M. In the classical problem M is taken to be the set of all siso controllable, observable linear systems with strictly proper transfer functions of the form g β(s) α(s) where g is a nonzero constant called the high frequency gain and α(s) and β(s) are monic, coprime polynomials. All g have the same sign and each transfer function is minimum phase {i.e., each β(s) is stable}. All transfer functions are required to have the same relative degree ¯ n {i.e., deg α(s) − deg β(s) = ¯ n.} and each must have a McMillan degree not exceeding some pre-specified integer n {i.e., deg α(s) ≤ n}. In the sequel we are going to discuss a simplified version of the problem in which all g = 1 and the reference model transfer function is of the form