Persistency of excitation and overparametrization in model reference adaptive control

The problem of overparameterization of the plant in model reference adaptive control (MRAC) is analyzed. Overparametrization may arise when an upper bound for the order of the plant is used in the design of the MRAC or when the transfer function of the modeled part of the plant has common zeros and poles. It is shown that under parametrization the parameters for model-plant transfer function matching lie in a linear variety. If the reference input signal has enough frequencies, the estimated parameters will converge to this variety. In the presence of modeling errors such as bounded disturbances, the parameters may drift to infinity along the linear variety independently of the number of frequencies in the reference input signal. In the case of exact parameterization but close zero-pole cancellation, the signal vector is shown to be persistently exciting with a low level of excitation. Such a low level of excitation may lead to poor rate of convergence and to loss of robustness with respect to modeling errors