Control of unknown plants in reduced state space

A method is proposed in this paper for the synthesis of an adaptive controller for a class of model reference systems in which the plant is not known exactly, but which is of the following type: single variable, time varying, either linear or nonlinear, of
n
th order, and capable of
m
th order input differentiation. The model is linear, stable, and of
n
'th order, where
(n - m) \leq n' \leq n
. The only knowledge of the plant that is required in this synthesis procedure is the form of the plant equation and the bounds of
b_{m}(t)
, the coefficient of the
m
th order plant input derivative. The synthesis procedure makes use of an unique function, called the characteristic variable, and Lyapunov type synthesis. The introduction of the characteristic variable reduces the synthesis problem to one that involves a known, linear time-invariant lower order plant. The control signal is generated by measuring the plant and model outputs, and their first
(n - m)
derivative signals. This ensures that the norm of the
(n - m)-
dimensional error vector is ultimately bounded by ε, an arbitrarily small positive number provided
\xi(t)
, the characteristic variable, is bounded. Two nontrivial simulation examples are included.