Finite dimensional systems of the form \[ \begin{aligned}\dot z(t) &= q\big(t,z(t),y(t)\big),\\ \dot x_i(t) &=x_{i+1}(t) + \Delta_i\big(t,z(t),y(t)\big),\qquad i=1, \ldots,n-1,\\ \dot x_n(t) &=u(t) + \Delta_n\big(t,z(t),y(t)\big),\\ y(t) &=x_1(t),\end{aligned} \] are considered. \(u(t)\) denotes the input, \(y(t)\) the output, and \(\Delta_i(\cdot)\), \(q(\cdot)\) are uncertain Lipschitz functions. Compared to previous results, the introduction of unmodelled dynamics by \(z(t)\) is the novelty.
If bounding functions for the \(\Delta_i(\cdot)\)’s are known, and if \(y\mapsto z\) is input-to-state practically stable, then an adaptive controller encompassing a partial state observer and \(y(t)\) only is designed, so that all signals of the closed-loop system are bounded and \(y(t)\) tends to a pre-specified neighbourhood of the origin.