## A combined backstepping and small-gain approach to adaptive output feedback control.(English)Zbl 0932.93045

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.

### MSC:

 93C40 Adaptive control/observation systems 93C10 Nonlinear systems in control theory
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