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Tools for semiglobal stabilization by partial state and output feedback. (English) Zbl 0843.93057
For a nonlinear system of the form (1) $$\begin{cases} \dot z= A(x,u)\\ y= C(z) \end{cases}$$, it is well known that gobal asymptotic stabilization at $$z=0$$ by means of full state feedback does not imply, in general, global stabilizability by means of dynamic output feedback. The authors consider therefore the weaker notion of semiglobal stabilizability. This means that by a suitable choice of the feedback law, the system can be stabilized at the origin and, at the same time, the region of attraction can be made larger than any prescribed compact set. They prove that if (1) is globally stabilizable (locally in the exponential sense) by a state feedback $$\overline {u} (z)$$ then (1) is semiglobally stabilizable by dynamic output feedback, provided that $$\overline {u}(z)$$ satisfies a technical assumption (uniform complete observability). A similar conclusion can be carried out for practical stabilizability. These results are obtained as applications of a number of technical tools which are developed in this paper. These include backstepping methods, a robust observer and a local nonlinear small gain theorem.

##### MSC:
 93D15 Stabilization of systems by feedback 93C10 Nonlinear systems in control theory
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