Inverse optimal design of input-to-state stabilizing nonlinear controllers.(English)Zbl 0910.93064

The systems considered in this paper are described by $\dot x= f(x)+g_1 (x)d+ g_2(x)u$ with the state $$x$$, the disturbance $$d$$ and the control input $$u$$. Input-to-state stabilizability is defined. Conditions for input-to-state stability with respect to the disturbance $$d$$ are derived. A differential game problem with a suitable chosen cost functional is defined. An inverse optimal gain assignment problem is said to be solvable if penality functions on the state, the control and the disturbance and a feedback control law exist which minimize the cost functional for worst case disturbances. It is shown that stabilizability is a necessary and sufficient condition for solvability of this problem.
Stability margins are derived by chosing suitable Lyapunov functions. Relations to the nonlinear $$H_\infty$$ problem are shown. Stability margins for unmodelled dynamics with passity properties are derived. The design of smooth control laws by integrator backstepping is described.
Reviewer: R.Tracht (Essen)

MSC:

 93D15 Stabilization of systems by feedback 93D30 Lyapunov and storage functions 49L99 Hamilton-Jacobi theories 93B36 $$H^\infty$$-control 93C10 Nonlinear systems in control theory
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