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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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