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Nonlinear state space $${\mathcal H}_ \infty$$ control theory. 2nd printing with correct. (English) Zbl 0839.93032
Trentelman, H. L. (ed.) et al., Essays on control: perspectives in the theory and its applications. Proceedings of the plenary lectures and mini-courses of the European Control Conference, ECC ’93, held in Groningen, Netherlands, June 28-July 1, 1993. Boston, MA: Birkhäuser. Prog. Syst. Control Theory. 14, 153-190 (1994).
The disturbance attenuation problem for a class of nonlinear systems is investigated. The paper is essentially based on the author’s work [IEEE Trans. Autom. Control 37, 770-784 (1992; Zbl 0755.93037)]. After a concise treatment of $$L_2$$-gain analysis of nonlinear systems, the state feedback $$H_\infty$$ suboptimal control problem is treated. Attention is payed to the so-called dynamic measurement feedback $$H_\infty$$ problem. The paper offers, apart from the derivation of the nonlinear central controller, an analysis of necessary conditions for the solvability of the problem and a generalized version of the separation principle. A key element in this approach is the strict relation between Hamilton-Jacobi equations and invariant manifolds of (hyperbolic) Hamiltonian vector fields. This relation provides insight about the global solvability of Hamilton-Jacobi equations, and is a very powerful tool for proving local existence of solutions. In fact, this allows to solve, in a very simple manner, the local nonlinear $$H_\infty$$ suboptimal control problem, based on the solution to the linear $$H_\infty$$ problem for the linearized system.
For the entire collection see [Zbl 0823.00028].

##### MSC:
 93B36 $$H^\infty$$-control 93C10 Nonlinear systems in control theory
Zbl 0755.93037