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].

For the entire collection see [Zbl 0823.00028].

Reviewer: V.Dragan (Bucureşti)

##### Keywords:

disturbance attenuation problem; nonlinear; \(H_ \infty\) problem; separation principle; Hamilton-Jacobi equations##### Citations:

Zbl 0755.93037
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\textit{A. J. van der Schaft}, in: 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. 2nd printing with correct. Boston, MA: Birkhäuser. 153--190 (1994; Zbl 0839.93032)

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