On a state space approach to nonlinear \(H_ \infty\) control. (English) Zbl 0737.93018

Van der Schaft, A. J.

Syst. Control Lett. 16, No. 1, 1-8 (1991).

Summary: We study the standard \(H_ \infty\) optimal control problem using state feedback for smooth nonlinear control systems. The main theorem obtained roughly states that the \(L_ 2\)-induced norm (from disturbances to inputs and outputs) can be made smaller than a constant \(\gamma >0\) if the corresponding \(H_ \infty\) norm for the system linearized at the equilibrium can be made smaller than \(\gamma\) by linear state feedback. Necessary and sufficient conditions for the latter problem are by now well-known e.g. from the state space approach to linear \(H_ \infty\) optimal control. Our approach to the nonlinear \(H_ \infty\) optimal control problem generalizes the state space approach to the linear \(H_ \infty\) problem by replacing the Hamiltonian matrix and corresponding Riccati equation as used in the linear context by a Hamiltonian vector field together with a Hamilton-Jacobi equation corresponding to its stable invariant manifold.

MSC:

93B36	\(H^\infty\)-control
93C10	Nonlinear systems in control theory
49L99	Hamilton-Jacobi theories
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

\(H_ \infty\) optimal control problem; Hamiltonian matrix; Hamilton- Jacobi equation

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