Nonlinear State Space H _∞ Control Theory

Abstract

Although the H _∞ control problem was originally formulated [55] as a linear design problem in the frequency domain (in fact, H _∞ stands for the Hardy space of complex functions bounded and analytic in the open righthalf complex plane), it can be naturally translated to the time-domain and extended to nonlinear state-space systems. Indeed, the standard H _∞ control problem can be equivalently formulated as the optimal attenuation of the L ₂-induced norm from exogenous inputs (inputs with unknown power spectrum) to the to-be-controlled outputs, under the constraint of internal stability. Also, although early research in H _∞ control was conducted solely using frequency domain methods, a satisfactory state space solution to the linear H _∞ (sub-)optimal control problem was reached by the end of the eighties (see especially [12], [31], [17], [46], [16], [30], [48], [49], [47]). Moreover, this state space solution relies on tools familiar from LQ and LQG theory, in particular Riccati equations and Hamiltonian matrices. In the classical paper by Willems on LQ control [52] the relations of these tools with the underlying notion of dissipativity were being stressed; while in [53] dissipativity was defined for general nonlinear systems, encompassing notions of passivity of physical systems and input-output stability of nonlinear (feedback) systems. The resulting dissipation inequalities were fruitfully explored in e.g. [35], [20], [21], also linking them to the Hamilton-Jacobi equation from classical nonlinear optimal control (see also [36]).