Minimax nonlinear control under stochastic uncertainty constraints

We consider in this paper a class of stochastic nonlinear systems in strict feedback form, where in addition to the standard Wiener process there is a norm-bounded unknown disturbance driving the system. The bound on the disturbance is in the form of an upper bound on its power in terms of the power of the output. Within this structure, we seek a minimax state-feedback controller, namely one that minimizes over all state-feedback controllers the maximum of a given class of integral costs, where the choice of the specific cost function is also part of the design problem as in inverse optimality. We derive the minimax controller by first converting the original constrained optimization problem into an unconstrained one (a stochastic differential game) and then making use of the duality relationship between stochastic games and risk-sensitive stochastic control. The state-feedback control law obtained is absolutely stabilizing. Moreover, it is both locally optimal and globally inverse optimal, where the first feature implies that a linearized version of the controller solves a linear quadratic risk-sensitive control problem, and the second feature says that there exists an appropriate cost function according to which the controller is optimal.