$$H^\infty $$ H ∞ -Optimal Control via Game-Theoretic Differential Dynamic Programming and Gaussian Processes

In this paper, we present a nonlinear $$H^\infty $$ H ∞ -optimal control algorithm for a system whose dynamics can be described by the summation of two terms: a known function obtained from system modeling and an unknown function that represents the model error induced by the disturbance and the noise that are not captured by the original model. A Gaussian Process (GP) is utilized as an alternative to a supervised artificial neural network to update the nominal dynamics of the system and provide disturbance estimates based on data gathered through interaction with the system. A soft-constrained two-player zero-sum differential game that is equivalent to the disturbance attenuation problem in nonlinear $$H^\infty $$ H ∞ -optimal control is then formulated to synthesis the $$H^\infty $$ H ∞ controller. The differential game is solved through the Game-Theoretic Differential Dynamic Programming (GT-DDP) algorithm in continuous time. In addition we provide a proof of quadratic convergence of the proposed GT-DDP algorithm. Simulation results on a quadcopter system demonstrate the efficiency of the learning-based control algorithm in handling model uncertainties and external disturbances.