Optimal Design of Disturbance Attenuation Feedback Controllers for Linear Dynamical Systems

This paper finds the optimal feedback controller for the discrete time, finite horizon disturbance attenuation problem under bounded disturbances. We consider a linear dynamical system and a quadratic objective function with a resulting nonlinear optimization and optimal nonlinear controller. In the space of initial states, two regions are identified. One region, containing the zero initial state, features the linear optimal
$H_{\infty}$
controller, while the other region features nonlinear optimal control, and converges to the linear quadratic regulator (LQR) controller for large initial states. The transition between the two regions offers a unified framework that spans from
$H_{\infty}$
control to LQR control as a function of the relative magnitude of the initial state and the imposed disturbance. This study enhances the versatility of disturbance attenuation feedback controllers and expands on the previous work on the dynamic game theory approach to optimal robust control design.