Disturbance Attenuation for Linear Systems: Optimal Solutions and Nonlinear Control

This study presents the optimal solutions to the state feedback finite horizon disturbance attenuation problem using a game theory approach. The proposed solution method is valid for linear dynamical systems and quadratic objective functions, under bounded disturbances. Two solution regions in the space of initial states define the optimal value of the controller. The first region, which contains the zero initial state, features the linear optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> controller, whereas the second region is characterized by a nonlinear optimal control, which converges to the linear quadratic regulator (LQR) for large initial states. The transition between the two regions provides a unified framework that spans from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> control to LQR control, depending on the relative sizes of the initial state and the bound on the disturbance. A novel solution algorithm is introduced, reducing the optimization of the disturbance attenuation problem to a linear algebra problem for the first region and a single, scalar nonlinear program for the second region. The performance of the algorithm is tested on a representative set of numerical examples. This paper enhances the versatility of disturbance attenuation state feedback controllers by introducing an efficient optimal solution strategy applicable to both zero and nonzero initial states. One consequence of these results is that optimal disturbance attenuation of even linear systems requires nonlinear control.