Policy Iterations on the Hamilton–Jacobi–Isaacs Equation for $H_{\infty}$ State Feedback Control With Input Saturation

An H
_infin
suboptimal state feedback controller for constrained input systems is derived using the Hamilton-Jacobi-Isaacs (HJI) equation of a corresponding zero-sum game that uses a special quasi-norm to encode the constraints on the input. The unique saddle point in feedback strategy form is derived. Using policy iterations on both players, the HJI equation is broken into a sequence of differential equations linear in the cost for which closed-form solutions are easier to obtain. Policy iterations on the disturbance are shown to converge to the available storage function of the associated L
₂
-gain dissipative dynamics. The resulting constrained optimal control feedback strategy has the largest domain of validity within which L
₂
-performance for a given gamma is guaranteed