Control of Constrained Discrete-Time Systems With Bounded $\ell _{2}$ Gain

We consider the problem of designing a control law for a constrained linear system with bounded disturbances that ensures constraint satisfaction over an infinite horizon, while also guaranteeing that the closed-loop system has bounded lscr₂ gain. To this end, we propose a receding horizon control strategy based on the repeated calculation of optimal finite horizon feedback policies. We parameterize these policies such that the input at each time is an affine function of current and prior states, and minimize a worst-case quadratic cost where the disturbance energy is negatively weighted as in H _infin control. We show that the resulting receding horizon controller has two advantages over previous results for this problem. First, the policy optimization problem to be solved at each time step can be rendered convex-concave, with a number of decision variables and constraints that grows polynomially with the problem size, thereby making its solution amenable to standard techniques in convex optimization. Second, the achievable lscr₂ gain of the resulting closed-loop system is bounded and non-increasing with increasing control horizon. A numerical example is included to demonstrate the improvement in achievable lscr₂ gain relative to existing methods.