Designing Safety Certificates for H ∞ Control of Unknown Linear Systems

This paper synthesizes safety barrier certificates for H∞ control of unknown linear systems. Inspired by the idea of expanding the domain of attraction (DoA) using control barrier functions (CBFs), we construct a region, called safe optimal DoA, in which both optimality and safety are simultaneously respected. The effect of unknown disturbances is attenuated by unifying the robustness property of CBFs and the H∞ control methods. To construct the control barrier certificates, a feasible optimization problem is developed using the relaxed game algebraic Riccati equation. A sum-of-squares (SOS)-based safe policy iteration algorithm is then proposed to solve the optimization problem iteratively. The resulting optimal policy minimizes a predefined cost function over an obtained maximum barrier-certified region for which there is no conflict between safety and optimality. Furthermore, to remedy the requirement of having the system dynamics, an online data-driven approach is presented using off-policy reinforcement learning. Finally, to demonstrate the efficacy of the proposed controller, a numerical example is given.