Constructing Safety Barrier Certificates for Unknown Linear Optimal Control Systems
Pouria Tooranjipour, Bahare Kiumarsi · 2022
This paper presents an online method to construct barrier certificates for linear optimal control systems with unknown dynamics. Inspired by the idea of expanding the domain of attraction (DoA) using control barrier certificates (CBCs), we construct a region, called safe optimal DoA, in which both stability and safety are equivalently respected without sacrificing the stability as well as performance in favor of safety. To this end, a feasible optimization problem is formalized using a relaxed algebraic Riccati equation (ARE) and safety constraints to find the maximum barrier-certified region under which a predefined cost function is also minimized. The proposed approach can cope with the possible conflicts between safety and stability without using a relaxation factor to simultaneously satisfy both stability and safety. Since this optimization problem is computationally hard to solve, a safe policy iteration method is implemented, and in each iteration, the problem is split into several smaller sum of squares (SOS) programs. Furthermore, an online data-driven approach is proposed to remedy the requirement of complete knowledge about the system dynamics by employing a safe off-policy reinforcement learning algorithm to solve the proposed optimization problem. Applying the off-policy method allows learning about safe optimal policy while using a different safe exploratory policy to collect data sets. In the end, to demonstrate the efficacy of this method, a numerical example is provided.