Minimally Conservative Controlled-Invariant Set Synthesis Using Control Barrier Certificates

Finding a controlled-invariant set for a system with state and control constraints is crucial for safety-critical applications. However, existing methods often produce overly conservative solutions. This paper presents a method for generating controlled-invariant (safe) sets for nonlinear polynomial control-affine systems using Control Barrier Certificates (CBCs). We formulate CBC conditions as Sum-of-Squares (SOS) constraints and solve them via an SOS Program (SOSP). First, we generalize existing SOSPs for CBC synthesis to handle environments with complex unsafe state representations. Then, we propose an iterative algorithm that progressively enlarges the safe set constructed by the synthesized CBCs by maximizing boundary expansion at each iteration. We theoretically prove that our method guarantees strict safe set expansion at every step. Finally, we validate our approach with numerical simulations in 2D and 3D for single-input and multi-input systems. Empirical results show that the safe set generated by our method covers in most part a larger portion of the state space compared to two state-of-the-art techniques.