Safety Verification and Controller Synthesis for Systems with Input Constraints

In this paper we consider the safety verification and safe controller synthesis problems for nonlinear control systems. The Control Barrier Certificates (CBC) approach is proposed as an extension to the Barrier certificates approach. Our approach can be used to characterize the control invariance of a given set in terms of safety of a general nonlinear control system subject to input constraints. From the point of view of controller design, the proposed method provides an approach to synthesize a safe control law that guarantees that the trajectories of the system starting from a given initial set do not enter an unsafe set. Unlike the related control Barrier functions approach, our formulation only considers the vector field within the tangent cone of the zero level set defined by the certificates, and is shown to be less conservative by means of numerical evidence. For polynomial systems with semi-algebraic initial and safe sets, CBCs and safe control laws can be synthesized using sum-of-squares decomposition and semi-definite programming. Examples demonstrate our method.