Assessing Safety for Control Systems Using Sum-of-Squares Programming

Abstract

In this chapter we introduce the concept of safety for control systems in both continuous and discrete time form. Given a system and a safe set, we say the system is safe if the system state remains inside the safe set for all initial conditions starting from the initial set. Control invariance can be employed to verify safety and design safe controllers. To this end, for general polynomial systems with semi-algebraic safe/initial sets, we show how Sum-of-Squares (SOS) programming can be used to construct invariant sets. For linear systems, evaluating invariance can be much more efficient by using ellipsoidal techniques and dealing with a series of SOS constraints. Following invariance analysis, safe control design and safety verification methods are proposed. We conclude this chapter by showing invariant set construction for both nonlinear and linear systems, and provide MATLAB code for reference.