Exploiting Structure in Safety Control

Abstract

For safety-critical systems such as autonomous vehicles, power systems, and robotics, it is important to guarantee the systems operate under given safety constraints. Numerous safety control methods have been proposed for this purpose, but many of them are developed for a wide range of systems and do not take full advantage of the structures inherent in dynamics, controllers, and disturbances. This dissertation focuses on enhancing scalability and reducing conservativeness in safety control by leveraging these structures.

The first part of the dissertation focuses on developing scalable safety controller synthesis algorithms. We begin with analyzing the convergence properties of the inside-out algorithm, a well-established method for computing inner approximations of the maximal robust controlled invariant set (RCIS). Under mild conditions, we show that the inside-out algorithm converges exponentially to the maximal RCIS for linear systems, filling an important gap in the literature. Following the analysis of the inside-out algorithm, we develop efficient methods for computing implicit RCISs for discrete-time controllable systems. By augmenting the original system with a periodic structure, our implicit RCISs are constructed in closed form, making the proposed methods more scalable than competing approaches. Leveraging the convergence analysis for the inside-out algorithm, we further prove that the proposed implicit RCIS converges exponentially to a well-defined maximal set with a tuning parameter. Finally, we investigate the safety control problem for input-delayed systems, which are very common in the real world and possess a special structure in the system dynamics. By exploiting this structure, we show that the maximal RCIS for systems with input delay is embedded in the maximal RCIS of an auxiliary system, whose dimension is independent of the delay time. Leveraging this property, we propose an efficient method for computing the maximal RCIS for input-delayed systems, which scales well with the delay time.

In the second part of the dissertation, we focus on reducing the conservativeness in safety control, by leveraging structure in disturbance. One such structure is preview on disturbance. To assess the value of preview information in safety control, we introduce a metric called safety regret that quantifies the variation of the maximal RCIS as the preview horizon changes. For discrete-time linear systems, we prove the exponential convergence of the safety regret with the preview horizon and offer numerical algorithms that estimate the convergence rate. Our analysis can provide valuable insights when it comes to selecting sensors or perception algorithms with different prediction horizons. It is worth noting that synthesizing safety controllers for systems with preview is in general a challenging task. In this dissertation, we present efficient methods for computing the maximal RCIS for three classes of systems with preview, for which we can again exploit special structures in system dynamics to improve scalability. Finally, we introduce a novel safety control framework called opportunistic safety control, enabling safe operation beyond the maximal RCIS. This framework identifies worst-case disturbance models for each state and constructs control inputs robust to these models. Such disturbance model and control inputs can be computed from the maximal RCIS of an auxiliary system. We show in both simulation and drone experiments that our approach outperforms the existing safety control framework, especially when the system operates beyond the maximal RCIS with unexpected disturbance.