Sampling and Quantization-Aware Control Barrier Functions for Safety-Critical Control of Cyber-Physical Systems

Safety is critical to a wide range of cyber-physical systems (CPS). Safety violations may damage CPS and cause harm to humans that co-exist in the operating environment. However, it is nontrivial to guarantee safety of complex CPS whose computation and control workload are shifted to the cloud. The reason is that the system states which evolve continuously are sampled periodically and quantized before being sent to the controller to compute control inputs. Moreover, the controller may operate with finite precision, making the coefficients involved in computation different from those of the actual system. Consequently, the synthesized control inputs to the system may lead to safety violations. In this paper, we study the co-design of quantizer and control inputs for such CPS. We construct a control barrier function (CBF) constraint for the digital controller and analyze how it differs from the CBF constraint formulated using the actual system states and dynamics. We observe that this difference is dependent on the sampling error, quantization error, and error induced by finite precision of the controller. We derive upper bounds of these errors and use the bounds to design a state quantizer. We show that the problem of designing a quantizer can be converted to a facility location problem. We prove the submodularity of the quantizer design problem, and leverage the submodularity property to develop an efficient greedy algorithm to construct the quantizer. Given the quantized states calculated by the quantizer, we modify the CBF constraint used by the controller to synthesize control inputs for the system at each sampling interval. We show that the synthesized inputs guarantee the system safety. We demonstrate the proposed approach using a numerical case study on a batch reactor system.