A N I NPUT- TO -S TATE S AFETY A PPROACH TO A NOMALY -R ESILIENT PARABOLIC PDE S :A PPLICATION TO C YBER -P HYSICAL BATTERY MODULES APREPRINT Tanushree Roy, Ashley Knichel and Satadru Dey * January 10, 2022 ABSTRACT Distributed Parameter Cyber-Physical Systems (DPCPSs), modelled by Partial Differential Equations (PDEs), are increasingly vulnerable to anomalies such as physical faults as well as cyber-attacks. This motivates the need for strategies towards anomaly-resilient control of these systems. Although anomaly detection and diagnostics in PDE systems have received considerable attention in existing literature, fault-tolerant or anomaly-resilient control for PDEs remains relatively under-explored. However, given the vulnerabilities of these systems against anomalies, it is essential that the control systems possess resilience against these disruptions. In this context, we explore a Practical Input-to- Safety (pISSf) based control design approach for a class of DPCPSs modelled by linear Parabolic PDEs. Specifically, we develop a design framework for anomaly-resilient control for this class of system with both safety and stability guarantees based on control Lyapunov functional and control barrier functional. To illustrate our methodology, we apply our strategy to design a thermal-anomaly resilient boundary coolant control system for a cyber-physical battery module. Several simulation studies are done to show the efficacy of our method under anomalies such as mechanical battery degradation and cyber-attack mediated overdischarge. 1 Introduction Many modern distributed parameter systems consist of wide-spread interconnections among their physical and cybernetic components, and are considered as Distributed Parameter Cyber-Physical Systems (DPCPSs). As a consequence, the vulnerability of these systems to anomalies such as physical faults and cyberattacks has increased considerably. This necessitates the design of controllers for such DPCPSs to be resilient against these anomalies. In this work, we explore control design of anomaly-resilient DPCPSs modelled by linear parabolic PDEs utilizing the practical Input-to-State Safety (pISSf) framework. We illustrate this approach by focusing on the application of thermal anomaly-resilient cyber-physical battery systems. Stability and safety verification in Ordinary Differential Equations (ODEs) has been explored widely and detailed list of works can be found in survey papers [1] and [2]. Input-to-State Stability (ISSt) in sense of Sontag has been investigated using Lyapunov functionals [3]. The notion of pratical ISSt (pISSt) has been explored in [3] that augments the original ISSt with certain practical considerations. On the other hand, Input-to-State Safety (ISSf) was introduced in [4]. Since then, two prominent methods are generally used for ISSf analysis/design/verification: reachable sets approximation [5–7] and barrier functionals [8–11]. Among the various works in this domain, [12, 13] introduces a notion of practical Input-to-State Safety (pISSf) in conjunction with pISSt with guaranteed safety for affine nonlinear ODEs using barrier functionals. Similar to ODEs, notions of ISSt for PDE systems have garnered a lot of attention recently (see survey paper [14]). For example, PDE ISSt have been explored for reaction-diffusion systems [15], hyperbolic systems [16], [17], parabolic * T. Roy, A. Knichel and S. Dey are with the Department of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA. {tbr5281,ark5514,skd5685}@psu.edu. arXiv:2201.02239v1 [eess.SY] 6 Jan 2022