  Cyber-physical systems (CPS) are required to operate safely under fault and
malicious attacks. The simplex architecture and the recently proposed cyber
resilient architectures, e.g., Byzantine fault tolerant++ (BFT++), provide
safety for CPS under faults and malicious cyber attacks, respectively. However,
these existing architectures make use of different timing parameters and
implementations to provide safety, and are seemingly unrelated. In this paper,
we propose an analytical framework to represent the simplex, BFT++ and other
practical cyber resilient architectures (CRAs). We construct a hybrid system
that models CPS adopting any of these architectures. We derive sufficient
conditions via our proposed framework under which a control policy is
guaranteed to be safe. We present an algorithm to synthesize the control
policy. We validate the proposed framework using a case study on lateral
control of a Boeing 747, and demonstrate that our proposed approach ensures
safety of the system.
