Resilient quantized control under Denial-of-Service: Variable bit rate quantization

In this paper, we investigate a networked control problem in the presence of Denial-of-Service (DoS) attacks, which prevent transmissions over the communication network. The communication between the process and controller is also subject to bit rate constraints. For mitigating the influences of DoS attacks and bit rate constraints, we develop a variable bit rate (VBR) encoding–decoding protocol and quantized controller to stabilize the control system. We show that the system's resilience against DoS under VBR is preserved comparing with those under constant bit rate (CBR) quantized control, with fewer bits transmitted. The proposed VBR quantized control framework in this paper is general enough so that the results of CBR quantized control under DoS and moreover the results of minimum bit rate in the absence of DoS can be recovered. In the last part of the paper, we propose two random VBR transmission protocols, in which not only the bit rate but also the interval between two transmission attempts are time-varying. These protocols enable us to prevent attackers from learning how transmissions are scheduled and to mitigate the influence of attacks specifically targeting at periodic transmission instants. A numerical example is given to illustrate the proposed solution approach.