One-stage control over an adversarial channel with finite codelength

We consider a model of stealthy attack on a networked control system by formulating a static zero-sum game among four players. Three of the players constitute a team of encoder, decoder and controller for a scalar discrete-time linear plant, while the fourth player is a jammer, who acts to flip the bits of the binary encoded observation signal of the communication channel between the plant and the controller. We assume that the observation and control signals have finite codelength. We characterize the encoding/decoding/control defense strategies available to the controller, and for simplicity in conveying the main ideas, we model it for a scalar discrete-time system with only one time step. We prove that there is no loss of generality in restricting our attention to binning-based encoding and control strategies. We determine the control and jamming strategies that are in saddle-point equilibrium for this game and show that the saddle-point value does not depend on the jamming policy. We also provide a necessary and sufficient condition on the minimum number of bits that are required to drive the cost to zero for this one-stage control problem in the presence of a jammer.