Remote State Estimation Under DoS Attacks in CPSs With Arbitrary Tree Topology: A Bayesian Stackelberg Game Approach

In this paper, we consider remote state estimation for an arbitrary tree topology in cyber-physical systems (CPSs) subject to Denial-of-Service (DoS) attacks. A sensor transmits its local estimation to the root node of the tree, and the root node transmits the optimal estimation to its child nodes until the leaf nodes are reached. In the meanwhile, a malicious attacker can jam all communication channels strategically connected to the attacked node. With the energy constraints in mind, both the defender and attacker adopt strategies that involve allocating energy to determine which nodes to protect or attack at each time step. A Bayesian Stackelberg game (BSG) framework with incomplete information is implemented, where the defender has no access to the available energy of the attacker exactly except for its probability distribution. In addition, a Markov decision process (MDP) and a Stackelberg Q-learning algorithm are presented to obtain the Stackelberg equilibrium (SE) policy over a finite time horizon. Finally, a numerical example is provided to demonstrate our main results.