  In this paper, we establish a zero-sum, hybrid state stochastic game model
for designing defense policies for cyber-physical systems against different
types of attacks. With the increasingly integrated properties of cyber-physical
systems (CPS) today, security is a challenge for critical infrastructures.
Though resilient control and detecting techniques for a specific model of
attack have been proposed, to analyze and design detection and defense
mechanisms against multiple types of attacks for CPSs requires new system
frameworks. Besides security, other requirements such as optimal control cost
also need to be considered. The hybrid game model we propose in this work
contains physical states that are described by the system dynamics, and a cyber
state that represents the detection mode of the system composed by a set of
subsystems. A strategy means selecting a subsystem by combining one controller,
one estimator and one detector among a finite set of candidate components at
each state. Based on the game model, we propose a suboptimal value iteration
algorithm for a finite horizon game, and prove that the algorithm results an
upper bound for the value of the finite horizon game. A moving-horizon approach
is also developed in order to provide a scalable and real-time computation of
the switching strategies. Both algorithms aims at obtaining a saddle-point
equilibrium policy for balancing the system's security overhead and control
cost. The paper illustrates these concepts using numerical examples, and we
compare the results with previously system designs that only equipped with one
type of controller.
