On the Value of Linear Quadratic Zero-sum Difference Games with Multiplicative Randomness: Existence and Achievability

We consider a wireless networked control system (WNCS) with multiple controllers and multiple attackers. The dynamic interaction between the controllers and the attackers is modeled as a linear quadratic (LQ) zero-sum difference game with multiplicative randomness induced by the the multiple-input and multiple-output (MIMO) wireless fading. The existence of the game value is of significant importance to the LQ zero-sum game. If it exists, the value characterizes the minimum loss that the controllers can secure against the attackers' best possible strategies. However, if the value does not exist, then there will be no Nash equilibrium policies for the controllers and attackers. Therefore, we focus on analyzing the existence and achievability of the value of the LQ zero-sum game. In stark contrast to existing literature, where the existence of the game value depends heavily on the controllability of the closed-loop systems, the multiplicative randomness induced by the MIMO wireless channel fading may destroy closed-loop controllability and introduce intermittent controllability or almost sure uncontrollability. We first establish a general sufficient and necessary condition for the existence of the game value. This condition relies on the solvability of a modified game algebraic Riccati equation (MGARE) under an implicit concavity constraint, which is generally difficult to verify due to the multiplicative randomness. We next introduce a novel positive semidefinite (PSD) kernel decomposition method induced by multiplicative randomness. A verifiable tight sufficient condition is then obtained by applying the proposed PSD kernel decomposition to the constrained MGARE. Under the existence condition, we finally construct a closed-form saddle-point policy based on the minimal solution to the constrained MGARE, which is able to achieve the game value in a certain class of admissible policies. We demonstrate that the proposed saddle-point policy is backward compatible to the existing strictly feedback stabilizing saddle-point policy. Our developed theory is also verified via numerical simulations.