Secure Control in Partially Observable Environments to Satisfy LTL Specifications

Abstract

This paper studies the synthesis of control policies for an agent that has to satisfy a temporal logic specification in a partially observable environment, in the presence of an adversary. The interaction of the agent (defender) with the adversary is modeled as a partially observable stochastic game. The goal is to generate a defender policy to maximize satisfaction of a given temporal logic specification under any adversary policy. The search for policies is limited to the space of finite state controllers, which leads to a tractable approach to determine policies. We relate the satisfaction of the specification to reaching (a subset of) recurrent states of a Markov chain. We present an algorithm to determine a set of defender and adversary finite state controllers of fixed sizes that will satisfy the temporal logic specification, and prove that it is sound. We then propose a value-iteration algorithm to maximize the probability of satisfying the temporal logic specification under finite state controllers of fixed sizes. Lastly, we extend this setting to the scenario where the size of the finite state controller of the defender can be increased to improve the satisfaction probability. We illustrate our approach with an example.

Privacy-Preserving Resilience of Cyber-Physical Systems to Adversaries

“…Therefore, the problem of maximizing the probability of satisfaction of ϕ under any adversary policy is equivalent to reaching a subset of states under these policies of the product game G ϕ that composes representations of the environment and the LTL objective. Moreover, results in [15]- [18] establish convergence of these policies to an equilibrium of the Stackelberg game between the defender and adversary.…”

“…Maximizing the probability of satisfaction in this setting corresponds to reaching an equilibrium of a zero-sum Stackelberg game between the defender and adversary. A careful justification of this assertion with detailed proofs for fully and partially observable environments has been studied in our earlier works [15]- [18]. We only state relevant results that will be useful in our goal to further establish differential privacy of trajectories in G ϕ that will satisfy ϕ under the respective agent policies.…”

“…The satisfaction of an LTL objective for two-player stochastic games when the players had competing objectives was presented in [15], [16] for the case when states were fully observable and in [17], [18] for the partially observable case. The authors of [19], [20] extended this to the case when an adversary could tamper with both actuators and clocks of the CPS to affect the satisfaction of a time-sensitive temporal constraint.…”

“…That is, a subset of recurrent states of the stochastic game which also satisfy ϕ. We note that in the terminology of [15], [16] this set is part of a generalized maximal accepting end component, while [17], [18] use the term ϕ− feasible recurrent set. Let P(reach E|s, µ, τ ) denote the probability of reaching the set of states E in G ϕ .…”