We consider a network of status updating sensors whose updates are collected and sent to a monitor by a gateway. The monitor desires as fresh as possible updates from the network of sensors. The gateway may either poll a sensor for its status update or it may transmit collected sensor updates to the monitor. We derive the average age at the monitor for such a setting. We observe that increasing the frequency of transmissions to the monitor has the upside of resetting sensor age at the monitor to smaller values. However, it increases the length of time that elapses before a sensor is polled again. This motivates our investigation of policies that fix the number of sensors s the gateway polls before transmitting to the monitor.For any s, we show that when sensor transmission times to the gateway are independent and identically distributed (iid), for independent but possibly non-identical transmission times to the monitor, it is optimal to poll a sensor with the maximum age at the gateway first. Also, under simplifying assumptions, the optimal value of s increases as the square root of the number of sensors. For non-identical sensor transmission times, we consider a policy that polls a sensor such that the resulting average change in age is minimized. We compare our policy proposals with other policies, over a wide selection of transmission time distributions.
We present the FlipDyn, a dynamic game in which two opponents (a defender and an adversary) choose strategies to optimally takeover a resource that involves a dynamical system. At any time instant, each player can take over the resource and thereby control the dynamical system after incurring a state-dependent and a control-dependent costs. The resulting model becomes a hybrid dynamical system where the discrete state (FlipDyn state) determines which player is in control of the resource. Our objective is to compute the Nash equilibria of this dynamic zero-sum game. Our contributions are four-fold. First, for any non-negative costs, we present analytical expressions for the saddle-point value of the FlipDyn game, along with the corresponding Nash equilibrium (NE) takeover strategies. Second, for continuous state, linear dynamical systems with quadratic costs, we establish sufficient conditions under which the game admits a NE in the space of linear state-feedback policies. Third, for scalar dynamical systems with quadratic costs, we derive the NE takeover strategies and saddle-point values independent of the continuous state of the dynamical system. Fourth and finally, for higher dimensional linear dynamical systems with quadratic costs, we derive approximate NE takeover strategies and control policies which enable the computation of bounds on the value functions of the game in each takeover state. We illustrate our findings through a numerical study involving the control of a linear dynamical system in the presence of an adversary.
We consider the classic motion planning problem defined over a roadmap in which a vehicle seeks to find an optimal path to a given destination from a given starting location in presence of an attacker who can launch attacks on the vehicle over any edge of the roadmap. The vehicle (defender) has the capability to switch on/off a countermeasure that can detect and permanently disable the attack if it occurs concurrently. We model this problem using the framework of a zero-sum dynamic game with a stopping state being played simultaneously by the two players. We characterize the Nash equilibria of this game and provide closed form expressions for the case of two actions per player. We further provide an analytic lower bound on the value of the game and characterize conditions under which it grows sub-linearly with the number of stages. We then study the sensitivity of the Nash equilibrium to (i) the cost of using the countermeasure, (ii) the cost of motion and (iii) the benefit of disabling the attack. We then apply these results to solve the motion planning problem and compare the benefit of our approach over a competing approach based on converting the problem to a shortest path problem using the expected cost of playing the game over each edge.
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