Optimal Scheduling of Uplink-Downlink Networked Control Systems with Energy Harvesting Sensor

Abstract

In this work, we consider a wireless networked control system (WNCS) consisting of a plant, a battery-operated sensor, a controller, and an actuator. The battery in the sensor harvests energy from the environment. The sensor then uses this energy for packet transmissions. There are two types of wireless communication channels, (i) sensor--controller channel (also called uplink channel), and (ii) controller--actuator channel (also called downlink channel). The controller is \emph{half-duplex}, and this prevents it from simultaneously receiving an update from the sensor, and also transmitting a control packet to the actuator. Though frequent transmissions via uplink channel improve controller's estimate of the plant state, but this also reduces the timely control of the plant. Hence, in order to strike a balance between these two, we consider the problem of designing an optimal scheduling policy that minimizes the expected cumulative infinite horizon discounted cost, where the instantaneous cost is equal to the square of the plant state. At each time $t$, the scheduler at the sensor has to decide whether it should activate the uplink channel, or downlink. We pose this dynamic optimization problem as a Markov decision process (MDP), in which the state at time $t$ is composed of (i) the plant state $x(t)$, (ii) the age of the data packet available at the controller, denoted by $τ(t)$, (iii) a binary variable $y(t)$ which indicates the availability of a control packet at the controller, and (iv) the energy level of the battery at the sensor $b(t)$. We show that there exists an optimal scheduling policy that exhibits a threshold structure, meaning that for each time $t$, if there is a control packet available with the controller, then the sensor activates the downlink channel in case $|x(t)|$ exceeds a threshold $x\ust(τ(t),b(t))$.