Transmit or Retransmit: A Tradeoff in Networked Control of Dynamical Processes Over Lossy Channels With Ideal Feedback

We study networked control of a dynamical process over a lossy channel with a hybrid automatic repeat request protocol that connects a sensor to an actuator. The dynamical process is modeled by a Gauss-Markov process, and the lossy channel by a packet-erasure channel with ideal feedback. We suppose that data is communicated in the format of packets with negligible quantization error. In such a networked control system, whenever a packet loss occurs, there exists a tradeoff between transmitting new sensory information with a lower success probability and retransmitting previously failed sensory information with a higher success probability. In essence, an inherent tradeoff between freshness and reliability. To address this tradeoff, we consider a linear-quadratic-regulator performance index, which penalizes state deviations and control efforts over a finite horizon, and jointly design optimal encoding and decoding policies for the encoder and the decoder, which are collocated with the sensor and the actuator, respectively. Our emphasis here lies specifically on designing switching and control policies, rather than error-correcting codes. We show that the optimal encoding policy is a threshold switching policy and the optimal decoding policy is a certainty-equivalent control policy. In addition, we determine the equations that the encoder and the decoder need to solve in order to implement the optimal policies. More specifically, we show that the encoder must solve the Kalman filtering equations, a mismatch linear equation, and a Bellman optimality equation, while the decoder must solve a linear filtering equation and an algebraic Riccati equation.