Limitations of Linear Control Over Packet Drop Networks

In this paper, we investigate control across stochastic dropout channels. In particular, we consider the Mean Square Stability of a SISO plant in the case there is only one channel in the feedback loop and the case where both actuator and sensor channels are present. We seek optimal networked control schemes that are memoryless functions of channel state information and for each channel state are otherwise linear and time invariant functions of channel output. We establish a fundamental limit on the dropout probability allowable for the Mean Square Stability of the closed loop system. The maximal tolerable dropout probability is only a function of the unstable eigenvalues of the plant. When either the actuator or the sensor channel is present, we propose a receiver structure that can stabilize the system under the worst dropout probability; moreover, we can simultaneously design the optimal controller and receiver and show that they can be implemented in physically separated locations (decentralized). When both actuator and sensor channels are present in the loop, the main result is a centralized stabilization technique that always achieves the fundamental bound via noiseless acknowledgement from the actuation receiver. Finally, we extend the results to the more general case where also the acknowledgements are lost with a given probability and compute how the unreliable delivery of the acknowledgements affects the minimal quality of service required of the actuator and sensor channels.