Optimal Output Feedback Control Using Two Remote Sensors Over Erasure Channels

Consider a discrete-time, linear time-invariant process, two sensors and one controller. The process state is observed in the presence of noise by the sensors, which are connected to the controller via links that feature erasure. If a link transmits successfully then a finite-dimensional vector of real numbers is conveyed from the sensor to the controller. If an erasure event occurs, then any information conveyed over the link is lost. This paper addresses the problem of designing the maps that specify the processing at the controller and at the sensors to minimize a quadratic cost function. When the information is lost over the links either in an independent and identically distributed (i.i.d.) or in a (time-homogeneous) Markovian fashion, we derive necessary and sufficient conditions for the existence of maps such that the process is stabilized in the bounded second moment sense. We also solve the optimal design problem in the presence of delayed noiseless acknowledgment signals at the sensors from the controller for an arbitrary packet drop pattern. We provide explicit recursive schemes to implement our solution. We also indicate how our approach can be extended to situations when more than two sensors are available and when the sensors can cooperate. The analysis also carries over to the case when each point-to-point erasure link connecting the sensors and the controller is replaced by a network of erasure links.