On optimal preview control across erasure communication links

Consider two identical discrete-time, finite-dimensional, linear and time-invariant systems denoted as leader and follower. The leader system is driven by a deterministic control input and by a zero-mean white Gaussian process noise. In this paper, we address the problem of designing a networked control scheme for the follower system that guarantees that the state of the follower system tracks the state of the leader system optimally, according to a mean squared cost. We adopt a networked control scheme featuring two erasure links and three design blocks: a controller acting on the follower system and two encoders, one at the output of each system. The controller has remote access to noisy measurements of the state of both systems, via two erasure links that are used to connect each encoder to the controller. We consider erasure links whose erasure events occur according to a Bernoulli process. If an erasure occurs then no information is transmitted, otherwise a finite vector of real numbers is conveyed through the link. The purpose of the encoders is to process noisy measurements of the output of each system prior to transmission over the corresponding erasure link. While the encoder of the follower system has access to current measurements, the encoder of the leader system has access to measurements that are advanced in time by a finite time interval (also denoted as preview). This paper describes a methodology for the design of controller and encoders that are jointly optimal, with respect to optimal tracking of the leader system by the follower system. We also obtain explicit necessary and sufficient conditions for the existence of a scheme that guarantees that the tracking error has finite second moment.