Linear Reformulation of Event-Triggered LQG Control under Unreliable Communication

We consider event-triggered linear-quadratic Gaussian (LQG) control when sensor updates are transmitted over an i.i.d. packet-erasure channel. Although the optimal controller in a standard LQG setup is available in closed form, choosing when to transmit remains computationally and analytically difficult because packet drops randomize packet delivery and couple scheduling decisions with the estimation-error dynamics, making direct dynamic-programming solutions impractical. By certainty equivalence, the co-design problem becomes choosing a binary send/skip sequence that balances control performance and communication cost. We derive a closed-form expansion of the error covariance as precomputable Gramian terms scaled by a survival factor that depends only on the number of transmission attempts on each interval. This converts the problem into an unconstrained binary program that we linearize exactly via running attempt counters and a one-hot encoding, yielding a compact MILP well suited to receding-horizon implementation. On the linearized Boeing-747 benchmark, a model predictive control (MPC) scheduler lowers cost while attempting far fewer transmissions than a one-shot baseline across channel success rates.