An LQG Optimal Linear Controller for Control Systems with Packet Losses

Motivated by control applications over lossy packet networks, this paper considers the Linear Quadratic Gaussian (LQG) optimal control problem in the discrete time setting and when packet losses may occur between the sensors and the estimation-control unit and between the latter and the actuation points. Previous work [1] shows that, for protocols where packets are acknowledged at the receiver (e.g. TCP- like protocols), the separation principle holds. Moreover, in this case the optimal LQG control is a linear function of the estimated state and there exist critical probabilities for the successful delivery of both observation and control packets, below which the optimal controller fails to stabilize the system. The existence of such critical values is determined by providing analytic upper and lower bounds on the cost functional, and stochastically characterizing their convergence properties in the infinite horizon. Finally, it turns out that when there is no feedback on whether a control packet has been delivered or not (e.g. UDP-like protocols), the LQG optimal controller is in general nonlinear, as shown in [2]. There exists a special case, i.e. the observation matrix C is invertible and there is no output noise. In this case this paper shows that the optimal control is linear and critical values for arrival probabilities exist and can be computed analytically.