Optimal control with limited control actions and lossy transmissions

We consider a class of non-standard networked control problems where (i) there is a limit on the number of times control signals can be transmitted to the plant, and (ii) the links over which the transmission takes place are lossy in the sense that there is a nonzero probability with which packets carrying control signals are dropped. The framework is that of discrete-time LQG optimal control where the control has access to noisy state measurements, and the objective is minimization of the expected value of a quadratic performance index under the nontraditional constraints introduced above. When there is a limitation on the number of control actions, we show that the optimal policy involves thresholding the optimal estimate of the state. When there is also a lossy link, and the control is allowed to receive acknowledgements from the plant as to whether the transmitted packets were received or not, we show that the optimal policy is again of the threshold type, involving off-line computation.