Simultaneous design of measurement and control strategies for stochastic systems with feedback

We consider stochastic dynamic decision problems where at each step two consecutive decisions must be taken, one being what information bearing signal(s) to transmit and the other what control action(s) to exert. For finite-horizon problems involving first-order ARMA models with Gaussian statistics and a quadratic cost criterion, we show that the optimal measurement strategy consists of transmitting the innovation linearly at each stage, which in turn leads to optimality of a linear control law. We then extend this result to infinite-horizon models with discounted costs, showing optimality of linear designs. Subsequently, we show that these appealing results do not necessarily carry over to higher order ARMA models, for which we first characterize the best designs within the affine class, and then derive instances of the problem for which there exist non-linear designs that outperform the optimal designs within the affine class. The paper also includes some illustrative numerical examples on the different classes of problems considered.