In this work we study mean-square (MS) stability and mean-square (MS) performance for discrete-time, finite-dimensional linear time-varying systems with dynamics subject to i.i.d. random variation.

We do so primarily in the context of networked control systems (NCS) where the network communication channels are unreliable and modeled as multiplicative stochastic uncertainties, e.g. wireless links subject to packet dropouts and modeled as Bernoulli processes.

We first focus on the analysis problem in general. We derive a convex feasibility problem and associated convex optimization problem

which can be used to determine the MS stability and MS performance respectively of a given system. Since this analysis theory is derived in terms of the feasibility of and optimization of a linear cost subject to linear matrix inequalities (LMIs), it serves as the foundation from which a solution methodology for numerous controller synthesis problems can be derived.

Next we formulate the main synthesis problem we consider in this work: a networked control system where both the sensor measurements for the plant(s) and the commands from the controller are transmitted via unreliable communication channels. We treat the unreliable communication links as i.i.d. random processes. We assume that the plant(s) and links are subject to additive exogenous noise, and that we have access to a reliable but delayed acknowledgment of whether or not the controller commands were received by the plant(s) on the previous time step. Finally we restrict the controller to be finite-dimensional, linear, have no structural dependence on the particular path history of the random processes, and scale in size and complexity linearly with the number of random channels.

We then show that this synthesis problem has a MS stabilizing solution if and only if two simpler convex problems have MS stabilizing solutions, and moreover that the optimal MS performance solution to this synthesis problem if it exists can be obtained by solving a sequence of these simpler convex problems. Additionally, we show that the overall optimal MS performance cost is the sum of two components which can be determined from the solutions to the special problems. That is, we derive a separation principle for our problem analogous to the classical H2 synthesis.