Control of a Multiplicative Observation Noise System: MAP-based Achievable Strategy

This paper considers the stabilization of a discrete-time linear system in the presence of multiplicative observation noise, i.e., when the controller observes the state multiplied by a random variable with a continuous density. In the case where this multiplicative observation noise has zero mean, the controller does not even have access to the sign of the state. Given this lack of sign information, a linear memoryless control strategy that optimizes the second moment would do nothing, i.e., would choose the control to be equal to zero. It is known that non-linear strategies can unboundedly outperform linear strategies for this system, yet the optimal strategy for this simple problem remains unknown.This paper provides a new achievable scheme based on the maximum a-posteriori (MAP) estimation of the state that provably stabilizes the system in any moment sense. Additionally, we can compute an explicit convergence rate for the system state. This MAP-controller emerges from a study of the evolution of the conditional density of the state. These densities illustrate the dual nature of the MAP-controller: it extracts information about the system while also driving the state towards zero. Simulations show that the MAP-controller outperforms neural-network-based control strategies as well as the previously best-known non-linear strategies.