We consider the problem of controlling a linear dynamical system from bilinear observations with minimal quadratic cost. Despite the similarity of this problem to standard linear quadratic Gaussian (LQG) control, we show that when the observation model is bilinear, neither does the Separation Principle hold, nor is the optimal controller affine in the estimated state. Moreover, the cost-to-go is non-convex in the control input. Hence, finding an analytical expression for the optimal feedback controller is difficult in general. Under certain settings, we show that the standard LQG controller locally maximizes the cost instead of minimizing it. Furthermore, the optimal controllers (derived analytically) are not unique and are nonlinear in the estimated state. We also introduce a notion of input-dependent observability and derive conditions under which the Kalman filter covariance remains bounded. We illustrate our theoretical results through numerical experiments in multiple synthetic settings.

In psychology and neuroscience it is common to describe cognitive systems as input/output devices where perceptual and motor functions are implemented in a purely feedforward, open-loop fashion. On this view, perception and action are often seen as encapsulated modules with limited interaction between them. While embodied and enactive approaches to cognitive science have challenged the idealisation of the brain as an input/output device, we argue that even the more recent attempts to model systems using closed-loop architectures still heavily rely on a strong separation between motor and perceptual functions. Previously, we have suggested that the mainstream notion of modularity strongly resonates with the separation principle of control theory. In this work we present a minimal model of a sensorimotor loop implementing an architecture based on the separation principle. We link this to popular formulations of perception and action in the cognitive sciences, and show its limitations when, for instance, external forces are not modelled by an agent. These forces can be seen as variables that an agent cannot directly control, i.e., a perturbation from the environment or an interference caused by other agents. As an alternative approach inspired by embodied cognitive science, we then propose a nonmodular architecture based on the active inference framework. We demonstrate the robustness of this architecture to unknown external inputs and show that the mechanism with which this is achieved in linear models is equivalent to integral control.