Structural Properties of Optimal Risk-Aware Controllers for Spatially Invariant Systems

We analyze the optimal Linear Exponential Quadratic Gaussian (LEQG) control synthesis of a spatially distributed system with a shift invariance in its spatial coordinate, perturbed by additive white Gaussian noise. We refer to such a system as spatially invariant. The LEQG framework accounts for the risk attitude of the controller in its synthesis by appropriate selection of the value of a free parameter, providing the possibility to continuously tune the degree of risk-awareness of the controller. We prove important structural properties of the optimal LEQG control problem for spatially invariant systems, namely that: (i) the optimal LEQG control gain is spatially invariant itself; (ii) the LEQG control synthesis problem is equivalent to a family of decoupled LEQG optimization problems of smaller dimension; and (iii) under some further assumptions, the optimal LEQG control gain is spatially localized. Through a case study, we illustrate how the risk attitude of the controller tunes the degree of spatial localization of the optimal control gain. We argue that the proven structural properties can be leveraged to reduce the computational complexity of obtaining the optimal LEQG control gain in large-scale systems and to design distributed risk-aware controller implementations.