Team Optimal Control of Stochastically Switched Systems with Local Parameter Knowledge

This paper considers the optimal decentralized control of linear systems with stochastically switched cost and system matrices depending on local parameters. Two types of dynamic switched problems are considered, with partially nested and one-step delayed sharing information structures. For the former case, parameters and measurements follow a partially nested structure with the parameters possibly being correlated across all stages. For the latter case, parameters are assumed to be Markov processes, with their values along with measurements available instantaneously to local controllers, but with a one time step delay to others. The solution to both these problems rely on the optimal solution to a static (one-stage) stochastic-parameter problem with local type dependent Gaussian measurements, and for this purpose the static quadratic team problem is examined in some generality. Using an operator theoretic framework, it is shown that the sequential update scheme always converges to the team optimal strategy for general static quadratic team setups. This, when applied to the static stochastic-parameter problem, yields the team optimal strategy. The corresponding strategy is affine in the measurements with the parameter dependent coefficients obtained by solving a set of linear equations. These equations are immediately solvable when the total number of parameter values is finite. However, for the case of infinite parameter values, the update scheme also provides a mechanism to determine an approximation to the team optimal strategy.