Probabilistic analysis and control of uncertain dynamic systems: Generalized polynomial chaos expansion approaches

Uncertainties are ubiquitous in mathematical models of complex systems and this paper considers the incorporation of generalized polynomial chaos expansions for uncertainty propagation and quantification into robust control design. Generalized polynomial chaos expansions are more computationally efficient than Monte Carlo simulation for quantifying the influence of stochastic parametric uncertainties on the states and outputs. Approximate surrogate models based on generalized polynomial chaos expansions are applied to design optimal controllers by solving stochastic optimizations in which the control laws are suitably parameterized, and the cost functions and probabilistic (chance) constraints are approximated by spectral representations. The approximation error is shown to converge to zero as the number of terms in the generalized polynomial chaos expansions increases. Several proposed approximate stochastic optimization problem formulations are demonstrated for a probabilistic robust optimal IMC control problem.