Fast stochastic model predictive control of high-dimensional systems

Probabilistic uncertainties and constraints are ubiquitous in complex dynamical systems and can lead to severe closed-loop performance degradation. This paper presents a fast algorithm for stochastic model predictive control (SMPC) of high-dimensional stable linear systems with time-invariant probabilistic uncertainties in initial conditions and system parameters. Tools and concepts from polynomial chaos theory and quadratic dynamic matrix control inform the development of an input-output formulation for SMPC with output constraints. Generalized polynomial chaos theory is used to enable efficient uncertainty propagation through the high-dimensional system model. Galerkin projection is used to construct the polynomial chaos expansion for a general class of linear differential algebraic equations (DAEs), so that the SMPC algorithm is applicable to both regular and singular/descriptor systems. The fast SMPC approach is applied for control of an end-to-end continuous pharmaceutical manufacturing process with approximately 8000 states. The on-line computational cost of the proposed probabilistic input-output SMPC algorithm is independent of the state dimension and, therefore, alleviates the prohibitive computational costs of control of uncertain systems with large state dimension.