(610b) Stochastic Model Predictive Control of High-Dimensional Systems: An End-to-End Continuous Pharmaceutical Manufacturing Case Study

Probabilistic uncertainties are ubiquitous in complex dynamical systems and can impair closed-loop performance of model predictive control (MPC) approaches, which are widely used to realize high-performance operation of complex systems. Stochastic MPC provides an alternative approach for conventional robust MPC strategies [1], which design the optimal control law with respect to worst-case system uncertainties that can have vanishingly small probability of occurrence. Stochastic MPC approaches shape the predicted probability density functions of system states and outputs in an optimal manner over a finite prediction horizon [2,3,4,5]. Stochastic MPC also enables considering state constraints in a probabilistic sense (so-called chance constraints) to alleviate the inherent conservatism of robust MPC approaches that merely deal with deterministic worst-case perturbations. The key challenge in stochastic MPC is however the propagation of probabilistic uncertainties. The commonly used methods (e.g., Monte Carlo and Markov Chain Monte Carlo methods [6]) for probabilistic uncertainty analysis are prohibitively expensive for real-time control. Additional complexity in stochastic MPC arises from chance constraints, which entail the computation of multivariable integrals [7].

In this talk, a fast stochastic MPC algorithm with chance constraints is presented for high-dimensional stable linear systems with time-invariant probabilistic uncertainties (e.g., uncertainties in initial conditions and system parameters). Generalized polynomial chaos (PC) theory [8,9] is used to enable efficient uncertainty propagation through the high-dimensional system model. In the PC approach, the implicit mappings between the uncertain variables/parameters and system outputs are replaced with an expansion of orthogonal polynomials, whose statistical moments can be determined efficiently from the expansion coefficients. To determine the expansion coefficients, Galerkin projection [9] is adapted to construct PC expansions 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 quadratic dynamic matrix control (QDMC) algorithm [10] is used to formulate an input-output framework for stochastic MPC with output constraints. The resulting probabilistic input-output framework is independent of the state dimension to circumvent the prohibitive computational costs of control of uncertain systems with a large state dimension. 

The proposed probabilistic input-output MPC approach with chance constraints is used for control of an end-to-end continuous pharmaceutical manufacturing process (with approximately 8000 states) [11] in the presence of time-invariant, probabilistic parametric uncertainties. The process has nine inputs and three outputs---the active pharmaceutical ingredient (API) dosage of tablets, the impurity content of tablets, and the production rate of the process [12]. The critical quality attributes (CQAs) of the manufactured tablets consist of the API dosage and impurity content of the tablets, which should be regulated consistently in a stochastic setting. The closed-loop simulation results reveal that the stochastic MPC approach leads to a much tighter distribution (i.e., lower variance) of the API dosage around its desired setpoint. In addition, the chance constraint imposed on impurity content ensures that the tablets meet the stringent drug specifications in the presence of system uncertainties. These results indicate the promising potential of the proposed fast stochastic MPC approach for application to control of high-dimensional systems (e.g., pharmaceutical manufacturing), in which stringent requirements on robust closed-loop performance should be met.               

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