(206a) Stochastic Output Feedback Control of Nonlinear Systems with Probabilistic Uncertainties: Application to Control of Polymorphic Transformations in Batch Crystallization

Measurement noise, parametric and structural uncertainties, and exogenous disturbances are ubiquitous in complex dynamical systems. Uncertainties can lead to undesired variability of the system outputs and, as a result, a notable degradation of closed-loop performance. In this talk, a general framework will be presented for stochastic model predictive control (MPC) and state estimation of nonlinear systems with probabilistic, time-invariant uncertainties. Stochastic MPC with chance constraints (e.g., [1,2,3]) uses probabilistic descriptions of system uncertainties to realize acceptable levels of risk during system operation. Stochastic MPC enables shaping the probability distribution of system states, while ensuring the satisfaction of constraints with a desired probability level in a stochastic setting. This flexibility would have significant economic and safety implications for high performance operation of complex chemical and biological systems, which is often achieved in the vicinity of constraints. On the other hand, nonlinear estimation of the uncertain state variables enables output feedback application of stochastic MPC to (partly) suppress the effect of model mismatches and time-varying disturbances on control performance.

The key challenges in state estimation and control of nonlinear systems are the propagation of probabilistic uncertainties and reformulation of chance constraints in terms of computationally tractable expressions. In this work, generalized polynomial chaos (PC) theory [4,5] is used to present a probabilistic framework for stochastic nonlinear state estimation and control. The PC approach enables replacing the implicit mappings between the uncertain variables/parameters and system states with an expansion of orthogonal polynomials, whose statistical moments can be determined efficiently from the expansion coefficients (e.g., see [6,7,8]). Hence, PC expansions are computationally efficient surrogates for Monte Carlo-based approaches to perform uncertainty analysis for real-time estimation and control. A nonlinear Bayesian state estimation algorithm is presented that utilizes PC expansions to describe the evolution of state uncertainty, which is then used to maximize the posterior probability density function of the random state variables. The stochastic MPC approach is formulated to control the predicted probability densities of system states over a finite prediction horizon, while ensuring the satisfaction of constraints with a desired probability level. To obtain a computationally tractable stochastic optimal control problem, the individual chance constraints are transformed into explicitly convex second-order cone constraints for a general class of probability distributions with known mean and covariance [9].     

The proposed stochastic output feedback control approach is applied for optimal control of the polymorphic transformation of L-glutamic acid in cooling batch crystallization [10]. Polymorphic transformation is of paramount importance in the specialty chemical and pharmaceutical industries, as such transformations can lead to undesired variations in the chemical and physical properties of the product crystals. The control problem in polymorphic transformations is to ensure consistent manufacturing of the desired polymorph in a stochastic setting due to time-invariant uncertainties in initial conditions and crystallization kinetic parameters. The simulation results indicate that the stochastic output feedback control approach enables effective shaping of the probability density of the performance index (i.e., the nucleation crystal mass to the seed crystal mass of crystals of the desired form). Stochastic MPC results in a significant reduction (approximately 30%) in the variance of probability density of the performance index, as compared to a nonlinear MPC approach that disregards the probabilistic system uncertainties. In addition, the chance constraints in stochastic MPC ensure that the system operation remains in the designated operating region defined in terms of state constraints, whereas in the nonlinear MPC approach the constraints are violated due to the system uncertainties.

[1] A. Schwarm and M. Nikolaou. Chance-constrained model predictive control. AIChE Journal, 45:1743-1752, 1999.

[2] D. Bernardini and A. Bemporad. Scenario-based model predictive control of stochastic constrained linear systems. In Proceedings of the 48^thIEEE Conference on Decision and Control, 6333-6338, Shanghai, 2009.

[3]  G. C. Calafiore and L. Fagiano. Robust model predictive control via scenario optimization. IEEE Transactions on Automatic Control, 58:219-224, 2013.

[4] N. Wiener. The homogeneous chaos. American Journal of Mathematics, 60:897-936, 1938.

[5] D. Xiu and G. E. Karniadakis. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal of Scientific Computation, 24:619-644, 2002.

[6] Z. K. Nagy and R. D. Braatz. Distributional uncertainty analysis using power series and polynomial chaos expansions. Journal of Process Control, 17:229-240, 2007.

[7] J. Fisher and R. Bhattacharya. Linear quadratic regulation of systems with stochastic parameter uncertainties. Automatica, 45:2831-2841, 2011.

[8] A. Mesbah, S. Streif, R. Findeisen, and R. D. Braatz. Stochastic nonlinear model predictive control with probabilistic constraints. In Proceedings of the American Control Conference, Portland, 2014, in press.

[9] G. C. Calafiore and L. El Ghaoui. On distributionally robust chance-constrained linear programs. Journal of Optimization Theory and Application, 130:1-22, 2006.

[10] M. W. Hermanto, N. C. Kee, R. B. H. Tan , M. Chiu, and R. D. Braatz. Robust Bayesian estimation of kinetics for the polymorphic transformation of L-Glutamic acid crsytals. AIChE Journal, 54:3248-3259, 2008.