Deterministic robust model predictive control approaches have been developed to account for worst-case uncertainties . These approaches can lead to conservative closed-loop performance due to the low probability of occurrence of worst-case uncertainties. Stochastic model predictive control approaches address this issue by incorporating the probabilistic description of stochastic system uncertainties into the optimal control problem (e.g., see  and the references therein). This work presents an output feedback stochastic nonlinear model predictive control (SNMPC) approach for a class of nonlinear systems with unbounded stochastic uncertainties. The SNMPC approach aims to shape (multivariate) probability density function (PDF) of the stochastic states in presence of input and joint state chance constraints. The output feedback control approach accounts for the fact that system states cannot be readily measured in most practical applications, which in turn necessitates the estimation of system states from available measurements.
This work uses the Fokker-Planck (FP) equation  to describe the dynamic evolution of the PDF of states. The FP equation characterizes the complete multivariate PDF arisen from the stochastic system uncertainties in initial states and disturbances. Characterizing the complete PDF of states allows for shaping the PDF with respect to any desired (multivariate) PDF, and computing chance constraints of any complexity directly without conservative approximations. The value function of the SNMPC approach is defined in terms of the Hellinger distance , which quantifies the degree of overlap between the target PDF and the predicted PDF of states. Asymptotic stability in the probabilistic sense is ensured by using a Lyapunov-based control law. Closed-loop stability is ensured by designing a stability constraint in terms of a stochastic control Lyapunov function, which explicitly characterizes stability in a probabilistic sense . To facilitate output feedback control, a continuous-discrete particle filter  based on Bayesian estimation is used for estimating the states. In the prediction step of the particle filter, the transition PDF of the states is obtained by solving the FP equation. This PDF is then substituted in the Bayes’ equation, along with the PDF of the measurements, to yield the posterior PDF of the states.
The proposed output feedback SNMPC approach is implemented on a continuous acetone-butanol-ethanol (ABE) fermentation process . The simulation results indicate that the control approach is able to shape the PDF of states, while ensuring the satisfaction of the state constraints with a (least) prespecified probability level. The simulation case study demonstrates the capability of the SNMPC approach to systematically seek tradeoffs between the closed-loop performance and robustness to system stochasticities.
- Bemporad A., Morari M. Robust model predictive control: A survey. In Robustness in Identification and Control, 1999; 207-226.
- Mesbah A., Streif S., Findeisen R., Braatz R. D. Stochastic nonlinear model predictive control with probabilistic constraints. Proceedings of the American Control Conference, 2014; 2413-2419.
- Risken H. The Fokker-Planck Equation: Methods of Solution and Applications. Springer-Verlag, Berlin, 1984.
- Gibbs A.L., Su F.E. On choosing and bounding probability metrics. International Statistical Review, 2002; 70:419-435.
- Deng H. and Krstic M. Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance. Systems and Control Letters, 2000, 39:173-182.
- Xia Y., Deng Z., Li L., Geng, X. A new continuous-discrete particle filter for continuous-discrete nonlinear systems. Information Sciences, 2013; 242:64-75.
- Haus S., Jabbari S., Millat T., Janssen H., Fischer R. J., Bahl H., King J.R., Wolkenhauer O. A systems biology approach to investigate the effect of pH-induced gene regulation on solvent production by Clostridium acetobutylicum in continuous culture. BMC Systems Biology, 2011, 5:1-13.