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This work has recently considered issues related to feedback control under uncertainty using nonlinear model predictive control methods and qualitative constraints. Specifically, Polynomial Chaos Theory was applied to a simple model system to examine the applicability of this model approach to control using qualitative constraints. Polynomial Chaos Theory allows for rapid determination of the evolution of the distribution of states in a nonlinear process response under assumed parametric uncertainty. Applying this modeling method to a closed-loop feedback control strategy allows the control engineer to pose qualitative constraints relative to uncertain process variables. Rather than a list of qualitative constraints in the form “keep pressure P2 below 450 psi” the constraint may be posed in the form “keep the 95% confidence limit on pressure P2 below 500 psi.” This allows for more aggressive control since process uncertainty is directly accommodated. However, the Polynomial Chaos Theory method results in a rather drastic explosion in the state dimension, limiting this approach to rather small nonlinear dynamic systems where the uncertainty is fairly well-understood. However, the results appear much more promising than traditional Monte Carlo methods for handling dynamic process uncertainty.

This work has also considered how to rapidly solve the related nonlinear model predictive control problem formulated lists of prioritized qualitative constraints. This type of formulation is beneficial in cases where the process has limited degrees of freedom due to actuator saturation or extra measured variables. The traditional formulation for prioritized qualitative model predictive control of nonlinear dynamic systems requires rapid solution of a nonconvex mixed-integer nonlinear programming problem. This type of problem formulation can be quite daunting to solve in real time. The proposed solution is to treat qualitative constraints as a list of soft constraints. The solution process proceeds by solving a sequence of nonlinear feasibility problems. For the highest priority control objective, the feasibility problem considers: Can the process meet this objective? If not, how close can the process get? The resulting answer is added as a hard constraint in the next iteration when a lower priority objective is considered. In this way, a simple sequence of nonlinear optimization problems can lead to reasonable solutions for this difficult numerical issue. Currently, global solution of the nonlinear optimization problem has not been considered. In past work, it has been shown that failure to consider deterministic global optimization in nonconvex optimization problems can lead to suboptimal closed-loop performance. The trade-offs between global solution and local solution leading to suboptimal performance may be considered in future work.

Dr. Gatzke was on sabbatical in Germany during the 2008-2009 year at the University of Stuttgart. During this period, he was also supported by the National Science Foundation and the Alexander von Humboldt foundation. Work related to this proposal was presented at a variety of European Universities as well as the 2008 NMPC Conference in Italy, the 2009 EUROPT Conference in Germany, and the 2009 Process Control Conference in Slovakia. Additionally, the grad student primarily supported by this work (Timur Aliyev) has successfully defended his PhD thesis.