(564d) Nonlinear Model Predictive Control for Zone Tracking

Model predictive control (MPC) is the most widely applied advanced control technique in the process industry due to its many advantages such as optimally handling process constraints and interaction. In the traditional paradigm, MPC fulfills the objective of set-point tracking by penalizing the deviation of the predicted state and input trajectories to the desired set-point. In practice, it is often beneficial to design MPC that tracks a zone region instead of a set-point. The superiority of zone control over set-point control lies in its ability to flexibly handle multiple objectives so that the degrees of freedom in the controller can be assigned to address more important objectives. Successful applications of zone MPC have been reported in various areas such as diabetes treatment [1], control of building heating system [2], and control of the FCC process in oil refinery [3].

Despite the widely successful industrial applications, there has not been a systematic approach for design and analysis of MPC that tracks a zone region. On the one hand, reported industrial zone MPC designs are heuristic in nature and lack stability guarantee. On the other hand, existing zone MPC designs with guaranteed stability essentially use ad hoc approaches to convert zone control to set-point control (e.g., [4]). Several difficulties arise when it comes to the design and analysis of MPC for zone tracking. First of all, the admissibility of a zone target needs to be carefully examined. Since the target zone is often user-specified based on the economic metrics of the process, it is not necessarily positive invariant. One might be tempted to find the largest positive invariant set in the target zone and convert the zone tracking objective to tracking of its largest positive invariant subset. Unfortunately, finding and characterizing the maximal positive invariant set in the target zone is very difficult for generic nonlinear systems, if possible at all. Even if such a set can be found, the aforementioned approach may not be desirable and may give deteriorated transient performance as shown by our simulation study.

In this talk, we propose a general framework for nonlinear model predictive zone control. The proposed zone MPC tracks a target zone characterized by coupled system state and input. A control invariant subset of the target zone is incorporated in the proposed zone MPC as the terminal constraint to ensure closed-loop stability. We resort to LaSalle's invariance principle and develop invariance-like theorem which is suitable for stability analysis of zone control. It is proved that under the proposed zone MPC design, the system converges to the largest positive invariant subset of the target zone. In the stability analysis, we focus on the evolvement of the state-input pair (x(n), u(n)) instead of merely the state trajectory x(n). This provides a better picture and more accurate description of the system behavior. Further discussions are made on enlargement of the region of attraction by employing an auxiliary controller as well as handling a secondary economic objective via multi-objective optimization.

We use two simple simulation examples to demonstrate the superiority of zone control over set-point control and the efficacy of our zone MPC design. The first example involves a discrete nonlinear scalar system. From its bifurcation diagram, it is clear how relaxing a set-point target to a zone target allows for more admissible operations in addition to steady state operations such as periodic orbits and chaotic attractors. The second example is a linear system with constrained state and input. In the simulation we compared three different zone MPC configurations: (1) a zone MPC that tracks the target zone, (2) a zone MPC that tracks the largest positive invariant set in the zone, and (3) the proposed zone MPC. Very interestingly, the results are analogous to the comparison between set-point MPC and economic MPC [5] where case (1), (2), (3), corresponds to economic MPC without terminal constraint, set-point MPC, and economic MPC with terminal constraint.

References

[1] Grosman, B., Dassau, E., Zisser, H.C., Jovanovič, L. and Doyle III, F.J., 2010. Zone model predictive control: a strategy to minimize hyper-and hypoglycemic events. Journal of diabetes science and technology, 4(4), pp.961-975.

[2] Privara, S., Široký, J., Ferkl, L. and Cigler, J., 2011. Model predictive control of a building heating system: The first experience. Energy and Buildings, 43(2), pp.564-572.

[3] Zanin, A.C., De Gouvea, M.T. and Odloak, D., 2002. Integrating real-time optimization into the model predictive controller of the FCC system. Control Engineering Practice, 10(8), pp.819-831.

[4] Ferramosca, A., Limon, D., González, A.H., Odloak, D. and Camacho, E.F., 2010. MPC for tracking zone regions. Journal of Process Control, 20(4), pp.506-516.

[5] Diehl, M., Amrit, R. and Rawlings, J.B., 2011. A Lyapunov function for economic optimizing model predictive control. IEEE Transactions on Automatic Control, 56(3), pp.703-707.