(756e) Fast-Nonlinear Model Predictive Control Implementation with Open-Source Tools

New modeling platforms and state of the art optimization solvers combined allow to build algorithms, where mathematical optimization is the basis, with relative low difficulty. For instance, Nonlinear Model Predictive Control (NMPC) that requires the solution of several parametric nonlinear optimization problems.

In this context, the resulting NLPs must be constrained by a DAE model of the plant. This can be approached in several ways of which the direct transcription method is possibly the most efficient for large DAE models. This implies that the DAE model of the plant is fully discretized with a high order implicit method like collocation, and then solved simultaneously during the optimization. The resulting NLP problem will have a large and sparse structure, which can be solved with Ipopt.

We have selected Pyomo as our modelling platform and solver interphase, because it has been founded on python, and therefore allows access to other packages and high-level programming capabilities. Consequently, the NMPC implementation allows for the controller input calculation and feedback from a plant model in the same platform. This; however, comes with a computational cost which can be circumvented by using sensitivity based NMPC.

The sensitivity NMPC relies on the first order curvature information of the optimality conditions that can be used to calculate the change of the solution of the optimization problem given a perturbation of the initial set of parameters in a much faster way than solving the NLP from the beginning. Then, the controller input can be calculated for a predicted state of the plant before the real state is known, and when the real state is known the solution can be updated via sensitivity. This strategy keeps most of the stability and robustness of the original NMPC while being faster. However, sometimes the sensitivity calculation will be not reliable if the matrix associated with the linearized optimality conditions is close to singularity; therefore, we propose re-scaling and restructuring if necessary, so the calculations become more robust.

A large problem was then used as example of the application of the framework. This was the bubbling fluidized bed reactor for CO2 capture which is partial differential algebraic model that consists of 11 PDEs, some of which contain both time and space differential terms, and around 80 algebraic equations that include boundary conditions, correlations for the hydrodynamics, and thermodynamic properties. The original optimal control problem for this model takes more than 100 seconds to solve, which makes the real time NMPC implementation infeasible. Nevertheless, by implementing the previously described sensitivity based NMPC, as well as further model reduction, the controller input can be calculated in a reasonable time frame.

Finally, as part of the ongoing work, we are developing the framework for state estimation using the Moving Horizon Estimation (MHE) approach with the same tools described before. These involve calculation of the arrival cost and advanced step approaches and will demonstrate the ability of open-source tools to combine optimization and modeling for large problems.