(299a) Stable Economic Nonlinear Model Predictive Control without a Pre-Calculated Steady-State Optimum

Model Predictive Control (MPC) has been widely applied to chemical processes in industry for several decades. Due to its success on optimization and control for dynamic systems, MPC has been treated as a mature process control tool. Nonlinear Model Predictive Control (NMPC), which handles nonlinear models and non-convex constraints, is getting more popular because it can realize high-performance control by capturing nonlinear details in the dynamics. Traditionally, economic real-time control is achieved in a two-stage process as depicted in the process control hierarchy (Yang, 2015), where the economic control goals are translated into state setpoints in the real-time optimization (RTO) layer and then they are passed to the advanced process control layer. Next, the advanced controller such as NMPC sends out the optimal control actions to the plant based on the current state and the setpoint. In recent studies, the concept of dynamic real-time optimization (DRTO) is proposed by coordinating the RTO and the advanced control layers, and the so-called economic NMPC (eNMPC) is constructed (Rawlings, 2012, Pontes, 2015). Instead of including the tracking stage cost in the NMPC objective, eNMPC considers the economic stage cost directly based on profit, operating cost, or some other performance measure, which eliminates the inconsistencies between two layers. However, steady-state stabilization is still an important characteristic for chemical processes and standard eNMPC does not guarantee asymptotic stability because the economic stage cost is not always a K function. This is in contrast to the setpoint tracking NMPC, which has stage costs that are K functions and thus guarantee asymptotic stability under terminal conditions, when the horizon is long enough.

Recent advances to eNMPC have led to key stability results. Diehl et al. (2011) proved that if the dynamic system is strictly dissipative with the existence of a storage function and satisfies strong duality at the equilibrium point, asymptotic stability can be constructed. However, dissipativity is a system-specific property that is difficult to implement for large-scale systems such as polymer plants and distillation columns. To ensure stability for more general systems, Jäschke et al. (2014) proposed to regularize the economic stage cost by adding a large enough tracking term in order to make the stage cost strongly convex, but this additional term may dominate the economic stage cost and lead to conservative performance. Griffith et al. (2017) derived a stable eNMPC by replacing the regularization term with a stabilizing constraint for better economic performance, while maintaining stability at the same time. Nonetheless, it still requires a specific steady-state optimum to stabilize the dynamic system. Therefore, if there are any parameter updates, off-line optimizations need to re-solve in the RTO layer, which can compromise the dynamic real-time optimization.

In this talk, we present a new eNMPC formulation that does not require any pre-calculated steady-state optimum while retaining asymptotic stability. We first derive the KKT conditions for the economic steady-state problem solved in the RTO layer and then define a new Lyapunov function and a new K-function stage cost that satisfy the Lyapunov Stability Theorem. In the NMPC formulation, we retain the economic stage cost in the NMPC objective to maintain the desired performance but add a stabilizing constraint with the new Lyapunov function to ensure asymptotic stability. We apply this eNMPC formulation to a CSTR case study. When the CSTR is controlled by the standard eNMPC, its states oscillate due to periodic negative cost values and never converge to the steady state. On the other hand, when it is controlled by our eNMPC formulation, the dynamic economic optimality is observed with the states ultimately converging to the steady state. In addition, our formulation includes a tuning parameter for the stabilizing constraint that adjusts the convergence rate for the dynamic system. Moreover, we investigate the impact of terminal conditions by implementing either the terminal constraint or the terminal cost. Finally, we demonstrate the benefit of this new eNMPC by updating one of the input parameters in the CSTR model. Our result shows that the system first converges to the optimal steady state and then, after the parameter update, it succeeds to converge to the new optimal steady state without any off-line optimization calculations. This truly realizes the integration dynamic real-time optimization and asymptotically stable control.

References

1. X. Yang, Advanced-multi-step and Economically Oriented Nonlinear Model Predictive Control. PhD thesis, Carnegie Mellon University (2015).
2. J. B. Rawlings, D. Angeli, C. N. Bates, Fundamentals of Economic Model Predictive Control, 2012 IEEE 51st Annual Conference in Decision and Control (CDC) (2012) 3851–3861.
3. K. V. Pontes, I. J. Wolf, M. Embiruu, W. Marquardt, Dynamic Real-time Optimization of Industrial Polymerization Processes with Fast Dynamics, Industrial & Engineering Chemistry Research 54 (47) (2015) 11881–11893.
4. M. Diehl, R. Amrit, J. B. Rawlings, A Lyapunov Function for Economic Optimizing Model Predictive Control, IEEE Transactions on Automatic Control 56 (3) (2011) 703–707.
5. J. Jäschke, X. Yang, L. T. Biegler, Fast Economic Model Predictive Control Based on NLP-sensitivities, Journal of Process Control 24 (2014) 1260–1272.
6. D. W. Griffith, V. M. Zavala, L. T. Biegler, Robustly Stable Economic NMPC for Non-dissipative Stage Costs, Journal of Process Control 57 (2017) 116–126.