(6c) Advanced Nonlinear Programming Formulations and Algorithms: Expanding the Scope of Industrial Nmpc Applications

During the last decades we have witnessed a widespread use of nonlinear programming (NLP) techniques and algorithms for the solution of a wide range of problems arising in engineering and science. In addition, it has also become clear that advances in optimization and the continuous increase of computational power are directly coupled to the application scope. In this work, we will present a complex problem posed by an industrial collaborator: the low-densitiy-polyethylene (LDPE) tubular reactor process. This application has served as a clear reference of the scope of current optimization capabilities. This has been useful in making a critical assessment of the potential of different optimization strategies and algorithms and, at the same time, has motivated the development of a number of novel NLP formulations and the expansion of current capabilities of NLP solvers.

In collaboration with Prof. Lorenz T. Biegler (CMU), Dr. Carl D. Laird (Texas A&M) and our industrial partners, we have expanded the current capabilities of the large-scale NLP solver (IPOPT) that has allowed for the solution of different types of optimization problems arising in Nonlinear Model Predictive Control frameworks. Some highlights of the results accomplished so far are the following:

1) Advanced Parameter Estimation Formulations: We have developed a general parameter estimation framework around IPOPT that allows for the analysis and solution very large-scale parameter estimation problems in both standard and parallel computer architectures [1][2]. We have tested our developments using an industrial low-density polyethylene (LDPE) tubular reactor model.

In this particular application, the LDPE reactor model is so complex that these estimation problems were previously thought intractable [3]. In our framework, we have been able to overcome these limitations and solve error-in-variables (EVM) parameter estimation problems with up to 32 data sets containing around 4,100 ordinary differential equations, 16,000 algebraic equations and 2,100 degrees of freedom in a distributed cluster in around 4 minutes of CPU time. The framework consists of a simultaneous collocation-based strategy and a parallel Schur complement decomposition strategy implemented IPOPT. In addition, we have established an efficient strategy to extract large-scale parameter covariance matrices from IPOPT which are essential for further statistical analysis. Significant improvements in the model predictions have been obtained and a systematic parameter inference analysis has been performed for the first time in this type of rigorous models. The rigorous LDPE reactor model has been tested against industrial plant data and an on-line parameter estimator has been developed and delivered to our industrial collaborator. Since the estimation framework uses general optimization techniques, it can also handle more general parameter estimation formulations.

2) Fast Nonlinear Model Predictive Control: When dealing with large-scale rigorous models in optimization-based control strategies such as NMPC, it is clear that not even the fastest optimization algorithms currently available will be able to provide fast feedback to the plant. Instead, what is usually observed is that there exists a long computational delay (several minutes) between the instant in which the new state of the system is received and the instant in which the optimization problem has been solved and the control action becomes available. This computational delay has been identified as one of the main limitations associated to NMPC since it has a direct impact on the stability and performance of the controller. On the other hand, the moving horizon optimal control problems (OCPs) solved in NMPC present an interesting property that can be exploited to obtain very fast approximate solutions: the problem is parametric in the initial conditions given by the moving state of the system. While fully explicit parametric control approaches cannot be currently applied to complex industrial applications, it is possible to exploit the parametric property of OCPs by computing and updating (on-line) explicit control laws using the concept of NLP sensitivity. We have developed different computational delay-free NLP sensitivity controller variants based on a simultaneous collocation-based approach and full-space optimization solvers (such as IPOPT) and tested our developments on a simplified industrial LDPE process model [4]. In this application, it has been possible to reduce the on-line computational tasks by 2-3 orders of magnitude and we are now able to provide instantaneous feedback in around 1 second. Finally, we have studied the nominal and robust stability properties of the proposed fast NMPC controllers to analyze their interactions with numerical errors, model-mismatch and state estimation errors [5].

3) Fast Moving Horizon Estimation: The parametric property of moving horizon problems arising in NMPC can also be exploited in the moving horizon estimation (MHE) case through appropriate formulations. With this, we have developed a fast MHE strategy that is able to provide state estimates without computational delay [6]. We have tested the developments on the LDPE process model were fast and accurate state estimates have been obtained. Interestingly, the numerical properties of the KKT matrix can be related to the observability of properties of the dynamic system. In addition, we are currently studying the theoretical properties of the fast MHE algorithm and analyzing the interactions of NLP sensitivity errors on the stability properties of the estimator.

4) Parallel Solution of DAE-constrained Optimization Problems with Loosely Coupled Algebraic Components: We have found that the main challenge posed by our industrial application is the size and complexity of the LDPE tubular reactor model. On the other hand, the LDPE process presents the interesting property that the dynamics of the reactor and those of the rest of the plant occur at significantly different time-scales. Because of this, the LDPE reactor model can usually be assumed to be at quasi-steady-state. This gives rise to a special class of DAE constrained problems were the proportion of algebraic states is much larger than that of differential states. This property is not specific to this application but is also present in some other complex systems such as metabolic networks, dynamic models with complex thermodynamic calculation routines, among others. We have developed a simple and practical reformulation of this type of DAE-constrained problems that leads to a general and efficient Schur complement parallel solution strategy. This decomposition required of some minor implementation modifications to the current parallel IPOPT capabilities and can be used in a wide range of optimization problems. Finally, these developments have been used to solve complicated multi-stage dynamic optimization problems arising in our industrial LDPE process in a few minutes [7].

Having this entire computational framework, we are currently incorporating all these developments on a stand-alone NMPC framework for the LDPE process. This will enable us to increase the profitability of the process through a more systematic decision-making. While the developments are directly applicable to many different industrial processes, our challenging application has also helped us in identifying potential improvements that will further expand the scope of nonlinear programming in some other industrial applications.

[1] Zavala, V. M. and Biegler, L.T. Large-Scale Parameter Estimation in Low-Density Polyethylene Tubular Reactors. Ind. Eng. Chem. Res. 45, 7867 -7881, 2006.

[2] Zavala, V. M.; Laird, C.D. and Biegler, L.T. Interior-Point Point Decomposition Approaches for Parallel Solution of Large-Scale Nonlinear Parameter Estimation Problems. Chem. Eng. Sci. Accepted for Publication, 2007.

[3] Bokis, C. P.; et.al. Physical Properties, Reactor Modeling, and Polymerization Kinetics in the Low-Density Polyethylene Tubular Reactor Process, Ind. Eng. Chem. Res. 41, 1017-1030, 2001.

[4] Zavala, V. M.; Laird, C.D. and Biegler, L.T. Fast Implementations and Rigorous Models: Can Both be Accommodated in NMPC? Int. J. Robust Nonlinear Control. Accepted for Publication, 2007.

[5] Zavala, V. M.; and Biegler, L.T. The Advanced Step NMPC Controller.

Proceedings of the IEEE Workshop on Advanced Process Control Applications for Industry, 2007.

[6] Zavala, V. M.; Laird, C.D. and Biegler, L.T. A Fast Computational Framework for Large-Scale Moving Horizon Estimation. Proceedings of the 8th International Symposium on Dynamics and Control of Process Systems, 2007.

[7] Laird, C.D.; Zavala, V. M. and Biegler, L.T. Efficient Parallel Solution Of DAE Constrained Optimization Problems With Loosely Coupled Algebraic Components. AIChE Annual Meeting, 2007.