(744a) Nonlinear Model Predictive Control for a Process with Steady-State Gain Inversion | AIChE

(744a) Nonlinear Model Predictive Control for a Process with Steady-State Gain Inversion


1. NMPC Methodology

Nonlinear Model Predictive Control (NMPC) is used infrequently for industrial applications compared to Linear Model Predictive Control (LMPC) [1] due to as­sociated modeling and computational burden, and most of these applications are in the field of polymerization control. Industrial use of NMPC is typically justified for processes where LMPC will not be able to achieve process control objectives due to severe nonlinearities in steady-state gains, dead times and dynamic response [2]. A robust NMPC formulation is required to control the output variable at the peak for an industrial application with steady-state gain inversion due to process dis­turbances and potential modeling errors. The measured feed-forward disturbances along with any unmeasured disturbances also have an effect on the location of the peak or optimum for the controlled variable. Researchers have attempted to address the problem of controlling a process that exhibits change in the sign of steady-state gain for output variable with respect to the manipulated input variable. Lack of stability for an unconstrained nonlinear controller at the peak for processes with steady-state gain inversion has been discussed [3, 4].

The proposed nonlinear model predictive controller augmented with a custom output measurement along with an appropriate disturbance model [5] is different and better because it solves the nonlinear constrained optimization problem explic­itly without formulating an unconstrained control law thereby ensuring robustness for controlling at the optimum point where steady-state gain changes sign. Previous simulation work focused on unconstrained controller formulation and appropriate tuning to prevent instability in presence of disturbances for a process with steady-state gain inversion. For ill-conditioned processes with steady-state gain inversion and the control objective of maximizing the output at the peak, the closed-loop controller will exhibit instability when the sign of the gain is different between the model and the plant [6]. In this work, the controller objective is to control at the peak where steady-state gain inversion happens in presence of disturbances instead of preventing instability that arises from operating at that point. Control­ling at the optimum in presence of measured and unmeasured disturbances will lead to frequent steady-state gain inversion and needs appropriate estimation of unmeasured plant-model mismatch for proper control action that maximizes the controlled output. Problems arise in applications to control actual industrial pro­cesses with gain inversion due to significant process disturbances and potential modeling errors thereby increasing the importance of a robust solution. The novel features of the proposed NMPC to maximize an output variable that has steady-state gain inversion with respect to the manipulated input are

  • Use of the optimum steady-state manipulated input as a custom output mea­surement that is available infrequently,
  • Input disturbance model that utilizes the infrequent custom output measure­ment to act as an integrator for effective offset free control

The optimum steady-state manipulated input is used as an infrequent custom mea­surement instead of the available output directly to provide a robust estimate of the input disturbance that accounts for plant-model mismatch and unmodeled distur­bances. Robust identification of the input disturbance is important to ensure effi­cient maximization of the output for noisy industrial data with potential modeling errors.

2. NMPC application

2.1 Process Description

Ethylene oxide is produced by using silver based catalysts for selective oxidation of ethylene to ethylene oxide thereby minimizing secondary reactions that decrease ethylene oxide (EO) selectivity. For conventional catalysts, EO selectivity does not reach values above 85.7 percent that had long been considered as the theoretical maximum selectivity for the process [7]. Modern industrial ethylene epoxidation reactors use co-fed chlorination promoters that are adsorbed on the catalyst to pro­vide moderate chlorine coverage thereby further increasing selective oxidation to ethylene oxide. These high efficiency industrial catalysts also tend to exhibit rela­tively steep parabolic curves for EO selectivity as a function of adsorbed chloride that is measured as a dimensionless chlorination parameter (Z). The location of the peak or optimum EO selectivity is also a strong function of reaction temper­ature that is used to control EO production rate. The objective of the feedback controller is to maximize EO selectivity by manipulating the chlorination param­eter (Z) in presence of disturbances due to EO production rate and inlet oxygen concentration. The location of peak where the steady-state gain for EO selectiv­ity with respect to chlorination parameter reaches a maximum and changes sign is dependent on measured disturbances and unmeasured disturbances. Unmeasured disturbances that also include the mismatch in predicted catalyst performance as per its age will cause the location of the optimized chlorination parameter for max­imizing EO selectivity to be different.

2.2 NMPC Results

EO selectivity has been maximized more effectively using NMPC than the old control scheme that used the steady-state non-linear process model to calculate open-loop targets for chlorination parameter (Z) resulting in an average gain of 0.5 − 1 percent in selectivity for the industrial process after taking catalyst aging and EO production rate effects into account. Three different scenarios have been shown with plant data for the NMPC application to maximize EO selectivity [5]

  • EO Selectivity control at high EO production rate
  • EO Selectivity control with increasing EO production rate
  • EO Selectivity control for a big unmeasured disturbance

The moves for the chlorination parameter, Z, are implemented using the proposed NMPC controller by passing them as targets to the secondary LMPC or PID con­troller that controls the chlorination parameter by manipulating ethyl chloride flow and rejects faster disturbances. Strong nonlinearities due to gain inversion for max­imization of EO selectivity along with noisy measurements for industrial data ne­cessitate the formulation of a novel robust NMPC controller. The computational burden for NMPC has been reduced to enable real-time industrial application by formulating a hierarchical controller for EO selectivity and chlorination parameter.

3. Conclusions

The proposed NMPC controller formulation for a process with steady-state gain inversion utilizes the optimum steady-state manipulated input as a custom infre­quent output measurement. The location of optimum steady-state manipulated input where controlled output is maximized depends on both measured and un­measured disturbances, and is used to update the unmeasured input disturbance estimate infrequently. Accurate identification of the input disturbance is problem­atic with using the controlled output directly due to associated input multiplicity. Robust identification of input multiplicity is important for a controller that has the goal of staying at the peak with noisy industrial data as measurements. The NMPC controller can also be used for processes that exhibit steady-state gain inversion at a minimum instead of a maximum with the control objective of minimizing at the trough. For a process that may have multiple gain inversions, it will be important to locate the global optimum to maximize the output for the associated non-convex optimization problem.

The NMPC controller has been applied successfully with results from indus­trial ethylene epoxidation reactor that show robustness in maximizing EO selec­tivity of the effluent ethylene oxide at the peak where steady-state gain inversion occurs [5]. The industrial NMPC has been in use since October 2018 to max­imize EO selectivity in presence of measured and unmeasured disturbances that affect the location of the peak where steady-state gain inversion occurs. Due to the success achieved in maximizing EO selectivity that resulted in significant com­mercial value associated with an average gain of 0.5 − 1 percent in EO selectivity for similar catalyst age and EO production rates, the business unit has made it a part of their strategy to deploy these NMPC applications at other EO production sites. A second application for maximizing EO selectivity at another Dow location has been implemented in July 2019 while another project to deploy a third NMPC application for maximization of EO selectivity will be implemented in May 2020.


  1. J. Qin, T. A. Badgwell, A survey of industrial model predictive technology, Control Engineering Practice 11 (2003) 733–764.
  2. Bindlish, Nonlinear model predictive control of an industrial polymerization process, Computers and Chemical Engineering 73 (2015) 43–48.
  3. T. Biegler, J. B. Rawlings, Optimization approaches to nonlinear model pre­dictive control, Fourth International Conference on Chemical Process Control, Padre Island, TX (1991) 543–571.
  4. Daoutidis, C. Kravaris, Dynamic output feedback control of minimum-phase nonlinear processes, Chemical Engineering Science 47 (7) (1992) 837–849.
  5. Bindlish, Nonlinear model predictive control of an industrial process with steady-state gain inversion, Computers and Chemical Engineering 135 (2020) accepted for publication.
  6. G. Pannocchia, J. B. Rawlings, Disturbance models for offset-free model-predictive control, AIChE Journal 49 (2) (2003) 426–437.
  7. L. Zhang, A. C. Liu, M. Habenschuss, Method of achieving and maintaining a specified alkylene oxide production parameter with a high efficiency catalyst, patent No. US 8,362,284 B2 (January 29 2013).


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