Synonyms

MPC

Introduction

Model predictive control (MPC) refers to a class of computer control algorithms that utilize an explicit mathematical model to predict future process behavior. At each control interval, in the most general case, an MPC algorithm solves a sequence of three nonlinear programs to answer the following essential questions: where is the process heading (state estimation), where should the process go (steady-state target optimization), and what is the best sequence of control (input) adjustments to send it to the right place (dynamic optimization). The first control (input) adjustment is implemented and then the entire calculation sequence is repeated at the subsequent control cycles.

MPC technology arose first in the context of petroleum refinery and power plant control problems (Cutler and Ramaker 1979; Richalet et al. 1978). Specific needs that drove the development of MPC technology include the requirement for economic optimization and strict enforcement of safety and equipment constraints. Promising early results led to a wave of successful industrial applications, sparking the development of several commercial offerings (Qin and Badgwell 2003) and generating intense interest from the academic community (Mayne et al. 2000). Today MPC technology permeates the refining and chemical industries and has gained increasing acceptance in a wide variety of areas including chemicals, automotive, aerospace, and food processing applications. The total number of MPC applications worldwide was estimated in 2003 to be 4,500 (Qin and Badgwell 2003).

MPC Control Hierarchy

In a modern chemical plant or refinery, MPC is part of a multilevel hierarchy, as illustrated in Fig. 1. Moving from the top level to the bottom, the control functions execute at a higher frequency but cover a smaller geographic scope. At the bottom level, referred to as Level 0, proportional-integral-derivative (PID) controllers execute several times a second within distributed control system (DCS) hardware. These controllers adjust individual valves to maintain desired flows, pressures, levels, and temperatures.

Model-Predictive Control in Practice, Fig. 1
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Hierarchy of control functions in a refinery/chemical plant

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At Level 1, MPC runs once a minute to perform dynamic constraint control for an individual processing unit, such as crude distillation unit or a fluid catalytic cracker (Gary et al. 2007). It typically utilizes a linear dynamic model identified directly from process step-test data. The MPC has the job of holding the unit at the best economic operating point in the face of dynamic disturbances and operational constraints.

At Level 2, a real-time optimizer (RTO) runs hourly to calculate optimal steady-state targets for a collection of processing units. It uses a rigorous first-principles steady-state model to calculate targets for key operating variables such as unit temperatures and feed rates. These are typically passed down to several MPCs for implementation.

At Level 3, planning and scheduling functions are carried out daily to optimize economics for an entire chemical plant or refinery. Simple steady-state models are typically used at this level, with some nonlinear but mostly linear connections between model inputs and outputs. Key operating targets and economic data are typically passed to several RTO applications for implementation.

Note that a different mathematical model of the process is used at each level of the hierarchy. These models must be reconciled in some manner with current plant operation and with each other in order for the overall system to function properly.

MPC Algorithms

MPC algorithms function in much the same way that an experienced human operator would approach a control problem. Figure 2 illustrates the flow of information for a typical MPC implementation. At each control interval, the algorithm compares the current model output prediction y p to the measured output y m and passes the prediction error e and control (input) u to a state estimator, which estimates the dynamic state x. The most commonly used methods for state estimation can be viewed as special cases of an optimization-based formulation called moving horizon estimation (MHE) (Rawlings and Mayne 2009). The state estimate \(\hat{x}\), which includes an estimate of the process disturbances \(\hat{d}\), is then passed to a steady-state optimizer to determine the best operating point for the unit. The steady-state optimizer must also consider operator-entered output and control (input) targets y t and u t . The steady-state state and control (input) targets x s and u s are then passed, along with the state estimate \(\hat{x}\), to a dynamic optimizer to compute the best trajectory of future control (input) adjustments. The first computed control (input) adjustment is then implemented and the entire calculation sequence is repeated at the next control interval. The various commercial MPC algorithms differ in such details as the mathematical form of the dynamic model and the specific formulations of the state estimation, steady-state optimization, and dynamic optimization problems (Qin and Badgwell 2003).

Model-Predictive Control in Practice, Fig. 2
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Information flow for MPC algorithm

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In the general case, the MPC algorithm must solve the three optimization problems outlined above at each control interval. For the case of linear models and reasonable tuning parameters, these problems take the form of a convex quadratic program (QP) with a constant, positive-definite Hessian. As such, they can be solved relatively easily using standard optimization codes. For the case of a linear state-space model, the structure can be exploited even further to develop a specialized solution algorithm using an interior point method (Rao et al. 1998).

For the case of nonlinear models, these problems take the form of a nonlinear program (NLP) for which the solution domain is no longer convex, greatly complicating the numerical solution. A typical strategy is to iterate on a linearized version of the problem until convergence (Bielger 2010).

Implementation

The combined experience of thousands of MPC applications in the process industries has led to a near consensus on the steps required for a successful implementation:

  • Justification – make the economic case for the application.

  • Pre-test – design the control and test sensors and actuators.

  • Step-test – generate process response data.

  • Modeling – develop model from process response data.

  • Configuration – configure the software and test preliminary tuning by simulation.

  • Commissioning – turn on and test the controller.

  • Post-audit – measure and certify economic performance.

  • Sustainment – monitor and maintain the application.

The most expensive of these steps, both in terms of engineering time and lost production, is the generation of process response data through the step test. This is accomplished, in principle, by making significant adjustments to each variable that will be adjusted by the MPC while operating open loop to prevent compensating control action. This will necessarily cause abnormal movement in key operating variables, which may lead to lower throughput and off-spec products. Significant progress has been made in recent years to minimize these difficulties through the use of approximate closed-loop step testing (Darby and Nikolaou 2012).

Once the application has been commissioned, it is critical to set up an aggressive monitoring and sustainment program. MPC application benefits can fall off quickly due to changes in the process operation and as new personnel interact with it. New constraint variables may need to be added and key sections of the model may need to be updated as time goes on. The mathematical problem of MPC monitoring remains a topic of current academic research (Zagrobelny et al. 2012).

Note that the implementation steps outlined above must be carried out by a carefully selected project team that typically includes, in addition to the MPC expert, an engineer with detailed knowledge of the process and an operator with significant relevant experience.

Summary and Future Directions

Model predictive control is now a mature technology in the process industries. A representative MPC algorithm in this domain includes a state estimator, a steady-state optimizer, and a dynamic optimizer, running once a minute. A successful MPC application usually starts with a careful economic justification, includes significant participation from process engineers and operators, and is maintained with an aggressive sustainment program. Many thousands of such applications are currently operating around the world, generating billions of dollars per year in economic benefits.

Likely future directions for MPC practice include increasing use of nonlinear models, improved state estimation through unmeasured disturbance modeling (Pannocchia and Rawlings 2003), and development of more efficient numerical solution methods (Zavala and Biegler 2009).

Cross-References

Recommended Reading

The first descriptions of MPC technology appear in papers by Richalet et al. (1978) and Cutler and Ramaker (1979). A detailed summary of the history of MPC technology development, as well as a survey of commercial offerings through 2003 can be found in the review article by Qin and Badgwell (2003). Darby and Nikolaou present a more recent summary of MPC practice (Darby and Nikolaou 2012). Textbook descriptions of MPC theory and design, suitable for classroom use, include Rawlings and Mayne (2009) and Maciejowski (2002). The book by Ljung (1999) provides a good summary of methods for identifying dynamic models from test data. Theoretical properties of MPC are analyzed in a highly cited paper by Mayne and coworkers (2000). Guidelines for designing disturbance models so as to achieve offset-free control can be found in Pannocchia and Rawlings (2003). Numerical solution strategies for the nonlinear programs found in MPC are discussed in the book by Biegler (2010). An efficient interior-point method for solving the linear MPC dynamic optimization is described in Rao et al. (1998). A promising algorithm for solving the nonlinear MPC dynamic optimization is outlined in Zavala and Biegler (2009). A data-based method for tuning Kalman Filters, which are often used for MPC state estimation, is described in Odelson et al. (2006). A new method for monitoring the performance of MPC is summarized in Zagrobelny et al. (2012). A readable summary of refining operations can be found in Gary et al. (2007).