Model Predictive Control, or MPC, is an advanced method of process control that has been in use in the process industries such as chemical plants and oil refineries since the 1980s. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification. The models are used to predict the behavior of dependent variables (ie., outputs) of a dynamical system with respect to changes in the process independent variables (ie., inputs). In chemical processes, independent variables are most often setpoints of regulatory controllers that govern valve movement (eg., valve positioners with or without flow, temperature or pressure controller cascades), while dependent variables are most often constraints in the process (eg., product purity, equipment safe operating limits). The model predictive controller uses the models and current plant measurements to calculate future moves in the independent variables that will result in operation that honors all independent and dependent variable constraints. The MPC then sends this set of independent variable moves to the corresponding regulatory controller setpoints to be implemented in the process. Process control is an engineering discipline that deals with architectures, mechanisms, and algorithms for controlling the output of a specific process. ... Typically, processing describes the act of taking something through an established and usually routine set of procedures to convert it from one form to another, as a manufacturing procedure (processing milk into cheese) or administrative procedure (processing paperwork to grant a mortgage loan). ... A Chemical plant is an industrial process plant that manufactures chemicals, usually on a large scale. ... View of the Shell/Valero Martinez oil refinery An oil refinery is an industrial process plant where crude oil is processed and refined into useful petroleum products. ... 1980 (MCMLXXX) was a leap year starting on Tuesday. ... Process (lat. ... Empirical is an adjective often used in conjunction with science, both the natural and social sciences, which means an observation or experiment based upon experience that is capable of being verified or disproved. ... System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. ... In experimental design, a dependent variable is a variable dependent on another variable (called the independent variable). ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... In an experimental design, the independent variable (also known as predictor or regressor or manipulated variable) is the variable which is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ...

Despite the fact that most real processes are approximately linear within only a limitted operating window, linear MPC approaches are used in the majority of applications with the feedack mechanism of the MPC compensating for prediction errors due to structural mismatch between the model and the plant. In model predictive controllers that consist only of linear models, the superposition principle of linear algebra enables the effect of changes in multiple independent variables to be added together to predict the response of the dependent variables. This simplifies the control problem to a series of direct matrix algrebra calculations that are fast and robust. In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ...

The major commercial suppliers of MPC software in the US are Honeywell, AspenTech, and Emerson. Honeywell (NYSE: HON) is a major American multinational corporation that produces electronic control systems and automation equipment. ... The DeltaV digital automation system, from Emerson Process Management, a business of Emerson is a process control system built with digital technologies to improve the performance of process manufacturers plants. ...

Theory behind MPC

In principle a non-linear (or linear but state constrained) control problem could be handled by solving the Hamilton-Jacobi-Bellman equation in an offline mode (that is before the plant goes into operation). This would yield the optimal control for every possible state of the plant. In practice this approach is completely impractical from a mathematical, computational and data storage point of view for a high order plant. MPC gets around this problem by restricting attention to the current plant state and to a relatively short time horizon in the future: [t,t + T]; an online or on-the-fly calculation is used to explore state trajectories that emanate from the current state and find (via the solution of Euler-Lagrange equations) a cost-minimizing control strategy until time t + T. Before that time the plant state is sampled again and the calculations are repeated starting from the now current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is also called receding horizon control. MPC gets around the massive computational requirements of a global HJB solution, but at the cost of requiring the calculations to be run live while the plant is operating. In practice, for those cases where such fast calculations are possible (including the linear case), MPC has given very good results. Much academic research has been done to find fast methods of solution of Euler-Lagrange type equations, to understand the global stability properties of MPC's local optimization, and in general to improve the MPC method. To some extent the theoreticians have been trying to catch up with the control engineers when it comes to MPC. The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ... The Euler-Lagrange Equation is the major formula of the Calculus of variations. ...

See also

• System identification
• Control theory
• Control engineering
• Feedforward

System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. ... In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ... Feed-forward is a term describing a kind of system which reacts to changes in its environment, usually to maintain some desired state of the system. ...

References

• Kwon, Bruckstein, Kailath: Stabilizing state feedback design via the moving horizon method, Intl. Journal of Control, 37, 1983, pp.631-643

• Garcia, Prett, Morari: Model predictive control: theory and practice, Automatica, 25, 1989, pp.335-348

• Mayne and Michalska: Receding horizon control of nonlinear systems, IEEE Transactions on Automatic Control, 35, 1990, pp.614-624

External link

• An Overview of Industrial Model Predictive Control Technology