in use in the process industries in chemical plants and oil refineries since

the 1980s. In recent years it has also been used in power system balancing

models. Model predictive controllers rely on dynamic models of the process,

most often linear empirical models obtained by system identification. The

main advantage of MPC is the fact that it allows the current timeslot to be

optimized, while keeping future timeslots in account. This is achieved

by optimizing a finite time-horizon, but only implementing the current timeslot.

MPC has the ability to anticipate future events and can take control actions

accordingly. PID and LQR controllers do not have this predictive ability. MPC is

nearly universally implemented as a digital control, although there is

research into achieving faster response times with specially designed analog

circuitry. Overview

The models used in MPC are generally intended to represent the behavior of

complex dynamical systems. The additional complexity of the MPC control

algorithm is not generally needed to provide adequate control of simple

systems, which are often controlled well by generic PID controllers. Common

dynamic characteristics that are difficult for PID controllers include

large time delays and high-order dynamics.

MPC models predict the change in the dependent variables of the modeled

system that will be caused by changes in the independent variables. In a chemical

process, independent variables that can be adjusted by the controller are often

either the setpoints of regulatory PID controllers or the final control

element. Independent variables that cannot be adjusted by the controller are

used as disturbances. Dependent variables in these processes are other

measurements that represent either control objectives or process

constraints. MPC uses the current plant measurements,

the current dynamic state of the process, the MPC models, and the process

variable targets and limits to calculate future changes in the dependent

variables. These changes are calculated to hold the dependent variables close to

target while honoring constraints on both independent and dependent

variables. The MPC typically sends out only the first change in each

independent variable to be implemented, and repeats the calculation when the

next change is required. While many real processes are not

linear, they can often be considered to be approximately linear over a small

operating range. Linear MPC approaches are used in the majority of applications

with the feedback mechanism of the MPC compensating for prediction errors due

to structural mismatch between the model and the process. 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

algebra calculations that are fast and robust.

When linear models are not sufficiently accurate to represent the real process

nonlinearities, several approaches can be used. In some cases, the process

variables can be transformed before and/or after the linear MPC model to

reduce the nonlinearity. The process can be controlled with nonlinear MPC that

uses a nonlinear model directly in the control application. The nonlinear model

may be in the form of an empirical data fit or a high-fidelity dynamic model

based on fundamental mass and energy balances. The nonlinear model may be

linearized to derive a Kalman filter or specify a model for linear MPC.

An algorithmic study by El-Gherwi, Budman, and El Kamel shows that

utilizing a dual-mode approach can provide significant reduction in online

computations while maintaining comparative performance to a non-altered

implementation. The proposed algorithm solves N convex optimization problems in

parallel based on exchange of information among controllers.

= Theory behind MPC = MPC is based on iterative,

finite-horizon optimization of a plant model. At time t the current plant state

is sampled and a cost minimizing control strategy is computed for a relatively

short time horizon in the future: . Specifically, an online or on-the-fly

calculation is used to explore state trajectories that emanate from the

current state and find a cost-minimizing control strategy until time . Only the

first step of the control strategy is implemented, then the plant state is

sampled again and the calculations are repeated starting from the new 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. Although this approach is not optimal, in practice it

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.

= Principles of MPC = Model Predictive Control is a

multivariable control algorithm that uses:

an internal dynamic model of the process a history of past control moves and

an optimization cost function J over the receding prediction horizon,

to calculate the optimum control moves. An example of a non-linear cost function

for optimization is given by: without violating constraints

With: = i -th controlled variable

= i -th reference variable = i -th manipulated variable

= weighting coefficient reflecting the relative importance of

= weighting coefficient penalizing relative big changes in

etc. Nonlinear MPC

Nonlinear Model Predictive Control, or NMPC, is a variant of model predictive

control that is characterized by the use of nonlinear system models in the

prediction. As in linear MPC, NMPC requires the iterative solution of

optimal control problems on a finite prediction horizon. While these problems

are convex in linear MPC, in nonlinear MPC they are not convex anymore. This

poses challenges for both NMPC stability theory and numerical solution.

The numerical solution of the NMPC optimal control problems is typically

based on direct optimal control methods using Newton-type optimization schemes,

in one of the variants: direct single shooting, direct multiple shooting

methods, or direct collocation. NMPC algorithms typically exploit the fact

that consecutive optimal control problems are similar to each other.

This allows to initialize the Newton-type solution procedure

efficiently by a suitably shifted guess from the previously computed optimal

solution, saving considerable amounts of computation time. The similarity of

subsequent problems is even further exploited by path following algorithms

that never attempt to iterate any optimization problem to convergence, but

instead only take one iteration towards the solution of the most current NMPC

problem, before proceeding to the next one, which is suitably initialized.

While NMPC applications have in the past been mostly used in the process and

chemical industries with comparatively slow sampling rates, NMPC is more and

more being applied to applications with high sampling rates, e.g., in the

automotive industry, or even when the states are distributed in space

Robust MPC Robust variants of Model Predictive

Control are able to account for set bounded disturbance while still ensuring

state constraints are met. There are three main approaches to robust MPC:

Min-max MPC. In this formulation, the optimization is performed with respect

to all possible evolutions of the disturbance. This is the optimal

solution to linear robust control problems, however it carries a high

computational cost. Constraint Tightening MPC. Here the

state constraints are enlarged by a given margin so that a trajectory can be

guaranteed to be found under any evolution of disturbance.

Tube MPC. This uses an independent nominal model of the system, and uses a

feedback controller to ensure the actual state converges to the nominal state.

The amount of separation required from the state constraints is determined by

the robust positively invariant set, which is the set of all possible state

deviations that may be introduced by disturbance with the feedback

controller. Multi-stage MPC. This uses a

scenario-tree formulation by approximating the uncertainty space with

a set of samples and the approach is non-conservative because it takes into

account that the measurement information is available at every time stages in the

prediction and the decisions at every stage can be different and can act as

recourse to counteract the effects of uncertainties. The drawback of the

approach however is that the size of the problem grows exponentially with the

number of uncertainties and the prediction horizon.

Commercially available MPC software Commercial MPC packages are available

and typically contain tools for model identification and analysis, controller

design and tuning, as well as controller performance evaluation.

A survey of commercially available packages has been provided by S.J. Qin

and T.A. Badgwell in Control Engineering Practice 11 733–764.

See also System identification

Control theory Control engineering

Feed-forward References

Further reading Kwon, W. H.; Bruckstein, Kailath.

"Stabilizing state feedback design via the moving horizon method".

International Journal of Control 37: pp.631–643.

doi:10.1080/00207178308932998. CS1 maint: Extra text

Garcia, C; Prett, Morari. "Model predictive control: theory and

practice". Automatica 25: pp.335–348. doi:10.1016/0005-1098(89)90002-2. CS1

maint: Extra text Mayne, D.Q.; Michalska. "Receding

horizon control of nonlinear systems". IEEE Transactions on Automatic Control

35: pp.814–824. doi:10.1109/9.57020. CS1 maint: Extra text

Mayne, D; Rawlings, Rao, Scokaert. "Constrained model predictive control:

stability and optimality". Automatica 36: pp.789–814.

doi:10.1016/S0005-1098(99)00214-9. CS1 maint: Extra text

Allgöwer; Zheng. Nonlinear model predictive control. Progress in Systems

Theory 26. Birkhauser. Camacho; Bordons. Model predictive

control. Springer Verlag. Findeisen; Allgöwer, Biegler. Assessment

and Future Directions of Nonlinear Model Predictive Control. Lecture Notes in

Control and Information Sciences 26. Springer.

Diehl, M; Bock, Schlöder, Findeisen, Nagy, Allgöwer. "Real-time optimization

and Nonlinear Model Predictive Control of Processes governed by

differential-algebraic equations". Journal of Process Control 12:

pp.577–585. doi:10.1016/S0959-1524(01)00023-3. CS1

maint: Extra text External links

Control Tuning and Best Practices P. Orukpe: Basics of Model Predictive

Control Case Study. Lancaster Waste Water

Treatment Works, optimisation by means of Model Predictive Control from

Perceptive Engineering ACADO Toolkit - Open Source Toolkit for

Automatic Control and Dynamic Optimization providing linear and

non-linear MPC tools. jMPC Toolbox - Open Source MATLAB

Toolbox for Linear MPC. Model Predictive Control Free book

edited by Tao Zheng, Publisher: Sciyo, 2010.

Study on application of NMPC to superfluid cryogenics.

Nonlinear Model Predictive Control Toolbox for MATLAB and Python

Model Predictive Control Toolbox from MathWorks for design and simulation of

model predictive controllers in MATLAB and Simulink

Pulse step model predictive controller - virtual simulator

Tutorial on MPC with Excel and MATLAB Examples