The idea of optimal control in the presence of constraints and the intuitive design of the control law as an optimization problem has made MPC interesting for many different tasks. Applications have spread wide recently throughout all fields of engineering. The following highlights main movements.
6.1 Process industry
For a long time, the process industry used MPC almost exclusively. This is not surprising as the petrochemical industry promoted the development decisively [
24,
97,
99,
105]. Motivated by its complex, multi-variable processes with time delay, MPC spread quickly since optimal control lead to significant economic benefit due to the large throughput. Darby et al. [
26] acknowledged that MPC is “
the standard approach for implementing constrained, multi-variable control in the process industries today”.
In the founding paper of MPC, [
105] described three applications: a distillation column of a catalytic cracker in oil refinery, a steam generator, and a polyvinyl chloride (PVC) plant. The catalytic cracker had two manipulated variables (mass flow rates) and three control variables (temperatures), of which only one was constrained. The plant was modeled through twelve impulse response functions and the sample time was
\(T_{s} = 3 \min \limits \) – manageable only because it used a heuristic control law.
With the control of the polyvinyl chloride (PVC) plant, they wanted to demonstrate the versatility of MPC by controlling five subprocesses. The results showed a severe reduction in variance of the controlled variables yielding to higher quality and energy savings. The impressive demonstration paved the way for the popularity of MPC. Richalet later also described how a distillation column and a vacuum unit was controlled in a refinery of
Mobil Oil [
104]. The objective function was already formulated as a quadratic
Lyapunov function, which—as was shown—is favorable for stability. He did not address robustness but mentioned a back-up control system in case of failure. The results showed that the controller reduced the variance in the quality criteria resulting in a payout time of less than a year.
Oil companies were the promoters of model-based advanced controllers. Cutler and Ramaker [
24] used a piecewise-linear model to control the furnace of a catalytic cracking unit at
Shell Oil. With a prediction horizon of
N2 = 30 and a control horizon of
Nu = 10, they exploited the predictive potential.
Prett and Gillette
97 used even longer horizons (
N2 = 35,
Nu = 15) with a sampling time of “
a few hours”. They successively linearized a non-linear process model determining the optimal operation point of the reactor and the regenerator of a catalytic cracker.
With distillation being one of the workhorses of the chemical process industry for the separation of molecules, it is still today a popular application examples for MPC, as in [
21,
80], which both were a simulation study on linear MPC. Only that [
80] successively linearized a non-linear model of a methanol/water mixture to apply linear MPC.
Piche et al. [
95] introduced a neural network (NN) in MPC to control the set point change in an polyethylene (PE) reactor. A neural network (NN) is a non-linear empirical model based on historic data. This type of machine learning model is experiencing extraordinary attention nowadays. Linear dynamic models were constructed from conventional (open-loop) plant tests to control the plant at its set points. Piche et al. achieved 30% faster transitions and an overall reduction in variation of the controlled variables. The idea is still under active research. Li et al. [
63] also explored successive linearization of a neural network (NN) in MPC but to control the temperature of a stirred reactor—a common application in process industry, e.g. for bioreactors. Shin et al. [
117] used a neural network (NN) (fully connected, 14-15-2) with MPC for a propane devaporizer (e.g. specialized distillation column). Although claiming that neural network (NN)–based non-linear MPC achieved better performance than linear MPC, they benchmarked the new controller on conventional PI control demonstrating a 60% quicker settling time (35 min with neural network (NN)-MPC to 92 min with PI control). They further stressed easier modeling of data-driven models as an additional benefit of using NNs in conjunction with MPC. Nunez et al. [
89] used a more complicated neural network (NN) structure, a recurrent neural network (RNN) (in fact, an attention-based encoder decoder recurrent neural network (RNN) with 23,000 free parameters) to model an industrial past thickening process. The sampling time was
Ts = 5
min giving the controller enough time to conduct a global optimization with particle swarm optimization (PSO) – a rather unusual choice – for a prediction and control horizon of
N2 = 10 and
Nu = 5 respectively. Presenting one rare example of an actual industrial deployment, they demonstrated the effectiveness of the control on an industrial plant for a working day. The recurrent neural network (RNN)–based MPC was capable of maintaining the target concentration of the paste thickener in spite of a severe disturbance when a pump failed. A recurrent neural network (RNN) structure was also used to control chained stirred reactors [
136]. There are applications with further network types with distinct features, such as echo state networks to model time delay of buffer tanks, e.g. for a refrigerator compressor test rig with (non-linear) MPC [
9].
In general, besides oil and gas, and the chemical industry, pharmaceutical and biology industry use MPC to manage the non-linearity coupled with large time-delays of their processes, e.g. in a fermentation process [
42]. Ławryńczuk [
6] compared linear MPC to non-linear MPC again for a stirred reactor and for a distillation column. He concluded that, in particular for the distillation process, the non-linear controller was more economic. On this background, he suggested to combine both approaches reducing the computational burden of pure non-linear MPC: applying non-linear optimization only for the first time instant k = 1 and using a linearized model for the other steps 1 < k <
N2. To the knowledge of the authors, such an approach has not been examined further.
Prasad et al. [
96] took a different route, preferring to use multiple linear models rather than a single non-linear one. They controlled the filled-height of a conical shaped tank. Since the diameter varies continuously with the height, they suggested to identify three separate linear models at different heights, to design one controller for each and combine the outputs as an ensemble to obtain a general output for the manipulation variable (the inlet flow rate).
In 2003, [
99] already counted over 4 600 industrial applications reviewing the available commercial software packages for MPC. They differed in the model structure, its identification, and in how constraints were implemented (as hard constraints or as an additional penalization term in the cost function). Nevertheless, all models were linear, time-invariant, and derived by empirical test data. Online adaption of the model was not supported by any software, although there had been (academic) works on this issue already from the beginning, e.g. [
105].
Although stability theory is at a mature level,
AspenTech as a major vendor of commercial MPC software assumed an infinite horizon control to ensure stability, which was implemented in practice by a prediction horizon much larger than the reaction time of the system [
33]. With regard to academia, the software MATLAB/Simulink from
The Mathworks is very popular, e.g. [
80,
96,
108].
Today, process industry is still the major user of MPC [
76] evolving towards faster, mechanical processes such as paper machines [
145] or stone mills [
108,
124].
Again, a report of an industrial application was presented by the
Anglo American Platium company, where a linear MPC (to be more precise: (DMC)) outperformed a back-than famous fuzzy controller [
124]. The power consumption of a large stone mill was reduced by 66% using the commercial system from
AspenTech. Nevertheless, no fully thrusting the novel control method, the established fuzzy controller was run as back-up option for abnormal states.
Olivier and Craig [
92] and coworkers [
55] detected faults of actuators within the process to update the available manipulated variables of the MPC maintaining the control performance. They used a particle filter in order to estimate whether a certain actuator could still be used or not (binary decision). Self-awareness was especially important for continuously-running large systems in rough environments. They simulated a mill of a mining facility to grind ore. The simulation demonstrated that the MPC can manage actuator failure if it knew about it.
Table
1 summarizes the key parameters of the discusses works in process industry. Only works are listed that provided their implementation details on MPC. The order has no significance besides order of publication.
Table 1
Overview of the tuning parameters of MPC in process industries
| 3 min | ? | ? | exp | L |
| ? | 30 | 10 | exp | L |
| few h | 35 | 15 | exp | L |
| 3 min | (10) | ? | sim | N |
| 1 min | 10 | 3 | sim | L+N |
| 1 min | 60 | (60) | exp | L |
| 1 s | 6 | 2 | sim | L |
| ? | 50 | 20 | sim | L |
| 1 min | 20 | 3 | sim | N |
| 1 min | 18 | 3 | sim | N |
| 200 ms | 50 | 5 | ? | N |
| 1 s | 76 | 16 | sim | L |
| 1 s | 10 | 2 | sim | L |
| 1 min | 10 | 2 | sim | N |
| 5 min | 10 | 5 | exp | N |
| 1 min | 15 | 3 | sim | N |
| 10 s | 150 | 2 | exp | L |
MPC often served as a supervisory control of classic PID controllers forming a cascaded control loop. Large multiple input multiple output (MIMO) systems, empirical models—mostly derived through step or impulse tests [
99]—and long calculation times
Ts > 1
h favored MPC in process industry. Today, the sampling times have largely decreased to the region of minutes and seconds [
26], Table
1. Complex couplings between process variables require empirical, nonlinear models, which are at the beginning often linearized.
6.2 Power electronics
Not until the mid 2000s, an opposite trend has taken shape in power electronics. These extremely fast single input single output (SISO) systems used pure analytical models to work at sampling frequencies below the ms-range [
15,
52,
65,
129]. The characteristics are diametrically different to process industry. Richalet [
104] foresaw this counter movement early reporting from an application to control a servo drive with a sampling time of
Ts < 1
ms. To achieve such short sample times, relatively simple models, short horizons, and often an explicit solution of the optimization problem were used. Explicit MPC solves the optimization in advance for a variety of cases to obtain a polytope of explicit (linear) control laws [
14]. This increases the overall computational effort but shifts it to offline optimization.
Linder and Kennel [
65] applied MPC for “field oriented control” of electrical AC drives using such an explicit MPC. The results were sobering: there was hardly any improvement to a conventional PID controller for large signal steps. For small steps, the MPC reached the new target value faster and better, but in summary, Linder and Kennel attributed potential of MPC more due to features like intuitive tuning and constraint satisfaction.
Nevertheless, Bolognani et al. [
15] saw MPC as being ideal for electric motor control since there existed analytical linear models describing the motor behavior accurately. They also used an explicit MPC formulation to achieve an sample time of
Ts = 83
ms. Since the prediction horizon
N2 = 5 was far from covering the complete drive dynamics, the assumption of an infinite prediction horizon did not hold, making stability a major (unconsidered) concern. The control was perfect if the load torque matched the design torque of the MPC design. Otherwise, there occurred an offset between the desired and the actual values (current, voltage, etc.). Nevertheless, the controller worked stable and enforced the current and voltage limits reliably.
Kouro et al. [
52] examined MPC regarding control of power converters. Power converters have only a finite number of discrete states
n. This handicaps an optimization requiring heuristic approaches (mixed-integer optimization). They took a brute force approach testing every possible control action resulting in an exponential increase of calculations:
\(n^{N_{2}}\). With
n = 7 converter states the prediction horizon was limited to
N2 = 2 in order to achieve a sample time of
Ts = 100
ms. Compared to a classic PID control, they concluded that the advantage of MPC is its flexibility regarding control variables and constraints—similar to [
65] before.
Geyer et al. [
38] used MPC for direct torque control of electrical drives. The control problem consisted of keeping the motor toque, the magnitude of the stator flux, and the inverter’s neutral point potential within their (hysteresis) bounds minimizing the switch frequency of the inverter. To reduce the computational complexity and to solve the MPC within
Ts = 25
ms, the control and prediction horizon were limited to
Nu =
N2 = 2. As a compromise between computational effort and system behavior, the value of the prediction horizon was extrapolated linearly 100 steps to roughly recognize future system behavior. The simulation results showed that MPC respected the constraints only slightly better but reduced the switching frequency on average by 25% thus reducing the power dissipation.
As an experimental validation for this, Papafotiou et al. [
93] implemented MPC for direct torque control on a 1.5 MW motor drive. Again, the major concern was on the computational speed, so that the control horizon was further reduced to a single step
Nu = 1. The two control tasks, motor flux and motor speed, were split into separate control tasks with different execution times (25 ms and 100 ms respectively). The results could not hold the euphoria of the simulation above. On average the control reduced the inverter’s switching frequency by 16.5% maintaining the same output quality as standard control. For motor drives of this size, the achieved faster torque response was even more valuable for certain applications. Especially high-voltage applications, such as motor control, must consider the time delay of the converter [
10]. Converters often exhibit a programmed time delay after switching in order to avoid a shoot-through. Model-based predictive control (MPC) can manage this naively, e.g. in the system model [
10].
The number of applications in power electronics increased so rapidly that Vazquez et al. [
129] felt impelled to give an extensive review of the academic implementations. They concluded that the lack of proper models is still the major obstacle towards an industrial application. And MPC for power converters and rectifiers (electrical devices that convert alternating current (AC) to direct current (DC)) is still subject of active research due to their ubiquity. It is likely to increase even further due to the transformation of society in the context of combating climate change and the accompanying electrification of whole industries. Efficiency is prime and researchers found MPC to provide valuable contribution, e.g. for determining optimal switching sequences of converters and rectifies already for mid-level voltage ratings [
40,
83]. Although computation is still an issue, e.g. [
2], both formulations are still competing in the this field of very fast control problems in power electronics: The standard implicit formulation of MPC with solving the control problem online and the explicit formulation where the optimization problem is solved a priori for all cases. A detailed general discussion on explicit MPC includes Section
8.1.
Again, Table
2 provides a condensed overview of the works on the application of MPC in power electronics. It emphasizes the diversity of the used parameters of MPC in this field. Having started with the control of individual electrical components, in particular converters, the application in electrical engineering has widened towards the control of systems of multiple components as the next section will show.
Table 2
Overview of the tuning parameters of MPC in power electronics
| 10 ms | 3 | (3) | sim | L |
| 20 ms | 40 | ? | sim | L |
| 1 ms | 1 | 1 | exp | N |
| 83 ms | 5 | 1 | exp | L |
| 24 ms | 1 | 1 | exp | L |
| 25 ms | 2/(100) | 2 | sim | L |
| 100 ms | 2 | ? | sim | L |
| 62.5 ms | ? | ? | exp | L |
| 1 ms | 1 | 1 | exp | L |
| 100 ms | 3 | 1 | sim | L |
| 200 ms | ? | ? | exp | L |
| 200 ms | 50 | 50 | sim | L |
6.3 Building climate and energy
Since 2010, MPC has attracted notice to the community of building climate control. Analytical and empirical models were combined in non-linear multiple input multiple output (MIMO) systems with long prediction horizons. Typical sample times were in the order of minutes to 1 h with prediction times usually smaller than 48 h [
113]. The objective was always to reduce the energy consumption while maintaining a certain (thermal) comfort. The success of MPC in this field was due to that it allows to incorporate statistical uncertainties and even weather forecasts [
5], e.g. as in [
90].
MPC for heating, ventilation and air conditioning (HVAC) had been applied to a broad range of buildings, starting from a single room to large spaces as airport buildings or multi-room problems as office buildings [
1]. The overwhelming majority of the works addressed non-residential buildings, where only 4% included residential buildings often as one energy sink among others in a micro-grid [
74]. In their latest review, they noted that heating, ventilation and air conditioning (HVAC) plays an important role in the field of building energy management systems with more than 50% of all publications; and that MPC is the most used strategy. The authors ascribed this to its native consideration of weather and occupation forecasts, e.g. demand forecasting. Google reported that MPC increased the efficiency of the air handling in one of their data centers so that they cut cooling costs by 9% [
54].
Most works in the field of climate and energy management were simulations due to the large implementation effort and the risk of discomfort. Gunay et al. [
43] actually demonstrated their findings on an actual room of their university offices; and Ma et al. [
69] implemented a MPC controller to the cooling system of their university building. The main component was a cold water storage tank, whose operation was controlled (when to fill, how fast to fill, how cold should the water input be—coming from the chillers, etc.). They reduced the energy costs by 19%, introducing the interesting idea of optimizing financial costs instead of pure energy consumption, [
1] later picked-up again in this filed. With “MPC”, nowadays a dedicated term for such formulations exist.
Yu et al. [
141] conducted a whole benchmark of different temperature control approaches on a small mock-up building in a thermal chamber. Model-based predictive control (MPC) outperformed the other approaches—including a commercial thermostat with a programmable schedule—and reduced the energy consumption by 43% compared to a constant temperature controller. However, the results suggested that for small buildings the main benefit came from an enhanced temperature measurement.
Industrial applications of MPC in building climate control are still rare, which is due to the enormous modeling effort (being up to 70% of the control effort) [
5,
94].
Often, individual rooms were modeled as capacity resistor elements [
82,
90,
91,
107]. Coupled resistance-capacitance models based on physical principles and pure empirical approaches are the two main types of modeling building energy systems for MPC [
113].
One way to approach the modeling effort and the related requirement of domain knowledge was to use black box modeling approaches, namely from the field of machine learning. Already Qin and Badgwell [
99] noted that NNs were popular to model unknown non-linear behavior for MPC. Afram et al. [
1] used NNs to model the individual subsystem of an energy management system, such as ventilation, heat storage, or a heat pump. The increase in model accuracy came at the cost of a non-linear optimization in the MPC. The system was tested on historic weather data—assuming an ideal weather forecast at every point as it is common practice, e.g. also in [
36,
90,
91]. Unfortunately, no details on the MPC parameters were given in [
1]. The objective was to optimize the cost of the energy consumption and not the amount of consumption itself. For this, the proposed neural network (NN)–based MPC shifted the energy consumption to the off-peak hours of the electricity price using the mass of the building as a storage. This worked excellent for moderate weather conditions but failed at extreme conditions as in midsummer when such passive thermal storage are not sufficient.
The interlaced individual models in building climate control let to a complex optimization problem, where gradient-based algorithms may fail and heuristic-based global optimization were more desirable [
82]. This increased the computational effort further and, thus, enlarged the sample time, which was seldom a problem due to the inertial nature of thermal behavior. If the number of rooms became large, the control problem was broken down into multiple decoupled MPCs achieving a near optimal solution at a lower computational cost [
82]. Shaltout et al. [
5] plead for a distributed network of MPC controllers cooperating with each other.
Gunay et al. [
43] claimed that shorter sample time favors temperature control (
Ts,short = 10
min compared to
Ts,long = 1
h, both
N2 = 6) since the model accuracy usually deteriorates with the predicted time. Furthermore, long horizons may be torpedoed by stochastic disturbances such as the occupancy behavior. They claimed that a short prediction horizon of
\(T_{N_{2}} = 6 h\) would have even eliminated the need for accurate weather forecasts and make the MPC more reactive. Yu et al. [
141] supported the finding that shorter horizons enabled for a more accurate tracking of a given temperature reference. In contrast, [
91] argued that
\(T_{N_{2}}=24 h\) should be used as a prediction horizon for heating, ventilation and air conditioning (HVAC) systems.
Park and Nagy [
94] identified MPC as recent trend in heating, ventilation and air conditioning (HVAC) control through mining the keywords of publications and predict that it will spread towards the control of smart grids. Another recent review on MPC for heating, ventilation and air conditioning (HVAC) systems [
113] stressed that it is importance will increase in step with the transformation in power generation towards renewable sources and its higher variability. And in fact, the increasing pressure to integrate flexible sources and sinks into power grids (introduced by renewable energy plants and PEVs) called for advanced control methods, e.g. [
126].
In particular, the ability to include stochastic models and, thus, modeling uncertainty explicitly was considered a unique feature especially in the field of energy management [
11]. Oldewurtel et al. [
91] formulated the MPC problem as a probability problem considering the uncertainty of weather forecast. Instead of using weather forecasts, Morrison et al. [
86] learned the day-to-day changes in solar radiation due to seasonal trends. The algorithm learned the behavior of humans in terms of hot water demand over days and weeks, while the MPC implements this learned reference on a lower-level (
\(T_{N_{2}} = 12 h\)). In a simulation study, they mimicked four weeks from midsummer to midwinter for the considered thermal-storage-tank system.
Also in the field of renewable energies, Dickler et al. [
27] applied a time-variant MPC for load alleviation and power leveling of wind turbines, where the model for the mechanical demand on the turbine was linearized at every control step for the current prediction and control horizon. The wind speed as one major load on the mechanical structure was handled by incorporating wind speed predictions. Sun et al. [
125] used MPC to smooth the effect of fluctuations in wind speed for wind turbines on resulting frequency of the power generation. The idea was to consider both, the dynamics of the turbine and of the wind itself, in a linearized MPC. Shaltout et al. [
114] picked up the same idea coupling the wind turbine with an energy storage system. Targeting multiple objectives, some with non-technical motivation, they formulated a so-called economic MPC. Adding fluctuating energy consumers to such a system, [
126] simulated a (connected) micro grid with an wind power supplier and 100 PEVs. The objective was to minimize the overall operation costs: maximizing the consumption of wind energy and minimizing the exchange to the main grid, i.e. balancing the energy consumption over consumption and production peaks. PEVs could be used as sources or sinks as long as they were fully charged at the end of a working day. The energy demand of the PEVs was modeled as a truncated
Gaussian model; the supply of a wind turbine in an auto-regressive integrated moving average model (ARIMA). They proposed a two-layer MPC where the top layer balanced the overall power demand aggregating the PEVs to a single value, while the underlying MPC handled the energy distribution to the individual PEVs. The top layer optimized the cost of the energy and the risk, which was determined through a
Monte Carlo simulation and stochastic models. A simulation showed that the costs was be reduced by more than 30% compared to an immediate maximum charge strategy, in which the batteries were charged to full capacity as soon as it was connected to the grid. This may exacerbate the energy imbalance of the micro grid at peak hours. Schmitt et al. [
109] optimized energy management for hybrid electric vehicles by establishing also a two-layer MPC. On the higher level non-linear MPC, the driving strategy including a rule-based gear selection was optimized, and the control and actuation of the physical system were realized on the faster lower level linear MPC.
In the advent of the electrification of the mobility, MPC experiences a new blossom, e.g. in balancing the fuel consumption of a hybrid-electric vehicle taking also the individual driving behavior into account [
61], or in health-aware battery charging [
147].
Again, the mega trend of energy transition and energy efficiency will lead to an increasing demand of intelligent strategies for energy balancing in (micro) grids and for building energy management systems. This in turn will call for more applications of advanced control strategies, especially MPC [
74,
113]. The field has developed from the control of pure heating, ventilation and air conditioning (HVAC) systems to entire consumer-producer systems (or grids). The complexity of the models represent this evolution, Table
3.
Table 3
Overview of the tuning parameters of MPC in building climate and energy
| 1 h | 24 | 24 | sim | N |
| 1 h | 24 | (24) | exp | N |
| 1 h | 24 | 24 | sim | N |
| 10 min, 1 h | 6 | ? | sim | L |
| 1 h | 8 | (8) | sim | N |
| 100 ms | 10 | 5 | sim | L |
| 15 min | (60) | (60) | exp | L |
| 30 s | ? | ? | exp | L |
| 5 min | ? | ? | sim | L |
| 1 s | 10 | ? | exp | L |
| 1 s | 20 | 5 | exp | N |
| 100 ms | 3 | 1 | sim | L |
| 100 ms | 40 | 8 | sim | L |
| 200 ms | 50 | 50 | sim | L |
| 40 ms, 1 s | 5 | 5 | sim | L+N |
6.4 Manufacturing
Manufacturing is a comparably new field for MPC and can be considered representative for a new development: MPC does not substitute existing controllers anymore but exploits new control tasks.
We want to emphasize the field of manufacturing in general and cutting technology in particular, where several papers already showed the potential benefit of advanced control, e.g. on a conceptual basis [
28].
Nevertheless first, fixed-gain controllers for the position control loop of machining centers were substituted to achieve higher precision [
122,
123]. Compensating the dynamics in high-precision milling with MPC is still an active field of research, e.g. [
73]. Nonetheless, the application evolved towards introducing additional high-level control with MPC. The control turned into process control rather than implementing machine tool settings, creating before unseen benefit. Mehta and Mears [
79] described a concept for controlling the deflection of slender bars in turning. And Zhang et al. [
142] examined MPC to avoid chatter—an undesired resonance phenomenon—in milling. The MPC used a linearized oscillation model assuming that mass, damping, and stiffness were given. The controller manipulated an external force actuator at the tool holder. In simulation, the system enlarged the chatter-free region by 60%.
The first constrained MPC for force control in milling was implemented at the RWTH Aachen University, Germany [
111,
112,
119,
120]. They manipulated the feed velocity in order to achieve a constant force in this highly dynamic process. Later, a black box model (support vector regression (SVR)) was added to consider non-linearities of machining centers [
7,
8].
Staying in the area of metal processing, Liu and Zhang [
67] introduced MPC-based control to welding. Predicting the
N2 =
Nu = 5 next steps (
Ts = 0.5
s), they controlled the penetration depth of the weld as a measure of quality. While the first approach relied on a dedicated vision system and a linearized model of the penetration depth, a newer approach dropped the vision system: [
148]. The feedback loop was closed by identifying a model online, which described the relation to the penetration depth. This was a similar set-up as for the milling process above. The approaches demonstrated the control of system variables that were hard to impossible to control without MPC.
Wehr et al. [
133] applied a linear MPC to control the gap during precision cold rolling of thin and narrow strips. The structure of the given process is anatomically overactuated by the existence of two redundant actuators for gap control. The overactuation and computational effort of the MPC are tackled at the same time by the introduction of a single time-varying optimization variable, which exploits the different availability of the actuators during the process.
A different field of production technology addressed Wu et al. [
135], who optimized the air-jet to insert the weft in weaving. This is the key to reduce the energy consumption (in terms of compressed air) of weaving machines.
And for injection molding of plastics, Reiter et al. [
103] (conceptual) and later Stemmler et al. [
121] built a MPC controlling the pressure within the mold. The idea was to obtain constant weight of the product as a quality criterion. It was standard to control the process with separate controllers for the different phases (injection and packing phases), while MPC was able to handle both phases and optimizing the transition (which was originally a switch of the controller) [
121]. The contribution to a higher usability of the MPC was the main driver in this work.
A bit more general, the field of “production” adds automation and handling systems to the scope. These are often graph or state-based modeled, e.g. by Petri Nets as Cataldo et al. [
20] did with a palette transportation and processing system. Using an MPC, they enabled the system to adapt to faults on the transportation line such as a blocked section. Automation applications with discrete states present mixed-integer optimization problems. They require dedicated solver, which often are heuristic-based and come with a larger computational burden than gradient-based optimizers.
Table
4 provides a quick overview on the chosen parameters. The sampling times are quite low with rather large prediction horizons compared to the early works on power electronics.
Table 4
Overview of the tuning parameters of MPC in manufacturing
| 100 ms | 50 | 4 | exp | L |
| 20 ms | 12 | 12 | exp | L |
| 10 ms | 13 | 13 | exp | L |
| 500 ms | 5 | 5 | exp | L |
| 8 ms | 25 | 1 | sim | L |
| 1 ms | 25∗ | 2 | sim | L |
| 8 ms | 12 | 3 | exp | L |
| ? | 7 | 2 | sim | N |
| 20 ms | (13) | (13) | exp | L |
| 1 ms | 8 | 6/1 | exp | L |
| ? | 8 | 8 | exp | L |
| 976 ms | 2 | (1) | exp | L |
| 20 ms | 10 | 10 | exp | L |
6.5 Further applications
Apart from these main movements, the range of applications in engineering is immense. From balancing walking robots [
134], hanging crane loads [
110], and cruise control for heavy duty trucks [
62,
140], to optimizing buffering and quality in video streaming [
138]. Even for path tracking of underwater robots, MPC was applied [
116]. In almost all applications, MPC outperforms classic controllers.
In particular, robotics is an emerging field of applications of MPC, e.g. [
47,
88,
134]. While humanoid robots are a special case [
134], industrial robots are ubiquitous in the shop floors today. The success of light-weight, economic, and collaborating robots has contributed to a significant increase of MPC related works in this field. Nubert et al. [
88] improved the tracking robustness in general with a robust MPC. While [
47] made use of the force feedback of a lightweight robot to polish the free-form surface of a metal workpiece. The MPC maintained a given pressure on a varying area while moving over the surface.
With the upcoming of new concepts of how vehicles are powered was accompanied with new applications of control strategies and applications of MPC. Be it traction control of in-wheel electric motors [? ], cruise control [
61,
62], or path planning for autonomous driving [
48]. The focus of advanced cruise control is yet on larger commercial vehicles, such as (hybrid) electric buses [
61,
137], due to its faster return on invest. It seems that the electrification of the power train spread electrical-engineering know-how to the development cycle of vehicles and with it, control engineering expertise.