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In the absence of reliable vital registration system of under-five mortalities in most of LMICs, it is difficult for stakeholders to track progress towards achieving the child survival targets of SDG-3, which is aimed at reducing U5MR to 25 deaths per live births by To adequately plan for child survival programmes in Nigeria, large investment is required. In the face of the current economic situation of Nigeria, accurate forecasts of childhood mortalities will guide effective use of the limited health resources.

On this note, sound modeling approach to improve childhood mortality estimates is needed in Nigeria. Considering the applicability of the traditional time series models for forecasting U5MR, there is little evidence to guide future planning of child health programmes in Nigeria.

The argument is that, it is challenging for researchers to choose appropriate time series modeling techniques that can detect non-linear patterns of mortality rates [ 3 — 5 ]. However, some authors have proposed artificial intelligence such as deep learning techniques e. ANN closely follows the structure and functionality of the human brain and its neurons to solve complex problems faster with minimal human interventions, hence reducing error rates [ 6 ].

As ANN is evolving with newer algorithms, few studies [ 9 — 11 , 15 — 18 ] have considered their applicability in population health studies. However, application of deep learning algorithm to forecast long-term childhood mortality is yet to be demonstrated in many LMICs including Nigeria. Since childhood mortality data from resource-limited countries are often non-linear, noisy, and associated with large degree of uncertainties [ 2 ], forecasting with conventional statistical methods is somewhat difficult.

In the fields of engineering, agriculture, finance, and urban planning, group method of data handling GMDH —a type of artificial neural net—was observed to improve forecasting, compared with other neural networks. On this basis, our study focuses on generating accurate estimates and observing the patterns of U5MR for Nigeria during the SDG-implementation era. As new approaches are needed for child health programming in resource-limited countries like Nigeria, identifying and demonstrating the use of an appropriate model will ease application of long, time series data for monitoring the attainment of global framework indicators such as SDGs.

GMDH algorithm is a self-organizing inductive modeling and forecasting technique that extracts important information from the data to build a multilayered model through supervised learning [ 27 ]. A well-known problem with all time series methods, is that inadequately preprocessed input data can result in poor forecasting.

Unlike the traditional statistical methods, no a priori knowledge of series stationarity and randomness is required for GMDH algorithm [ 28 ]. GMDH neural network can automatically learn from the data and uncover hidden processes not detectable by the conventional methods [ 29 ]. On the other hand, implementation of GMDH ANN turns out to be tricky because there is currently no theoretical guidelines for designing GMDH architectural layers in order to improve prediction accuracy [ 7 ].

This study was exempted from ethical review by the University of Saskatchewan Behavioural Ethics Committee ID as it relied on a publicly available aggregated de-identified dataset [ 30 ]. The dataset used is the historical aggregated yearly U5MR of Nigeria for — Supplementary file 1. The dataset was obtained from the official website of the World Bank [ 31 ]. The historical mortality data span from to , giving a total of 54 observations, which was adequate to fit ARIMA regression i. GMDH-type ANN was purposefully selected from the class of deep learning algorithms because of its robustness against incorrect, noisy, and small dataset [ 33 ].

The model construction is in four iterative steps: model identification, parameter estimation, diagnostic checking, and prediction. As the first step, data preprocessing geared towards understanding the underlying patterns in the data and data transformation was ensured. The stationarity of the aggregated U5MR was assessed by plotting line graph Fig. It was observed that the assumption of stationarity for time series analysis was violated as evident by the non-seasonal downward trend of the overall under-five mortality rates.

After different calibrations, third-order differencing was appropriate in removing the observed trend Fig. The autocorrelation function ACF and partial autocorrelation PACF plots were also checked to determine the structure of the correlation between time lags of the differenced data Fig. The model with smallest possible number of parameters principle of parsimony was selected to represent the distribution of the data. The adequacy of the fitted model was determined by the randomness of the model residuals Fig.

Also, all the eigenvalues for stability of estimates were less than one and the inverse roots of MA polynomial visually indicates that the eigenvalues were within the unit circle modulus of 0. This suggests that the MA parameters satisfied invertibility condition Fig. Holt-Winters non-seasonal smoothing often referred to as triple exponential smoothing was used to predict the overall under-five mortality rates. According to Chatfield [ 38 ], it is the most advanced method in the category of smoothing methods.

The residual plots after fitting under-five mortality rates for Nigeria, using Holt-Winters exponential model are shown in Fig. Residual plots for Holt-Winters exponential smoothing for overall under-five mortality rates, Nigeria — We used the built-in time series pre-processing features of GMDH-type algorithm [ 40 ] to automatically remove the under-five mortality trend.

The target variable U5MR was automatically transformed into cube root, with a minimum of zero lag and maximum of 6 lags. The input variables included time and lags of the transformed mortality rates. The polynomial neuron function of GMDH-type model is as follows:. We designed an optimal neural-type time series model based on best performing hyper parametrization with polynomial neural networks of GMDH-type [ 43 ] Fig.

Following the rule of thumb that the number of hidden neurons should be less than twice the input layer size, we developed the neural architecture [ 44 , 45 ]. After different calibrations of neural architecture, the parameters for the network was configured with maximum number of network layers of 60 and initial neurons of Similar to a method used by Banica et al. Adequacy of the model was further confirmed with criterion value and residual plots Fig.

With low criterion value of 1. The parameters and coefficients of equation for GMDH-type model are given as:. Residual plots for GMDH-type neural network for overall under-five mortality rates, Nigeria — The foremost problem with measuring prediction accuracy is the identification of key performance indicators. MAPE is generally not considered as a good performance indicator because of its disadvantages— 1 only accurate for ratio-scaled data, and 2 it disfavors models when the predicted values are more than the actual historical values [ 48 ].

On the other hand, a benefit of using RMSE is that it is more appropriate if large errors are anticipated. Even though strength of performance measurements vary, we selected root mean absolute error RMAE because of its robustness against outliers [ 49 ]. In addition, RMSE was chosen because it minimizes the effects of bias, and measures dispersion of prediction errors i.

Furthermore, modified Nash-Sutcliffe model efficiency coefficient NSE was calculated to address the challenges of overestimating extreme values, arising from squared differences of actual and predicted values in original Nash-Sutcliffe efficiency equation [ 50 ]. DM test is a statistical test for comparing two competing forecasts based on loss-differentials, given the historical observed values.

In estimating the predictive accuracy, squared error was not used because of its tendency to overestimate errors [ 53 ]. Also for the long-run variance of the differenced series from its autocovariance function, a maximum lag order of 9 was selected by Schwert criterion and the weights of Bartlett kernel i. The measurements are expressed as [ 54 ]:. To further test the equivalence of the in-sample predictions from the individual methods with the observed historical values, Deming regression—an extension of errors-in-variables regression was performed.

The mean annual U5MR was From Fig. Also, similar out-of-sample rates were observed from to for the three models Fig. However, the out-of-sample forecasts from to for each model were different Fig. Holt-Winters method generated smallest mortality rate for , Observed historical , predicted and forecasted under-five mortality rates by modeling techniques a in-sample prediction — b out-of-sample forecasting — GMDH: Group method of data handling, ARIMA: autoregressive integrated moving averages, Holt-Winters: Holt-Winters exponential smoothing method.

All the lines basically overlap in Plot A. Performance measures of time series techniques for under-five mortality rates in Nigeria. As shown in Table 2 , the coefficients of slopes and intercepts suggest there were no proportional and systematic differences between the predicted rates for the three models and the observed historical rates. While the slopes proportional difference for the three methods were similar, the intercepts systematic difference and standard errors were different.

This study compared the predictive ability of artificial intelligence technique with traditional statistical methods in view of forecasting U5MR for Nigeria from to For the period from to in-sampling prediction and out-of-sample forecasting from to , all three models had similar results, however, for the longer out-of-sample forecasting period — , the rates were significantly different.

Also, Nigeria will not achieve child survival targets of SDG by Further analysis with age-specific mortality rates suggests that the surge in U5MR from to is due to increasing trend in neonatal mortality rates between and , and child mortality rates from to results are not shown.

According to Koutsoyiannis [ 58 , 59 ], ARIMA regression may not be ideal for data that exhibit long-range dependencies because of its slow decay of autocorrelation structure with lag time, making it less sensitive to tipping-points. In addition, ARIMA and Holt-Winters models assume normality of time series data, whereas under-five mortality data for Nigeria showed a non-linear trend.

As opposed to ARIMA and Holt-Winters methods, GMDH time series also allows for detection of recent changes in data arising from natural behavior, policy changes and interventions , and weighs recent data more than past data during model training [ 29 ]. These detailed patterns might be easily missed by the conventional methods.

Given that more accurate results were obtained with the GMDH-type algorithm, projecting childhood mortality rates based on neural network would provide better evidence to guide prevention strategies to accelerate gains in child survival for Nigeria.

A similar pattern of results was obtained from previous studies that predicted health outcomes with other artificial neural networks. Purwanto et al [ 60 ] and Zernikow et al [ 61 ] showed that multilayer perceptron ANN was superior to linear regression for predicting infant and preterm neonatal deaths, respectively.

More generally, this study indicates that, though U5MR in Nigeria continues to decline from More importantly, the government of Nigeria needs policy innovations to address the observed rise in U5MR by On evidence such as indicated in this paper, the government of Nigeria should use reliable estimates to improve the design and accelerate the implementation of child health programmes in order to attain the SDG-3 targets for under-five mortality by Given the high validation accuracy Although more datapoints are needed to generate more stable models, the forecasts from this GMDH-ANN model seem adequate because of non-seasonality of the dataset [ 63 ].

As often encountered with ANN modeling, a major gap is paucity of evidence for optimization of neurons for generating ANN architecture [ 7 ]. We relied on calibrations that could give maximum predictive power.

In addition, we observed that RNN and LSTM algorithms might be less suitable because of the few data points available for this study, coupled with the problems of gradient vanishing and gradient explosion i. There was no indication to suggest under over-fitting of data. The unexpected rise in U5MR from to warrants further investigation.

It is also important to note that it is somewhat challenging to accurately estimate data preprocessing time for time series models because they are based on trial and error approach. In addition, computational time heavily relies on computer hardware efficiency such as central processing unit CPU and random-access memory RAM. To generate interventions for improving child survival programmes in Nigeria, we prioritized model accuracy over time. Also, it does not require complicated assumptions needed for traditional time series models.

The content is solely the responsibility of the authors and does not necessarily reflect the official views of the university. DAA conceived the study, analyzed and interpreted the data, and wrote the first draft of the paper. NM assisted in the design, data interpretation, and critically reviewed the manuscript. NM supervised this study. All authors read and approved the final manuscript. This study is a secondary data analysis of publicly available de-identified dataset. The dataset for this study is open and publicly available at the official website of the World Bank.

In addition, this study was exempted from ethical review by the University of Saskatchewan Research Ethics Committee. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Published online Dec 3. Daniel Adedayo Adeyinka 1, 2 and Nazeem Muhajarine 1, 3. Author information Article notes Copyright and License information Disclaimer. Daniel Adedayo Adeyinka, Email: ac. Corresponding author. Received Jul 17; Accepted Nov 9. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

This article has been cited by other articles in PMC. Aggregated under-five mortality rates for Nigeria, — Abstract Background Accurate forecasting model for under-five mortality rate U5MR is essential for policy actions and planning. Supplementary Information The online version contains supplementary material available at Background Childhood mortality has traditionally been used as an important health indicator for assessing population well-being and consistently gained visibility in the Millennium Development Goals MDGs and Sustainable Development Goals SDGs [ 1 ].

Methods This study was exempted from ethical review by the University of Saskatchewan Behavioural Ethics Committee ID as it relied on a publicly available aggregated de-identified dataset [ 30 ]. Open in a separate window. Fitting Holt-Winters exponential smoothing model Holt-Winters non-seasonal smoothing often referred to as triple exponential smoothing was used to predict the overall under-five mortality rates.

Where Y corresponds to year of forecast , and N indicates neurons 2 and Model comparison The foremost problem with measuring prediction accuracy is the identification of key performance indicators. Results The mean annual U5MR was Table 1 Performance measures of time series techniques for under-five mortality rates in Nigeria. Discussion This study compared the predictive ability of artificial intelligence technique with traditional statistical methods in view of forecasting U5MR for Nigeria from to Supplementary Information Additional file 1 Table S1.

Funding The authors received no specific funding for this work. Availability of data and materials Dataset for this study is attached in the Supplementary file 1. There exists an extensive stability theory for linear MPCs. For systems in state space form, the stability analysis is based on eigenvalues and on the unit disk as it is familiar from the stability analysis of conventional linear control [ ].

However, optimization problems with hard input constraints are often non-linear [ ]. Establishing stability—especially robust stability—is extremely difficult for non-linear problems. This is mainly due to the lack of an explicit functional description of the control algorithm, which is required for most stability analysis [ 84 ]. Today, stability of non-linear, constrained, finite-horizon MPC is achieved by formulating the cost function as a Lyapunov function and introducing a terminal set constraint [ 75 , 77 ].

Using a terminal set links the stability problem with the constraint satisfaction problem [ 17 ]—ironically, additional constraints stabilize a constrained , non-linear MPC. Robustness is a trade-off to performance. Several approaches increase robustness at the cost of computation and optimality e.

Nevertheless, it can only be achieved if the amount of uncertainty can be quantified. A practical compromise to maintain optimality—the key feature of MPC—is to add the requirement the the worst-case prediction must contract [ 85 , ]. Once again, motivated by the chemical process industry, [ 58 ] integrated a MPC into an iterative learning control ILC building a controller dedicated for batch processing.

In this way, it approaches the ideal profile incrementally from cycle-to-cycle and may react to trends over multiple cycles. In contrast to this, MPC is a closed-loop controller but considers repetitive tasks as independent of each other. Splitting both techniques, let the iterative learning control ILC work as an upper-level reference governor for the MPC as was conducted, e. Li et al. The resulting system determined an optimal profile of the manipulated variable s for each cycle.

In a subsequent work, [ 60 ] suggested to smooth the commands over cycles. This essentially states that the optimal solution is not entirely trusted. Such systems only touch MPC in general, because they lack of a receding horizon and effectively filter their optimal control recursively. Among the works of [ 66 ] and [ ] lies the combination of iterative learning MPC and the uprising field of data-based learning in control theory.

The former extracts new trajectories of a linear-quadratic regulator LQR based on overall objectives and data of previous trajectories with the help of the k-nearest-neighborhood algorithm. The latter extends the idea of an iterative, data-driven adjustment of trajectories to the application of MPC.

Although also applied to a repetitive task, [ 78 ] focused on learning a model of the system dynamics rather than a trajectory. The authors took advantage of data and weighted linear Bayesian regression to model uncertainties of vehicle dynamics on a repeating path. The same way [ 50 ] applied Gaussian process modeling to elaborate confidence intervals on possible trajectories to guarantee safety. Data-driven modeling, such as machine learning, can be used for the system model that the MPC uses in its optimization, or to approximate the solution space of an explicit MPC, as e.

The possibilities of learning are enhanced especially for multi-agent systems, where every single agent contributes to the data-acquisition and policy exploration. The learning problem for this purpose was defined as a quadratic optimization problem under the condition of collision avoidance as constraint. 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. 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 , ]. 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. In the founding paper of MPC, [ ] 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.

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 [ ].

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. 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. Piche et al. 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.

The idea is still under active research. Shin et al. They further stressed easier modeling of data-driven models as an additional benefit of using NNs in conjunction with MPC. Nunez et al. 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 [ ]. There are applications with further network types with distinct features, such as echo state networks to model time delay of buffer tanks, e. 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.

He concluded that, in particular for the distillation process, the non-linear controller was more economic. To the knowledge of the authors, such an approach has not been examined further. Prasad et al. 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 , [ 99 ] already counted over 4 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.

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 ]. Today, process industry is still the major user of MPC [ 76 ] evolving towards faster, mechanical processes such as paper machines [ ] or stone mills [ , ].

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 [ ]. 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. MPC often served as a supervisory control of classic PID controllers forming a cascaded control loop. 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.

Not until the mid s, 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 , ]. The characteristics are diametrically different to process industry. 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. 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. 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. Power converters have only a finite number of discrete states n. This handicaps an optimization requiring heuristic approaches mixed-integer optimization. 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. As a compromise between computational effort and system behavior, the value of the prediction horizon was extrapolated linearly steps to roughly recognize future system behavior.

As an experimental validation for this, Papafotiou et al. The two control tasks, motor flux and motor speed, were split into separate control tasks with different execution times 25 ms and ms respectively. The results could not hold the euphoria of the simulation above. 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. The number of applications in power electronics increased so rapidly that Vazquez et al. 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. Although computation is still an issue, e.

A detailed general discussion on explicit MPC includes Section 8. 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.

Since , 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 [ ]. 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. 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 authors ascribed this to its native consideration of weather and occupation forecasts, e. 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. 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. Yu et al. However, the results suggested that for small buildings the main benefit came from an enhanced temperature measurement.

Often, individual rooms were modeled as capacity resistor elements [ 82 , 90 , 91 , ]. Coupled resistance-capacitance models based on physical principles and pure empirical approaches are the two main types of modeling building energy systems for MPC [ ]. 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.

Afram et al. 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. 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. Furthermore, long horizons may be torpedoed by stochastic disturbances such as the occupancy behavior. 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 [ ] 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. 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. Instead of using weather forecasts, Morrison et al. 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. The wind speed as one major load on the mechanical structure was handled by incorporating wind speed predictions. Sun et al. The idea was to consider both, the dynamics of the turbine and of the wind itself, in a linearized MPC. Targeting multiple objectives, some with non-technical motivation, they formulated a so-called economic MPC.

Adding fluctuating energy consumers to such a system, [ ] simulated a connected micro grid with an wind power supplier and 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. PEVs could be used as sources or sinks as long as they were fully charged at the end of a working day. The top layer optimized the cost of the energy and the risk, which was determined through a Monte Carlo simulation and stochastic models.

This may exacerbate the energy imbalance of the micro grid at peak hours. Schmitt et al. 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.

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 , ]. 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.

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. Nevertheless first, fixed-gain controllers for the position control loop of machining centers were substituted to achieve higher precision [ , ].

Compensating the dynamics in high-precision milling with MPC is still an active field of research, e. 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. 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. 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. 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: [ ]. 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. 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. 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. 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 [ ].

The contribution to a higher usability of the MPC was the main driver in this work. These are often graph or state-based modeled, e. 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. Apart from these main movements, the range of applications in engineering is immense. From balancing walking robots [ ], hanging crane loads [ ], and cruise control for heavy duty trucks [ 62 , ], to optimizing buffering and quality in video streaming [ ]. Even for path tracking of underwater robots, MPC was applied [ ].

In almost all applications, MPC outperforms classic controllers. In particular, robotics is an emerging field of applications of MPC, e. While humanoid robots are a special case [ ], 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. 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 [? The focus of advanced cruise control is yet on larger commercial vehicles, such as hybrid electric buses [ 61 , ], 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.

While many researchers show an extraordinary meticulousness when describing the models they have used, some miss to provide basic information on MPC tuning. We want to emphasize that at least the sample time T s and all horizons lower prediction horizon N 1 , upper prediction horizon N 2 , and the control horizon N u should be listed, as Table 1 to Table 4 demonstrate. With the horizons given, applications can be compared and and the computational effort can be estimated.

The exact cost function is required to reproduce the results ensuring good scientific practice. The initial hurdle to use MPC is relatively small—provided you have an adequate model describing the process in question. The effort is shifted from controller design towards modeling [ 35 , , ].

Nonetheless, the MPC offers an enormous flexibility regarding its design and tuning [ 37 ]. The most significant effect have:. The model is the essence of a MPC. Both, theory and commercial application software favor linear models or a linear MPC. To apply linear control even to non-linear systems successive lineraization can be used, e. The idea is to take advantage of a linear optimization, i. Few applications use non-linear MPC meeting the fact that often the available models are non-linear.

However, not all check stability. Others focus explicitly on the stability aspect in their applications, e. In particular with the popularity of machine learning model, non-linear MPC applications increase. A sometimes ignored drawback of non-linear MPC is the larger computation of non-linear optimization. However, there was a new computation scheme introduced recently: RTI. Gros et al. The main idea is as simple as it is charming, making use of the previous solution.

Thus, one can limit the number of iterations of each optimization assuming that the next optimizations continue improving the solution of the trajectory of the manipulation variable. Because the RTI scheme implements one single full Newton step per time step, it generally works better if the non-linearity between time steps is mild and if the prediction horizon is longer. Controlling large multiple input multiple output MIMO systems with a single MPC may be difficult [ 32 ], that is why cascaded or hierarchical MPC structures are some times suggested, e.

Slack variables soften constraints moving it to the cost function where the amount of its violation is penalized. It is usually an identity matrix, whose entries are several orders higher than the weight matrix of the control error W w.

A trade-off between accurate tracking of the reference and smooth control behavior can be performed by considering the change of the manipulated variable in the cost function:. The same constraints apply as before in Eq. The cost function minimizes the deviation from the reference r over the prediction horizon N 2. It must be tuned manually until the controller reflects the desired behavior. Typical solvers are based on linear programming LP or quadratic programming QP [ 26 ].

If one uses the commercial tools, i. But, for deeper dives into the design, a good option for a solver is quadratic programming online active set strategy qpOASIS. The choice of the solver influences the demand of computational resources. Besides those major design building blocks, the MPC exhibits a whole slew of tuning parameters: the horizons N 1 , N 2 , N u , the weights in the cost function, Eq. It is unique for every case but this review can provide tips and best practices for the other tuning parameters.

The prediction horizon N 2 must be long enough to capture the effect of a change of the manipulated variable u. In this way the minimum length of the manipulation horizon N u can be estimated by. The effect on computation is small if the time delay of the system is small in terms of multiples of the sampling time. The lower prediction horizon describes the time delay of the system. This considers that the manipulated variable is not implemented instantly, which would make the exact moment indeterministic as it depends on the time the MPC requires for solving the optimization problem.

Instead, the obtained optimal command u is implemented at the next time step. These considerations reduce the problem of finding suitable prediction horizons to the problem of determining the necessary prediction horizon N 2. Its choice can be estimated using the system model by simulating all possible step changes in the manipulated variable s.

If the combination that has the longest effect on the control variable is known, it is sufficient to simulate this. It does not help to talk about MPC, i. Nevertheless, there are more sophisticated strategies to reduce computation than wrecking prediction.

Morari [ 84 ] argued that computational effort was irrelevant based on the computing power in Overview of the evolution of the computation power data taken from [ 49 , ]. That usually implies that computational performance doubles too — and prices dropped in sync, Fig. This comfortable development may not continue forever; in fact, special-purpose chips are on the advance think of low energy CPUs that power smartphones letting the microprocessor landscape diverge.

The tremendous success of machine learning techniques and the increasing parallelization in software were paved by the replacement of CPUs for GPU chips. At the same time, the clock speed had been limited because of the heat dissipation in the resistors. With this in mind, strategies to reduce the computational load become very well important again.

With increasing computational resources, more demanding systems were controlled that were not even imaginable before. In the year , [ 14 ] still claimed that MPC was only applicable to slow or small systems due to the computational effort that solving an optimization problem imposes. Parallel to the increasing computational power, many dedicated approaches have been introduced bringing MPC towards more efficiency.

They combine an offline solved optimization problem with online control. The optimization problem—and thereby the control law—is solved for a multitude of possible situations and stored in a look-up table. This shifts the task of computation to a non-time-critical offline calculation.

Essentially, MPC in this was becomes an online gain-scheduling algorithm. The advantage is that closed-loop control can be performed at higher rates which, in some cases, made closed-loop control feasible in the first place and, in other cases, improved the control behavior due to quicker feedback. The major drawback is the increasing computational effort solving the problem for all possible situations in conjunction with the increasing memory demand.

It lacks of flexibility regarding unexpected disturbances and of the opportunity to adjust the process model. Explicit MPC increases the overall computation because every possible state needs to be calculated a priori. This might be the reason why it emerged from the control of power converters with simple mostly binary problems, short horizons, and almost no time for calculation [ ]. For complex systems, the advantage at execution is somewhat diminished if searching the a priori solved result takes long [ ].

One way to reduce the general computational effort is to approximate the solution-space by a non-linear function. Recent studies suggested to use NNs for this [ 64 , ]. This sped up the required online computation by a factor of 65— in [ ]. Approximating the solution space by a function let the MPC work with near optimal solutions but shifts the computational burden may allow to decrease the online computation time. The charm of an approximation through machine learning is that the training can be flexibly stopped if a defined accuracy is reached.

Hertneck et al. They quantified the probability of a wrong approximation. In this way, they were able to adjust and extend the training until it reached the desired quality. The procedure was demonstrated on a simple numerical example reducing the computation time by a factor of —at the cost of a training effort of 20 days.

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