Embedded Model Predictive Control Research Articles

Flexible development and evaluation of machine‐learning‐supported optimal control and estimation methods via HILO‐MPC

AbstractModel‐based optimization approaches for monitoring and control, such as model predictive control and optimal state and parameter estimation, have been used successfully for decades in many engineering applications. Models describing the dynamics, constraints, and desired performance criteria are fundamental to model‐based approaches. Thanks to recent technological advancements in digitalization, machine‐learning methods such as deep learning, and computing power, there has been an increasing interest in using machine learning methods alongside model‐based approaches for control and estimation. The number of new methods and theoretical findings using machine learning for model‐based control and optimization is increasing rapidly. However, there are no easy‐to‐use, flexible, and freely available open‐source tools that support the development and straightforward solution to these problems. This article outlines the basic ideas and principles behind an easy‐to‐use Python toolbox that allows to solve machine‐learning‐supported optimization, model predictive control, and estimation problems quickly and efficiently. The toolbox leverages state‐of‐the‐art machine learning libraries to train components used to define the problem. Machine learning can be used for a broad spectrum of problems, ranging from model predictive control for stabilization, set point tracking, path following, and trajectory tracking to moving horizon estimation and Kalman filtering. For linear systems, it enables quick generation of code for embedded model predictive control applications. HILO‐MPC is flexible and adaptable, making it especially suitable for research and fundamental development tasks. Due to its simplicity and numerous already implemented examples, it is also a powerful teaching tool. The usability is underlined, presenting a series of application examples.

Embedded Model Predictive Control for Torque Distribution Optimization of Electric Vehicles

Embedded Model Predictive Control for Torque Distribution Optimization of Electric Vehicles

Code generation for embedded predictive control of gas water heaters

Conventional control strategies usually employed in tankless gas water heaters present difficulty in controlling the hot water temperature when subjected to sudden changes in water flow rate. Inadequate control leads to temperature overshoots and undershoots with long settling times that severely affect the user comfort, increasing water and energy wastage and associated gas emissions. A strategy based on model predictive control is presented to reduce the impact of changes in hot water demand. A semi-empirical model, parameterized with experimental data and compatible with real-time simulation, is used for the heat cell. A tailored state observer is proposed, considering time-varying delays characterizing this thermal process. An automatic code generation software tool was developed for the embedded implementation of gas water heater predictive controllers. Numerical simulations and hardware-in-the-loop experiments were established to evaluate conventional and predictive control strategies. It was demonstrated that embedded model predictive control could be successfully implemented on computationally limited microcontrollers, even for thermal systems with extensive varying time delays. Predictive control has shown significant performance improvements, with decreased temperature fluctuations, a gain in comfort index from 36% to 75% and a reduction of up to 32 s in the settling time.

Co-Design of a Controller and Its Digital Implementation: The MOBY-DIC2 Toolbox for Embedded Model Predictive Control

Several software tools are available in the literature for the design and embedded implementation of linear model predictive control (MPC), both in its implicit and explicit (either exact or approximate) forms. Most of them generate C code for easy implementation on a microcontroller, and the others can convert the C code into hardware description language code for implementation on a field programmable gate array (FPGA). However, a unified tool allowing one to generate efficient embedded MPC for an FPGA, starting from the definition of the plant and its constraints, was still missing. The MOBY-DIC2 toolbox described in this brief bridges this gap. To illustrate its functionalities, the tool is exploited to embed the controller and observer for a real buck power converter in an FPGA. This implementation achieves a latency of about 30 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $</tex-math> </inline-formula> s with the implicit controller and 240 ns with the approximate explicit controller.

Embedded Model Predictive Control of Tankless Gas Water Heaters to Enhance Users’ Comfort

Water heating is a significant part of households’ energy consumption, and tankless gas water heaters (TGWHs) are commonly used. One of the limitations of these devices is the difficulty of keeping hot water temperature setpoints when changes in water flow occur. As these changes are usually unexpected, the controllers typically used in these devices cannot anticipate them, strongly affecting the users’ comfort. Moreover, considerable water and energy waste are associated with the long-time response to cold starts. This work proposes the development of a model predictive control (MPC) to be deployed in low-cost hardware, such that the users’ thermal comfort and water savings can be improved. Matlab/Simulink were used to develop, validate and automatically generate C code for implementing the controller in microcontroller-based systems. Hardware-in-the-loop simulations were performed to evaluate the performance of the MPC algorithm in 8-bit and 32-bit microcontrollers. A 6.8% higher comfort index was obtained using the implementation on the 32-bit microcontroller compared to the current deployments; concerning the 8-bit microcontroller, a 4.2% higher comfort index was achieved. These applications in low-cost hardware highlight that users’ thermal comfort can be successfully enhanced while ensuring operation safety. Additionally, the environmental impact can be significantly reduced by decreasing water and energy consumption in cold starts of TGWHs.

Influence of Discretization in Dynamically Embedded Model Predictive Control

The paper analyzes the closed-loop stability of Dynamically Embedded Model Predictive Control for input-constrained continuous-time nonlinear systems. Given a stabilizing continuous-time optimal control problem, the proposed method performs a discrete approximation to obtain a finite number of optimization variables. The resulting optimization problem is then embedded into a dynamic feedback law that evolves in parallel to the system. Using Input-to-State Stability, it is shown that the dynamic interconnection between the ideal continuous-time model predictive controller and the dynamically embedded solver is asymptotically stable for a sufficiently small discretization step and sufficiently fast solver dynamics. Numerical results, however, highlight a counter-intuitive behavior: as the discretization step decreases, the stability of the closed-loop system tends to deteriorate. This suggests that, although the discretization should be sufficiently accurate to correctly capture the behavior of the system, oversampling the system dynamics may be just as harmful as undersampling.

A Dual Active-Set Solver for Embedded Quadratic Programming Using Recursive LDL$^{T}$ Updates

In this technical article, we present a dual active-set solver for quadratic programming that has properties suitable for use in embedded model predictive control applications. In particular, the solver is efficient, can easily be warm started, and is simple to code. Moreover, the exact worst-case computational complexity of the solver can be determined offline and, by using outer proximal-point iterations, ill-conditioned problems can be handled in a robust manner.

Implementation of an embedded model predictive controller for a novel medical oxygen concentrator

Medical Oxygen Concentrators (MOCs) produce high purity oxygen from ambient air for medical therapies, and can range in size from large stationary units to small ultra-portable devices. These devices use a complex Rapid Pressure Swing Adsorption (RPSA) cyclic process which is subject to many process disturbances. Feedback control is required to operate a MOC reliably in different operating conditions and in presence of disturbances. Recently, a multivariable Model Predictive Controller (MPC) for a single-bed MOC device was developed in simulation, and we now present an implementation of this MPC for a lab-scale MOC device. The single-bed MOC uses a four-step RPSA cycle which can be controlled in real-time by adjusting the four cycle step durations to control the product oxygen concentration and product storage tank pressure. The MPC uses a linear model, identified using experimental sub-space system identification techniques, and an embedded convex quadratic optimization problem for making control decisions. This work presents the implementation of this MPC algorithm using Raspberry Pi hardware for the single-bed MOC device. The complete closed-loop system is evaluated using various set point tracking and disturbance case studies.

最適経路生成器を内包したモデル予測制御による障害物回避走行

Research and development are actively conducted for recent advances in the self-driving and unmanned vehicles, along with further efficiency and safety. In such fields, obstacle avoidance is frequently addressed in control applications through the concept of active safety technology. This paper investigates the real-time optimal obstacle avoidance control that the optimal path generator embedded model predictive controller. By offline computation, a large number of optimal obstacle avoidance control solutions are calculated to generate the supervised learning data for the deep-learning neural network used for the optimal path generator. Then, constructed optimal path generator is embedded in the model predictive controller to generate the reference command for the tracking task during the optimization process. The utility of the proposed controller is evaluated by both the numerical simulation and experimental testing of the obstacle avoidance maneuver of the micro-unmanned vehicle. Results show that the proposed real-time optimal controller successfully performs obstacle avoidance.

Embedded Model Predictive Control With Certified Real-Time Optimization for Synchronous Motors

Model predictive control (MPC) is a very attractive candidate to replace standard field-oriented control algorithms for electrical motors. We demonstrate that it is possible to implement an MPC algorithm for continuous control set (CCS-MPC), with both inputs and states constraints, in which the associated quadratic programming (QP) problem is solved online, even on the computationally limited platforms used in control of electrical motors. We detail the implementation of an active-set algorithm to solve efficiently the associated QP problem. Moreover, by exploiting recent results on active-set solver certification, we are able to assess the computational complexity of the online optimization algorithm, providing the exact worst-case solution time. The controller is experimentally tested on an embedded control unit for the torque regulation of a permanent magnet synchronous motor and benchmarked against explicit MPC. Computational feasibility, low-memory occupancy, and worst-case certification are achieved, fulfilling all the requirements of embedded control.

An Embedded FPGA Architecture for Real-Time Model Predictive Control

We develop a custom hardware architecture for real-time implementation of embedded Model Predictive Control (MPC) on a Field Programmable Gate Array (FPGA). We propose a novel modular framework that allows for easy and rapid prototyping of control with capability for analog-to-digital conversion, numerical scaling, and digital-to-analog conversion. We demonstrate the effectiveness of the proposed framework for real-time control on a quadruple water tank system.

Machine learning-based warm starting of active set methods in embedded model predictive control

We propose to apply artificial intelligence approaches in a warm-starting procedure to accelerate active set methods that are used to solve strictly convex quadratic programs in the context of embedded model predictive control (MPC). The proposed warm-starting is based on machine learning where a good initialization of the active set method is learned from training data. Two approaches to generate the training data set are discussed, one based on gridding the feasibility domain, and one based on closed-loop simulations with typical initial conditions. The training data are then processed by machine learning-based classification algorithms that yield a good estimate of the initial active set for the iterative active set algorithm. By means of extensive case studies we demonstrate that the proposed approach is superior to existing warm-starting procedures in that it considerably reduces the number of active set iterations, thus allowing embedded MPC to be implemented using less computational effort.

Efficient Embedded Model Predictive Vibration Control via Convex Lifting

This paper presents an efficient real-time implementation of embedded model predictive control, adopted in the context of active vibration control with the objective of minimizing the tip deflection of lightly damped cantilever beams. In particular, we focus on memory and time-efficient explicit solutions of the associated constrained optimal control problems that are easily implementable on low-end embedded hardware. To this end, we exploit the concept of convex lifting and show how it can be used to devise low-complexity, regionless piecewise affine controllers without any loss of optimality and performance. The efficiency of this constructive procedure is quantified via an extensive complexity analysis, evidenced by a successful practical deployment and optimal vibration control performance using a family of 32-bit ARM Cortex-M-based microcontroller platforms.

A Numerically Stable Solver for Positive Semidefinite Quadratic Programs Based on Nonnegative Least Squares

This paper proposes a new algorithm for solving convex quadratic programs (QP) subject to linear inequality and equality constraints. The method extends an approach recently proposed by the author for solving strictly convex QP's using nonnegative least squares, by making it numerically more robust and able to handle also the nonstrictly convex case, equality constraints, and warm starting from an initial guess. Robustness is achieved by introducing an outer proximal-point iteration scheme that regularizes the problem without altering the solution, and by adaptively weighting the least squares problems encountered while solving the problem. The performance of the resulting QP solver in terms of speed and robustness in the single precision arithmetic is assessed against other optimization algorithms tailored to embedded model predictive control applications.

Efficient implementation of step response models for embedded Model Predictive Control

This paper proposes efficient step response model implementation strategies that lead to accurate control and high computational performance in an embedded Model Predictive Control (MPC) scheme. Different implementations of the step response prediction model are examined, and inherent properties that directly affect control performance in the presence of disturbances are discussed. Model errors that are inconsistent with bias updates (i.e. the model of unknown disturbances commonly used in step response MPC) are identified, and it is shown that the bias updates may worsen the effect of the errors in some cases. Particular attention is paid to the robustness of the prediction models to small truncation errors and errors in the input or measured disturbance history. Several implementation aspects that are crucial for embedded targets with limited resources are discussed. The findings are illustrated by simple simulation examples and an industrial case-study involving hardware-in-the-loop simulation of a subsea compact separation process.

A Quadratic Programming Algorithm Based on Nonnegative Least Squares With Applications to Embedded Model Predictive Control

This technical note proposes an active set method based on nonnegative least squares (NNLS) to solve strictly convex quadratic programming (QP) problems, such as those that arise in Model Predictive Control (MPC). The main idea is to rephrase the QP problem as a Least Distance Problem (LDP) that is solved via a NNLS reformulation. While the method is rather general for solving strictly convex QP's subject to linear inequality constraints, it is particularly useful for embedded MPC because (i) is very fast, compared to other existing state-of-the-art QP algorithms, (ii) is very simple to code, requiring only basic arithmetic operations for computing $LDL^{T}$ decompositions recursively to solve linear systems of equations, (iii) contrary to iterative methods, provides the solution or recognizes infeasibility in a finite number of steps.

Automatically generated embedded model predictive control: Moving an industrial PC‐based MPC to an embedded platform

SummaryAn embedded model predictive controller (eMPC) based on the linear MPC module in Statoil Estimation and Prediction Tool for Identification and Control has been developed, and it is intended to facilitate the MPC application on embedded hardware. The control design approach illustrates a viable way of migrating from an industrial PC‐based MPC technology to an embedded platform. This work also provides the transformations and the real‐time considerations necessary to achieve a functional high‐performance predictive controller on a low‐cost embedded hardware, with limited computational resources. The eMPC was implemented for rapid prototyping on the Ethernut 5 open‐source hardware, and its performance was tested for a simulated process. The concept was tested using a real MPC application developed for a prototype of a new industrial process that requires an advanced embedded multivariable, constraint handling control solution. The results confirm the viability of the embedded MPC design approach. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Improved Fast Dual Gradient Methods for Embedded Model Predictive Control

Recently, several authors have suggested the use of first order methods, such as fast dual ascent and the alternating direction method of multipliers, for embedded model predictive control. The main reason is that they can be implemented using simple arithmetic operations only. In this paper, we present results that enable for significant improvements when using fast dual ascent for embedded model predictive control. These improvements rely on a new characterization of the set of matrices that can be used to describe a quadratic upper bound to the negative dual function. For many interesting formulations, it is shown that the provided bound cannot be improved and that it is tighter than bounds previously presented in the literature. The improved quadratic upper bound is used in fast dual gradient methods as an approximation of the dual function. Since it better approximates the dual function than previous upper bounds, the convergence is improved. The performance enhancement is most significant for ill-conditioned problems. This is illustrated by a numerical evaluation on a AFTI-16 aircraft model where the algorithm outperforms previous proposals of first order methods for embedded control with one to three orders of magnitude.

A cost effective high-speed auto-coded embedded model predictive controller

Compared to PID control, model predictive controllers place high demands on the computing environment in terms of precision, speed, and memory, take time to design due to the embedded dynamic model, and do not scale particularly well. This paper describes a way to seamlessly implement a flexible MPC controller designed to operate in embedded hardware starting from a dynamic model in Matlab. The resultant controller runs stand-alone on the embedded hardware, is extremely fast, exhibits a modest memory footprint and best of all, requires no particular embedded programming experience from the user.

Comparison of Traditional and Bayesian Calibration Techniques for Gray-Box Modeling

Bayesian and nonlinear least-squares methods of calibration were evaluated and compared for gray-box modeling of a retail building. Gray-box model calibration is one form of system identification and is examined here with perturbations to the simple yet popular European Committee for Standardization (CEN)-ISO thermal network model. The primary objective was to understand whether the computational expense of probabilistic Bayesian techniques is required to provide robustness to signal noise, specifically with regard to lower dimensional problems (physical or semiphysical), where model calibration is preferred over uncertainty quantification. The Bayesian approach allows parameter interactions and trade-offs to be revealed, one form of sensitivity analysis, but its full power for uncertainty quantification cannot be harnessed with gray-box or other simplified models. Surrogate data from a detailed building energy simulation program were used to ensure command over latent variables, whereas a range of signal-to-noise and noise colors were considered in the experimental study. The fidelity to the building zone temperature and thermal load was the basis for comparing results. Utilization of uniform priors showed that both methods performed similarly well. Bayesian calibration outperformed traditional methods on noisy data sets; however, traditional methods were adequate up to an approximately 25% noise level. The thermal gray-box model calibration has the intended application of embedded model predictive control, where speed, accuracy, and robustness are crucial. Traditional methods required approximately 100 times less CPU time and are recommended given the model simplicity, application, and expected system noise levels.