Model Predictive Control Formulation Research Articles

Polynomial NARX-based nonlinear model predictive control of modular chemical systems

The design of control systems for modular chemical systems typically requires the identification of nonlinear dynamic models. Mechanistic models for modular chemical systems are typically of high order, which results in high online computational cost when the models are incorporated into the nonlinear model predictive control (NMPC) formulations developed for explicitly taking constraints into account. This article proposes the use of a particular class of nonlinear input–output models, polynomial nonlinear-autoregressive-with-exogenous-inputs (NARX) models, in the NMPC formulations. A machine learning algorithm, elastic net, is used to select which terms to include within the NARX polynomial series representation. The approach for constructing sparse predictive models and their use in real-time implementable NMPC are demonstrated in a two-input two-output chemical reactor case study. The Julia programming language is used to solve the NMPC optimization problem, resulting in low online computational cost.

Nonlinear Hierarchical MPC With Application to Aircraft Fuel Thermal Management Systems

A nonlinear hierarchical model predictive control (MPC) framework is proposed and applied to maximize the thermal endurance of aircraft. Effectively controlling the fuel temperatures in a nonlinear multitimescale aircraft fuel thermal management system (FTMS) requires controllers capable of long-term planning and fast update rates. In this article, a two-level hierarchical MPC controller is formulated using successive linearization (SL) that directly accounts for the multitimescale and nonlinear system dynamics to achieve accurate predictive capabilities and computational efficiency. Detailed simulation results show that the proposed hierarchical structure can increase aircraft thermal endurance by at least 21% compared to a centralized approach while significantly reducing the computational cost. The results also show that SL provides a valuable framework for efficiently accounting for nonlinear system dynamics within both levels of the hierarchical MPC formulation.

Model predictive control of an active ankle-foot orthosis with non-linear actuation constraints

The goal of active ankle-foot orthoses (AAFO) is to assist the user in recreating a nominal gait motion. Previously proposed control systems for AAFOs have been reactive, with controllers based on the current tracking error. However, the optimal AAFO controller must ensure adequate trajectory tracking while minimizing the amount of assistance provided to the user. To this end, a model predictive controller (MPC) can be considered to determine the optimal control sequence for a given trajectory. The MPC minimizes the control effort, thereby determining the minimal assistance required for a user to regain a natural gait. This work outlines the formulation of a non-linear MPC for an AAFO and its combination with an extended state observer (ESO) for tracking of an AAFO.In this work, a MPC controller is proposed for optimal control of an AAFO, a linear plant subject to non-linear actuation constraints. A novel method is expressed to determine the initial control horizon selection in combination with variable step Newton-based optimization to enable real-time implementation. The MPC control effort is then combined with the estimated disturbance from an ESO to accurately track the desired gait.Simulations and experimental results prove the tracking capabilities of the proposed method. Compared to a tuned proportional–derivative controller, the simulated controller reduced the trajectory tracking error by up to 65.1% and 15.3% in an undisturbed and disturbed environment, respectively. The MPC and ESO combination allows the controller to identify the optimal control action for accurate trajectory tracking. Concurrently, the ESO rejects both external and internal disturbances, with superior tracking capabilities in both undisturbed and heavily disturbed systems, making it the optimal choice for AAFO control.

Model predictive control of short-term winter natural ventilation in a smart building using machine learning algorithms

Natural ventilation in buildings has the potential to reduce building energy consumption and the incidence of sick building syndrome. However, controlling natural ventilation is often challenging. For instance, natural ventilation through operable windows may cause cold draughts for the occupants in winter. This paper presents the key findings of a novel machine learning-based model predicted control of single-sided natural ventilation in a retrofitted four-story smart building in Cambridge, Massachusetts, in the northeast region of the United States. A data-driven methodology is used to realize this natural ventilation modeling, including data collection, system dynamics identification, and control formulation. The prediction model of indoor CO2 and room air temperature is built through a learning process. The model predictive control (MPC) formulation step combines thermal comfort and indoor air quality requirements. Finally, the natural ventilation MPC is tested in the real building, and the results showed that indoor air quality and thermal comfort are maintained successfully, without increasing significantly the heating energy consumption in winter. This scalable data-driven methodology can potentially be applied to natural ventilation in smart buildings.

Model predictive impedance control with Gaussian processes for human and environment interaction

Robotic tasks which involve uncertainty – due to variation in goal, environment configuration, or confidence in task model – may require human input to instruct or adapt the robot. In tasks with physical contact, several existing methods for adapting robot trajectory or impedance according to individual uncertainties have been proposed, e.g., realizing intention detection or uncertainty-aware learning from demonstration. However, isolated methods cannot address the wide range of uncertainties jointly present in many tasks.To improve generality, this paper proposes a model predictive control (MPC) framework which plans both trajectory and impedance online, can consider discrete and continuous uncertainties, includes safety constraints, and can be efficiently applied to a new task. This framework can consider uncertainty from: contact constraint variation, uncertainty in human goals, or task disturbances. An uncertainty-aware task model is learned from a few (≤3) demonstrations using Gaussian Processes. This task model is used in a nonlinear MPC problem to optimize robot trajectory and impedance according to belief in discrete human goals, human kinematics, safety constraints, contact stability, and frequency-domain disturbance rejection. This MPC formulation is introduced, analyzed with respect to convexity, and validated in co-manipulation with multiple goals, a collaborative polishing task, and a collaborative assembly task.

Hybrid FES-exoskeleton control: Using MPC to distribute actuation for elbow and wrist movements.

Individuals who have suffered a cervical spinal cord injury prioritize the recovery of upper limb function for completing activities of daily living. Hybrid FES-exoskeleton systems have the potential to assist this population by providing a portable, powered, and wearable device; however, realization of this combination of technologies has been challenging. In particular, it has been difficult to show generalizability across motions, and to define optimal distribution of actuation, given the complex nature of the combined dynamic system. In this paper, we present a hybrid controller using a model predictive control (MPC) formulation that combines the actuation of both an exoskeleton and an FES system. The MPC cost function is designed to distribute actuation on a single degree of freedom to favor FES control effort, reducing exoskeleton power consumption, while ensuring smooth movements along different trajectories. Our controller was tested with nine able-bodied participants using FES surface stimulation paired with an upper limb powered exoskeleton. The hybrid controller was compared to an exoskeleton alone controller, and we measured trajectory error and torque while moving the participant through two elbow flexion/extension trajectories, and separately through two wrist flexion/extension trajectories. The MPC-based hybrid controller showed a reduction in sum of squared torques by an average of 48.7 and 57.9% on the elbow flexion/extension and wrist flexion/extension joints respectively, with only small differences in tracking accuracy compared to the exoskeleton alone. To realize practical implementation of hybrid FES-exoskeleton systems, the control strategy requires translation to multi-DOF movements, achieving more consistent improvement across participants, and balancing control to more fully leverage the muscles' capabilities.

A Nonlinear MPC Scheme for Output Tracking Without Terminal Ingredients

In this paper, we propose a novel nonlinear model predictive control (MPC) scheme without terminal ingredients for output tracking. The control objective is to steer the output of the controlled system towards some desired set, or towards the best reachable set, in case of unreachable targets. The considered MPC formulation combines a stabilizing MPC scheme without terminal ingredients with an output tracking formulation using an artificial steady-state. In addition, we propose an offset-cost that is applicable to general non-convex steady-state manifolds. The resulting MPC scheme is simple to implement since it does not require the design of terminal ingredients (i.e., terminal controller, terminal cost, and terminal region) and is applicable to general (possibly unreachable) output specifications. We show the benefits of the proposed approach by means of a numerical example where we compare it to the state of art tracking MPC formulation.

Grey-box model and neural network disturbance predictor identification for economic MPC in building energy systems

This paper presents a novel model identification procedure to treat the occupancy induced unmeasured heat disturbance in the building model identification step for a model predictive control (MPC) implementation. First, we present a grey-box identification framework to estimate a dynamic building model that systematically accounts for the large unmeasured disturbance in the training data. Then, we discuss an approach to compute the confidence intervals (CIs) on the parameters estimated in the building model. Next, we develop a framework to identify a neural network (NN) using historical operational data to give the feedforward predictions of the heat disturbance in the MPC optimization problem. Further, we discuss the application of the composite building model and the NN disturbance predictor for energy cost optimization using an economic MPC formulation. We present case studies using a simple two time scale based building model, and demonstrate the effectiveness of the proposed grey-box model and the NN disturbance predictor identification procedures. We perform closed-loop simulations to highlight that additional energy cost savings of up to 4–5% are provided by including the feedforward NN disturbance predictions in the MPC controller. Finally, we also demonstrate the suitability of the proposed two-step model identification procedure to estimate accurate and reliable models using an industrial data set collected from a real office building.

A decentralized robust control approach for virtually coupled train sets

AbstractIn this paper, a novel approach to train control systems based on virtual coupling is presented. Virtual coupling is a concept that has evolved from platooning of vehicles and allows to reduce the distance and headway between trains without constructing new lines while ensuring safe operation. With this objective in mind, we propose a decentralized robust model predictive control (MPC) framework for a virtually coupled train set based on a min–max approach. Unlike the nominal MPC, robust MPC is designed to consider external undetermined disturbances and errors to improve robustness in real‐world applications. Therefore, in this study, we present the formulation of a robust MPC based on solving a finite‐horizon optimization problem with bounded uncertainties. The bounds consider resistive modeling errors, positioning errors, communication delays, and a possible adhesion loss of up to 10%. We then performed four simulations to compare the behavior of the robust MPC with the equivalent nominal MPC. In these simulations, we simulated a metro line, main line, and high‐speed line. The simulations also analyzed the behavior of the robust MPC under the considered perturbations and different communication delays. The results show that the robust MPC ensures safer operation than nominal MPC in subways, conventional lines, and high‐speed lines. Future research can focus on centralized MPC and artificial intelligence.

Distributed Nonlinear Trajectory Optimization for Multi-Robot Motion Planning

This work presents a method for multi-robot coordination based on a novel distributed nonlinear model predictive control (NMPC) formulation for trajectory optimization and its modified version to mitigate the effects of packet losses and delays in the communication among the robots. Our algorithms consider that each robot is equipped with an onboard computation unit to solve a local control problem and communicate with neighboring autonomous robots via a wireless network. The difference between the two proposed methods is in the way the robots exchange information to coordinate. The information exchange can occur in a following: 1) synchronous or 2) asynchronous fashion. By relying on the theory of the nonconvex alternating direction method of multipliers (ADMM), we show that the proposed solutions converge to a (local) solution of the centralized problem. For both algorithms, the communication exchange preserves the safety of the robots; that is, collisions with neighboring autonomous robots are prevented. The proposed approaches can be applied to various multi-robot scenarios and robot models. In this work, we assess our methods, both in simulation and with experiments, for the coordination of a team of autonomous vehicles in the following: 1) an unsupervised intersection crossing and 2) the platooning scenarios.

Incorporating Sparse and Quantized Carbohydrates Suggestions in Model Predictive Control for Artificial Pancreas in Type 1 Diabetes

People with type 1 diabetes (T1D) face the challenge of administering exogenous insulin to maintain blood glucose (BG) levels in a safe physiological range, so as to avoid (possibly severe) complications. By automatizing insulin infusion, the artificial pancreas (AP) assists patients in this challenge. While insulin can decrease BG, having another input inducing glucose increase could further improve BG control. Here, we develop a model predictive control (MPC) algorithm that, in addition to insulin infusion, also provides suggestions of carbohydrates (CHOs) as a second, glucose-increasing, control input. Since CHO consumption has to be manually actuated, great care is paid in limiting the extra burden that may be caused to patients. By resorting to a mixed logical-dynamical MPC formulation, CHO intake is designed to be sparse in time and quantized. The algorithm is validated on the UVa/Padua T1D simulator, a well-established large-scale model of T1D metabolism, accepted by Food and Drug Administration (FDA). Compared with an insulin-only MPC, the new algorithm ensures increased time spent in the safe physiological range in 75% of patients. The improvement is limited for those already well controlled by the state-of-art strategy but relevant for the others: the 25th percentile of this metric is increased from 74.75% to 79.06% in the population. This is achieved while simultaneously decreasing time spent in hypoglycemia (from 0.5% to 0.12% in median) and with limited manual interventions (2.86 per day in median).

Model predictive control with implicit terminal ingredients

This paper proposes a new formulation of model predictive control for linear discrete time systems under state and control constraints. The main novelty of the proposed formulation is the introduction, and a computationally adequate utilization, of the implicit terminal constraint set and the implicit terminal cost function. From a numerical point of view, the deployment of the implicit terminal ingredients enables a computationally tractable use of model predictive control for higher dimensional control problems. From a practical point of view, the design of the proposed model predictive control algorithm is very constructive for a class of linear systems with polyhedral constraints. From a structural point of view, the proposed model predictive control retains a priori guarantees of the highly desirable positive invariance and exponential stability properties.

Design and Comparative Analysis of Several Model Predictive Control Strategies for Autonomous Vehicle Approaching a Traffic Light Crossing

Recent advancements in automated driving technology and vehicle connectivity are associated with the development of advanced predictive control systems for improved performance, energy efficiency, safety, and comfort. This paper designs and compares different linear and nonlinear model predictive control strategies for a typical scenario of urban driving, in which the vehicle is approaching a traffic light crossing. In the linear model predictive control (MPC) case, the vehicle acceleration is optimized at every time instant on a prediction horizon to minimize the root-mean-square error of velocity tracking and RMS acceleration as a comfort metric, thus resulting in a quadratic program (QP). To tackle the vehicle-distance-related traffic light constraint, a linear time-varying MPC approach is used. The nonlinear MPC formulation is based on the first-order lag description of the vehicle velocity profile on the prediction horizon, where only two parameters are optimized: the time constant and the target velocity. To improve the computational efficiency of the nonlinear MPC formulation, multiple linear MPCs, i.e., a parallel MPC, are designed for different fixed-lag time constants, which can efficiently be solved by fast QP solvers. The performance of the three MPC approaches is compared in terms of vehicle velocity tracking error, root-mean-square acceleration, traveled distance, and computational time. The proposed control systems can readily be implemented in future automated driving systems, as well as within advanced driver assist systems such as adaptive cruise control or automated emergency braking systems.

Model predictive control of legged and humanoid robots: models and algorithms

ABSTRACT Model predictive control (MPC) of legged and humanoid robotic systems has been an active research topic in the past decade. While MPC for robotic systems has a long history, its paradigm such as problem formulations and algorithms has changed along with the recent drastic progress in robot hardware, computational processors, and algorithms. This survey paper reviews recent progress on MPC for legged and humanoid robotics from the following three points of view. First, we review a variety of dynamical models of robotic systems used in the MPC formulation. Second, we give an overview of MPC algorithms, particularly focusing on suitable ones for robotic problems. Finally, we introduce methods and applications of MPC for practical robotic problems from MPC based on reduced-order models to recent progress on MPC based on whole-body models.

Model predictive control for path tracking in cable driven parallel robots with flexible cables: collocated vs. noncollocated control

This paper proposes a novel control scheme for precise path tracking control in cable driven parallel robots (CDPRs) with axially-flexible cables, with particular focus to the challenging case of cable suspended parallel robots (CSPRs). To handle model nonlinearities while ensuring small computational effort, a controller made by two sequential control actions is developed. The first term is a position-dependent, model predictive control (MPC) with embedded integrator to compute the optimal cable tensions ensuring accurate path tracking and fulfilling the feasibility constraints; bounds on the feasible tensions are also included. The second control term transforms the optimal tensions into the commanded motor torques, and hence currents, that are evaluated through the kinetostatic model of the electric motors used for winding and unwinding the cables. Control design is performed through the robot dynamics model, formulated with the assumption of rigid cables. Moreover, the proposed control strategy is presented in two different architectures, collocated control and noncollocated control. Flexibility is handled by penalizing large tension variations in the cost function adopted in the controller design, plus some hard constraints on the maximum tension derivatives. These features, together with the embedding of the integrator within the MPC formulation, ensure smooth control tensions that allow handling the axial flexibility of the cables, although it is not explicitly considered in the controller design.To assess the performances of the proposed control algorithm, a kinematically-determined robot with a suspended, lumped end-effector is simulated by also adopting very flexible cables. Additionally, a simplified dynamic model of the electrical dynamics and the sensor quantization are included to provide a realistic representation of the real environments. The results, together with the fair comparison with a benchmark, corroborate the effectiveness of the proposed approach, its robustness, and its feasibility in real-time controllers due to the wise reduction of the computational effort.

Variable Elimination in Model Predictive Control Based on K-SVD and QR Factorization

For linearly constrained least-squares problems that depend on a vector of parameters, this article proposes techniques for reducing the number of involved optimization variables. After first eliminating equality constraints in a numerically robust way by QR factorization, we propose a technique based on singular value decomposition (SVD) and unsupervised learning, that we call <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> -SVD, and neural classifiers to automatically partition the set of parameter vectors in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> nonlinear regions in which the original problem is approximated by using a smaller set of variables. For the special case of parametric constrained least-squares problems that arise from model predictive control (MPC) formulations, we propose a novel and very efficient QR factorization method for eliminating equality constraints. Together with SVD or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> -SVD, the method provides a numerically robust alternative to standard condensing and move blocking, and to other complexity reduction methods for MPC based on basis functions. We show the good performance of the proposed techniques in numerical tests and in a problem of linearized MPC of a nonlinear benchmark process.

Distributionally Robust Risk Map for Learning-Based Motion Planning and Control: A Semidefinite Programming Approach

In this article, we propose a novel safety specification tool, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributionally robust risk map</i> (DR-risk map), for a mobile robot operating in a learning-enabled environment. Given the robot's position, the map aims to reliably assess the conditional value-at-risk (CVaR) of collision with obstacles whose movements are inferred by Gaussian process regression (GPR). Unfortunately, the inferred distribution is subject to errors, making it difficult to accurately evaluate the CVaR of collision. To overcome this challenge, our tool measures the risk under the worst-case distribution in a so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ambiguity set</i> that characterizes allowable distribution errors. To resolve the infinite-dimensionality issue inherent in the construction of the DR-risk map, we derive a tractable semidefinite programming formulation that provides an upper bound of the risk, exploiting techniques from modern distributionally robust optimization. As a concrete application for motion planning, a distributionally robust RRT* algorithm is considered using the risk map that addresses distribution errors caused by GPR. Furthermore, a motion control method is devised using the DR-risk map in a learning-based model predictive control (MPC) formulation. In particular, a neural network approximation of the risk map is proposed to reduce the computational cost in solving the MPC problem. The performance and utility of the proposed risk map are demonstrated through simulation studies that show its ability to ensure the safety of mobile robots despite learning errors.

Hierarchical model predictive control for an off-highway diesel engine with SCR catalyst

Diesel engines become more and more complex systems, especially due to increasingly restrictive emission laws. The selective catalytic reduction (SCR) is a promising way to lower nitrogen oxide emissions. The slow dynamics of the catalyst, however, have to be compensated by the engine, for example to comply with emission laws during transients. This work presents a bi-level nonlinear model predictive control (MPC) concept for a diesel engine with an upper MPC layer to account for the SCR catalyst and a lower MPC layer to control the engine. The resulting hierarchical MPC structure is able to solve the overall control problem in real-time on ECU level using suitable MPC formulations as well as interconnecting variables. The numerical solution is based on an augmented Lagrangian MPC algorithm and control-oriented models. Simulation results against a detailed reference model of the overall system demonstrate the high performance of the hierarchical MPC concept.

Deep Transfer Learning for Approximate Model Predictive Control

Transfer learning is a machine learning technique that takes a pre-trained model that has already been trained on a related task, and adapts it for use on a new, related task. This is particularly useful in the context of model predictive control (MPC), where deep transfer learning is used to improve the training of the MPC by leveraging the knowledge gained from related controllers. One way in which transfer learning is applied in the context of MPC is by using a pre-trained deep learning model of the MPC, and then fine-tuning the controller training for a new process automation task. This is similar to how an equipment operator quickly learns to manually control a new processing unit because of related skills learned from controlling the prior unit. This reduces the amount of data required to train the approximate MPC controller, and also improves the performance on the target system. Additionally, learning the MPC actions alleviates the computational burden of online optimization calculations, although this approach is limited to learning from systems where an MPC has already been developed. The paper reviews approximate MPC formulations with a case study that illustrates the use of neural networks and transfer learning to create a multiple-input multiple-output (MIMO) approximate MPC. The performance of the resulting controller is similar to that of a controller trained on an existing MPC, but it requires less than a quarter of the target system data for training. The main contributions of this paper are a summary survey of approximate MPC formulations and a motivating case study that includes a discussion of future development work in this area. The case study presents an example of using neural networks and transfer learning to create a MIMO approximate MPC and discusses the potential for further research and development in this area. Overall, the goal of this paper is to provide an overview of the current state of research in approximate MPC, as well as to inspire and guide future work in transfer learning.

A Dual-Control Effect Preserving Formulation for Nonlinear Output-Feedback Stochastic Model Predictive Control With Constraints

We propose a formulation for approximate constrained nonlinear output-feedback stochastic model predictive control. Starting from the ideal but intractable stochastic optimal control problem (OCP), which involves the optimization over output-dependent policies, we use linearization with respect to the uncertainty to derive a tractable approximation which includes knowledge of the output model. This allows us to compute the expected value for the outer functions of the OCP exactly. Crucially, the dual control effect is preserved by this approximation. In consequence, the resulting controller is aware of how the choice of inputs affects the information available in the future which in turn influences subsequent controls.