Model Predictive Control Formulation Research Articles

Tube-Based Robust MPC for Two-Timescale Systems Using Reduced-Order Models

A tube-based robust Model Predictive Control (MPC) formulation is presented for systems with slow and fast timescale dynamics. The controller uses a reduced-order model that approximates the slow timescale dynamics and is implemented with a relatively large time step size. By analyzing the error between the full- and reduced-order models, the MPC optimization problem is proven to be recursively feasible and to produce closed-loop trajectories that satisfy state and input constraints. By relating bounds on all model errors to bounds on the change of the input in time, input change bounds are included as decision variables in the MPC problem to achieve a time-varying balance between fast transients with large model error and steady-state operation with zero model error. Zonotopes are used to make the approach practical and a numerical example demonstrates the benefits of optimizing time-varying input change bounds as part of the MPC formulation.

Lifted Bilinear Model-based Linear Model Predictive Control with Scalability

We propose a novel linear model predictive control (MPC) using a lifted bilinear model based on Koopman theory, which is computationally scalable against the dimension of the target system and the prediction horizon length. In MPC, the accuracy of the prediction model determines control performance, but it is a challenge to reduce the computational cost especially when considering nonlinearity of the model. To address this, a method has been proposed which represents the nonlinear input affine system as a lifted bilinear model and utilizes linear approximation and prediction error correction regarding the lifted state to achieve a low-computational-cost linear MPC with equivalent performance to nonlinear MPC. However, although the previous studies have shown its effectiveness for relatively low-order systems, it has not been applied to practical systems with higher dimensions. In this study, we extend the conventional method and propose a scalable linear MPC using a lifted bilinear model and apply it to higher-order nonlinear systems. In this paper, a quadrotor system operating in three-dimensional space is considered and its analytical lifted bilinear model is derived. In the formulation of linear MPC using the lifted bilinear model, an error correction method is newly introduced to feedback error for adjustment of the numerical relationships among the elements in the lifted state. The effectiveness of the proposed method is demonstrated through numerical simulations.

Long Duration Stochastic MPC With Mission-Wide Probabilistic Constraints Using Waysets

A stochastic Model Predictive Control (MPC) formulation is presented for systems operating for a finite time subject to constraints on the Mission-Wide Probability of Safety (MWPS). For linear discrete-time systems subject to unknown disturbances, the goal is to formulate an MPC controller to achieve a desired probability of mission success, where the entire closed-loop state and input trajectories stay within the state and input constraint sets and the final state reaches the desired terminal set. To enable longer missions under greater uncertainty, a wayset-based approach is proposed that allows for the prediction horizon of the MPC to be significantly shorter than the length of the mission. Using a scenario-based approach to stochastic MPC, the use of constrained zonotopes makes the computation of these waysets efficient and practical. Numerical results demonstrate the utility of the waysets for increasing the feasibility of MWPS constraints to longer missions and that the percentage of successful missions asymptotically converges above the desired probability.

Towards advanced bioprocess optimization: A multiscale modelling approach

Mammalian cells produce up to 80 % of the commercially available therapeutic proteins, with Chinese Hamster Ovary (CHO) cells being the primary production host. Manufacturing involves a train of reactors, the last of which is typically run in fed-batch mode, where cells grow and produce the required protein. The feeding strategy is decided a priori, from either past operations or the design of experiments and rarely considers the current state of the process. This work proposes a Model Predictive Control (MPC) formulation based on a hybrid kinetic-stoichiometric reactor model to provide optimal feeding policies in real-time, which is agnostic to the culture, hence transferable across CHO cell culture systems. The benefits of the proposed controller formulation are demonstrated through a comparison between an open-loop simulation and closed-loop optimization, using a digital twin as an emulator of the process.

NMPC of Unmanned Aerial Manipulators Considering Obstacle Avoidance and Manipulability

Trajectory tracking and obstacle avoidance are essential features to ensure the safe navigation of unmanned aerial manipulators (UAMs) in unstructured environments. However, these techniques may not work effectively in complex scenarios; therefore, this work presents a novel method to control the end effector of an aerial manipulator that simultaneously maximizes the manipulability of the robotic arm and guarantees obstacle avoidance of the entire system using non-linear model predictive control (NMPC) formulations. The NMPC includes trajectory tracking, obstacle avoidance, and the manipulability index in the cost function, resulting in fast algorithm convergence. The controller considers system constraints, such as the actuators’ limitations, including the aerial platform's maneuverability velocity and joint limits in the articulated manipulator. The results of the simulations demonstrate that the NMPC formulation successfully achieves the desired tasks, which validates the effectiveness of our approach ensuring safe and flexible navigation of aerial manipulators in complex scenarios.

Practical considerations in reinforcement learning-based MPC for mobile robots*

In mobile robot applications, the trajectory tracking task hides several difficulties, including the choice of the setpoint and the search for an acceptable trade-off between performance and computational constraints. In this work, we discuss practical issues of a Reinforcement Learning (RL) based Model Predictive Control (MPC) tuning approach by focusing on a specific mobile robot application, where the objective is to maximize the velocity, while keeping the robot within the track bounds. Among others, we show that softening the latter constraints allows us to obtain a RL-tuned tracking controller with the same performance of an economic nonlinear MPC formulation, but requiring significantly less computational resources.

Diffusing-Horizon Model Predictive Control

We analyze a time-coarsening strategy for model predictive control (MPC) that we call diffusing-horizon MPC. This strategy seeks to overcome the computational challenges associated with optimal control problems that span multiple timescales. The coarsening approach uses a time discretization grid that becomes exponentially more sparse as one moves forward in time. This design is motivated by a recently established property of optimal control problems that is known as exponential decay of sensitivity (EDS). This property states that the impact of a parametric perturbation at a future time decays exponentially as one moves backward in time. We establish conditions under which this property holds for a constrained MPC formulation with linear dynamics and costs. Moreover, we show that the proposed coarsening scheme can be cast as a parametric perturbation of the MPC problem and, thus, the exponential decay condition holds. We use a HVAC plant case study with real data to demonstrate the proposed approach. Specifically, we show that computational times can be reduced by two orders of magnitude while increasing the closed-loop cost by only 3%.

Nonlinear MPC for Tracking Piecewise-Constant Reference Signals: the Positive Semidefinite Stage Cost Case

Model Predictive Control (MPC) is a successful control strategy, with solid theoretical and practical backgrounds. Currently, several stabilizing MPC formulations are available to deal with tracking of piecewise constant references. In particular, it is well understood that, in many cases, the use of artificial reference variables in the optimisation problem allows to sensibly extend the region of attraction of the controller. This work proposes a modified MPC for tracking formulation which is able to guarantee nominal stability also in presence of positive semidefinite stage cost. This can be particularly useful when dealing with high order and/or black-box models, as it allows penalizing the outputs or a subset of states of the system without compromising stability. The algorithm design is based on terminal ingredients and a cost detectability assumption which is explicitly accounted for in the algorithm formulation. Such assumption can be verified by means of input-output-to-state stability arguments, as well as dissipativity ones, thus exploiting techniques already available in the literature.

A New MPC Formulation Based on Suboptimal Voltage Vectors for Multilevel Inverters

Different new model predictive control (MPC) formulations have been recently proposed to reduce the real-time computation load, which makes the MPC algorithm promising for multilevel power converters. Unlike the conventional MPC formulation, which searches for the optimal switching state at each sampling time, the existing computationally efficient MPC formulations are to search for the optimal output voltage in the first stage. In the second stage, only the switching state redundancy under this optimal output voltage established in the first stage will be employed to achieve multiobjective. These computationally efficient MPC formulations based on searching an optimal voltage vector are usually validated on the power converter topologies with abundant switching redundancy or without floating capacitors. However, for the emerging topologies with less switching redundancy and floating capacitors, such as the five-level (5L) T-type nested neutral point clamped (T-NNPC) converters topology, the existing computationally efficient formulations based on optimal output voltage vector can lead to potential capacitor control failure due to the sacrificed multiobjective control performance. To address this issue, this article presents a novel MPC formulation based on suboptimal output voltage vectors considering both the system’s multiobjective control performance and computation burden reduction. With the determined suboptimal voltage vectors, a small group of the switching state candidate can be established to improve the system’s multiobjective control performance and efficiency. The proposed MPC formulation is finally validated on a 5L T-NNPC topology.

A MPC Combined Decision Making and Trajectory Planning for Autonomous Vehicle Collision Avoidance

Increasing focus is being paid to ensuring safety in autonomous driving. The current paper addresses the challenge of collision avoidance with dynamic surrounding vehicles in different driving situations. The established solution formulated utilizing Model Predictive Control (MPC) includes decision making and trajectory planning. A simplified prediction model is used, which takes into account the relative positions and velocities of the surrounding vehicles and the ego vehicle. Depending on traffic conditions, which are stated as constraints in the MPC formulation, the ego vehicle may perform lane keeping, lane shift, overtaking or braking to avoid collision with the road participants. The decision making constraints are included into the MPC in a mixed integer formulation-like manner. The safety constraints are defined using the Sigmoid function and the braking barrier to define the navigable zone of the ego vehicle. The proposed algorithm has been evaluated through simulation, with different scenarios revealing its effectiveness.

Predictive management of the hybrid generation dispatch and the dispatchable demand response in microgrids with heating, ventilation, and air-conditioning (HVAC) systems

Heating, ventilation, and air-conditioning (HVAC) systems are considered essential technologies for modern human life for many different purposes, such as providing human comfort or increasing agricultural production. However, the energy consumption of HVAC systems is very high, and usually leads to peak power demand issues that require adequate mitigation measures to ensure grid stability and not to exceed the maximum power demand allowed. In this work, a model predictive control (MPC) scheme is proposed for the joint management of hybrid generation dispatch and dispatchable demand response of HVAC systems in microgrids. The energy generation, the storage devices, and the controllable loads are managed simultaneously to improve the performance of the microgrids without losing thermal comfort. Demand response strategies are applied explicitly (constraining controllable loads) and implicitly (optimizing system operation by means of its thermal inertia). This simple MPC scheme is based on linear parameter-varying (LPV) modeling for HVAC system and battery bank, and rough predictions for external temperature, renewable generation and non-dispatchable demand. As MPC tolerates large plant-model mismatches, simple forecasting models are suitable for the required rough predictions. The LPV model scheduling signals are forecast to predict the time variable parameters and thus improve feedback and feedforward control. Viable MPC formulations for embedded processors are derived for off-grid and grid-connected operation. The developed scheme is applied to two case studies of interest in Argentina, an automated poultry farm (isolated from the grid), and a health facility (connected to the grid) with an expensive bill due to penalties. The simulation results demonstrate the effectiveness and the economic benefit of the management strategies proposed. Indeed, the isolated case provided 55% fuel saving and better use of the resources compared to the conventional rule-based controller. Furthermore, the connected case provided 26% monthly bill saving and 750% investment saving. • A simple MPC scheme to improve feedback and feedforward control of power flows in microgrids. • It based on the joint management of generation dispatch and HVAC demand response. • Viable MPC formulations for embedded processors are derived using simple forecasting models. • Framework for seamless integration and optimization of HVAC, storage and renewable systems. • Two applications of this complete MPC scheme for off-grid and grid-connected operation are studied.

Adaptive personalized prior-knowledge-informed model predictive control for type 1 diabetes

This work considers the problem of adaptive prior-informed model predictive control (MPC) formulations that explicitly incorporate prior knowledge in the model development and is robust to missing data in the output measurements. The proposed prediction model is based on a latent variables model to extract glycemic dynamics from highly-correlated data and incorporates prior knowledge of exponential stability to improve the prediction ability. Missing data structures are formulated to enable model predictions when output measurements are missing for short periods of time. Based on the latent variables model, the MPC strategy and adaptive rules are developed to automatically tune the aggressiveness of the MPC. The adaptive prior-knowledge-informed MPC is evaluated with computer simulations for the control of blood glucose concentrations in people with Type 1 diabetes (T1D) using simulated virtual patients. Due to the variability among people with T1D, the hyperparameters of the prior-knowledge-informed model are personalized to individual subjects. The percentage of time spent in the target range is 76.48% when there are no missing data and 76.52% when there are missing data episodes lasting up to 30mins (6 samples). Incorporating the adaptive rules further improves the percentage of time in target range to 84.58% and 84.88% for cases with no missing data and missing data, respectively. The proposed adaptive prior-informed MPC formulation provides robust, effective, and safe regulation of glucose concentration in T1D despite disturbances and missing measurements.

Multi‐rate observer design and optimal control to maximize productivity of an industry‐scale fermentation process

AbstractChemical industries focus primarily on profitable operations, resulting in growing attention and advances in the field of digital twins and optimal control algorithms. However, most industries still struggle due to a lack of physical sensors, infrequent measurements, and asynchronous sampling. Thus, in this work, we have designed a multi‐rate state observer for state estimation from plant measurements and developed a model predictive controller (MPC) that maximized the profitability of an industry‐scale fermentation process (fermenter volume < 378,500 L). Additionally, as the fermentation process is complex due to the use of microorganisms, which cannot be accurately captured using a first‐principles model, we utilize a previously developed hybrid model in the proposed MPC formulation. The MPC uses a GAMS‐MATLAB framework to determine the optimal input profiles while considering practical process constraints. It is shown using multiple datasets, that the MPC can increase productivity while also decreasing the plant operating cost.

Trajectory tracking in an underactuated, non-minimum phase two-link multibody system through model predictive control with embedded reference dynamics

This paper proposes a novel formulation of Model Predictive Control (MPC), based on linearized models, for tip trajectory tracking of time-varying references in an underactuated, non-minimum phase, two-link multibody system. Indeed, tip trajectory tracking in such systems is challenging due to the unstable internal dynamics. Additionally, when traditional MPC is used, the reference is usually tracked with delay, since MPC is intrinsically devoted to regulation or to step tracking. To overcome these issues, the proposed approach embeds an autonomous state-space model of the time-varying reference into the constrained optimization process of MPC. The controller is therefore named MPC with Embedded Reference Dynamics, MPC-ERD. By embedding the reference in the controller design, tracking with negligible error and delay is achieved, and no feedforward control is required, thus overcoming the difficulties in model inversion.The MPC-ERD is experimentally validated and the results are compared with those provided by standard MPC with embedded integrator (also including output error constraints). Besides inheriting the benefits of standard MPC, MPC-ERD ensures precise tracking of the tip time-varying reference, with negligible delay and error.

Peak Shaving in District Heating Utilizing Adaptive Predictive Control

District heating systems (DHS) are driven by the heat demands of their consumers, with higher demands giving a higher load on the heat production. While heat demands are human-dependent, they contain diurnal behaviors and weather dependencies. The diurnal behaviors contain periods with high demands causing peak loads on the heat production, which is operationally costly. This is especially true for heat pumps, a solution for DHS to include green energy, as the cost depends directly on the needed temperature. This paper presents a formulation of adaptive model predictive control (MPC) for inducing peak shaving on the production load to handle the peak load problem by using the DHS distribution network as a heat storage. It also presents a simulator model to describe the DHS. The MPC was applied to data from a case study of the DHS in Brønderslev, Denmark, showing a peak reduction of around 8%.

Neural-Network-Based Nonlinear Model Predictive Control of Multiscale Crystallization Process

The purpose of this study was to develop an integrated control strategy for multiscale crystallization processes. An image analysis method using a deep learning neural network is used to measure the fine-scale information of the crystallization process, and the mathematical statistical method is adopted to obtain the mean size of the crystal population. A feedforward neural network is subsequently trained and employed in a nonlinear model predictive control formulation to obtain the optimal profile of the manipulated variable. The effectiveness of the proposed nonlinear model predictive control method is evaluated using alum cooling crystallization experiments. Experimental results demonstrate benefits of the proposed combination of feedforward neural network and nonlinear model predictive control method for the multiscale crystallization process.

Nonlinear extension of battery constrained predictive charging control with transmission of Jacobian matrix

Battery charging operation limits can be considered systematically in the optimal constrained control framework of model predictive control (MPC). Hence, MPC-based battery charge control techniques are very promising for solving health-aware optimal charging problem. However, the battery’s nonlinear state of charge (SOC) - open circuit voltage (OCV) relationship prevents the direct application of standard MPC design process. Therefore, in this paper, we present a nonlinear extension of MPC charge control structure for lithium-ion batteries based on one of the most commonly used equivalent circuit models (ECMs). In the proposed charge control structure, extended Kalman filter (EKF) is adopted for battery state estimation. Furthermore, the Jacobian matrix obtained through first-order Taylor approximation in EKF is reused in MPC formulation. This can effectively handle the nonlinear battery charging control problem, while simultaneously make full use of the linearization result of EKF. Consequently, the linear assumption of SOC–OCV relationship in the literature is unlocked by embedding EKF into MPC formulation in a systematic way. Numerical validation with real experimental data has shown that the battery’s electrical constraints are respected during the whole charging process. Furthermore, the prediction horizon of MPC has much more impact on battery limit operation time reduction than the control horizon of MPC. The ratio between weighting factors λ and γ in MPC’s cost function is important to ensure the objective of controlling the minimum charging time.

Deep Learning-Based Long-Horizon MPC: Robust, High Performing, and Computationally Efficient Control for PMSM Drives

This article presents a computationally efficient and high performing approximate long-horizon model predictive control (MPC) for permanent magnet synchronous motors (PMSMs). Two continuous control set MPC (CCS-MPC) formulations are considered: the classical current tracking delta MPC (Del-MPC) and the torque tracking economic MPC (EMPC). To achieve offset-free torque tracking under model uncertainties and in all regions of operation, a disturbance observer and a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> -current reference generator are used. To enable real-time implementation of the long-horizon CCS-MPC, the development of a real-time capable solver is not required, since MPC approximation based on deep neural networks (DNNs) is considered and utilized for controller’s evaluation at run time. The approximation is done by training the DNN to learn the MPC functionality based on offline-generated training data and in an open-loop manner. The robust and offset-free tracking performance of the proposed DNN-based approximate long-horizon Del-MPC and EMPC has been validated through simulation and real-time implementation at test bench and compared to the state-of-the-art field oriented control (FOC) using internal model controller with field-weakening (FW) part and to the exact short-horizon MPC based on the fast gradient method (FGM-MPC). Results show that the long-horizon MPC can achieve significantly faster torque transient responses in comparison with the short-horizon FGM-MPC and the conventional FOC, especially in FW region.

Real-time motion planning and decision-making for a group of differential drive robots under connectivity constraints using robust MPC and mixed-integer programming

This work is concerned with the problem of planning trajectories and assigning tasks for a Multi-Agent System (MAS) comprised of differential drive robots. We propose a multirate hierarchical control structure that employs a planner based on robust Model Predictive Control (MPC) with mixed-integer programming (MIP) encoding. The planner computes trajectories and assigns tasks for each element of the group in real-time, while also guaranteeing the communication network of the MAS to be robustly connected at all times. Additionally, we provide a data-based methodology to estimate the disturbances sets required by the robust MPC formulation. The results are demonstrated with experiments in two obstacle-filled scenarios.

On nonlinear model predictive direct yaw moment control for trailer sway mitigation

In car–trailer combinations, the hitch angle is the relative yaw angle between towing car and trailer. The literature has shown that the inclusion of the hitch angle measurement for the feedback control of trailer oscillations can bring safety benefits, compared with conventional trailer sway mitigation algorithms based on the yaw rate of the car. Given the nonlinearity of the vehicle system in the typical conditions requiring the hitch angle control function intervention, nonlinear model-based controllers could be an effective solution. This paper presents four real-time implementable nonlinear model predictive control (NMPC) formulations, using the hitch angle measurement for the torque-vectoring (TV) control of an electric front-wheel drive car towing a trailer. The simulation results show that: (i) the active safety is enhanced by the proposed NMPC TV formulations, with respect to a benchmarking NMPC TV controller only based on the control of the towing car; (ii) the NMPC formulations that directly constrain the hitch angle error, or perform continuous hitch angle tracking, outperform those that modify the reference yaw rate or yaw rate error based on the hitch angle error; and (iii) the NMPC approaches including a dynamic model of the trailer are robust with respect to variations of trailer parameters.