Explicit Model Predictive Control Research Articles

On Prediction Model Fidelity in Explicit Nonlinear Model Predictive Vehicle Stability Control

This study discusses vehicle stability control based on explicit nonlinear model predictive control (NMPC) and investigates the influence of prediction model fidelity on controller performance. The explicit solutions are generated through an algorithm using multiparametric quadratic programming (mp-QP) approximations of the multiparametric nonlinear programming (mp-NLP) problems. Controllers with different prediction models are assessed through objective indicators in sine-with-dwell tests. The analysis considers the following prediction model features: 1) nonlinear lateral tire forces as functions of slip angles, which are essential for the operation of the stability controller at the limit of handling. Moreover, a simple nonlinear tire force model with saturation is shown to be an effective alternative to a more complex model based on a simplified version of the Magic Formula; 2) longitudinal and lateral load transfers, playing a crucial role in the accurate prediction of the lateral tire forces and their yaw moment contributions; 3) coupling between longitudinal and lateral tire forces, which has a significant influence on the front-to-rear distribution of the braking forces generated by the controller; and 4) nonlinear peak and stiffness factors of the tire model, with visible yet negligible effects on the results.

Comparison of Anti-Jerk Controllers for Electric Vehicles With On-Board Motors

Anti-jerk controllers actively suppress the torsional oscillations of automotive drivetrains, caused by abrupt variations of the traction torque. The main benefits are: i) enhanced passengers’ comfort; and ii) increased component life. Extensive literature deals with the design of anti-jerk controllers for electric powertrains with on-board motors, i.e., in which the electric motor is part of the sprung mass of the vehicle, and transmits torque to the wheels through a transmission, half-shafts and constant velocity joints. Nevertheless, a complete and structured comparison of the performance of the different control options is still missing. This study addresses the gap through the assessment of six anti-jerk controllers – five exemplary formulations from the literature, and one novel formulation based on explicit nonlinear model predictive control (eNMPC). All proposed control structures have the potential to be implemented on production vehicles. A set of objective performance indicators is defined to assess the controllers, which are tuned through an optimization-based routine. Results show that the wheel speed input is critical to enhance controller performance, but may lead to reduced robustness.

Explicit multiobjective model predictive control for nonlinear systems under uncertainty

SummaryIn real‐world problems, uncertainties (eg, errors in the measurement, precision errors, among others) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where in the case of uncertainties, optimal solutions can degrade in quality or they can even become unfeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multiobjective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. In order to reduce the numerical cost of the offline phase—which grows exponentially with the parameter dimension—we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multiobjective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multiobjective framework allows for online adaptations of the desired objective. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.

Design of Line-of-Sight Guidance Law and a Constrained Optimal Controller for an Autonomous Underwater Vehicle

A line-of-sight (LOS) guidance law with an explicit model predictive control (MPC) is proposed for an underactuated autonomous underwater vehicle (AUV). The derived LOS law is based on quaternion angles thus avoids the problem of singularity inherited in Euler angles. The proposed guidance and control scheme is computationally inexpensive and generates a robust optimal control signal for AUV dynamics. In view of the practical implementation, the constraints on control planes and parametric uncertainties are taken into consideration while designing the proposed control law. The effectiveness of the proposed controller is verified by both simulation and experimentation using a prototype AUV developed in the laboratory. Subsequently, multiple experiments were conducted to ascertain the applicability of the proposed algorithm in a practical scenario.

MPC-Based Fast Frequency Control of Voltage Source Converters in Low-Inertia Power Systems

A rapid deployment of renewable generation has led to significant reduction in the rotational system inertia and damping, thus making frequency control in power systems more challenging. This paper proposes a novel control scheme based on Model Predictive Control (MPC) for converter-interfaced generators operating in a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">grid-forming</i> mode, with the goal of exploiting their fast response capabilities to provide fast frequency control service to the system. The controller manipulates converter power injections to limit the frequency nadir and rate-of-change-of-frequency after a disturbance. Both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">centralized</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decentralized</i> MPC approaches are considered and compared in terms of performance and practical implementation. Special attention is given to the decentralized controller by generating an explicit MPC solution to enhance computational efficiency and reduce hardware requirements. Simulation results obtained from a detailed dynamic model of the IEEE 39-bus system demonstrate the effectiveness of the proposed control schemes.

Parametric optimization and control for a smart Proton Exchange Membrane Water Electrolysis (PEMWE) system

Hydrogen production from Proton Exchange Membrane Water Electrolysis (PEMWE) system is attracting attention due to its ability to circumvent the effect of the intermittent nature of variable renewal energy. However, the relatively high operating temperatures of the PEMWE system exacerbates the degradation of polymer membranes in PEMWE system. This paper describes the development of an optimal thermal management strategy for a PEMWE system that can attenuate the long-term effects of high operating temperatures or rapid temperature changes on the polymer membranes. The thermal management strategy is tested on a laboratory scale smart PEMWE system. First, a high fidelity mathematical model of the smart PEMWE is developed and validated based on which the application of the PARameteric Optimization and Control (PAROC) framework results in the design of an explicit model predictive controller which is deployed into the laboratory PEMWE system. It is shown that the smart PEMWE prototype system empowered with the embedded control policy achieves effective temperature control across the electrolyzer maintaining the integrity of the membrane electrode assembly.

Real-Time Optimal Lithium-Ion Battery Charging Based on Explicit Model Predictive Control

The rapidly growing use of lithium-ion batteries across various industries highlights the pressing issue of optimal charging control, as charging plays a crucial role in the health, safety and life of batteries. The literature increasingly adopts model predictive control (MPC) to address this issue, taking advantage of its capability of performing optimization under constraints. However, the computationally complex online constrained optimization intrinsic to MPC often hinders real-time implementation. This paper is thus proposed to develop a framework for real-time charging control based on explicit MPC (eMPC), exploiting its advantage in characterizing an explicit solution to an MPC problem, to enable real-time charging control. The study begins with the formulation of MPC charging based on a nonlinear equivalent circuit model. Then, multi-segment linearization is conducted to the original model, and applying the eMPC design to the obtained linear models leads to a charging control algorithm. The proposed algorithm shifts the constrained optimization to offline by precomputing explicit solutions to the charging problem and expressing the charging law as piecewise affine functions. This drastically reduces not only the online computational costs in the control run but also the difficulty of coding. Extensive numerical simulation and experimental results verify the effectiveness of the proposed eMPC charging control framework and algorithm. The research results can potentially meet the needs for real-time battery management running on embedded hardware.

Learning Robustly Stabilizing Explicit Model Predictive Controllers: A Non-Regular Sampling Approach

Off-line supervised learning from data of robustly-stabilizing nonlinear explicit model predictive controllers (EMPC) is dealt with in this letter. The learning procedure relies on the construction of a suitably large set of specifically chosen sampling points of the state space in which the values of the optimal EMPC control function have to be computed. When bounding the magnitude of approximation errors is important for stability or performance specifications, regular gridding techniques are not feasible due to the curse of dimensionality arising from the structural exponential growth of the number of points with the state dimension. In this note, we consider non-regular sampling techniques - namely, i.i.d. sampling with uniform distribution, low-discrepancy sequences and lattice point sets - that offer a good covering of the state space without suffering from an unfeasible growth of the number of points, while preserving at the same time the method guarantees in terms of robustness and stability. Some theoretical properties of the proposed sampling schemes are briefly discussed, and their successful application is showcased in a practically-relevant optimal heating problem involving a 21-dimensional state space that rules out the use of regular gridding techniques.

A Dynamic Control System for Server Processor Direct Liquid Cooling

Direct liquid cooling enables the development of IC, especially processors in servers, toward higher density, which demands more efficient means of heat removal. To exploit the capability, the control system that dynamically balances between cooling power and energy consumption is critical to operational safety and cost. This article presents the design, implementation, and characteristic of a dynamic control system for server processor direct liquid cooling in a high-density data center. The controller was trained and deployed in an online–offline structure. The loss function to train this model was built based on the model predictive control (MPC) principle. To simplify the complex physical model, a hybrid model with a nonlinear steady-state term and a linearized transient term was extracted. We used both averaged overheat and ratio of pumping power consumption to evaluate the performance of controllers. Benchmark tests on open- and closed-loop controllers were conducted. The results showed the advantages of the deep explicit MPC (DEMPC) in terms of cooling effectiveness and energy efficiency. The normalized pumping energy consumption of DEMPC reached $2.3\times 10^{-4}$ with the mean overheat level of −0.96 °C and the overheat time percentage of 6.5%. Comparing with open-loop control that has similar normalized energy costs, the mean overheat level was reduced by 1.2 °C and the time percentage of overheat was reduced by eight times. Meanwhile, normalized energy consumption was reduced by 21.7% than closed-loop proportional control who has a similar average overheat level and overheat time percentage.

Explicit model predictive control of semi-active suspension systems with magneto-rheological dampers subject to input constraints

This article presents vibration control of a semi-active quarter-car suspension system equipped with a magneto-rheological damper that provides the physical constraint of a damping force. In this study, model predictive control was designed to handle the constraints of control input (i.e. the limited damping force). The explicit solution of model predictive control was computed using multi-parametric programming to reduce the computational time for real-time implementation and then adopted in the semi-active suspension system. The control performance of model predictive control was compared with that of a clipped linear-quadratic optimal controller, where the damping force was bound using a standard saturation function. Two types of road conditions (bump and random excitation) were applied to the suspension system, and the vibration control performance was evaluated through both simulations and experiments.

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.

A Low Voltage Ride Through Strategy of DFIG based on Explicit Model Predictive Control

In this paper, an improved demagnetization control, based on Explicit Model Predictive Control (E-MPC) is proposed to improve Low Voltage Ride Through (LVRT) capability of Doubly Fed Induction Generators (DFIGs). By injecting an additional rotor current component, the demagnetization control can efficiently eliminate the free and negative flux to avoid saturation of the rotor converter. The conventional demagnetization control is based on fixed scaling factors, whose control performance can’t be guaranteed for different fault conditions. The proposed E-MPC approach can fully explore the potential of rotor side converter. Besides, the control parameters of E-MPC are derived offline and very efficient for online control. In addition, the proposed E-MPC structure is simple and easy to be implemented. The mechanism of proposed E-MPC is presented in detail and verified in Matlab/Simulink. The results show that the proposed control scheme has a good performance and can improve the LVRT capability of DFIGs under various fault conditions, especially unbalanced faults.

A Backpropagation Neural Network-Based Explicit Model Predictive Control for DC–DC Converters With High Switching Frequency

The explicit model predictive control (MPC) can solve the piecewise control laws offline to save online implementation burden. However, many offline control laws have to be stored to adapt the operating point variation, the correct control law needs to be searched, and the control parameter needs to be calculated. The large storage and computational burdens make the explicit MPC difficult to be applied to the scenarios with high switching and control frequencies. To solve these problems, this article proposes to utilize a backpropagation neural network (BPNN) to fit the input-output relationship of the offline control laws under different operating points. It not only guarantees the control performance but also reduces the storage and computational burden. Such a BPNN method directly calculates the control parameter in a parallel way and thus eliminates serial evaluation of the searching process. Simulation results are provided and compared with the state-of-the-art controls to show the effectiveness of the proposed method. Experimental results demonstrate that a BPNN with 49 parameters can fit more than 10 000 offline control laws, and its implementation can be completed within three clock cycles by field-programmable gate array (FPGA) or application-specific integrated circuit (ASIC), so the 1-MHz switching and control frequency can be achieved with 4-MHz clock frequency.

Nonlinear Model Predictive Stabilization of DC–DC Boost Converters With Constant Power Loads

This article addresses the problem of stabilization of dc–dc converters feeding constant power loads (CPLs). An explicit model predictive control is developed based on an average model of a boost converter. The optimal control law is obtained offline by solving a multiparametric nonlinear programming problem over a simplex partition of the state space. The performance is validated through simulation and experimental studies. A delay compensation scheme is included for online implementation. The control is able to drive the output voltage to the desired level and stabilize it despite the instability effects produced by the presence of CPLs. It also features robustness against abrupt changes on the input voltage and the load.

Some manifold learning considerations toward explicit model predictive control

AbstractModel predictive control (MPC) is a de facto standard control algorithm across the process industries. There remain, however, applications where MPC is impractical because an optimization problem is solved at each time step. We present a link between explicit MPC formulations and manifold learning to enable facilitated prediction of the MPC policy. Our method uses a similarity measure informed by control policies and system state variables, to “learn” an intrinsic parametrization of the MPC controller using a diffusion maps algorithm, which will also discover a low‐dimensional control law when it exists as a smooth, nonlinear combination of the state variables. We use function approximation algorithms to project points from state space to the intrinsic space, and from the intrinsic space to policy space. The approach is illustrated first by “learning” the intrinsic variables for MPC control of constrained linear systems, and then by designing controllers for an unstable nonlinear reactor.

On Some Possible Explanations for the Continuing Denial of the Existence of Keynes’s IS-LM Model in Section 4 of Chapter 21 of the General Theory: Propaganda (from Joan Robinson), Ignorance (of the Keynes-Harrod Correspondence in Volume 13 Of the CWJMK in Late 1935), Anchoring (on Keynes’s Mention of Robinson’s Name in the Preface to the General Theory), Groupthink, and Cognitive

On Some Possible Explanations for the Continuing Denial of the Existence of Keynes’s IS-LM Model in Section 4 of Chapter 21 of the General Theory: Propaganda (from Joan Robinson), Ignorance (of the Keynes-Harrod Correspondence in Volume 13 Of the CWJMK in Late 1935), Anchoring (on Keynes’s Mention of Robinson’s Name in the Preface to the General Theory), Groupthink, and Cognitive

Accelerating Explicit Model Predictive Control by Constraint Sorting

Explicit MPC represents one of the fastest ways of real-time MPC implementation. As the explicit MPC policy is optimization-free in real-time control, its efficiency is determined by solving a point location problem. This paper proposes the novel concept of accelerating explicit MPC that significantly speeds up the real-time evaluation of the point location problem. The introduced strategy has two layers: (i) an offline phase determines a smart order of the regions to be explored, and (ii) an online phase removes further regions to be explored on the fly based on the current value of the value function. The main advantage of layer (i) is that the order is evaluated offline, therefore, it does not increase the real-time implementation of explicit MPC. The implementation of layer (ii) slightly increases the real-time evaluation but leads to further speed-up of the point location problem. As the proposed layers are based just on the evaluation of some appropriate value function, the main benefit is that these layers are fully applicable also for higher-dimensional systems. Although the accelerated explicit MPC variant does not reduce the worst-case time of solving the point location problem, an extensive case study demonstrates the efficiency of the proposed strategy.

An explicit model predictive controller for constrained stochastic linear systems

In this paper we introduce an explicit Model Predictive Controller (eMPC) for a linear system subject to an additive stochastic disturbance with bounded support. The finite horizon control problem that is solved to determine the eMPC consists in minimizing an average quadratic cost subject to robust linear constraints involving state and input. By resorting to a control law parametrization that is affine in the disturbance, the finite horizon control problem is reformulated as a convex quadratic optimization program and solved via multiparametric quadratic programming. The resulting eMPC is piecewise affine as a function of the state. The proposed approach is compared with an alternative min-max approach from the literature on a numerical example.

Simulation-based trajectory tracking coordination of intelligent vehicle with explicit model prediction

In this paper a novel approach to trajectory tracking problem of intelligent vehicle is presented. A model-based optimisation algorithm for steering systems with multiple physical constraints is designed. A simplified linear vehicle model is designed in order to improve its real-time ability. Moreover, an offline solving method based on explicit model predictive control (EMPC) is presented to improve the computation speed. The simulation results show the offline solving method has the same tracking performance compared with the online solving problem by YALMIP toolbox. The computation speed is significantly increased.

Some Necessary and Sufficient Conditions for Correctness of Linear Machine in Presence of Numerical Errors

Linear machine (LM) has been recently proposed (Airan et al., 2017) for solving the point location problem which arises in explicit model predictive control (e-MPC). LM associates a linear discriminant function with each critical region identified in the offline phase in e-MPC. The solution to the online point location problem in the LM approach then simply corresponds to the region whose discriminant function attains the largest value amongst all the discriminant functions. LM involves two steps: (i) identification of neighbouring critical regions, and (ii) finding the discriminant functions by writing constraints involving discriminant functions of neighbouring pairs of regions. Both these steps involve solving linear programming (LP) problems. Similar to any other optimization problem, the constraints of the LP are satisfied with some tolerances. Even though theoretically sound, the resulting LM may not accurately identify the critical region due to the numerical errors arising from these tolerances. In the current work, we identify some conditions which can be used as an aid by the user to judge the accuracy of LM results. In particular, we give a necessary condition for step (i) whose violation will yield incorrect misclassification for some point location problems. We also propose a sufficient condition whose satisfaction guarantees the accuracy of linear machine solution despite numerical errors which may have crept in during step (ii) of the LM design. This condition needs to be evaluated for each specified point during the point location phase. We illustrate these ideas on the well known quadruple tank system.