Real-time Nonlinear Model Predictive Control Research Articles

A Terminal State Feasibility Governor for Real-Time Nonlinear Model Predictive Control Over Arbitrary Horizons

A Terminal State Feasibility Governor for Real-Time Nonlinear Model Predictive Control Over Arbitrary Horizons

Design and Control of the “TransBoat”: A Transformable Unmanned Surface Vehicle for Overwater Construction

This article presents the TransBoat, a novel omnidirectional unmanned surface vehicle (USV) with a magnet-based docking system for overwater construction with wave disturbances. This is the first such USV that can build overwater structures by transporting modules. The TransBoat incorporates two features designed to reject wave disturbances. First, the TransBoat's expandable body structure can actively transform from a mono-hull into a multihull for stabilization in turbulent environments by extending its four outrigger hulls. Second, a real-time nonlinear model predictive control (NMPC) scheme is proposed for all shapes of the TransBoat to enhance its maneuverability and resist disturbance to its movement, based on a nonlinear dynamic model. An experimental approach is proposed to identify the parameters of the dynamic model, and a subsequent trajectory tracking test validates the dynamics, NMPC controller, and system mobility. Further, docking experiments identify improved performance in the expanded form of the TransBoat compared with the contracted form, including an increased success rate (of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\sim } 10\%$</tex-math></inline-formula> ) and reduced docking time (of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\sim } 40$</tex-math></inline-formula> s on average). Finally, a bridge construction test verifies our system design and the NMPC control method.

Energy-efficient lane-change motion planning for personalized autonomous driving

With the aim of realizing energy-efficient, personalized, and safe mobility, a novel lane-changing motion planning strategy for personalized energy-efficient autonomous driving is proposed in this research. The key technologies consist of trajectory planning and trajectory tracking. Taking the quintic polynomials as the general trajectory cluster generator, the overall trajectory planning is converted into a constrained optimization problem using the lane-changing duration. The feasible and safe lane-changing trajectories can be extracted from the general trajectory cluster by introducing a stable handling envelope and a safe lane-changing area considering the constraints of vehicle dynamics limitation and surrounding traffic vehicles. A driving style identification module is developed based on multi-class Gaussian process classification utilizing real driving data to determine the trajectories that can characterize personalized features. Reflecting the constraints of feasibility, safety, and personalization on the boundaries of lane-changing duration, an energy-optimal lane-changing trajectory representing a specific driving style can be found and regarded as a reference. To precisely and rapidly control the vehicle to track the reference trajectory, a real-time nonlinear model predictive controller is designed and solved utilizing the parallel method. The algorithms proposed above are integrated and Driver-in-the-Loop experimental verifications are conducted. Experiment results demonstrated that the proposed strategy is able to realize lane change with an energy saving rate of 2.87% to 5.73% compared with human drivers’ maneuver. Comparative simulation with a typical automatic lane-change model also shows the effectiveness of the proposed approach, which is capable of not only accomplishing the energy-efficient lane change but also satisfying human driver’s personalized driving preferences.

Real-time nonlinear model predictive control for optimizing the ammonia coverage ratio of an SCR system based on TC-QPSO

SCR systems with complex chemical reaction dynamics are important components of emission aftertreatment systems. Their contradictory emission requirements of higher NOx conversion efficiency and lower NH3 slip can be simultaneously achieved by optimizing the ammonia coverage ratio, which can be accomplished with a nonlinear model predictive control (NMPC) framework. However, how to find a reasonable NMPC approach for optimizing the ammonia coverage ratio and utilize an appropriate method to solve it remain a formidable challenge. In this paper, a two-layer hierarchical control framework for an SCR system is developed to decouple the optimization and tracking control of the ammonia coverage ratio. In the upper layer, an NMPC controller with a long prediction horizon is designed to account for the emission objectives and constraints and generate optimal trajectories for the ammonia coverage ratio. In the lower layer, a robust observing and tracking controller with a short horizon is employed which takes the planned trajectories as the tracking target. Moreover, an improved thermonuclear-initiation and cross-boundary quantum-behaved particle swarm optimization (TC-QPSO) method for optimizing the ammonia coverage ratio based on its dynamic characteristics is proposed to solve the NMPC problem. The simulation results show that the proposed TC-QPSO emission optimization effects of NOx and NH3 are closest to those of the SQP method (the benchmark controller) with a slight computation time penalty. The experimental results under the three transient cycles show that the average NOx conversion efficiency of the proposed two-layer hierarchical control framework with the TC-QPSO algorithm can reach more than 98% with an average NH3 slip of less than 11 ppm, which is able to satisfy emission requirements, and that compared to the traditional QPSO algorithm, the average NH3 slip of the proposed TC-QPSO algorithm is improved by 45.6% with an average NOx conversion efficiency penalty of 1.09% and its emission optimization effects have better robustness for uncertainties in transient cycles.

Wave condition preview assisted real-time nonlinear predictive control of point-absorbing wave energy converter based on long short-term memory recurrent neural identification

A real-time nonlinear model predictive controller is presented for point-absorbing wave energy converter (PA-WEC) to maintain the optimal wave energy extraction by tracking the reference PA-WEC velocity. In order to reduce computational cost, the predictive controller is designed using wave condition preview and the Taylor series expansion based nonlinear optimization to determine the q-axis current control signals while the constraints on both the device velocity and q-axis current are respected. The dynamics of the PA-WEC is modeled and details of the nonlinear model predictive controller are presented. A pragmatic method is proposed to obtain the optimal reference PA-WEC velocity by observing the time instants when the excitation force passes a threshold. A long short-term memory recurrent neural network (LSTM-RNN) identifier is designed to identify the wave condition term for implementing the controller. The overall stability of the closed control system including the predictive control and the LSTM-RNN identifier is proved. The proposed nonlinear predictive controller is validated under realistic wave conditions. The results indicate that the proposed control allows higher PA-WEC power production than the conventional control under representative irregular wave conditions.

Design for Real-Time Nonlinear Model Predictive Control With Application to Collision Imminent Steering

Model predictive control (MPC) is a competitive option in modern control systems due to its ability to account for future response and incorporation of complex control objectives. As applications become more intricate, nonlinearities limit the utility of linear control strategies, thus requiring more sophisticated architectures, often at a significant computational cost. This article investigates the computational cost of solving nonlinear MPC problems and provides a framework for designing nonlinear MPC architectures compatible with real-time performance. To motivate the computational complexity associated with nonlinear MPC, the design of an automotive collision imminent steering system and the controller is considered. Various trajectory optimization strategies are examined and compared for this application, identifying multiple-shooting-based Runge–Kutta explicit integration as the most suitable. The control algorithm is then mapped into a graphics processor unit-based hardware system, where special considerations of the parallel hardware architecture are discussed. Compared to the single-shooting solution as the benchmark, multiple shootings on parallel hardware achieve three orders of magnitude improvement in wall time, supporting real-time implementation.

A Real-Time NMPC Strategy for Electric Vehicle Stability Improvement Combining Torque Vectoring With Rear-Wheel Steering

On low-friction surfaces, vehicle stability is significantly influenced by nonlinear vehicle dynamics. To realize stability improvement, a real-time nonlinear model predictive control (NMPC) strategy that combines torque vectoring control (TVC) and rear-wheel steering (RWS) is proposed. A multitarget optimization objective function that considers system state tracking, stability, control constraints, and actuation energy consumption is established. Then, a fast-solving algorithm based on Pontryagin’s minimum principle (PMP) is proposed to reduce the heavy computational burden of NMPC and satisfy the real-time computation requirement of vehicular applications. The effectiveness of the proposed fast-solving algorithm is verified by comparing its closed-loop performance with interior point optimization (IPOPT). A series of hardware-in-the-loop (HIL) experiments with the real human driver is conducted to verify the effectiveness of the proposed fast-solving algorithm and combined control strategy. Furthermore, the HIL results show that the proposed combined control strategy can improve a vehicle’s handling stability on a low-friction surface and has a better performance and lower actuation energy consumption than the TVC-only method.

Real-Time Nonlinear Model Predictive Control of Active Power Filter Using Self-Feedback Recurrent Fuzzy Neural Network Estimator

A real-time nonlinear model predictive control using a self-feedback recurrent fuzzy neural network (SFRFNN) estimator for an active power filter is developed to improve the performance of harmonic compensation. First, an SFRFNN with a recurrent structure and fuzzy rules is proposed as a prediction model for nonlinear systems. The SFRFNN merges the advantages of the fuzzy system and the recurrent neural network with a self-feedback structure, which can significantly improve the dynamic performance. Second, the optimization method based on gradient descent is employed to solve the optimal control problem. In addition, the convergence of the proposed SFRFNN and the stability of RT-NMPC are guaranteed using Lyapunov stability theory. Finally, the hardware experiment demonstrated that the proposed method has better performance in both steady and dynamic states compared with existing methods and RT-NMPC using a radial basis function neural network.

A Method of Optimal Control for Class 8 Vehicle Platoons Over Hilly Terrain

Abstract This work develops and implements a nonlinear model predictive control (NMPC) control system to facilitate fuel-optimal platooning of class 8 vehicles over challenging terrain. Prior research has shown that cooperative adaptive cruise control (CACC), which allows multiple class 8 vehicles to follow in close succession, can save between 3% and 8% in overall fuel consumption on flat terrain. However, on more challenging terrain, e.g., rolling hills, platooning vehicles can experience diminished fuel savings, and, in some cases, an increase in fuel consumption relative to individual vehicle operation. This research explores the use of NMPC with predefined route grade profiles to allow platooning vehicles to generate an optimal velocity trajectory with respect to fuel consumption. In order to successfully implement the NMPC system, a model relating vehicle velocity to fuel consumption was generated and validated using experimental data. Additionally, the predefined route grade profiles were created by using the vehicle's GPS velocity over the desired terrain. The real-time NMPC system was then implemented on a two-truck platoon operating over challenging terrain, with a reference vehicle running individually. The results from NMPC platooning are compared against fuel results from a classical proportional-integral-derivative (PID) headway control method. This comparison yields the comparative fuel savings and energy efficiency benefit of the NMPC system. In the final analysis, significant fuel savings of greater than 14 and 20% were seen for the lead and following vehicles relative to their respective traditional cruise control and platooning architectures.

Real-Time Nonlinear Model Predictive Controller for Multiple Degrees of Freedom Wave Energy Converters with Non-Ideal Power Take-Off

An increase in wave energy converter (WEC) efficiency requires not only consideration of the nonlinear effects in the WEC dynamics and the power take-off (PTO) mechanisms, but also more integrated treatment of the whole system, i.e., the buoy dynamics, the PTO system, and the control strategy. It results in an optimization formulation that has a nonquadratic and nonstandard cost functional. This article presents the application of real-time nonlinear model predictive controller (NMPC) to two degrees of freedom point absorber type WEC with highly nonlinear PTO characteristics. The nonlinear effects, such as the fluid viscous drag, are also included in the plant dynamics. The controller is implemented on a real-time target machine, and the WEC device is emulated in real-time using the WECSIM toolbox. The results for the successful performance of the design are presented for irregular waves under linear and nonlinear hydrodynamic conditions.

Validation of Real-Time Nonlinear Model Predictive Control on Point Absorber Type Wave Energy Converter by Tank Test

Real-time implementation of a nonlinear model predictive control (NMPC) for a point absorber type wave energy converter (PAWEC) with a tubular air cored permanent magnet generator is experimentally validated through tank model tests. The control inputs of the NMPC were updated by the differential equation to trace the solution of an associated two-point boundary-value problem. Two conventional control strategies, the resistive load control and the approximate complex-conjugate control with considering the copper loss (ACL), and the NMPC were implemented to the PAWEC model. The floater heave motion, control force of the generator, and power generation characteristics were compared in these control strategies. The power generation performance of the NNPC was the same level of that of the ACL in regular and irregular waves. The sensitivities to the linear viscous damping estimation error and to the wave excitation force estimation error in the NMPC were experimentally evaluated by the tank model tests.

Model Predictive Control With Environment Adaptation for Legged Locomotion

Re-planning in legged locomotion is crucial to track the desired user velocity while adapting to the terrain and rejecting external disturbances. In this work, we propose and test in experiments a real-time Nonlinear Model Predictive Control (NMPC) tailored to a legged robot for achieving dynamic locomotion on a variety of terrains. We introduce a mobility-based criterion to define an NMPC cost that enhances the locomotion of quadruped robots while maximizing leg mobility and improves adaptation to the terrain features. Our NMPC is based on the real-time iteration scheme that allows us to re-plan online at $25\,\mathrm{Hz}$ with a prediction horizon of $2$ seconds. We use the single rigid body dynamic model defined in the center of mass frame in order to increase the computational efficiency. In simulations, the NMPC is tested to traverse a set of pallets of different sizes, to walk into a V-shaped chimney,and to locomote over rough terrain. In real experiments, we demonstrate the effectiveness of our NMPC with the mobility feature that allowed IIT's $87\, \mathrm{kg}$ quadruped robot HyQ to achieve an omni-directional walk on flat terrain, to traverse a static pallet, and to adapt to a repositioned pallet during a walk.

Design and Implementation of a Real-Time Nonlinear Model Predictive Controller for a Lower Limb Exoskeleton with Input Saturation

In this paper, a nonlinear model predictive controller (NMPC) with input saturation is designed and modified for a rehabilitative exoskeleton for paraplegic individuals. An analytical solution for the NMPC optimization problem is obtained for small prediction horizons ( $$N < 3$$ ). Additionally, an iterative solution for longer horizon problems ( $$N \ge 3$$ ) is performed by employing the linear time-varying approach and using the active set method to include the constraints. Real-time guarantee for the implementation of both NMPC solutions is derived, and the robustness and stability of the closed-loop system are discussed. Finally, the proposed controller is successfully simulated and implemented on a real exoskeleton robot with 1 ms sampling time. The results show that the proposed controller is more effective than PID and adaptive fuzzy controllers.

A Real-Time Nonlinear Model Predictive Controller for Yaw Motion Optimization of Distributed Drive Electric Vehicles

This paper proposes a real-time nonlinear model predictive control (NMPC) strategy for direct yaw moment control (DYC) of distributed drive electric vehicles (DDEVs). The NMPC strategy is based on a control-oriented model built by integrating a single track vehicle model with the Magic Formula (MF) tire model. To mitigate the NMPC computational cost, the continuation/generalized minimal residual (C/GMRES) algorithm is employed and modified for real-time optimization. Since the traditional C/GMRES algorithm cannot directly solve the inequality constraint problem, the external penalty method is introduced to transform inequality constraints into an equivalently unconstrained optimization problem. Based on the Pontryagin's minimum principle (PMP), the existence and uniqueness for solution of the proposed C/GMRES algorithm are proven. Additionally, to achieve fast initialization in C/GMRES algorithm, the varying predictive duration is adopted so that the analytic expressions of optimally initial solutions in C/GMRES algorithm can be derived and gained. A Karush-Kuhn-Tucker (KKT) condition based control allocation method distributes the desired traction and yaw moment among four independent motors. Numerical simulations are carried out by combining CarSim and Matlab/Simulink to evaluate the effectiveness of the proposed strategy. Results demonstrate that the real-time NMPC strategy can achieve superior vehicle stability performance, guarantee the given safety constraints, and significantly reduce the computational efforts.

FPGA Implementation Framework for Low Latency Nonlinear Model Predictive Control

Embedded implementation of real-time Nonlinear Model Predictive Control (NMPC) is extremely challenging and complex. This paper presents a framework for implementation of NMPC on Field Programmable Gate Array (FPGA). We show the step-by-step procedure of FPGA implementation framework design of NMPC for a case study of 2D-crane system. In the implementation, we used GRAMPC software to construct NMPC and subsequently generate an FPGA specific low-level C/C++ code of the optimization solver. Generated C/C++ code is optimized for memory, speed, and resource utilization by the customized approach of applying pipelining and directives using Xilinx Vivado HLS toolchain. The NMPC is implemented on a Xilinx’s ZYNQ-7000 SoC ZC706 FPGA board. The detailed analysis of the controller computational complexity in terms of memory, resource utilization, clock, and power consumption is presented. The performance of implemented NMPC is verified through Hardware-in-the-Loop (HIL) co-simulation using system generator tool. The presented results show the feasibility of FPGA-based GRAMPC framework for ultra-fast applications of NMPC.

Least Conservative Linearized Constraint Formulation for Real-Time Motion Generation

Abstract Today robotics has shown many successful strategies to solve several navigation problems. However, moving into a dynamic environment is still a challenging task. This paper presents a novel method for motion generation in dynamic environments based on real-time nonlinear model predictive control (NMPC). At the core of our approach is a least conservative linearized constraint formulation built upon the real-time iteration (RTI) scheme with Gauss-Newton Hessian approximation. We demonstrate that the proposed constraint formulation is less conservative for planners based on Newton-type method than for those based on a fully converged NMPC method. Additionally, we show the performance of our approach in simulation, in a scenario where the Crazyflie nanoquadcopter avoids balls and reaches its desired goal in spite of the uncertainty about when the balls will be thrown. The numerical results validate our theoretical findings and illustrate the computational efficiency of the proposed scheme.

A computationally efficient model predictive control scheme for space debris rendezvous

Abstract We propose a non-linear model predictive scheme for planning fuel efficient maneuvers of small spacecrafts that shall rendezvous space debris. The paper addresses the specific issues of potential limited on-board computational capabilities and low-thrust actuators in the chasing spacecraft, and solves them by using a novel MatLab-based toolbox for real-time non-linear model predictive control (MPC) called MATMPC. This tool computes the MPC rendezvous maneuvering solution in a numerically efficient way, and this allows to greatly extend the prediction horizon length. This implies that the overall MPC scheme can compute solutions that account for the long time-scales that usually characterize the low-thrust rendezvous maneuvers. The so-developed controller is then tested in a realistic scenario that includes all the near-Earth environmental disturbances. We thus show, through numerical simulations, that this MPC method can successfully be used to perform a fuel-efficient rendezvous maneuver with an uncontrolled object, plus evaluate performance indexes such as mission duration, fuel consumption, and robustness against sensor and process noises.

Nonlinear Model Predictive Control for Disturbance Rejection in Isoenergetic-isochoric Flash Processes

Abstract We present a nonlinear model predictive control (NMPC) algorithm for semi-explicit index-1 stochastic differential-algebraic equations. It is natural to model isoenergetic-isochoric (constant energy-constant volume) flash processes with such equations. The algorithm uses the continuous-discrete extended Kalman filter (EKF) for state estimation, and it uses a single-shooting method to solve the involved optimal control problems. It computes the gradients with an adjoint method. The isoenergetic-isochoric flash is also called the UV flash, and it is important to rigorous models of phase equilibrium processes because it is a mathematical statement of the second law of thermodynamics. NMPC algorithms for UV flash processes are therefore relevant to both safe and economical operation of phase equilibrium processes such as flash separation, distillation, two-phase flow in pipes, and oil production. We design the NMPC algorithm for disturbance rejection, and we therefore augment the state vector with the unknown disturbance variables in the continuous-discrete EKF. We present a numerical example of economical NMPC of a UV flash separation process. It involves output constraints and the estimation of an unknown and unmeasured disturbance. The computation time of the NMPC algorithm does not exceed 3.14 s in any of the 5 min control intervals. This indicates that real-time NMPC of UV flash separation processes is computationally feasible.

Nonlinear Model Predictive Control for Ecological Driver Assistance Systems in Electric Vehicles

Battery Electric Vehicle (BEV) has one of the most promising drivetrain technology. However, the BEVs are facing the limited cruising range which generally reduces their share in the automotive market. Velocity profile, acceleration characteristics, road gradients, and drive techniques around curves have significant impacts on the energy consumption of the BEVs. A semi-autonomous ecological driver assistance system to regulate the velocity with energy-efficient techniques is proposed to address the limitation. The main contribution of this paper is the design of a real-time nonlinear model predictive controller with improved inequality constraints handling and economic penalty function to plan the online cost-effective cruising velocity. This system is based on the extended cruise control driver assistance system which controls the longitudinal velocity of the BEV in a safe and energy efficient manner by taking advantage of road slopes, effective drive around curves, and respecting the traffic regulation. A real-time optimisation algorithm is adapted and extended with economic objective function. Instead of the conventional Euclidean norms, deadzone penalty functions are proposed to achieve the economic objectives. In addition, the states inequality constraints are handled based on the proposed soft nonlinear complementarity function aimed to preserve the relaxed complementary slackness to enhance the Pontryagin’s Minimum Principle (PMP) method. Obtained numerical simulation and field experimental results demonstrate the effectiveness of the proposed method for a semi-autonomous electric vehicle in terms of real-time energy-efficient velocity regulation and constraints satisfaction intended to improve the cruising range capability of the BEVs.

A Real-Time Nonlinear Model Predictive Control Strategy for Stabilization of an Electric Vehicle at the Limits of Handling

In this paper, we propose a real-time nonlinear model predictive control (NMPC) strategy for stabilization of a vehicle near the limit of lateral acceleration using the rear axle electric torque vectoring configuration of an electric vehicle. A nonlinear four-wheel vehicle model that neglects the wheel dynamics is coupled with a nonlinear tire model to design three MPC strategies of different levels of complexity that are implementable online: one that uses a linearized version of the vehicle model and then solves the resulting quadratic program problem to compute the necessary longitudinal slips on the rear wheels, a second one that employs the real-time iteration scheme on the NMPC problem, and a third one that applies the primal dual interior point method on the NMPC problem instead until convergence. Then, a sliding mode slip controller is used to compute the necessary torques on the rear wheels based on the requested longitudinal slips. After analyzing the relative tradeoffs in performance and computational cost between the three MPC strategies by comparing them against the optimal solution in a series of simulation studies, we test the most promising solution in a high-fidelity environment.