Nonlinear Model Predictive Control Formulation Research Articles

Application of terminal region enlargement approach for discrete time quasi infinite horizon nonlinear model predictive control

SummaryEnsuring nominal asymptotic stability of the nonlinear model predictive control (NMPC) controller is not trivial. Stabilizing ingredients such as terminal penalty term and Terminal Region (TR) are crucial in establishing the asymptotic stability. Approaches available in the literature provide limited degrees of freedom for the characterization of the TR for the discrete time quasi infinite horizon NMPC formulation. Current work presents alternate approaches namely arbitrary controller based approach and linear quadratic regulator (LQR) based approach, which provide larger degrees of freedom for enlarging the TR. Both the approaches are scalable to system of any dimension. Approach from the literature provides a scalar whereas proposed approaches provide two additive matrices as tuning parameters for shaping of the TR. Proposed approaches involve solving modified Lyapunov equations to compute terminal penalty term, followed by explicit characterization of the TR. Efficacy of the proposed approaches is demonstrated using benchmark two state system. TR obtained using the arbitrary controller based approach and LQR based approach are approximately 10.4723 and 9.5055 times larger by area measure when compared to the largest TR obtained using the approach from the literature. As a result, there is significant reduction in the prediction horizon length while retaining the feasibility of the controller.

Nonlinear Model Predictive Control-based Collision Avoidance for Mobile Robot

This work proposes an efficient and safe single-layer Nonlinear Model Predictive Control (NMPC) system based on LiDAR to solve the problem of autonomous navigation in cluttered environments with previously unidentified static and dynamic obstacles of any shape. Initially, LiDAR sensor data is collected. Then, the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm, is used to cluster the (Lidar) points that belong to each obstacle together. Moreover, a Minimum Euclidean Distance (MED) between the robot and each obstacle with the aid of a safety margin is utilized to implement safety-critical obstacle avoidance rather than existing methods in the literature that depend on enclosing the obstacles with a circle or minimum bounding ellipse. After that, to impose avoidance constraints with feasibility guarantees and without compromising stability, an NMPC for set-point stabilization is taken into consideration with a design strategy based on terminal inequality and equality constraints. Consequently, numerous obstacles can be avoided at the same time efficiently and rapidly through unstructured environments with narrow corridors. Finally, a case study with an omnidirectional wheeled mobile robot (OWMR) is presented to assess the proposed NMPC formulation for set-point stabilization. Furthermore, the efficacy of the proposed system is tested by experiments in simulated scenarios using a robot simulator named CoppeliaSim in combination with MATLAB which utilizes the CasADi Toolbox, and Statistics and Machine Learning Toolbox. Two simulation scenarios are considered to show the performance of the proposed framework. The first scenario considers only static obstacles while the second scenario is more challenging and contains static and dynamic obstacles. In both scenarios, the OWMR successfully reached the target pose (1.5m, 1.5m, 0°) with a small deviation. Four performance indices are utilized to evaluate the set-point stabilization performance of the proposed control framework including the steady-state error in the posture vector which is less than 0.02 meters for position and 0.012 for orientation, and the integral of norm squared actual control inputs which is 19.96 and 21.74 for the first and second scenarios respectively. The proposed control framework shows a positive performance in a narrow-cluttered environment with unknown obstacles.

Distributed control for flock navigation using nonlinear model predictive control

We present a novel approach that employs nonlinear model predictive control (NMPC) to address flock control while incorporating obstacle avoidance within a leader-follower framework. This strategy utilises each agent's predictive capabilities to navigate effectively. Building upon Reynolds' foundational flocking principles of cohesion, separation, and alignment, we adapt them to serve a navigation objective rather than focusing solely on formation tasks. In doing so, we introduce novel concepts that enhance our approach's robustness and flexibility. These concepts include the assessment of information credibility and significance received from neighbouring agents, along with the dynamic adjustment of trade-offs between reference values. We incorporate these features into our NMPC formulation and illustrate their advantages through numerical simulations and laboratory experiments.

Stability of Progressively Tightening Model Predictive Control in Continuous Time

We consider a continuous-time nonlinear model predictive control formulation that is progressively tightening in path costs and constraints. Under standard assumptions, we prove asymptotic stability of the origin for the corresponding closed-loop system and extend this result to formulations employing an auxiliary dynamic system. The theoretical results are illustrated on a numerical example.

Versatile Telescopic-Wheeled-Legged Locomotion of Tachyon 3 via Full-Centroidal Nonlinear Model Predictive Control

This paper presents a nonlinear model predictive control (NMPC) toward versatile motion generation for the telescopic-wheeled-legged robot Tachyon 3, the unique hardware structure of which poses challenges in control and motion planning. We apply the full-centroidal NMPC formulation with dedicated constraints that can capture the accurate kinematics and dynamics of Tachyon 3. We have developed a control pipeline that includes an internal state integrator to apply NMPC to Tachyon 3, the actuators of which employ high-gain position-controllers. We conducted simulation and hardware experiments on the perceptive locomotion of Tachyon 3 over structured terrains and demonstrated that the proposed method can achieve smooth and dynamic motion generation under harsh physical and environmental constraints.

Time-certified Input-constrained NMPC via Koopman Operator

Determining solving-time certificates of nonlinear model predictive control (NMPC) implementations is a pressing requirement when deploying NMPC in production environments. Such a certificate guarantees that the NMPC controller returns a solution before the next sampling time. However, NMPC formulations produce nonlinear programs (NLPs) for which it is very difficult to derive their solving-time certificates. Our previous work, Wu and Braatz (2023), challenged this limitation with a proposed input-constrained MPC algorithm having exact iteration complexity but was restricted to linear MPC formulations. This work extends the algorithm to solve input-constrained NMPC problems, by using the Koopman operator and a condensing MPC technique. We illustrate the algorithm performance on a high-dimensional, nonlinear partial differential equation (PDE) control case study, in which we theoretically and numerically certify the solving time to be less than the sampling time.

Multi-scenario and multi-stage robust NMPC with state estimation application on the Tennessee-Eastman process

This study presents the implementation of two discrete robust approaches to Non-linear Model Predictive Control (NMPC), multi-scenario NMPC (MSc-NMPC) and multi-stage NMPC (MS-NMPC), to the benchmark Tennessee-Eastman (TE) challenge, with Extended Kalman Filter (EKF) and Moving Horizon Estimation (MHE) as state estimators. The robust NMPC formulation results in closed-loop responses that prevent constraint violation and closely track the process set-point under parameter uncertainty, even in scenarios where traditional NMPC results in an unstable response for this process. Additionally, unconstrained state estimators such as EKF are unsuitable because the parameter uncertainty may cause estimates to fall outside the feasible region of the process, which ultimately destabilizes the process. MHE was able to overcome this challenge because it considers process constraints in its formulation. The additional computational time required to solve the robust NMPC formulations and MHE does not cause significant delays for the sampling time considered, demonstrating their applicability to challenging large-scale industrial chemical processes.

Integrated Control of Motion Actuators for Enhancing Path Following and Yaw Stability of Over-Actuated Autonomous Vehicles

Advanced active safety systems play a crucial role in ensuring the safe driving of vehicles in critical conditions such as an obstacle avoidance manoeuvre. However, conventional techniques relying mainly on braking interventions may not result in the desired vehicle response in such situations. Over-actuation through the control of individual motion actuators could potentially improve the safety performance of vehicles. This study evaluates various configurations of motion actuators for path following and yaw stability control of vehicles in critical driving scenarios. The configurations include active front steering (S), active front steering + torque vectoring (ST), active front steering + active camber (SC) and active front steering + torque vectoring + active camber (STC). The evaluation is achieved based on a nonlinear model predictive control formulation, which considers yaw stability and the physical limits of motion actuators. This problem formulation uses a double-track vehicle model, combined with the Dugoff tyre model and its variant with the camber effect, to model the vehicle dynamics. The actuator configurations are evaluated regarding the passing velocity, tracking accuracy, safety distance and robustness to reference trajectory variation. The results indicate that the integrated control of STC performs the best among all the four configurations while S performs the worst. Furthermore, SC is generally superior to ST.

Predictive Anti-Jerk and Traction Control for V2X Connected Electric Vehicles With Central Motor and Open Differential

V2X connectivity and powertrain electrification are emerging trends in the automotive sector, which enable the implementation of new control solutions. Most of the production electric vehicles have centralized powertrain architectures consisting of a single central on-board motor, a single-speed transmission, an open differential, half-shafts, and constant velocity joints. The torsional drivetrain dynamics and wheel dynamics are influenced by the open differential, especially in split- scenarios, i.e., with different tire-road friction coefficients on the two wheels of the same axle, and are attenuated by the so-called anti-jerk controllers. Although a rather extensive literature discusses traction control formulations for individual wheel slip control, there is a knowledge gap on: a) model based traction controllers for centralized powertrains; and b) traction controllers using the preview of the expected tire-road friction condition ahead, e.g., obtained through V2X, for enhancing the wheel slip tracking performance. This study presents nonlinear model predictive control formulations for traction control and anti-jerk control in electric powertrains with central motor and open differential, and benefitting from the preview of the tire-road friction level. The simulation results in straight line and cornering conditions, obtained with an experimentally validated vehicle model, as well as the proof-of-concept experiments on an electric quadricycle prototype, highlight the benefits of the novel controllers.

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.

A Comparative Study between NMPC and Baseline Feedback Controllers for UAV Trajectory Tracking

Transport, rescue, search, surveillance, and disaster relief tasks are some applications that can be developed with unmanned aerial vehicles (UAVs), where accurate trajectory tracking is a crucial property to operate in a cluttered environment or under uncertainties. However, this is challenging due to high nonlinear dynamics, system constraints, and uncertainties presented in cluttered environments. Hence, uncertainties in the form of unmodeled dynamics, aerodynamic effects, and external disturbances such as wind can produce unstable feedback control schemes, introducing significant positional tracking errors. This work presents a detailed comparative study between controllers such as nonlinear model predictive control (NMPC) and non-predictive baseline feedback controllers, with particular attention to tracking accuracy and computational efficiency. The development of the non-predictive feedback controller schemes was divided into inverse differential kinematics and inverse dynamic compensation of the aerial vehicle. The design of the two controllers uses the mathematical model of UAV and nonlinear control theory, guaranteeing a low computational cost and an asymptotically stable algorithm. The NMPC formulation was developed considering system constraints, where the simplified dynamic model was included; additionally, the boundaries in control actions and a candidate Lyapunov function guarantees the stability of the control structure. Finally, this work uses the commercial simulator DJI brand and DJI Matrice 100 UAV in real-world experiments, where the NMPC shows a reduction in tracking error, indicating the advantages of this formulation.

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.

Nonlinear model predictive control of a DISI turbocharged engine with virtual engine co-simulation and real-time experimental validation

An increasing number of control actuators are being added to internal combustion engines to reduce fuel consumption and meet with emission standards. This research utilizes Nonlinear Model Predictive Control (NMPC) to manage a spark ignition engine equipped with a turbocharger, low pressure Exhaust Gas Recirculation (EGR), and Variable Valve Timing (VVT). This paper focuses on discussing the experimental setup and validation of the proposed NMPC strategy, while an overview of the control-oriented engine model and NMPC formulation is provided. The proposed NMPC-based engine control was implemented into a rapid prototype control system and successfully deployed during dynamometer tests. With a minimum amount of calibration effort, the NMPC exhibits sophisticated transient actuator control, which would otherwise require many calibration maps to achieve. A detailed discussion of these transient maneuvers is provided and reveals that these control actions indeed optimize the control objective function without violation of combustion and hardware capacity constraints.

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.

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.

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.

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.

A Nonlinear Model Predictive Control for the Position Tracking of Cable-Driven Parallel Robots

This article proposes a nonlinear model predictive control (NMPC) strategy for the position tracking of cable-driven parallel robots (CDPRs). The NMPC formulation handles explicitly the cable tensions and their limits. Accordingly, the cable tension distribution is performed as an integral part of the NMPC feedback control strategy, which notably allows the CDPR to operate on the wrench-feasible workspace boundaries without failure. In order to integrate the cable tension minimization within the NMPC formulation, the concept of wrench equivalent optimality (WEO) is introduced. The WEO is a nonnegative measure able to evaluate if the wrench generated by a given cable tension vector can be generated by an alternative tension vector with smaller 2-norm. The redundancy resolution performed by means of the minimization of the WEO enables the stability of the closed-loop system to be proved. More precisely, sufficient conditions for the uniform asymptotic stability are deduced using results from the analysis of NMPC schemes without terminal constraints and costs. Furthermore, the proposed NMPC strategy is validated experimentally on a fully constrained 6 degree-of-freedom CDPR.

An MPC Formulation on $SO(3)$ for a Quadrotor With Bidirectional Thrust and Nonlinear Thrust Constraints

We examine the problem of agile trajectory tracking for a quadrotor vehicle with bidirectional thrust capabilities. A time-dependent linearization of the dynamics is presented based on the Lie algebra of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SO(3)$</tex-math></inline-formula> . The obtained linear dynamics are used as the basis for a nonlinear model predictive control algorithm that does not require operating around equilibrium state-input pairs. Our approach allows the problem to be transcribed as a quadratic program with an analytical Hessian. A method for ensuring the controller satisfies the nonlinear motor constraints involved with bidirectional thrust is then suggested, and the modelling of the thrust dynamics is discussed. Simulations for the performance of the controller are carried out on a basic half-flip maneuver as well as challenging aggressive turnaround and barrel roll maneuvers. The proposed controller performs well on all three maneuvers, significantly outperforming a reference Euler angle-based nonlinear MPC formulation and improving upon the simplified handling of the thrust constraints over the prediction horizon of the MPC.

Nonlinear Model Predictive Control for Integrated Energy-Efficient Torque-Vectoring and Anti-Roll Moment Distribution

This study applies nonlinear model predictive control (NMPC) to the torque-vectoring (TV) and front-to-total anti-roll moment distribution control of a four-wheel-drive electric vehicle with in-wheel-motors, a brake-by-wire system, and active suspension actuators. The NMPC cost function formulation is based on energy efficiency criteria, and strives to minimize the power losses caused by the longitudinal and lateral tire slips, friction brakes, and electric powertrains, while enhancing the vehicle cornering response in steady-state and transient conditions. The controller is assessed through simulations using an experimentally validated high-fidelity vehicle model, along ramp steer and multiple step steer maneuvers, including and excluding the direct yaw moment and active anti-roll moment distribution actuations. The results show: 1) the substantial enhancement of energy saving and vehicle stabilization performance brought by the integration of the active suspension contribution and TV; 2) the significance of the power loss terms of the NMPC formulation on the results; and 3) the effectiveness of the NMPC with respect to the benchmarking feedback and rule based controllers.