Nonlinear Model Predictive Trajectory Tracking Control of Underactuated Marine Vehicles: Theory and Experiment

<div class="section abstract"><div class="htmlview paragraph">This paper investigates the problem of nonlinear model predictive control (NMPC) strategy for a class of nonlinear systems with multiple actuators’ response time-delays. Conventional approaches that incorporate these time-delays into the NMPC formulation typically result in a significant increase in the optimization problem's scale. To address these problems, we propose a novel NMPC strategy. In the first stage, the NMPC strategy is designed for the nonlinear system without considering actuator’s response time-delay, thereby maintaining the original scale of the optimization problem. The optimal control sequence derived from this NMPC is then fitted to a time-continuous polynomial function, serving as a reference signal for the actuators' response time-delay models. In the second stage, combining inverse model and inverse Laplace transform techniques, a novel inverse model compensation control (IMCC) strategy is designed for actuators’ response time-delays. This IMCC strategy enables tracking of the reference signal without phase time-delay or amplitude deviation. For comparative analysis, we also implement a model augmentation NMPC strategy that directly incorporates actuators’ time-delays, inevitably increasing the scale of the optimization problem. By quantitative analysis, the model augmentation NMPC strategy will increase the number of optimal variables and equality constraints of the optimization problem. Finally, vehicle control of transport vehicle in open-pit mine is taken as simulation example, the simulation results show that both the proposed novel NMPC and IMCC algorithms and model augmentation NMPC algorithm can achieve high precision control performance, the maximum and average calculation time of the proposed novel NMPC and IMCC algorithms are 31.9% and 46.2% lower than that of model augmentation NMPC algorithm, respectively.</div></div>

This article presents the design, simulation and real-time implementation of a constrained non-linear model predictive controller for a coupled tank system. A novel wavelet-based function neural network model and a genetic algorithm online non-linear real-time optimisation approach were used in the non-linear model predictive controller strategy. A coupled tank system, which resembles operations in many chemical processes, is complex and has inherent non-linearity, and hence, controlling such system is a challenging task. Particularly important is low-level control where often instability and oscillatory responses are observed. This article designs a wavelet neural network with high predicting precision and time–frequency localisation characteristics for an online prediction model in the non-linear model predictive controller to show the effectiveness of this approach in controlling the liquid at low level. To speed up the training process, a fast global search stochastic non-linear conjugate wavelet gradient algorithm is initially used to train the wavelet neural network structure before the genetic algorithm optimisation technique is utilised to tune adaptively the wavelet neural network parameters. The non-linear model predictive controller algorithm is tested for both approaches: first, in a simulation using identified models, and second, in a real-time practical application to a single-input single-output system coupled tank system. The results show an excellent control performance with respect to mean square error and average control energy values obtained.

<div class="section abstract"><div class="htmlview paragraph">Autonomous vehicle performance is increasingly highlighted in many highway driving scenarios, which leads to more priorities to vehicle stability as well as tracking accuracy. In this paper, a nonlinear model predictive controller for autonomous vehicle trajectory tracking is designed and verified through a real-time simulation bench of a virtual test track. The dynamic stability constraints of nonlinear model predictive control (NLMPC) are obtained by a novel quadrilateral stability region criterion instead of the conventional phase plane method using the double-line region. First, a typical lane change scene of overtaking is selected and a new composited trajectory model is proposed as a reference path that combines smoothness of sine wave and comfort of linear functional path. Reference lateral velocity, azimuth angle, yaw rate, and front wheel steering angle are subsequently taken into account. Then, by establishing a nonlinear vehicle dynamics model where Magic Formula of nonlinear tire model is adapted, the quadrilateral vehicle stability region is defined in consideration of designed velocity, road adhesion coefficient, and front wheel steering angle. Working condition-variant constraints determined by the boundaries of the quadrilateral region are subsequently obtained to guarantee the stability and vehicle performance. Finally, a nonlinear motion state space model with measured and unmeasured disturbance for NLMPC tracking maneuver is proposed, Meanwhile, a multi-objective cost function based on track error, ride comfort, and the smoothness of control derivative is established. Laguerre functions are applied to design optimal control trajectory and Hildreth’s quadratic programming procedure is introduced to find converged solutions meeting constraints derived from previously investigated quadrilateral stability region for sake of lightening computation load and finding better numerically conditioned solutions of control when NLMPC is implemented online. The configuration of a real-time virtual test track is explained and the NLMPC algorithm is validated. The simulation and experiment results are illustrated to show the effectiveness of the designed nonlinear model predictive control scheme under the test of the overtaking scene compared with the conventional driver control. This work may provide a useful basis for researches of autonomous vehicle lane change in terms of track accuracy, ride comfort as well as stability.</div></div>

In this work, a nonlinear model predictive control (NMPC) strategy is presented for trajectory tracking of nonholonomic wheeled mobile robots (WMR). The problem is formulated in the receding horizon framework using the measured information. A detailed kinematic model of WMR is presented to understand the motion of the robot in a plane environment and then the proposed nonlinear model is used for the trajectory tracking control in the NMPC framework. A receding horizon optimization problem is solved in a novel way to design the control inputs (translational and angular velocities) which are used to force the system trajectories to track the reference signals. Simulation results are presented to show the effectiveness of the proposed strategy.

In this article, we present a quasi-infinite horizon nonlinear model predictive control (MPC) scheme for tracking of generic reference trajectories. This scheme is applicable to nonlinear systems, which are locally incrementally stabilizable. For such systems, we provide a reference generic offline procedure to compute an incrementally stabilizing feedback with a continuously parameterized quadratic quasi-infinite horizon terminal cost. As a result, we get a nonlinear reference tracking MPC scheme with a valid terminal cost for general reachable reference trajectories without increasing the online computational complexity. The practicality of this approach is demonstrated with a benchmark example.

This paper presents a nonlinear model predictive tracking control scheme for a six-wheeled nonholonomic unmanned ground vehicles (UGVs). It is employed as a high-level guidance control with kinematic approximation for UGV motion. A nonlinear model predictive control algorithm solves trajectory planning and optimal control problems by sequentially solving an online numerical optimization problem. The optimal control inputs for the UGV are obtained with a gradient descent optimization algorithm considering constraints of UGV motion as well as its input constraints. The characteristics of the proposed controller in terms of tracking performance and collision avoidance were investigated. The real-time performance of the proposed numerical optimization algorithm was verified with an experimental six-wheeled UGV platform in indoor and outdoor environments.

This thesis aims at the investigation and development of the control of waste heat recovery systems (WHR) for heavy duty trucks based on the organic Rankine cycle. It is desired to control these systems in real time so that they recover as much energy as possible, but this is no trivial task since their highly nonlinear dynamics are strongly affected by external inputs (disturbances). Additionally, nonlinear operational constraints must be satisfied. To deal with this problem, in this thesis a dynamic model of a WHR that is based on first principles and empirical relationships from thermodynamics and heat transfer is formulated. This model corresponds to a DAE of index 1. In view of the requirements of the employed numerical methods, it includes a spline-based evaluation method for the thermophysical properties needed to evaluate the model. Therewith, the continuous differentiability of the state trajectories with respect to controls and states on its domain of evaluation is achieved. Next, an optimal control problem (OCP) for a fixed time horizon is formulated. From the OCP, a nonlinear model-predictive control (NMPC) scheme is formulated as well. Since NMPC corresponds to a state feedback strategy, a state estimator is also formulated in the form of a moving horizon estimation (MHE) scheme. In this thesis, we make use of efficient numerical methods based on the direct multiple shooting (DMS) method for optimal control, backward differentiation formulae for the solution of initial value problems for DAE, and the corresponding versions of the real-time iteration (RTI) scheme in order to approximately solve the OCP and implement the MHE and NMPC schemes. The simultaneous implementation of NMPC and MHE schemes based on RTI has been already proven to be stable in the control literature. Several numerical instances of the DMS method for the proposed OCP, NMPC and MHE schemes are tested assuming a given real-world operation scenario consisting of truck exhaust gas data recorded during a real trip. These data have been kindly provided by our industry cooperation partner Daimler AG. Additionally, the PI and LQGI control strategies, of wide-spread use in the literature of control of WHR, are also considered for comparison with the proposed scheme. An important result of this thesis is that, considering the highest energy recovery obtained from both strategies as a reference for the given operation scenario, the proposed NMPC scheme is able to reach an additional energy generation of around 3% when the full state vector is assumed to be known, and its computational speed allows it to update the control function in times shorter than the considered sampling time of 100 [ms], which makes it a suitable candidate for real-time implementation. In a more realistic scenario in which the state has to be estimated from noisy measurements, a combination of both aforementioned NMPC and MHE schemes yields an additional energy generation of around 2%. Concretely, this thesis presents novel results and advances in the following areas: • A first principles DAE model of the WHR is presented. The model is derived from the energy and mass conservation considerations and empirical heat transfer relationships; and features a tailored evaluation method of thermophysical properties with which it possesses the property of being at least continuously differentiable with respect to its controls and states on its whole domain of evaluation. • A new real-time optimization control strategy for the WHR is developed. It consists of an NMPC strategy based on efficient simulation, optimization and control tools developed in previous works. The scheme is able to explicitly handle nonlinear constraints on controls and states. In contrast to other NMPC instances for the WHR found in the literature, our scheme's efficient numerical treatment make it real-time feasible even if the full nonlinear WHR dynamics are considered. • To the author's knowledge, this is the first implementation that considers both the NMPC and the MHE approaches used simultaneously in the control of the WHR. The combination of NMPC and MHE produces a closed-loop, model-based implementation that can treat realistic measurements as inputs and calculates the corresponding control functions as outputs.

Nonlinear model predictive control of the Tennessee Eastman challenge process

A Nonlinear model predictive control (NMPC) for trajectory tracking with the obstacle avoidance of autonomous road vehicles traveling at realistic speeds is presented in this paper, with a focus on the performance of those controllers with respect to the look-ahead horizon of the NMPC. Two different methods of obstacle avoidance are compared and then the NMPC is tested in several simulated but realistic tracking scenarios involving static obstacles on constrained roadways. In order to simplify the vehicle dynamics, a bicycle model is used for the prediction of future vehicle states in the NMPC framework. However, a high-fidelity, nonlinear CarSim vehicle model is used to evaluate the vehicle performance and test the controllers in the simulation results. The CPU time is also analyzed to evaluate these schemes for real-time applications. The results show that the NMPC controller provides satisfactory online tracking performance in a realistic scenario at normal road speeds while still satisfying the real-time constraints. In addition, it is shown that the longer prediction horizons allow for better responses of the controllers, which reduce the deviations while avoiding the obstacles, as compared with shorter horizons.

Event triggered nonlinear model predictive control for a wastewater treatment plant

In this paper, we present the design, modeling, and real-time nonlinear model predictive control (NMPC) of an autonomous robotic boat. The robot is easy to manufacture, highly maneuverable, and capable of accurate trajectory tracking in both indoor and outdoor environments. In particular, a cross type four-thruster configuration is proposed for the robotic boat to produce efficient holonomic motions. The robot prototype is rapidly 3D-printed and then sealed by adhering several layers of fiberglass. To achieve accurate tracking control, we formulate an NMPC strategy for the four-control-input boat with control input constraints, where the nonlinear dynamic model includes a Coriolis and centripetal matrix, the hydrodynamic added mass, and damping. By integrating “GPS” modules and an inertial measurement unit (IMU) into the robot, we demonstrate accurate trajectory tracking of the robotic boat along preplanned paths in both a swimming pool and a natural river. Furthermore, the code generation strategy employed in our paper yields a two order of magnitude improvement in the run time of the NMPC algorithm compared to similar systems. The robot is designed to form the basis for surface swarm robotics testbeds, on which collective algorithms for surface transportation and self-assembly of dynamic floating infrastructures can be assessed.

This paper presents an online Nonlinear Model Predictive Control (NMPC) framework for trajectory tracking of autonomous vehicles. The operating environment is assumed to be unknown with various different types of obstacles. A bicycle model is used for the prediction of the future states in the NMPC framework, and a fully nonlinear CarSim vehicle model is used for the simulations. Real-time analysis is presented for a particular situation and the effect of warm initialization of optimization process on the computation time is elaborated. Simulation results show that the NMPC controller provides satisfactory online tracking performance while satisfying the real-time constraints, and warm initialization reduces the optimizer computational load significantly.

Model predictive control (MPC) refers to the class of computer control algorithms in which a dynamic process model is used to predict and optimize process performance. Since its lower request of modeling accuracy and robustness to complicated process plants, MPC for linear systems has been widely accepted in the process industry and many other fields. But for highly nonlinear processes, or for some moderately nonlinear processes with large operating regions, linear MPC is often inefficient. To solve these difficulties, nonlinear model predictive control (NMPC) attracted increasing attention over the past decade (Qin et al., 2003, Cannon, 2004). Nowadays, the research on NMPC mainly focuses on its theoretical characters, such as stability, robustness and so on, while the computational method of NMPC is ignored in some extent. The fact mentioned above is one of the most serious reasons that obstruct the practical implementations of NMPC. Analyzing the computational problem of NMPC, the direct incorporation of a nonlinear process into the linear MPC formulation structure may result in a non-convex nonlinear programming problem, which needs to be solved under strict sampling time constraints and has been proved as an NP-hard problem (Zheng, 1997). In general, since there is no accurate analytical solution to most kinds of nonlinear programming problem, we usually have to use numerical methods such as Sequential Quadric Programming (SQP) (Ferreau et al., 2006) or Genetic Algorithm (GA) (Yuzgec et al., 2006). Moreover, the computational load of NMPC using numerical methods is also much heavier than that of linear MPC, and it would even increase exponentially when the predictive horizon length increases. All of these facts lead us to develop a novel NMPC with analytical solution and little computational load in this chapter. Since affine nonlinear system can represents a lot of practical plants in industry control, including the water-tank system that we used to carry out the simulations and experiments, it has been chosen for propose our novel NMPC algorithm. Follow the steps of research work, the chapter is arranged as follows: In Section 2, analytical one-step NMPC for affine nonlinear system will be introduced at first, then, after description of the control problem of a water-tank system, simulations will be carried out to verify the result of theoretical research. Error analysis and feedback compensation will be discussed with theoretical analysis, simulations and experiment at last. Then, in Section 3, by substituting reference trajectory for predicted state with stair-like control strategy, and using sequential one-step predictions instead of the multi-step

Motion control is one of the three core modules of autonomous driving, and nonlinear model predictive control (NMPC) has recently attracted widespread attention in the field of motion control. Vehicle dynamics equations, as a widely used model, have a significant impact on the solution efficiency of NMPC due to their stiffness. This paper first theoretically analyzes the limitations on the discretized time step caused by the stiffness of the vehicle dynamics model equations when using existing common numerical methods to solve NMPC, thereby revealing the reasons for the low computational efficiency of NMPC. Then, an A-stable controller based on the finite element orthogonal collocation method is proposed, which greatly expands the stable domain range of the numerical solution process of NMPC, thus achieving the purpose of relaxing the discretized time step restrictions and improving the real-time performance of NMPC. Finally, through CarSim 8.0/Simulink 2021a co-simulation, it is verified that the vehicle dynamics model equations are with great stiffness when the vehicle speed is low, and the proposed controller can enhance the real-time performance of NMPC. As the vehicle speed increases, the stiffness of the vehicle dynamics model equation decreases. In addition to the superior capability in addressing the integration stability issues arising from the stiffness nature of the vehicle dynamics equations, the proposed NMPC controller also demonstrates higher accuracy across a broad range of vehicle speeds.

Nonlinear model predictive control (NMPC) has proved its competency in controlling constrained nonlinear processes. Although NMPC can achieve exemplary tracking performance, the related computation effort as well as guaranteeing tracking convergence are its main drawbacks. Indeed, constrained NMPC is a nonlinear and nonconvex optimization problem where it is difficult to find a feasible solution within a reasonable time. Motivated by these difficulties, this paper proposes a procedure, using Euler approximation, to transform the nonlinear optimization problem of NMPC into a constrained quadratic optimization problem. The proposed tracking controller is applied to the autonomous navigation problem of a wheeled mobile robot (WMR) in a constrained space. Under certain assumptions, we prove the closed-loop system stability and the boundedness of the tracking error. Further, we show the recursive feasibility of the solution. Simulations are performed, first to determine the adequate control parameters, and secondly to show the effectiveness of the proposed algorithm, while its real-time implementation is experimentally verified using a differential drive mobile robot.