Explicit MPC controller structure.

Structure of an explicit MPC controller for path-following control constructed using a vehicle model from CarSim. Based on the values of the parameter vector θ, the block “critical regions” selects a critical region CRi; then, the block “MPC feedback law” calculates the control action by applying the first MPC feedback law to the selected critical region CRi.</p

Lateral displacement of the vehicle with different prediction horizons.

This figure shows the tracking ability of the proposed controller with prediction horizons of 20, 40, and 60. When the prediction horizon is 20, a relatively large error in the lateral displacement appears as the prediction ability of the controller is degraded. On the other hand, when the prediction ability increases overwhelmingly a prediction horizon of 60 in this case, an extremely early steering is observed. Therefore, the prediction horizon is set to 40 for the controller.</p

Explicit model predictive control for linear time-variant systems with application to double-lane-change maneuver - Fig 5

Frameworks of eMPC for LTI systems (A) and LTV systems (B). In terms of parameter variation, there is no alternative way in Fig 5A to adjust the variation because the critical regions cannot be changed with respect to variation. In contrast, in Fig 5B, compensating for the state vector with an error compensator, enables the controller to be robust against such parameter variation. The main advantage of this approach is that, by simply adding a compensator, where only matrix multiplication is taken in its process, no modification of the critical regions is necessary.</p

Reference road path for DLC maneuver.

This path is a built-in path in CarSim, having 200 m of longitudinal distance, and the vehicle is supposed to change the lane twice within this distance. The orange line represents the trajectory of the vehicle. Performance is evaluated by checking whether the vehicle collides with the traffic cones. During driving, the longitudinal velocity of the vehicle is assumed to be constant.</p

States of the vehicle model during DLC maneuver.

In the implementation of the eMPC controller for the LTV system, the states are compensated regarding the parameter variation to improve the robustness of the controller. The proposed controller not only improves the tracking ability of the controller, but also enhances ride comfort as the lateral velocity is more restricted compared to the eMPC controller for the LTI system.</p

Critical regions.

An example of critical regions and the associated cost function are illustrated in this figure. In this figure, , where x(t), u2(t), and ysp(t) are the state vector, second input shown in Eq (2), and set point of the output within the prediction horizon Ny, respectively. The proposed controller predicts the second input, which can be obtained from the desired path, along Ny.</p

Analysis of explicit model predictive control for path-following control

<div><p>In this paper, explicit Model Predictive Control(MPC) is employed for automated lane-keeping systems. MPC has been regarded as the key to handle such constrained systems. However, the massive computational complexity of MPC, which employs online optimization, has been a major drawback that limits the range of its target application to relatively small and/or slow problems. Explicit MPC can reduce this computational burden using a multi-parametric quadratic programming technique(mp-QP). The control objective is to derive an optimal front steering wheel angle at each sampling time so that autonomous vehicles travel along desired paths, including straight, circular, and clothoid parts, at high entry speeds. In terms of the design of the proposed controller, a method of choosing weighting matrices in an optimization problem and the range of horizons for path-following control are described through simulations. For the verification of the proposed controller, simulation results obtained using other control methods such as MPC, Linear-Quadratic Regulator(LQR), and driver model are employed, and CarSim, which reflects the features of a vehicle more realistically than MATLAB/Simulink, is used for reliable demonstration.</p></div

Simulation result of eMPC controller for the LTI system at a longitudinal velocity of 60 km/h.

This figure shows the fulfillment of the constraint set (14) of the eMPC controller. The vertical range of each window is identical with the constraints of each variable and input.</p

Comparison of optimization controllers.

<p>This figure shows the dynamics of the states of the LQR controllers, MPC controller, and explicit MPC controller. It is proved that LQR₁ cannot fulfil the constraints as set <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194110#pone.0194110.e029" target="_blank">Eq (12)</a> and that the MPC controller consumes more time than the explicit MPC controller in the first 100 simulation runs (0.51 s in the case of the explicit MPC controller and 35.71 s in the case of the MPC controller). Moreover, LQR₂ is designed to limit the maximum values of the state dynamics in the constraints by adjusting the weighting matrices; nevertheless, a high steering wheel angular velocity, which reduces ride comfort, persist.</p

Simulation results with different ranges of prediction horizon.

<p>This figure shows the simulation results when the range of the prediction horizon <i>N</i>_<i>y</i> is varied while the input horizon <i>N</i>_<i>u</i> is fixed at 3. As <i>N</i>_<i>y</i> increases, the input dynamics, i.e., the steering wheel angle, changes in advance; this consequently reduces the lateral position error because a longer <i>N</i>_<i>y</i> improves the prediction ability of the controller. However, we found that an extremely long <i>N</i>_<i>y</i> leads to an increase in the steering wheel angular velocity, which deteriorates ride comfort.</p