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 results for optimization controllers and the driver model.

<p>Simulation results for the path-following controllers obtained using the LQR and explicit MPC methods as well as those obtained using the driver model are shown in this figure. Both controllers use the same weighting matrices to solve the optimization problem. From the results of the error variables, in particular, from the result of the lateral position error, the superiority of the explicit MPC controller over the LQR controller is demonstrated.</p

Tire cornering stiffness.

<p>(A) Corresponding lateral tire force, <i>F</i>_<i>y</i>, as a function of the slip angle of the tire, <i>α</i> with different vertical tire loads. (B) the initial slope of the function (red line) when the vertical tire load is 3187.16 N, considering the vehicle mass, i.e., 967 N/deg or 55405 N/rad, which are both values of tire cornering stiffness <i>C</i>_<i>r</i> and <i>C</i>_<i>f</i>, respectively.</p

Paths of controllers and driver model.

<p>This figure shows the paths of the LQR controller, explicit MPC controller, and driver model. It can be observed that in the case of the LQR controller, the deviation from the center line of the desired path is larger than that in the case of the explicit MPC controller, whereas path-following control performed using the explicit MPC controller is similar to that performed using the driver model in CarSim. The details of the error variables are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194110#pone.0194110.g009" target="_blank">Fig 9</a>.</p

Determination of weighting factors for the state.

<p>(A) and (B) The effects of <i>q</i>₁ and <i>q</i>₂, respectively, which are weighting factors of the state, as given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194110#pone.0194110.e030" target="_blank">Eq (13)</a>, where <i>e</i>₁(<i>t</i>) indicates the lateral position error of the vehicle with respect to the desired path. It is demonstrated that for achieving path following control with error variables, the weighting factor of the position error must be large, whereas the weighting factor of the position error derivative must be small. (C) The lateral acceleration of the sprung mass with different values of <i>q</i>₁. As ride comfort is typically evaluated according to the sprung mass of the vehicle, this figure shows a very large <i>q</i>₁ deteriorates ride comfort.</p

Parameters of vehicle model for path-following control.

<p>Parameters of vehicle model for path-following control.</p

Lateral position errors at different vehicle speeds.

<p>This figure shows the lateral position error that occurred when the vehicle speed was 18 m/s, 20 m/s, 22 m/s, and 24 m/s. The position error increases as the vehicle speed increases.</p

Desired path.

<p>Desired path for path-following control is plotted in this figure. This path comprises of four parts: a straight part, two curves, and a clothoid part. The straight part is used to prove the fulfilment of the proposed controller in this paper by setting the starting point, which is deviated from the desired path. By using the two curves and the clothoid part, the ability of the controller to perform path-following control will be demonstrated.</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