Dynamics of the time-varying first-digit probability <i>p</i>_<i>t</i>.

Time-varying first-digit probability for digits 1 to 9 (p1,t, …, p9,t) are presented from top left to bottom right. The chart further includes the posterior mean (solid line) and 95% credible bands (shaded areas) of pd,t, the sample empirical distribution (point), and Benford’s distribution (dashed line).</p

Map of the global tropical cyclones tracks from International Best Track Archive for Climate Stewardship (IBTrACS).

(Top) paths from 1842 to 1960; (Bottom) paths from 1961 to 2017;(Left) Map projection on longitude 10°E ∼170°W; (Right) Map projection on Longitude 170°W ∼10°E.</p

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

Dynamics of sum of squared deviations (SSD).

The chart gives the posterior mean of SSD (solid blue line) and 95% credible bands (shaded blue area) in each year. The time horizons suggested in the previous study is labeled for reference: Two long-term division (Period 1 and 2) and four short-term episodes (Episode A, B, C, and D).</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

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

Dynamics of the time-varying second-digit probability .

Time-varying second-digit probability for digits 0 to 9 are presented from top left to bottom right. The chart further includes the posterior mean (solid line) and 95% credible bands (shaded areas) of , the sample empirical distribution (point), and Benford’s distribution (dashed line).</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

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