Online adaptive optimal tracking control for model-free nonlinear systems via a dynamic neural network

Figures & data

Figure 1. Structural diagram of the control scheme.

Figure 2. State identification error.

Figure 3. State tracking for x₁.

Figure 4. State tracking for x₂.

Figure 5. Convergence property for the critic NN weight $x=\left[\text{β}\gamma \right],$.

Figure 6. Single-track vehicle model.

Table 1. Description of vehicle model parameters.

Parameters	Description
M	Vehicle mass
Iz	Yaw inertia
lf,lr	Distance from CG to the front axle and rear axle
Cf0, Cr0	Corner stiffness of the front and rear wheel
Cx	Longitudinal tire stiffness
H	CG height
vx,vy	Vehicle longitudinal and lateral speed
$RMS=\sqrt{\sum _{i=1}^n{e}^2\left(i\right)/n}$	Vehicle longitudinal and lateral acceleration
Fxi, Fyi	Longitudinal and lateral force, i = 1,2,3,4.
Fxwi, Fywi	Longitudinal and lateral tire force
Fzi	The normal force of the ith wheel
g	Gravity acceleration
Rw,Jw	Wheel rolling radius, the moment of inertia
${\omega }_{Wi}$	Wheel angular speed
Twi	Active brake torque
T	Wheel track width
$\mu$	The friction coefficient between tire and road
$\gamma$	Yaw rate about the z-axis
${\alpha }_i,{\sigma }_i$	The ith wheel slip angle, slip ratio
${\delta }_f$	Steer angle of the front wheel

Figure 7. Side-slip angle.

Figure 8. Yaw rate.