Research on Intelligent Platoon Formation Control Based on Kalman Filtering and Model Predictive Control

By inergency On Apr 2, 2024

[ad_1]

The intelligent vehicle platoon system investigated in this paper is a nonlinear system, which can give rise to intricate nonlinear constraints that impact the real-time performance and stability of vehicle control. In this paper, the approximate linearization method is employed to linearize Equation (14) and depict the nonlinear system as a linear system for analysis, as shown in Equation (15):

$\{\begin{array}{l} {\dot{ζ}}_{t} = B_{t} ζ_{t} + E_{t} u_{t} \\ η_{t} = G_{t} ζ_{t} \end{array}$

(15)

where $B_{t}$ is the constant matrix related to vehicle parameters, $E_{t}$ is the nonlinear input part of the intelligent vehicle platoon Equation of state, $G_{t}$ is the output variable constant matrix, as shown in Equations (17)–(19) for details, and $u_{t}$ is the input parameter of the intelligent vehicle platoon.

During platoon driving, the inevitable presence of input and measurement disturbances can adversely affect the estimation of system variable parameters, leading to a reduction in the control effectiveness of the platoon. This paper considers these perturbations and employs the Euler method for discretizing Equation (15), as described in Equation (16):

$\{\begin{array}{l} ζ (k + 1) = B (k) ζ (k) + E (k) u (k) + ω (k) \\ Z (k) = H (k) ζ (k) + ν (k) \\ η (k) = G (k) ζ (k) \end{array}$

(16)

$B_{t} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{C_{y f} (\dot{y} + \dot{φ} l_{f}) - C_{y r} (\dot{φ} l_{r} - \dot{y})}{m {(\dot{x})}^{2}} - \dot{φ} & - \frac{C_{y f} + C_{y r}}{m \dot{x}} & 0 & \frac{C_{y r} l_{r} - C_{y f} l_{f}}{m \dot{x}} - \dot{x} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \frac{C_{y f} l_{f} (\dot{y} + \dot{φ} l_{f}) + C_{y r} l_{r} (\dot{φ} l_{r} - \dot{y})}{{I_{z} (\dot{x})}^{2}} & \frac{C_{y r} l_{r} - C_{y f} l_{f}}{I_{z} \dot{x}} & 0 & - \frac{C_{y f} {l_{f}}^{2} + C_{y r} {l_{r}}^{2}}{I_{z} \dot{x}} & 0 & 0 & 0 & 0 \\ 1 & - φ & - \dot{y} & 0 & 0 & 0 & 0 & 0 \\ φ & 1 & \dot{x} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \ddot{y} & - \dot{Y} & 0 & 0 & 0 & - \dot{φ} \\ 0 & 0 & \ddot{x} & \dot{X} & 0 & 0 & \dot{φ} & 0 \end{matrix}]$

(17)

$E_{t} = [\begin{matrix} 1 & 0 \\ 0 & \frac{C_{x f} s_{f} + C_{y f}}{m} \\ 0 & 0 \\ 0 & \frac{C_{x f} s_{f} l_{f} + C_{y f} l_{f}}{I_{z}} \\ 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ φ & 0 \end{matrix}]$

(18)

$G_{t} = [\begin{matrix} 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]$

(19)

$H (k) = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

(20)

where $B (k) = I + T B_{t}$ , $E (k) = T E_{t}$ , $G (k) = G_{t}$ , $T$ is the sampling time, and $H (k)$ is the observation matrix. $ω (k)$ and $ν (k)$ are process noise and measurement noise, respectively; the two are independent of each other and accord with normal distribution. The process noise covariance of $Q (k) = Ε [ω (k) ω {(k)}^{T}]$ , and the measurement noise covariance $R (k) = Ε [υ (k) υ {(k)}^{T}]$ . The initial state value of the system is $x (0)$ , and the initial covariance matrix is $ρ_{0}$ . This paper calculates the estimated state according to the Kalman filter, as shown in Equation (21):

$\{\begin{array}{l} \hat{ζ} (k + 1| k) = B (k) \hat{ζ} (k) + E (k) u (k) \\ ϱ (k + 1) = \hat{Z} (k + 1) - H (k) \hat{ζ} (k + 1| k) \\ \overset{´}{P} (k + 1| k) = B (k + 1) {P (k| k) B (k + 1)}^{T} + Q (k) \\ K (k + 1) = \frac{\overset{´}{P} (k + 1) {H (k + 1)}^{T}}{H (k + 1) \overset{´}{P} (k + 1| k) {H (k + 1)}^{T} + R (k)} \\ \hat{ζ} (k + 1| k + 1) = \hat{ζ} (k + 1| k) + K (k + 1) ϱ (k + 1) \\ P (k + 1) = (I - K (k + 1) H (k + 1)) \overset{´}{P} (k + 1| k) \\ \hat{ζ} (k + 1| k + 1) = \hat{ζ} (k + 1| k) + K (k + 1) ϱ (k + 1) \end{array}$

(21)

where $ϱ$ is the difference between the observed and predicted values of the system, $\overset{´}{P}$ is the predicted value of the system covariance, $K$ is the Kalman gain, and $P$ is the updated value of the system covariance. After de-perturbation, the state parameter value shown in Equation (21) is finally obtained and input to the controller.

In the process of tracking the vehicle in front, the system needs to predict the decision-making behavior of the vehicle within the specified prediction interval and reduce the difference between the predicted value and the target value. Due to the mechanical constraints of the vehicle itself, the changes in the control quantity before and after the time cannot be too big, so the change increment of the control quantity needs to be constrained. This paper establishes a new state Equation based on Equation (16), selects

∆ u

as the new control variable, and adds

u (k - 1)

to the system state variable, as shown in Equation (22):

$\{\begin{array}{l} \tilde{ζ} (k + 1) = \tilde{B} (k) \tilde{ζ} (k) + \tilde{E} (k) \tilde{u} (k) \\ \tilde{η} (k) = \tilde{G} (k) \tilde{ζ} (k) \end{array}$

(22)

where

$\{\begin{matrix} \tilde{ζ} (k) = [\begin{matrix} \hat{ζ} (k) \\ u (k - 1) \end{matrix}] \\ \tilde{u} (k) = ∆ u (k) = u (k) - u (k - 1) \\ \tilde{B} (k) = [\begin{matrix} B (k) & E (k) \\ 0_{1 \times 8} & I \end{matrix}] \\ \tilde{E} (k) = [\begin{matrix} E (k) \\ I \end{matrix}] \\ \tilde{G} (k) = [\begin{matrix} G (k) & 0 \end{matrix}] \end{matrix}$

(23)

Set the prediction time domain and control time domain of the MPC controller to

N_{p}

and

N_{c}

, and specify

N_{p} > N_{c}

. Specifying that the current moment is

k

and the initial moment state

\tilde{ζ} (k)

can be obtained by sensor measurement or state estimation. Equation (22) can be written in the following form, as shown in Equation (24):

${\tilde{η}}_{a} (k) = ψ (k) {\tilde{ζ}}_{a} (k) + Φ (k) ∆ {\tilde{u}}_{a} (k)$

(24)

where ${\tilde{η}}_{a} (k)$ , $ψ$ , $Φ$ , $∆ {\tilde{u}}_{a} (k)$ See Equations (25)–(28) for details:

${\tilde{η}}_{a} (k) = {[\begin{matrix} η (k + 1| k) \\ η (k + 2| k) \\ ⋮ \\ η (k + N_{c}| k) \\ ⋮ \\ η (k + N_{p}| k) \end{matrix}]}_{N_{p} \times 1}$

(25)

$ψ (k) = \tilde{G} (k) [\begin{matrix} \tilde{B} (k) \\ {\tilde{B}}^{2} (k) \\ ⋮ \\ {\tilde{B}}^{N_{p}} (k) \end{matrix}]$

(26)

$Φ (k) = \tilde{G} (k) [\begin{matrix} \tilde{E} (k) & 0 & \dots & 0 \\ \tilde{B} (k) \tilde{E} (k) & \tilde{E} (k) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{B}}^{N_{c - 1}} (k) \tilde{E} (k) & {\tilde{B}}^{N_{c - 2}} (k) \tilde{E} (k) & \dots & \tilde{E} (k) \\ {\tilde{B}}^{N_{c}} (k) \tilde{E} (k) & {\tilde{B}}^{N_{c - 1}} (k) \tilde{E} (k) & \dots & \tilde{B} (k) \tilde{E} (k) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{B}}^{N_{p - 1}} (k) \tilde{E} (k) & {\tilde{B}}^{N_{p - 2}} (k) \tilde{E} (k) & \dots & {\tilde{B}}^{N_{p} - N_{c}} (k) \tilde{E} (k) \end{matrix}]$

(27)

$∆ {\tilde{u}}_{a} (k) = {[\begin{matrix} ∆ u (k) \\ ∆ u (k + 1) \\ ⋮ \\ ∆ u (k + N_{c} - 1| k) \end{matrix}]}_{N_{c} \times 1}$

(28)

[ad_2]