Application of asymmetric barrier Lyapunov function using ADRC approach for development of autonomous vehicle lateral control

Abstract

This paper presents a control scheme for performance improvement in lane-keeping activity of autonomous vehicle, within guaranteed specified deviations. Such feature will provide better and safe maneuvering for autonomous vehicle. The proposed control scheme consists of Asymmetric Barrier Lyapunov Function (ABLF) based backstepping method for ensuring stability and constraints satisfaction. Since autonomous vehicle dynamics is subjected to unknown disturbances and also the states, it has been shown that the vehicle performance can be improved by the application of Active Disturbance Rejection Control (ADRC) which uses the Extended State Observer (ESO). For the proposed control scheme, the asymptotic stability and convergence of error dynamics have been established. The simulation results have been presented to illustrate the performance robustness and accuracy provided by the proposed control scheme. Further, it has been show that the vehicle motion performance is better in comparison to the performances obtained from two existing control methods.

Keywords

autonomous vehicle lateral offset error backstepping active disturbance rejection control extended state observer asymmetric barrier Lyapunov function

1. Introduction

According to road crash statistics, improper steering wheel movement is responsible for an estimated 40% of all fatalities on the road. As a result, the lateral control support system is crucial in enhancing safety (Jiang and Astolfi, 2017).

A lateral dynamics is required to solve the lateral control problem of the autonomous vehicle. In the past, the lateral control methods were simply considered look-down approaches for lateral reference sensing systems, and, in this approach, inductive cables were used for marking the reference lane. We refer to the following works regarding this approach such as (Guldner et al., 1999) and (Tan et al., 2000).

At present time, the look-ahead approach is being used because of the advancements in the vision system, radar, and camera sensor-based technology. And the sensor used for measuring lateral offset at a look-ahead distance has a typically wide range. We cite the following works (Sorniotti et al., 2017), (Khodayari et al., 2010) regarding this look-head approach. The main purpose of using a look-ahead-based approach sensing system is to enhance the damping effect as given in the paper (Son et al., 2014). Based on the lateral offset error at a look-ahead distance, the following control schemes have been studied: an adaptive controller, that is, self-tuning regulator-based lateral control (Netto et al., 2004), nested PID control (Marino et al., 2011), expert fuzzy controller (Yang and Zheng, 2007), a robust controller, that is, lane-keeping using $H_{\infty}$ control method (Roselli et al., 2017), sliding mode and ISS property-based lane-keeping method (Parkash, and Swarup, 2020), and the backstepping and forwarding design control-based method (Jiang and Astolfi, 2018).

However, the above-reported control methods have solved the lateral control problem, but the lateral offset error cannot be guaranteed within constraints. In order to handle the system’s constraint, recently, in the paper (Tee et al., 2009), a novel Asymmetric Barrier Lyapunov Function (ABLF) based backstepping method has been developed. This method has been applied to various applications (e.g., flexible beam and marine vessel) (He and Ge, 2015), (He et al., 2016), and this method has not yet been developed for an autonomous vehicle. Although the existing Model Predictive Control (MPC) based method has been solved in the paper (Kamat, 2019) to handle the constraints for the lane-keeping problem of the autonomous vehicle, the computational burdens using such kinds of methods may sluggish/delay the system response to achieve the desired performance.

The disturbance is a major issue in the control system, for example, road disturbances while vehicle maneuvering. As stated in the research work (Zhou, 2007), in order to reject disturbances (input disturbances, output disturbances, generalized disturbances, and model uncertainties), there are two disturbance rejection methods, namely, Passive Disturbance Rejection (PDR) method and the Active Disturbance Rejection (ADR) method. In the PDR method, the controller suppresses the disturbances via the feedback regulation. Passive Disturbance Rejection based controllers cannot react fast enough in presence of a strong disturbance although it can be eventually suppressed, whereas, ADR-based methods have great capabilities to deal with rejecting even the strong disturbances actively, that is, very fast in real-time. Few examples of PDR-based methods are an adaptive control, that is, a self-tuning regulator for based lateral control (Netto et al., 2004) and a robust control, that is, $H_{\infty}$ control-based lateral control of autonomous vehicles (Roselli et al., 2017). The detailed descriptions about all possible ADR methods such as Internal Model Control (IMC), Disturbance Observer (DoB), Unified Input-Output (UIO) observer, Perturbation Observer (PoB), and augmented state observer is given in research work (Zhou, 2007), of which, IMC can only estimate the output disturbance. The limitation of these methods is that they cannot estimate the generalized disturbance.

Considering the advantages of ADR-based method, this paper adopts the Active Disturbance Rejection Control (ADRC). The superiority of the ADRC strategy over all other DOBs is that it can estimate the generalized disturbance as well as estimate the states of the system via Extended State Observer (ESO) using only knowing the input and output information of the system. Both the ARRC strategy and ESO were first proposed by Prof. Han (Han, 2009). Moreover, in this paper, the modified ESO is utilized, in which only one tuning parameter is needed for converging the linear ESO as given in the paper (Zheng et al., 2008). Whereas, in the nonlinear ESO as proposed by Prof. Han (Han, 2009) and even the linear ESO as used in the paper (Hwang et al., 2020), the number of tuning parameters equal to the order of the system dynamics were required to converge the ESO. And the paper (Hwang et al., 2020) has solved the lateral control problem using linear ESO to improve lane-keeping performance. Recently, in a paper (Chu et al., 2018), the ADRC scheme has been implemented experimentally for autonomous vehicle dynamics (based on look-ahead error) to improve the lateral control performance. Although this paper has a great deal with rejecting the parameter uncertainties and external disturbance, the output cannot be guaranteed within constraints.

Kang et al. first proposed the reduced-order vehicle dynamics in 2018 (Kang et al., 2018) to achieve the same objective (to keep the vehicle in its lane) as given in (Rajamani, 2011). Based on this dynamics, the papers (Hwang et al., 2020), (Kang et al., 2018), (Kang et al., 2021) have solved the lane-keeping problems. The paper (Chu et al., 2018) uses the ADRC strategy (with classical PID controller) without guaranteeing the constraining of output (lateral offset error), as well as, up to now, this problem has not been applied to reduced-order vehicle dynamics as given in the paper (Hwang et al., 2020). And the paper (Hwang et al., 2020) only solves the problem of symmetric lateral offset error constraint using the Symmetric Barrier Lyapunov Function (SBLF) based backstepping method, in which the number of tuning parameters equal to the number of states of the ESO is required to the convergence of linear ESO; up to now, this problem has not been solved for ABLF-based method to provide guaranteed asymmetric lateral offset error within constraint.

Based on the above discussions, and considering the following points, the potential of ABLF (Tee et al., 2009) for providing the asymmetric lower and upper bounds for lateral offset error, advantages of ADRC via modified linear ESO (Zheng et al., 2008), and reduced-order dynamics (Kang et al., 2018) for easy implementation of backstepping method, the main contributions of this paper are summarized as follows:

• This paper proposes an ABLF-based backstepping control using the ADRC approach for lateral control of the autonomous vehicle, in which the proposed strategy using the ADRC concept consists of two components viz. linear ESO and ABLF-based backstepping controller.

• To improve the damping effect and overcome the problem related to strict-order dynamics, a look-ahead error-based reduced-order vehicle dynamics is considered. And the desired trajectories (based on desired yaw rate profile), smooth and non-smooth, are solved to test the maneuver in its lane.

• The proposed control scheme is robust against any kind of uncertainties (whether known, unknown, internal, external, or hybrid of these) and satisfies the asymmetric lateral offset error constraint while improving the lateral control performances.

The structure of the paper is as follows: In Section 2, the vehicle dynamics is described and defined as the problem of the paper. Section 3 describes the preliminaries and definitions for preparing the result for control design and trajectory generations. In Section 4, the desired trajectory for two cases, smooth and non-smooth, are solved. In Section 5, the controller design and ESO design are described. Simulation results are presented in Section 6. Finally, Section 7 concludes with the analysis of simulation results and future work.

2. Vehicle dynamics and problem definition

This section describes the vehicle dynamics, and then reduced-order dynamics is derived by defining the deviation variable (based on look-ahead distance). Further, the research problem is defined for this paper.

2.1. Vehicle dynamics

A vehicle dynamic is uncertain, complex, and highly nonlinear with many unknowns and uncertainties.

Considering the vehicle dynamics (Rajamani, 2011) for simplification in the development of control, the following assumptions have been made.

A1. Road bank angle is ignored.

A2. Longitudinal velocity, $v_{x}$ , is strictly positive and constant.

A3. Only the front wheel is used to steer the vehicle.

A4. Approximation of $\tan (θ) \approx θ$ for small slip angles of the tires.

A5. The lateral forces ( $F_{f}$ and $F_{r}$ ) of the tires are proportional to the slip angles ( $α_{f}$ and $α_{r}$ ) of the tires for small slip angles.

The vehicle lateral dynamics is described in the following manner (Rajamani, 2011).

2.2.1. Translational part

By applying Newton’s second law for motion along $y$ axis with the assumption of (A1), the translation dynamics is obtained as

m a_{y} = F_{f} + F_{r},

(1)

where

m

denotes the mass,

a_{y}

is the acceleration along the vertical axis. Further,

a_{y}

is composed of two terms as follows

a_{y} = \ddot{y} + v_{x} \dot{ψ},

where

\ddot{y}

is the lateral acceleration, the term

v_{x} \dot{ψ}

denotes the centripetal acceleration. Substituting the lateral acceleration,

a_{y}

, in equation (1), the dynamics become as

m (\ddot{y} + v_{x} \dot{ψ}) = F_{f} + F_{r} .

(2)

2.2.2. Angular part

The yaw dynamics by considering the moment along $z$ axis is obtained as

I_{z} \ddot{ψ} = F_{f} l_{f} + F_{r} l_{r} .

(3)

where

I_{z}

denotes the moment of inertia of the vehicle,

\ddot{ψ}

is derivative of the yaw rate,

\dot{ψ}

l_{r}

and

l_{f}

are the distances of the vehicle form c. g. of the vehicle.

2.2.3. Definitions of angles

The slip angles $α_{f}$ and $α_{r}$ for the front and rear tires from Figure 1. are defined as

α_{f} = δ - θ_{f}, α_{r} = - θ_{r},

(4)

where

δ

is the steering wheel angle, where the angles (Figure 1)

θ_{f}

and

θ_{r}

are the angles between longitudinal speed and the total velocity,

v

The slip angles can be calculated as

\tan (θ_{f}) = \frac{v_{y} + l_{f} \dot{ψ}}{v_{x}}, \tan (θ_{f}) = \frac{v_{y} - l_{r} \dot{ψ}}{v_{x}},

Applying the assumption (A4), this becomes simplified and the dynamics becomes linear. The angles $θ_{f}$ and $θ_{r}$ , with replacing $v_{y} = \dot{y}$ , are calculated as

θ_{f} = \frac{\dot{y} + l_{f} \dot{ψ}}{v_{x}}, θ_{r} = \frac{\dot{y} - l_{r} \dot{ψ}}{v_{x}} .

(5)

2.2.4. Lateral forces

The lateral forces using the assumption of (A5) and (4) are defined as

{\begin{cases} F_{f} = 2 C_{f} α_{f} = 2 C_{f} (δ - θ_{f}), \\ F_{r} = 2 C_{r} α_{r} = 2 C_{r} (- θ_{r}) . \end{cases}

(6)

2.2.5. Error dynamics with respect to road

Let us define new variables (in Figure 2) as follows

e_{y} = y - y_{d e s}, e_{ψ} = ψ - ψ_{d e s},

and

{\ddot{e}}_{y} = \ddot{y} + v_{x} (\dot{ψ} - {\dot{ψ}}_{d e s}),

(7)

e_{ψ} = ψ - ψ_{d e s} .

(8)

By two times differentiating of $e_{ψ}$ , that is, ${\ddot{e}}_{ψ} = \ddot{ψ} - {\ddot{ψ}}_{D}$ , and using (2), (6), and (7)

{\ddot{e}}_{y} = a_{22} {\dot{e}}_{y} + a_{23} e_{ψ} + a_{24} {\dot{e}}_{ψ} + (a_{24} - v_{x}) {\dot{ψ}}_{d e s} + b_{21} δ,

(9)

and using (3), (6), and (8)

{\ddot{e}}_{ψ} = a_{42} {\dot{e}}_{y} + a_{43} e_{ψ} + a_{44} {\dot{e}}_{ψ} + a_{44} {\dot{ψ}}_{d e s} + b_{41} δ .

(10)

2.2.6. Error dynamics in state variable form

By defining state vector, $s = {[e_{y} {\dot{e}}_{y} e_{ψ} {\dot{e}}_{ψ}]}^{T}$ , $u = δ$ , and ${\dot{ψ}}_{d} = {\dot{ψ}}_{d e s}$ , the dynamics (9) and (10) are defined as

\dot{s} = A s + B u + B_{φ} ψ_{d}

(11)

where

A = [\begin{array}{l} 0 1 0 0 \\ 0 a_{22} a_{23} a_{24} \\ 0 0 0 1 \\ 0 a_{42} a_{43} a_{44} \end{array}], B = [\begin{array}{l} 0 \\ b_{21} \\ 0 \\ b_{41} \end{array}], B_{φ} = [\begin{array}{l} 0 \\ a_{24} - v_{x} \\ 0 \\ a_{44} \end{array}],

with

\begin{array}{l} a_{22} = - \frac{2 (C_{f} + C_{r})}{m v_{x}}, a_{23} = - a_{22} v_{x}, a_{24} = - \frac{2 (C_{f} l_{f} - C_{r} l_{r})}{m v_{x}^{2}}, a_{42} = \frac{- 2 (C_{f} l_{f} - C_{r} l_{r})}{I_{z} v_{x}}, \\ a_{43} = - a_{42} v_{x}, a_{44} = \frac{- 2 (C_{f} l_{f}^{2} + C_{r} l_{r}^{2})}{I_{z} v_{x}}, \end{array}

b_{21} = \frac{2 C_{f}}{m v_{x}}, b_{41} = \frac{2 C_{f} l_{f}}{I_{z}} .

2.2.7. Reduced-order state dynamics

Let us define the state $x = {[x_{1} x_{2}]}^{T}$ as follows (Hwang et al., 2020), (Kang et al., 2018)

x_{1} = s_{1} + L s_{2}, x_{2} = s_{3} + L s_{4},

(12)

where

s_{1} \in ℝ

is the lateral offset error at c. g. of the vehicle, which is very important state variable to know how much far the vehicle is from road center. The

x_{1} \in ℝ

is the lateral offset error at a look-ahead distance

L

, and this distance mimics as vehicle driver’s eyes;

s_{2} \in ℝ

is the rate of change offset error (or lateral velocity error) at c. g. of the vehicle, which has great significance for slip and sliding point of view. The

x_{2} \in ℝ

is the rate of change offset error at look-ahead distance

L

. The dynamics of

x_{2}

using (11) and (12) with the assumptions of (A2) and (A3) is defined as

\begin{array}{l} {\dot{x}}_{2} = a_{22} s_{2} + a_{23} s_{3} + a_{24} s_{4} + b_{21} u + (a_{24} - v_{x}) {\dot{ψ}}_{d} \\ + L a_{42} s_{2} + L a_{43} s_{3} + L a_{44} s_{4} + L b_{41} u + L a_{44} {\dot{ψ}}_{d} \\ = (a_{22} + L a_{42}) s_{2} + (a_{23} + L a_{43}) s_{3} + (a_{24} + L a_{44}) s_{4} \\ + (a_{24} + L a_{44} - v_{x}) {\dot{ψ}}_{d} + (b_{21} + L b_{41}) u, \end{array}

(13)

As a result, the reduced second-order dynamics is defined as

\begin{array}{l} {\dot{x}}_{1} = x_{2}, \\ {\dot{x}}_{2} = g_{0} u + f (s_{2}, s_{3}, s_{4}, {\dot{ψ}}_{d}), \end{array}

(14)

where

\begin{array}{l} g_{0} = b_{21} + L b_{41}, f (\cdot) = (a_{22} + L a_{42}) s_{2} + (a_{23} + L a_{43}) s_{3} \\ + (a_{24} + L a_{44}) s_{4} + (a_{24} + L a_{44} - v_{x}) {\dot{ψ}}_{d} \end{array}

(15)

where

f (\cdot)

denotes the generalized disturbance and

{\dot{ψ}}_{d}

is the desired yaw rate as disturbance in the system.

2.2. Problem definition

Given the system (14) with partially unknown states and generalized disturbance $f (\cdot)$ as given in (15), and desired yaw rate ${\dot{ψ}}_{d} (t)$ as disturbance, design the control law such that the closed-loop system has the following properties:

P1. The lateral offset error and its first derivative, and the disturbance $f (\cdot)$ converge to its zero reference value, that is, $\lim_{t \to \infty} (x_{1}, x_{2}, f (\cdot)) = 0$ , where the lateral offset error $x_{1}$ is constrained, that is, $x_{1} (0) \in (k_{a}, k_{b})$ , with $(k_{a}, k_{b}) : {x_{1} : - k_{a} < x_{1} < k_{b}}$ for all $t \geq 0$ , where $k_{a}$ and $k_{b}$ are the positive constants.

P2. The system states and disturbance are estimated, that is,

\lim_{t \to \infty} ((x_{1} - ◯_{1}), (x_{2} - ◯_{2}), (f (\cdot) - \hat{f} (\cdot))) = 0,

where

◯_{1}

and

◯_{2}

are the estimated states,

\hat{f} (\cdot)

is the estimated generalized disturbance.

3. Preliminaries

Considering the following,

System

\begin{array}{l} {\dot{ξ}}_{1} = q_{11} ξ_{2}, \\ {\dot{ξ}}_{2} = f_{ξ} (ξ_{1}, ξ_{2}) + λ_{A} u_{A}, \end{array}

{\dot{ξ}}_{2} = f_{ξ} (ξ_{1}, ξ_{2}) + λ_{A} u_{A},

(16)

where the

ξ_{1}

and

ξ_{2}

are states of the system with the constraint:

$ξ_{1}$ , that is, $ξ_{1} (0) \in (k_{a}, k_{b})$ , where $(k_{a}, k_{b}) : {ξ_{1} : - k_{a} < ξ_{1} < k_{b}}$ for all $t \geq 0$ , where $k_{a}$ and $k_{b}$ are the positive constants and $u_{A}$ is the control input.

Assumptions:

AS1. $λ_{A} \neq 0$ , $q_{11} > 0$ ;

AS2. The function $f_{ξ} (ξ_{1}, ξ_{2})$ is continuous.

Using the assumptions AS1 and AS2, a result has been used in formulating the controller involving the ABLF (Tee et al., 2009) is given as under

Theorem 1

There exists a nonlinear state feedback controller such that the states of the system (16) are exponentially stable and $ξ_{1}$ , $ξ_{2}$ converge to zero equilibrium without violating the constraint. The state feedback controller, $u_{A}$ based on ABLF is obtained as

\begin{array}{l} u_{A} = - \frac{1}{λ_{A}} (\frac{1 - q (ξ_{1})}{k_{a}^{p} - ξ_{1}^{p}} + \frac{q (ξ_{1})}{k_{b}^{p} - ξ_{1}^{p}}) ξ_{1}^{p - 1} \\ - \frac{1}{λ_{A}} (k_{2} z_{2} - f_{ξ} (ξ_{1}, ξ_{2}) + {\dot{α}}_{A}) \end{array}

(17)

where

z_{2} = ξ_{2} - α_{A}

α_{A} = - \frac{1}{λ_{A}} [(1 - q (ξ_{1})) (k_{a}^{p} - ξ_{1}^{p}) + q (ξ_{1}) (k_{b}^{p} - ξ_{1}^{p})] k_{1} ξ_{1}^{h}

k_{1}

and

k_{2}

are the controller gains,

q (ξ_{1})

is defined in (19),

{\dot{α}}_{A}

is time derivative of

α_{A}

h \in ℤ^{+}

and

p \in ℤ^{+}

are the optimum values (see below in Remark 1).

Note: Here, the $z_{1}$ is defined as $z_{1} = ξ_{1}$ instead of $z_{1} = ξ_{1} - ξ_{1 d}$ .

Proof of Theorem 1

Step 1: Consider the asymmetric barrier Lyapunov candidate as

V_{1} = \frac{1}{p} (1 - q (ξ_{1})) \log \frac{k_{a}^{p}}{k_{a}^{p} - ξ_{1}^{p}} + \frac{1}{p} q (ξ_{1}) \log \frac{k_{b}^{p}}{k_{b}^{p} - ξ_{1}^{p}}

(18)

where

q (ξ_{1}) = {\begin{cases} 1, if ξ_{1} > 0 \\ 0, if ξ_{1} \leq 0 \end{cases}

(19)

we define

z_{2}

and virtual control

α_{A}

z_{2} = ξ_{2} - α_{A},

(20)

α_{A} = \frac{- 1}{λ_{A}} [(1 - q (ξ_{1})) (k_{a}^{p} - ξ_{1}^{p}) + q (ξ_{1}) (k_{b}^{p} - ξ_{1}^{p})] k_{1} ξ_{1}^{h}

(21)

Substituting the value of $z_{2}$ and $α_{A}$ , ${\dot{V}}_{1}$ becomes

{\dot{V}}_{1} = - k_{1} ξ_{1}^{h + p - 1} + (\frac{1 - q (ξ_{1})}{k_{a}^{p} - ξ_{1}^{p}} + \frac{q (ξ_{1})}{k_{b}^{p} - ξ_{1}^{p}}) ξ_{1}^{p - 1} z_{2}

(22)

Step 2: From equation (20), the derivative of $z_{2}$ with respect to time is defined as

\begin{array}{l} {\dot{z}}_{2} = {\dot{ξ}}_{2} - {\dot{α}}_{A} \\ = f_{ξ} (ξ_{1}, ξ_{2}) + λ_{A} u_{A} - {\dot{α}}_{A} \end{array}

(23)

Further, consider the $V_{2}$ as

V_{2} = V_{1} + \frac{1}{2} z_{2}^{2}

(24)

The derivative of $V_{2}$ with respect to time is given by

\begin{array}{l} {\dot{V}}_{2} = {\dot{V}}_{1} + z_{2} {\dot{z}}_{2} \\ = - k_{1} ξ_{1}^{h + p - 1} + (\frac{1 - q (ξ_{1})}{k_{a}^{p} - ξ_{1}^{p}} + \frac{q (ξ_{1})}{k_{b}^{p} - ξ_{1}^{p}}) ξ_{1}^{p - 1} z_{2} \\ + z_{2} (f_{ξ} (ξ_{1}, ξ_{2}) + λ_{A} u_{A} - {\dot{α}}_{A}) \end{array}

(25)

The control law, $u_{A}$ , is written as

\begin{array}{l} u_{A} = - \frac{1}{λ_{A}} (\frac{1 - q (ξ_{1})}{k_{a}^{p} - ξ_{1}^{p}} + \frac{q (ξ_{1})}{k_{b}^{p} - ξ_{1}^{p}}) ξ_{1}^{p - 1} \\ - \frac{1}{λ_{A}} (k_{2} z_{2} - f_{ξ} (ξ_{1}, ξ_{2}) + {\dot{α}}_{A}) \end{array}

(26)

Substituting of control input, $u_{A}$ , in (25), ${\dot{V}}_{2}$ becomes

{\dot{V}}_{2} = - k_{1} ξ_{1}^{h + p - 1} - k_{2} z_{2}^{2} .

(27)

By (AS1) and (AS2), for $k_{1} > 0$ , $k_{2} > 0$ and the optimum values of $h$ and $p$

{\dot{V}}_{2} = 0 \Leftarrow \Rightarrow ξ_{1} = ξ_{2} = 0

Hence, the claims hold

Remark 1

To ensure the stability and guarantee of output within constraint, the conditions for requiring the optimum values for $h$ and $p$ are given in (Tee et al., 2009), respectively.

In order to formulate the desired trajectory, following definition has been used.

Definition 1

(Hjelmstad, 2007): The piecewise continuous linear function is a function that is composed of straight-line segments. The function is linear between the points $t_{n - 1}$ and $t_{n}$ , the points where the slope changes are called the breakpoints as illustrated in Figure 3. The function $x (t)$ is described as given in (28).

The piecewise continuous linear function is defined as

x (t) = - v_{n - 1} (\frac{t - t_{n}}{t_{n} - t_{n - 1}}) + v_{n} (\frac{t - t_{n - 1}}{t_{n} - t_{n - 1}}), t_{n - 1} \leq t \leq t_{n}

(28)

where

v_{n}

is the ordinate of the function at a point

t_{n}

4. Desired trajectory formulation

Considering the following, types of desired trajectories based on desired yaw rate profile for obtaining the control are given as

4.1. Smooth trajectory

Given that the predefined coordinates $(x_{n}, t_{n})$ in Figure 4(a), the desired yaw rate, ${\dot{ψ}}_{d} (t)$ , using (28) is defined as

{\dot{ψ}}_{d} (t) = {\begin{cases} v_{0}; t < t_{0} \\ v_{1} \frac{t - t_{1}}{t_{2} - t_{1}}; t_{0} \leq t < t_{1} \\ v_{2}; t_{1} \leq t < t_{2} \\ - v_{3} \frac{t - t_{3}}{t_{4} - t_{3}}; t_{2} \leq t < t_{3} \\ v_{3}; t \geq t_{3} \end{cases}

(29)

where

(v_{0}, t_{0}) = (0,8)

, that is,

v_{0} = 0

t_{0} = 8

seconds; likewise,

$(v_{1}, t_{1}) = (0.9, 10)$ , $(v_{2}, t_{2}) = (0.9, 15)$ , $(v_{2}, t_{2}) = (0.9, 15)$ , and $(v_{3}, t_{3}) = (0,17)$ . Then, the desired trajectory is generated using (29) and the following formulas as given in (Rajamani, 2011)

x_{d} = \int_{0}^{t} V \cos ({\dot{ψ}}_{d}) d t, y_{d} = \int_{0}^{t} V \sin ({\dot{ψ}}_{d}) d t .

(30)

as given in Figure 4(b), where

V

is actual velocity of the vehicle,

x_{d}

and

y_{d}

are the desired longitudinal and lateral distances, respectively.

4.2. Non-smooth trajectory

Given that the predefined coordinates $(v_{n}, t_{n})$ in Figure 5(a), that is, $\begin{array}{l} (v_{0}, t_{0}) = (0,8), (v_{1}, t_{1}) = (0.9, 9), (v_{2}, t_{2}) = (0.4, 9.01), (v_{3}, t_{3}) = (0.4, 15) (v_{4}, t_{4}) = (0,20), \\ (v_{5}, t_{5}) = (0.8, 25), (v_{6}, t_{6}) = (0.8, 30), (v_{7}, t_{7}) = (0.2, 30.01), and (v_{8}, t_{8}) = (0,40) \end{array}$ , then the trajectory is generated as shown in Figure 5(b) using the desired yaw rate, ${\dot{ψ}}_{d} (t)$ [solved by utilizing the (28)], and the trajectory generation formula (30).

5 Proposed control design

The proposed control for the autonomous vehicle has been developed for using the ADRC approach. The schematic approach is shown in Figure 6. The vehicle dynamics is observed using ESO. The proposed controller has been derived with the help of the ABLF under the output constraint.

5.1. Extended State Observer

As the vehicle dynamics is subjected to disturbance and the output state is not precisely measurable, for the application of the proposed control scheme (Figure 6), the performance depends on the measured quality of the state $x_{2}$ which is derivative of lateral offset error and the generalized disturbance $x_{3}$ . Therefore, an ESO has been used for obtaining estimates of these two states and explained below.

Considering the vehicle dynamics (14), and defining the extended state, that is, $x_{3} = f (\cdot)$ , the augmented system is defined as

\begin{array}{l} {\dot{x}}_{1} = x_{2}, \\ {\dot{x}}_{2} = g_{0} u + x_{3}, \\ {\dot{x}}_{3} = d, \end{array}

(31)

y = x_{1}

(32)

where

x = {[x_{1}, x_{2}, x_{3}]}^{T} \in ℝ^{3}

u \in ℝ

and

y \in ℝ

are the input and output of the system, respectively, and

d = \dot{f} (\cdot)

is the derivative of generalized disturbance

f (\cdot)

Assumption 1

The derivative of generalized disturbance is bounded, that is, $d = | \dot{f} (\cdot) |$ .

The ESO design for the system (31) and (32) is defined as

\begin{array}{l} {\dot{◯}}_{1} = ◯_{2} + l_{1} (x_{1} - ◯_{1}), \\ {\dot{◯}}_{2} = ◯_{3} + l_{2} (x_{1} - ◯_{1}) + g_{0} u, \\ {\dot{◯}}_{3} = l_{3} (x_{1} - ◯_{1}), \end{array}

(33)

where the observer gains,

[l_{1}, l_{2}, l_{3}] = [ω_{0} η_{1}, ω_{0}^{2} η_{2}, ω_{0}^{3} η_{3}],

are chosen such that characteristics polynomial, that is,

λ_{0} (s) = s^{3} + ω_{0} η_{1} s^{2} + ω_{0} η_{2} s + ω_{0} η_{2} = {(s + ω_{0})}^{3},

(34)

is Hurwitz, where

η_{1} = 3,

η_{2} = 3,

η_{2} = 1,

ω_{0} > 0

, and now,

ω_{0}

is the observer bandwidth, which becomes the only parameter of the observer (Zheng et al., 2008).

5.2. Convergence of ESO

To prove the convergence of ESO and boundedness, the following Lemma has been made

Lemma 1

(Zheng et al., 2008): Consider the dynamics (31), (33) with Assumption-1, and by defining the error variables ${\tilde{x}}_{1} = x_{1} - ◯_{1}$ , ${\tilde{x}}_{2} = x_{2} - ◯_{2}$ and ${\tilde{x}}_{3} = x_{3} - ◯_{4}$ , the observation estimation error dynamics is obtained as

\begin{array}{l} {\dot{\tilde{x}}}_{1} = {\tilde{x}}_{2} - l_{1} {\tilde{x}}_{1}, \\ {\dot{\tilde{x}}}_{2} = {\hat{x}}_{3} - l_{2} {\tilde{x}}_{1}, \\ {\dot{\tilde{x}}}_{3} = - l_{3} {\tilde{x}}_{1} + d, \end{array}

(35)

Let us scale the observation estimation error

ε_{1} = \frac{{\tilde{x}}_{1}}{ω_{0}^{0}}, ε_{2} = \frac{{\tilde{x}}_{2}}{ω_{0}^{1}}, ε_{3} = \frac{{\tilde{x}}_{3}}{ω_{0}^{2}}

(36)

the dynamics (34) using (35) is rewritten as

\dot{ε} = ω_{0} F_{ε} ε + H_{ε} \frac{\dot{f} (\cdot)}{ω_{0}^{2}},

(37)

where

F_{ε} = [\begin{matrix} - 3 & 1 & 0 \\ - 3 & 0 & 1 \\ - 1 & 0 & 0 \end{matrix}], H_{ε} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}],

(38)

and using the Assumption-1, then there exists a constant

σ_{i} > 0

and a finite time

T_{1} > 0

such that

| {\tilde{x}}_{i} | \leq σ_{i}

for all

t \geq T_{1} > 0

and the observer bandwidth

ω_{0} > 0

, where

i = 1,2,3

. Furthermore, the constant

σ_{i} = O (1 / ω_{0}^{k})

for some positive integer

k

Proof of Lemma-1:

After solving the (36), we have

ε (t) = e^{ω_{0} F_{ε} t} ε (0) + Θ (t),

(39a)

where

Θ (t) = \int_{0}^{t} e^{ω_{0} F_{ε} (t - τ)} H_{ε} \frac{f (\cdot)}{ω_{0}^{2}} d τ,

(39b)

where

Θ (t) = {[\begin{matrix} Θ_{1} (t) & Θ_{2} (t) & Θ_{3} (t) \end{matrix}]}^{T}

, since the generalized disturbance, that is,

| \dot{f} (\cdot) | \leq d

is bounded, then

Θ (t)

becomes

| Θ_{i} (t) | \leq \frac{d}{ω_{0}^{3}} [| {(F_{ε}^{- 1} H_{ε})}_{i} | + | {(F_{ε}^{- 1} e^{ω_{0} F_{ε} t} H_{ε})}_{i} |],

(40)

since

$F_{ε}^{- 1} = [\begin{matrix} 0 & 0 & - 1 \\ 1 & 0 & - 3 \\ 0 & 1 & - 3 \end{matrix}]$ , we get

| {(F_{ε}^{- 1} H_{ε})}_{i} | = {\begin{cases} 1, for i=1 \\ 3, for i=2,3 \end{cases}

(41)

Since $F_{ε}$ is Hurwitz, there exist a finite time $T_{1} > 0$ such that

| {[e^{ω_{0} F_{ε} t}]}_{i j} | \leq \frac{1}{ω_{0}^{3}},

(42)

For all $t \geq T_{1}$ , where $i, j = 1,2,3$ . Then, we get

| {[e^{ω_{0} F_{ε} t} H_{ε}]}_{i} | \leq \frac{1}{ω_{0}^{3}} .

(43)

For all $t \geq T_{1}$ , where $i = 1,2,3$ . Here, the value $T_{1}$ depends on the $ω_{0} F_{ε}$ .

Let us consider

F_{ε}^{- 1} = [\begin{matrix} Δ_{11} & Δ_{12} & Δ_{13} \\ Δ_{21} & Δ_{22} & Δ_{23} \\ Δ_{31} & Δ_{32} & Δ_{33} \end{matrix}], e^{ω_{0} F_{ε} t} = [\begin{matrix} Φ_{11} & Φ_{12} & Φ_{13} \\ Φ_{21} & Φ_{23} & Φ_{23} \\ Φ_{31} & Φ_{32} & Φ_{33} \end{matrix}],

(44)

From (44) and (37), we have

\begin{array}{l} | {(F_{ε}^{- 1} e^{ω_{0} F_{ε} t} B_{ε})}_{i} | = | Δ_{i 1} Φ_{13} + Δ_{i 2} Φ_{23} + Δ_{i 3} Φ_{33} |, \\ \leq {\begin{cases} 1 / ω_{0}^{3}, for i=1 \\ 4 / ω_{0}^{3},   for i=2,3 \end{cases} \end{array}

(45)

Using (40), (41), and (45), we obtain

| Θ_{i} (t) | \leq \frac{3 d}{ω_{0}^{3}} + \frac{4 d}{ω_{0}^{6}},

(46)

for all

t \geq T_{1}

, where

i = 1,2,3

Now, let us define

$ε_{sum} (0) = | ε_{1} (0) | + | ε_{2} (0) | + | ε_{3} (0) |,$ it yields that

| {(e^{ω_{0} F_{ε} t} ε (0))}_{i} | \leq \frac{ε_{sum} (0)}{ω_{0}^{3}},

(47)

for all

t \geq T_{1}

, where

i = 1,2,3

Using (39), we have

| ε_{i} (t) | = | {[e^{ω_{0} F_{ε} t} ε (0)]}_{i} | + | Θ_{i} (t) | .

(48)

Let ${\tilde{x}}_{sum} (0) = | {\tilde{x}}_{1} (0) | + | {\tilde{x}}_{2} (0) | + | {\tilde{x}}_{3} (0) |,$ and according to the observation estimation error from (35), and (46)–(48), we obtain

| {\tilde{x}}_{i} (t) | = | \frac{ε_{sum} (0)}{ω_{0}^{3}} | + \frac{3 d}{ω_{0}^{4 - i}} + \frac{4 d}{ω_{0}^{7 - i}} = σ_{i},

(49)

for all

t \geq T_{1}

, where

i = 1,2,3

Hence, the claims hold

Remarks 2

In this paper, the proof of Lemma-1 is presented, where only one tuning parameter is required for the convergence of linear ESO. Whereas, in the paper (Hwang et al., 2020), the stability has been shown for convergence of linear ESO using three tuning parameters, that is, $l_{1}$ , $l_{2}$ , and $l_{3}$ .

5.3. Active Disturbance Rejection Control based ABLF backstepping controller

Considering the augmented system (31)–(32) with ESO (33), the proposed control law for the overall system (see Figure 6) is defined as

u = \frac{(u_{0} - ◯_{3})}{g_{0}},

(50)

where the control component

u_{0}

, using Theorem-1, is given by

\begin{array}{l} u_{0} = - \frac{1}{g_{0}} (\frac{1 - q}{k_{a}^{p} - x_{1}^{p}} + \frac{q}{k_{b}^{p} - x_{1}^{p}}) x_{1}^{p - 1} \\ - \frac{1}{g_{0}} (k_{2} ê_{2} - f (\cdot) + {\dot{α}}_{1}) \end{array}

(51)

with

ê_{2} = ◯_{2} - α_{1},

α_{1} = - \frac{1}{g_{0}} [(1 - q) (k_{a}^{p} - x_{1}^{p}) + q (k_{b}^{p} - x_{1}^{p})] k_{1} x_{1}^{h} .

where

{\dot{α}}_{1}

is the derivative of

α_{1}

f (\cdot)

is given in (15),

q (\cdot)

is given in (19).

h

and

p

are the optimum values (see Remark 1).

Remark 2

The design concept and stability proof of ADRC strategy are given in (Zheng et al., 2008), respectively.

Remark 3

Although the controller (50) is designed to track the center lane of the road for lane-keeping cases, this controller can also be used for lane-changing cases.

6. simulation results and discussions

In this section, two case studies, maneuver test for (i) smooth trajectory and (ii) non-smooth trajectory, are discussed to verify the validity of the proposed lateral control method. The fixed longitudinal speed,

v_{x} = 30

m/s, look-ahead distance,

L = 3

m, for both cases, have been assumed. The system (14) has been considered for the simulation study. The computational simulation is performed through MATLAB/SIMULINK to solve the dynamics as given in Figure 6. The trajectories for the two cases are generated according to desired yaw rate profile as given in Figure 4 and Figure 5. The nominal parameters of the vehicle are given in Table 1 (Rajamani, 2011).

Table 1.

Vehicle Parameters (Rajamani, 2011).

The symbols with their value and unit

m = 1573 kg,

l_{f}

= 1.1 m,

l_{r}

= 1.58 m

I_{z}

= 2873 kg m²,

C_{f}

C_{r}

= 80,000 N/rad

6.1. Performance measure index

To measure the performance for ESO and smoothness of the control effort, the Root Means Square Error (RMSE) and the Total Variation (TV) following performance indexes are defined respectively as

{[R S M E_{k}]}^{T} = {[\sqrt{\sum_{i = 1}^{N} {\frac{(x_{k} (i) - {\hat{x}}_{k} (i))}{N_{1}}}^{2}}]}^{T},

(52)

and

T V_{u} = \sum_{i = 1}^{N_{1}} | (u (i + 1) - u (i)) |,

(53)

where

x_{k}

denotes the

k^{t h}

observed state of the system,

◯_{k}

denotes the

k^{t h}

estimated states of the system,

u

is the control input,

k = 1,2,3

;

i = 1,2,3..... N_{1}

, where

N_{1}

denotes the number of observations.

6.2. Vehicle test for smooth trajectory

The proposed ADRC control scheme has been applied to the system of error dynamics (14) for desired smooth trajectory (based on yaw rate profile) as given in Figure 4. The vehicle passes through the points where the road curvature changes at $t_{0} = 8 s$ , $t_{1} = 10 s$ , $t_{2} = 15 s$ , and $t_{3} = 17 s$ . And the values of asymmetric constraint (lower and upper bound), that is, $k_{a} = 0.01 m$ and $k_{b} = 0.03$ m are considered, respectively.

In the Figure 7, the observed and estimated states for the vehicle and generalized disturbance are demonstrated in Figure 7, respectively. The ESO performs well; also, it is easy to figure out that all states converge to zero within constraint, and the RSME values of these states are given in Table 2.

Table 2.

RMSE and TV values.

Cases	RMSE_k (for ADRC-ABLF) (equation (52))			$T V_{u}$ (equation (53))
Cases	$k = 1$	$k = 2$	$k = 3$	QLF	SBLF	ADRC-ABLF
Smooth	$0.99 \times 10^{- 5}$	0.0027	0.2365	0.4737	0.4600	0.2783
Non-smooth	$1.22 \times 10^{- 5}$	0.0030	0.2660	1.3308	0.7624	0.4345

Figure 8 shows the performance comparison of the proposed method with conventional QLF and SBLF-based control, respectively, in which the “QLF-Bksp” represents QLF-based method using augmented state observer (Kang et al., 2018), “SBLF-Bksp” represents SBLF-based method using ESO (Hwang et al., 2020), and “ADRC-ABLF-Bksp” represents the proposed method. The symmetric constraint $(k_{b} = 0.03 m)$ of SBLF-based method and asymmetric constraint ( $k_{a} = 0.01 m$ and $k_{b} = 0.03 m$ ) of ABLF-based control are predefined, respectively. It is figured out that the proposed method performs well and does not violates the constraint in comparison to the other two methods.

Figure 9 shows the robustness of the proposed method to variations $(\mp 10 %)$ in longitudinal speed and mass of the vehicle while keeping the vehicle in its center lane within constraint.

Figure 10 shows the steering wheel effort; it is shown that the proposed control has shown the smooth performance without inducing the jerks, and hence, it provides a smooth and comfortable passenger ride. The values of TV of control effort for measuring the smoothness are given in Table 2.

6.3. Vehicle test for non-smooth trajectory

In this case, the vehicle passes through the points where the road curvature changes at $t_{0} = 8 s$ , $t_{1} = 9 s$ , $t_{2} = 9.01 s$ , and $t_{3} = 15 s$ , $t_{4} = 20 s$ , $t_{5} = 25 s$ , $t_{6} = 30 s$ , $t_{7} = 30.01 s$ , $t_{8} = 40 s$ . And the values of asymmetric constraint are the same as previously given in the smooth trajectory case.

Figure 11 shows that the ESO performance is still well in spite of the vehicle passing through a non-smooth trajectory (i.e., road curvature changes accidentally within a very short time). The observed and estimated states for vehicle and generalized disturbance are given in Figure 11, respectively. The RSME values for all vehicle states and augmented states are given in Table 2.

Figure 12 shows that the proposed method still performs well even considering the non-smooth trajectory (with the setting of the same constraint as taken in the smooth trajectory case) and does not violate the constraint in comparison to the other two methods.

Figure 13 shows the robustness of the proposed method to $\mp 10 %$ variations (considering the same constraint as given for smooth trajectory case) in longitudinal speed and mass of the vehicle while not violating the constraint. Also, it shows that the deviation of lateral offset error $x_{1}$ is more sensitive as it decreases the vehicle speed in comparison to increasing the vehicle speed.

The proposed method has performed better without obtaining the jerks in comparison to QLF and SBLF-based methods as given in Figure 14. And the values for measuring the smoothness (TVs) of steering wheel performance are given in Table 2.

7. Conclusion

In this paper, the asymmetric output (lateral offset error) constraint problem of autonomous vehicle dynamics has been solved using ABLF-based backstepping control via the ADRC approach, which involves ESO. The reduced-order state dynamics has been considered to improve the damping response and easy implementation of the backstepping method. The designed ESO has an advantageous feature that only one design parameter optimizes to bring observed states very close to vehicle states. The comparative responses and the performance analysis have been presented through a simulation study. The proposed control method contributes (a) to satisfy the asymmetric lateral offset error constraint, (b) to provide smoothness for the steering wheel performance (even without having any jerks); rather, it is better in comparison to QLF (even used the augmented state observer) method (Kang et al., 2018) and SBLF (based on ESO) method (Hwang et al., 2020), and (c) to provide robustness under system parameters uncertainties maintaining the vehicle in its lane within constraint. The application of the proposed method in the case of lane-changing dynamics introducing both the front and rear-wheel angle with optimal trajectories will be considered in future work.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research study is financially supported by Institute Ph. D. Fellowship from National Institute of Technology, Kurukshetra-136119, India.

ORCID iD

Aman Parkash

References

Chu

Sun

, et al. (2018) Active disturbance rejection control applied to automated steering for lane keeping in autonomous vehicles. Control Engineering Practice 74: 13–21.

Guldner

Sienel

Tan

, et al. (1999) Robust automatic steering control for look-down reference systems with front and rear sensors. IEEE Transactions on Control Systems Technology 7(1): 2–11.

Guo

B-Z

Zhao

Z-l

(2011) On the convergence of an extended state observer for nonlinear systems with uncertainty. Systems & Control Letters 60(6): 420–430.

Han

(2009) From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics 56(3): 900–906.

(2015) Vibration control of a flexible beam with output constraint. IEEE Transactions on Industrial Electronics 62(8): 5023–5030.

Yin

Sun

(2016) Adaptive neural network control of a marine vessel with constraints using the asymmetric barrier Lyapunov function. IEEE Transactions on Cybernetics 47(7): 1641–1651.

Hjelmstad

(2007) Fundamentals of Structural Mechanics. Boston, MA: Springer Science & Business Media.

Hwang

Kang

Kim

(2020) Robust nonlinear control using barrier Lyapunov function under lateral offset error constraint for lateral control of autonomous vehicles. IEEE Transactions on Intelligent Transportation Systems 23(2): 1–7.

Jiang

Astolfi

, (2017) A lateral control assistant for the dynamic model of vehicles subject to state constraints. In: IEEE 56th Annual Conference on Decision and Control (CDC), Melbourne, VIC, Australia, 2017, pp. 244–249.

10.

Jiang

Astolfi

, (2018) Lateral control of an autonomous vehicle. IEEE Transactions on Intelligent Vehicles 3(2): 228–237.

11.

Kamat

, (2019) Lane keeping of vehicle using model predictive control. In: IEEE 5th International Conference for Convergence in Technology (I2CT), Bombay, India, 2019, pp. 1–6.

12.

Kang

Kim

Baek

(2021) Cascade backstepping control with augmented observer for lateral control of vehicle. IEEE Access 9: 45367–45376.

13.

Kang

Kim

Chung

(2018) Observer-based backstepping control method using reduced lateral dynamics for autonomous lane-keeping system. ISA Transactions 83: 214–226.

14.

Khodayari

Ghaffari

Ameli

, et al. (2010) A historical review on lateral and longitudinal control of autonomous vehicle motions. In: International Conference on Mechanical and Electrical Technology, 2010, 421–429.

15.

Marino

Scalzi

Netto

(2011) Nested PID steering control for lane keeping in autonomous vehicles. Control Engineering Practice 19(12): 1459–1467.

16.

Netto

Chaib

Mammar

(2004) Lateral adaptive control for vehicle lane keeping. In: Proceedings of the 2004 American Control Conference, Boston, MA, USA, 2004, vol. 3, 2693–2698.

17.

Parkash

Swarup

(2020) Control of autonomous vehicle for lateral dynamics using sliding mode and input-to state stability methods. In: International Conference on Emerging Trends in Information Technology and Engineering (ic-ETITE), Vellore, India, 2020, 1–7.

18.

Rajamani

(2011) Vehicle Dynamics and Control. Boston, MA: Springer Science & Business Media.

19.

Roselli

Corno

Savaresi

, et al. (2017) H∞ control with look-ahead for lane keeping in autonomous vehicles. In: 2017 IEEE Conference on Control Technology and Applications (CCTA), Maui, HI, USA, 2017, 2220–2225.

20.

Son

Kim

Lee

, et al. (2014) Robust multirate control scheme with predictive virtual lanes for lane-keeping system of autonomous highway driving. IEEE Transactions on Vehicular Technology 64(8): 3378–3391.

21.

Sorniotti

Barber

De Pinto

(2017) Path tracking for automated driving: a tutorial on control system formulations and ongoing research. Automated Driving: 71–140.

22.

Tan

H-S

Guldner

Chen

, et al. (2000) Lane changing with look-down reference systems on automated highways. Control Engineering Practice 8(9): 1033–1043.

23.

Tee

Tay

(2009) Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4): 918–927.

24.

Yang

Zheng

(2007) An expert fuzzy controller for vehicle lateral control. In: IECON 2007 - 33rd Annual Conference of the IEEE Industrial Electronics Society, Taipei, Taiwan, 2007, 880–885.

25.

Zheng

Dong

Lee

, et al. (2008) Active disturbance rejection control for MEMS gyroscopes. In: 2008 American Control Conference, Seattle, USA, 2008, 4425–4430.

26.

Zhou

(2007) Robust and Decentralized Control of Web Winding Systems. Cleveland, Ohio: Cleveland State University.