Cascade Backstepping Control With Augmented Observer for Lateral Control of Vehicle

This paper proposes a cascaded backstepping control method with an augmented observer for the lateral control of an autonomous vehicle. The proposed cascaded backstepping control structure consists of an inner-loop electric power steering (EPS) system and an outer-loop lane-keeping system (LKS). The outer-loop controller for LKS calculates and provides the desired steering angle to the inner-loop EPS system for maintaining the vehicle at the center of the lane. Subsequently, the inner-loop controller for the EPS system generates steering torque to track the desired steering angle from the outer loop. The proposed method can guarantee the stability of the vehicle considering both inner-loop and outer-loop dynamics. In addition, an augmented observer affords robustness against unknown model parameters and external disturbances. The stability of the closed loop of the overall system, including the lateral dynamics and the EPS system, is proven using the input-to-state stable property. Controller design with consideration of the overall system can improve the lateral control performance.

INDEX TERMS Autonomous vehicle, backstepping control, lateral motion, state observer.

NOMENCLATURE
{XYZ }: Local map coordinate frame {xyz}: Vehicle coordinate frame V x : Longitudinal velocity at the center of gravity (c.g.) of the vehicle m: Total mass of the vehicle I z : Yaw moment of the inertia of the vehicle l: (l f , l r ) Longitudinal distance from c.g. to (front, rear) tires α: (α f , α r ) Slip angle at (front, rear) tires C α : (C αf , C αr ) Cornering stiffness of (front, rear) tires F y : (F yf , F yr ) Lateral tire force on (front, rear) tires e y : Lateral position error with respect to (w.r.t.) referencė e y : Time derivative of lateral position error w.r.t. reference The associate editor coordinating the review of this manuscript and approving it for publication was Sara Dadras . e ψ = ψ d − ψ: Yaw angle error with respect to road ψ: Yaw and heading angle of vehicle on global axiṡ ψ: Yaw rate of vehicle δ: Steering angle θ h : Steering wheel angular position θ h d : Desired steering wheel angular position ω h : Steering wheel angular velocity θ m : Motor angular position ω m : Motor angular velocity i: Current input of motor T : Input torque of system (T = K t i) T EPS :

I. INTRODUCTION
Advanced driver assistant systems (ADASs) provide ride comfort and stability for drivers. Recently, these ADASs have been developed and improved from driver assistance levels to vehicle control levels. In the vehicle-control-level ADASs, the control objective can be divided into two types: longitudinal and lateral motion. The longitudinal motion of a vehicle is controlled by considering engine dynamics and aerodynamics. The fuel efficiency of the vehicle's longitudinal safety from the front vehicle can be improved with longitudinal motion control. Therefore, many studies on longitudinal motion control have focused on fuel economy and maintaining a safe distance. A longitudinal controller featuring better properties than existing ones with respect to performance and impact on fuel economy and pollution during traffic disturbances was introduced in [1]. In [2], [3], a controller that guaranteed significant stability of vehicle platoons was presented. A decentralized controller design scheme with a limited communication structure for heterogeneous vehicles was proposed and validated in [4]. An adaptive cruise control (ACC) system that could address tracking capability and fuel economy was proposed in [5], [6].
The lateral control of a vehicle, instead of longitudinal control, is examined to ensure stability. The lateral control structure consists of an outer-loop (upper-level) controller for lateral dynamics and an inner-loop (lower-level) controller for the electric power steering (EPS) system. In the outer-loop controller, the desired steering angle is obtained to track the desired lateral position for the lateral position. Various methods have been studied for the outer-loop control. A linear quadratic controller and model predictive control method were previously proposed [7]- [9]. Fuzzy-logicbased lateral motion controllers were presented in [10]- [12]. An online estimation of inertial parameters was proposed for lightweight electric vehicles by using a dual unscented Kalman filter approach [13]. In [14], a backstepping control scheme using a simple second-order model was introduced for LKS. In the outer loop (lateral dynamics), the steering angle is regarded as the system input, although the input torque is the actual input of the overall system. Thus, the desired steering angle is obtained by the outer-loop controller. Then, the steering angle is controlled to track the desired steering angle by using an inner-loop controller. In the inner-loop controller, the torque of the EPS system is calculated to track the desired steering angle.
Several methods have been proposed to improve the steering wheel tracking performance. A nonlinear hybrid impedance approach was proposed in [15]. In [16], a secondorder sliding-mode controller with a fuzzy neural network was studied. A model predictive control approach for steering systems was proposed in [17]. Although these previous methods improved the control performance, only the outer loop or the inner loop was considered. In other words, the overall system, including the lateral dynamics and the EPS system, was not considered in the controller design. Because the previous outer-loop controller was designed without considering the inner-loop system (EPS system), the mismatch between the desired steer angle and the actual steering angle can result in poor lateral control performance. Thus, a control method for the overall system, including the lateral dynamics and the EPS system, is required to improve the lateral control performance. Recently, in [18], a two-layer controller approach for the lateral control of vehicles was proposed; however, the stability of the overall system was not mathematically proven.
In this paper, we propose a cascaded backstepping control method with an augmented observer for the lateral control of an autonomous vehicle. The proposed method is designed to improve the lateral control performance, while guaranteeing the stability of the overall system via a cascaded backstepping procedure. The proposed method consists of an outer-loop controller and an inner-loop controller. The outer-loop controller calculates the desired steering angle for lateral control in lateral dynamics. From the inner-loop controller, the torque of the EPS system required to track the desired steering angle is obtained. Augmented observers are designed to estimate the complete state and disturbances, including unknown model parameters and external disturbances. Accordingly, augmented observers are implemented in the outer-loop and inner-loop controllers to improve the robustness against unknown model parameters and external disturbances. The stability of the closed loop of the overall system is proven using the input-to-state stable property. Controller design with consideration of the overall system can improve the lateral control performance. The control performance of the proposed controller is validated via the vehicle dynamic simulator CarSim and MATLAB/Simulink. In the simulation, the proposed cascaded backstepping approach for a hierarchical system exhibits a stable and reasonable lane-keeping performance, even in the presence of unknown disturbances.

II. CASCADE BACKSTEPPING CONTROLLER A. CONTROLLER
Let us consider second-order dynamics given bẏ where d denotes an unknown disturbance including uncertain system parameters, g is a system parameter, and u is the control input. Now, let us define the tracking error as 45368 VOLUME 9, 2021 Then, we can represent the tracking error dynamics as follows:ė The control input u and the desired value of the second state x 2 d can be designed as follows to guarantee stability: where k 1 and k 2 are positive constants, andd is the bounded disturbance estimated by the observer that is yet to be designed. When (4) is applied to (1), the error dynamics can be expressed as (5)ė Theorem 1: The tracking error dynamics in equation (5) represent the serial interconnected system with the following input-to-stable (ISS) property such that Proof: From (5), the dynamics of e 2 2 are written as d dt Using Lemma 6.20 and Theorem C.2 in [19], we derive the following result: Equation (8) shows that the relationship between e 2 andd has the ISS property. From (5), the dynamics of e 2 1 are obtained as d dt Then, Equation (10) guarantees that the relationship between e 1 and e 2 has the ISS property. From (8) and (10), the ISS property of the overall tracking error system is represented by (6).
Thus the tracking error dynamics in (5) represent the serial interconnected system of the ISS system. Note that from the ISS property (6), as t → ∞ In this section, the augmented observer is designed to estimate the unmeasured state and unknown disturbance. By designing an augmented observer, the backstepping controller can compensate for the effect of unknown disturbances. Let us define an augmented state, x a and its observer state,x a as We assume that the disturbance of the given system varies slowly. With this assumption, the dynamics of d are represented as the following unknown nonlinear function κ(·): The dynamics of augmented observer can be written aṡ where Here, C o denotes the measurement matrix depending on the sensor configuration. ϕ and B ϕ represent additional available external signals and matrice depending on the system. Let us define the estimated error asx a = x a −x a . Then, the estimated state error dynamics becomė where B d = 0 0 1 T . In this paper, we assume that the values of disturbances in the system are physically bounded. Therefore, κ is numerically bounded. In addition, there exists an unknown positive upper boundedness κ max of |κ| such that ḋ = |κ| ≤ κ max .
By substituting x byx in (4), we get instead of (4). Consequently, substituting the estimated state and disturbance yield the following tracking error dynamicṡ where The overall closed-loop system becomeṡ There exists γ > 0 such that Theorem 2: The tracking error dynamics (17) have the ISS property such that Proof: In (5), the e 2 dynamics are written aṡ Then, we have d dt where Equation (8) is then rewritten as

III. CASCADE APPROACH
Let us consider the cascade system depicted in Fig. 1. The overall dynamics can be represented as follows: where the superscript s ∈ {i, o} denotes the inner-loop and outer-loop, respectively. Here, the input signal of the outerloop, oû , becomes the desired value of the inner-loop. The actual input of the outer-loop system becomes the output signal of the inner-loop. Further, the actual input of the outer-loop that includes the error, oû a , can be represented as follows: From (3), the outer-loop and inner-loop of the cascade system can be represented as follows: • Inner-loop Here, the outer-loop dynamic equation e 2 becomes Applying the control law (4) yields the following equation: Now we can represent the overall cascade system as follows with the backstepping controller (4) Note that od + g i e 1 is bounded because both i e 1 and g are bounded. Then, where ρ = od + g i e 1 . Finally, the tracking error dynamics (31) represent the serial interconnected system with the following ISS property from Theorem 1.

IV. APPLICATION
In this paper, we applied the proposed cascade control to the lateral control of autonomous vehicles such as in a lane keeping system (LKS). Lateral control of the vehicle is composed of an upper (outer) controller LKS to follow the given path, and a lower (inner) EPS controller to follow the target steering angle from the upper controller. The simplified second-order model for applying the proposed controller is covered in the next section.

A. INNER-LOOP ELECTRIC POWER STEERING SYSTEM
We consider a column-mounted EPS (C-EPS) system shown in Fig. 2. The C-EPS dynamic model can be represented with motor dynamics and a rack-bar motion dynamics as Further we have the following torque equation at the pinion To represent the EPS model in a second-order form, we assume that the steering column is considerable rigid. Then, we have θ m = N θ p between the motor and pinion with the gear ratio, N . In addition, it can be assumed that the relation between the rack bar, x r and the rotation of pinion, θ p , is linear, i.e., θ p = 1 R p x r with (34), (35) and (36). Consequently, the second-order C-EPS dynamics can be modeled as:θ where u is the input torque. Here, F r is the load force applied to the rack bar in Fig. 2. We define the following parameters for the readability of the dynamic equation: Consequently, the inner-loop second-order dynamics of EPS is given by Here, the subscript i denotes the inner-loop. Now we can implement the proposed controller (4) using a simplified second-order EPS model (39).

B. OUTER-LOOP VEHICLE LATERAL MOTION MODEL
In this section, a simplified second-order dynamic model of the vehicle is derived using the well-known bicycle model. The bicycle model is used for lateral vehicle dynamics [20], as described in Fig. 3. The equation for lateral motion is where In addition, the equation for the yaw dynamics is described as (42) VOLUME 9, 2021 To track the desired lane on the road, both the lateral position error and heading angle with respect to the road are defined as follows Then, from (43) Now, let us define outer- follows [14]: Here, the convergence of o x 1 and o x 2 guarantees that both e y and e ψ are uniformly and ultimately bounded [14]. In other words, an LKS that satisfies the bounded error even with a simple second-order vehicle model can be designed. Finally, a cascade backstepping controller with an augmented observer using an inner-loop (39) and outer-loop (47) can be designed.

V. COMPUTATIONAL SIMULATIONS A. SIMULATION SETUP
In the simulations, both the dynamic vehicle model in CarSim and the 4th-order nominal EPS model in [21] were used to validate the proposed algorithm. The proposed controller based on a simplified second-order model was implemented via MATLAB/Simulink. The sample times of the inner-loop EPS model and the outer-loop vehicle model were 10 ms and 100 ms, respectively. The parameters of a C-class hatchback in CarSim were used for the vehicle controller. We applied a sine function with a magnitude of 5 Nm as the inner-loop EPS disturbance in consideration of the driver's disturbance and the uncertainty of system parameters. In addition, we considered that the difference between CarSim which contains high-order vehicle dynamics, and the simplified second-order model is the disturbance of the outer-loop vehicle model. Fig. 4 shows the reference roads for the high-speed and low-speed simulations, where the vehicle traveled at speeds of 80 km/h and 30 km/h, respectively. In this simulation, we set two different fixed look-ahead distances (L = 1 m for 80 km/h and L = 4 m for 30 km/h) for two cases depending on the longitudinal velocity and maximum road curvature. For comparison, the previous method (denoted as BTSP+BLF), which consists of barrier Lyapunov function (BLF) algorithm [22] for the outer-loop system and a backstepping controller for the inner-loop system was used. We can verify the lane keeping performance of the proposed algorithm through lateral offset error and heading angle error. The estimation performance of the proposed method for the high speed case is shown in Fig. 5. The overall state of the inner-loop system is shown in Fig. 5(a). Here, i x 1 denotes the motor angle, i x 2 denotes the rate of the motor angle, and i x 3 denotes the disturbance of the EPS. The proposed augmented observer shows reasonable performance in the estimation of i x 1 . It is also evident that the motor angle i x 1 changed as the vehicle entered the curved road at 12 s. The steering wheel angle at the first curve was approximately 15 • . Owing to the difference between the 4th-order steering model for the EPS system and the 2nd-order steering model for the controller, there was a disturbance when the vehicle entered or exited the curved road. Note that the disturbance also included the sinusoidal disturbances that we added.
In Fig. 5(b), the state of the outer-loop system is described. Here, o x 1 denotes the lateral offset at the look-ahead distance, o x 2 denotes its rate, and o x 3 denotes disturbance of lateral dynamics. In particular, o x 3 includes the nonlinear characteristics of vehicle dynamics and the parameter uncertainties. The estimation of o x 1 is found to be highly accurate. The nonlinearity of the vehicle increases as the radius of curvature decreases. This nonlinearity became a disturbance for the second-order lateral dynamics. We observe that the magnitude of the disturbance increases when the vehicle travels on a curved road.
The lane keeping performances of both the proposed method and the previous method can be validated using Fig. 6. The tracking errors of both methods are shown in Fig. 6(a). The proposed method maintained the vehicle at the center of the lane within approximately e y = ±0.1 m. In addition, the maximum absolute value of the heading angle error was less than 0.012 rad on a severe curved road. In the high-speed simulation, the road radius gradually decreased. The difference between the lateral control performances of the two methods is the evident when the curvature of the road was the most severe at 45 s. Near 45 s, the tracking errors of the previous method increased compared to the proposed method. Furthermore, the oscillations occurred. At the end of the curve, the proposed method shows a more stable response without oscillations. On the other hand, the previous method had oscillations. Consequently, the input of the previous method also had oscillations compared to the proposed method as shown in Fig. 6(b). The yaw rate and lateral acceleration of the vehicle during the simulation are depicted in Fig. 6(c) We confirmed that even if disturbances occur in the inner and outer systems, the proposed algorithm achieves stable and reasonable lane keeping performance unlike the previous method.

C. RESULT OF LOW-SPEED CASE, 30 KM/H
The estimation performance of the proposed method for the low speed case is shown in Fig. 7. The overall state of the inner-loop system is shown in Fig. 7(a). The proposed augmented observer shows reasonable performance in the estimation of i x 1 . The motor angle i x 1 changed as the vehicle entered the curved road at 12 s. The steering wheel angle at the first curve was approximately 35 • . Owing to the difference between the 4th-order steering model for EPS system and the 2nd-order steering model for controller, there was a disturbance when the vehicle entered or exited the curved road. Note that the disturbance also included the sinusoidal disturbances that we added.
In Fig. 7(b), the state of the outer-loop system is described. In particular, o x 3 includes the nonlinear characteristics of vehicle dynamics and parameter uncertainties. The estimation of o x 1 was found to be highly accurate. The nonlinearity of the vehicle increases as the radius of curvature decreases. This nonlinearity became a disturbance for the second-order lateral dynamics. We observed that the magnitude of the disturbance increases when the vehicle travels on a curved road.
The lane keeping performances of both the proposed method and the previous method can be validated through Fig. 8. The tracking errors of both methods are shown in Fig. 8(a).
The proposed method maintained the vehicle at the center of the lane within approximately e y = ±0.1 m. In addition, the maximum absolute value of the heading angle error was less than 0.2 rad on a severe curved road. In the low-speed simulation, the road radius continuously varied. The difference between the lateral control performances of the two methods is evident where the road curve was relatively severe from 60 s to 69 s and from 208 s to 213 s. From 60 s to 69 s and from 208 s to 213 s, the tracking errors of the proposed method were less than those of the previous method. Furthermore, the input of the previous method also had oscillations compared to the proposed method as shown in Fig. 8(b). The yaw rate and lateral acceleration of the vehicle during the simulation are depicted in Fig. 8(c). We confirmed that, even if disturbances occur in the inner and outer systems, the proposed algorithm achieved stable and reasonable lane-keeping performance, unlike the previous method.

VI. CONCLUSION
In this paper, we proposed a cascaded backstepping control method with an augmented observer for the lateral control of an autonomous vehicle. The proposed method was designed to improve the lateral control performance, while guaranteeing the stability of the overall system via a cascaded backstepping procedure. The proposed method consists of an outer-loop controller for lateral dynamics and an inner-loop controller for the EPS system. Augmented observers were implemented in both the outer-loop and inner-loop controllers to improve robustness against unknown model parameters and external disturbances. Through simulations, it was shown that the proposed method guaranteed stability and improved lateral control performance under unknown model parameters and external disturbances. In future works, we plan to study lateral control methods by considering the time delay owing to communication.