Abstract
This paper examines the problem of designing a robust output-feedback yaw controller with both input and output constraints for four-wheel independently driven in-wheel electric vehicles (EVs) with differential steering. Specifically, the controller aims are to ensure the stability and improve the performance of the EV despite variations in the road adhesion coefficient, longitudinal velocity, and external disturbance. Based on the linear matrix inequalities approach, sufficient conditions for the existence of an output-feedback controller for linear systems with polytopic uncertainties, and input and control output constraints, are derived. Then those sufficient conditions are utilized to design an output-feedback yaw controller that guarantees the robust performance and stability of an EV over a wider range of road conditions. Finally, the capability of the developed controller is simulated on a vehicle model with uncertain road conditions and longitudinal velocities.
1. Introduction
Active vehicle control (AVC) is a system that continuously observes the vehicle’s dynamic responses and implements a corrective control action when necessary to improve drivability and stability of the vehicle. These subtle corrective actions can be a blend of regulating the steering angle and optimizing the transmitted tyre forces to the road [1–4]. Examples of AVC systems are active four-wheel steering (4WS), direct yaw moment control (DYC), active roll-over protection, anti-lock brake system (ABS), electronic stability control (ESC), active front-wheel steering (AFS), and active suspension system [2, 5]. The main aim of AVC is to improve vehicle safety, vehicle handling, and ride comfort and reduce the driving stress on poor road conditions.
In recent years, studies addressing active yaw moment control for physically steerable wheels to maintain the vehicle stability are enormous. In [4], stability control strategies of the vehicle via DYC are studied. Braking forces and traction controls are used to establish the stability threshold of the vehicle. Similarly, robust controller design [6–9] and optimal distribution of tyre forces [10–13] have been developed for steerable wheels. However, there are currently no studies, as far as we know, addressing the robust control of EVs without physically steerable wheels.
Vehicles are subject to uncertainties such as the crosswind disturbances, payload, vehicle’s longitudinal velocity, and, most importantly, friction differences in the tyre-ground interaction among many other factors. Several authors have adopted robust controller design for yaw stability control in an independently driven electric vehicle. In [14], a -synthesis robust controller design improves the yaw motion and chassis sideslip. In a similar study, in [15], the authors designed a controller with the aim of achieving both a neutral steer and a reduction in the sideslip of a four-wheel EV. These studies do not consider the tire force saturation.
In [16, 17], the authors consider the tire force saturation, parametric uncertainties, and external disturbances in solving the lateral-plane motion and yaw stability problem of an independently driven electric vehicle. Reference [18] investigated an active steering failure mechanism in a differential assisted steering for an autonomous system. The authors consider the tyre force saturation and the use of a robust multiple-disturbances observer-based controller for a path following control.
In [19], a dynamic output-feedback controller is designed to realize a differential speed steering control strategy. However, in the previous design, the controller design was performed by linearizing the vehicle under a single operating condition tested for robustness. As one knows, the vehicle needs to operate over a wide range of guaranteed operating conditions, especially at different vehicle speed and road conditions. Based on the LMI approach, this paper designs a robust dynamic output-feedback controller for four-wheel independently driven electric vehicles (FWIDEV) under bounded uncertainties and external environmental disturbance. The bounded uncertainties under consideration are the road coefficients of friction, vehicle’s longitudinal speed, and the presence of lateral wind disturbances.
The main contributions of this paper are the derivation of sufficient conditions for the existence of an output-feedback controller for linear systems with polytopic uncertainties regarding input and output constraints and the design of an robust yaw controller that can achieve both robust stability and performance for EVs in the presence of uncertainties considering both input and output constraints.
The rest of the paper is organized as follows. Section 2 provides the mathematical model and the uncertain linear system of a four-wheel independently driven in-wheel electric vehicle. Section 3 presents the control problem and the robust controller design. Simulation results and discussion are given in Section 4. Concluding remarks are given in Section 5.
2. Modelling of a Four-Wheel Independently Driven In-Wheel Electric Vehicle
The four-wheel independently driven in-wheel electric vehicle under consideration is shown in Figure 1. Each wheel in the EV is independently driven. Schematically, the longitudinal, lateral, and yaw equations of motion are as follows:where , , and are the vehicle’s velocities along the longitudinal motion, lateral motion, and the yaw motion, respectively. is the vehicle mass, is the distance between the centre of gravity (COG) and the centre point of the front wheel, is the distance between the COG and the rear wheel, is half of the vehicle width, is the yaw moment of inertia, and is an external disturbance.
The tyre forces, and with , are the result of the pneumatic deflections of the tire properties due to the weight of the vehicle, tyre pressure distribution, and also the tyre-ground interaction. To determine the forces generated by the tyres as a result of the tyre-road interaction, the tyre nonlinear model for the combined lateral and longitudinal forces will be used as in [20–23].
The longitudinal and lateral forces components arewherewhere is the total slip for each tyre and and are the limiting values of slip.
Parameters and are, respectively, tyre lumped lateral stiffness and the coefficient of tyre-road friction. The lateral tyre slip, , isand the longitudinal tyre slip, , is is the effective wheel radius, is the angular velocity of each wheel, and is the tyre side slip angle. The normal load, , is based on the vehicle’s geometry as
Here a linear form of the combined tyre force modelled by (4) and (5) is used to describe the tyre frictional forces on each wheel [24]. We assume the vertical load of the vehicle is evenly distributed on each wheel. So that when the tyre side slip angle is small, by [21, 25], the tyre forces can be approximated aswhere is the longitudinal tyre stiffness; the tyre side slip angles, , are approximated asAnd the longitudinal slip ratio, , is
In this paper, we assume the longitudinal velocity, , is an unknown constant but bounded as where and , are, respectively, the lower and upper bounds of . is a constant; hence (1) is not considered here.
Moreover, we also assume that the coefficient of road friction, , is an unknown constant but bounded as where and are, respectively, the lower and upper bounds of .
We model the uncertainties in and as polytopic uncertainties; hence (2) and (3) can be expressed as follows:where , , , and is the measurement noise. The uncertain matrices and belong to the polytopic uncertain domain where and , and In the next section, a robust dynamic output-feedback controller design technique will be developed for general linear uncertain systems with polytopic uncertainties.
3. Robust Controller Design
Consider the following uncertain system:where is the state vector, is the control input, is the disturbance, is the performance output, is the control output, is the measured output, and exists in a convex hull. The uncertain matrices , , , and belong to the polytopic uncertain domain where and . is the number of vertices in the polytopic system. The matrices , and build the vertex of the polytope.
Without loss of generality, we assume
Since the sensors required to measure the lateral velocity are costly, in this paper, we will consider the dynamic output-feedback controller of the formwhere , , and are the parameters of the controller.
Robust Controller Formulation. Given , find a controller of the form (22) such that(1)system (19) with (22) is stable and satisfying the following inequality(2)under zero initial conditions, the following inequality is satisfied:(3)the following control output and input constraints are met: where , and is the number of rows in and .
The closed-loop system of (19) with (22) iswhere
Theorem 1. Given , , and , suppose that there exist symmetric matrices , , , , and and matrices , , , , , and such that for and the following inequalities conditions hold:where , Then, system (19) with (22) is stable and the performance (23) and the input and output constraints (24) hold. Moreover, the controller parameters are given as follows:
Proof. Refer to the Appendix for the proof.
4. Simulation Results
In this section, by using the parameters given in Table 1, the proposed control design is verified by stabilizing the EV along a straight path. The uncertainties in the coefficient of road friction and the longitudinal velocity are varying parameters from and km/hr, respectively. For comparison, the nominal operating points of 50km/hr and .8 are selected for the design. The uncertainty in and , respectively, represents to and to of the nominal points.
In this simulation, the maximum speed of each wheel is assumed to be or in relation to vehicle’s speed. Since the maximum operating speed of the vehicle is 120 (), the input constraint is given as For the comfort of the driver and passengers, the output constraint for the yaw acceleration is chosen as ; that is, where is the centroid of the vehicle.
The disturbance term used in the simulation is shown in Figure 2. The yaw rate measurement white noise’s power density is 0.02 with a maximum amplitude of .
With this disturbance and the measurement noise, and .
By Theorem 1, with , , and , the following robust control parameters are obtained by solving the LMIs in (27), (28), (29), (30), and (31) using YALMIP toolbox (MOSEK solver) [26].
In comparison, we design an with , , and ; the control parameters are
To test the effectiveness of the robust controller design and control strategy, different simulations are performed under multiple vehicle operating conditions. The nonlinear vehicle dynamic model in Section 2 is used in the final simulations. Each simulation ran for 50 seconds and generated 505 data points per second. The total displayed distance for the vehicle trajectory depends on the road condition () and the vehicle’s longitudinal velocity ().
Figure 3 shows the trajectories of the EV with its longitudinal speeds at 120kmhr and coefficient of road friction at . The simulation starts with all the three vehicles operating without the crosswind disturbance, Figure 2; the trajectories of the vehicles can be seen to be on track. At the situation when the disturbance is injected from 10 sec to 30 sec, the vehicle without a controller begins to swerve off the desired path while the vehicles with a controller stay the course. The vehicle with the controller shows a slight lateral deviation of less than 2 cm. After the vehicles exit the disturbance, the vehicles with a controller are still able to continue on the trajectory, but the vehicle with the robust controller shows an improved performance by maintaining a straight path.
The controller output, , and are shown in Figures 4 and 5, respectively. The robust controller shows a fast and good damping of the disturbance while still being within the constraints.
Figure 6 shows the vehicle’s trajectories at the operating condition, and . Both of the vehicles with a controller indicated a strong performance, but an improvement can be seen with the robust controller as shown in the input, Figure 7, and control output, Figure 8. We can observe that the robust controller can quickly attenuate the disturbance upon entry and exit of the disturbance without violating the constraints. Figure 7 shows an improved performance of the robust controller. It is interesting to note the fast dynamics of the robust controller and its corresponding output constraint in Figure 8, for a straight-ahead driving, Figure 6. Figure 8 shows the vehicle’s yaw acceleration at and an entry speed of 20kmhr. The controllers are still able to stabilize the vehicle on the straight path with minimal lateral deviation.
From the simulations, we observe that the vehicle with the robust controller is able to attenuates disturbance while satisfying both the input and output constraints.
5. Conclusion
This paper has designed a robust dynamic output-feedback controller that attenuates external disturbances of four-wheel independently driven in-wheel electric vehicles. The electric vehicle has been modelled as an uncertain polytopic system with bounded uncertainties and external disturbance. We have developed sufficient conditions for the existence of a robust controller with input and output constraints that ensures the vehicle stays on course despite different road conditions and different longitudinal speeds. The performance of the robust controller has been tested on a complete nonlinear vehicle dynamic model. The simulation results have shown that despite significant differences in the operating conditions, the proposed robust controller ensures that the vehicle’s trajectory stays on the desired path and satisfies both the input and control output constraints.
Appendix
Proof of Theorem 1. For the closed-loop system (25), consider the following parameter Lyapunov function:where Taking the time derivative of (A.1) along with (25) leads toAddingto (A.3) by [27], for robust stability and performance, yieldswhere and Adding and subtracting () to and from (A.5) results in whereSuppose the conditions in Theorem 1 hold; then . This condition implies that and are nonsingular matrices. Without the loss of generality, let partition be We assume . Define Multiplying the left-hand side of (A.8) by and its right-hand side by reads wherewithwithWith the notations given in Theorem 1, can be rewritten as whereandAlso rewrite as and as Then Multiplying to the left of and to the right of , we obtain with The condition in (30) implies . Applying Schur complement with respect to the last two rows of (31), it can be shown that . Therefore, Integrating both sides and using the fact that , we have When is zero, This implies that (25) is stable.
For the control output and input constraints, With the zero initial condition,Note that , where . Hence, Using (A.27),From (A.29), ifApplying the Schur complement on (A.30) givesMultiplying left and right of (A.31) by givesThen multiplying the left and right of (A.32) by respectively, gives (28).
Note that , where . Therefore,Using (A.27), From (A.35), ifApplying the Schur complement on (A.36) givesMultiplying left and right of (A.37) by givesThen multiplying the left and right of (A.38) by respectively, gives (29). This completes the proof.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant 61573315.