Robust Control for Active Suspension of Hub-Driven Electric Vehicles Subject to in-Wheel Motor Magnetic Force Oscillation

Due to the aggravation of environmental pollution and the energy crisis, automotive products are required to be energy-saving, highly efficient, and environmentally-friendly [1,2,3]. Compared with traditional diesel locomotives, electric vehicles (EVs) which have several advantages such as no CO₂ emission and energy efficiency have caught widespread attention in recent years [4,5]. The propulsion configurations of EVs can be classified as a distributed motor driven layout and a centralized driven layout. Compared with centralized driven layout EVs, in-wheel motor (IWM) propelling EVs, as the distributed propulsion system, enjoy various additional structure and control merits [6,7]. As in the IWM the brake and the reducer are integrated into the rim, the mechanical transmission system of hub-driven EVs is simpler and more efficient. Moreover, by controlling the torque of each electric motor precisely and independently, hub-driven EVs have played a significant role in enhancing the performance of existing vehicle stability/motion control systems, such as electronic stability control systems (ESC) and traction control systems (TCS) [8]. However, the development of hub-driven EVs has introduced some new technical problems. The integration of the driving system into the rim leads to an increase in the unsprung mass, which deteriorates road handling capacities and the ride comfort of vehicles [9,10]. What is more, the road surface roughness excitation and the coupling effect in IWM make the magnet gap deformed. The deformed magnet gap then leads to magnetic force oscillation which is a critical vibration source for vehicle dynamics [11,12].

In the past decades, some passive and active methods, such as optimization of the vibration isolator, redesign of the IWM, and active control for suspension systems, have been adopted to suppress magnetic force oscillation and improve the ride comfort. However, there are some deficiencies which remain to be solved. Luo et al. [12] proposed a novel IWM topology scheme and studied the coupling effect of the IWM system. By mounting rubber bushings in the IWM device, the vibration energy from the road surface was absorbed and the deformation of magnet gap was restricted. However, the life of the rubber bushing was short due to the severe operating environment of the IWM, making it difficult for practical use. Moreover, due to neglect of the permanent magnet brushless direct current (PMBDC) motor model which could provide the phase current for calculating the unbalanced electromagnetic force (UEF), the established coupling model was simple and inaccurate. The transient characteristics of electromechanical coupling effect cannot be effectively studied under vehicle running conditions. Zhao et al. [13] designed the motor mount, making the motor mass as dynamic vibration absorber. The results show that the ride comfort was improved. However, there was a distinct axis relative displacement between the rotor and the stator, prompting magnet gap deformed and magnetic force oscillation. The active suspension control methods have been proposed to suppress the magnetic force oscillation. Nevertheless, the majority of the active suspension system of hub-driven EVs are based on switched reluctance in-wheel driven motors [14,15] or neglect the negative electromagnetic coupling effect in IWM [16,17,18,19]. Different from switched reluctance motors (SRM), the magnetic field in permanent magnet synchronous motors (PMSM) is provided by permanent magnets. Consequently, as a permanent magnet synchronous IWM, more complex multi-field coupling effects need to be studied to enhance the ride comfort of EVs. In brief, it is meaningful to suppress magnetic force oscillation and improve the ride comfort. However, for the PMSM, the passive methods to suppress magnetic force oscillation have some defects and the active methods are rarely studied. Furthermore, the electromagnetic coupling model of hub-driven electric vehicle based on PMSM that has been established by Luo is inaccurate [12]. It is necessary to establish the complete electromagnetic coupling model, investigate the coupling mechanism of PMSM and study the active control methods for the suspension system of hub-driven electric vehicle to solve the negative coupling effect of the PMSM.

Besides, failures of electrical components, sensors, and the actuator are fairly common to active suspension systems, which will lead to a series of problems, such as system instability, performance degradation, and even disaster traffic accidents. Therefore, it is of great significance to design the active suspension considering actuator failures. Choi et al. [20] proposed a robust controller for a semi-active suspension system with actuator saturation. The vehicle vibration attenuation problem under two cases was addressed, namely, without actuator fault and with base oil leakage in MR damper. Liu et al. [21] proposed a new adaptive fault tolerant control scheme by employing adaptive feed-back stepping technique which ensured the boundedness in probability of the considered systems. Alain et al. [22] presented an application of a diagnosis and a fault-tolerant control method for an active suspension system. This method could improve the vehicle performance in the presence of road disturbances by identifying an actuator fault and reconfiguring the controller. In the literature [23,24,25], the fault-tolerant control was proposed for an active suspension system with an actuator fault. The results showed that the control effects of the active suspension system deteriorated without considering the actuator failure. Similarly, the control effects will be unsatisfactory if the model parameter uncertainties are neglected. In the literature [26,27,28], the robust control was proposed for vehicle active suspensions. The comparative simulations presented the availability of the designed controllers considering the model uncertainties. In addition, some other control methods were also proposed to suppress disturbance and obtain the better performance of the system, such as u-synthesis [29], neural network method [30], mixed H_ꝏ/H₂ [31], slide-mode control [32,33,34,35,36], adaptive control [37], and nonlinear control [38,39]. However, most of the control strategies are proposed for conventional vehicles and rarely for PMSM-driven EVs. From the existing studies, the issues of active suspension systems associated with actuator fault tolerance, model parameter uncertainties, and coupling effect in IWM based on PMSM have not been well studied in previous research.

This paper focuses on the coupling effect in permanent magnet synchronous IWM and the robust control for active suspension of hub-driven EVs. The challenges and technical contributions are summarized as follows: (1) unlike studies in Ref. [11,12], in which the established electromechanical coupling model was simple and inaccurate, in this paper, according to the electromagnetic field theory, the complete electromechanical coupling model of the hub-driven EVs composed of four sub-models is established. The transient characteristics of electromechanical coupling effect can be effectively studied under vehicle running conditions; (2) unlike studies in Ref. [15,16,24], in which the authors proposed active suspension controller for the SRM driven electric vehicle, the electric vehicle neglecting the coupling effect or the conventional vehicle, in this study, a multi-objective control is designed for the PMSM driven EV to obtain better ride comfort and restrict coupling effect of in-wheel PMSM. Meanwhile, the other performances such as actuator limitation, small suspension deflection and road holding capability are also maintained; (3) the suspension parameter uncertainties and actuator fault are simultaneously considered and a reliable robust H∞ controller is designed to attenuate the effects caused by the magnetic force oscillation, parameter uncertainties, the actuator fault and road disturbances.

The rest of this paper is organized as follows. In Section 2, the mathematical model of a hub-driven electric vehicle is established and the coupling effect is analyzed. In Section 3, a reliable robust H∞ controller is presented. The simulation results are described in Section 4 and key conclusions are given in Section 5.

As magnetic force oscillation worsens the performance of EV, the electric vehicle model considering the coupling effect of IWM is established to explore the influence of the coupling effect on vehicle performance and provide a theoretical basis for designing active suspension controller later.

Traditionally, most studies mainly consider three requirements, namely, road-holding stability, suspension deflection, and the ride comfort, when designing a control law for suspension systems. Some studies also take energy-saving into consideration. However, few papers have covered the coupling performance of IWMs in suspension design, especially for PMSM. The issues for active suspension system associated with actuator fault tolerance, model parameter uncertainties and coupling effect in IWM based on PMSM have not been well studied in any previous research. Therefore, the reliable robust H∞ control law for active suspension of hub-driven EVs is designed. Considering factors mentioned above, the following suspension performances should be taken into account.

(1). Maximum actuator force. The active control force provided by the suspension system should be constrained by a threshold due to the limited power of the actuator, that is where u_max is the maximum force of the actuator.

(2). Ride comfort. To provide comfort for passengers, a control u is designed to isolate the sprung mass from the road and magnetic force induced vibration, i.e., to minimize the vertical acceleration ${\ddot{z}}_b$ in the presence of parameter uncertainties m_b and k_t, unknown dynamics F_r, and road speed excitation.

(3). Suspension deflection. The suspension deflection should not exceed its travel limit to avoid ride comfort degradation and vehicle component damage. i.e., where z_max is the maximum deflection of the suspension.

(4). Motor unbalance dynamic force. The UEF in IWM promotes the rotor eccentricity which intensifies the bearing wear, shortening the motor life. Furthermore, it aggravates the sprung mass acceleration and provokes cacophony, deteriorating the vehicle comfort significantly. As previously analyzed, the UEF is closely related to the rotor eccentricity. Thus, another primary objective is to minimize the rotor eccentricity when the controller is designed. That is, $\left|z_r-z_s\right|$.

(5). Road-holding stability. The firm uninterrupted contact of wheels to road should be ensured to make sure the vehicle safety, that is to say, the dynamic load of the tire should not exceed its corresponding static load, i.e.,

To analyze the performances mentioned above, for the sake of convenience the controlled outputs are defined as

Thus, active suspension control system can be described by a state-space equation as: where where γ and β are weight coefficients. To obtain a better control performance of the suspension, a state-feedback controller as u_d = k(λ)x(t) is designed. Then, by considering the actuator fault, a state-feedback controller can be modelled as where $k\left(\lambda \right)$ is a gain matrix of the feedback controller that needs to be determined, ξ represents the possible actuator fault, u_d is the desired force calculated by the controller, and u_actual is the actual force generated by the actuator. Assuming that ξ is constrained by its maximum ξ_max value and minimum value ξ_min, the control law considering actuator faults and parameter uncertainties can be denoted as where ${\xi }_0=\left({\xi }_{max}+{\xi }_{min}\right)/2$ and $L=\left(\xi -{\xi }_0\right)/{\xi }_0$. Defining $\mathrm{J}=\left({\xi }_{max}-{\xi }_{min}\right)/\left({\xi }_{max}+{\xi }_{min}\right)$, then, we have $L^{T}L\le {\mathrm{J}}^T\mathrm{J}\le \mathrm{I}$.

Therefore, the closed-loop system with state-feedback controller is rewritten as: where the system matrices A(λ), B₁(λ), B₂(λ), C₁(λ), D₂(λ) and C₂(λ), which are dependent on the sprung mass m_b and the tire stiffness k_t, are functions of λ. It is assumed that A(λ), B₁(λ), B₂(λ), C₁(λ), D₂(λ) and C₂(λ) are constrained within the polytope Ω where

Additionally, the relationship between the uncertain masses (m_b and k_t) and the vector $a\left(\lambda \right)=\left(a_1\left(\lambda \right),a_2\left(\lambda \right),a_3\left(\lambda \right),a_4\left(\lambda \right)\right)$ is expressed by where

To establish a system in Equation (29) that is robust asymptotically stable with disturbance attenuation $\gamma >0$, the following Hamiltonian should be less than zero where

To achieve $\mathrm{H}\left(\mathrm{x},\mathsf{\lambda },\mathrm{w},t\right)<0$ for all $\mathsf{\Gamma }\left(t\right)\ne 0$, the following inequality should hold

By using the Schur complement, inequality (37) can be converted into inequality (38). where

Based on Lemma 1, inequality (38) can be converted into inequality (39). where

By the Schur complement, we get

Pre- and post-multiplying diag $\left\{\mathrm{P}{\left(\lambda \right)}^{-1} \mathrm{I} \mathrm{I} \mathrm{I} \mathrm{I}\right\}$, the congruent transformation of the matrix is achieved. Subsequently, by defining $\overline{P}\left(\lambda \right)=\mathrm{P}{\left(\lambda \right)}^{-1}$ and $\overline{k}\left(\lambda \right)=k\left(\lambda \right)\mathrm{P}{\left(\lambda \right)}^{-1}$, the inequality (40) is equivalent to the first LMI in inequality (32). Then we will show that hand constrains in inequalities (22)–(24) are guaranteed. From the Lyapunov function in inequality (35), it can be known that $x^T\mathrm{Px}<\rho$, with $\rho ={\gamma }^2{\mathbf{w}}_{\mathit{max}}+\mathrm{v}\left(0\right)$. Similarly, the following inequalities hold: where ${\theta }_{max}(\cdot )$ represents the maximal eigenvalue. Using the Schur complement and Lemma 1, inequality (41) can be written as

Pre- and post-multiplying inequality (43) by diag $\left\{\mathrm{I} \mathrm{P}{\left(\lambda \right)}^{-1} \mathrm{I} \right\}$, defining $\overline{P}\left(\lambda \right)=\mathrm{P}{\left(\lambda \right)}^{-1}$ and $\overline{k}\left(\lambda \right)=k\left(\lambda \right)\mathrm{P}{\left(\lambda \right)}^{-1}$, the inequality is equivalent to the second LMI in inequality (33). Furthermore, based on the Schur complement, the inequality matrix in inequality (42) can be change to

Pre- and post-multiplying inequality (44) by diag $\left\{\mathrm{I} \mathrm{P}{\left(\lambda \right)}^{-1}\right\}$, using $\overline{P}\left(\lambda \right)$ to replace $\mathrm{P}{\left(\lambda \right)}^{-1}$, the LMI inequality (44) is equal to the third LMI in inequality (34). Thus, the proof is completed.

According to the inner property of the polytopic tire stiffness uncertainties and sprung mass uncertainties, Equations (32)–(34) in Theorem 1 are equivalent to the inequalities as follows: where