Robust Speed Control in Nonlinear Electric Vehicles Using HInfinity Control and the LMI Approach
Abstract:
In this investigation, the robust H$\infty$ control of nonlinear electric vehicles (EVs), powered by permanent magnet synchronous motors (PMSM), was examined. Emphasis was placed on enhancing the accuracy and robustness of the vehicle speed regulation by incorporating a meticulous H$\infty$ method, supplemented by the proficient integration of Linear Matrix Inequality (LMI). A solution predicated on the LMI approach was devised, encompassing two distinct H$\infty$ controllers for both current and speed control. Subsequent to an extensive analysis of the mathematical and control model of the EV, weighting functions were judiciously selected to optimize stability and performance. The proposed methodology offers significant advancements in the domain of EV control strategies and proffers insights into the application of robust control methods. Through comprehensive simulations, the effectiveness of the outlined method was validated, revealing impeccable speed control and ensuring steadfast performance when applied to the dynamic model of an EV equipped with a PMSM motor. This research elucidates the progressive strides made in the realm of EV control tactics and offers profound understandings of robust control methodologies.
1. Introduction
Over the past several decades, challenges within control systems have been the focus of extensive research. Such challenges, including the introduction of undesirable variables like modeling uncertainty, the influence of measurement noise, and effects stemming from overlooked nonlinear dynamics, have necessitated the development of highly effective synthesis techniques. It has been observed that these techniques necessitate the synthesis of controllers which strike a balance between robust loop system stability and optimal performance [1]. Importantly, this balance is not merely restricted to the nominal operating speed of the system under control but also extends to numerous conditions wherein the synthetic model's parameters can exhibit significant variability.
Prominent among the applied robust methodologies, the H$\infty$ control utilizing RICATTI equations and those leveraging the Linear Matrix Inequalities (LMI) approach have been cited [2] [3] [4] [5] [6]. It was in the early 1980s that this robust methodology found its origins, primarily attributed to the pioneering efforts of researchers like DOYLE and SAFONOV, wherein novel mathematical tools were introduced. Currently, this method stands as a primary research direction, particularly within the domain of robust techniques [2] [3] [5].
The central challenge identified in this study pertains to the development of more efficient control laws, especially concerning trajectory tracking, disturbance negation, stability, and robustness visavis parameter uncertainty [7] [8].
An EV, typically characterized by an electric motor operating exclusively on electrical energy, has at its core a PMSM. This motor, while bringing benefits like a substantial reduction in weight and volume for a specific output power (high power density), also offers increased efficiency and effective heat dissipation. However, challenges such as pitch disturbance, augmented torque, vibration, and noise serve as significant impediments, particularly impacting EV performance in roadspeed contexts [9] [10].
For ensuring consistent vehicular speed without frequent adjustments, and considering the aforementioned challenges, the cruise control system was designed, primarily focusing on facilitating safety, comfort, and ease of drive for users. In this context, the robust Hinfinity control, implemented via the LMI approach, was utilized for designing a controller specific to EVs powered by a PMSM [11]. Through this approach, stabilization and effective speed tracking were achieved.
Subsequent sections are structured as follows: Section 2 delves into the longitudinal mathematical model of EVs. Sections 3 and 4 focus on the design approach and its subsequent implementation for speed tracking control specific to EVs. Performance evaluations of the controller, facilitated by comprehensive numerical simulations, are presented in Section 5, with concluding remarks featured in Section 6.
2. Problem Description
As depicted in Figure 1, the dynamics of the EV system can be primarily segmented into two components: motor system dynamics and vehicle dynamics. The motor system is interfaced with the EV system through a transmission unit, encompassing the transmission system. In a standard EV setup, accelerator/brake pedals are employed by the driver to relay acceleration/deceleration command signals to the controller. A PMSM motor, which is integrated with the EV system through a transmission unit equipped with a gear system, is utilized to propel the aforementioned EV system. In essence, the speed of the PMSM motor is modulated to control the overall EV system [9] [12].
The dynamic model of the vehicle, based on the cumulative forces exerted upon it (refer to Figure 2), is governed by Newton’s second law [9] [10] [13] [14] [15] [16].
The summation of all active forces yields the total resultant force, given by:
(1) The gravitational force, which acts upon a vehicle during incline traversal, is determined by the slope angle and is proportional to the vehicle's mass. It is represented as:
(2) The aerodynamic resistance force, created by air flowing over the moving vehicle's body, is influenced by the vehicle's frontal area orientation. Its mathematical representation is:
(3) The primary source of rolling resistance arises from the friction between the vehicle’s tires and the road surface, opposing the direction of vehicle movement. This force is defined by:
The resultant force, F, produced due to the friction between the vehicle tires and the road, generates a torque that impacts the drive motor. Termed as the traction force, it plays a pivotal role in vehicle propulsion and is expressed as:
The association between the motor's angular velocity, $\Omega_{m o t}$, and the EV's linear speed, $V$, is elucidated as:
where, $T_L$ is the torque generated by the driving motor, $R_{\text {wheel }}$ is the wheel radius, and $\Omega_{m o t}$ is the rotation speed of the motor.
• Gearbox
The inclusion of a gearbox in an EV is noted to augment engine efficiency by modulating the relationship between engine velocity and wheel speed. Such modulation of torque in alignment with driving conditions contributes to an enhanced driving range and experience. The mechanical conversion value is procured from the ensuing equation [17] [18]:
where, $T_{\text {gear }}$ and $\Omega_{\text {gear }}$ are torque and speed rotation after reduction, $\Omega_{m o t}$ is the motor rotation speed and $K_{\text {gear }}$ is the gearbox reduction coefficient.
• Differential mechanical
The mechanical differential in EVs is imperative for power distribution between wheels, ensuring optimal traction and stability. Moreover, it modulates wheel rotation speeds during turns to prevent skidding. The even torque distribution to both the left and right wheels is depicted in Eq. (8) [17] [18]:
where, $T_{\text {diff }_{\text {left }}}$, $T_{\text {diff }_{\text {right }}}$ and $T_{\text {diff }_{\text {Tot }}}$ are left, right and total torques after differential, respectively.
• The traction forces
Using wheel rotation, differential torque, and vehicle speed, the traction forces can be calculated:
where, $R_{\text {wheel }}$ is the wheel radius, $T_{\text {diff }_\text { right} }$ and $T_{\text {diff }_{T o t}}$ are left, right and total torques after differential, respectively.
The representation of the equations describing the PMSM model is situated in the rotor frame (dq) [19] [20] [21].
The subsequent equations can be derived:
where, the terms $u_d, u_q, i_d$ and $i_q$ are defined as stator voltages and currents in the dq frame respectively. $R_s$ signifies the stator resistance, while $L$ denotes the inductance in the dq frame. $P_p$ stands for the pole pairs and $\Phi$ is the permanent magnet flux. $J$ represents the rotor moment of inertia, $B$ is the viscous friction coefficient, and $T_L$ is the load torque.
By incorporating Eq. (1), Eq. (5), and Eq. (6) into Eq. (10), the dynamic model of the entire EV system is defined. This integration combines the PMSM model with the vehicle dynamics model, given by:
From Eq. (10), it can be discerned that the model is nonlinear. However, for effective tracking of the desired vehicle speed with the robust H$\infty$ controller, a linear model is imperative. To mitigate this challenge, nonlinear effects, such as friction, were neglected, and a compensation method that simplifies torque control was applied. This approach involved the linearization of the relationship between torque and current, thus establishing a linear and singleinput model for the permanent magnet synchronous machine.
Subsequent computations yielded the transfer functions for the daxis current and the speed system:
3. H$\infty$ Control Analysis
H$\infty$ techniques have been recognized as robust control approaches, prominently designed to bolster stability while optimizing performance. Their significance, particularly within dynamic systems such as EV speed control, cannot be overstressed [22] [23].
LMI techniques were employed in the design of the H$\infty$ suboptimal controllers. Within the paradigm of H$\infty$ control design, the LMI strategy was adopted to discern controller gains, ensuring the stability of the closedloop system and concurrently minimizing a defined H$\infty$ norm from disturbance input to the output signal. The efficacy of the LMI technique in transforming the H$\infty$ control issue into a LMI set was substantiated through numerical methods [11] [21] [24].
A schematic representation incorporating the envisaged H$\infty$ controller can be discerned in Figure 3 [2] [3] [5] [25]. Within this configuration:
• P(s) typifies the generalized plant.
• K(s) delineates the controller.
• Variables $\omega$, $Z$, $y$ and $u$ represent exogenous inputs, minimized output signals, measurement outputs, and control signals, respectively.
Generalized plant $P(s)$ encompasses the plant model $G(s)$, the interconnection structure, and the weighting functions predetermined by the designer, formulated as:
Given the variables $x \in \Re^n$, $\omega \in \mathfrak{R}^{n_\omega}$, $u \in \Re^{n_u}$, $e \in \Re^{n_e}$, $y \in \Re^{n_y}$, the system's stabilizability of $\left(A, B_2\right)$ and the detectability of $\left(C_2, A\right)$ were taken into account. The transfer matrix in a closed loop from $\omega$ to the performance output $Z$ is articulated as:
The crux of the H$\infty$ control issue lies in identifying a controller that ensures the H$\infty$ norm of the $H_{z w}(s)$ transfer function remains below $\gamma$. The bounded lemma, integral to the H$\infty$ evaluation, establishes that the H$\infty$ norm of $\left.H_{z w}(s\right)$ is less than $\gamma$ if, and only if, a positive definitive matrix $\mathbb{P}$ meets the prescribed criteria.
$$\left[\begin{array}{ccc} A_c^T \mathbb{P}+\mathbb{P} A_c & \mathbb{P} B_c & C_c^T \\ B_c^T \mathbb{P} & \gamma I & D_c^T \\ C_c & D_c & \gamma I \end{array}\right]<0$$
The solvability of the H$\infty$ control dilemma is contingent upon the existence of symmetric matrices $R$, $S$ that satisfy the prescribed LMI system [11] [21] [24] [25]:
(1) LMI
Has a positive definite solution $X$.
(2) LMI
Has a positive definite solution $Y$.
(3) Moreover, $X$ and $Y$ are bound by specific conditions.
In which $n_K$ is the degree of controller $K(s)$.
Where $N_X$ and $N_Y$ respectively represents bases of the null spaces of $\left(\begin{array}{llll}B_2^T & D_{12}^T\end{array}\right)\left(\begin{array}{lll}B_2^T, & D_{\mathrm{inf}2}^T\end{array}\right)$ and $\left(\begin{array}{ll}C_2 & D_{21}\end{array}\right)\left(C_2, D_{01}\right)$.
In crafting a suboptimal controller, two pivotal steps were followed [26]. Firstly, the aforementioned triad of LMIs was resolved. Then, using the identified matrices $R$ and $S$ in conjunction with the plant statespace data, the controller was calculated.
Consequently, a stabilizing suboptimal controller was realized that ensures $\left\H_{z w}(s)\right\_{\infty} \leq \gamma$. By streamlining the controller LMI formula and adhering to stipulated assumptions, a suboptimal controller was derived [26]. This controller can be expressed as:
where,
$\begin{aligned} & A_k=A+\gamma^{2} B_1 B_1^T X_{\infty}B_2 B_2^T X_{\infty}\left(I\gamma^{2} Y_{\infty} X_{\infty}\right)^{1} Y_{\infty} C_2^T C_2 \\& B_k=\left(I\gamma^{2} Y_{\infty} X_{\infty}\right)^{1} Y_{\infty} C_2^T \\& C_k=B_2^T X_{\infty}\end{aligned}$
It is noteworthy that the derivation of the H$\infty$ controller can be efficiently executed utilizing Matlab's LMI toolbox.
4. Application
In Figure 4, the overarching structure of the nonlinear EV drive system, predicated on the permanents magnets synchronous motor (PMSM), is illustrated. The motor's control strategy, which was investigated in this study, is also highlighted therein.
The synthesis of controller design commenced with the selection of weight functions, as depicted in Figure 5 [7] [27] [28]. A paramount challenge associated with H$\infty$ control pertains to the proper frequencybased selection of transfer functions and the order of weighting. It is understood that these weighting functions serve the purpose of defining system performance prerequisites, disturbance impacts, actuator boundaries, and system uncertainties.
Two distinctive loops can be discerned within the proposed control schema:
• The stator direct current control loop focuses on flux control. This loop is governed by an H$\infty$ regulator. Through this regulator, the system's nonlinearity is sought to be negated by maintaining the direct current id at a null value.
• Another loop, managed by an H$\infty$ regulator, is dedicated to speed control, specifically overseeing the speed and torque.
With the operation of the PMSM being under the purview of fieldoriented control, its modeling, analogous to a DC motor, is evident.
Figure 5 delineates the speed loop block diagram encompassing weighting filters. Notably, the weight functions, symbolized as $W_{1 V}(s)$, $W_{2 V}(s)$, $W_{3 V}(s)$, play a pivotal role in finetuning the controller’s performance and robustness. Such functions can either represent a gain or be frequencydependent.
Adherence to the foundational precepts of mixed design is witnessed in the weighting functions $W_1$, $W_2$ and $W_3$. Given that $W_1$ pertains to the objective of error sensitivity function performance S, it is fundamentally designed as a low pass filter to diminish error sensitivity in lower range frequencies for mitigating output disturbances. Conversely, $W_2$ is integral for ascertaining the controlled system's stability across varying operations, rendering it as a highpass filter. An auxiliary disturbance weighting function, $W_3$, delineates the disturbance limits, allowing it to be chosen as a high pass filter or retained as a constant.
For the issue at hand, which is speed control, the selected weighting functions aimed at crafting a controller that adheres to stipulated specifications (pertaining to tracking performance, antidisturbance properties, and robustness) are defined as:
Upon utilizing the design showcased in Figure 5, the $K_V(s)$ controller can be derived by addressing the Hinfinity problem via the LMI method. The ensuing expression for the controller is:
$K_V(s)=\frac{5.373 \times 10^8 s^3+3.646 \times 10^{11} s^2+1.028 \times 10^{13} s+1.862 \times 10^{14}}{s^4+53970 s^3+3970 s^2+1.067 \times 10^8 s+4.759 \times 10^{10} s+2.38 \times 10^8}$
As Figure 6 demonstrates, the synthesis of the direct current loop controller uses weighting functions that are reminiscent of those in speed tracking synthesis. These functions are formulated as:
The Hinfinity controller, represented by $K_I(s)$ for the current loop, takes the form:
$K_{i d}(s)=\frac{1.027 \times 10^6 s^2+7.244 \times 10^9 s+3.698 \times 10^{12}}{s^3+1.019 \times 10^6 s^2+6.382 \times 10^9 s+3.191 \times 10^7}$
5. Simulation Results
To assess the perturbation rejection capabilities of the proposed H$\infty$ controller, simulations were performed on an EV, utilizing the PMSM and EV parameters outlined in Table 1 and Table 2.
Initiating from a static state, the vehicle was observed to steadily accelerate to a reference speed of 80 km/h. Within this phase, two directional changes were enacted: the initial leftward shift occurred at t = 38.89 s, followed by a rightward shift at t = 83.33s. A return to the original trajectory was recorded at t = 111.11s, as visualized in Figure 7. Throughout these tests, the evolution of several electrical and mechanical variables, including currents, voltages, torque, and velocity, was monitored.
The Parameter  Symbol  Value  Unit 
daxis Inductances  $L_d$  0.17  mH 
qaxis Inductances  $L_q$  0.29  mH 
Flux linkage  $\Phi$  0.071  wb 
Statorwinding resistance  $R_S$  0.0083  Ω 
Number of poles  $P_p$  8 

Moment of inertia  $J$  0.089  Kg.m^{2} 
Moment of inertia  $B$  0.005  Nm/rad/s 
The Parameter  Symbol  Value  Unit 
The mass of the vehicle  $M$  1450  Kg 
Vehicle Frontal area  $A_f$  2.711  m^{2} 
Wheel radius  $R_{\text {wheel }}$  0.29  m 
Aerodynamic drag coefficient  $C_D$  0.29 

Air density  $\rho$  1.204  Kg/m^{3} 
Rolling resistance coefficient  $f_r$  0.013 

Total inertia 
 5.209  Kg.m^{2} 
Total gear ratio  $K_{g e a r}$  8.75 

The results presented in Figure 8 depict the longitudinal speed of the vehicle. It was evident that the vehicle's speed remained stable, illustrating an agile performance devoid of overshoot, with no persistent errors identified. Notably, the system consistently matched the reference signal with remarkable precision.
During the turning phases, it was observed in Figure 9 that each wheel responded differently, rotating at varying speeds in the same direction. The H$\infty$ controller's critical role was apparent, as it facilitated precise, realtime adjustments to the motor during these maneuvers, ensuring vehicular stability. Concurrently, a mechanical differential was engaged to equilibrate the speed disparity between wheels. This mechanism slowed the inner wheel, allowing the exterior wheel to rotate more rapidly, thus preventing skidding and control loss. This combined strategy enhanced traction, stability, and overall vehicle performance during turns. Evidently, the robust H$\infty$ control mechanism presented superior tracking and control capabilities.
Figure 10 reveals the daxis current response of the PMSM under two distinct control methods. Rapid alignment of the daxis current response with the reference was noted across all trajectories.
The qaxis current responses and the electromagnetic torque of the PMSM in the vehicle system were highlighted in Figure 11 and Figure 12. The qaxis current component, proportional to the requisite torque, exhibited outstanding dynamic response. The required electromagnetic torque was efficiently developed by the PMSM motor at different speed reference stages, emphasizing the efficacy of the H$\infty$ technique in PMSM control systems.
Figure 13 and Figure 14 present the phase voltages and phase currents throughout the operation. Both were observed to maintain a sinusoidal pattern, with frequency alterations in response to speed fluctuations.
Lastly, Figure 15 demonstrates the tractive forces exerted by the wheels. Strong tractive forces were observed initially, a necessary condition to overcome resistive forces opposing vehicle movement. Notably, a disparity in tractive forces was registered during the vehicle's navigations through turns, signifying the requirement for varied traction levels to facilitate efficient vehicle maneuvering during trajectory shifts.
6. Conclusions
In this investigation, an LMIbased solution for robust H$\infty$ control of a nonlinear EV model powered by a PMSM was elucidated. The primary objective of this study was identified as the mitigation of interference effects, aiming to enhance the overall system performance.
Distinctively, the focus was on the peculiar attributes of EVs equipped with PMSM using robust H$\infty$ control. The integration of control methodologies tailored for nonlinearities, in conjunction with linear matrix inequalities, was found to be more adeptly matched to the demands of EVs than traditional techniques.
Within the proposed methodology, two H$\infty$ controllers were employed: the first was identified as pivotal for the Id current loop control, and the second was crucial for precise speed regulation. Through the application of the vector control technique and judicious selection of weight functions, the stability and performance of the comprehensive system were observed to be enhanced.
The robust H$\infty$ controller's efficacy was assessed through simulations utilizing a longitudinal model of the EV. The outcomes depicted superior dynamic and steadystate performances, thereby indicating the controller's competency in managing disturbances and ensuring meticulous velocity control.
The adoption of LMIbased synthesis for robust H$\infty$ control in EVs revealed multiple benefits, encompassing augmented performance, stability, and robustness. Such findings suggest this method holds promise in refining EV control tactics, presenting a viable solution for practical implementations.
Future explorations may consider additional optimizations and parameter adjustments to further bolster the control system's efficacy. It would be beneficial to conduct experimental validations on tangible EV platforms to corroborate the effectiveness of the proposed strategy. Extending this methodology to diverse control targets and system configurations may pave the way for addressing varied EV control conundrums, aiming to elevate the overall system operation.
The data used to support the findings of this study are available from the corresponding author upon request.
The DG RSDT (Direction Générale de la Recherche Scientifique et du Développement Technologique) provided support for this work.
The authors declare that they have no conflicts of interest.
$E V$  Electric Vehicle 
$P M S M$  Permanent Magnet Synchronous Motor 
$NEDC$  New European Driving Cycle 
$L M I$  Linear Matrix Inequalities 