# Active Control for an Electric Vehicle with an Observer for Torque Energy-Saving

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{z}) and power steering (${\delta}_{ce}$). Furthermore, it presents a pioneering method for estimating rear tire torque based on the M

_{z}control input in active state feedback control, a novelty not found in previous literature. This estimate is then compared to the rear tire torque achieved through PID control. By harnessing both PID and state feedback approaches, this methodology enables the computation of power (${P}_{mot}$) in the rear axle motor. This calculation, when applied to trajectory analysis, yields valuable insights into power in watts and vehicle energy. This innovative proposal holds promise for application in research concerning autonomy, efficiency, and greenhouse gas emissions, thus making a significant contribution to the current body of knowledge.

## 2. Mathematical Model of the Vehicle

#### 2.1. Definition of the Mathematical Model of the Vehicle

- -
- When studying stability and maneuverability, the analysis focuses on the dynamics of lateral velocity and yaw angular velocity.
- -
- The active brake actuator, denoted M
_{z}, is responsible for creating differential braking forces, generating a moment around the z-axis, which further impacts the dynamics of lateral velocity. - -
- The dynamics of the pitch angle are not considered. Since the system is a rigid body, Figure 1 can be utilized to depict the nonlinear dynamics of the system.
- -
- ${\omega}_{z}\cong {\dot{\delta}}_{dact}\xb7R$ is the angular velocity of turn (rad/s), which is in synchrony with ${\delta}_{d}$ [39].
- -
- R > 0 is a constant gain that is chosen so that the angular velocity of the turn is not saturated. It relates the input voltage on the actuator with the angular velocity obtained from [40].
- -
- ${\delta}_{d}$, ${\delta}_{ce}$ are the tire angle components imposed by the driver and controller (rad), ${\dot{\delta}}_{dact}=({u}_{m}-{R}_{m}\xb7I)/{k}_{b}$ is the angular velocity response of the actuator on the power steering wheel (rad/s), where ${u}_{m}$ is the input voltage to actuator (V), ${k}_{b}>0$ is an estimated back electromotive force constant (V/(rad/s)), ${R}_{m}$ is the resistance of the actuator (Ω), and I is the current (A), considering the simplified mathematical model of the cc motor where its values are obtained experimentally [41].

- ➢
- m is the mass of the car (kg).
- ➢
- J expresses the moment of inertia of the vehicle (kg·m
^{2}). - ➢
- h is the height of the center of gravity (C.G) with respect to the ground (m).
- ➢
- β denotes the chassis side-slip angle (rad).
- ➢
- l
_{f}, l_{r}, are the lengths from the center of the vehicle to the front and rear tires (m). - ➢
- r indicates the rolling radius of the tire (m).
- ➢
- v
_{x}signifies the longitudinal velocity (m/s). - ➢
- v
_{y}represents the lateral velocity (m/s). - ➢
- ω
_{z}is the angular rate of turn (rad). - ➢
- x = [v
_{x}, v_{y}, ω_{z}] is the compact vector of the vehicle state. - ➢
- α
_{fy}= α_{fyl}= α_{fyr}, α_{ry}= α_{ryl}= α_{ryr}denote the front and rear side-slip angles of the left and right tires (rad). - ➢
- α
_{f,}_{0}= α_{rx}_{0}are the uncontrolled front lateral and rear longitudinal slip angles respectively (rad). - ➢
- α
_{fx}= α_{fxl}= α_{fxr}, α_{rx}= α_{rxl}= α_{rxr}signify the front and rear longitudinal slip angles of the left and right wheels (rad). - ➢
- M
_{z}means the turning moment resulting from the active brakes (N·m). - ➢
- μ is the coefficient of friction between the tire and the ground.
- ➢
- F
_{dx}, F_{dy}are the longitudinal and lateral aerodynamic forces (N). - ➢
- M
_{dz}is the external aerodynamic yaw disturbance (N·m). - ➢
- F
_{f,x}, F_{r,x}are the front and rear longitudinal forces on the wheels (N). - ➢
- F
_{f,y}, F_{r,y}the front and rear lateral forces (N) as a function of the angle imposed on the front tires (δ = δ_{d}+ δ_{ce}).

_{fxl}, ω

_{fxr}as the angular velocities of the front left and right controlled tires (rad) and ω

_{rx}

_{0l}, ω

_{rx}

_{0r}as the uncontrolled rear left and right wheel revolution speeds (rad), T

_{n}represents the applied torque for driving and braking at the tire T

_{n}= M

_{z}$\xb7$r/T (N$\xb7$m), J

_{n}being the moment of inertia of the tire (kg$\xb7$m

^{2}), T is the tire tread width (m), with F

_{rxl}F

_{rxr}denoting the rear left and right longitudinal forces, respectively. (F

_{rxl}+ F

_{rxr}= F

_{r,x}) and F

_{rozl}, F

_{rozr}represent the friction forces for the tires in the z-axis (N) [42]:

_{roz}= m·g·cos ($\alpha $), where $\alpha \approx $ 0 to consider the EV on a flat road. However, the influence of gravity (g = 9.81 m/s

^{2}) and the coefficient ${\mu}_{0}$ are suggested to represent the average asphalt conditions. Additionally, ${\mu}_{1}$ = $\mu $ is used as the coefficient to account for the velocity effect on each rear tire.

#### 2.2. Aerodynamics

_{f,x}= F

_{fxl}+ F

_{fxr}, F

_{r,x}), comprising the longitudinal air disturbance force F

_{dx}, acting in the opposite direction, and lateral forces (F

_{f,y}, F

_{r,y}), as well as a counteracting lateral air disturbance force, F

_{dy}. Depending on the disturbance caused by the wind, an aerodynamic turning disturbance denoted M

_{dz}is created on the automobile chassis. Analyzing the wind impacts, it is essential to establish a coordinate system parallel to the coordinates of the automobile’s suspension mass, with the lateral displacement shown in Figure 1.

_{s}is the surface area of the vehicle (m

^{2}), ρ indicates the air mass density (kg/m

^{3}), ${c}_{\psi}$ means the constant aerodynamic coefficient, and v

_{w}signifies the air velocity (m/s). Therefore, the air disturbances are obtained as [43]:

_{s,f}, and A

_{s,l}, along with the longitudinal and lateral aerodynamic coefficients c

_{a,x}, c

_{a,y}, respectively, influenced by the wind velocity components along the x and y axes (v

_{a,w,x}, v

_{a,w,y}in m/s), acting on the vehicle’s suspension mass.

_{s,f}= 2.59 m

^{2}, A

_{s,l}= 5.10 m

^{2}, c

_{a,x}= 0.3, c

_{a,y}= 0.6, ρ = 1.206 kg/m

^{3}, v

_{a,w,z}= 0, $\psi $ = 225°, Vair = 185 km/h.

#### 2.3. State Feedback Controller Design

_{j,i}, C

_{j,i}, D

_{j,}

_{i}in Equations (1)–(3) are determined experimentally (see the end of subtopic 2). The control inputs will be considered as the result of the active brake torque moment M

_{z}, the torque T

_{n}applied to the rear tires, and the difference in lateral and longitudinal forces between the front and rear tires of the EV:

_{ce}, which can be determined by Equation (22):

_{x}, v

_{y}, $\omega $

_{z}) asymptotically follows a reference system characterized by having bounded derivatives. The significance of the reference system is established under ideal conditions, which assume a new vehicle, new tires, and new mechanical and electrical components with parameters, as exemplified in [34], is defined by:

_{,ref}(v

_{x,ref}, v

_{y,ref}, $\omega $

_{z,ref}) is the compact reference vector of the vehicle state. J

_{,ref}= J, $\mu $

_{,ref}= $\mu $ are appropriate parameters, and F

_{fy,ref}, F

_{ry,ref}, F

_{fx,ref}, F

_{rx,ref}are ideal curves depending on:

_{r,x}, $\u2206$F

_{f,y}and M

_{z}, using the following control law:

_{ii}> 0, ii = 1, 2, 3. Therefore, the control inputs will be:

_{VE}is defined as:

_{1}, $\phi $

_{2}and $\phi $

_{3}are:

_{VE}, where the first column depends on the parameter μ, which will be invertible if $\mu \ne $ 0:

#### 2.4. Design of the State Feedback Observer

_{x}, a

_{y}, ω

_{z}, v

_{x}, ω

_{fxl}, ω

_{fxr}are measured, which is a reasonable assumption in modern vehicles typically equipped with the necessary sensors, and measurements can also be taken using instruments such as the Autel Otofix Scanner and Hantek Oscilloscope-Multimeter to observe variables and dynamics. The proposal for the nonlinear observer for lateral velocity is performed as a copy of the plant in Equations (1) and (2):

_{o}

_{1}, k

_{o}

_{2}are the observer gains determined by Equations (1) and (2). Since the angular turn rate is approximated as ${\omega}_{z}\cong {\dot{\delta}}_{dact}\xb7R$, which is assumed to be measured, the variable ${\omega}_{z}$ is implicit in the estimation of the variable ${\dot{\widehat{v}}}_{x}$, thus facilitating the estimation of longitudinal velocity.

_{z}. To ensure exponential convergence of the error estimation, it is assumed that the yaw angular velocity ω

_{z}$\ne 0$, considering $\left|{\omega}_{z}\right|\le \omega $

_{z,max}, with a maximum value (ω

_{z,max}), at all time t $\ge $ 0. These physical considerations are reasonable because the vehicle possesses finite energy, thereby bounding the maximum yaw angular velocity. In summary, during driver steering input maneuvers, ω

_{z}may pass through zero but cannot be zero within a finite time interval, and neither can the lateral forces of the vehicle’s tires.

_{o}

_{1}, k

_{o}

_{2}are obtained using the following candidate Lyapunov function:

_{1}> ${\kappa}_{1}^{2}$ > 0, ${\kappa}_{1}\ne $ 0, and sign (·) the signum function:

_{D}(ω

_{z}), for $\left|{\omega}_{z}\right|$ > 0, is considered zero ($\delta $

_{D}(ω

_{z}) = 0), yielding to:

#### 2.5. Design of the Active PID Controller

_{fxlpid}, ω

_{fxrpid}as the angular velocities of the front left and right controlled tires (rad) and ω

_{rx}

_{0lpid}, ω

_{rx}

_{0rpid}as the uncontrolled rear left and right wheel revolution speeds (rad), T

_{npid}represents the applied torque for driving and braking at the tire (N·m), with F

_{rxlpid}, F

_{rxrpid}denoting the rear left and right longitudinal forces, respectively (N). (F

_{rxlpid}+ F

_{rxrpid}= F

_{r,xpid}) and F

_{rozlpid}, F

_{rozrpid}represent the friction forces for the tires in the z-axis (N):

_{rozpid}= m·g·cos (α). The values for k

_{p}, T

_{i}, T

_{d}(established experimentally) are considered in the following parameters: m = 1550 kg, J = 3352 kg$\xb7$m

^{2}, l

_{r}= 1.53 m, l

_{f}= 1.38 m, J

_{n}= 62 kg$\xb7$m

^{2}, r = 0.20 m, T = 0.16 m, ${\mu}_{0}$ = 0.9, h = 0.5 m, B

_{f,x}= 6.9, C

_{f,x}= 1.3, D

_{f,x}= 7500 N, B

_{r,x}= 7.1, C

_{r,x}= 1.6, D

_{r,x}= 7500 N, B

_{r,y}= 10, C

_{r,y}= 1.32, D

_{r,y}= 7834 N, B

_{f,y}= 6.9, C

_{f,y}= 1.78, D

_{f,y}= 7240 N, K

_{p}= 100.475, T

_{d}= 0.00073, T

_{i}= 45.84383, B

_{f,x,ref}= 6.9, C

_{f,x,ref}= 1.3, D

_{f,x,ref}= 10,000 N, B

_{r,x,ref}= 7.1, C

_{r,x,ref}= 1.6, D

_{r,x,ref}= 10,000 N, B

_{r,y,ref}= 10, C

_{r,y,ref}= 1.32, D

_{r,y,ref}= 10,000 N, B

_{f,y,ref}= 6.9, C

_{f,y,ref}= 1.78, D

_{f,y,ref}= 10,000 N. Moreover, the analysis includes F

_{f,ypid}, F

_{r,ypid}in Equation (22), but with the dynamics of the PID controller, along with T

_{npid}= M

_{zpid}·r/T. These parameters are obtained from [37].

## 3. Experimental Results and Discussion

#### 3.1. Maneuver ISO 7401

_{1}= 100, k

_{2}= 100, k

_{3}= 100 for Equations (35)–(37). For ko

_{1}, ko

_{2}in Equations (62) and (63) is considered ${\gamma}_{1}$ = 0.023, ${\kappa}_{1}$ = 0.15, ${\lambda}_{s}$ = 1, which are carried out experimentally in simulations. The results for the longitudinal velocity also start with v

_{x}= 28 m/s. Additionally, the values obtained with the PID controller deviate from the reference value, in contrast to the state feedback control, which is accurately estimated by the proposed state feedback observer. As a result, an estimated torque ${\widehat{T}}_{n}$ is achieved (Figure 4) for the rear tires of the electric vehicle with significant energy-saving efficiency, even under different tire-ground friction coefficient conditions. The state feedback control consistently shows positive power and torque values during the simulation, as it remains unaffected by changes in the tire-to-ground friction coefficient. In contrast, the PID control exhibits both positive and negative values due to the coefficient’s variability.

_{total}= T

_{front}+ T

_{rear}, where the input torques can be defined as T

_{iii}= (F

_{j,iiis}·r)·k

_{T}·4 = T

_{total}, j = f,r, iii = x, s = left, right, k

_{T}$\ge $ 15, with an adjustment gain (Figure 4), the system efficiency can be calculated as n

_{system}= ${\widehat{T}}_{n}$/T

_{total}, or n

_{system}= ${T}_{npid}$/T

_{total}, for the state feedback controller and the PID controller, respectively. The motor power, P

_{mot}= n

_{system}$\xb7\sum _{iiii=1}^{4}{T}_{iii}\xb7{\omega}_{j,iiis}$, iiii = corresponds to the vehicle’s wheels, as shown in Figure 5.

#### 3.2. Maneuver ISO 3888-2

## 4. Conclusions

_{ce}, δ

_{cpid}) for the automotive power steering system. Therefore, it can be concluded that with ISO 7401 [44] and the ISO 3888-2 [46] standard, the PID controller encounters some issues in controlling the variables and dynamics of the vehicle in Section 3. Finally, it is established that the recommended control modes for obtaining a control observer for automotive power steering, studied in this research article, can be the state feedback approach, as a proposal for certain vehicle maneuvering situations. Furthermore, future work includes comparing the Matlab-Simulink platform with others, such as the National Instrument HiL, Carsim or dSpace MicroAutoBox.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

González-López, J.M.; Pérez, S.S.; Betancourt, R.O.J.; Barreto, G.
Active Control for an Electric Vehicle with an Observer for Torque Energy-Saving. *World Electr. Veh. J.* **2023**, *14*, 288.
https://doi.org/10.3390/wevj14100288

**AMA Style**

González-López JM, Pérez SS, Betancourt ROJ, Barreto G.
Active Control for an Electric Vehicle with an Observer for Torque Energy-Saving. *World Electric Vehicle Journal*. 2023; 14(10):288.
https://doi.org/10.3390/wevj14100288

**Chicago/Turabian Style**

González-López, Juan Miguel, Sergio Sandoval Pérez, Ramón O. Jiménez Betancourt, and Gilberto Barreto.
2023. "Active Control for an Electric Vehicle with an Observer for Torque Energy-Saving" *World Electric Vehicle Journal* 14, no. 10: 288.
https://doi.org/10.3390/wevj14100288