Robust integrated control for an automotive electric power steering system based on observed state variables

Abstract

This article presents a robust control strategy for Electric Power Steering (EPS) to enhance overall performance. The proposed approach integrates Backstepping Control (BSC) and Sliding Mode Control (SMC) techniques, with the input signal finely tuned using a Proportional-Integral-Derivative (PID) controller. An Extended State Observer (ESO) is utilized to estimate the system’s state variables, including the effects of external disturbances, through an augmented state variable. This work’s novelty lies in designing an integrated control mechanism that effectively eliminates steady-state error and sensor noise while reducing overshoot and chattering, which commonly affect traditional control methods. The numerical simulation method is conducted to compare the proposed controller’s performance with existing algorithms. The results show that the controlled state variables closely track their desired values with minimal errors. The Root Mean Square Error (RMSE) for most state variables is insignificant (below 0.37%), except for motor current signals (9.93% for v₁ and 25.65% for v₂). The ESO demonstrates high accuracy in estimating both the state variables and disturbances. In conclusion, the proposed control approach significantly outperforms conventional methods in EPS system control.

Introduction

Power-assisted steering systems have been developed since the 1950s of the last century. Today, they are equipped with most car models. They are known as three main types: electric power steering (EPS), hydraulic power steering, and electrohydraulic power steering, according to Xia, and Jiang. ¹ We use the EPS system to reduce steering effort and improve steering safety across a wide speed range. ² Baharom et al. ³ confirmed that the EPS system provided superior performance compared to the hydraulic system. They claimed that the EPS system used less energy than other systems. At the same time, this system was environmentally friendly because it avoided using hydraulic oil. A study conducted by Ramasamy showed that the EPS system had a positive impact on vehicle handling. ⁴ A recent publication by Park et al. showed that energy consumption efficiency was significantly improved when the EPS system was applied instead of other traditional systems. ⁵ Nowadays, EPS systems are commonly used on family cars, ⁶ light trucks, or all-terrain vehicles. ⁷

Many studies on EPS system control have been published recently. In, ⁸ Kim et al. designed a PI controller to control the automotive EPS system. Computational and experimental results showed that the output signal was chattered, which was caused by disturbances. A combination of PID and PI was performed by Manca et al. ⁹ Simulation results showed that the motor current signal was strongly chattered. In addition, the steering wheel position signal was out of phase at high frequencies. In Ref., ¹⁰ Zheng and Wei designed a fuzzy PI algorithm to eliminate the phase shift phenomenon partially. However, this caused the overshoot phenomenon to increase by 1.5%. A backpropagation neural network technique was proposed in Ref. ¹¹ by Li et al. to tune the PID controller. The structure of this algorithm was formed based on three layers: input, output, and hidden layers. The system’s stability improved when driving at high speed; however, phase lag still existed when moving at low speed. ¹¹ Turan presented a new method for tuning PID control parameters in Ref. ¹² This method aims to minimize the steady-state error of multiple outputs. A neuro-fuzzy controller based on PD was introduced in Ref. ¹³ by Ramos-Fernández et al. The simulation results illustrated that the systematic error was quite large, caused by the previous training process. Regarding large vehicles, the assisted torque generated by an electric motor is not enough. Therefore, it is necessary to combine two electric motors, which operate independently and are controlled by two different controllers. In Ref., ¹⁴ Li et al. proposed applying a dual PID algorithm to control electric motors. An improvement of this mechanism was made by Fu et al. ¹⁵ Generally, a PID controller has many advantages, such as low cost, high reliability, and being systematic. However, its performance is not high compared to other robust controllers. ¹⁶ In addition, this algorithm can only be applied to linear systems with a single input and a single output.

The traditional PID technique should be replaced by robust control techniques that apply to nonlinear systems, such as sliding mode control (SMC), backstepping control (BSC), and active disturbance rejection control (ADRC) to improve system performance. In Ref., ¹⁷ Khasawneh and Das designed a robust sliding mode controller to control the automotive EPS system. The stability of the system was evaluated according to the Lyapunov criterion. Simulation results in Ref. ¹⁷ showed that the measured signal strongly chattered. Furthermore, phase shifts occurred in simulated conditions. In Ref., ¹⁸ Lee et al. designed an adaptive SMC controller for steering wheel torque tracking. The steering wheel angle value tracked the estimated value with no small error. Furthermore, there was significant chatter in the steering wheel velocity and acceleration. This could degrade the quality of the system. In Ref., ¹⁹ Nguyen proposed a new approach for controlling automotive EPS systems. This strategy was formed by combining SMC and PID techniques to eliminate the chattering phenomenon. According to Ref., ¹⁹ the output signals from two distinct controllers were amplified by optimal coefficients before being summed to become the final control signal. However, the stability of the system was not fully demonstrated. Additionally, when disturbances were present, uncontrollable objects continued to chatter. This method was improved in Ref. ²⁰ by Nguyen and Iqbal. The parameters of the PID control were tuned by a genetic-fuzzy algorithm. This helped reduce chattering under the influence of disturbances. In some travel conditions, however, the influence of sensor noise was significant. In order to improve the quality of the SMC technique, the influence of disturbances should be estimated by observers. ²¹ In Ref., ²² Kim et al. designed a new sliding mode controller to track steering wheel torque. System signals were estimated by an extended state observer (ESO) to eliminate the effects of sensor noise. Simulation results in Ref. ²² showed that the output signals followed the reference signal with no significant errors. However, the chattering phenomenon still existed. Some other observers for EPS system control should be referred to Refs.^23,24

The chattering phenomenon can be reduced by replacing the SMC technique with BSC. In Ref., ²⁵ Shi et al. designed an adaptive robust backstepping controller to control the electrohydraulic steering system. Simulation results in Ref. ²⁵ showed that the algorithm’s performance was strongly degraded under extreme steering conditions (high frequency). A combination of BSC and PID techniques to become a BSPID controller was introduced in Ref. ²⁶ The output value obtained from this algorithm tended to follow the reference value better than the conventional PID. ²⁷ presented a method to tune the coefficients for the BSPID controller based on a dynamic fuzzy algorithm. This was intended to eliminate signal phase delay. A more prominent improvement of this algorithm was mentioned in Ref. ²⁸ by Nguyen. Simulation results illustrated that the controlled object tracked the reference value with insignificant errors. However, other output signals were strongly affected by disturbances. ADRC is an effective solution that ensures the stability of the EPS system when subjected to external influences. However, the effect of chattering cannot be eliminated.^29,30

Several advanced robust control methods have recently been developed for automotive mechatronic systems. In Ref., ³¹ Alan et al. designed control barrier functions to control automated vehicles. A yaw moment control framework based on a robust T-S fuzzy approach was presented in ³² by Liang et al. In Ref., ³³ Wong et al. introduced a robust finite-time fault-tolerant control mechanism for controlling an air suspension system subject to external disturbances and uncertainties. Alqudsi et al. designed optimal and robust control methodologies for a suspension system with four degrees of freedom. ³⁴ An application of the combination of ADRC and SMC techniques in steering performance control was presented in Ref. ³⁵ by Nguyen and Nguyen. In addition, several other robust control methods, which have been utilized for controlling automotive mechatronic systems, have been presented in some previous publications.^36,37 In general, these methods effectively control the system under certain conditions. However, most of these applications have not been integrated into the EPS system.

Research gaps

Although the algorithms mentioned above can effectively control the EPS system under various conditions, several issues persist. Firstly, phase lag occurs when only a single BSC or traditional PID control techniques are applied.^9,27 Additionally, the overshoot phenomenon remains a concern when traditional PID controllers are used, as noted in studies.^10–12 While SMC and ADRC techniques provide stability despite strong external disturbances, they still exhibit chattering phenomena, which negatively impact the system.^{8,9,17–19,29,30} This represents the second issue. Thirdly, even with optimal parameter selection, the system’s power consumption remains high.^38,39 Furthermore, the influence of external disturbances needs to be accurately calculated instead of assuming it to be predetermined.^19,20,40

Most of the algorithms discussed primarily control a single object. When applying robust nonlinear techniques such as BSC, SMC, or ADRC to Multi-Input and Multi-Output (MIMO) systems, interactions can become exceedingly complex. Although the linear quadratic technique can be used in these systems,^41,42 its performance is often insufficient, constituting the fifth issue. Finally, several other techniques result in significant steady-state errors.^13,18

All of the above are considered key limitations of existing EPS control methods, which must be overcome in future works. Addressing the remaining drawbacks is considered the motivation for the work done in this article.

New contributions

In this work, we propose designing a robust integrated algorithm to address the identified issues. This method combines BSC, SMC, and PID techniques based on the ESO. Two key variables, steering column angle, and steering motor angle, are managed by two independent, robust techniques to resolve the second, fifth, and sixth issues. The first issue is addressed by adjusting the controller’s input signals using the PID technique. This new combination aims to enhance energy efficiency, tackling the third issue. Finally, the fourth issue is resolved by employing the ESO to estimate system disturbances instead of ignoring them or assuming they are known.

Although the aforementioned control techniques have been extensively studied in previous works, most of these studies focus on the design of individual (standalone) controllers. The key contribution of this article lies in developing an integrated control scheme, which combines multiple control strategies in a unified framework to address existing limitations and enhance overall system performance.

The article is structured into four sections. The first section includes the introduction and literature review, while the second outlines the mathematical method. The third section discusses numerical calculations and the simulation process. Lastly, the conclusion section presents observations based on the work.

The symbols are listed in the notation subsection.

Notation

Symbol	Meaning
ψ	Yaw angle
β	Heading angle
δ	Steering angle
θ	Pitch angle
ϕ	Roll angle
ϕ c	Column angle
θ cas	Caster angle
θ kin	Kingpin angle
ϕ m	Motor angle
B c	Column damping
B eq	Equivalent damping
B m	Motor damping
B r	Rack damping
C	Damping coefficient
d c	Caster trail
d k	Kingpin distance
d n	Arm length
F C	Damping force
F g	Gravitational force
F K	Spring force
F T	Tire force
F x	Longitudinal force
F y	Lateral force
F z	Vertical force
G m	Motor ratio
G s	Steering ratio
h θ	Pitch height
h ϕ	Roll height
I ϕ	Roll inertia moment
I θ	Pitch inertia moment
I ψ	Yaw inertia moment
I c	Column inertia moment
I eq	Equivalent inertia
i m	Motor current
I m	Motor inertia moment
K	Spring coefficient
K c	Column stiffness
K r	Rate coefficient
K t	Torque coefficient
L	Axle distance
L m	Motor inductance
M r	Rack mass
m s	Sprung mass
m u	Unsprung mass
M z	Self-aligning torque
R m	Motor resistance
r p	Pinion radius
T	Tire spring coefficient
T a	Assisted torque
T d	Driver torque
T ed	External disturbance
T id	Internal disturbance
T r	Reaction torque
t w	Trackwidth
U	Control signal
v x	Longitudinal velocity
v y	Lateral velocity
z r	Road displacement
z s	Sprung mass displacement
z u	Unsprung mass displacement

Mathematical method

System model

Figure 1(a) depicts the structure of a C-EPS system. The system’s overall structure includes a steering wheel, a steering column, an electric motor, an ECU, and sensors. A pair of gears amplify the motor-generated torque, while a rack and pinion amplify the combined steering torque. The relationship between the steering column angle (φ _c ) and steering motor angle (φ _m ) is described according to (1).

$$I_c\frac{d^2}{d{t}^2}{\varphi }_c+B_c\frac{d}{\mathrm{dt}}{\varphi }_c+K_c{\varphi }_c=\frac{K_c}{G_m}{\varphi }_m+T_d$$

(1)

Figure 1.

System models. (a) EPS model. (b) Nonlinear model. (c) Spatial model.

The effect of assisted torque through motor current (i _m ) is illustrated by (2).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{K_c}{G_m}{\varphi }_c+K_t{i}_m-\frac{T_r}{G_m}=I_{\mathrm{eq}}\frac{d^2}{d{t}^2}{\varphi }_m+B_{\mathrm{eq}}\frac{d}{\mathrm{dt}}{\varphi }_m}\\ \multicolumn{1}{c}{+\frac{K_c+K_r{r}_p^2}{G_m^2}{\varphi }_m}\end{array}$$

(2)

Equation (3) explains the relationship between the control signal u and the motor current. The equations mentioned include the following symbols: I _c is the inertia moment of the steering column, B _c is steering column damping, K _c is steering column stiffness, G _m is motor gear ratio, T _d is driver torque, K _t is motor torque coefficient, T _r is road reaction torque, r _p is pinion radius, L _m is electric motor inductance, R _m is electric motor resistance, K _r is rate coefficient, and u is control signal, B _eq is equivalent damping, and I _eq is equivalent moment of inertia.

$$K_t\frac{d}{\mathrm{dt}}{\varphi }_m+L_m\frac{d}{\mathrm{dt}}{i}_m+R_m{i}_m=u$$

(3)

The values of B _eq and I _eq are determined according to equations (4) and (5), respectively, where B _m is motor damping, B _r is rack damping, I _m is the inertia moment of the electric motor, and M _r is rack mass.

$$B_{\mathrm{eq}}=B_m+\frac{r_p^2}{G_m^2}{B}_r$$

(4)

$$I_{\mathrm{eq}}=I_m+\frac{r_p^2}{G_m^2}{M}_r$$

(5)

The system disturbance is also known as road reaction torque (T _r ). According to (6), system disturbances consist of two components: external disturbance (T _ed ) and internal disturbance (T _id ).

$$T_r=T_{\mathrm{id}}+T_{\mathrm{ed}}$$

(6)

External disturbance refers to the influence of environmental conditions when the vehicle moves, such as road surface roughness, crosswind, weather, and others. Conversely, the steering resistance torque represents the internal disturbance when the vehicle steers. The variation of T _id is approximately described by equation (7), where d _n is knuckle arm length, d _c is caster trail, θ _cas is caster angle, d _k is kingpin distance, θ _kin is kingpin angle, F _y is lateral tire force, F _x is longitudinal tire force, and M _z is self-aligning torque.

$$T_{\mathrm{id}}\approx r_p\frac{\mathrm{co}{s}^2\left({\theta }_{\mathrm{kin}}\right)\mathrm{co}{s}^2\left({\theta }_{\mathrm{cas}}\right)}{d_n}(\pm d_k{F}_{\mathrm{xij}}\pm d_c{F}_{\mathrm{yij}}\pm M_{\mathrm{zij}})$$

(7)

To enhance the accuracy of the calculation, we consider the tire to be nonlinearly deformed. The Pacejka nonlinear tire model, designed according to equations (8)–(10), is a reliable tool for determining tire forces and moments. The coefficients D, C, B, κ, and S _v are calculated according to. ⁴³

$$F_x=D_x\sin \left[C_x\arctan \left(B_x{\kappa }_x\right)\right]+S_{\mathrm{vx}}$$

(8)

$$F_y=D_y\sin \left[C_y\arctan \left(B_y{\kappa }_y\right)\right]+S_{\mathrm{vy}}$$

(9)

$$M_z=D_z\sin \left[C_z\arctan \left(B_z{\kappa }_z\right)\right]+S_{\mathrm{vz}}$$

(10)

The load variation (F _z ) is the input of the Pacejka nonlinear tire model, which is determined by (11).

$$F_{\mathrm{zij}}=F_{\mathrm{gij}}\pm F_{\mathrm{Tij}}$$

(11)

Gravitational force (F _g ) depends on the vehicle roll angle (φ) and is calculated according to (12), where m _s is sprung mass, t _w is trackwidth, mu is unsprung mass, h_φ is roll distance, and g is gravitational acceleration.

$$F_{\mathrm{gij}}=\frac{m_{\mathrm{si}}(t_{\mathrm{wi}}-h_{\varphi }\sin \varphi )+2t_{\mathrm{wi}}{m}_{\mathrm{uij}}}{2t_{\mathrm{wi}}}g$$

(12)

Spring force (F _K ), damping force (F _C ), and tire force (F _T ) are determined by (13), (14), and (15), respectively, where K is spring coefficient, C is damping coefficient, T is tire spring coefficient, and l is axle distances (Figure 1(c)).

Spring force:

$$F_{\mathrm{Kij}}=K_{\mathrm{ij}}[z_s-z_{\mathrm{uij}}+{(-1)}^{j+1}{t}_{\mathrm{wi}}\varphi +{(-1)}^{i+1}{l}_i\theta ]$$

(13)

Damping force:

$$F_{Cij}=C_{ij}\frac{d}{dt}[z_s-z_{uij}+{\left(-1\right)}^{j+1}{t}_{wi}\varphi +{\left(-1\right)}^{i+1}{l}_i\theta ]$$

(14)

Tire force:

$$F_{\mathrm{Tij}}=T_{\mathrm{ij}}(z_{\mathrm{rij}}-z_{\mathrm{uij}})$$

(15)

Sprung mass displacement (z _s ) and unsprung mass displacement (z _u ) are determined by equations (16)–(19), where Iφ is roll inertia moment and I_θ is pitch inertia moment.

Sprung mass motion:

$$m_s\frac{d^2}{d{t}^2}{z}_s=\sum _{i,j=1}^2(F_{\mathrm{Kij}}+F_{\mathrm{Cij}})$$

(16)

Roll motion:

$$\begin{array}{c}\multicolumn{1}{c}{(I_{\varphi }+m_s{h}_{\varphi }^2)\frac{d^2}{d{t}^2}\varphi =\sum _{i,j=1}^2\left[{(-1)}^{j-1}(F_{\mathrm{Kij}}+F_{\mathrm{Cij}})t_{\mathrm{wi}}\right]}\\ \multicolumn{1}{c}{+\{\mathrm{gsin}\varphi +[\frac{d}{\mathrm{dt}}{v}_y+v_x\frac{d}{\mathrm{dt}}(\beta +\psi )]\cos \varphi \}{m}_s{h}_{\varphi }}\end{array}$$

(17)

Pitch motion:

$$(I_{\theta }+m_s{h}_{\theta }^2)\frac{d^2}{d{t}^2}\theta =\sum _{i,j=1}^2{(-1)}^{i-1}(F_{\mathrm{Kij}}+F_{\mathrm{Cij}})l_i$$

(18)

Unsprung mass motion:

$$m_{\mathrm{uij}}\frac{d^2}{d{t}^2}{z}_{\mathrm{uij}}=F_{\mathrm{Tij}}-F_{\mathrm{Kij}}-F_{\mathrm{Cij}}$$

(19)

Lateral velocity (v _y ), longitudinal velocity (v _x ), are the solutions of equations (20) and (21); Figure 1(b)).

$$\begin{array}{c}\multicolumn{1}{c}{(m_s+\sum _{i,j=1}^2{m}_{\mathrm{uij}})[\frac{d}{\mathrm{dt}}{v}_x-v_y\frac{d}{\mathrm{dt}}(\beta +\psi )]}\\ \multicolumn{1}{c}{=\sum _{i,j=1}^2(F_{\mathrm{xij}}\cos {\delta }_{\mathrm{ij}}-F_{\mathrm{yij}}\sin {\delta }_{\mathrm{ij}})}\end{array}$$

(20)

$$\begin{array}{c}\multicolumn{1}{c}{(m_s+\sum _{i,j=1}^2{m}_{\mathrm{uij}})[\frac{d}{\mathrm{dt}}{v}_y+v_x\frac{d}{\mathrm{dt}}(\beta +\psi )]}\\ \multicolumn{1}{c}{=\sum _{i,j=1}^2(F_{\mathrm{xij}}\sin {\delta }_{\mathrm{ij}}+F_{\mathrm{yij}}\cos {\delta }_{\mathrm{ij}})}\end{array}$$

(21)

The yaw angle (ψ) is defined according to (22), where I_ψ is the yaw inertia moment.

$$\begin{array}{c}\multicolumn{1}{c}{I_{\psi }\frac{d^2}{d{t}^2}\psi =\sum _{i,j=1}^2}\\ \multicolumn{1}{c}{\left[\begin{array}{c}{(-1)}^j(F_{\mathrm{xij}}\cos {\delta }_{\mathrm{ij}}-F_{\mathrm{yij}}\sin {\delta }_{\mathrm{ij}})t_{\mathrm{wi}}\hfill \\ \multicolumn{1}{c}{+{(-1)}^{i+1}(F_{\mathrm{xij}}\sin {\delta }_{\mathrm{ij}}+F_{\mathrm{yij}}\cos {\delta }_{\mathrm{ij}})l_i-M_{\mathrm{zij}}}\end{array}\right]}\end{array}$$

(22)

The relationship between steering column angle (φ _c ) and steering angle (δ) is expressed by (23), where G _s is the steering ratio.

$$\delta =\frac{{\varphi }_c}{G_s}$$

(23)

The ideal steering characteristic curve (T_{a_ideal}) is illustrated in Figure 2. The value of T_{a_ideal} depends on driver torque and vehicle speed, according to (24), where C _i are empirical coefficients. According to this description, the value of T_{a_ideal} reaches saturation when T _d ≥T_{d_max} and is zero when T _d ≤T_{d_min}. The change in assisted torque is considered linear with driver torque when T_{d_min} < T _d < T_{d_max}. One thing to remember is that if the vehicle speed increases, the value of T_{a_ideal} will suddenly decrease. Several other characteristic curves should be referred to Ref. ⁴⁴

$$T_{a\_\mathrm{ideal}}=(T_d-T_{d\_\min })(C_1{v}^2+C_{2}v+C_3)$$

(24)

Figure 2.

Ideal assisted torque.

Control method

This work controls two objects: the steering motor and column angles. These two objects are controlled using two different techniques. Set the state variables to (25).

$${\left[x_i\right]}^T={\left[\begin{array}{ccccc}{\varphi }_c\hfill & \frac{d}{\mathrm{dt}}{\varphi }_c\hfill & {\varphi }_m\hfill & \frac{d}{\mathrm{dt}}{\varphi }_m\hfill & i_m\hfill \end{array}\right]}^T$$

(25)

Taking the derivative of (25), we get equations from (26) to (30).

$$\frac{d}{\mathrm{dt}}{x}_1=x_2$$

(26)

$$\frac{d}{\mathrm{dt}}{x}_2=-\frac{K_c}{I_c}{x}_1-\frac{B_c}{I_c}{x}_2+\frac{K_c}{I_c{G}_m}{x}_3+\frac{T_d}{I_c}$$

(27)

$$\frac{d}{\mathrm{dt}}{x}_3=x_4$$

(28)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}{x}_4=\frac{K_c}{I_{\mathrm{eq}}{G}_m}{x}_1-\frac{K_c+K_r{r}_p^2}{I_{\mathrm{eq}}{G}_m^2}{x}_3-\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}}\\ \multicolumn{1}{c}{x_4+\frac{K_t}{I_{\mathrm{eq}}}{x}_5-\frac{T_r}{I_{\mathrm{eq}}{G}_m}}\end{array}$$

(29)

$$\frac{d}{\mathrm{dt}}{x}_5=-\frac{K_t}{L_m}{x}_4-\frac{R_m}{L_m}{x}_5+\frac{1}{L_m}u$$

(30)

In this article, only signal x₃ is measured by the sensor, while an ESO determines the remaining signals. An augmented variable is shown in (31). This state variable is only used to observe system disturbances.

$$x_6=T_r$$

(31)

Sliding mode control design

The estimated state variables are described by equations from (32) to (37).

$$\frac{d}{\mathrm{dt}}{\hat{x}}_1={\hat{x}}_2-{\beta }_{1}e$$

(32)

$$\frac{d}{\mathrm{dt}}{\hat{x}}_2=-\frac{K_c}{I_c}{\hat{x}}_1-\frac{B_c}{I_c}{\hat{x}}_2+\frac{K_c}{I_c{G}_m}{\hat{x}}_3+\frac{T_d}{I_c}-{\beta }_{2}e$$

(33)

$$\frac{d}{\mathrm{dt}}{\hat{x}}_3={\hat{x}}_4-{\beta }_{3}e$$

(34)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}{\hat{x}}_4=\frac{K_c}{I_{\mathrm{eq}}{G}_m}{\hat{x}}_1-\frac{K_c+K_r{r}_p^2}{I_{\mathrm{eq}}{G}_m^2}{\hat{x}}_3-\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}{\hat{x}}_4+\frac{K_t}{I_{\mathrm{eq}}}}\\ \multicolumn{1}{c}{{\hat{x}}_5-\frac{1}{I_{\mathrm{eq}}{G}_m}{\hat{x}}_6-{\beta }_{4}e}\end{array}$$

(35)

$$\frac{d}{\mathrm{dt}}{\hat{x}}_5=-\frac{K_t}{L_m}{\hat{x}}_4-\frac{R_m}{L_m}{\hat{x}}_5+\frac{1}{L_m}u-{\beta }_{5}e$$

(36)

$$\frac{d}{\mathrm{dt}}{\hat{x}}_6=-{\beta }_{6}e$$

(37)

The error (e) between the signal estimated by the ESO and the signal received by the sensor is described by (38), and β _i are observer gains.

$$e={\hat{x}}_3-x_3$$

(38)

Taking the derivative of (32), we obtain equations from (39) to (42).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d^2}{d{t}^2}{\hat{x}}_1=-\frac{K_c}{I_c}{\hat{x}}_1-\frac{B_c}{I_c}{\hat{x}}_2+\frac{K_c}{I_c{G}_m}{\hat{x}}_3+\frac{T_d}{I_c}}\\ \multicolumn{1}{c}{-{\beta }_{2}e-{\beta }_1\frac{d}{\mathrm{dt}}e}\end{array}$$

(39)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d^3}{d{t}^3}{\hat{x}}_1=\sum _{i=1}^4{a}_i{\hat{x}}_i+(-\frac{B_c{T}_d}{I_c^2}+\frac{\frac{d}{\mathrm{dt}}{T}_d}{I_c})}\\ \multicolumn{1}{c}{+(\frac{K_c}{I_c}{\beta }_1+\frac{B_c}{I_c}{\beta }_2-\frac{K_c}{I_c{G}_m}{\beta }_3)e}\\ \multicolumn{1}{c}{-{\beta }_2\frac{d}{\mathrm{dt}}e-{\beta }_1\frac{d^2}{d{t}^2}e}\end{array}$$

(40)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d^4}{d{t}^4}{\hat{x}}_1=\sum _{i=1}^6{b}_i{\hat{x}}_i+(\frac{a_2{T}_d}{I_c}-\frac{B_c\frac{d}{\mathrm{dt}}{T}_d}{I_c^2}+\frac{\frac{d^2}{d{t}^2}{T}_d}{I_c})}\\ \multicolumn{1}{c}{-\sum _{i=1}^4{a}_i{\beta }_{i}e+(\frac{K_c}{I_c}{\beta }_1+\frac{B_c}{I_c}{\beta }_2-\frac{K_c}{I_c{G}_m}{\beta }_3)}\\ \multicolumn{1}{c}{\frac{d}{\mathrm{dt}}e-{\beta }_2\frac{d^2}{d{t}^2}e-{\beta }_3\frac{d^3}{d{t}^3}e}\end{array}$$

(41)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d^5}{d{t}^5}{\hat{x}}_1=\sum _{i=1}^6{c}_i{\hat{x}}_i}\\ \multicolumn{1}{c}{+(\frac{b_2{T}_d}{I_c}+\frac{a_2\frac{d}{\mathrm{dt}}{T}_d}{I_c}-\frac{B_c\frac{d^2}{d{t}^2}{T}_d}{I_c^2}+\frac{\frac{d^3}{d{t}^3}{T}_d}{I_c})}\\ \multicolumn{1}{c}{+(\frac{a_4}{I_{\mathrm{eq}}{G}_m}-b_6){\beta }_{6}e-\sum _{i=1}^4{a}_i{\beta }_i\frac{d}{\mathrm{dt}}e}\\ \multicolumn{1}{c}{+(\frac{K_c}{I_c}{\beta }_1+\frac{B_c}{I_c}{\beta }_2-\frac{K_c}{I_c{G}_m}{\beta }_3)\frac{d^2}{d{t}^2}e}\\ \multicolumn{1}{c}{-{\beta }_2\frac{d^3}{d{t}^3}e-{\beta }_1\frac{d^4}{d{t}^4}e+\frac{b_5{u}_1}{L_m}=f\left(\hat{x}\right)}\\ \multicolumn{1}{c}{+f\left(T_d\right)+f\left(e\right)+f\left(u_1\right)}\end{array}$$

(42)

where

$a_1=\frac{B_c{K}_c}{I_c^2}$ $b_1=\frac{a_4{K}_c}{I_{\mathrm{eq}}G}-\frac{a_2{K}_c}{I_c}$ $c_1=\frac{b_4{K}_c}{I_{\mathrm{eq}}{G}_m}-\frac{b_2{K}_c}{I_c}$

$a_2=-\frac{K_c}{I_c}+\frac{B_c^2}{I_c^2}$ $b_2=a_1-\frac{a_2{B}_c}{I_c}$ $c_2=b_1-\frac{b_2{B}_c}{I_c}$

$a_3=-\frac{B_c{K}_c}{I_c^2{G}_m}$ $b_3=\frac{a_2{K}_c}{I_c{G}_m}-\frac{a_4(K_c+K_r{r}_p^2)}{I_{\mathrm{eq}}{G}_m^2}$ $c_3=\frac{b_2{K}_c}{I_{c}N}-\frac{b_4(K_c+K_r{r}_p^2)}{I_{\mathrm{eq}}{G}_m^2}$

$a_4=\frac{K_c}{I_c{G}_m}$ $b_4=a_3-\frac{a_4{B}_{\mathrm{eq}}}{I_{\mathrm{eq}}}$ $c_4=b_3-\frac{b_4{B}_{\mathrm{eq}}}{I_{\mathrm{eq}}}-\frac{b_5{K}_t}{L_m}$

$b_5=\frac{a_4{K}_t}{I_{\mathrm{eq}}}$ $c_5=\frac{b_4{K}_t}{I_{\mathrm{eq}}}-\frac{b_5{R}_m}{L_m}$

$b_6=-\frac{a_4}{I_{\mathrm{eq}}{G}_m}$ $c_6=-\frac{b_4}{I_{\mathrm{eq}}{G}_m}$

The systematic error between the controlled object (steering column angle) and its reference value (x_{1_ref}) is denoted as e₁, according to (43).

$$e_1=x_{1\_\mathrm{ref}}-{\hat{x}}_1$$

(43)

The nth derivative of e₁ is described according to (44).

$$\frac{d^n}{d{t}^n}{e}_1=\frac{d^n}{d{t}^n}{x}_{1\_\mathrm{ref}}-\frac{d^n}{d{t}^n}{\hat{x}}_1$$

(44)

A linear sliding surface (σ) is selected according to (45).

$$\sigma =\sum _{i=0}^4{k}_i\frac{d^{4-i}}{d{t}^{4-i}}{e}_1$$

(45)

The coefficients k _i are chosen from (46) such that k₀ = 1 and F(s) is a Hurwitz polynomial.

$$F\left(s\right)=\sum _{i=0}^4{k}_i{s}^{4-i}$$

(46)

Taking the derivative of the sliding surface, we get (47).

$$\frac{d}{\mathrm{dt}}\sigma =\frac{d^5}{d{t}^5}{x}_{1\_\mathrm{ref}}-\frac{d^5}{d{t}^5}{\hat{x}}_1+\sum _{i=1}^4{k}_i\frac{d^{5-i}}{d{t}^{5-i}}{e}_1$$

(47)

Equation (48) is established by substituting (42) into (47).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}\sigma =\frac{d^5}{d{t}^5}{x}_{1\_\mathrm{ref}}-[f\left(\hat{x}\right)+f\left(T_d\right)+f\left(e_r\right)+f\left(u_1\right)]}\\ \multicolumn{1}{c}{+\sum _{i=1}^4{k}_i\frac{d^{5-i}}{d{t}^{5-i}}{e}_1}\end{array}$$

(48)

The sliding mode control law will be selected in the following subsections once the control mechanism design process is complete.

Backstepping control design

While the SMC technique controls the steering column angle, the steering motor angle is controlled by the BSC technique. Equation (49) describes the systematic error (e₂) between the controlled object and its reference value.

$$e_2=x_3-x_{3\_\mathrm{ref}}{\to }^{e\to 0}{\hat{x}}_3-x_{3\_\mathrm{ref}}$$

(49)

Equation (50) is obtained by taking the derivative of (49).

$$\frac{d}{\mathrm{dt}}{e}_2={\hat{x}}_4-{\beta }_{3}e-\frac{d}{\mathrm{dt}}{x}_{3\_\mathrm{ref}}$$

(50)

Let e₄ and e₅ be the virtual errors of the system, which are described by (51) and (52), respectively.

$$e_3={\hat{x}}_4-{\lambda }_1$$

(51)

$$e_4={\hat{x}}_5-{\lambda }_2$$

(52)

The first virtual control variable (λ₁) is selected according to (53).

$${\lambda }_1=-d_1{e}_2+\frac{d}{\mathrm{dt}}{x}_{3\_\mathrm{ref}}+{\beta }_{3}e$$

(53)

Substituting equations (50) and (53) into (51), we get (54). The results (54) show that selecting the first virtual control variable is reasonable.

$$\begin{array}{c}\multicolumn{1}{c}{e_3={\hat{x}}_4-(-d_1{e}_2+\frac{d}{\mathrm{dt}}{x}_{3\_\mathrm{ref}}+{\beta }_{3}e)=\frac{d}{\mathrm{dt}}{e}_2}\\ \multicolumn{1}{c}{+d_1{e}_2{\to }^{e_2\to 0}\frac{d}{\mathrm{dt}}{e}_2}\end{array}$$

(54)

Equation (55) is obtained by combining equations (50), (51), and (53).

$$\frac{d}{\mathrm{dt}}{e}_2=e_3-d_1{e}_2$$

(55)

Taking the derivative of (51), we get (56).

$$\frac{d}{\mathrm{dt}}{e}_3=\frac{d}{\mathrm{dt}}{\hat{x}}_4-\frac{d}{\mathrm{dt}}{\lambda }_1$$

(56)

Similarly, equation (57) is obtained by taking the derivative of (53).

$$\frac{d}{\mathrm{dt}}{\lambda }_1=-d_1\frac{d}{\mathrm{dt}}{e}_2+\frac{d^2}{d{t}^2}{x}_{3\_\mathrm{ref}}+{\beta }_3\frac{d}{\mathrm{dt}}e$$

(57)

Equation (58) is formed by substituting equations (55) and (57) into (56).

$$\frac{d}{\mathrm{dt}}{e}_3=\frac{d}{\mathrm{dt}}{\hat{x}}_4+d_1(e_3-d_1{e}_2)-\frac{d^2}{d{t}^2}{x}_{3\_\mathrm{ref}}-{\beta }_3\frac{d}{\mathrm{dt}}e$$

(58)

Substituting equations (35), (49), (51), and (53) into (58), we obtain (59). The symbol d₂ in equation (59) is described according to (60).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}{e}_3=\frac{K_c}{I_{\mathrm{eq}}{G}_m}{\hat{x}}_1+(d_1\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}-\frac{K_c+K_r{r}_p^2}{I_{\mathrm{eq}}{G}_m^2}){\hat{x}}_3}\\ \multicolumn{1}{c}{+\frac{K_t}{I_{\mathrm{eq}}}{\hat{x}}_5-\frac{1}{I_{\mathrm{eq}}{G}_m}{\hat{x}}_6}\\ \multicolumn{1}{c}{-(d_1\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}{x}_{3\mathrm{ref}}+\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}\frac{d}{\mathrm{dt}}{x}_{3\_\mathrm{ref}}+\frac{d^2}{d{t}^2}{x}_{3\_\mathrm{ref}})}\\ \multicolumn{1}{c}{-(\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}{\beta }_{3}e+{\beta }_{4}e+{\beta }_3\frac{d}{\mathrm{dt}}e+d_1^2{e}_2)}\\ \multicolumn{1}{c}{-d_2{e}_3=f_1\left(\hat{x}\right)-f_1\left(e\right)-d_2{e}_3}\end{array}$$

(59)

$$d_2=\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}-d_1$$

(60)

Taking the derivative of e₄, we get (61).

$$\frac{d}{\mathrm{dt}}{e}_4=\frac{d}{\mathrm{dt}}{\hat{x}}_5-\frac{d}{\mathrm{dt}}{\lambda }_2$$

(61)

The second virtual control variable (λ₂) is selected according to (62), where K₁ is a proportional coefficient.

$${\lambda }_2=K_1{x}_{3\_\mathrm{ref}}$$

(62)

Equation (63) is formed based on the combination of equations (36), (52), (61), and (62).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}{e}_4=-\frac{K_t}{L_m}{\hat{x}}_4-\frac{K_1{R}_m}{L_m}{x}_{3\_\mathrm{ref}}-K_1\frac{d}{\mathrm{dt}}{x}_{3\_\mathrm{ref}}+\frac{1}{L_m}{u}_2}\\ \multicolumn{1}{c}{-\frac{R_m}{L_m}{e}_4-{\beta }_{5}e=f_2\left(\hat{x}\right)+\frac{1}{L_m}}\\ \multicolumn{1}{c}{u_2-\frac{R_m}{L_m}{e}_4-{\beta }_{5}e}\end{array}$$

(63)

Stability proof based on control law selection

A Lyapunov function V(x) proposed in (64) satisfies the positive definiteness condition.

$$V\left(x\right)=\frac{1}{2}({\sigma }^2+\sum _{i=2}^4{e}_i^2)>0$$

(64)

The derivative of V(x) is described according to (65).

$$\frac{d}{\mathrm{dt}}V\left(x\right)=\sigma \frac{d}{\mathrm{dt}}\sigma +\sum _{i=2}^4{e}_i\frac{d}{\mathrm{dt}}{e}_i$$

(65)

Equation (66) is obtained based on the combination of equations (48), (55), (59), (63), and (65), where the symbol d₃ is defined by (67).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}V\left(x\right)=\sigma }\\ \multicolumn{1}{c}{[\frac{d^5}{d{t}^5}{x}_{1\_\mathrm{ref}}-f\left(\hat{x}\right)-f\left(T_d\right)-f\left(e\right)+\sum _{i=1}^4{k}_i\frac{d^{5-i}}{d{t}^{5-i}}{e}_1]}\\ \multicolumn{1}{c}{-e_3{f}_1\left(e\right)-{\beta }_5{e}_{4}e+[e_2{e}_3+e_3{f}_1\left(\hat{x}\right)+e_4{f}_2\left(\hat{x}\right)]}\\ \multicolumn{1}{c}{-\sum _{i=1}^3{d}_i{e}_{i+1}^2+u}\end{array}$$

(66)

$$d_3=\frac{R_m}{L_m}$$

(67)

The combination control signal u is selected according to (68), which includes both u₁ and u₂.

$$u=-\sigma \frac{b_5}{L_m}{u}_1+\frac{e_4}{L_m}{u}_2+e_3{f}_1\left(e\right)+{\beta }_5{e}_{4}e$$

(68)

The component control signal of sliding mode control is selected according to (69). A selection of backstepping control signal is mentioned in (70). This proposal aims to make the system stable; that is, the derivative of V(x) is negative definite.

$$\begin{array}{c}\multicolumn{1}{c}{u_1=-\frac{L_m}{b_5}[f\left(\hat{x}\right)+f\left(T_d\right)+f\left(e\right)-\sum _{i=1}^4{k}_i\frac{d^{5-i}}{d{t}^{5-i}}{e}_1}\\ \multicolumn{1}{c}{-\frac{d^5}{d{t}^5}{x}_{1\_\mathrm{ref}}-K_2\mathrm{sat}\left(\sigma \right)]}\end{array}$$

(69)

$$u_2=-L_m[\frac{e_3(e_2+f_1\left(\hat{x}\right))}{e_4}+f_2\left(\hat{x}\right)]$$

(70)

Equation (66) is rewritten as (71). This result is achieved by combining equations (66), (68), (69), and (70). The derivative of the Lyapunov function is negative-definite when the coefficients K₂ and d₁ are chosen according to (72). In conclusion, the system is considered stable according to the Lyapunov criterion.

$$\frac{d}{\mathrm{dt}}V\left(x\right)=-\sigma K_2\mathrm{sat}\left(\sigma \right)-\sum _{i=1}^3{d}_i{e}_{i+1}^2<0$$

(71)

$$\begin{array}{cc}0<K_2\hfill & 0<d_1<\frac{B_{\mathrm{eq}}}{I_{\mathrm{eq}}}\hfill \end{array}$$

(72)

In practice, model uncertainties such as parameter variations (including the change of vehicle mass, stiffness or damping coefficients, and others) are inevitable. To assess the robustness of the proposed control strategy, consider the Lyapunov function V(x), whose time derivative was shown to be strictly negative under nominal conditions.

Suppose minor bounded uncertainties are introduced into the system dynamics. In that case, the structure of (71) may be perturbed, but as long as the uncertain terms remain sufficiently small and the controller dominates their effects, (71) can still be rendered negative definite or negative semi-definite. This means that even if there are minor uncertainties or disturbances, the system states will not diverge. They will stay bound and continue to operate near the desired state.

The ESO proposed in equations (32)–(37) is rewritten in a general form as in (73), where A is the state matrix, B is input matrix, C is output matrix, and β is observer gain matrix.

$$\frac{d}{\mathrm{dt}}\hat{x}=A\hat{x}+\mathrm{Bu}\left(t\right)+\beta \mathrm{Ce}$$

(73)

The observed error (e) is defined according to (74).

$$e=x-\hat{x}$$

(74)

Taking the derivative of (74), we get (75).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}e=\frac{d}{\mathrm{dt}}x-\frac{d}{\mathrm{dt}}\hat{x}=\mathrm{Ax}+\mathrm{Bu}\left(t\right)}\\ \multicolumn{1}{c}{-[A\hat{x}+\mathrm{Bu}\left(t\right)+\beta \mathrm{Ce}]=(A-\beta C)e}\end{array}$$

(75)

A candidate Lyapunov function representing the ESO is proposed according to (76), where P is a positive definite matrix.

$$V_E\left(e\right)=\frac{1}{2}{e}^T\mathrm{Pe}$$

(76)

Taking the derivative of (76), we get (77).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d}{\mathrm{dt}}{V}_E\left(e\right)={\left[(A-\beta C)e\right]}^T\mathrm{Pe}+e^{T}P(A-\beta C)e}\\ \multicolumn{1}{c}{=e^T[{(A-\beta C)}^{T}P+P(A-\beta C)]e}\end{array}$$

(77)

The coefficients of the observed gain matrix are selected by the pole placement method to ensure that the eigenvalue of ( A –β C ) lies on the left side of the complex plane, satisfying the Hurwitz stability condition. As a result, a matrix P exists such that (77) is negative definite. The required matrix P is the solution of (78), where Q is a positive definite matrix.

$${(A-\beta C)}^{T}P+P(A-\beta C)=-Q$$

(78)

In conclusion, the proposed ESO is stable when the gains of the observed matrix are selected by the pole placement method.

A flowchart, which describes the system’s control mechanism, is illustrated in Figure 3. The ideal signals are referenced from the ideal model (the ideal assigned torque controls the ideal model). These ideal signals are used as inputs to the BSC and SMC controllers. The input of the backstepping control is adjusted through a PID controller. Finally, the SMC and BSC output signals are synthesized to become the ultimate control signal, which is used to control the electric motor. In addition, ESO is utilized to estimate the changes in the state variables instead of measuring them directly by sensors.

Figure 3.

Control scheme.

Using a conventional BSC often introduces a phase delay, which can result in significant system errors, as noted in. ²⁷ To address this issue, a PID controller is employed to recalibrate the input signal of the BSC, thereby mitigating the phase delay effect. The PID controller generates the reference steering motor angle, denoted as x_{3_ref}, which serves as the input to the BSC. The mathematical formulation of the PID controller is presented in equation (79), where k _p , k _i , and k _d represent the proportional, integral, and derivative gains, respectively. These parameters are optimized using an iterative algorithm inspired by natural genetic processes to minimize the root mean square error.

$$x_{3\_\mathrm{ref}}=k_p{e}_5+k_i\int e_5\mathrm{dt}+k_d\frac{d}{\mathrm{dt}}{e}_5$$

(79)

The input to the PID controller is the systematic error (e₅), defined as the difference between the actual steering motor angle (x₃) and the ideal angle (x_{3_ideal}), as expressed in equation (80). Integrating the PID controller into the proposed control framework demonstrates the potential for effectively compensating phase lag and reducing tracking errors. This improvement will be validated through simulation results presented in the following section.

$$e_5=x_{3\_\mathrm{ideal}}-x_3$$

(80)

Simulation results

Simulation conditions

Simulation conditions are covered in this subsection. Driver torque is the problem’s input, while the state variable (x _i ) is the output. The change in driver torque over time is illustrated in Figure 4(a). According to this description, two driver torque signals correspond to two investigated cases: J-turn (step) for the first case and sine wave for the second case. Figure 4(b) provides information about external disturbances (T _ed ) mentioned in (6). This type of random nonlinear disturbance describes the influence of factors on the system. The system specifications are listed in Table 1.

Figure 4.

Simulation inputs. (a) Driver torque. (b) External disturbances.

Table 1.

The system specifications.

Symbol	Unit	Value	Symbol	Unit	Value
I c	kgm2	0.075	R m	Ω	0.42
B c	Nms/rad	0.072	L m	H	0.005
K c	Nm/rad	113	I m	kgm2	0.0005
G m	-	18.5	B m	Nms/rad	0.0045
K t	Nm/A	0.06	M r	kg	32
r p	m	0.006	G s	-	21
C α	N/rad	35,500	I ψ	kgm2	4050
B r	Ns/m	3970	l i	m	1.08/1.62
Iφ	kgm2	606	K	N/m	39,000/38,200
m s	kg	1700	C	Ns/m	3350/3200
I θ	kgm2	4192	T	N/m	180,000
m u	kg	40	d c	m	0.03
d k	m	0.04	d n	m	0.28

Two conditions are examined in each case: v₁ = 20 km/h and v₂ = 60 km/h. This is to evaluate the algorithm’s response to different motion conditions.

Result and discussion

The simulation results are presented in two specific cases. The results obtained from the proposed control will be compared with those obtained from other algorithms, including PID-GA, BSC, SMC, and switch ADRC (sADRC). There are six subplots listed in each Figure below, including (a) Column angle, (b) Motor angle, (c) Column speed, (d) Motor speed, (e) Motor current, and (f) Assisted torque.

Since the torque driver in the first case is the J-turn signal (similar to a step signal), the output values will be compared regarding Maximum Error (ME), overshoot, and settling time. The tracking error for the periodically varying signal in the second case (the sine wave signal) is evaluated in terms of ME, Root Mean Square Error (RMSE), Integral Absolute Error (IAE), and Integral Square Error (ISE). In addition, the motor current’s Total Harmonic Distortion (THD) is also considered in this case.

The first case (J-turn steering)

V₁ = 20 km/h

The change in state variables (outputs) over time is illustrated in Figure 5. These results are achieved when steering at a speed of v₁ = 20 km/h. Figure 5(a) shows the change in column angle over 10 s. According to this description, the column angle signals increase abruptly to a peak within a short period, consistent with the proposed steering law. The ME of the proposed control is the smallest (0.002 rad), while the values obtained from sADRC, SMC, BSC, and PID-GA are 0.285, 0.203, 1.204, and 0.453 rad, respectively. Looking at these subplots more closely, it is clear that the overshoot of BSC is the largest, reaching 14.95%, while the overshoot of SMC is only 3.84%. A significant improvement in results is observed as the overshoot of the proposed control and PID-GA is only about 1.70%, slightly higher than that of sADRC. According to the simulation results, the settling time of the proposed control is only 6.73 s, lower than that of other controllers (PID-GA: 7.07 s, BSC: 7.60 s, sADRC: 9.55 s), except for SMC (5.20 s).

Figure 5.

Simulation results (1st case, v₁ = 75 km/h): (a) Column angle; (b) Motor angle; (c) Column speed; (d) Motor speed; (e) Motor current; and (f) Assisted torque.

Figure 5(b) shows the variation in motor angle over time. The most significant tracking error is for BSC (up to 8.319 rad). The errors of the PID-GA and sADRC controllers are lower, at 8.319 and 5.175 rad, respectively. If the SMC mechanism replaces the above techniques, the ME is reduced to 3.732 rad. On the contrary, the accuracy of the results is ensured once the proposed control is applied instead of the existing control methods, which reduces the ME to 0.038 rad. The overshoot in the motor angle is similar to that in the column angle, with negligible error. The same is true for settling time. These results are listed in Table 2 below.

Table 2.

Simulation results with J-turn steering (v₁ = 20 km/h).

Objects	PID-GA	BSC	SMC	sADRC	Proposed control
Column angle
ME (rad)	0.453	1.204	0.203	0.285	0.002
Overshooting (%)	1.7	14.95	3.84	0.88	1.71
Settling time (s)	7.07	7.6	5.2	9.55	6.73
Column speed
ME (rad/s)	1.582	2.190	0.347	1.033	0.043
Motor angle
ME (rad)	8.319	22.269	3.732	5.175	0.038
Overshooting (%)	1.69	15.06	3.83	0.87	1.71
Settling time (s)	7.07	7.63	5.22	9.55	6.73
Motor speed
ME (rad/s)	28.812	39.692	6.105	18.475	0.603
Motor current
ME (A)	4.894	5.917	2.813	4.611	2.809

The external disturbance strongly affects the column speed signals (Figure 5(c)). In this study, the highest tracking error for column speed using the new control method is just 0.043 rad/s, which is significantly less than the errors from PID-GA (1.582 rad/s), BSC (2.190 rad/s), SMC (0.347 rad/s), and sADRC (1.033 rad/s). Similarly, the ME in the motor speed signal obtained from the proposed control is much smaller than that of existing control methods (Figure 5(d)). Looking closely at the window plot in Figure 5(c), one can see that the chattering phenomenon still exists when applying the single SMC technique to control the system. In contrast, this phenomenon does not happen for the proposed control.

The actuator’s performance is evaluated by the variation in the motor current (Figure 5(e)). The ME of motor current is small when the proposed control is applied to control the system. In contrast, the tracking error of other control techniques is significant. The chattering effect still exists when the classical SMC technique controls the system. The relationship between hand torque (driver torque) and assisted torque is shown in Figure 5(f). Generally, the signal obtained by the proposed control closely follows the reference signal, while the errors of PID-GA, BSC, and sADRC are significant.

V₂ = 60 km/h

As the speed increases, the assisted performance will decrease. Therefore, investigating a high speed (v₂ = 60 km/h) is necessary.

The changes in the output state variables are illustrated by the subplots in Figure 6. In general, their values decrease because of the degradation in the assisted power performance when steering at high speeds. The maximum tracking error of column and motor angles is the smallest when the system is controlled by the proposed control algorithm (Figure 6(a) and (b)). In addition, the overshoot and settling time are also significantly improved compared with the existing control methods. Finally, the influence of chattering is largely eliminated compared with the conventional SMC mechanism (Figure 6(e)). The specific simulation results are listed in Table 3 (rounded).

Figure 6.

Simulation results (1st case, v₂ = 60 km/h): (a) Column angle; (b) Motor angle; (c) Column speed; (d) Motor speed; (e) Motor current; (f) Assisted torque.

Table 3.

Simulation results with J-turn steering (v₂ = 60 km/h).

Objects	PID-GA	BSC	SMC	sADRC	Proposed control
Column angle
ME (rad)	0.210	0.589	0.023	0.119	0.002
Overshooting (%)	0.23	3.78	0.71	1.25	0.04
Settling time (s)	9.68	9.5	7.69	9.66	8.36
Column speed
ME (rad/s)	0.692	0.725	0.128	0.898	0.043
Motor angle
ME (rad)	3.849	10.875	0.418	2.133	0.035
Overshooting (%)	0.29	3.76	0.73	1.14	0.06
Settling time (s)	9.68	9.5	7.7	9.67	8.37
Motor speed
ME (rad/s)	12.396	13.388	2.097	16.043	0.603
Motor current
ME (A)	3.752	1.860	2.823	3.295	2.816

The second case (sine wave steering)

A sinusoidally varying periodic steering is shown in the second case (Figure 4(a)).

V₁ = 20 km/h

The variation in the state variables in this case is illustrated in Figure 7. The column angle signals vary periodically according to the proposed law (Figure 7(a)). The maximum tracking error of column angle is 1.373 rad (PID-GA), 4.426 rad (BSC), 0.344 (SMC), 0.658 (sADRC), and 0.003 rad (proposed control), respectively. For signals that vary continuously over time, the RMSE value is considered to evaluate the accuracy of the results over a specific period. According to the article’s findings, the RMSE of the proposed control is only 0.040 rad, while the figures belonging to PID-GA, and BSC can be up to 19.783 and 59.704 rad.

Figure 7.

Simulation results (2nd case, v₁ = 20 km/h): (a) Column angle; (b) Motor angle; (c) Column speed; (d) Motor speed; (e) Motor current; and (f) Assisted torque.

In this case, IAE and ISE are used to evaluate the performance of the controllers instead of overshooting the index and settling time. The simulation results indicate that the largest IAE belongs to BSC (58.662%). This value is slightly reduced to 19.903% when utilizing PID-GA to control the system. Compared with the two control methods above, the IAE of sADRC and SMC are 6.629% and 2.794%, respectively, much higher than the results obtained from the proposed control (0.040%).

The trend of the motor angle tracking error (Figure 7(b)) is similar to that of the column angle (Figure 7(a)), but their values are more significant. The proposed control’s maximum tracking error is 0.051 rad, and its RMSE and IAE are 0.040%, much lower than other methods.

The ISE value of the proposed control is only about 0.01%, while the figures for other controllers are significantly more prominent (Figure 7(c)). Overall, the error in motor speed signals is the smallest when the EPS system is controlled by the control algorithm proposed in this work (Figure 7(d)).

The results in Figure 7(e) indicate that the motor current experiences chattering when the EPS system uses traditional SMC, leading to an increase in RMSE to 14.962 A, significantly higher than the proposed control’s RMSE of 9.929 A. In this case, the THD index is considered to evaluate the control mechanisms’ performance comprehensively. According to the calculation results, the largest THD belongs to sADRC (47.630%), while the values obtained from SMC and the proposed control are smaller (35.400% and 33.840%, respectively). The main reason for the increase in THD value is the saturation state in the assisted torque (Figure 7(f)), which is reached when the driver torque exceeds 7 Nm. The simulation results under this condition are fully listed in Table 4.

Table 4.

Simulation results with sine wave steering (v₁ = 20 km/h).

Objects	PID-GA	BSC	SMC	sADRC	Proposedcontrol
Column angle
ME (rad)	1.373	4.426	0.344	0.658	0.003
RMSE (rad)	19.783	59.704	3.374	7.122	0.040
IAE (%)	19.903	58.662	2.794	6.629	0.040
ISE (%)	3.913	35.633	0.114	0.507	0.000
Column speed
ME (rad)	3.895	7.310	0.788	3.608	0.070
RMSE (rad)	22.829	61.566	3.629	19.960	0.258
IAE (%)	23.141	59.831	3.175	18.578	0.230
ISE (%)	5.215	37.918	0.132	3.979	0.001
Motor angle
ME (rad)	25.142	81.693	6.300	12.053	0.051
RMSE (rad)	19.882	60.057	3.393	7.002	0.040
IAE (%)	20.012	59.034	2.811	6.523	0.040
ISE (%)	3.952	36.055	0.115	0.490	0.000
Motor speed
ME (rad)	71.341	135.482	12.743	64.430	0.866
RMSE (rad)	22.839	61.931	3.367	19.303	0.214
IAE (%)	23.127	60.243	2.995	18.043	0.191
ISE (%)	5.219	38.370	0.113	3.721	0.000
Motor current
ME (rad)	7.354	13.917	4.270	8.712	2.788
RMSE (rad)	34.415	89.160	14.962	37.179	9.929
IAE (%)	32.170	92.625	13.599	35.834	8.286
ISE (%)	11.852	79.486	2.241	13.824	0.987
THD (%)	42.170	26.090	35.400	47.630	33.840

V₂ = 60 km/h

A final investigation is conducted at a higher speed of v₂ = 60 km/h. Observations from the subplots in Figure 8 indicate that the output values decline sharply as speed increases. The tracking error observed under these conditions is slightly reduced compared to the previously examined speed (v₁ = 20 km/h). The proposed algorithm demonstrates stable system control across various operating conditions, even when external disturbances significantly influence the system. Consequently, the outputs closely follow the reference values with minimal errors (as shown in Table 5).

Figure 8.

Simulation results (2nd case, v₁ = 25 km/h): (a) Column angle; (b) Motor angle; (c) Column speed; (d) Motor speed; (e) Motor current; and (f) Assisted torque.

Table 5.

Simulation results with sine wave steering (v₂ = 60 km/h).

Objects	PID-GA	BSC	SMC	sADRC	Proposedcontrol
Column angle
ME (rad)	0.698	1.669	0.206	0.405	0.002
RMSE (rad)	16.932	35.284	3.178	7.691	0.042
IAE (%)	17.289	34.125	2.388	7.231	0.040
ISE (%)	2.867	12.437	0.101	0.591	0.000
Column speed
ME (rad)	1.851	2.853	0.509	2.646	0.054
RMSE (rad)	18.898	35.814	3.348	26.292	0.367
IAE (%)	17.842	36.194	2.888	25.186	0.337
ISE (%)	3.573	12.838	0.112	6.919	0.001
Motor angle
ME (rad)	12.786	30.803	3.813	7.230	0.031
RMSE (rad)	17.130	35.728	3.217	7.516	0.041
IAE (%)	17.498	34.571	2.418	7.084	0.040
ISE (%)	2.934	12.752	0.103	0.565	0.000
Motor speed
ME (rad)	34.266	52.000	8.657	47.683	0.825
RMSE (rad)	19.048	36.231	3.107	25.517	0.299
IAE (%)	18.008	36.619	2.708	24.464	0.275
ISE (%)	3.630	13.139	0.097	6.517	0.001
Motor current
ME (rad)	5.102	4.516	3.406	6.829	2.777
RMSE (rad)	47.936	83.427	30.970	80.062	25.646
IAE (%)	42.380	85.620	27.721	76.529	21.396
ISE (%)	22.999	69.610	9.600	64.136	6.583
THD (%)	46.220	29.230	39.380	87.370	36.150

Figure 8(e) shows that the motor current obtained from sADRC has a significant error, while the signal obtained from traditional SMC is affected by chattering, which increases the tracking error. Under these investigated conditions, the system’s stability is still guaranteed when the proposed mechanism is utilized to control the system.

Observed disturbances

This work uses ESO to estimate system disturbances instead of measuring them with sensors. System disturbances are considered an extended state variable. Figure 9(a) provides information about the estimated system disturbance in the first case, while Figure 9(b) describes the results obtained in the second case. Looking at this Figure more closely, we can see that the estimated value closely follows the actual value with negligible error. This is true even when the vehicle speed and driver torque change.

Figure 9.

Observed disturbances. (a) J-turn steering (1st case). (b) Sine wave steering (2nd case).

The calculation results show that when steering in the second case, the RMSE between the estimated and actual values is only 0.1176% (v₁) and 0.1322% (v₂). The maximum tracking errors for the remaining cases are 0.4612% and 0.4612%, respectively.

Some comments are made based on the results obtained, as follows:

• + The output signals tend to change according to the law of steering torque. These signals are affected by external disturbances.

• + When driver torque increases (vehicle speed does not change), the value of state variables also increases. The state variable values also increase when the speed decreases (driver torque remains constant). This results from the increased assisted torque, depicted in Figure 2. However, assisted torque will reach saturation when the driver torque exceeds its limit, that is, T _a  = T_{a_max} when T _d  > T_{d_max}.

• + System errors are large when the system is controlled by traditional techniques. The tracking error will decrease sharply once the proposed algorithm replaces these techniques. This is true under the conditions investigated above.

• + The power consumption efficiency of the proposed controller is significantly improved compared to other conventional controllers.

• + The phase delay phenomenon is almost eliminated when the BSC technique is combined with SMC and PID to become an integrated nonlinear technique. In addition, the chattering phenomenon does not occur even when the integrated controller is formed based on the SMC technique.

• + The signals estimated by the ESO are highly accurate. As a result, the output signals obtained from the proposed algorithm follow the reference value with negligible error.

The ESO proposed in this work offers functionality equivalent to the observer introduced in Ref. ²² but with a simpler mathematical formulation. This simplification facilitates more straightforward implementation and integration into the control framework without compromising estimation accuracy. Moreover, the proposed ESO exhibits lower estimation error than the lead-lag control-based observer presented in Ref. ²³ In addition, its structure is simpler than the Kalman filter and delivers better performance than the Luenberger observer. A comparison between ESO and other observers is presented in Table 6.

Table 6.

Observers comparison.

Criteria	Extended state observer (ESO)	Kalman filter	Luenberger observer
Primary Objective	Estimates both system states and unknown disturbances	Provides optimal state estimation under stochastic noise	Estimates internal states based on output measurements
Typical Applications	Linear or nonlinear systems with uncertainties	Navigation, tracking, signal processing, systems with Gaussian noise	Linear control systems, state feedback implementation
Model Requirements	Operates effectively with inaccurate or incomplete models	Requires an accurate model and known noise statistics	Requires a known linear or linearized model
Disturbance Handling	Excellent—explicitly estimates and compensates for unknown disturbances	Good—assuming disturbances are modeled as stochastic processes	Limited—does not inherently account for unknown disturbances
Computational Complexity	Moderate to high, depending on system order	High	Low

The simulation results demonstrate that the proposed algorithm provides superior control performance to conventional control methods under specific conditions. Specifically, applying the proposed control strategy reduces the phase delay observed in the system output, in contrast to the traditional PID and BSC controllers, as referenced in Refs.^9,27 Moreover, the overshoot phenomenon is notably mitigated, highlighting a key advantage of the proposed approach over those presented in Refs.^10–12 Regarding energy efficiency, the proposed controller exhibits lower energy consumption than the methods described in Refs.^38,39 Although the control law is derived from the SMC framework, the proposed algorithm effectively suppresses the chattering phenomenon, which remains a limitation in conventional SMC, and ADRC approaches.^8,17,19,30 Lastly, the proposed use of the Extended State Observer (ESO) to estimate system states significantly reduces the impact of sensor noise in MIMO systems. This issue persists in the Linear Quadratic Tracking (LQT) technique discussed in Ref. ³⁵

Conclusion

A nonlinear integrated control algorithm is introduced in this work. This algorithm is formed by combining BSC and SMC techniques to eliminate the influence of chattering phenomena. At the same time, the phase delay problem is solved by adjusting the input signal based on the PID technique. State variables are estimated by the ESO instead of directly measured by sensors.

According to the research findings, the values of the state variables closely follow the desired values with negligible errors if and only if the proposed algorithm controls the EPS system. Chattering and phase lag are almost eliminated, even when moving conditions change. In addition, power consumption is significantly improved when the proposed technique is applied instead of traditional controllers such as PID or BSC. Finally, the signals estimated by the ESO (including state variables and disturbances) are equivalent to the actual values with slight deviations.

Although the proposed algorithm provides outstanding performance in controlling the EPS system, some drawbacks still exist. Firstly, column and motor speed signals and the motor current signal are affected by external disturbances. Secondly, the motor current value error is still quite large when driving high speed. Thirdly, some equations are described as rough approximations rather than as exact. These issues will be addressed in future studies. Additionally, conducting experiments is necessary to demonstrate the effectiveness of the proposed algorithm.

In this work, the proposed sliding surface is only linear in form to make the calculation simpler. As mentioned in Ref.,^45–48 the advanced SMC mechanism may enhance the efficiency of controlling the EPS system through fixed-time and predefined-time approaches. This idea should be considered and developed in future works.