An electronically controlled hydraulic power steering system for heavy vehicles

Abstract

To remedy the existing problems concerning vehicle handling for the heavy vehicle hydraulic power steering system, this article investigates a type of electronically controlled hydraulic power steering system, which can realize variable assistance characteristics and improve high-speed steering feel. To this end, a controlled electrohydraulic proportional valve is mounted in parallel with the rotary valve to regulate the flow rate through the rotary valve on demand according to vehicle speed and steering rate. After a system description in terms of structure and working principle, mathematical modeling of the electronically controlled hydraulic power steering system, vehicle, and steering resistance is conducted. To realize accurate tracking of valve spool displacement under parameter uncertainty and external disturbance, an adaptive dynamic surface control scheme is proposed for the electrohydraulic proportional valve. Numerical simulations and field tests are performed to validate the effectiveness and applicability of the electronically controlled hydraulic power steering system. The results show that the electronically controlled hydraulic power steering system can significantly improve high-speed steering feel compared with the hydraulic power steering system while maintaining the similar performance of low-speed steering portability.

Keywords

Mathematical modeling controller design power steering system heavy vehicle

Introduction

Nowadays, almost every modern vehicle uses a power steering system to help the driver steer. The most popular power steering systems in the automotive market can be divided into three types, namely, hydraulic power steering (HPS) system, electrohydraulic power steering (EHPS) system, and electric power steering (EPS) system.^1–6 The HPS system was developed by American companies around the 1950s and has achieved a high level of acceptance during the last six decades due to its high power density and reliability. To date, heavy vehicles like buses and trucks have been widely equipped with the recirculating ball-type HPS system.⁷ Nevertheless, there exist some problems concerning vehicle handling for the HPS system, which comprises a constant flow pump driven by the engine, an open-center rotary valve, and an assistance cylinder. In addition to the valve geometry, the cylinder pressure for generating steering assistance is positively correlated to the flow rate through the rotary valve that is nearly equal to the pump flow rate. Since the engine-driven pump outputs constant flow rate, the assistance performance cannot be changed.⁸ Moreover, to ensure low-speed steering portability, the HPS system is dimensioned for the parking situation where the required steering assistance is the highest considering the greatest resistance.⁹ As a result, it reduces the driver’s steering feel during high-speed driving when the need for steering assistance is comparatively low.

With the development of mechatronics technology, the EHPS and EPS systems, which were developed by Japanese companies around the 1980s, have been incorporated in many passenger cars in recent years. In the EHPS system, the pump is driven by a controlled electric motor to provide the flow rate on demand according to vehicle speed and steering rate, which makes it possible to realize variable steering assistance.¹⁰ However, when it comes to heavy vehicles, the EHPS system may be problematic for application because the 24 V on-board power supply is insufficient to actuate the electric motor for steering assistance due to the greater axle loads of heavy vehicles compared with passenger cars. In the EPS system, a controlled electric motor is directly mounted on the steering column or the steering gear to provide variable steering assistance, eliminating the hydraulic components and simplifying manufacturing and maintenance.¹¹ Nevertheless, considering the utilization of electric motor, the EPS system has the same limitation as the EHPS system for heavy vehicle application.

To remedy the aforementioned problems for the heavy vehicle HPS system, a type of electronically controlled hydraulic power steering (ECHPS) system was proposed.¹² Based on a slight modification to the recirculating ball-type HPS system, the ECHPS system can realize variable assistance characteristics and improve high-speed steering feel. To this end, a controlled electrohydraulic proportional valve (EHPV) is mounted in parallel with the rotary valve to regulate the flow rate through the rotary valve on demand according to vehicle speed.¹³ Guo et al.¹⁴ researched the parameter optimization of the rotary valve in the ECHPS system to increase the alterable assist effort range. Geng et al.¹⁵ presented the alterable assist characteristics of ECHPS system under different speeds by simulation. However, few of previous studies were verified by experiments. Moreover, the factor of steering rate also influences the demand of the flow rate through the rotary valve, which will be considered in this article.

In general, the comprehensive performance of the ECHPS system is very much dependent on accurate control for the EHPV. Due to inherent uncertainty and strong nonlinearity, EHPVs can be regarded as a class of uncertain nonlinear systems. Moreover, EHPVs will be inevitably affected by disturbances such as friction forces and forces exerted by the fluid in practical applications. During the past 15 years, the dynamic surface control (DSC) technique has proved to be efficient for dealing with large time-varying parameter uncertainty and external disturbances.¹⁶ By now, various adaptive DSC controllers have been developed for a variety of nonlinear systems, such as flexible-joint robots, servo systems, DC-DC buck converter, electrohydraulic actuator, hypersonic flight vehicle, and permanent magnet synchronous motors.^17–22 To the best of our knowledge, the adaptive DSC method has been rarely used for the control of EHPVs yet.

Inspired by previous work, this article presents an analysis of the ECHPS system in terms of modeling, control, and experiment study. First of all, a system description is given. Then, mathematical modeling of the ECHPS system, vehicle, and steering resistance is conducted. Moreover, an adaptive DSC scheme is proposed for the EHPV to realize accurate tracking of valve spool displacement under parameter uncertainty and external disturbance. Finally, numerical simulations and field tests are performed to validate the effectiveness and applicability of the ECHPS system.

System description

The ECHPS system under consideration is illustrated in Figure 1. It is composed of three subsystems: (1) a mechanical steering subsystem comprising a steering wheel, a steering column, a torsion bar, a recirculating ball steering gear, and steering linkages; (2) a hydraulic assistance subsystem composed of a constant flow pump driven by the engine, an open-center rotary valve, and an assistance cylinder; and (3) an electronic control subsystem consisting of an EHPV, an electronic control unit (ECU), and vehicle speed and steering rate sensors.

Figure 1.

ECHPS system.

The rotary valve is mechanically connected to the torsion bar. Moreover, the mechanical part of the recirculating ball steering gear consists of a steering screw, a steering nut, a sector, and recirculating balls. To simplify the structure for manufacturing, the rotary valve and assistance cylinder are integrated into the recirculating ball steering gear to constitute the integral power steering gear, where the steering nut acts as the piston of the assistance cylinder.

It should be noted that the modification to the recirculating ball-type HPS system lies in the electronic control subsystem including the EHPV. The schematic of the EHPV being used is shown in Figure 2. It is made up of a valve body, a valve spool with integrated armature, an amplifier, a spring, an inlet nut, an adjustable bolt, solenoid coils, and magnetic-isolated rings. The EHPV is controlled by the ECU. By varying the input voltage of the amplifier, the current of the solenoid coils is regulated, changing the magnetic force that determines the valve spool displacement, opening area of the EHPV orifice and flow rate through the EHPV. Note that the greater the input voltage of the amplifier, the greater will be the valve spool displacement and flow rate through the EHPV.

Figure 2.

Schematic of EHPV.

When the driver applies a steering wheel torque, the twisting of the torsion bar will activate the rotary valve, creating the assistance cylinder pressure that helps the driver steer. As mentioned previously, the cylinder pressure for generating steering assistance is positively correlated to the flow rate through the rotary valve, and the need for steering assistance decreases with vehicle speed. Therefore, considering the factor of steering rate, the flow rate through the rotary valve should be regulated on demand, which gradually decreases with vehicle speed and increases with steering rate at a certain vehicle velocity since more flow is needed to fill the cylinder. Note that the pump’s constant flow rate is equal to the summation of the flow rate through the rotary valve and EHPV. To ensure on-demand flow rate regulation, the desired valve spool displacement gradually increases with vehicle speed and decreases with steering rate at a certain vehicle velocity. In this article, the pump flow rate is 18 L/min and the maximum valve spool displacement is 2.5 mm. To realize the assistance characteristic curve shown in Figure 3, the desired flow rate through the rotary valve and desired valve spool displacement are listed in Tables 1 and 2, respectively.

Figure 3.

Assistance characteristic curve.

Table 1.

Desired flow rate through rotary valve.

Flow rate through rotary valve (L/min)		Steering rate (°/s)
		0	90	180	270	360	450
Vehicle speed (km/h)	0	18	18	18	18	18	18
	20	14.1	15.5	16.9	18	18	18
	40	11.5	13.2	15	16.9	18	18
	60	9.2	11.1	13.2	15.5	18	18
	80	7.5	9.2	11.5	14.1	16.9	18
	100	7.5	7.5	9.9	12.8	15.9	18

Table 2.

Desired valve spool displacement.

Valve spool displacement (mm)		Steering rate (°/s)
		0	90	180	270	360	450
Vehicle speed (km/h)	0	0	0	0	0	0	0
	20	0.8	0.5	0.2	0	0	0
	40	1.4	1	0.6	0.2	0	0
	60	2	1.5	1	0.5	0	0
	80	2.5	2	1.4	0.8	0.2	0
	100	2.5	2.5	1.8	1.1	0.4	0

Mathematical modeling

ECHPS system

The equations of motion for the mechanical steering subsystem can be described as

J_{s w} {\overset{\cdot\cdot}{θ}}_{s w} + B_{s w} {\dot{θ}}_{s w} + K_{t b} (θ_{s w} - θ_{s s}) = T_{s w}

(1)

J_{s s} {\overset{\cdot\cdot}{θ}}_{s s} + B_{s s} {\dot{θ}}_{s s} + K_{t b} (θ_{s s} - θ_{s w}) = - F L

(2)

M_{sn} {\overset{\cdot\cdot}{x}}_{sn} + B_{sn} {\overset{\cdot}{x}}_{sn} = F + p_{ac} A_{p} - F_{s}

(3)

J_{s} {\overset{\cdot\cdot}{θ}}_{s} + B_{s} {\overset{\cdot}{θ}}_{s} = F_{s} r_{s} - T_{p}

(4)

where J_sw and B_sw are, respectively, the moment of inertia and viscous damping coefficient of the steering wheel and column. K_tb is the torsion bar stiffness. θ_sw and T_sw are, respectively, the steering wheel angle and torque. J_ss, B_ss, θ_ss, F, and L are, respectively, the moment of inertia, viscous damping coefficient, angle, axial force, and center distance of the steering screw. M_sn, B_sn, and x_sn are, respectively, the mass, viscous damping coefficient, and displacement of the steering nut. p_ac and A_p are, respectively, the pressure and piston area of the assistance cylinder. J_s, B_s, θ_s, F_s, and r_s are, respectively, the moment of inertia, viscous damping coefficient, angle, force, and reference radius of the sector. T_p is the steering resistance.

The hydraulic assistance subsystem is illustrated in Figure 4. The rotary valve is modeled as a Wheatstone bridge with opposite orifices assumed to be equal. The flow rate and pressure equations are described as

Q_{pump} = Q_{EHPV} + Q_{rv}

(5)

Q_{rv} = Q_{1} + Q_{2} = Q_{3} + Q_{4}

(6)

Q_{ac} = Q_{1} - Q_{4} = Q_{3} - Q_{2} = v_{p} A_{p} = ω_{sw} s A_{p}

(7)

Q_{1} = C_{d} A_{1} \sqrt{\frac{2 (p_{rv} - p_{left})}{ρ}} = Q_{3} = C_{d} A_{1} \sqrt{\frac{2 p_{right}}{ρ}}

(8)

Q_{2} = C_{d} A_{2} \sqrt{\frac{2 (p_{rv} - p_{right})}{ρ}} = Q_{4} = C_{d} A_{2} \sqrt{\frac{2 p_{left}}{ρ}}

(9)

p_{ac} = p_{left} - p_{right}

(10)

where Q_pump is the pump flow rate. Q_EHPV, Q_rv, and Q_i (i = 1, 2, 3, 4) are, respectively, the flow rate through the EHPV, rotary valve, and rotary valve orifices. Q_ac and v_p are, respectively, the flow rate and piston velocity of the assistance cylinder. ω_sw is the steering rate, and s is the lead of the steering screw. C_d is the flow coefficient, A_i (i = 1, 2) are the opening areas of rotary valve orifices, and ρ is the oil density. p_rv is the inlet pressure of the rotary valve. p_left and p_right are, respectively, the left and right chamber pressures of the assistance cylinder.

Figure 4.

Hydraulic assistance subsystem.

Figure 5 shows the schematic of the EHPV orifice, which is chosen as L-shaped. Based on the working principle of the EHPV, the following mathematical models are obtained

u_{c} = k_{u} u_{e} = R_{s} i + L_{s} \overset{\cdot}{i} + k_{e} {\overset{\cdot}{x}}_{v}

(11)

F_{m} = k_{i} i = M_{v} {\overset{\cdot\cdot}{x}}_{v} + B_{v} {\overset{\cdot}{x}}_{v} + K_{v} x_{v} + F_{f}

(12)

\bar{O_{2} O_{9}} = \bar{O_{2} O_{4}} = r_{o} = \frac{h_{o}^{2} + L_{o}^{2}}{2 h_{o}}

(13)

\bar{O_{6} O_{9}} = r_{o} - \sqrt{{(r_{o} - h_{o})}^{2} + {(L_{o} - x_{v})}^{2}}

(14)

A_{EHPV} = b_{o} \bar{O_{6} O_{9}}

(15)

Q_{EHPV} = C_{d} A_{EHPV} \sqrt{\frac{2 p_{EHPV}}{ρ}}

(16)

where k_u is the amplifier gain, and u_e is the input voltage of the amplifier. u_c, i, R_s, L_s, and k_e denote the voltage, current, resistance, inductance, and back-electromotive force (EMF) coefficient of the solenoid coils, respectively. x_v is the valve spool displacement. F_m is the magnetic force, and k_i is the ratio of magnetic force to current. M_v is the valve spool mass. B_v is the viscous damping coefficient, K_v is the spring stiffness, and F_f denotes external disturbances. A_EHPV denotes the opening area of the EHPV orifice. p_EHPV is the inlet pressure of the EHPV that is equal to p_rv.

Figure 5.

Schematic of EHPV orifice.

Vehicle

A 2-degree-of-freedom (2-DOF) bicycle model in the yaw plane is adopted for vehicle dynamics, as shown in Figure 6. In the model, vehicle lateral and yaw motions are mainly considered, ignoring the longitudinal, vertical, roll, and pitch motions. The vehicle has the mass m, moment of inertia I_z, sideslip angle β, yaw rate γ, and velocity v at the center of gravity (CG). l_f and l_r are the distances from the CG to the front and rear axles, respectively. δ_f is the front wheel steering angle. F_yf and F_yr are the tire lateral forces of the front and rear wheels, respectively. α_f and α_r are the tire slip angles of the front and rear wheels, respectively. The vehicle motions are modeled as

{\begin{matrix} m v_{x} (\overset{\cdot}{β} + γ) = F_{yf} + F_{yr} \\ I_{z} \overset{\cdot}{γ} = F_{yf} l_{f} - F_{yr} l_{r} \end{matrix}

(17)

where v_x is the longitudinal velocity that is equal to v.cos β.

Figure 6.

Bicycle model for vehicle dynamics.

Given the assumption that the vehicle lateral acceleration is less than 0.4g (acceleration of gravity), the tire lateral force can be modeled in linear form²³

F_{yf} = C_{f} α_{f}

(18)

F_{yr} = C_{r} α_{r}

(19)

where C_f and C_r are the tire cornering stiffness of the front and rear wheels, respectively. α_f and α_r can be expressed as

α_{f} = δ_{f} - \frac{l_{f} γ}{v_{x}} - β

(20)

α_{r} = \frac{l_{r} γ}{v_{x}} - β

(21)

Steering resistance

When the driver steers, the moments generated between the tires and ground are delivered to the driver through the ECHPS system and act as steering resistance. The steering resistance mainly consists of two moments, namely, the self-aligning moment and the moment by king pin inclination.²⁴ Therefore, the steering resistance can be approximated by

T_{p} = F_{yf} r \tan (λ_{1} + t_{p}) - F_{z} λ_{2} \sin λ_{3} \sin δ_{f}

(22)

where r is the tire radius. λ₁ denotes the caster angle and t_p represents the pneumatic trail. F_z is the normal force. λ₂ represents the king pin offset at the ground and λ₃ denotes the king pin inclination.

Controller design

Problem formulation

Select the state variables as $x_{1} = x_{v}$ , $x_{2} = {\overset{\cdot}{x}}_{v}$ , $x_{3} = {\overset{\cdot\cdot}{x}}_{v}$ , and the control input as $u = u_{e}$ . Then, equations (11) and (12) can be transformed into the following state-space system

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} \\ {\overset{\cdot}{x}}_{3} = \sum_{i = 1}^{3} θ_{i} x_{i} + bu + d \\ y = x_{1} \end{matrix}

(23)

where $θ_{1} = - (K_{v} R_{s}) / (M_{v} L_{s})$ , $θ_{2} = - (B_{v} R_{s} + K_{v} L_{s} + k_{i} k_{e}) / (M_{v} L_{s})$ , $θ_{3} = - (M_{v} R_{s} + B_{v} L_{s}) / (M_{v} L_{s})$ , $b = (k_{i} k_{u}) / (M_{v} L_{s})$ , and $d = - (R_{s} F_{f} + L_{s} {\overset{\cdot}{F}}_{f}) / (M_{v} L_{s})$ are unknown.

For system (23), the control objective is to design an adaptive feedback control u such that the system output y = x₁ tracks the desired valve spool displacement x_1d under parameter uncertainties and external disturbances.

The following assumptions are required:

Assumption 1. All state variables of system (23) are measurable.

Assumption 2. The disturbance d is assumed to be bounded by a known positive constant.

Assumption 3. The reference trajectory vector $Π = [x_{1 d}, {\overset{\cdot}{x}}_{1 d}, {\overset{\cdot\cdot}{x}}_{1 d}]^{T} \in Ω_{d}$ is known, where $Ω_{d} = {Π : x_{1 d}^{2} + {\overset{\cdot}{x}}_{1 d}^{2} + {\overset{\cdot\cdot}{x}}_{1 d}^{2} \leq A_{0}}$ , and A₀ is a known positive constant.

Adaptive DSC scheme

The adaptive DSC scheme consists of three steps:

Step 1. Define the first surface error as

e_{1} = x_{1} - x_{1 d}

(24)

From system (23), we obtain

{\overset{\cdot}{e}}_{1} = {\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{1 d} = x_{2} - {\overset{\cdot}{x}}_{1 d}

(25)

Choose the virtual control function ${\bar{x}}_{2}$ as

{\bar{x}}_{2} = - k_{1} e_{1} + {\overset{\cdot}{x}}_{1 d}

(26)

where k₁ is a positive design parameter.

Let ${\bar{x}}_{2}$ pass through a first-order filter with time constant τ₂ to obtain x_2d

τ_{2} {\overset{\cdot}{x}}_{2 d} + x_{2 d} = {\bar{x}}_{2}, x_{2 d} (0) = {\bar{x}}_{2} (0)

(27)

Step 2. Define the second surface error as

e_{2} = x_{2} - x_{2 d}

(28)

According to system (23), we obtain

{\overset{\cdot}{e}}_{2} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{2 d} = x_{3} - {\overset{\cdot}{x}}_{2 d}

(29)

Choose the virtual control function ${\bar{x}}_{3}$ as

{\bar{x}}_{3} = - k_{2} e_{2} + {\overset{\cdot}{x}}_{2 d}

(30)

where k₂ is a positive design parameter.

Let ${\bar{x}}_{3}$ pass through a first-order filter with time constant τ₃ to obtain x_3d

τ_{3} {\overset{\cdot}{x}}_{3 d} + x_{3 d} = {\bar{x}}_{3}, x_{3 d} (0) = {\bar{x}}_{3} (0)

(31)

Step 3. Define the third surface error as

e_{3} = x_{3} - x_{3 d}

(32)

Considering system (23), we obtain

{\overset{\cdot}{e}}_{3} = {\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{3 d} = \sum_{i = 1}^{3} θ_{i} x_{i} + bu + d - {\overset{\cdot}{x}}_{3 d}

(33)

Define $p = 1 / b$ , and the adaptive feedback control u can be designed as

u = \hat{p} x_{d} = \hat{p} (- k_{3} e_{3} - \sum_{i = 1}^{3} {\hat{θ}}_{i} x_{i} - \hat{d} + {\overset{\cdot}{x}}_{3 d})

(34)

where k₃ is a positive design parameter, x_d is a state variable, and $\hat{p}$ , ${\hat{θ}}_{1}$ , ${\hat{θ}}_{2}$ , ${\hat{θ}}_{3}$ , and $\hat{d}$ are the estimations of p, θ₁, θ₂, θ₃, and d.

The adaptation laws of ${\hat{θ}}_{1}$ , ${\hat{θ}}_{2}$ , ${\hat{θ}}_{3}$ , $\hat{p}$ , and $\hat{d}$ are employed as

{\overset{\cdot}{\hat{θ}}}_{i} = ρ_{i} (e_{3} x_{i} - r_{i} {\hat{θ}}_{i}), i = 1, 2, 3

(35)

\overset{\cdot}{\hat{p}} = ρ_{b} (- e_{3} x_{d} - r_{b} \hat{p})

(36)

\overset{\cdot}{\hat{d}} = ρ_{d} (e_{3} - r_{d} \hat{d})

(37)

where ρ₁, ρ₂, ρ₃, ρ_b, ρ_d, r₁, r₂, r₃, r_b, and r_d are design parameters.

Stability analysis

The stability analysis of the closed-loop control system is conducted via Lyapunov synthesis.

From equations (27) and (31), the boundary layer errors are defined as

y_{i} = x_{id} - {\bar{x}}_{i} = - τ_{i} {\overset{\cdot}{x}}_{id}, i = 2, 3

(38)

The estimate errors are given as

{\tilde{θ}}_{i} = {\hat{θ}}_{i} - θ_{i}, i = 1, 2, 3

(39)

\tilde{p} = \hat{p} - p

(40)

\tilde{d} = \hat{d} - d

(41)

Considering equations (25) and (26), (28)–(30), (32)–(34), and (38)–(41), the derivatives of the surface errors can be deduced as

\begin{matrix} {\overset{\cdot}{e}}_{1} = e_{2} + x_{2 d} - {\overset{\cdot}{x}}_{1 d} \\ = e_{2} + y_{2} + {\bar{x}}_{2} - {\overset{\cdot}{x}}_{1 d} \\ = e_{2} + y_{2} - k_{1} e_{1} \end{matrix}

(42)

\begin{matrix} {\overset{\cdot}{e}}_{2} = e_{3} + x_{3 d} - {\overset{\cdot}{x}}_{2 d} \\ = e_{3} + y_{3} + {\bar{x}}_{3} - {\overset{\cdot}{x}}_{2 d} \\ = e_{3} + y_{3} - k_{2} e_{2} \end{matrix}

(43)

\begin{matrix} {\overset{\cdot}{e}}_{3} = \sum_{i = 1}^{3} θ_{i} x_{i} + b (\tilde{p} + p) x_{d} + d - {\overset{\cdot}{x}}_{3 d} \\ = - k_{3} e_{3} - \sum_{i = 1}^{3} {\tilde{θ}}_{i} x_{i} - \tilde{d} + b \tilde{p} x_{d} \end{matrix}

(44)

From equations (26), (27), (30), (31), and (38), the derivatives of the boundary layer errors can be obtained as

\begin{matrix} {\overset{\cdot}{y}}_{2} = {\overset{\cdot}{x}}_{2 d} - {\overset{\cdot}{\bar{x}}}_{2} \\ = - \frac{x_{2 d} - {\bar{x}}_{2}}{τ_{2}} - (- k_{1} {\overset{\cdot}{e}}_{1} + {\overset{\cdot\cdot}{x}}_{1 d}) \\ = - \frac{y_{2}}{τ_{2}} + B_{2} ({\overset{\cdot}{e}}_{1}, {\overset{\cdot\cdot}{x}}_{1 d}) \end{matrix}

(45)

\begin{matrix} {\overset{\cdot}{y}}_{3} = {\overset{\cdot}{x}}_{3 d} - {\overset{\cdot}{\bar{x}}}_{3} \\ = - \frac{x_{3 d} - {\bar{x}}_{3}}{τ_{3}} - (- k_{2} {\overset{\cdot}{e}}_{2} + {\overset{\cdot\cdot}{x}}_{2 d}) \\ = - \frac{y_{3}}{τ_{3}} + B_{3} ({\overset{\cdot}{e}}_{2}, {\overset{\cdot\cdot}{x}}_{2 d}) \end{matrix}

(46)

where $B_{2} ({\overset{\cdot}{e}}_{1}, {\overset{\cdot\cdot}{x}}_{1 d})$ and $B_{3} ({\overset{\cdot}{e}}_{2}, {\overset{\cdot\cdot}{x}}_{2 d})$ are simplified as B₂ and B₃ in the following discussions, which are assumed to be bounded by positive constants M₂ and M₃, respectively.

Define the Lyapunov function candidate as

V = \frac{1}{2} (\sum_{i = 1}^{3} e_{i}^{2} + \sum_{i = 2}^{3} y_{i}^{2} + \sum_{i = 1}^{3} \frac{{\tilde{θ}}_{i}^{2}}{ρ_{i}} + \frac{b {\tilde{p}}^{2}}{ρ_{b}} + \frac{{\tilde{d}}^{2}}{ρ_{d}})

(47)

Invoking equations (35)–(37) and (39)–(46), the derivative of V is given by

\begin{matrix} \overset{\cdot}{V} = \sum_{i = 1}^{3} e_{i} {\overset{\cdot}{e}}_{i} + \sum_{i = 2}^{3} y_{i} {\overset{\cdot}{y}}_{i} + \sum_{i = 1}^{3} \frac{{\tilde{θ}}_{i} {\overset{\cdot}{\hat{θ}}}_{i}}{ρ_{i}} + \frac{b \tilde{p} \overset{\cdot}{\hat{p}}}{ρ_{b}} + \frac{\tilde{d} \overset{\cdot}{\hat{d}}}{ρ_{d}} \\ = e_{1} e_{2} + e_{2} e_{3} + e_{1} y_{2} + e_{2} y_{3} + y_{2} B_{2} + y_{3} B_{3} \\ - k_{1} e_{1}^{2} - k_{2} e_{2}^{2} - k_{3} e_{3}^{2} - \frac{y_{2}^{2}}{τ_{2}} - \frac{y_{3}^{2}}{τ_{3}} \\ - \sum_{i = 1}^{3} r_{i} {\tilde{θ}}_{i} {\hat{θ}}_{i} - b r_{b} \tilde{p} \hat{p} - r_{d} \tilde{d} \hat{d} \end{matrix}

(48)

Employing the following inequalities

{\begin{matrix} e_{i} e_{i + 1} \leq | e_{i} | | e_{i + 1} | \leq e_{i}^{2} + \frac{e_{i + 1}^{2}}{4}, i = 1, 2 \\ e_{i} y_{i + 1} \leq | e_{i} | | y_{i + 1} | \leq e_{i}^{2} + \frac{y_{i + 1}^{2}}{4}, i = 1, 2 \\ y_{i} B_{i} \leq | y_{i} | M_{i} \leq y_{i}^{2} M_{i}^{2} + \frac{1}{4}, i = 2, 3 \\ {\tilde{θ}}_{i} {\hat{θ}}_{i} = {\tilde{θ}}_{i} ({\tilde{θ}}_{i} + θ_{i}) \geq \frac{{\tilde{θ}}_{i}^{2} - θ_{i}^{2}}{2}, i = 1, 2, 3 \\ \tilde{p} \hat{p} = \tilde{p} (\tilde{p} + p) \geq \frac{{\tilde{p}}^{2} - p^{2}}{2} \\ \tilde{d} \hat{d} = \tilde{d} (\tilde{d} + d) \geq \frac{{\tilde{d}}^{2} - d^{2}}{2} \end{matrix}

(49)

The derivative of V can be rewritten as

\begin{matrix} \overset{\cdot}{V} \leq (2 - k_{1}) e_{1}^{2} + (\frac{9}{4} - k_{2}) e_{2}^{2} + (\frac{1}{4} - k_{3}) e_{3}^{2} \\ + (\frac{1}{4} - \frac{1}{τ_{2}} + M_{2}^{2}) y_{2}^{2} + (\frac{1}{4} - \frac{1}{τ_{3}} + M_{3}^{2}) y_{3}^{2} \\ - \sum_{i = 1}^{3} \frac{r_{i} {\tilde{θ}}_{i}^{2}}{2} - \frac{b r_{b} {\tilde{p}}^{2}}{2} - \frac{r_{d} {\tilde{d}}^{2}}{2} \\ + \frac{1}{2} (\sum_{i = 1}^{3} r_{i} θ_{i}^{2} + b r_{b} p^{2} + r_{d} d^{2} + 1) \end{matrix}

(50)

Let

{\begin{matrix} 2 - k_{1} = \frac{9}{4} - k_{2} = \frac{1}{4} - k_{3} = \frac{1}{4} - \frac{1}{τ_{2}} + M_{2}^{2} \\ = \frac{1}{4} - \frac{1}{τ_{3}} + M_{3}^{2} = - \frac{1}{2} k \\ r_{i} = \frac{k}{ρ_{i}}, i = 1, 2, 3, b, d \\ M = \frac{1}{2} (\sum_{i = 1}^{3} r_{i} θ_{i}^{2} + b r_{b} p^{2} + r_{d} d^{2} + 1) \end{matrix}

(51)

where k is a positive design parameter.

Substituting equation (51) into equation (50), we obtain

\overset{\cdot}{V} \leq - kV + M

(52)

Integrating equation (52) over [0, t], we have

0 \leq V \leq \frac{M}{k} + (V (0) - \frac{M}{k}) e^{- kt}

(53)

Therefore, it is concluded that the closed-loop control system is semi-globally uniformly ultimately bounded.

Simulation

To verify the effectiveness of the ECHPS system, numerical simulations of vehicle handling and stability test are conducted in MATLAB Simulink^® environment. Table 3 lists the major parameters used in the numerical simulations.

Table 3.

Simulation parameters.

Parameter	Value
Moment of inertia of steering wheel and column (kg m²)	0.009
Torsion bar stiffness (N m/rad)	170
Moment of inertia of steering screw (kg m²)	5.5 × 10⁻³
Center distance of steering screw (m)	0.01
Mass of steering nut (kg)	3.5
Piston area of assistance cylinder (m²)	1.13 × 10⁻²
Moment of inertia of sector (kg m²)	0.028
Reference radius of sector (m)	4.675 × 10⁻²
Lead of steering screw (m)	0.014
Flow coefficient	0.62
Oil density (kg/m³)	900
Amplifier gain	40
Resistance of solenoid coils (Ω)	305
Inductance of solenoid coils (H)	0.03
Back-EMF coefficient of solenoid coils (V/(m/s))	40.11
Ratio of magnetic force to current (N/A)	24.6
Valve spool mass (kg)	1.16 × 10⁻²
Spring stiffness (N/m)	2.6 × 10⁴
Vehicle mass (kg)	1.6 × 10⁴
Vehicle moment of inertia (kg m²)	1.3 × 10⁵
Distance from CG to front axle (m)	3.69
Distance from CG to rear axle (m)	1.86
Tire cornering stiffness of front wheels (N/rad)	6.36 × 10⁴
Tire cornering stiffness of rear wheels (N/rad)	1.2 × 10⁵
Tire radius (m)	0.524

CG: center of gravity.

The simulation of lemniscate test is performed to check low-speed steering portability. In the lemniscate test, the vehicle moves with a constant speed of 10 km/h. The type of steering wheel angle input is sinusoidal, with a frequency of 0.02 Hz and an amplitude around 800°. Figure 7 shows the curve of steering wheel torque versus steering wheel angle in the lemniscate simulation. For ease of comparison, the steering wheel torque indices using the HPS and ECHPS systems in the lemniscate simulation are listed in Table 4. As can be seen in Table 4, the peak value of steering wheel torque using the ECHPS system is 5.32 N m, which increases by 6.83% compared with 4.98 N m using the HPS system. Moreover, the standard deviation of steering wheel torque using the ECHPS system is 3.08, which increases by 5.48% compared with 2.92 using the HPS system. In conclusion, the ECHPS system achieves nearly the same level of steering portability performance as the HPS system.

Figure 7.

Steering wheel torque versus steering wheel angle in lemniscate simulation.

Table 4.

Steering wheel torque indices in lemniscate simulation.

	HPS	ECHPS	Comparison
Peak value (N m)	4.98	5.32	6.83%
Standard deviation	2.92	3.08	5.48%

HPS: hydraulic power steering; ECHPS: electronically controlled hydraulic power steering.

The simulation of on-center handling test is conducted to evaluate high-speed steering feel. The on-center handling test simulates the situation when driving a vehicle on a highway under low lateral acceleration without road or wind disturbances. In the test, the vehicle moves at a high speed of 80 km/h, and the steering wheel angle input is sinusoidal with a frequency of 0.2 Hz and a bounded amplitude to ensure the vehicle lateral acceleration under 0.2g. The curve of steering wheel torque versus lateral acceleration in the on-center handling simulation is shown in Figure 8. Table 5 lists the comparison of steering wheel torque gradient at lateral acceleration of 0g, which is an important index to evaluate steering feel. As can be seen in Table 5, the steering wheel torque gradient using the ECHPS system is 30.1 N m/g, which increases by 80.2% compared with 16.7 N m/g using the HPS system. It is concluded that the ECHPS system can significantly improve high-speed steering feel compared with the HPS system.

Figure 8.

Steering wheel torque versus lateral acceleration in on-center handling simulation.

Table 5.

Steering wheel torque gradient in on-center handling simulation.

	HPS	ECHPS	Improvement
Gradient (N m/g)	16.7	30.1	80.2%

HPS: hydraulic power steering; ECHPS: electronically controlled hydraulic power steering.

Field test

To implement the adaptive DSC scheme, a rapid control prototyping (RCP) system is developed as illustrated in Figure 9. The developed RCP system consists of the MATLAB Simulink for control system modeling, the dSPACE MicroAutoBox as virtual controller for real-time simulation, the dSPACE RapidPro as power drive unit for actuating the EHPV, the linear displacement sensor for measuring valve spool displacement, and the EHPV as controlled plant. Moreover, Figure 10 shows the integration of the EHPV and recirculating ball steering gear.

Figure 9.

Rapid control prototyping system.

Figure 10.

Integration of EHPV and recirculating ball steering gear.

The test vehicle and measuring instruments are shown in Figures 11 and 12, respectively. The test vehicle is SLK6118 bus manufactured by Shanghai Shenlong Bus Company Limited. The measuring instruments include a steering wheel sensor, a gyroscope, a vehicle speed sensor, and an LMS data acquisition device. The steering wheel sensor measures steering wheel angle, steering wheel torque, and steering rate. The gyroscope provides signals of vehicle yaw rate and lateral acceleration, and the vehicle speed sensor measures vehicle speed. The LMS data acquisition device collects and monitors those signals.

Figure 11.

Test vehicle.

Figure 12.

Measuring instruments.

Figure 13 shows the tracking of valve spool displacement for desired value of 1.5 mm in simulation and experiment. The external disturbances in simulation is set to be 100(1 + sin 10πt) N. As shown in Figure 13, the tracking of valve spool displacement in simulation exhibits an overshoot of 0.6% and the settling time is 0.42 s. Moreover, the valve spool displacement in experiment tracks the desired value with acceptable overshoot of 8.3% and the settling time is 0.7 s, which is basically consistent with that in simulation. It can be seen that the adaptive DSC scheme can realize accurate tracking of valve spool displacement.

Figure 13.

Tracking of valve spool displacement.

Field tests are performed to validate the applicability of the ECHPS system. The lemniscate test is conducted to check low-speed steering portability. Figure 14 shows the curve of steering wheel torque versus steering wheel angle in the lemniscate test. Table 6 lists the steering wheel torque indices using the HPS and ECHPS systems in the lemniscate test. As can be seen in Table 6, the peak value of steering wheel torque using the ECHPS system is 5.04 N m, which increases by 4.78% compared with 4.81 N m using the HPS system. The standard deviation of steering wheel torque using the ECHPS system is 2.90, which increases by 4.69% compared with 2.77 using the HPS system. In conclusion, the ECHPS system maintains the similar performance of low-speed steering portability as the HPS system. Moreover, it should be noted that the differences between simulation and experimental results are all within 10%, which indicates the consistency between numerical simulations and field tests.

Figure 14.

Steering wheel torque versus steering wheel angle in lemniscate test.

Table 6.

Steering wheel torque indices in lemniscate test.

	HPS	ECHPS	Comparison
Peak value (N m)	4.81	5.04	4.78%
Standard deviation	2.77	2.90	4.69%

HPS: hydraulic power steering; ECHPS: electronically controlled hydraulic power steering.

The on-center handling test is performed to evaluate high-speed steering feel. The curve of steering wheel torque versus lateral acceleration in the on-center handling test is shown in Figure 15. Table 7 lists the comparison of steering wheel torque gradient at lateral acceleration of 0g. As can be seen in Table 7, the steering wheel torque gradient using the ECHPS system is 28.9 N m/g, which increases by 80.6% compared with 16 N m/g using the HPS system. It is concluded that the ECHPS system can significantly improve high-speed steering feel compared with the HPS system.

Figure 15.

Steering wheel torque versus lateral acceleration in on-center handling test.

Table 7.

Steering wheel torque gradient in on-center handling test.

	HPS	ECHPS	Improvement
Gradient (N m/g)	16	28.9	80.6%

HPS: hydraulic power steering; ECHPS: electronically controlled hydraulic power steering.

Conclusion

This article has presented an analysis of the ECHPS system that can realize variable assistance characteristics and improve high-speed steering feel for heavy vehicles. A system description is given and mathematical modeling of the ECHPS system, vehicle, and steering resistance is conducted. An adaptive DSC scheme is proposed for the EHPV to realize accurate tracking of valve spool displacement under parameter uncertainty and external disturbance. Numerical simulations and field tests are performed to validate the effectiveness and applicability of the ECHPS system. The simulation and experimental results are consistent, which show that the ECHPS system can significantly improve high-speed steering feel compared with the HPS system while maintaining the similar performance of low-speed steering portability.

Footnotes

Academic Editor: Francesco Mass

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 51275211) and the Special Fund of Jiangsu Province for the Transformation of Scientific and Technological Achievements (No. BR2015168).

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