# HIS-Based Semiactive Suspension Dual-Frequency-Range Switching Control to Improve Ride Comfort and Antiroll Performance

Authors: Xiaojian Wu; Xiang Qiu; Bing Zhou; Juhua Huang; Tingfang Zhang
Journal: Shock and Vibration
 (2019)

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## Abstract

The parameter sensitivity analysis of a hydraulically interconnected suspension (HIS) system shows that the sensitivity of the vibration responses in the bounce and roll modes to the hydraulic parameters are complementary. A novel HIS-based semiactive control method was thereby proposed to improve ride comfort and antiroll performance. In addition, the classic sky-hook max-min damping switched strategy provides significant benefits around the body resonance, but otherwise performs similarly to, or sometimes even worse than, passive suspension. Therefore, a dual-frequency-range switching strategy, which has optimal max-min damping in both frequency ranges, was developed for improving the ride comfort in a wider frequency bandwidth. In this study, a 9-DOF HIS system dynamics model was established, and the hydraulically interconnected subsystem model was validated experimentally. Subsequently, the elastic and damping characteristics of the hydraulically interconnected subsystem, as well as the parameter sensitivity in bounce mode and roll mode, were analyzed. Next, the sensitive parameters were optimized under sinusoidal excitation at various frequencies, and a frequency-range selector used to determine the excitation frequency range and adjust the shock absorber damping was designed. Finally, simulations in the frequency domain and time domain show that the proposed HIS-based semiactive dual-frequency-range switching control suspension improves the ride comfort in a wider frequency bandwidth and enhances the antiroll performance in the transient and steady steering process.


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## Body

## 1. Introduction

A semiactive suspension system is characterized by low energy consumption, economical cost, and achieving the majority performance of the active suspension system within a certain frequency bandwidth. Thus far, a vast amount of academic and industrial research activities have been undertaken to study such systems [1–4].As far as the research of semiactive suspensions is concerned, it can be summed up in two mainstream directions: the first focuses on developing advanced control strategies and algorithms, while the second revolves around developing reliable, high-performance adjustable shock absorbers (electrorheological, ER; magnetorheological dampers, MR) and emphatically analyzes the damping adjusting mechanism, delay response, and compensation control. In terms of the first of these directions, the most widely used classic control scheme, known as two-state sky-hook damping, is an on/off strategy that switches between high and low damping coefficients to improve ride comfort characterized by the body vertical acceleration [5]. Meanwhile, the road holding performance represented by wheel dynamic deformation is improved by developing a ground-hook control strategy [6]. A sky-hook and ground-hook combined control strategies were proposed by Ahmadian and Vahdati to take advantage of their respective benefits [7]. A hybrid sliding mode controller, including two separate model reference controllers designed to respectively force the plant to follow the ideal sky-hook and the ideal ground-hook models, was developed by Assadsangabi et al. [8] to enhance both ride comfort and road holding performance. In addition, some intelligent control algorithms, such as fuzzy, adaptive-fuzzy with Kalman filter [9], and neural networks [10] algorithms, were also applied to improve the performance of semiactive suspension in terms of ride comfort and road holding.However, regardless of adjusting the damping coefficient by variable orifices, electrorheological fluids or magnetorheological fluids, the semiactive control suspension could improve only the antiroll performance during the transient steering process, but not the steady steering process. Furthermore, as the most widely applied control strategy, the sky-hook max-min damping switched method provides large benefits around the body resonance, but in other frequency ranges, it performs similarly to or sometimes even worse than passive suspension [11]. Therefore, studies on improving ride comfort in a wider frequency bandwidth and enhancing the antiroll performance during the transient and steady steering processes are a meaningful and challenging undertaking.It is noted that research on hydraulically interconnected suspension (HIS) equipped with cross-connected hydraulic circuits and high-pressure accumulators is gradually becoming a hot topic. Swift et al. [12] designed a fluid inerter and established its mathematical model and analyzed the damping characteristics in various passive network layouts. Zhang et al. [13–15], Du et al. [16, 17], and Smith et al. [18, 19] have engaged in research on the antiroll of the HIS system, and they pointed out that the HIS system can effectively improve the antiroll performance without impacting the ride comfort. To improve the dynamic performance of different vibration modes, Yao et al. [20] proposed a dual-mode interconnected suspension that can be switched between two passive modes of roll-plane connected and pitch-plane connected. Based on the hydraulically interconnected architecture, Zhang et al. [15] proposed a switchable suspension system according to the motion-mode energy method. Wang et al. [21] developed a dual-mode switched suspension. Here, when the vehicle is traveling straight, the hydraulically interconnected inerter-spring-damper mode is used to improve the ride comfort, and when the vehicle turns, the passive roll-plane interconnected mode is selected to improve the antiroll performance.In our research, we found that the sensitivity of the vibration responses in the bounce mode and roll mode to the parameters of hydraulically interconnected subsystem is complementary, which provides an idea for developing a novel semiactive control to improve ride comfort and antiroll performance. Namely, we can adjust the two modes’ sensitive parameters to improve the ride comfort and antiroll performance, respectively.At the outset, it must be stressed that, because the development of a novel control theory is beyond the scope of this research, referring to the superiorities of the sky-hook control strategy, this method is adopted to focus on the following: (1) to analyze parameter sensitivities for the bounce mode and roll mode, thereby developing a novel HIS-based semiactive control method and (2) to propose a switching control method based on excitation frequency-range identification, thereby improving the ride comfort in a wider frequency bandwidth.The paper is organized as follows. In Section2, a full-vehicle 9-DOF dynamics model and the “Pressure-Flow” equations for the hydraulically interconnected subsystem are established and validated by experimental tests. In Section 3, the parameter sensitivity analysis of a hydraulically interconnected subsystem is performed, and the dynamic characteristics in the bounce mode and roll mode are analyzed. On this basis, a novel HIS-based semiactive control method is proposed to improve ride comfort and antiroll performance. In Section 4, a HIS-based semiactive control strategy is designed using sky-hook algorithm, and a dual-frequency-range switching control method based on approximately identifying excitation frequency-range is proposed. In Section 5, the amplitude-frequency characteristic analyses of “body vertical acceleration-road excitation” for several semiactive control systems are compared, and numerical simulations including step input and the fishhook test are undertaken to illustrate the effectiveness of the developed approach. Finally, conclusions are summarized in Section 6.


## 2. Dynamics Modeling and Experimental Verification

### 2.1. 9-DOF Full-Vehicle Dynamics Modeling

As shown in Figure1, based on the traditional suspension system, the HIS system adds a hydraulically interconnected subsystem plotted in Figure 2. Accordingly, a 9-DOF dynamics model, including body bounce motion, body roll motion, body pitch motion, unsprung mass vertical motion, lateral motion, and yaw motion, is established, as shown in the following equations (1)–(9):Figure 1
Vehicle dynamics model for the HIS system.
(a)
(b)
(c)Figure 2
Hydraulically interconnected subsystem.Body bounce motion:(1)msx¨c=∑j0=fl,fr,rl,rrFj0−kj0xsj0−xuj0−cj0x˙sj0−x˙uj0.Body roll motion:(2)Iθθ¨=∑j1=fl,rlkj1xsj1−xuj1+cj1x˙sj1−x˙uj1−Fj1−∑j2=fr,rrkj2xsj2−xuj2+cj2x˙sj2−x˙uj2−Fj2bs2+Mθ.Body pitch motion:(3)Iφφ¨=a∑j3=fr,flkj3xsj3−xuj3+cj3x˙sj3−x˙uj3−Fj3−b∑j4=rl,rrkj4xsj4−xuj4+cj4x˙sj4−x˙uj4−Fj4.Unsprung mass vertical motion:(4)muj0x¨uj0=kj0xsj0−xuj0+cj0x˙sj0−x˙uj0+ktj0xtj0−xuj0−Fj0.Yaw motion:(5)Izr˙=a∑j3=fl,frFyj3cosδ−b∑j4=rl,rrFyj4+bs2Fyfr−Fyflsinδ.Lateral motion:(6)may=∑j3=fl,fr;j4=rl,rrFyj3cosδ+Fyj4.Roll moment:(7)Mθ=msayh+msghθ,(8)ay=v˙+ur,(9)xsfl=xc−bs2θ+aφ,xsfr=xc+bs2θ+aφ,xsrl=xc−bs2θ−bφ,xsrr=xc+bs2θ−bφ,where j is equal to fl, fr, fr, and rr, and the subscripts fl, fr, fr, and rr represent the front left, front right, rear left, and rear right sides, respectively. m is the vehicle mass. ms and muj represent the sprung mass and the unsprung mass. xc is the displacement at the center of gravity of the body. θ and φ are the body roll angle and pitch angle, respectively. xsj and xuj are the displacements of the sprung mass and the unsprung mass. xtj is the road excitation. Iθ, Iφ, and Iz are roll moment of inertia, pitch moment of inertia, and yaw moment of inertia, respectively. a and b are the distances from vehicle’s center of gravity to the front and rear axles, respectively. bs is the track width. h is the height of the sprung mass center of gravity from the roll center. kj is the spring stiffness. cj is the damping coefficient. ktj is the tire vertical stiffness. r is the yaw rate, u is the vehicle longitudinal velocity, and v is the lateral velocity. δ is the wheel steering angle. ay is the vehicle lateral acceleration. Fyj is the wheel lateral force, which can be calculated by the tire model. Fj is the vertical force generated by the hydraulic interconnection subsystem, which is expressed as equation (27).Note that the body vertical accelerationx¨c and body roll angle θ are usually used as indicators of ride comfort and antiroll performance, respectively, as well as wheel dynamic deformation xuj0−xtj0 is adopted to reflect road holding performance.


### 2.2. Tire Model

The Magic formula tire model is adopted in this study to reflect the coupling relationship between the vertical load and lateral force as follows:(10)Fy=DsinCarctanBx−EBx−arctanBx,where x is the tire slip angle, B is the stiffness factor, C is the shape factor, D is the peak factor, and E is the curvature factor. B, D, and E, relating with the wheel load Fz, which is calculated by Fzj3=0.5bmg/a+b+ktj3x0j3−xuj3 and Fzj4=0.5amg/a+b+ktj4x0j4−xuj4, can be expressed as follows:(11)D=a1Fz2+a2Fz,E=a6Fz2+a7Fz+a8,B=B1/C×D,B1=a3sina4×arctana5×Fz.The values of parametersa1–a8 are listed in Table 1.Table 1
Magic formula tire model parameters [22].
a
1
a
2
a
3
a
4
a
5
a
6
a
7
a
8
C
−22.1
1011
1078
1.82
0.208
0
−0.35
0.707
1.3


### 2.3. Hydraulically Interconnected Subsystem Modeling

As shown in Figure2, for the hydraulically interconnected subsystem, the chambers of the left and right hydraulic cylinders are cross-connected by hydraulic circuits, and two accumulators with a certain pressure are linked in the hydraulic circuits. Taking the hydraulic circuit “1-2-3-4-5-6” plotted in Figure 2 as an example, the “Pressure-Flow” model of the hydraulically interconnected subsystem is established as follows.Ignoring friction loss in the pipeline, it can be derived thatp2=p4=p5, p2′=p4′=p5′, p1=p1l, p6′=p2l, p1′=p1r, p6=p2r, and p3=p3l, and p3′=p3r. Here, the subscripted numbers correspond to the position of the markers in Figure 2.(1)
The “Pressure-Displacement” relationship in the accumulator:(12)Δv3l=A2xsr−xur−A1xsl−xul,Δv3r=A2xsl−xul−A1xsr−xur,Q3l=A2x˙sr−x˙ur−A1x˙sl−x˙ul,Q3r=A2x˙sl−x˙ul−A1x˙sr−x˙ur,where Δv3l and Δv3r are the volume that flows in or out of the liquid chamber of the left and right accumulators, respectively. A1 and A2 are the liquid action areas of the rod chamber and the rodless chamber, respectively.The instantaneous pressure of gas in the accumulator is derived assuming the polytropic process of the gas as follows:(13)p0v0n=p3lv3ln,wheren is the polytropic exponent of gas which is usually between 1.2 and 1.4 [23], v3l is the gas volume in the gas chamber of the accumulator, and v0 and p0 are the initial gas volume and pressure in the accumulator, respectively.Combining (12) and (13) yields the following equation:(14)p3l=p0v0v0−Δv3ln,p3r=p0v0v0−Δv3rn.(2)
The relationship between throttle valve flow and pressure in segment “1-2” is given by(15)Q12=cdA1l2Δp12ρsgnΔp12=−A1x˙sl−x˙ul,where sgn represents the signum function.It can be rewritten as follows:(16)p1l=p2+ρ2A1x˙sl−x˙ulcdA1l2sgnQ12.(3)
The relationship between throttle valve flow and pressure in segment “3-4” is given by(17)Q34=cdA3l2Δp34ρsgnΔp34=−Q3lDerivingp4 as(18)p4=p3−ρ2Q3lcdA3l2sgn−Q3l.(4)
The relationship between throttle valve flow and pressure in segment “5-6” is given by(19)Q56=cdA2r2Δp56ρsgnΔp56=−A2x˙sr−x˙ur.Therefore,(20)p6=p5−ρ2Q56cdA2r2sgnQ56.The final expressions of pressure become as follows:(21)p1l=p3l+ρ2Q3lcdA3l2sgnQ3l−ρ2A1x˙sl−x˙ulcdA1l2sgnA1x˙sl−x˙ul,(22)p2l=p3r+ρ2Q3rcdA3r2sgnQ3r+ρ2A2x˙sl−x˙ulcdA2l2sgnA2x˙sl−x˙ul,(23)p1r=p3r+ρ2Q3rcdA3r2sgnQ3r−ρ2A1x˙sr−x˙urcdA1r2sgnA1x˙sr−x˙ur,(24)p2r=p3l+ρ2Q3lcdA3l2sgnQ3l+ρ2A2x˙sr−x˙urcdA2r2sgnA2x˙sr−x˙ur,where cd is the discharge coefficient and ρ is the fluid density. A1l, A2l, A1r, A2r, A3l, and A3r are the flow areas of throttle valves.Based on equations (21)–(24), it can be determined that the hydraulically interconnected subsystem consists of the elastic system and damping system. The elastic force comes from the accumulator, and the damping force is caused by the throttle valve. Taking the left side as an example, the elastic force can be calculated by(25)Fl_k=p3l−p0A1−p3r−p0A2.The damping force can be computed by(26)Fl_c=p1l−p3lA1−p2l−p3rA2.The vertical forces generated by the hydraulic interconnection subsystem shown in Figure2 are derived to be as follows:(27)Fl=p1l−p0A1−p2l−p0A2,Fr=p1r−p0A1−p2r−p0A2.


### 2.4. Experimental Verification for Hydraulically Interconnected Subsystem

A test bench used to validate the established hydraulically interconnected subsystem dynamics model is shown in Figure3. In the experimental test scheme, the hydraulic cylinder base is fixed, and the sinusoidal displacement excitations are loaded on the hydraulic rods by actuators. Subsequently, sensors measure the displacements, and actuator forces with a sampling frequency of 102.4 Hz. In Figures 4(a)–4(c), a pair of sinusoidal excitations with the same phase, a magnitude of 30 mm, and a period of 2 seconds were applied to perform a bounce motion. Meanwhile, in Figures 4(d)–4(f), the sinusoidal excitations are changed to a phase difference of 180° to simulate a roll motion.Figure 3
Test bench for the hydraulically interconnected subsystem. A–F: throttle valves;G1, G2: accumulators; H1, H2: hydraulic cylinders; S1, S2: displacement sensors; and F1, F2: actuation cylinders.Figure 4
Comparisons between simulation and experimental test. (a) Sinusoidal displacement excitation for bounce modal. (b) Hydraulic force for bounce modal. (c) Indicator diagram for bounce modal. (d) Sinusoidal displacement excitation for roll modal. (e) Hydraulic force for roll modal. (f) Indicator diagram for roll modal.
(a)
(b)
(c)
(d)
(e)
(f)It can be seen from Figure4 that, although there are errors between the simulated and the experimental curves that may be caused by the run-in period of hydraulic cylinders, the trend and the main characteristics of the curves are consistent. Therefore, the established dynamic model has the potential to describe the characteristics of the hydraulically interconnected subsystem.


## 2.1. 9-DOF Full-Vehicle Dynamics Modeling

As shown in Figure1, based on the traditional suspension system, the HIS system adds a hydraulically interconnected subsystem plotted in Figure 2. Accordingly, a 9-DOF dynamics model, including body bounce motion, body roll motion, body pitch motion, unsprung mass vertical motion, lateral motion, and yaw motion, is established, as shown in the following equations (1)–(9):Figure 1
Vehicle dynamics model for the HIS system.
(a)
(b)
(c)Figure 2
Hydraulically interconnected subsystem.Body bounce motion:(1)msx¨c=∑j0=fl,fr,rl,rrFj0−kj0xsj0−xuj0−cj0x˙sj0−x˙uj0.Body roll motion:(2)Iθθ¨=∑j1=fl,rlkj1xsj1−xuj1+cj1x˙sj1−x˙uj1−Fj1−∑j2=fr,rrkj2xsj2−xuj2+cj2x˙sj2−x˙uj2−Fj2bs2+Mθ.Body pitch motion:(3)Iφφ¨=a∑j3=fr,flkj3xsj3−xuj3+cj3x˙sj3−x˙uj3−Fj3−b∑j4=rl,rrkj4xsj4−xuj4+cj4x˙sj4−x˙uj4−Fj4.Unsprung mass vertical motion:(4)muj0x¨uj0=kj0xsj0−xuj0+cj0x˙sj0−x˙uj0+ktj0xtj0−xuj0−Fj0.Yaw motion:(5)Izr˙=a∑j3=fl,frFyj3cosδ−b∑j4=rl,rrFyj4+bs2Fyfr−Fyflsinδ.Lateral motion:(6)may=∑j3=fl,fr;j4=rl,rrFyj3cosδ+Fyj4.Roll moment:(7)Mθ=msayh+msghθ,(8)ay=v˙+ur,(9)xsfl=xc−bs2θ+aφ,xsfr=xc+bs2θ+aφ,xsrl=xc−bs2θ−bφ,xsrr=xc+bs2θ−bφ,where j is equal to fl, fr, fr, and rr, and the subscripts fl, fr, fr, and rr represent the front left, front right, rear left, and rear right sides, respectively. m is the vehicle mass. ms and muj represent the sprung mass and the unsprung mass. xc is the displacement at the center of gravity of the body. θ and φ are the body roll angle and pitch angle, respectively. xsj and xuj are the displacements of the sprung mass and the unsprung mass. xtj is the road excitation. Iθ, Iφ, and Iz are roll moment of inertia, pitch moment of inertia, and yaw moment of inertia, respectively. a and b are the distances from vehicle’s center of gravity to the front and rear axles, respectively. bs is the track width. h is the height of the sprung mass center of gravity from the roll center. kj is the spring stiffness. cj is the damping coefficient. ktj is the tire vertical stiffness. r is the yaw rate, u is the vehicle longitudinal velocity, and v is the lateral velocity. δ is the wheel steering angle. ay is the vehicle lateral acceleration. Fyj is the wheel lateral force, which can be calculated by the tire model. Fj is the vertical force generated by the hydraulic interconnection subsystem, which is expressed as equation (27).Note that the body vertical accelerationx¨c and body roll angle θ are usually used as indicators of ride comfort and antiroll performance, respectively, as well as wheel dynamic deformation xuj0−xtj0 is adopted to reflect road holding performance.


## 2.2. Tire Model

The Magic formula tire model is adopted in this study to reflect the coupling relationship between the vertical load and lateral force as follows:(10)Fy=DsinCarctanBx−EBx−arctanBx,where x is the tire slip angle, B is the stiffness factor, C is the shape factor, D is the peak factor, and E is the curvature factor. B, D, and E, relating with the wheel load Fz, which is calculated by Fzj3=0.5bmg/a+b+ktj3x0j3−xuj3 and Fzj4=0.5amg/a+b+ktj4x0j4−xuj4, can be expressed as follows:(11)D=a1Fz2+a2Fz,E=a6Fz2+a7Fz+a8,B=B1/C×D,B1=a3sina4×arctana5×Fz.The values of parametersa1–a8 are listed in Table 1.Table 1
Magic formula tire model parameters [22].
a
1
a
2
a
3
a
4
a
5
a
6
a
7
a
8
C
−22.1
1011
1078
1.82
0.208
0
−0.35
0.707
1.3


## 2.3. Hydraulically Interconnected Subsystem Modeling

As shown in Figure2, for the hydraulically interconnected subsystem, the chambers of the left and right hydraulic cylinders are cross-connected by hydraulic circuits, and two accumulators with a certain pressure are linked in the hydraulic circuits. Taking the hydraulic circuit “1-2-3-4-5-6” plotted in Figure 2 as an example, the “Pressure-Flow” model of the hydraulically interconnected subsystem is established as follows.Ignoring friction loss in the pipeline, it can be derived thatp2=p4=p5, p2′=p4′=p5′, p1=p1l, p6′=p2l, p1′=p1r, p6=p2r, and p3=p3l, and p3′=p3r. Here, the subscripted numbers correspond to the position of the markers in Figure 2.(1)
The “Pressure-Displacement” relationship in the accumulator:(12)Δv3l=A2xsr−xur−A1xsl−xul,Δv3r=A2xsl−xul−A1xsr−xur,Q3l=A2x˙sr−x˙ur−A1x˙sl−x˙ul,Q3r=A2x˙sl−x˙ul−A1x˙sr−x˙ur,where Δv3l and Δv3r are the volume that flows in or out of the liquid chamber of the left and right accumulators, respectively. A1 and A2 are the liquid action areas of the rod chamber and the rodless chamber, respectively.The instantaneous pressure of gas in the accumulator is derived assuming the polytropic process of the gas as follows:(13)p0v0n=p3lv3ln,wheren is the polytropic exponent of gas which is usually between 1.2 and 1.4 [23], v3l is the gas volume in the gas chamber of the accumulator, and v0 and p0 are the initial gas volume and pressure in the accumulator, respectively.Combining (12) and (13) yields the following equation:(14)p3l=p0v0v0−Δv3ln,p3r=p0v0v0−Δv3rn.(2)
The relationship between throttle valve flow and pressure in segment “1-2” is given by(15)Q12=cdA1l2Δp12ρsgnΔp12=−A1x˙sl−x˙ul,where sgn represents the signum function.It can be rewritten as follows:(16)p1l=p2+ρ2A1x˙sl−x˙ulcdA1l2sgnQ12.(3)
The relationship between throttle valve flow and pressure in segment “3-4” is given by(17)Q34=cdA3l2Δp34ρsgnΔp34=−Q3lDerivingp4 as(18)p4=p3−ρ2Q3lcdA3l2sgn−Q3l.(4)
The relationship between throttle valve flow and pressure in segment “5-6” is given by(19)Q56=cdA2r2Δp56ρsgnΔp56=−A2x˙sr−x˙ur.Therefore,(20)p6=p5−ρ2Q56cdA2r2sgnQ56.The final expressions of pressure become as follows:(21)p1l=p3l+ρ2Q3lcdA3l2sgnQ3l−ρ2A1x˙sl−x˙ulcdA1l2sgnA1x˙sl−x˙ul,(22)p2l=p3r+ρ2Q3rcdA3r2sgnQ3r+ρ2A2x˙sl−x˙ulcdA2l2sgnA2x˙sl−x˙ul,(23)p1r=p3r+ρ2Q3rcdA3r2sgnQ3r−ρ2A1x˙sr−x˙urcdA1r2sgnA1x˙sr−x˙ur,(24)p2r=p3l+ρ2Q3lcdA3l2sgnQ3l+ρ2A2x˙sr−x˙urcdA2r2sgnA2x˙sr−x˙ur,where cd is the discharge coefficient and ρ is the fluid density. A1l, A2l, A1r, A2r, A3l, and A3r are the flow areas of throttle valves.Based on equations (21)–(24), it can be determined that the hydraulically interconnected subsystem consists of the elastic system and damping system. The elastic force comes from the accumulator, and the damping force is caused by the throttle valve. Taking the left side as an example, the elastic force can be calculated by(25)Fl_k=p3l−p0A1−p3r−p0A2.The damping force can be computed by(26)Fl_c=p1l−p3lA1−p2l−p3rA2.The vertical forces generated by the hydraulic interconnection subsystem shown in Figure2 are derived to be as follows:(27)Fl=p1l−p0A1−p2l−p0A2,Fr=p1r−p0A1−p2r−p0A2.


## 2.4. Experimental Verification for Hydraulically Interconnected Subsystem

A test bench used to validate the established hydraulically interconnected subsystem dynamics model is shown in Figure3. In the experimental test scheme, the hydraulic cylinder base is fixed, and the sinusoidal displacement excitations are loaded on the hydraulic rods by actuators. Subsequently, sensors measure the displacements, and actuator forces with a sampling frequency of 102.4 Hz. In Figures 4(a)–4(c), a pair of sinusoidal excitations with the same phase, a magnitude of 30 mm, and a period of 2 seconds were applied to perform a bounce motion. Meanwhile, in Figures 4(d)–4(f), the sinusoidal excitations are changed to a phase difference of 180° to simulate a roll motion.Figure 3
Test bench for the hydraulically interconnected subsystem. A–F: throttle valves;G1, G2: accumulators; H1, H2: hydraulic cylinders; S1, S2: displacement sensors; and F1, F2: actuation cylinders.Figure 4
Comparisons between simulation and experimental test. (a) Sinusoidal displacement excitation for bounce modal. (b) Hydraulic force for bounce modal. (c) Indicator diagram for bounce modal. (d) Sinusoidal displacement excitation for roll modal. (e) Hydraulic force for roll modal. (f) Indicator diagram for roll modal.
(a)
(b)
(c)
(d)
(e)
(f)It can be seen from Figure4 that, although there are errors between the simulated and the experimental curves that may be caused by the run-in period of hydraulic cylinders, the trend and the main characteristics of the curves are consistent. Therefore, the established dynamic model has the potential to describe the characteristics of the hydraulically interconnected subsystem.


## 3. Parameter Sensitivity Analysis

Understanding the effect of each throttle damping on the vibration response in bounce and roll modes is an important basis for the optimization of HIS and the development of semiactive control. We thereby analyzed the sensitivity of hydraulically interconnected subsystem parameters including damping coefficient of throttle valves and precharge pressure in accumulators, adopting the experimental design method of parameter study in Isight software. The parameter interval for sensitivity analysis and the vibration response function under the two modes are listed in Table2.Table 2
Experimental test scheme for sensitivity analysis.
Parameters
Symbol
Scale
Remark
Vibration response
Bounce mode
Roll mode
The diameter of throttle valve connecting to the rodless chamber
D
1l
[2,8] mm
A
i
=
π
D
i
2
/
4
i
=
1
l
,
2
l
,
3
l
Root mean square (RMS) value of sprung mass acceleration:1/N∑i=1Nx¨ci2
Root mean square (RMS) value of roll angle:1/N∑i=1Nθi2
The diameter of throttle valve connecting to the rod chamber
D
2l
[2,8] mm
The diameter of throttle valve connecting to the accumulator
D
3l
[2,8] mm
Precharge pressure in accumulator
p
0
[5 × 105, 25 × 105] pa
whereN is the sampling pointAssume that the throttle valve flow aperture is a regular round hole. Given that the damping coefficient of throttle valve depends on the flow cross-sectional area, the parameter of damping coefficient of throttle valve can be replaced by flow diameter. With consideration of the symmetrical arrangement of the hydraulically interconnected system, we use the left-side parameters of the hydraulically interconnected subsystem as representative to illustrate the experimental test scheme and analyze sensitivity results.The following conclusions can be drawn from the Pareto charts, and the main effect graph is illustrated in Figures5–7:(1)
In the bounce mode, the body vertical acceleration is sensitive to the flow aperture of the throttle valve connecting to the hydraulic cylinder, but insensitive to the accumulator precharge pressure and flow aperture of the throttle valve connecting to the accumulator.(2)
In the roll mode, the roll vibration is sensitive to the flow aperture of the throttle valve connecting to the accumulator, but insensitive to the rest of the parameters.Figure 5
Pareto chart in the bounce mode.Figure 6
Pareto chart in the roll mode.Figure 7
Main effect graph.
(a)
(b)It is interesting to note that the sensitivity of the ride comfort and the antiroll performance of the vehicle equipped with the HIS system to the hydraulic system parameters are exactly complementary.(3)
Although the vibration responses in the two modes are insensitive to precharge pressure in accumulators, it can be found from Figure7 that, as the precharge pressure increases, the root mean square (RMS) of the roll angle is effectively reduced.In addition, according to equations (25) and (26), the curves of elastic and damping characteristics of the hydraulically interconnected subsystem in the two modes are obtained in Figure 8.Figure 8
Elastic force and damping force in the two modes.
(a)
(b)From Figure8, it can be found that the elastic force and the damping force in the roll mode are much larger than that in the bounce mode. Thus, two conclusions can be drawn: first, the elastic force and damping force negligibly affect the ride comfort in the bounce mode and second, the elastic force and the damping force in roll mode will promptly increase to provide antiroll moment to improve the antiroll performance.For the result of sensitivity analysis, and the results of elastic and damping characteristics analysis, the main reason can be explained as follows: in the bounce mode, most of the liquid cross flows between the hydraulic cylinder chambers, and only a small part flows into the accumulators. As a result, the damping effect of the throttle valves connecting to the hydraulic cylinder chambers is fully played. By contrast, the contribution of the throttle valves below the accumulators is small, and the gas pressure change in the accumulator is very limited. In the roll mode, the situation is just the opposite; most of the liquid flows into accumulators, which makes full use of the damping effect of the throttle valves under the accumulator and causes the gas pressure in the accumulator to increase rapidly.Thus far, the above analysis provides a scientific basis for developing a HIS-based semiactive control method to improve the ride comfort and antiroll performance. The principle of the proposed novel HIS-based semiactive control system can be seen as Figure9: to improve the ride comfort by adjusting the diameters of throttle valves connecting to the hydraulic cylinders and to improve the antiroll performance by adjusting the diameters of throttle valves connecting to the accumulators.Figure 9
Novel semiactive control system based on HIS system.


## 4. HIS-Based Semiactive Control and Optimization

### 4.1. Semiactive Control for Improving Ride Comfort and Antiroll Performance

As previously stated, the sky-hook control method is adopted in this study. Therefore, the HIS-based semiactive control strategy can be summarized as follows:To improve ride comfort:ifx˙sjx˙sj−x˙uj≥0,D1j=Dmin1,D2j=Dmin2ifx˙sjx˙sj−x˙uj<0,D1j=Dmax1,D2j=Dmax2To improve antiroll performance:ifay≥ay0,D3j=Dmin3if ay<ay0,D3j=Dmax3where ay0 is the switched threshold of lateral acceleration and Dmin and Dmax are the minimize and maximize values of throttle valves, respectively.


### 4.2. Dual-Frequency-Range Switching Strategy and Optimization for HIS-Based Semiactive with Constraints

The research by Savaresi and Spelta [11] showed that the control effect of the sky-hook max-min damping switched strategy is remarkable when the excitation frequency nears the lower resonant frequency of sprung mass; otherwise, it performs similarly to or sometimes even worse than a passive suspension. Accordingly, a dual-frequency-range switching strategy is proposed in this study for the HIS-based semiactive control system. Specifically, when excitation frequency is lower than a threshold frequency, the HIS-based semiactive suspension adopts a group of maximum and minimum damping coefficients corresponding to low-frequency control strategy, and when the excitation frequency increases to a threshold, then the HIS-based semiactive suspension will switch to another group of maximum and minimum damping coefficients corresponding to high-frequency control strategy, where the appropriate values of the two groups of maximum and minimum damping coefficients are implemented by parameters optimization based on the sensitivity analysis.For the optimization of ride comfort, the diameters of throttle valves connecting to hydraulic cylinders are selected as optimization variables, the algebraic sum of weighted RMS value of body vertical acceleration and weighted max absolute value of body vertical acceleration is taken as the optimization objective function, and suspension deflection and tire dynamic load are used as constraints. On the basis of ride comfort optimization, we select the diameters of throttle valves connecting to accumulators as optimization variables and treat the RMS value of roll angle as the optimization objective function to improve the antiroll performance for the HIS-based semiactive control system. Finally, the optimization scheme is listed in Table3.Table 3
Parameter optimization for HIS-based semiactive control suspension system.
Parameter
Symbol
Remark
Scale
Optimization objective function
Constraints
The diameter of throttle valve connecting to the rodless chamber in front axle
D
1fl
D
1fl = D1fr
[2,8] (mm)
Bounce mode:minf11/N∑i=1Nx¨ci2+minf2⋅maxx¨cwhere f1 and f2 are weighted coefficients and N is the sampling point
x
s
j
0
−
x
u
j
0
≤
0.08
x
u
j
0
−
x
t
j
0
k
t
j
0
≤
G
s
j
0
where Gsj0 is the static load of every wheel
D
1fr
The diameter of throttle valve connecting to the rodless chamber in rear axle
D
1rl
D
1rl = D1rr
[2,8] (mm)
D
2rr
The diameter of throttle valve connecting to the rod chamber in front axle
D
2fl
D
2fl = D2fr
[2,8] (mm)
D
2fr
The diameter of throttle valve connecting to the rod chamber in rear axle
D
2rl
D
2rl = D2rr
[2,8] (mm)
D
2rr
The diameter of throttle valve connecting to the accumulator in front axle
D
3fl
D
3fl = D3fr
[2,8] (mm)
Roll mode:min1/N∑i=1nθi2
P
max ≤ 10 Mpa
D
3fr
The diameter of throttle valve connecting to the accumulator in rear axle
D
3rl
D
3rl = D3rr
[2,8] (mm)
D
3rr
Precharge pressure in accumulator
p
0
[5e5,25e5] (pa)Using the data listed in Table4, in accordance with the requirements in Table 3, the optimizations are performed in the joint environment of Isight and MATLAB. The optimized results for HIS-based semiactive control using sky-hook max-min damping switched method are listed in Table 5.Table 4
Parameter values for the HIS-based semiactive control suspension.
Parameter
Symbol
Value
Vehicle mass (kg)
m
1390
Sprung mass (kg)
m
s
1210
Unsprung mass (kg)
m
uj0
45
Spring stiffness (N/m)
k
j
18000
Tire stiffness (N/m)
k
tj
200000
Roll moment of inertia (kg·m2)
I
θ
390
Pitch moment of inertia (kg·m2)
I
φ
1920
Yaw moment of inertia (kg·m2)
I
z
2000
Height of the sprung mass center of gravity from the roll center (m)
h
0.3
Track width (m)
b
s
1.48
Distance from vehicle center of gravity to the front axle (m)
a
1.13
Distance from vehicle center of gravity to the rear axle (m)
b
1.47
Discharge coefficient
c
d
0.6
Liquid action areas of the rodless chamber (m2)
A
1
3.8 × 10−4
Liquid action areas of the rod chamber (m2)
A
2
3.5 × 10−4Table 5
Optimized results.
Parameter
Symbol
Initial value
Bounce mode: optimized value
Roll mode: optimized value
Low frequency
High frequency
Min
Max
Min
Max
Min
Max
The diameter of throttle valve connecting to the rodless chamber in front axle (mm)
D
1fl
5
2
4.8
4
6
—
D
1fr
The diameter of throttle valve connecting to the rodless chamber in rear axle (mm)
D
1rl
5
2
4
4
6
D
1rr
The diameter of throttle valve connecting to the rod chamber in front axle (mm)
D
2fl
5
2
5.4
5
5
D
2fr
The diameter of throttle valve connecting to the rod chamber in rear axle (mm)
D
2rl
5
2
5.6
5
5
D
2rr
The diameter of throttle valve connecting to the accumulator in front axle (mm)
D
3fl
5
—
2.5
6
D
3fr
The diameter of throttle valve connecting to the accumulator in rear axle (mm)
D
3rl
5
2.5
6
D
3rr
Precharge pressure in accumulator (pa)
p
0
5 × 105
2.5 × 106


### 4.3. Frequency-Range Selector for the HIS-Based Semiactive Dual-Frequency Switching Control System

In Section4.2, the parameter optimization for the low-frequency and high-frequency strategies was completed. Then, the critical issue of the proposed dual-frequency switching control becomes to how to identify the excitation frequency range.Considering that online identification of road excitation frequency has been a difficult research problem, especially for random road excitation with multiple frequency components, Savaresi and Spelta [11] suggested the optimal-control problem cannot be solved in a real-time implementation but can be solved numerically offline by making the simplifying assumption that the road profile is a pure sinusoid. We also assume the road profile is a pure sinusoid but modifies a frequency-range selector to identify the frequency range of the sinusoidal road excitation online.For a step excitation damping system, the transient response will disappear after a period of time, and the vibration frequency of the steady-state response will be the same as that of the excitation. Therefore, the road excitation frequency may be determined by the body vertical acceleration steady-state response. Based on this principle, Savaresi and Spelta [11] proposed an approach for frequency-selector; he constructed a symbolic decision function as shown in equation (28), and the relationship between the excitation frequency ω and the threshold frequency ω0 used for switching the ride comfort control strategy is judged by the sign of ft:(28)ft=x¨t2−ω02x˙t2,where xt and x˙t are the sprung mass vertical displacement and vertical velocity, whose frequency is regarded as the road excitation frequency and assumed as ω. ω0 is the threshold frequency; when ft>0, then ω>ω0; otherwise, ω<ω0.The approach developed by Savaresi is novel and meaningful. Based on the work of Savaresi, a further analysis was carried out: for the sinusoidal road input, assuming sprung mass displacement response asxt=Asinωt, then x˙t=Awcosωt and x¨t=−Aω2sinωt, thus ft=A2ω2ω2sin2ωt−ω02cos2ωt. Apparently, there is a phase difference of 90 degrees between sinωt and cosωt, and thus, the sign of ft cannot always represent the relationship between the excitation frequency ω and the threshold frequency ω0.We proposed an improved method, and the function is defined asFt in the following equation:(29)Ft=x¨t−ω02xt.According to the above assumptions,Ft=Aω2sinωt−Aω02sinωt. Because sinωt≥0, Ft will have the same sign with ω2−ω02; namely, if Ft>0, then ω>ω0, and otherwise, ω<ω0.Taking the vibration response of common passive suspension under road excitations of0.04sin2πt and 0.04sin6πt as examples, setting the threshold frequency as ω0 = 3.5π, which will be specially explained in Section 5.1, the accuracy of the Ft frequency-range selector and the ft frequency-selector is analyzed below.Figure10(a) shows that the ft frequency-range selector is accurate during most of each cycle, but it can only approximate the relationship between ω and ω0, which means that the semiactive control in some period cannot switch in accordance with designed requirement.Figure 10
Results of frequency-range selector for passive suspension. Road excitation frequency of (a) 2π, (b) 2π, (c) 6π, and (d) 6π.
(a)
(b)
(c)
(d)As shown in Figure10, it can be determined that, compared to the ft frequency-range selector, the Ft frequency-range selector can be more accurate, which provides a more reliable basis for dual-frequency switching control.However, the above comparisons are based on the offline data of the passive suspension system. It should be noted that, for the semiactive suspension using the sky-hook algorithm, the damping coefficient will be frequently switched between the maximum and minimum values, and there will be a transient response period after every switching, which may lead to incorrect judgment for theFt frequency-range selector.The online judgment results of theFt frequency-range selector for the proposed semiactive control system in the excitation frequencies of 2π and 6π are shown in Figures 11 and 12. Owing to interference by the transient response, the judgment results in errors in the two different excitation frequencies, causing the vibration response appearing to abruptly change.Figure 11
Results ofFt frequency-range selector for the semiactive suspension when excitation frequency is 2π.Figure 12
Results ofFt frequency-range selector for the semiactive suspension when excitation frequency is 6π.Considering that, because of the existence of the damping, the transient response caused by damping coefficient switching will be soon attenuated, it can be known that the proportion of the vibration energy of the transient response caused by damping coefficient switching is relatively small. Thereafter, we further improve theFt frequency-selector by using the energy moving average (EMA) method to filter out the impact of transient response. The specific process of EMA method is as follows:(1)
Definingk as a fixed number, the average energy of k points according to equations (30) and (31) is calculated.Note thatn1 is the number of test points; when it is less than k, the low-frequency control strategy is adopted by default:(30)Hn1−k=1k∑n1−kn1−1x¨ci2,Pn1−k=1k∑n1−kn1−1ω02xci2,(31)Fn1−k=Hn1−k−Pn1−k=1k∑n1−kn1−1x¨ci2−1k∑n1−kn1−1ω02xci2.(2)
ComparingHn1−k with Pn1−k, if Hn1−k<Pn1−k, namely, Fn1−k<0, it can be deduced that ω<ω0, and the semiactive control adopts the low-frequency strategy. Otherwise, the semiactive control switches to the high-frequency strategy.The judgment results of EMA frequency-selector for the HIS-based semiactive control system, under the sinusoidal road excitations with frequency of 2π and 6π rad/s, are shown in Figure 13.Figure 13
Judgment results of EMA frequency-range selector. Road excitation frequency of (a) 2π and (b) 6π.
(a)
(b)Figure13 shows that, when the excitation frequencies are, respectively, 2π and 6π and the threshold frequency is 3.5π, the EMA frequency-range selector accurately judges the relationship between the excitation frequency and the threshold frequency, which indicates that the EMA frequency-selector proposed in this paper is accurate and feasible.Thus far, the principle of the proposed HIS-based semiactive dual-frequency-range switching control system can be described, as shown in Figure14.Figure 14
Proposed HIS-based semiactive dual-frequency-range switching control system schedule.


## 4.1. Semiactive Control for Improving Ride Comfort and Antiroll Performance

As previously stated, the sky-hook control method is adopted in this study. Therefore, the HIS-based semiactive control strategy can be summarized as follows:To improve ride comfort:ifx˙sjx˙sj−x˙uj≥0,D1j=Dmin1,D2j=Dmin2ifx˙sjx˙sj−x˙uj<0,D1j=Dmax1,D2j=Dmax2To improve antiroll performance:ifay≥ay0,D3j=Dmin3if ay<ay0,D3j=Dmax3where ay0 is the switched threshold of lateral acceleration and Dmin and Dmax are the minimize and maximize values of throttle valves, respectively.


## 4.2. Dual-Frequency-Range Switching Strategy and Optimization for HIS-Based Semiactive with Constraints

The research by Savaresi and Spelta [11] showed that the control effect of the sky-hook max-min damping switched strategy is remarkable when the excitation frequency nears the lower resonant frequency of sprung mass; otherwise, it performs similarly to or sometimes even worse than a passive suspension. Accordingly, a dual-frequency-range switching strategy is proposed in this study for the HIS-based semiactive control system. Specifically, when excitation frequency is lower than a threshold frequency, the HIS-based semiactive suspension adopts a group of maximum and minimum damping coefficients corresponding to low-frequency control strategy, and when the excitation frequency increases to a threshold, then the HIS-based semiactive suspension will switch to another group of maximum and minimum damping coefficients corresponding to high-frequency control strategy, where the appropriate values of the two groups of maximum and minimum damping coefficients are implemented by parameters optimization based on the sensitivity analysis.For the optimization of ride comfort, the diameters of throttle valves connecting to hydraulic cylinders are selected as optimization variables, the algebraic sum of weighted RMS value of body vertical acceleration and weighted max absolute value of body vertical acceleration is taken as the optimization objective function, and suspension deflection and tire dynamic load are used as constraints. On the basis of ride comfort optimization, we select the diameters of throttle valves connecting to accumulators as optimization variables and treat the RMS value of roll angle as the optimization objective function to improve the antiroll performance for the HIS-based semiactive control system. Finally, the optimization scheme is listed in Table3.Table 3
Parameter optimization for HIS-based semiactive control suspension system.
Parameter
Symbol
Remark
Scale
Optimization objective function
Constraints
The diameter of throttle valve connecting to the rodless chamber in front axle
D
1fl
D
1fl = D1fr
[2,8] (mm)
Bounce mode:minf11/N∑i=1Nx¨ci2+minf2⋅maxx¨cwhere f1 and f2 are weighted coefficients and N is the sampling point
x
s
j
0
−
x
u
j
0
≤
0.08
x
u
j
0
−
x
t
j
0
k
t
j
0
≤
G
s
j
0
where Gsj0 is the static load of every wheel
D
1fr
The diameter of throttle valve connecting to the rodless chamber in rear axle
D
1rl
D
1rl = D1rr
[2,8] (mm)
D
2rr
The diameter of throttle valve connecting to the rod chamber in front axle
D
2fl
D
2fl = D2fr
[2,8] (mm)
D
2fr
The diameter of throttle valve connecting to the rod chamber in rear axle
D
2rl
D
2rl = D2rr
[2,8] (mm)
D
2rr
The diameter of throttle valve connecting to the accumulator in front axle
D
3fl
D
3fl = D3fr
[2,8] (mm)
Roll mode:min1/N∑i=1nθi2
P
max ≤ 10 Mpa
D
3fr
The diameter of throttle valve connecting to the accumulator in rear axle
D
3rl
D
3rl = D3rr
[2,8] (mm)
D
3rr
Precharge pressure in accumulator
p
0
[5e5,25e5] (pa)Using the data listed in Table4, in accordance with the requirements in Table 3, the optimizations are performed in the joint environment of Isight and MATLAB. The optimized results for HIS-based semiactive control using sky-hook max-min damping switched method are listed in Table 5.Table 4
Parameter values for the HIS-based semiactive control suspension.
Parameter
Symbol
Value
Vehicle mass (kg)
m
1390
Sprung mass (kg)
m
s
1210
Unsprung mass (kg)
m
uj0
45
Spring stiffness (N/m)
k
j
18000
Tire stiffness (N/m)
k
tj
200000
Roll moment of inertia (kg·m2)
I
θ
390
Pitch moment of inertia (kg·m2)
I
φ
1920
Yaw moment of inertia (kg·m2)
I
z
2000
Height of the sprung mass center of gravity from the roll center (m)
h
0.3
Track width (m)
b
s
1.48
Distance from vehicle center of gravity to the front axle (m)
a
1.13
Distance from vehicle center of gravity to the rear axle (m)
b
1.47
Discharge coefficient
c
d
0.6
Liquid action areas of the rodless chamber (m2)
A
1
3.8 × 10−4
Liquid action areas of the rod chamber (m2)
A
2
3.5 × 10−4Table 5
Optimized results.
Parameter
Symbol
Initial value
Bounce mode: optimized value
Roll mode: optimized value
Low frequency
High frequency
Min
Max
Min
Max
Min
Max
The diameter of throttle valve connecting to the rodless chamber in front axle (mm)
D
1fl
5
2
4.8
4
6
—
D
1fr
The diameter of throttle valve connecting to the rodless chamber in rear axle (mm)
D
1rl
5
2
4
4
6
D
1rr
The diameter of throttle valve connecting to the rod chamber in front axle (mm)
D
2fl
5
2
5.4
5
5
D
2fr
The diameter of throttle valve connecting to the rod chamber in rear axle (mm)
D
2rl
5
2
5.6
5
5
D
2rr
The diameter of throttle valve connecting to the accumulator in front axle (mm)
D
3fl
5
—
2.5
6
D
3fr
The diameter of throttle valve connecting to the accumulator in rear axle (mm)
D
3rl
5
2.5
6
D
3rr
Precharge pressure in accumulator (pa)
p
0
5 × 105
2.5 × 106


## 4.3. Frequency-Range Selector for the HIS-Based Semiactive Dual-Frequency Switching Control System

In Section4.2, the parameter optimization for the low-frequency and high-frequency strategies was completed. Then, the critical issue of the proposed dual-frequency switching control becomes to how to identify the excitation frequency range.Considering that online identification of road excitation frequency has been a difficult research problem, especially for random road excitation with multiple frequency components, Savaresi and Spelta [11] suggested the optimal-control problem cannot be solved in a real-time implementation but can be solved numerically offline by making the simplifying assumption that the road profile is a pure sinusoid. We also assume the road profile is a pure sinusoid but modifies a frequency-range selector to identify the frequency range of the sinusoidal road excitation online.For a step excitation damping system, the transient response will disappear after a period of time, and the vibration frequency of the steady-state response will be the same as that of the excitation. Therefore, the road excitation frequency may be determined by the body vertical acceleration steady-state response. Based on this principle, Savaresi and Spelta [11] proposed an approach for frequency-selector; he constructed a symbolic decision function as shown in equation (28), and the relationship between the excitation frequency ω and the threshold frequency ω0 used for switching the ride comfort control strategy is judged by the sign of ft:(28)ft=x¨t2−ω02x˙t2,where xt and x˙t are the sprung mass vertical displacement and vertical velocity, whose frequency is regarded as the road excitation frequency and assumed as ω. ω0 is the threshold frequency; when ft>0, then ω>ω0; otherwise, ω<ω0.The approach developed by Savaresi is novel and meaningful. Based on the work of Savaresi, a further analysis was carried out: for the sinusoidal road input, assuming sprung mass displacement response asxt=Asinωt, then x˙t=Awcosωt and x¨t=−Aω2sinωt, thus ft=A2ω2ω2sin2ωt−ω02cos2ωt. Apparently, there is a phase difference of 90 degrees between sinωt and cosωt, and thus, the sign of ft cannot always represent the relationship between the excitation frequency ω and the threshold frequency ω0.We proposed an improved method, and the function is defined asFt in the following equation:(29)Ft=x¨t−ω02xt.According to the above assumptions,Ft=Aω2sinωt−Aω02sinωt. Because sinωt≥0, Ft will have the same sign with ω2−ω02; namely, if Ft>0, then ω>ω0, and otherwise, ω<ω0.Taking the vibration response of common passive suspension under road excitations of0.04sin2πt and 0.04sin6πt as examples, setting the threshold frequency as ω0 = 3.5π, which will be specially explained in Section 5.1, the accuracy of the Ft frequency-range selector and the ft frequency-selector is analyzed below.Figure10(a) shows that the ft frequency-range selector is accurate during most of each cycle, but it can only approximate the relationship between ω and ω0, which means that the semiactive control in some period cannot switch in accordance with designed requirement.Figure 10
Results of frequency-range selector for passive suspension. Road excitation frequency of (a) 2π, (b) 2π, (c) 6π, and (d) 6π.
(a)
(b)
(c)
(d)As shown in Figure10, it can be determined that, compared to the ft frequency-range selector, the Ft frequency-range selector can be more accurate, which provides a more reliable basis for dual-frequency switching control.However, the above comparisons are based on the offline data of the passive suspension system. It should be noted that, for the semiactive suspension using the sky-hook algorithm, the damping coefficient will be frequently switched between the maximum and minimum values, and there will be a transient response period after every switching, which may lead to incorrect judgment for theFt frequency-range selector.The online judgment results of theFt frequency-range selector for the proposed semiactive control system in the excitation frequencies of 2π and 6π are shown in Figures 11 and 12. Owing to interference by the transient response, the judgment results in errors in the two different excitation frequencies, causing the vibration response appearing to abruptly change.Figure 11
Results ofFt frequency-range selector for the semiactive suspension when excitation frequency is 2π.Figure 12
Results ofFt frequency-range selector for the semiactive suspension when excitation frequency is 6π.Considering that, because of the existence of the damping, the transient response caused by damping coefficient switching will be soon attenuated, it can be known that the proportion of the vibration energy of the transient response caused by damping coefficient switching is relatively small. Thereafter, we further improve theFt frequency-selector by using the energy moving average (EMA) method to filter out the impact of transient response. The specific process of EMA method is as follows:(1)
Definingk as a fixed number, the average energy of k points according to equations (30) and (31) is calculated.Note thatn1 is the number of test points; when it is less than k, the low-frequency control strategy is adopted by default:(30)Hn1−k=1k∑n1−kn1−1x¨ci2,Pn1−k=1k∑n1−kn1−1ω02xci2,(31)Fn1−k=Hn1−k−Pn1−k=1k∑n1−kn1−1x¨ci2−1k∑n1−kn1−1ω02xci2.(2)
ComparingHn1−k with Pn1−k, if Hn1−k<Pn1−k, namely, Fn1−k<0, it can be deduced that ω<ω0, and the semiactive control adopts the low-frequency strategy. Otherwise, the semiactive control switches to the high-frequency strategy.The judgment results of EMA frequency-selector for the HIS-based semiactive control system, under the sinusoidal road excitations with frequency of 2π and 6π rad/s, are shown in Figure 13.Figure 13
Judgment results of EMA frequency-range selector. Road excitation frequency of (a) 2π and (b) 6π.
(a)
(b)Figure13 shows that, when the excitation frequencies are, respectively, 2π and 6π and the threshold frequency is 3.5π, the EMA frequency-range selector accurately judges the relationship between the excitation frequency and the threshold frequency, which indicates that the EMA frequency-selector proposed in this paper is accurate and feasible.Thus far, the principle of the proposed HIS-based semiactive dual-frequency-range switching control system can be described, as shown in Figure14.Figure 14
Proposed HIS-based semiactive dual-frequency-range switching control system schedule.


## 5. Simulations and Analysis

### 5.1. Response Analysis in the Frequency Domain

The frequency response of “Body vertical acceleration-Road excitation” is usually used to depict the improvement of vehicle ride comfort, but the notion of frequency response is not well defined for a strongly nonlinear semiactive control caused by damping switch. Thus, the amplitude-frequency characteristic is obtained by counting the ratio of output energy to input energy in different excitation frequencies, as shown in the following equation:(32)Hjωi=1/N1∑q=1N1x¨cq−x¨c¯21/N1∑q=1N1xtq−xt¯2,where j is a complex symbol, xtt is the road excitation, x¨c¯ is the average body vertical acceleration, xt¯ is the mean value of road excitation, N1 is the number of sampling points, and ωi represents the excitation frequency.The amplitude-frequency characteristics of “Body vertical acceleration-Road excitation” for 4 cases listed in Table6 are obtained in Figure 15.Figure 15
Amplitude-frequency characteristics of “Body vertical acceleration-Road excitation.”Table 6
Four cases of suspensions for frequency response analysis.
Case 1
The proposed HIS-based semiactive dual-frequency-range switching control strategy
Case 2
The HIS-based semiactive low-frequency control strategy
Case 3
The HIS-based semiactive high-frequency control strategy
Case 4
Passive suspensionFrom Figure15, compared with passive suspension corresponding to Case 4, it can be found that (1) for Case 2, the first resonance peak obviously reduces, but the amplitude ratio increases in the frequency-range higher than the first resonance frequency, which indicates that the ride comfort is improved in the first resonance frequency but deteriorates in the second resonance frequency; (2) for Case 3, the first resonance peak obviously increases, and the amplitude ratio reduces in the frequency range between the first and second resonance frequencies; and (3) Case 1 combines the advantages of Case 2 and Case 3, indicating that the proposed control system improves the ride comfort in a wider frequency range.Incidentally, from Figure15, the intersection frequency of the amplitude-frequency characteristics curves for low-frequency and the high-frequency strategies is approximately 11 rad/s, which is the reason we select the threshold frequency as 3.5π.


### 5.2. Time-Domain Simulations and Analysis

#### 5.2.1. Straight Forward with Sinusoidal Excitation

To test the effectiveness and superiority of the proposed semiactive control system in improving the ride comfort, the time-domain simulations of passive suspension, common semiactive control suspension, and the proposed HIS-based semiactive dual-frequency switching control suspension are analyzed and compared.Because the first resonance frequency is the natural frequency of the vehicle body, which significantly affects the ride comfort, semiactive control should ensure that it has a good attenuation effect on body vibration when road excitation frequency nears the first-order resonant frequency. On this basis, in order to make the common semiactive suspension comparable with the proposed HIS-based semiactive suspension, we obtain the maximum and minimum damping coefficients for the common semiactive suspension by linearizing the maximum and minimum damping curves of the proposed semiactive suspension. Figure16 shows the damping force curves of the proposed semiactive control suspension under a sinusoidal road excitation of 2π rad/s; the solid blue curves show the damping coefficient switches between the max and min values. Fitting the two curves by the least square method, the equivalent maximum and minimum damping coefficients of the common semiactive suspension are obtained as 2810 N·s/m and 292 N·s/m, respectively.Figure 16
Damping force curves comparison between common semiactive control suspension and the proposed semiactive control suspension.Figure17 shows the body vertical acceleration curves of the proposed and the common semiactive control under a road excitation of 0.04sin2πt. The two curves are basically coincident, indicating the two semiactive control systems can achieve nearly the same effect under the low-frequency road excitation condition, which provides a comparable foundation for the higher road excitation frequency comparison.Figure 17
Body vertical acceleration comparison between the common semiactive control suspension and the proposed semiactive control suspension.Figure18 shows that, under a road excitation of 0.04sin6πt, the proposed semiactive control system switches from low-frequency strategy to high-frequency strategy according to the EMA frequency-selector, and the amplitude of the body vertical acceleration is well restrained. As a result, the ride comfort is obviously improved. However, even though most of time, the common semiactive control system can achieve the same control effect as the proposed semiactive control system, there are obvious mutations, causing deterioration of the ride comfort, which is consistent with the conclusions of Savaresi and Spelta [11].Figure 18
Body vertical acceleration comparisons between the proposed and the common semiactive control suspension under the road excitation period of 6π.


#### 5.2.2. Simulations for Fishhook Test and Step Input Steering

The fishhook test and step input steering simulations for validating the antiroll performance are carried out at a speed of 60 km/h. Antiroll performance comparisons of passive suspension, common semiactive control suspension, and the proposed semiactive control suspension are shown in Figure19.Figure 19
Comparison of roll angle-roll rate phase-plane for the fishhook test. (a) Steering input. (b) Roll angle for fishhook test. (c) Roll angle for step input steering. (d) “Roll angle-Roll rate” phase-plane for fishhook test.
(a)
(b)
(c)
(d)As shown in Figure19, the common semiactive suspension significantly improves the antiroll performance during the transient steering process, but it cannot improve the antiroll performance during the steady steering process. By contrast, the proposed HIS-based semiactive dual-frequency switching control suspension can improve antiroll performance in both transient and steady-state steering processes.Figure19(d) depicts the “Body roll angle-Body roll rate” phase-plane, the common semiactive control, whose roll stability domain is superior to passive suspension. By contrast, the roll stability domain of the proposed semiactive control suspension has been further improved.


## 5.1. Response Analysis in the Frequency Domain

The frequency response of “Body vertical acceleration-Road excitation” is usually used to depict the improvement of vehicle ride comfort, but the notion of frequency response is not well defined for a strongly nonlinear semiactive control caused by damping switch. Thus, the amplitude-frequency characteristic is obtained by counting the ratio of output energy to input energy in different excitation frequencies, as shown in the following equation:(32)Hjωi=1/N1∑q=1N1x¨cq−x¨c¯21/N1∑q=1N1xtq−xt¯2,where j is a complex symbol, xtt is the road excitation, x¨c¯ is the average body vertical acceleration, xt¯ is the mean value of road excitation, N1 is the number of sampling points, and ωi represents the excitation frequency.The amplitude-frequency characteristics of “Body vertical acceleration-Road excitation” for 4 cases listed in Table6 are obtained in Figure 15.Figure 15
Amplitude-frequency characteristics of “Body vertical acceleration-Road excitation.”Table 6
Four cases of suspensions for frequency response analysis.
Case 1
The proposed HIS-based semiactive dual-frequency-range switching control strategy
Case 2
The HIS-based semiactive low-frequency control strategy
Case 3
The HIS-based semiactive high-frequency control strategy
Case 4
Passive suspensionFrom Figure15, compared with passive suspension corresponding to Case 4, it can be found that (1) for Case 2, the first resonance peak obviously reduces, but the amplitude ratio increases in the frequency-range higher than the first resonance frequency, which indicates that the ride comfort is improved in the first resonance frequency but deteriorates in the second resonance frequency; (2) for Case 3, the first resonance peak obviously increases, and the amplitude ratio reduces in the frequency range between the first and second resonance frequencies; and (3) Case 1 combines the advantages of Case 2 and Case 3, indicating that the proposed control system improves the ride comfort in a wider frequency range.Incidentally, from Figure15, the intersection frequency of the amplitude-frequency characteristics curves for low-frequency and the high-frequency strategies is approximately 11 rad/s, which is the reason we select the threshold frequency as 3.5π.


## 5.2. Time-Domain Simulations and Analysis

### 5.2.1. Straight Forward with Sinusoidal Excitation

To test the effectiveness and superiority of the proposed semiactive control system in improving the ride comfort, the time-domain simulations of passive suspension, common semiactive control suspension, and the proposed HIS-based semiactive dual-frequency switching control suspension are analyzed and compared.Because the first resonance frequency is the natural frequency of the vehicle body, which significantly affects the ride comfort, semiactive control should ensure that it has a good attenuation effect on body vibration when road excitation frequency nears the first-order resonant frequency. On this basis, in order to make the common semiactive suspension comparable with the proposed HIS-based semiactive suspension, we obtain the maximum and minimum damping coefficients for the common semiactive suspension by linearizing the maximum and minimum damping curves of the proposed semiactive suspension. Figure16 shows the damping force curves of the proposed semiactive control suspension under a sinusoidal road excitation of 2π rad/s; the solid blue curves show the damping coefficient switches between the max and min values. Fitting the two curves by the least square method, the equivalent maximum and minimum damping coefficients of the common semiactive suspension are obtained as 2810 N·s/m and 292 N·s/m, respectively.Figure 16
Damping force curves comparison between common semiactive control suspension and the proposed semiactive control suspension.Figure17 shows the body vertical acceleration curves of the proposed and the common semiactive control under a road excitation of 0.04sin2πt. The two curves are basically coincident, indicating the two semiactive control systems can achieve nearly the same effect under the low-frequency road excitation condition, which provides a comparable foundation for the higher road excitation frequency comparison.Figure 17
Body vertical acceleration comparison between the common semiactive control suspension and the proposed semiactive control suspension.Figure18 shows that, under a road excitation of 0.04sin6πt, the proposed semiactive control system switches from low-frequency strategy to high-frequency strategy according to the EMA frequency-selector, and the amplitude of the body vertical acceleration is well restrained. As a result, the ride comfort is obviously improved. However, even though most of time, the common semiactive control system can achieve the same control effect as the proposed semiactive control system, there are obvious mutations, causing deterioration of the ride comfort, which is consistent with the conclusions of Savaresi and Spelta [11].Figure 18
Body vertical acceleration comparisons between the proposed and the common semiactive control suspension under the road excitation period of 6π.


### 5.2.2. Simulations for Fishhook Test and Step Input Steering

The fishhook test and step input steering simulations for validating the antiroll performance are carried out at a speed of 60 km/h. Antiroll performance comparisons of passive suspension, common semiactive control suspension, and the proposed semiactive control suspension are shown in Figure19.Figure 19
Comparison of roll angle-roll rate phase-plane for the fishhook test. (a) Steering input. (b) Roll angle for fishhook test. (c) Roll angle for step input steering. (d) “Roll angle-Roll rate” phase-plane for fishhook test.
(a)
(b)
(c)
(d)As shown in Figure19, the common semiactive suspension significantly improves the antiroll performance during the transient steering process, but it cannot improve the antiroll performance during the steady steering process. By contrast, the proposed HIS-based semiactive dual-frequency switching control suspension can improve antiroll performance in both transient and steady-state steering processes.Figure19(d) depicts the “Body roll angle-Body roll rate” phase-plane, the common semiactive control, whose roll stability domain is superior to passive suspension. By contrast, the roll stability domain of the proposed semiactive control suspension has been further improved.


## 5.2.1. Straight Forward with Sinusoidal Excitation

To test the effectiveness and superiority of the proposed semiactive control system in improving the ride comfort, the time-domain simulations of passive suspension, common semiactive control suspension, and the proposed HIS-based semiactive dual-frequency switching control suspension are analyzed and compared.Because the first resonance frequency is the natural frequency of the vehicle body, which significantly affects the ride comfort, semiactive control should ensure that it has a good attenuation effect on body vibration when road excitation frequency nears the first-order resonant frequency. On this basis, in order to make the common semiactive suspension comparable with the proposed HIS-based semiactive suspension, we obtain the maximum and minimum damping coefficients for the common semiactive suspension by linearizing the maximum and minimum damping curves of the proposed semiactive suspension. Figure16 shows the damping force curves of the proposed semiactive control suspension under a sinusoidal road excitation of 2π rad/s; the solid blue curves show the damping coefficient switches between the max and min values. Fitting the two curves by the least square method, the equivalent maximum and minimum damping coefficients of the common semiactive suspension are obtained as 2810 N·s/m and 292 N·s/m, respectively.Figure 16
Damping force curves comparison between common semiactive control suspension and the proposed semiactive control suspension.Figure17 shows the body vertical acceleration curves of the proposed and the common semiactive control under a road excitation of 0.04sin2πt. The two curves are basically coincident, indicating the two semiactive control systems can achieve nearly the same effect under the low-frequency road excitation condition, which provides a comparable foundation for the higher road excitation frequency comparison.Figure 17
Body vertical acceleration comparison between the common semiactive control suspension and the proposed semiactive control suspension.Figure18 shows that, under a road excitation of 0.04sin6πt, the proposed semiactive control system switches from low-frequency strategy to high-frequency strategy according to the EMA frequency-selector, and the amplitude of the body vertical acceleration is well restrained. As a result, the ride comfort is obviously improved. However, even though most of time, the common semiactive control system can achieve the same control effect as the proposed semiactive control system, there are obvious mutations, causing deterioration of the ride comfort, which is consistent with the conclusions of Savaresi and Spelta [11].Figure 18
Body vertical acceleration comparisons between the proposed and the common semiactive control suspension under the road excitation period of 6π.


## 5.2.2. Simulations for Fishhook Test and Step Input Steering

The fishhook test and step input steering simulations for validating the antiroll performance are carried out at a speed of 60 km/h. Antiroll performance comparisons of passive suspension, common semiactive control suspension, and the proposed semiactive control suspension are shown in Figure19.Figure 19
Comparison of roll angle-roll rate phase-plane for the fishhook test. (a) Steering input. (b) Roll angle for fishhook test. (c) Roll angle for step input steering. (d) “Roll angle-Roll rate” phase-plane for fishhook test.
(a)
(b)
(c)
(d)As shown in Figure19, the common semiactive suspension significantly improves the antiroll performance during the transient steering process, but it cannot improve the antiroll performance during the steady steering process. By contrast, the proposed HIS-based semiactive dual-frequency switching control suspension can improve antiroll performance in both transient and steady-state steering processes.Figure19(d) depicts the “Body roll angle-Body roll rate” phase-plane, the common semiactive control, whose roll stability domain is superior to passive suspension. By contrast, the roll stability domain of the proposed semiactive control suspension has been further improved.


## 6. Conclusions

In vehicles equipped with an HIS system, sensitivity analysis results show that the sensitivities of ride comfort and antiroll performance to the hydraulic system parameters are complementary. Therefore, a novel HIS-based semiactive control method was developed by adjusting the damping aperture of throttle valves connecting to the hydraulic cylinders to improve ride comfort and regulate the flow aperture of throttle valves connecting to the accumulators to enhance antiroll performance.Thereafter, a dual-frequency-range switching control strategy for improving ride comfort was proposed. The hydraulic system parameters for the low-frequency and the high-frequency sinusoidal road excitations were respectively optimized, and an EMA frequency-range selector was designed to identify the relationship between the excitation frequency and the threshold frequency to switch the ride comfort control strategy.The analysis in the frequency domain and time domain showed that the proposed HIS-based semiactive dual-frequency-range switching control method can improve vehicle ride comfort in a wider frequency bandwidth and have better antiroll performance than common semiactive suspension.



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*Source: 5193593-2019-05-12.xml*