Research ArticleControl and Stability Analysis of Double Time-Delay ActiveSuspension Based on Particle Swarm Optimization

Kaiwei Wu and Chuanbo Ren

School of Transportation and Vehicle Engineering Shandong University of Technology Zibo 255000 Shandong China

Correspondence should be addressed to Chuanbo Ren chuanborsduteducn

Received 29 March 2020 Revised 12 August 2020 Accepted 21 August 2020 Published 10 September 2020

Academic Editor Mohammad Rafiee

Copyright copy 2020 Kaiwei Wu and Chuanbo Ren ampis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

With the application of an active control unit in the suspension system the phenomenon of time delay has become an importantfactor in the control system Aiming at the application of time-delay feedback control in vehicle active suspension systems thispaper has researched the dynamic behavior of semivehicle four-degree-of-freedom structure including an active suspension withdouble time-delay feedback control focusing on analyzing the vibration response and stability of the main vibration system of thestructureampe optimal objective function is established according to the amplitude-frequency characteristics of the system and theoptimal time-delay control parameters are obtained by using the particle swarm optimization algorithm ampe stability for activesuspension with double time-delay feedback control by frequency-domain scanning method is analyzed and the simulationmodel of active suspension with double time delay based on feedback control is finally established ampe simulation results showthat the active suspension with double time-delay feedback control could reduce the bodyrsquos vertical vibration acceleration pitchacceleration and other indicators significantly whether under harmonic excitation or random excitation So it is indicating thatthe active suspension with double time-delay feedback control has a better control effect in improving the ride comfort of the carand it has important reference value for further research on suspension performance optimization

1 Introduction

With the rapid improvement of modern automobile tech-nology more and more consumers have higher require-ments for car ride comfort and operational stability In theoverall structure of the car the biggest relationship betweencomfort and operation stability is the suspension system ofthe car Suspension as a part of the elastic connection be-tween the body and the axle bears the force between theunsprung weight and the sprung weight buffers the impactof road surface excitation on the body and attenuates thevibration of various loads on the body It is an important partof the vehicle Compared with passive suspensions activesuspensions are highly adaptive [1 2] ampey can adjust theoptimal damping in real time for the movement and roadconditions of cars Research on active suspension systemshas become a focus in the field of vehicle engineering Withthe application of the active control link in the actual

engineering system control the control system needs acertain time from the signal collection and transmissioncomputer analysis and response of the actuator and timedelay has become an inevitable factor in the suspensionsystem control process [3]

Within this work it is found that the time-delay controlsystem has strong damping characteristics and high effi-ciency in the wide frequency bandwidth of the externalexcitations under the condition of system stability In ad-dition the proposed controller requires external energy lessthan the control of stiffness and damping Time delay has agreat influence on the dynamic characteristics of the activesuspension system and even leads to the instability of thefeedback control system [4] However designing effectivetime delay and feedback gain canmake the main system get agood damping effect Many scholars have done a lot ofresearch on the time-delay problem in vibration controlsystems For example Olgac et al [5 6] proposed a time-

HindawiShock and VibrationVolume 2020 Article ID 8873701 12 pageshttpsdoiorg10115520208873701

delay dynamic vibration absorber and started a series ofbasic researches namely installing a shock absorber withtime-delay displacement feedback on the main vibrationsystem Vyhlidal et al [7] studied and analyzed the stabilityof acceleration-feedback time-delay dynamic vibration ab-sorber in the full time-delay region by using spectral analysismethod and provided an effective method for the design ofvibration absorber Hu and Wang [8 9] in order to analyzethe stability interval of the time-delay dynamical systemused the stability switching idea to study the influence of thetime-delay positive feedback on the system Su and Tang[10 11] studied the design of active suspension vibrationcontroller with time delay by using the quarter-car modelunder random excitation Xu and Li [12 13 14] studied thedynamic behavior of two-degree-of-freedom structure withtime delay by using a direct method and analyzed the dy-namic characteristics and stability of the system Under thepremise of ensuring stability the vibration response of themain system was obtained through simulation Saeed et al[15ndash18] studied the influence of time delay on the controlsystem dynamics and obtained the time-delay stability re-gion ampen they applied the time-delay feedback control tothe vibration control of the Jeffcott-rotor system and per-formed a numerical simulation ampe simulation results showthat time-delay feedback control can effectively suppresssystem vibration From the existing research the time-delaycontrol despite the vibration analysis method has made greatdevelopment most of these studies are the work of the basictheory of the structure with few degrees of freedom ampesuspension system of the vehicle is a very complex multi-degree of freedom vibration system with many uncertaintiesand complexities For the quarter-car model with singletime-delay feedback control although the model is simplethe dynamic response obtained is not comprehensive amperesearch on the stability of the delayed feedback system is notperfect so this paper analyzes the dynamic response of ahalf-car four-degree-of-freedom suspension model withdouble time-delay feedback control under the premise ofstability

According to the dynamic characteristics of a four-de-gree-of-freedom half-vehicle suspension this paper appliesthe active suspension theory with stable double time-delayfeedback control to a half-vehicle model and innovates afrequency-domain scanning method to determine the sta-bility interval of double time delay ampe active suspensionwith double delay feedback control under random excitationis simulated

2 Half-Care Mathematical Model

According to the characteristics of the vehicle suspensionthe physical model of the vehicle suspension system issimplified from the perspective of scientific research Whenthe vehicle is symmetrical to its longitudinal axis only thevertical vibration and pitch vibration of the vehicle bodyhave the greatest impact on ride comfort which is simplifiedas a two-axis four-degree-of-freedom physical model m isthe mass of the half-car body I is the moment of inertia ofthe axis perpendicular to the centroid of the half-car modelmtf is the unsprung mass of the front wheels mtr is theunsprung mass of the rear wheels ksf is the spring stiffnesscoefficient of the front suspension ksr is the spring stiffnesscoefficient of the rear suspension csf is the damping of thefront suspension coefficient csr is the damping coefficientgenerated by the rear suspension ktf is the stiffness coef-ficient generated by the front tire ktr is the stiffness coef-ficient generated by the rear tire φ is the longitudinal pitchangle of the half body of the vehicle xtf xtr are the massdisplacement of front and rear sprung xgf xgr are theexcitation displacement input of front and rear road surfaceff fr are the active control force of the front and rearsuspension Lf Lr are the distance from the front and rearsuspension to the center of mass of the half-car the model isshown in Figure 1

According to Newtonrsquos second law the dynamic dif-ferential equation of the half-car four-degree-of-freedommodel can be obtained as follows [19]

mtfeuroxtf minus ksf xsf minus xtf1113872 1113873 minus csf _xsf minus _xtf1113872 1113873 + ktf xtf minus xgf1113872 1113873 + ff 0

mtreuroxtr minus ksr xsr minus xtr( 1113857 minus csr _xsr minus _xtr( 1113857 + ktr xtr minus xgr1113872 1113873 + fr 0

meuroxc + ksf xsf minus xtf1113872 1113873 + csf _xsf minus _xtf1113872 1113873 minus ff + ksr xsr minus xtr( 1113857 + csr _xsr minus _xtr( 1113857 minus fr 0

Ieuroφ minus Lf ksf xsf minus xtf1113872 1113873 + csf _xsf minus _xtf1113872 1113873 minus ff1113960 1113961 + Lr ksr xsr minus xtr( 1113857+csr _xsr minus _xtr( 1113857 minus fr1113859 01113858

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where ff g1xtf(t minus τ1) and fr g2xtr(t minus τ2) representthe active control force of front suspension and rear suspen-sion respectively xgf and xgr are the excitation displacementinput of the front and rear road surface respectively g1 and g2are the time-delay feedback gain coefficients of the activecontrol force of the front and rear suspensions respectively τ1and τ2 are the time delay of the active control force of the frontand rear suspensions respectively

As in Figure 1 when the pitch angle φ is small φ asymp tanφthe approximate values are as follows

xsf xc minus Lfφ

xsr xc + Lrφ1113896 (2)

Equations of motion (1) and (2) can be expressed asfollows

2 Shock and Vibration

euroxtf ksf xsf minus xtf1113872 1113873 + csf _xsf minus _xtf1113872 1113873 minus ktf xtf minus xgf1113872 1113873 minus ff1113872 1113873

mtf

euroxtr ksr xsr minus xtr( 1113857 + csr _xsr minus _xtr( 1113857 minus ktr xtr minus xgr1113872 1113873 minus fr1113872 1113873

mtr

euroxc minusksf xsf minus xtf1113872 1113873 minus csf _xsf minus _xtf1113872 1113873 + ff minus ksr xsr minus xtr( 1113857 minus csr _xsr minus _xtr( 1113857 + fr1113872 1113873

m

euroφ Lf ksf xsf minus xtf1113872 1113873 + csf _xsf minus _xtf1113872 1113873 minus ff1113960 1113961 minus Lr ksr xsr minus xtr( 1113857 + csr _xsr minus _xtr( 1113857 minus fr1113858 1113859

I

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

3 Optimization Control Analysis ofSuspension System

Figure 2 shows the amplitude-frequency characteristics of thebody when the time- delay feedback control parameters aredifferent It can be seen from Figure 2 that when the timedelay and the feedback gain are zero the system reaches thehighest point of amplitude at 59Hz ampis indicates that whenthe external excitation frequency is equal to the naturalfrequency of the system the system is forced to vibrate at themaximum amplitude due to resonance effects When thefeedback control parameters are τ03 0434 τ02 0452gf01 19287 gr01 23930 the systemrsquos vibration responsedecreases significantly around 59Hz ampis shows that thetime-delay feedback control can attenuate vibration and theremust be a maximum damping point in a certain intervalWhen the feedback gain parameters are τ05 0906τ06 0546 gf02 25333 gr02 23371 there are multiple peaksin the response curve of the system ampis indicates that thepresence of a time delay factor can also destabilize the system

amprough analysis it can be seen that the time delayfeedback control can change the vibration response of thesystem amperefore in this paper the optimal time delayand feedback gain coefficient are obtained by particleswarm optimization and the frequency-domain scanningmethod is used to ensure the stability of the time-delayfeedback system so as to achieve the best vibration re-duction effect

Fourier transform the dynamic differential equation ofthe half-car four-degree-of-freedom model transform thetime-domain characteristics to the frequency-domainrange for research and analysis and rewrite it into theform of a matrix ampe Fourier change of equation (1) is asfollows

A44X B (4)

where X Xc φ Xtf Xtr1113960 1113961T B (ktfXgfmtf)0(ktr1113960

Xgrmtr)01113961T Xgr XgfeminussΔt

Lf Lr

xc

fr

ktf

xtf xtr

xsrxsf

csrcsfksf ksr

ktr

mtf

xgf xgr

mtr

m I φ

ff

Figure 1 Four-degree-of-freedom suspension model with active control with double time- delay

Shock and Vibration 3

A44

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

a11 minusmtfω2

+ iωcsf + ksf + ktf1113872 1113873 + gfeminusiτ1ω

a12 0 a13

a14 iωcsfLf + ksfLf

a21 0 a22

a23 minusksr minus iωcsr

a24 ksrLr minus iωcsrLr

a31 minusksf minus iωcsf minus gfeminus iωτ1

a32 minusksr minus iωcsr minus greminusiωτ2

a33 minusmcω2

+ iω csf + csr1113872 1113873 + ksf + ksr

a34 iω csrLr minus csfLf1113872 1113873 + ksrLr minus ksfLf

a41 Lf iωcsf + ksf + gfeminusiωτ11113872 1113873

a42 minusLr iωcSr + kSr+greminusiωτ211138731113872

a43 minusLf iωcsf + ksf1113872 1113873 + Lr iωcsr + ksr( 1113857

a44 minusIω2+ L

2f ksf + iωcsf1113872 1113873 + L

2r iωcSr + kSr( 1113857

(5)

External excitation input to front and rear wheels has atime difference of ∆t of xgr(t) xgf(t minus Δt) whereΔt (Lf + Lr)v

In equation (4) the amplitude-frequency characteristicfunction |H _Xc sim _Xgf

(ω)| ω|Xc(ω)Xgf(ω)| of the verticalacceleration of the body centroid and the amplitude-fre-quency characteristic function |Heuroφsim _Xgf

(ω)| ω|φ(ω)Xgf(ω)| of the acceleration of the body pitch angle areobtained respectively

ampe vertical centroid acceleration of the vehicle bodyand the vertical pitch acceleration of the vehicle body areimportant indicators to measure the ride comfort of thevehicle amperefore the vertical acceleration of the center ofmass of the vehicle body and the vertical accelerationangle of the vehicle body are used as the main evaluationindicators to establish a weighted objective function forthe optimization of the control parameters At the sametime according to the engineering background the searchrange is set the feedback gain is no more than twice thepassive stiffness ampe smaller the delay the smaller theovershoot

min J gf gr τ1 τ21113872 1113873 n1 H euroXcsim _Xgf(ω)

11138681113868111386811138681113868

11138681113868111386811138681113868 + n2 Heuroφsim _Xgf(ω)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

stminus2klegle 2k

0lt τ le 11113896

(6)

where the weighting coefficients n1 and n2 are 07 and 03which measures the importance of each amplitude-fre-quency function in the objective function

ampe objective function is optimized based on theestablished objective function and the characteristics ofthe particle swarm optimization algorithm [20] Due tothe large difference in the magnitude of the optimizedfeedback gain and time delay a four-dimensional searchspace is assumed to represent the two feedback gains andtime delays respectively ampe individual positions areupdated by tracking individual extreme values Pbest andgroup extreme values Gbest Once the position isupdated the fitness value is calculated By comparing thefitness value of the new particle with the individualextreme value the fitness value of the group extremevalue updates the individual extreme value Pbest and thegroup extreme value Gbest position in each iterationprocess the particle passes the individual extreme valueand group extreme value update their speed and posi-tion ampe formula is updated to the following equation

Vk+1ik αV

kid + ε1r1 P

kid minus X

kid1113872 1113873 + ε2r2 P

kgd minus X

kid1113872 1113873

Xk+11 X

ki1 + rV

k+1i1

(7)

where α is the inertial weight d = 1 2 i = 1 2 n k is thecurrent iteration time Vi d is the particle update speed ε1and ε2 are nonnegative acceleration factor r1 and r2 aregenerated from [0 1] random constant We select 60particles for iterative optimization randomly in order tofind the optimal individual extremum and group extremummore quickly during optimization ε1 = ε2 = 2 after 200iterations we obtain the change graph of the number ofiterations of the fitness function of the suspension per-formance index as in Figure 3 With reference to a vehiclersquossuspension parameters (as in Table 1) the global optimalcontrol parameters under random excitation and harmonicexcitation are obtained respectively after optimizationgf 19345Nm gr 26150Nm τ1 05530 s τ2

05186 s gff 29653Nm grr minus4416Nm τ11 03255 sτ22 05438 s

0

05

1

15

2

25

3

35

4

45

5

|XcXg

f|

18166 8 10 200 1242 14Frequency (Hz)

τ05 = 0906 τ06 = 0546 gf02 = 25333 gr02 = 23371τ03 = 0434 τ04 = 0452 gf01 = 19287 gr01 = 23930τ01 = 0 τ02 = 0 gf0 = 0 gr0 = 0

Figure 2 Amplitude-frequency response curve

4 Shock and Vibration

4 Stability Analysis

ampe existence of time delay has a great impact on the dy-namic performance of the active suspension system In orderto ensure the stability of the feedback control system withdouble time delay the frequency-domain scanning methodis proposed in this paper to analyze the stability of theoptimized control parameters [21ndash24]

First equation (1) is rewritten as the form of stateequation

_x(t) Ax(t) + Bx(t minus τ) (8)

where A and B are constant matrices and τ ge 0 is constant

x xtf _xtf xtr _xtr xc _xc j _j1113960 1113961T

A

0 1 0 0 0 0 0 0

minusksf + ktfmtf minuscsf + csrmtf 0 0 msfmtf csfmtf minusLfksfmtf Lfccfmtf

0 0 0 1 0 0 0 0

0 0 minusksr + ktrmtr minuscsrmtr ksrmtr minuscsrmtr Lrksrmtr Lrcsrmtr

0 0 0 0 0 1 0 0

ksfm csfm ksrm csrm minusksf + ktfm minuscsf + csrm Lfksf minus Lrksrm Lfcsf minus Lrcsrm

0 0 0 0 0 0 0 1

minusLfksfI minusLfccfI LrksrI LrcsrI Lfksf minus LrksrI Lfcsf minus LrcsrI minusL2fksf + L

2rKsrI minusL

2fcsf + L

2rcsrI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

0 0 0 0 0 0 0 0

1m 0 1m 0 0 0 0 0

0 0 0 0 0 0 0 0

minuslfI 0 lfI 0 0 0 0 0

0 0 0 0 0 0 0 0

minus1mtf 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 minus1mtr 0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

Optimal individual fitness

0

02

04

06

08

1

12

14

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(a)

Optimal individual fitness

002

0025

003

0035

004

0045

005

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(b)

Figure 3 Iterative optimization of the fitness function (a) ampe fitness function of optimization under harmonic excitation (b) ampe fitnessfunction of optimization under random excitation

Shock and Vibration 5

ampe characteristic equation of (1) of time-delay controlsystem is as follows

det sI minus A minus Beminusτs

( 1113857 0 (10)

ampe specific form is as follows

CE(s z) an(s)zn

+ anminus1(s)znminus 1

+ middot middot middot + a0(s) 1113944n

k0ak(s)z

k 0

(11)

where

a0 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a1 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a2 c8s8

+ c7s7

+ c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

(12)

z eminusτs n 2 c is the coefficient of the characteristicequation

Furthermore for a given ak(s)(k 0 1 n) thereexists a continuous polynomial of Gl(s)(1 1 middot middot middot n) thatsatisfies the systemrsquos equation of state that can be equivalentto the following

CE(s z) an(s) 1113945n

l1z + Gl(s)( 1113857

k (13)

where an(s) is a constant Gl(s) satisfies the following

1113944

n

k0

ak(s)

an(s)1113888 1113889 minusGl(s)( 1113857

k 0 (14)

It is not necessary to find polynomial Gl(s) because s

and z are complex numbers CE(s z) 0 if and only if asubquasipolynomial (z + Gl(s))(1 1 n) is equal tozero

Calculate the time delay substitute s wlm forCE(s z) 0 calculate z at zero choose the root of 1 x + yiby calculating the θ value in eminusiθ x + yi whereθ arctan(minusyx) + 2π when xgt 0 ygt 0 the time delay isθw + 2rπw(r 0 1 infin)

We made stability analysis of time-delay control pa-rameters gf gr τ1 and τ2 under random excitation byfrequency-domain scanning method ampe control system isstable when the time delay is equal to zero (the characteristicroots of the equation are as follows)

λ1 minus20267857911405181608535831306952 + 17090553240960446610326055558861lowasti

λ2 minus18708774367416789822443106075043 minus 70205741985817738371855015300647lowasti

λ3 minus37274143773467233351129630918096 + 61657802483584587870216137245193lowasti

λ4 minus18708774367416789822443106075043 + 70205741985817738371855015300647lowasti

λ5 minus21213896670092750906944528230251 + 72731773377804970851980013939354lowasti

λ6 minus20267857911405181608535831306952 minus 17090553240960446610326055558861lowasti

λ7 minus21213896670092750906944528230251 minus 72731773377804970851980013939354lowasti

λ8 minus37274143773467233351129630918096 minus 61657802483584587870216137245193lowasti

(15)

It can be obtained that the characteristic roots of thesystem are in the left half-plane of the complex planeamperefore the active suspension system with double time-delay feedback control at τ 0 is stable Figure 4 shows therelationship between |z| and w generated by the frequency-domain scanning method ampe system has two crossoverfrequencies w1 207651 w2 207653 ampe system hastwo positive imaginary roots corresponding to two sub-quasipolynomials and the crossing directions of the twoimaginary roots are from left to rightampus as the time delayincreases once the system has characteristic virtual roots

the system will no longer be stable the system is asymp-totically stable when τ2 isin (0 06312) Similarly the stabilityanalysis of the time-delay control parameters under har-monic excitation is performed and the stability interval ofthe time delay is τ3 isin (0 05817)

5 Establishing a Simulation Model andResult Analysis

51 Simulation Analysis under Harmonic Excitation Takethe optimized double time-delay control parameters into

Table 1 Vehicle suspension model parameters

Vehicle parameters Valuem(kg) 690I(kgm2) 1200mtf(kg) 405mtr(kg) 454ktf(ktr)(Nmiddotmminus1) 192000ksf(Nmiddotmminus1) 17000ksr(Nmiddotmminus1) 22000Lf(m) 125Lr(m) 151csf(csr)(Nmiddotsmiddotmminus1) 1500

6 Shock and Vibration

equation (1) Taking the harmonic excitationxgf 005 sin(7t) as the road input excitation the vibrationresponse characteristics of the vehicle in the time domain ofthe passive suspension system and the active suspensionsystem with double time-delay feedback control are ana-lyzed We made a time-domain simulation of suspensionbody acceleration pitch acceleration suspension dynamicdeflection and tire dynamic displacement ampe simulationcurve is as in Figure 5

ampe RMS value of the vehicle ride comfort index iscalculated (as in Table 2) according to the 20 s simulationdata Compared with the passive suspension Figures 5(a)and 5(b) give a comparison of body acceleration and bodypitch acceleration response respectively ampe active sus-pension with double time-delay feedback control reducesthe bodyrsquos center of mass acceleration and pitch accel-eration significantly ampe RMS value drops from 30647and 23646 to 05026 and 11162 and the damping effi-ciency is as high as 8360 and 5280 It can be seen fromFigures 5(c) and 5(d) that the dynamic deflection of thefront and rear suspensions has also been reduced sig-nificantly ampe RMS values have decreased from 00383and 00772 to 00275 and 00362 correspondingly and thedamping efficiency is as high as 978 and 2820 It canbe seen from (e) and (f ) of Figure 5 that the front and reartire dynamic relative displacements have also been re-duced significantly and the corresponding RMS value ofthe tire dynamic displacements have decreased from00108 and 97508e minus 4 to 00020 and 33363e minus 4 Itdropped by 8148 and 6878 ampe simulation resultsshow that the active suspension system with double time-delay feedback control reduces the body acceleration andpitch acceleration without increasing the tire deformationand dynamic load ensuring the safety of vehicle drivingand vehicle handling stability ampis shows the effectivenessof the active control method with double time delay underharmonic excitation

52 Simulation Analysis under Random Excitation In orderto further study the damping effect of active suspension withdouble time-delay feedback control the time-delay pa-rameters optimized by particle swarm optimization in thispaper are applied to the vehicle active suspension modelwith double time delay in the actuator In order to verify thedamping effect of active suspensions with double time delaythe vehicle is simulated to travel at a speed of 20ms ampeparameters of an automobile suspension system are shownin Table 1 Random excitation is selected as the verticaldisturbance to the wheel axle Here a sine function su-perposition method is used to establish a time-domainmodel of random excitation as in Figure 6

xr(t) 1113944n

i1ξi sin ωit + δi( 1113857 (16)

where ξ is the amplitude ω is the equivalent frequency and δis the value randomly distributed on (0 2π)

ampe optimized parameters are brought into equation (1)and the random excitation is selected as the vertical dis-turbance to the wheel and shaft to analyze the vibrationresponse characteristics of the vehicle in the time-domainstate of the passive suspension system and the active sus-pension system with double time-delay feedback controlTime-domain simulation is performed for the body accel-eration pitching acceleration suspension dynamic deflec-tion and tire dynamic displacement of the suspension andthe simulation curves are as in Figure 7

From the time-domain simulation in Figure 7 and theroot mean square value of the vehicle ride comfort indexcalculated from the 20 s simulation data as in Table 3compared with the passive suspension the body accelerationand pitch acceleration are as in Figures 7(a) and 7(b)Corresponding comparison graphs are given respectivelyand their corresponding root mean square values havedropped from 14898 and 23858 to 12578 and 18610

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250N

orm

of r

oot 1

10 20 30 40 500Frequency

(a)

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

Nor

m o

f roo

t 2

10 20 30 40 500Frequency

(b)

Figure 4 Modulus of Z in the characteristic equation (a) Norm of root1 (b) Norm of root2

Shock and Vibration 7

ndash6

ndash4

ndash2

0

2

4

6Bo

dy ac

cele

ratio

n (m

middotsndash2)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(a)

Passive suspensionActive suspension with time delay

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Body

pith

acce

lera

tion

(mmiddotsndash2

)

18166 8 10 200 122 144Time (tmiddotsndash1)

(b)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

ndash006

ndash004

ndash002

0

002

004

006

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

(c)

ndash015

ndash01

ndash005

0

005

01

015D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(d)

Figure 5 Continued

8 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

euroxtf ksf xsf minus xtf1113872 1113873 + csf _xsf minus _xtf1113872 1113873 minus ktf xtf minus xgf1113872 1113873 minus ff1113872 1113873

mtf

euroxtr ksr xsr minus xtr( 1113857 + csr _xsr minus _xtr( 1113857 minus ktr xtr minus xgr1113872 1113873 minus fr1113872 1113873

mtr

euroxc minusksf xsf minus xtf1113872 1113873 minus csf _xsf minus _xtf1113872 1113873 + ff minus ksr xsr minus xtr( 1113857 minus csr _xsr minus _xtr( 1113857 + fr1113872 1113873

m

euroφ Lf ksf xsf minus xtf1113872 1113873 + csf _xsf minus _xtf1113872 1113873 minus ff1113960 1113961 minus Lr ksr xsr minus xtr( 1113857 + csr _xsr minus _xtr( 1113857 minus fr1113858 1113859

I

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

3 Optimization Control Analysis ofSuspension System

Figure 2 shows the amplitude-frequency characteristics of thebody when the time- delay feedback control parameters aredifferent It can be seen from Figure 2 that when the timedelay and the feedback gain are zero the system reaches thehighest point of amplitude at 59Hz ampis indicates that whenthe external excitation frequency is equal to the naturalfrequency of the system the system is forced to vibrate at themaximum amplitude due to resonance effects When thefeedback control parameters are τ03 0434 τ02 0452gf01 19287 gr01 23930 the systemrsquos vibration responsedecreases significantly around 59Hz ampis shows that thetime-delay feedback control can attenuate vibration and theremust be a maximum damping point in a certain intervalWhen the feedback gain parameters are τ05 0906τ06 0546 gf02 25333 gr02 23371 there are multiple peaksin the response curve of the system ampis indicates that thepresence of a time delay factor can also destabilize the system

amprough analysis it can be seen that the time delayfeedback control can change the vibration response of thesystem amperefore in this paper the optimal time delayand feedback gain coefficient are obtained by particleswarm optimization and the frequency-domain scanningmethod is used to ensure the stability of the time-delayfeedback system so as to achieve the best vibration re-duction effect

Fourier transform the dynamic differential equation ofthe half-car four-degree-of-freedom model transform thetime-domain characteristics to the frequency-domainrange for research and analysis and rewrite it into theform of a matrix ampe Fourier change of equation (1) is asfollows

A44X B (4)

where X Xc φ Xtf Xtr1113960 1113961T B (ktfXgfmtf)0(ktr1113960

Xgrmtr)01113961T Xgr XgfeminussΔt

Lf Lr

xc

fr

ktf

xtf xtr

xsrxsf

csrcsfksf ksr

ktr

mtf

xgf xgr

mtr

m I φ

ff

Figure 1 Four-degree-of-freedom suspension model with active control with double time- delay

Shock and Vibration 3

A44

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

a11 minusmtfω2

+ iωcsf + ksf + ktf1113872 1113873 + gfeminusiτ1ω

a12 0 a13

a14 iωcsfLf + ksfLf

a21 0 a22

a23 minusksr minus iωcsr

a24 ksrLr minus iωcsrLr

a31 minusksf minus iωcsf minus gfeminus iωτ1

a32 minusksr minus iωcsr minus greminusiωτ2

a33 minusmcω2

+ iω csf + csr1113872 1113873 + ksf + ksr

a34 iω csrLr minus csfLf1113872 1113873 + ksrLr minus ksfLf

a41 Lf iωcsf + ksf + gfeminusiωτ11113872 1113873

a42 minusLr iωcSr + kSr+greminusiωτ211138731113872

a43 minusLf iωcsf + ksf1113872 1113873 + Lr iωcsr + ksr( 1113857

a44 minusIω2+ L

2f ksf + iωcsf1113872 1113873 + L

2r iωcSr + kSr( 1113857

(5)

External excitation input to front and rear wheels has atime difference of ∆t of xgr(t) xgf(t minus Δt) whereΔt (Lf + Lr)v

In equation (4) the amplitude-frequency characteristicfunction |H _Xc sim _Xgf

(ω)| ω|Xc(ω)Xgf(ω)| of the verticalacceleration of the body centroid and the amplitude-fre-quency characteristic function |Heuroφsim _Xgf

(ω)| ω|φ(ω)Xgf(ω)| of the acceleration of the body pitch angle areobtained respectively

ampe vertical centroid acceleration of the vehicle bodyand the vertical pitch acceleration of the vehicle body areimportant indicators to measure the ride comfort of thevehicle amperefore the vertical acceleration of the center ofmass of the vehicle body and the vertical accelerationangle of the vehicle body are used as the main evaluationindicators to establish a weighted objective function forthe optimization of the control parameters At the sametime according to the engineering background the searchrange is set the feedback gain is no more than twice thepassive stiffness ampe smaller the delay the smaller theovershoot

min J gf gr τ1 τ21113872 1113873 n1 H euroXcsim _Xgf(ω)

11138681113868111386811138681113868

11138681113868111386811138681113868 + n2 Heuroφsim _Xgf(ω)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

stminus2klegle 2k

0lt τ le 11113896

(6)

where the weighting coefficients n1 and n2 are 07 and 03which measures the importance of each amplitude-fre-quency function in the objective function

ampe objective function is optimized based on theestablished objective function and the characteristics ofthe particle swarm optimization algorithm [20] Due tothe large difference in the magnitude of the optimizedfeedback gain and time delay a four-dimensional searchspace is assumed to represent the two feedback gains andtime delays respectively ampe individual positions areupdated by tracking individual extreme values Pbest andgroup extreme values Gbest Once the position isupdated the fitness value is calculated By comparing thefitness value of the new particle with the individualextreme value the fitness value of the group extremevalue updates the individual extreme value Pbest and thegroup extreme value Gbest position in each iterationprocess the particle passes the individual extreme valueand group extreme value update their speed and posi-tion ampe formula is updated to the following equation

Vk+1ik αV

kid + ε1r1 P

kid minus X

kid1113872 1113873 + ε2r2 P

kgd minus X

kid1113872 1113873

Xk+11 X

ki1 + rV

k+1i1

(7)

where α is the inertial weight d = 1 2 i = 1 2 n k is thecurrent iteration time Vi d is the particle update speed ε1and ε2 are nonnegative acceleration factor r1 and r2 aregenerated from [0 1] random constant We select 60particles for iterative optimization randomly in order tofind the optimal individual extremum and group extremummore quickly during optimization ε1 = ε2 = 2 after 200iterations we obtain the change graph of the number ofiterations of the fitness function of the suspension per-formance index as in Figure 3 With reference to a vehiclersquossuspension parameters (as in Table 1) the global optimalcontrol parameters under random excitation and harmonicexcitation are obtained respectively after optimizationgf 19345Nm gr 26150Nm τ1 05530 s τ2

05186 s gff 29653Nm grr minus4416Nm τ11 03255 sτ22 05438 s

0

05

1

15

2

25

3

35

4

45

5

|XcXg

f|

18166 8 10 200 1242 14Frequency (Hz)

τ05 = 0906 τ06 = 0546 gf02 = 25333 gr02 = 23371τ03 = 0434 τ04 = 0452 gf01 = 19287 gr01 = 23930τ01 = 0 τ02 = 0 gf0 = 0 gr0 = 0

Figure 2 Amplitude-frequency response curve

4 Shock and Vibration

4 Stability Analysis

ampe existence of time delay has a great impact on the dy-namic performance of the active suspension system In orderto ensure the stability of the feedback control system withdouble time delay the frequency-domain scanning methodis proposed in this paper to analyze the stability of theoptimized control parameters [21ndash24]

First equation (1) is rewritten as the form of stateequation

_x(t) Ax(t) + Bx(t minus τ) (8)

where A and B are constant matrices and τ ge 0 is constant

x xtf _xtf xtr _xtr xc _xc j _j1113960 1113961T

A

0 1 0 0 0 0 0 0

minusksf + ktfmtf minuscsf + csrmtf 0 0 msfmtf csfmtf minusLfksfmtf Lfccfmtf

0 0 0 1 0 0 0 0

0 0 minusksr + ktrmtr minuscsrmtr ksrmtr minuscsrmtr Lrksrmtr Lrcsrmtr

0 0 0 0 0 1 0 0

ksfm csfm ksrm csrm minusksf + ktfm minuscsf + csrm Lfksf minus Lrksrm Lfcsf minus Lrcsrm

0 0 0 0 0 0 0 1

minusLfksfI minusLfccfI LrksrI LrcsrI Lfksf minus LrksrI Lfcsf minus LrcsrI minusL2fksf + L

2rKsrI minusL

2fcsf + L

2rcsrI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

0 0 0 0 0 0 0 0

1m 0 1m 0 0 0 0 0

0 0 0 0 0 0 0 0

minuslfI 0 lfI 0 0 0 0 0

0 0 0 0 0 0 0 0

minus1mtf 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 minus1mtr 0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

Optimal individual fitness

0

02

04

06

08

1

12

14

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(a)

Optimal individual fitness

002

0025

003

0035

004

0045

005

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(b)

Figure 3 Iterative optimization of the fitness function (a) ampe fitness function of optimization under harmonic excitation (b) ampe fitnessfunction of optimization under random excitation

Shock and Vibration 5

ampe characteristic equation of (1) of time-delay controlsystem is as follows

det sI minus A minus Beminusτs

( 1113857 0 (10)

ampe specific form is as follows

CE(s z) an(s)zn

+ anminus1(s)znminus 1

+ middot middot middot + a0(s) 1113944n

k0ak(s)z

k 0

(11)

where

a0 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a1 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a2 c8s8

+ c7s7

+ c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

(12)

z eminusτs n 2 c is the coefficient of the characteristicequation

Furthermore for a given ak(s)(k 0 1 n) thereexists a continuous polynomial of Gl(s)(1 1 middot middot middot n) thatsatisfies the systemrsquos equation of state that can be equivalentto the following

CE(s z) an(s) 1113945n

l1z + Gl(s)( 1113857

k (13)

where an(s) is a constant Gl(s) satisfies the following

1113944

n

k0

ak(s)

an(s)1113888 1113889 minusGl(s)( 1113857

k 0 (14)

It is not necessary to find polynomial Gl(s) because s

and z are complex numbers CE(s z) 0 if and only if asubquasipolynomial (z + Gl(s))(1 1 n) is equal tozero

Calculate the time delay substitute s wlm forCE(s z) 0 calculate z at zero choose the root of 1 x + yiby calculating the θ value in eminusiθ x + yi whereθ arctan(minusyx) + 2π when xgt 0 ygt 0 the time delay isθw + 2rπw(r 0 1 infin)

We made stability analysis of time-delay control pa-rameters gf gr τ1 and τ2 under random excitation byfrequency-domain scanning method ampe control system isstable when the time delay is equal to zero (the characteristicroots of the equation are as follows)

λ1 minus20267857911405181608535831306952 + 17090553240960446610326055558861lowasti

λ2 minus18708774367416789822443106075043 minus 70205741985817738371855015300647lowasti

λ3 minus37274143773467233351129630918096 + 61657802483584587870216137245193lowasti

λ4 minus18708774367416789822443106075043 + 70205741985817738371855015300647lowasti

λ5 minus21213896670092750906944528230251 + 72731773377804970851980013939354lowasti

λ6 minus20267857911405181608535831306952 minus 17090553240960446610326055558861lowasti

λ7 minus21213896670092750906944528230251 minus 72731773377804970851980013939354lowasti

λ8 minus37274143773467233351129630918096 minus 61657802483584587870216137245193lowasti

(15)

It can be obtained that the characteristic roots of thesystem are in the left half-plane of the complex planeamperefore the active suspension system with double time-delay feedback control at τ 0 is stable Figure 4 shows therelationship between |z| and w generated by the frequency-domain scanning method ampe system has two crossoverfrequencies w1 207651 w2 207653 ampe system hastwo positive imaginary roots corresponding to two sub-quasipolynomials and the crossing directions of the twoimaginary roots are from left to rightampus as the time delayincreases once the system has characteristic virtual roots

the system will no longer be stable the system is asymp-totically stable when τ2 isin (0 06312) Similarly the stabilityanalysis of the time-delay control parameters under har-monic excitation is performed and the stability interval ofthe time delay is τ3 isin (0 05817)

5 Establishing a Simulation Model andResult Analysis

51 Simulation Analysis under Harmonic Excitation Takethe optimized double time-delay control parameters into

Table 1 Vehicle suspension model parameters

Vehicle parameters Valuem(kg) 690I(kgm2) 1200mtf(kg) 405mtr(kg) 454ktf(ktr)(Nmiddotmminus1) 192000ksf(Nmiddotmminus1) 17000ksr(Nmiddotmminus1) 22000Lf(m) 125Lr(m) 151csf(csr)(Nmiddotsmiddotmminus1) 1500

6 Shock and Vibration

equation (1) Taking the harmonic excitationxgf 005 sin(7t) as the road input excitation the vibrationresponse characteristics of the vehicle in the time domain ofthe passive suspension system and the active suspensionsystem with double time-delay feedback control are ana-lyzed We made a time-domain simulation of suspensionbody acceleration pitch acceleration suspension dynamicdeflection and tire dynamic displacement ampe simulationcurve is as in Figure 5

ampe RMS value of the vehicle ride comfort index iscalculated (as in Table 2) according to the 20 s simulationdata Compared with the passive suspension Figures 5(a)and 5(b) give a comparison of body acceleration and bodypitch acceleration response respectively ampe active sus-pension with double time-delay feedback control reducesthe bodyrsquos center of mass acceleration and pitch accel-eration significantly ampe RMS value drops from 30647and 23646 to 05026 and 11162 and the damping effi-ciency is as high as 8360 and 5280 It can be seen fromFigures 5(c) and 5(d) that the dynamic deflection of thefront and rear suspensions has also been reduced sig-nificantly ampe RMS values have decreased from 00383and 00772 to 00275 and 00362 correspondingly and thedamping efficiency is as high as 978 and 2820 It canbe seen from (e) and (f ) of Figure 5 that the front and reartire dynamic relative displacements have also been re-duced significantly and the corresponding RMS value ofthe tire dynamic displacements have decreased from00108 and 97508e minus 4 to 00020 and 33363e minus 4 Itdropped by 8148 and 6878 ampe simulation resultsshow that the active suspension system with double time-delay feedback control reduces the body acceleration andpitch acceleration without increasing the tire deformationand dynamic load ensuring the safety of vehicle drivingand vehicle handling stability ampis shows the effectivenessof the active control method with double time delay underharmonic excitation

52 Simulation Analysis under Random Excitation In orderto further study the damping effect of active suspension withdouble time-delay feedback control the time-delay pa-rameters optimized by particle swarm optimization in thispaper are applied to the vehicle active suspension modelwith double time delay in the actuator In order to verify thedamping effect of active suspensions with double time delaythe vehicle is simulated to travel at a speed of 20ms ampeparameters of an automobile suspension system are shownin Table 1 Random excitation is selected as the verticaldisturbance to the wheel axle Here a sine function su-perposition method is used to establish a time-domainmodel of random excitation as in Figure 6

xr(t) 1113944n

i1ξi sin ωit + δi( 1113857 (16)

where ξ is the amplitude ω is the equivalent frequency and δis the value randomly distributed on (0 2π)

ampe optimized parameters are brought into equation (1)and the random excitation is selected as the vertical dis-turbance to the wheel and shaft to analyze the vibrationresponse characteristics of the vehicle in the time-domainstate of the passive suspension system and the active sus-pension system with double time-delay feedback controlTime-domain simulation is performed for the body accel-eration pitching acceleration suspension dynamic deflec-tion and tire dynamic displacement of the suspension andthe simulation curves are as in Figure 7

From the time-domain simulation in Figure 7 and theroot mean square value of the vehicle ride comfort indexcalculated from the 20 s simulation data as in Table 3compared with the passive suspension the body accelerationand pitch acceleration are as in Figures 7(a) and 7(b)Corresponding comparison graphs are given respectivelyand their corresponding root mean square values havedropped from 14898 and 23858 to 12578 and 18610

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250N

orm

of r

oot 1

10 20 30 40 500Frequency

(a)

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

Nor

m o

f roo

t 2

10 20 30 40 500Frequency

(b)

Figure 4 Modulus of Z in the characteristic equation (a) Norm of root1 (b) Norm of root2

Shock and Vibration 7

ndash6

ndash4

ndash2

0

2

4

6Bo

dy ac

cele

ratio

n (m

middotsndash2)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(a)

Passive suspensionActive suspension with time delay

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Body

pith

acce

lera

tion

(mmiddotsndash2

)

18166 8 10 200 122 144Time (tmiddotsndash1)

(b)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

ndash006

ndash004

ndash002

0

002

004

006

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

(c)

ndash015

ndash01

ndash005

0

005

01

015D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(d)

Figure 5 Continued

8 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

A44

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

a11 minusmtfω2

+ iωcsf + ksf + ktf1113872 1113873 + gfeminusiτ1ω

a12 0 a13

a14 iωcsfLf + ksfLf

a21 0 a22

a23 minusksr minus iωcsr

a24 ksrLr minus iωcsrLr

a31 minusksf minus iωcsf minus gfeminus iωτ1

a32 minusksr minus iωcsr minus greminusiωτ2

a33 minusmcω2

+ iω csf + csr1113872 1113873 + ksf + ksr

a34 iω csrLr minus csfLf1113872 1113873 + ksrLr minus ksfLf

a41 Lf iωcsf + ksf + gfeminusiωτ11113872 1113873

a42 minusLr iωcSr + kSr+greminusiωτ211138731113872

a43 minusLf iωcsf + ksf1113872 1113873 + Lr iωcsr + ksr( 1113857

a44 minusIω2+ L

2f ksf + iωcsf1113872 1113873 + L

2r iωcSr + kSr( 1113857

(5)

External excitation input to front and rear wheels has atime difference of ∆t of xgr(t) xgf(t minus Δt) whereΔt (Lf + Lr)v

In equation (4) the amplitude-frequency characteristicfunction |H _Xc sim _Xgf

(ω)| ω|Xc(ω)Xgf(ω)| of the verticalacceleration of the body centroid and the amplitude-fre-quency characteristic function |Heuroφsim _Xgf

(ω)| ω|φ(ω)Xgf(ω)| of the acceleration of the body pitch angle areobtained respectively

ampe vertical centroid acceleration of the vehicle bodyand the vertical pitch acceleration of the vehicle body areimportant indicators to measure the ride comfort of thevehicle amperefore the vertical acceleration of the center ofmass of the vehicle body and the vertical accelerationangle of the vehicle body are used as the main evaluationindicators to establish a weighted objective function forthe optimization of the control parameters At the sametime according to the engineering background the searchrange is set the feedback gain is no more than twice thepassive stiffness ampe smaller the delay the smaller theovershoot

min J gf gr τ1 τ21113872 1113873 n1 H euroXcsim _Xgf(ω)

11138681113868111386811138681113868

11138681113868111386811138681113868 + n2 Heuroφsim _Xgf(ω)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868

stminus2klegle 2k

0lt τ le 11113896

(6)

where the weighting coefficients n1 and n2 are 07 and 03which measures the importance of each amplitude-fre-quency function in the objective function

ampe objective function is optimized based on theestablished objective function and the characteristics ofthe particle swarm optimization algorithm [20] Due tothe large difference in the magnitude of the optimizedfeedback gain and time delay a four-dimensional searchspace is assumed to represent the two feedback gains andtime delays respectively ampe individual positions areupdated by tracking individual extreme values Pbest andgroup extreme values Gbest Once the position isupdated the fitness value is calculated By comparing thefitness value of the new particle with the individualextreme value the fitness value of the group extremevalue updates the individual extreme value Pbest and thegroup extreme value Gbest position in each iterationprocess the particle passes the individual extreme valueand group extreme value update their speed and posi-tion ampe formula is updated to the following equation

Vk+1ik αV

kid + ε1r1 P

kid minus X

kid1113872 1113873 + ε2r2 P

kgd minus X

kid1113872 1113873

Xk+11 X

ki1 + rV

k+1i1

(7)

where α is the inertial weight d = 1 2 i = 1 2 n k is thecurrent iteration time Vi d is the particle update speed ε1and ε2 are nonnegative acceleration factor r1 and r2 aregenerated from [0 1] random constant We select 60particles for iterative optimization randomly in order tofind the optimal individual extremum and group extremummore quickly during optimization ε1 = ε2 = 2 after 200iterations we obtain the change graph of the number ofiterations of the fitness function of the suspension per-formance index as in Figure 3 With reference to a vehiclersquossuspension parameters (as in Table 1) the global optimalcontrol parameters under random excitation and harmonicexcitation are obtained respectively after optimizationgf 19345Nm gr 26150Nm τ1 05530 s τ2

05186 s gff 29653Nm grr minus4416Nm τ11 03255 sτ22 05438 s

0

05

1

15

2

25

3

35

4

45

5

|XcXg

f|

18166 8 10 200 1242 14Frequency (Hz)

τ05 = 0906 τ06 = 0546 gf02 = 25333 gr02 = 23371τ03 = 0434 τ04 = 0452 gf01 = 19287 gr01 = 23930τ01 = 0 τ02 = 0 gf0 = 0 gr0 = 0

Figure 2 Amplitude-frequency response curve

4 Shock and Vibration

4 Stability Analysis

ampe existence of time delay has a great impact on the dy-namic performance of the active suspension system In orderto ensure the stability of the feedback control system withdouble time delay the frequency-domain scanning methodis proposed in this paper to analyze the stability of theoptimized control parameters [21ndash24]

First equation (1) is rewritten as the form of stateequation

_x(t) Ax(t) + Bx(t minus τ) (8)

where A and B are constant matrices and τ ge 0 is constant

x xtf _xtf xtr _xtr xc _xc j _j1113960 1113961T

A

0 1 0 0 0 0 0 0

minusksf + ktfmtf minuscsf + csrmtf 0 0 msfmtf csfmtf minusLfksfmtf Lfccfmtf

0 0 0 1 0 0 0 0

0 0 minusksr + ktrmtr minuscsrmtr ksrmtr minuscsrmtr Lrksrmtr Lrcsrmtr

0 0 0 0 0 1 0 0

ksfm csfm ksrm csrm minusksf + ktfm minuscsf + csrm Lfksf minus Lrksrm Lfcsf minus Lrcsrm

0 0 0 0 0 0 0 1

minusLfksfI minusLfccfI LrksrI LrcsrI Lfksf minus LrksrI Lfcsf minus LrcsrI minusL2fksf + L

2rKsrI minusL

2fcsf + L

2rcsrI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

0 0 0 0 0 0 0 0

1m 0 1m 0 0 0 0 0

0 0 0 0 0 0 0 0

minuslfI 0 lfI 0 0 0 0 0

0 0 0 0 0 0 0 0

minus1mtf 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 minus1mtr 0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

Optimal individual fitness

0

02

04

06

08

1

12

14

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(a)

Optimal individual fitness

002

0025

003

0035

004

0045

005

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(b)

Figure 3 Iterative optimization of the fitness function (a) ampe fitness function of optimization under harmonic excitation (b) ampe fitnessfunction of optimization under random excitation

Shock and Vibration 5

ampe characteristic equation of (1) of time-delay controlsystem is as follows

det sI minus A minus Beminusτs

( 1113857 0 (10)

ampe specific form is as follows

CE(s z) an(s)zn

+ anminus1(s)znminus 1

+ middot middot middot + a0(s) 1113944n

k0ak(s)z

k 0

(11)

where

a0 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a1 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a2 c8s8

+ c7s7

+ c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

(12)

z eminusτs n 2 c is the coefficient of the characteristicequation

Furthermore for a given ak(s)(k 0 1 n) thereexists a continuous polynomial of Gl(s)(1 1 middot middot middot n) thatsatisfies the systemrsquos equation of state that can be equivalentto the following

CE(s z) an(s) 1113945n

l1z + Gl(s)( 1113857

k (13)

where an(s) is a constant Gl(s) satisfies the following

1113944

n

k0

ak(s)

an(s)1113888 1113889 minusGl(s)( 1113857

k 0 (14)

It is not necessary to find polynomial Gl(s) because s

and z are complex numbers CE(s z) 0 if and only if asubquasipolynomial (z + Gl(s))(1 1 n) is equal tozero

Calculate the time delay substitute s wlm forCE(s z) 0 calculate z at zero choose the root of 1 x + yiby calculating the θ value in eminusiθ x + yi whereθ arctan(minusyx) + 2π when xgt 0 ygt 0 the time delay isθw + 2rπw(r 0 1 infin)

We made stability analysis of time-delay control pa-rameters gf gr τ1 and τ2 under random excitation byfrequency-domain scanning method ampe control system isstable when the time delay is equal to zero (the characteristicroots of the equation are as follows)

λ1 minus20267857911405181608535831306952 + 17090553240960446610326055558861lowasti

λ2 minus18708774367416789822443106075043 minus 70205741985817738371855015300647lowasti

λ3 minus37274143773467233351129630918096 + 61657802483584587870216137245193lowasti

λ4 minus18708774367416789822443106075043 + 70205741985817738371855015300647lowasti

λ5 minus21213896670092750906944528230251 + 72731773377804970851980013939354lowasti

λ6 minus20267857911405181608535831306952 minus 17090553240960446610326055558861lowasti

λ7 minus21213896670092750906944528230251 minus 72731773377804970851980013939354lowasti

λ8 minus37274143773467233351129630918096 minus 61657802483584587870216137245193lowasti

(15)

It can be obtained that the characteristic roots of thesystem are in the left half-plane of the complex planeamperefore the active suspension system with double time-delay feedback control at τ 0 is stable Figure 4 shows therelationship between |z| and w generated by the frequency-domain scanning method ampe system has two crossoverfrequencies w1 207651 w2 207653 ampe system hastwo positive imaginary roots corresponding to two sub-quasipolynomials and the crossing directions of the twoimaginary roots are from left to rightampus as the time delayincreases once the system has characteristic virtual roots

the system will no longer be stable the system is asymp-totically stable when τ2 isin (0 06312) Similarly the stabilityanalysis of the time-delay control parameters under har-monic excitation is performed and the stability interval ofthe time delay is τ3 isin (0 05817)

5 Establishing a Simulation Model andResult Analysis

51 Simulation Analysis under Harmonic Excitation Takethe optimized double time-delay control parameters into

Table 1 Vehicle suspension model parameters

Vehicle parameters Valuem(kg) 690I(kgm2) 1200mtf(kg) 405mtr(kg) 454ktf(ktr)(Nmiddotmminus1) 192000ksf(Nmiddotmminus1) 17000ksr(Nmiddotmminus1) 22000Lf(m) 125Lr(m) 151csf(csr)(Nmiddotsmiddotmminus1) 1500

6 Shock and Vibration

equation (1) Taking the harmonic excitationxgf 005 sin(7t) as the road input excitation the vibrationresponse characteristics of the vehicle in the time domain ofthe passive suspension system and the active suspensionsystem with double time-delay feedback control are ana-lyzed We made a time-domain simulation of suspensionbody acceleration pitch acceleration suspension dynamicdeflection and tire dynamic displacement ampe simulationcurve is as in Figure 5

ampe RMS value of the vehicle ride comfort index iscalculated (as in Table 2) according to the 20 s simulationdata Compared with the passive suspension Figures 5(a)and 5(b) give a comparison of body acceleration and bodypitch acceleration response respectively ampe active sus-pension with double time-delay feedback control reducesthe bodyrsquos center of mass acceleration and pitch accel-eration significantly ampe RMS value drops from 30647and 23646 to 05026 and 11162 and the damping effi-ciency is as high as 8360 and 5280 It can be seen fromFigures 5(c) and 5(d) that the dynamic deflection of thefront and rear suspensions has also been reduced sig-nificantly ampe RMS values have decreased from 00383and 00772 to 00275 and 00362 correspondingly and thedamping efficiency is as high as 978 and 2820 It canbe seen from (e) and (f ) of Figure 5 that the front and reartire dynamic relative displacements have also been re-duced significantly and the corresponding RMS value ofthe tire dynamic displacements have decreased from00108 and 97508e minus 4 to 00020 and 33363e minus 4 Itdropped by 8148 and 6878 ampe simulation resultsshow that the active suspension system with double time-delay feedback control reduces the body acceleration andpitch acceleration without increasing the tire deformationand dynamic load ensuring the safety of vehicle drivingand vehicle handling stability ampis shows the effectivenessof the active control method with double time delay underharmonic excitation

52 Simulation Analysis under Random Excitation In orderto further study the damping effect of active suspension withdouble time-delay feedback control the time-delay pa-rameters optimized by particle swarm optimization in thispaper are applied to the vehicle active suspension modelwith double time delay in the actuator In order to verify thedamping effect of active suspensions with double time delaythe vehicle is simulated to travel at a speed of 20ms ampeparameters of an automobile suspension system are shownin Table 1 Random excitation is selected as the verticaldisturbance to the wheel axle Here a sine function su-perposition method is used to establish a time-domainmodel of random excitation as in Figure 6

xr(t) 1113944n

i1ξi sin ωit + δi( 1113857 (16)

where ξ is the amplitude ω is the equivalent frequency and δis the value randomly distributed on (0 2π)

ampe optimized parameters are brought into equation (1)and the random excitation is selected as the vertical dis-turbance to the wheel and shaft to analyze the vibrationresponse characteristics of the vehicle in the time-domainstate of the passive suspension system and the active sus-pension system with double time-delay feedback controlTime-domain simulation is performed for the body accel-eration pitching acceleration suspension dynamic deflec-tion and tire dynamic displacement of the suspension andthe simulation curves are as in Figure 7

From the time-domain simulation in Figure 7 and theroot mean square value of the vehicle ride comfort indexcalculated from the 20 s simulation data as in Table 3compared with the passive suspension the body accelerationand pitch acceleration are as in Figures 7(a) and 7(b)Corresponding comparison graphs are given respectivelyand their corresponding root mean square values havedropped from 14898 and 23858 to 12578 and 18610

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250N

orm

of r

oot 1

10 20 30 40 500Frequency

(a)

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

Nor

m o

f roo

t 2

10 20 30 40 500Frequency

(b)

Figure 4 Modulus of Z in the characteristic equation (a) Norm of root1 (b) Norm of root2

Shock and Vibration 7

ndash6

ndash4

ndash2

0

2

4

6Bo

dy ac

cele

ratio

n (m

middotsndash2)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(a)

Passive suspensionActive suspension with time delay

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Body

pith

acce

lera

tion

(mmiddotsndash2

)

18166 8 10 200 122 144Time (tmiddotsndash1)

(b)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

ndash006

ndash004

ndash002

0

002

004

006

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

(c)

ndash015

ndash01

ndash005

0

005

01

015D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(d)

Figure 5 Continued

8 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

4 Stability Analysis

ampe existence of time delay has a great impact on the dy-namic performance of the active suspension system In orderto ensure the stability of the feedback control system withdouble time delay the frequency-domain scanning methodis proposed in this paper to analyze the stability of theoptimized control parameters [21ndash24]

First equation (1) is rewritten as the form of stateequation

_x(t) Ax(t) + Bx(t minus τ) (8)

where A and B are constant matrices and τ ge 0 is constant

x xtf _xtf xtr _xtr xc _xc j _j1113960 1113961T

A

0 1 0 0 0 0 0 0

minusksf + ktfmtf minuscsf + csrmtf 0 0 msfmtf csfmtf minusLfksfmtf Lfccfmtf

0 0 0 1 0 0 0 0

0 0 minusksr + ktrmtr minuscsrmtr ksrmtr minuscsrmtr Lrksrmtr Lrcsrmtr

0 0 0 0 0 1 0 0

ksfm csfm ksrm csrm minusksf + ktfm minuscsf + csrm Lfksf minus Lrksrm Lfcsf minus Lrcsrm

0 0 0 0 0 0 0 1

minusLfksfI minusLfccfI LrksrI LrcsrI Lfksf minus LrksrI Lfcsf minus LrcsrI minusL2fksf + L

2rKsrI minusL

2fcsf + L

2rcsrI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B

0 0 0 0 0 0 0 0

1m 0 1m 0 0 0 0 0

0 0 0 0 0 0 0 0

minuslfI 0 lfI 0 0 0 0 0

0 0 0 0 0 0 0 0

minus1mtf 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 minus1mtr 0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

Optimal individual fitness

0

02

04

06

08

1

12

14

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(a)

Optimal individual fitness

002

0025

003

0035

004

0045

005

Fitn

ess

20 40 60 80 100 120 140 160 180 2000Generation

(b)

Figure 3 Iterative optimization of the fitness function (a) ampe fitness function of optimization under harmonic excitation (b) ampe fitnessfunction of optimization under random excitation

Shock and Vibration 5

ampe characteristic equation of (1) of time-delay controlsystem is as follows

det sI minus A minus Beminusτs

( 1113857 0 (10)

ampe specific form is as follows

CE(s z) an(s)zn

+ anminus1(s)znminus 1

+ middot middot middot + a0(s) 1113944n

k0ak(s)z

k 0

(11)

where

a0 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a1 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a2 c8s8

+ c7s7

+ c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

(12)

z eminusτs n 2 c is the coefficient of the characteristicequation

Furthermore for a given ak(s)(k 0 1 n) thereexists a continuous polynomial of Gl(s)(1 1 middot middot middot n) thatsatisfies the systemrsquos equation of state that can be equivalentto the following

CE(s z) an(s) 1113945n

l1z + Gl(s)( 1113857

k (13)

where an(s) is a constant Gl(s) satisfies the following

1113944

n

k0

ak(s)

an(s)1113888 1113889 minusGl(s)( 1113857

k 0 (14)

It is not necessary to find polynomial Gl(s) because s

and z are complex numbers CE(s z) 0 if and only if asubquasipolynomial (z + Gl(s))(1 1 n) is equal tozero

Calculate the time delay substitute s wlm forCE(s z) 0 calculate z at zero choose the root of 1 x + yiby calculating the θ value in eminusiθ x + yi whereθ arctan(minusyx) + 2π when xgt 0 ygt 0 the time delay isθw + 2rπw(r 0 1 infin)

We made stability analysis of time-delay control pa-rameters gf gr τ1 and τ2 under random excitation byfrequency-domain scanning method ampe control system isstable when the time delay is equal to zero (the characteristicroots of the equation are as follows)

λ1 minus20267857911405181608535831306952 + 17090553240960446610326055558861lowasti

λ2 minus18708774367416789822443106075043 minus 70205741985817738371855015300647lowasti

λ3 minus37274143773467233351129630918096 + 61657802483584587870216137245193lowasti

λ4 minus18708774367416789822443106075043 + 70205741985817738371855015300647lowasti

λ5 minus21213896670092750906944528230251 + 72731773377804970851980013939354lowasti

λ6 minus20267857911405181608535831306952 minus 17090553240960446610326055558861lowasti

λ7 minus21213896670092750906944528230251 minus 72731773377804970851980013939354lowasti

λ8 minus37274143773467233351129630918096 minus 61657802483584587870216137245193lowasti

(15)

It can be obtained that the characteristic roots of thesystem are in the left half-plane of the complex planeamperefore the active suspension system with double time-delay feedback control at τ 0 is stable Figure 4 shows therelationship between |z| and w generated by the frequency-domain scanning method ampe system has two crossoverfrequencies w1 207651 w2 207653 ampe system hastwo positive imaginary roots corresponding to two sub-quasipolynomials and the crossing directions of the twoimaginary roots are from left to rightampus as the time delayincreases once the system has characteristic virtual roots

the system will no longer be stable the system is asymp-totically stable when τ2 isin (0 06312) Similarly the stabilityanalysis of the time-delay control parameters under har-monic excitation is performed and the stability interval ofthe time delay is τ3 isin (0 05817)

5 Establishing a Simulation Model andResult Analysis

51 Simulation Analysis under Harmonic Excitation Takethe optimized double time-delay control parameters into

Table 1 Vehicle suspension model parameters

Vehicle parameters Valuem(kg) 690I(kgm2) 1200mtf(kg) 405mtr(kg) 454ktf(ktr)(Nmiddotmminus1) 192000ksf(Nmiddotmminus1) 17000ksr(Nmiddotmminus1) 22000Lf(m) 125Lr(m) 151csf(csr)(Nmiddotsmiddotmminus1) 1500

6 Shock and Vibration

equation (1) Taking the harmonic excitationxgf 005 sin(7t) as the road input excitation the vibrationresponse characteristics of the vehicle in the time domain ofthe passive suspension system and the active suspensionsystem with double time-delay feedback control are ana-lyzed We made a time-domain simulation of suspensionbody acceleration pitch acceleration suspension dynamicdeflection and tire dynamic displacement ampe simulationcurve is as in Figure 5

ampe RMS value of the vehicle ride comfort index iscalculated (as in Table 2) according to the 20 s simulationdata Compared with the passive suspension Figures 5(a)and 5(b) give a comparison of body acceleration and bodypitch acceleration response respectively ampe active sus-pension with double time-delay feedback control reducesthe bodyrsquos center of mass acceleration and pitch accel-eration significantly ampe RMS value drops from 30647and 23646 to 05026 and 11162 and the damping effi-ciency is as high as 8360 and 5280 It can be seen fromFigures 5(c) and 5(d) that the dynamic deflection of thefront and rear suspensions has also been reduced sig-nificantly ampe RMS values have decreased from 00383and 00772 to 00275 and 00362 correspondingly and thedamping efficiency is as high as 978 and 2820 It canbe seen from (e) and (f ) of Figure 5 that the front and reartire dynamic relative displacements have also been re-duced significantly and the corresponding RMS value ofthe tire dynamic displacements have decreased from00108 and 97508e minus 4 to 00020 and 33363e minus 4 Itdropped by 8148 and 6878 ampe simulation resultsshow that the active suspension system with double time-delay feedback control reduces the body acceleration andpitch acceleration without increasing the tire deformationand dynamic load ensuring the safety of vehicle drivingand vehicle handling stability ampis shows the effectivenessof the active control method with double time delay underharmonic excitation

52 Simulation Analysis under Random Excitation In orderto further study the damping effect of active suspension withdouble time-delay feedback control the time-delay pa-rameters optimized by particle swarm optimization in thispaper are applied to the vehicle active suspension modelwith double time delay in the actuator In order to verify thedamping effect of active suspensions with double time delaythe vehicle is simulated to travel at a speed of 20ms ampeparameters of an automobile suspension system are shownin Table 1 Random excitation is selected as the verticaldisturbance to the wheel axle Here a sine function su-perposition method is used to establish a time-domainmodel of random excitation as in Figure 6

xr(t) 1113944n

i1ξi sin ωit + δi( 1113857 (16)

where ξ is the amplitude ω is the equivalent frequency and δis the value randomly distributed on (0 2π)

ampe optimized parameters are brought into equation (1)and the random excitation is selected as the vertical dis-turbance to the wheel and shaft to analyze the vibrationresponse characteristics of the vehicle in the time-domainstate of the passive suspension system and the active sus-pension system with double time-delay feedback controlTime-domain simulation is performed for the body accel-eration pitching acceleration suspension dynamic deflec-tion and tire dynamic displacement of the suspension andthe simulation curves are as in Figure 7

From the time-domain simulation in Figure 7 and theroot mean square value of the vehicle ride comfort indexcalculated from the 20 s simulation data as in Table 3compared with the passive suspension the body accelerationand pitch acceleration are as in Figures 7(a) and 7(b)Corresponding comparison graphs are given respectivelyand their corresponding root mean square values havedropped from 14898 and 23858 to 12578 and 18610

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250N

orm

of r

oot 1

10 20 30 40 500Frequency

(a)

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

Nor

m o

f roo

t 2

10 20 30 40 500Frequency

(b)

Figure 4 Modulus of Z in the characteristic equation (a) Norm of root1 (b) Norm of root2

Shock and Vibration 7

ndash6

ndash4

ndash2

0

2

4

6Bo

dy ac

cele

ratio

n (m

middotsndash2)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(a)

Passive suspensionActive suspension with time delay

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Body

pith

acce

lera

tion

(mmiddotsndash2

)

18166 8 10 200 122 144Time (tmiddotsndash1)

(b)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

ndash006

ndash004

ndash002

0

002

004

006

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

(c)

ndash015

ndash01

ndash005

0

005

01

015D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(d)

Figure 5 Continued

8 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

ampe characteristic equation of (1) of time-delay controlsystem is as follows

det sI minus A minus Beminusτs

( 1113857 0 (10)

ampe specific form is as follows

CE(s z) an(s)zn

+ anminus1(s)znminus 1

+ middot middot middot + a0(s) 1113944n

k0ak(s)z

k 0

(11)

where

a0 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a1 c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

a2 c8s8

+ c7s7

+ c6s6

+ c5s5

+ c4s4

+ c3s3

+ c2s2

+ c1s1

+ c0

(12)

z eminusτs n 2 c is the coefficient of the characteristicequation

Furthermore for a given ak(s)(k 0 1 n) thereexists a continuous polynomial of Gl(s)(1 1 middot middot middot n) thatsatisfies the systemrsquos equation of state that can be equivalentto the following

CE(s z) an(s) 1113945n

l1z + Gl(s)( 1113857

k (13)

where an(s) is a constant Gl(s) satisfies the following

1113944

n

k0

ak(s)

an(s)1113888 1113889 minusGl(s)( 1113857

k 0 (14)

It is not necessary to find polynomial Gl(s) because s

and z are complex numbers CE(s z) 0 if and only if asubquasipolynomial (z + Gl(s))(1 1 n) is equal tozero

Calculate the time delay substitute s wlm forCE(s z) 0 calculate z at zero choose the root of 1 x + yiby calculating the θ value in eminusiθ x + yi whereθ arctan(minusyx) + 2π when xgt 0 ygt 0 the time delay isθw + 2rπw(r 0 1 infin)

We made stability analysis of time-delay control pa-rameters gf gr τ1 and τ2 under random excitation byfrequency-domain scanning method ampe control system isstable when the time delay is equal to zero (the characteristicroots of the equation are as follows)

λ1 minus20267857911405181608535831306952 + 17090553240960446610326055558861lowasti

λ2 minus18708774367416789822443106075043 minus 70205741985817738371855015300647lowasti

λ3 minus37274143773467233351129630918096 + 61657802483584587870216137245193lowasti

λ4 minus18708774367416789822443106075043 + 70205741985817738371855015300647lowasti

λ5 minus21213896670092750906944528230251 + 72731773377804970851980013939354lowasti

λ6 minus20267857911405181608535831306952 minus 17090553240960446610326055558861lowasti

λ7 minus21213896670092750906944528230251 minus 72731773377804970851980013939354lowasti

λ8 minus37274143773467233351129630918096 minus 61657802483584587870216137245193lowasti

(15)

It can be obtained that the characteristic roots of thesystem are in the left half-plane of the complex planeamperefore the active suspension system with double time-delay feedback control at τ 0 is stable Figure 4 shows therelationship between |z| and w generated by the frequency-domain scanning method ampe system has two crossoverfrequencies w1 207651 w2 207653 ampe system hastwo positive imaginary roots corresponding to two sub-quasipolynomials and the crossing directions of the twoimaginary roots are from left to rightampus as the time delayincreases once the system has characteristic virtual roots

the system will no longer be stable the system is asymp-totically stable when τ2 isin (0 06312) Similarly the stabilityanalysis of the time-delay control parameters under har-monic excitation is performed and the stability interval ofthe time delay is τ3 isin (0 05817)

5 Establishing a Simulation Model andResult Analysis

51 Simulation Analysis under Harmonic Excitation Takethe optimized double time-delay control parameters into

Table 1 Vehicle suspension model parameters

Vehicle parameters Valuem(kg) 690I(kgm2) 1200mtf(kg) 405mtr(kg) 454ktf(ktr)(Nmiddotmminus1) 192000ksf(Nmiddotmminus1) 17000ksr(Nmiddotmminus1) 22000Lf(m) 125Lr(m) 151csf(csr)(Nmiddotsmiddotmminus1) 1500

6 Shock and Vibration

equation (1) Taking the harmonic excitationxgf 005 sin(7t) as the road input excitation the vibrationresponse characteristics of the vehicle in the time domain ofthe passive suspension system and the active suspensionsystem with double time-delay feedback control are ana-lyzed We made a time-domain simulation of suspensionbody acceleration pitch acceleration suspension dynamicdeflection and tire dynamic displacement ampe simulationcurve is as in Figure 5

ampe RMS value of the vehicle ride comfort index iscalculated (as in Table 2) according to the 20 s simulationdata Compared with the passive suspension Figures 5(a)and 5(b) give a comparison of body acceleration and bodypitch acceleration response respectively ampe active sus-pension with double time-delay feedback control reducesthe bodyrsquos center of mass acceleration and pitch accel-eration significantly ampe RMS value drops from 30647and 23646 to 05026 and 11162 and the damping effi-ciency is as high as 8360 and 5280 It can be seen fromFigures 5(c) and 5(d) that the dynamic deflection of thefront and rear suspensions has also been reduced sig-nificantly ampe RMS values have decreased from 00383and 00772 to 00275 and 00362 correspondingly and thedamping efficiency is as high as 978 and 2820 It canbe seen from (e) and (f ) of Figure 5 that the front and reartire dynamic relative displacements have also been re-duced significantly and the corresponding RMS value ofthe tire dynamic displacements have decreased from00108 and 97508e minus 4 to 00020 and 33363e minus 4 Itdropped by 8148 and 6878 ampe simulation resultsshow that the active suspension system with double time-delay feedback control reduces the body acceleration andpitch acceleration without increasing the tire deformationand dynamic load ensuring the safety of vehicle drivingand vehicle handling stability ampis shows the effectivenessof the active control method with double time delay underharmonic excitation

52 Simulation Analysis under Random Excitation In orderto further study the damping effect of active suspension withdouble time-delay feedback control the time-delay pa-rameters optimized by particle swarm optimization in thispaper are applied to the vehicle active suspension modelwith double time delay in the actuator In order to verify thedamping effect of active suspensions with double time delaythe vehicle is simulated to travel at a speed of 20ms ampeparameters of an automobile suspension system are shownin Table 1 Random excitation is selected as the verticaldisturbance to the wheel axle Here a sine function su-perposition method is used to establish a time-domainmodel of random excitation as in Figure 6

xr(t) 1113944n

i1ξi sin ωit + δi( 1113857 (16)

where ξ is the amplitude ω is the equivalent frequency and δis the value randomly distributed on (0 2π)

ampe optimized parameters are brought into equation (1)and the random excitation is selected as the vertical dis-turbance to the wheel and shaft to analyze the vibrationresponse characteristics of the vehicle in the time-domainstate of the passive suspension system and the active sus-pension system with double time-delay feedback controlTime-domain simulation is performed for the body accel-eration pitching acceleration suspension dynamic deflec-tion and tire dynamic displacement of the suspension andthe simulation curves are as in Figure 7

From the time-domain simulation in Figure 7 and theroot mean square value of the vehicle ride comfort indexcalculated from the 20 s simulation data as in Table 3compared with the passive suspension the body accelerationand pitch acceleration are as in Figures 7(a) and 7(b)Corresponding comparison graphs are given respectivelyand their corresponding root mean square values havedropped from 14898 and 23858 to 12578 and 18610

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250N

orm

of r

oot 1

10 20 30 40 500Frequency

(a)

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

Nor

m o

f roo

t 2

10 20 30 40 500Frequency

(b)

Figure 4 Modulus of Z in the characteristic equation (a) Norm of root1 (b) Norm of root2

Shock and Vibration 7

ndash6

ndash4

ndash2

0

2

4

6Bo

dy ac

cele

ratio

n (m

middotsndash2)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(a)

Passive suspensionActive suspension with time delay

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Body

pith

acce

lera

tion

(mmiddotsndash2

)

18166 8 10 200 122 144Time (tmiddotsndash1)

(b)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

ndash006

ndash004

ndash002

0

002

004

006

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

(c)

ndash015

ndash01

ndash005

0

005

01

015D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(d)

Figure 5 Continued

8 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

equation (1) Taking the harmonic excitationxgf 005 sin(7t) as the road input excitation the vibrationresponse characteristics of the vehicle in the time domain ofthe passive suspension system and the active suspensionsystem with double time-delay feedback control are ana-lyzed We made a time-domain simulation of suspensionbody acceleration pitch acceleration suspension dynamicdeflection and tire dynamic displacement ampe simulationcurve is as in Figure 5

ampe RMS value of the vehicle ride comfort index iscalculated (as in Table 2) according to the 20 s simulationdata Compared with the passive suspension Figures 5(a)and 5(b) give a comparison of body acceleration and bodypitch acceleration response respectively ampe active sus-pension with double time-delay feedback control reducesthe bodyrsquos center of mass acceleration and pitch accel-eration significantly ampe RMS value drops from 30647and 23646 to 05026 and 11162 and the damping effi-ciency is as high as 8360 and 5280 It can be seen fromFigures 5(c) and 5(d) that the dynamic deflection of thefront and rear suspensions has also been reduced sig-nificantly ampe RMS values have decreased from 00383and 00772 to 00275 and 00362 correspondingly and thedamping efficiency is as high as 978 and 2820 It canbe seen from (e) and (f ) of Figure 5 that the front and reartire dynamic relative displacements have also been re-duced significantly and the corresponding RMS value ofthe tire dynamic displacements have decreased from00108 and 97508e minus 4 to 00020 and 33363e minus 4 Itdropped by 8148 and 6878 ampe simulation resultsshow that the active suspension system with double time-delay feedback control reduces the body acceleration andpitch acceleration without increasing the tire deformationand dynamic load ensuring the safety of vehicle drivingand vehicle handling stability ampis shows the effectivenessof the active control method with double time delay underharmonic excitation

52 Simulation Analysis under Random Excitation In orderto further study the damping effect of active suspension withdouble time-delay feedback control the time-delay pa-rameters optimized by particle swarm optimization in thispaper are applied to the vehicle active suspension modelwith double time delay in the actuator In order to verify thedamping effect of active suspensions with double time delaythe vehicle is simulated to travel at a speed of 20ms ampeparameters of an automobile suspension system are shownin Table 1 Random excitation is selected as the verticaldisturbance to the wheel axle Here a sine function su-perposition method is used to establish a time-domainmodel of random excitation as in Figure 6

xr(t) 1113944n

i1ξi sin ωit + δi( 1113857 (16)

where ξ is the amplitude ω is the equivalent frequency and δis the value randomly distributed on (0 2π)

ampe optimized parameters are brought into equation (1)and the random excitation is selected as the vertical dis-turbance to the wheel and shaft to analyze the vibrationresponse characteristics of the vehicle in the time-domainstate of the passive suspension system and the active sus-pension system with double time-delay feedback controlTime-domain simulation is performed for the body accel-eration pitching acceleration suspension dynamic deflec-tion and tire dynamic displacement of the suspension andthe simulation curves are as in Figure 7

From the time-domain simulation in Figure 7 and theroot mean square value of the vehicle ride comfort indexcalculated from the 20 s simulation data as in Table 3compared with the passive suspension the body accelerationand pitch acceleration are as in Figures 7(a) and 7(b)Corresponding comparison graphs are given respectivelyand their corresponding root mean square values havedropped from 14898 and 23858 to 12578 and 18610

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250N

orm

of r

oot 1

10 20 30 40 500Frequency

(a)

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

Nor

m o

f roo

t 2

10 20 30 40 500Frequency

(b)

Figure 4 Modulus of Z in the characteristic equation (a) Norm of root1 (b) Norm of root2

Shock and Vibration 7

ndash6

ndash4

ndash2

0

2

4

6Bo

dy ac

cele

ratio

n (m

middotsndash2)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(a)

Passive suspensionActive suspension with time delay

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Body

pith

acce

lera

tion

(mmiddotsndash2

)

18166 8 10 200 122 144Time (tmiddotsndash1)

(b)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

ndash006

ndash004

ndash002

0

002

004

006

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

(c)

ndash015

ndash01

ndash005

0

005

01

015D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(d)

Figure 5 Continued

8 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

ndash6

ndash4

ndash2

0

2

4

6Bo

dy ac

cele

ratio

n (m

middotsndash2)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(a)

Passive suspensionActive suspension with time delay

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Body

pith

acce

lera

tion

(mmiddotsndash2

)

18166 8 10 200 122 144Time (tmiddotsndash1)

(b)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

ndash006

ndash004

ndash002

0

002

004

006

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

(c)

ndash015

ndash01

ndash005

0

005

01

015D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

Passive suspensionActive suspension with time delay

18166 8 10 200 122 144Time (tmiddotsndash1)

(d)

Figure 5 Continued

8 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

respectively and the damping efficiency is 1557 and2199 ampis illustrates the active suspension pair withdouble time-delay feedback control Both the body

acceleration and pitch acceleration have been significantlyoptimized which has greatly improved the ride comfort ofthe vehicle Still the ride comfort of the vehicle has increased

Table 2 RMS value of ride comfort index under harmonic excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (m middot sminus2) 30647 05026 minus8360RMS acceleration of vehicle pitch (radsminus2) 23646 11162 minus5280RMS of dynamic deflection of front suspension (m) 00383 00275 978RMS of dynamic deflection of rear suspension (m) 00772 00362 2820RMS of dynamic displacement of front tire (m) 00108 00020 minus8148RMS of dynamic displacement of the rear tire (m) 97508eminus 4 33363eminus 4 minus6878

ndash01

ndash005

0

005

01

Disp

lace

men

t (x(

m))

1510 200 5Time (s)

Figure 6 Disturbance change curve of random excitation displacement

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

002

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

Passive suspensionActive suspension with time delay

2 4 6 8 10 12 14 16 18 200Time (tmiddotsndash1)

(e)

times10ndash3

ndash2

ndash15

ndash1

ndash05

0

05

1

15

Rear

susp

ensio

n tir

e disp

lace

men

t (m

)

18166 8 10 200 122 144Time (tmiddotsndash1)

Passive suspensionActive suspension with time delay

(f )

Figure 5 Simulation comparison of ride comfort index under harmonic excitation (a) Body acceleration (b) Body pitch acceleration(c) Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f )Rear suspension tire displacement

Shock and Vibration 9

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4Bo

dy ac

cele

ratio

n (m

s2 )

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(a)

Passive suspensionActive suspension with time delay

ndash8

ndash6

ndash4

ndash2

0

2

4

6

Body

pitc

h ac

cele

ratio

n re

spon

se (m

s2 )

1510 200 5Time (s)

(b)

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

Dyn

amic

defl

ectio

n of

fron

t sus

pens

ion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(c)

ndash01

ndash008

ndash006

ndash004

ndash002

0

002

004

006

008

01D

ynam

ic d

eflec

tion

of re

ar su

spen

sion

(m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(d)

Figure 7 Continued

10 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

while the dynamic deflection of the front and rear sus-pensions has increased as in Figures 7(c) and 7(d) ampedynamic deflection of the front and rear suspensions hasincreased and the root mean square values have increasedfrom 00276 and 00341 to 00303 and 00384 but the in-crease is within the range of our design (plusmn100mm) and thelimit stroke of the dynamic deflection has not been exceededAs in Figures 7(e) and 7(f) the corresponding root meansquare values of the relative displacement of the front andrear tires have been reduced from 00060 and 00067 to00055 and 00047 and the optimized efficiency is 833 and2985 ampe passive suspension has also been reduced to acertain extent indicating that double time-delay feedbackcontrol active suspension can significantly improve vehicleride comfort and vehicle driving safety

6 Conclusions

Under the premise of stability this paper researches thedamping effect of the active suspension system with doubletime-delay feedback control on the semicar model Simulatethe vibration characteristics of the vehicle under randomexcitation and harmonic excitation Use the amplitude-frequency characteristic function as the objective function to

obtain the time-delay feedback gain and time delay byparticle swarm optimization and analyze the stability of thesystem to ensure the stability of the system ampe belowconclusions are obtained from the simulation and analyzingthe semicar model with double time-delay feedback control

(1) Aiming at the four-degree-of-freedom vehicle sus-pension system use the time-delay dynamic shockabsorber theory to bring in the front and rear doubletime-delay tire state feedback control and proposethe frequency-domain scanning method to deter-mine the stability of the double time-delay feedbackcontrol system

(2) ampe center of mass acceleration and pitch acceler-ation of the vehicle body are improved significantlyby using the active suspension with double time-delay feedback control under harmonic excitationand random excitation which also improves thecomfort and maneuverability of the vehicle signifi-cantly Although the dynamic deflection of the frontand rear suspensions increases under random ex-citation the increasing range is within the designpermission and the dynamic displacement of thefront and rear wheels is also clearly controlled to

Table 3 RMS value of ride comfort index under random excitation

Performance indicators Passive suspension Active suspension with time delay Reduced proportion ()RMS acceleration of body centroid (mmiddotsminus2) 14898 12578 minus1557RMS acceleration of vehicle pitch (radsminus2) 23858 18610 minus2199RMS of dynamic deflection of front suspension (m) 00276 00303 978RMS of dynamic deflection of rear suspension (m) 00341 00384 1261RMS of dynamic displacement of front tire (m) 00060 00055 minus833RMS of dynamic displacement of the rear tire (m) 00067 00047 minus2985

ndash002

ndash0015

ndash001

ndash0005

0

0005

001

0015

Fron

t sus

pens

ion

tire d

ispla

cem

ent (

m)

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(e)Re

ar su

spen

sion

tire d

ispla

cem

ent (

m)

ndash0015

ndash001

ndash0005

0

0005

001

0015

1510 200 5Time (s)

Passive suspensionActive suspension with time delay

(f )

Figure 7 Simulation comparison of smoothness index under complex excitation (a) Body acceleration (b) Body pitch acceleration (c)Dynamic deflection of the front suspension (d) Dynamic deflection of the rear suspension (e) Front suspension tire displacement (f ) Rearsuspension tire displacement

Shock and Vibration 11

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration

ensure the grounding of the tires and the drivingsafety of the vehicle ampe results show that the activesuspension vehicle with double time-delay feedbackcontrol has a significant damping control effectwhich can improve the vehiclersquos comfort and ma-neuverability very much

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China (Grant no 51275280)

References

[1] J D J Lozoya-Santos R Morales-Menendez andR A Ramırez Mendoza ldquoControl of an automotive semi-active suspensionrdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash21 2012

[2] L Chen R C Wang H B Jiang L K Zhou and S H WangldquoTime delay on semi-active suspension and control systemrdquoChinese Journal of Mechanical Engineering vol 42 no 1pp 130ndash133 2006

[3] D M Zhang and L Yu ldquoReview of stability analysis of lineartime delay systemsrdquo Control and Decision vol 23 no 8pp 841ndash849 2008

[4] W Z Zhang B Zhang X HWu andQ Sun ldquoAnalysis of theinfluence of time delay on the control effect of active vibrationcontrol systemrdquo China Mechanical Engineering vol 24 no 3pp 317ndash321 2013

[5] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5pp 793ndash797 2002

[6] N Olgac and R Sipahi ldquoampe cluster treatment of charac-teristic roots and the neutral type time delayed systemsrdquoDynamic Systems and Control Parts A and B vol 127 no 12005

[7] T Vyhlıdal N Olgac and V Kucera ldquoDelayed resonator withacceleration feedback - complete stability analysis by spectralmethods and vibration absorber designrdquo Journal of Sound andVibration vol 333 no 25 pp 6781ndash6795 2014

[8] H Y Hu and Z H Wang ldquoResearch progress in dynamics ofcontrolled mechanical systems with time delayrdquo Progress inNatural Science vol 10 no 7 pp 577ndash585 2000

[9] H Y Hu and Z H Wang ldquoResearch progress of nonlineartime delay dynamical systemsrdquo Mechanics Progress vol 29no 4 pp 501ndash512 1999

[10] H Su and G Y Tang ldquoVibration control of active suspensionsystem with input delayrdquo Control 8eory and Applicationvol 33 no 4 pp 552ndash558 2016

[11] J Zhang H Su L K Wang and G Y Tang ldquoApproximateoptimal tracking control for discrete time systems with state

and input delaysrdquo Control and Decision vol 32 no 1pp 157ndash162 2017

[12] J Xu and J P Li8e Recent Research Progress and Prospect ofTime Delay Systems Dynamics Springer Berlin Germany2006

[13] Y Y Zhao and J Xu ldquoTime delay dynamic vibration absorberand its influence on the vibration of main systemrdquo Journal ofVibration Engineering vol 19 no 4 pp 548ndash552 2006

[14] Y Y Zhao and R M Yang ldquoSaturation control of dampingfrequency band of self-parameter vibration system using timedelay feedback controlrdquo Acta Physica Sinica vol 60 no 10p 104304 2011

[15] N A Saeed W A El-Ganini and M Eissa ldquoNonlinear timedelay saturation-based controller for suppression of nonlinearbeam vibrationsrdquo Applied Mathematical Modelling vol 37no 20-21 pp 8846ndash8864 2013

[16] N A Saeed and W A El-Ganaini ldquoTime-delayed control tosuppress the nonlinear vibrations of a horizontally suspendedJeffcott-rotor systemrdquo Applied Mathematical Modellingvol 44 pp 523ndash539 2017

[17] N A Saeed and W A El-Ganaini ldquoUtilizing time delays toquench the nonlinear vibrations of a two-degree-of-freedomsystemrdquo Meccanica vol 52 no 11-12 pp 2969ndash2990 2017

[18] N A Saeed and H A El-Gohary ldquoInfluences of time-delayson the performance of a controller based on the saturationphenomenonrdquo European Journal of Mechanics - ASolidsvol 66 pp 125ndash142 2017

[19] K W Wu C B Ren J S Cao and Z C Sun ldquoReach ondamping control and stability analysis of vehicle with doubletime-delay and five degrees of freedomrdquo Journal of LowFrequency Noise Vibration and Active Control 2020

[20] W Hu and L I Zhi-Shu ldquoA simpler and more effectiveparticle swarm optimization algorithmrdquo Journal of Softwarevol 18 no 4 pp 861ndash868 2007

[21] X G Li Several Studies on the Stability of Time Delay SystemsDoctoral dissertation Shanghai Jiao Tong UniversityShanghai China 2007

[22] W Q Fu H Pang and K Liu ldquoModeling and stabilityanalysis of semi-active suspension with time delayrdquo Journal ofMechanical Science and Technology vol 18 no 4 pp 213ndash2182017

[23] R Sipahi and N Olgac ldquoKernel and offspring concepts for thestability robustness of multiple time delayed systems(MTDS)rdquo Journal of Dynamic Systems Measurement ampControl vol 129 no 3 pp 245ndash251 2007

[24] R Sipahi and N Olgac ldquoStability robustness of retarded LTIsystems with single delay and exhaustive determination oftheir imaginary spectrardquo SIAM Journal on Control and Op-timization vol 45 no 5 pp 1680ndash1696 2006

12 Shock and Vibration