Performance Benefits of Using Inerter in Semiactive Suspensions

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Performance Benefits of Using Inerter in Semiactive SuspensionsMichael Z. Q. Chen, Yinlong Hu, Chanying Li, and Guanrong Chen

Abstract This brief investigates the performance benefits ofusing inerter in semiactive suspensions. A novel structure forsemiactive suspensions with inerter, which consists of a passivepart and a semiactive part, is proposed. Six semiactive suspensionstruts, each of which employs an inerter in the passive partand a semiactive damper in the semiactive part, are introduced.Two suboptimal control laws named clipped optimal control andsteepest gradient control laws are derived to control the dampingcoefficient in the semiactive part. Extensive simulations withrespect to different choices of weighting factors and suspensionstatic stiffnesses are conducted based on both a quarter carmodel and a full car model. The results show that, comparedwith the conventional semiactive suspension strut, the overallsuspension performance can be significantly improved usinginerters, including ride comfort, suspension deflection, and roadholding. Comparative studies between these two suboptimalcontrol laws and between these semiactive struts are also carriedout to facilitate future practical application of the proposedsemiactive suspensions with inerter.

Index Terms Full car model, inerter, quarter car model,semiactive suspension, suboptimal control.

I. INTRODUCTION

INERTER is a two-terminal mechanical device with theproperty that the applied force at the two terminals is pro-portional to the relative acceleration between them [1], whosesymbolic representation is shown in Fig. 1. The inerter expandsthe class of mechanical realizations of complex impedancescompared with the ones using only springs and dampers,and has been applied to various mechanical systems, such asvehicle suspensions [2][6]. It has also rekindled interest inpassive network synthesis [7][14].

Semiactive suspension has attracted much attention becauseof its low energy consumption compared with the activeone [15][17] and its high performance compared with

Manuscript received May 27, 2014; accepted October 19, 2014. Manuscriptreceived in final form October 20, 2014. This work was supported in part bythe National Natural Science Foundation of China under Grant 61374053and Grant 61203067, in part by the Hong Kong University Committeeon Research and Conference Grants under Grant 201211159112, and inpart by the Research Grants Council, Hong Kong, through the GeneralResearch Fund under Grant CityU 1120/14. Recommended by AssociateEditor S. M. Savaresi.(Corresponding author: Michael Z. Q. Chen.)

M. Z. Q. Chen is with the Department of Mechanical Engineering, Uni-versity of Hong Kong, Hong Kong, and also with the Shenzhen Instituteof Research and Innovation, University of Hong Kong, Hong Kong (e-mail:[email protected]).

Y. Hu is with the School of Automation, Nanjing University of Science andTechnology, Nanjing 210044, China (e-mail: [email protected]).

C. Li is with the Academy of Mathematics and Systems Sciences, ChineseAcademy of Sciences, Beijing 100190, China (e-mail: [email protected]).

G. Chen is with the Department of Electronic Engineering, City Universityof Hong Kong, Hong Kong (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2014.2364954

Fig. 1. Symbolic representation of inerter with F = b(v2 v1), where b iscalled inertance, which has units of kilograms [1].

the passive one. The conventional semiactive suspensionconfiguration, that is, a spring in parallel with a variableshock absorber (may contain a passive damper), has beenextensively investigated with a large number of meaningfulresults obtained [18][23].

This brief proceeds to applying inerter in semiactivesuspensions, showing its significant performance benefits aswell. These are demonstrated by proposing a novel struc-ture for semiactive suspensions employing inerters, whichconsists of a passive part and a semiactive part. Six pas-sive networks, each of which employs an inerter, constitutethe passive part, whereas the semiactive part incorporatesa semiactive damper. Two suboptimal control laws namedclipped optimal control (COC) law and steepest gradientcontrol (SGC) law [21] are employed to control the semiactivedamper, and extensive simulations based on both a quartercar model and full car model are conducted to demonstratethe significant improvements of using inerter in semiactivesuspensions. All these constitute the main contributions ofthis brief.

The organization of this brief is as follows. In Section II, theproposed semiactive suspensions with inerter are introducedbased on a quarter car model and the semiactive suspensioncontrol problem is formulated. In Section III, two suboptimalcontrol laws for these six semiactive suspension struts arederived, and in Section IV, numerical simulation in terms of aquarter car model is carried out. A study of the proposed semi-active suspensions in a full car model is reported in Section V.Conclusions are drawn in Section VI.

The notation used throughout this brief is standard. I and 0are used to denote the identity matrix and zero matrix ofappropriate dimensions, respectively, and diag are used todenote the diagonal matrix.

II. QUARTER CAR MODEL ANDPROBLEM FORMULATION

A. Quarter Car ModelConsider the quarter car model shown in Fig. 2, which

consists of the sprung mass ms , unsprung mass mu , and tirevertical stiffness kt [20]. The suspension system consists of

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2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

Fig. 2. Semiactive quarter car model.

Fig. 3. Configurations as W (s) of the suspensions in Fig. 2. (a) C1. (b) C2.(c) C3. (d) C4. (e) C5. (f) C6. (g) C7.

two parts: 1) the passive part and 2) semiactive part. The pas-sive part contains a spring in parallel with a passive networkshown in Fig. 3, where the admittance of the passive part isYi (s) = (ks/s) + Wi (s), i = 1, . . . , 7, and Wi (s) is shownin Table I. Here, admittance is defined as the ratio of forceto velocity according to the force-current analogy [1], [7].The semiactive part involves a variable shock absorber, suchas electrohydraulic dampers (EH dampers), magnetorheolog-ical dampers (MR dampers), or electrorheological dampers(ER dampers) [19].

Denote cv as the controllable damping coefficient, whichcan be adjusted by modifying the electronic valves forEH dampers or changing the physical properties of fluidsfor MR dampers and ER dampers (see [19, Ch. 2] fordetails). The force generated by the semiactive part Fd canbe represented as Fd = cv (zs zu), where cv [cmin, cmax].In this brief, a nominal damping coefficient cv0 is consideredfor the semiactive damper so that a stable open-loop plantcan be obtained, which means that the semiactive dampingcoefficient cv is adjusted based on cv0. In this way, the lowerand upper bounds for cv become cmin cv0 and cmax cv0,respectively.

TABLE IW (s) FOR EACH CONFIGURATION IN FIG. 3, WHERE s

DENOTES THE LAPLACE VARIABLE

A state space model for the semiactive quarter car systemcan easily be obtained as

xm = Am xm + Bm Fp + Bm Fd + Bmrzr (1)where xm = [zs zu zs zu]T , Fp denotes the force generatedby the networks W (s), and

Am =[

0 IM1 K M1Cv0

], Bm =

[0

M1 E

]

Bmr =[

0M1 Kt

], M = diag{ms, mu}, E = [1 1]T

Kt = [0 kt ]T, Cv0 =[

cv0 cv0cv0 cv0

], K =

[ks ks

ks ks +kt].

Define Fp = sW (s)(zs zu), where Fp , zs , and zu denotethe Laplace transforms of Fp , zs , and zu , respectively. A statespace model for the passive part W (s) can be obtained as

x p = A px p + Bp(zs zu) (2)Fp = Cpx p + Dp(zs zu) (3)

where zs zu is the input and Fp is the output. Combining(1)(3), the semiactive quarter car system can be rewritten as

xg = Agxg + Bg Fd + Bgrzr (4)where xg = [x Tm x Tp ]T and

Ag =[

Am + Bm Dp L BmCpBp L A p

], Bg =

[Bm0

]

Bgr =[

Bmr0

], L = [0 0 1 1].

B. Road ModelTypically, random road profiles can be described using a

power spectral density (PSD) function in the form of

() = (0)(

0

)

where is the wave number with the unit [rad/m] and0 = (0) in [m2/(rad/m)] is the value of the PSD at0 = 1 rad/m. In addition, denotes the waviness, where = 2 for most of the road surfaces. To realize such a PSDfunction in the time domain, the shaping filter method [25]can be employed as follows:

zr (t) = V zr (t) + w(t) (5)where the white noise process w(t) with the spectral densityw = 2V 2 and V is the vehicle forward speed.


CHEN et al.: PERFORMANCE BENEFITS OF USING INERTER IN SEMIACTIVE SUSPENSIONS 3

TABLE IIROAD DESCRIPTION CLASSIFIED BY ISO 8608 [24], [25]

As suggested by the International Organization for Stan-dardization (ISO) [24], typical road profiles based on thevalue of 0 can be classified as Class AClass E, as shownin Table II. The values of and in the linear filterthat correspond to the ISO standard road profiles are shownin Table II (see [25] for details).

C. Augmented Model and Problem FormulationCombining (4) and (5), and denoting x = [x Tg , zr ]T , one

can obtain an augmented model as

x = Ax + B Fd + Bww (6)where

A =[

Ag Bgr0 V

], B =

[Bg0

], Bw =

[01

].

The controlled output y involving the sprung mass acceler-ation, suspension deflection, and tire deflection is defined as

y = [zs (zs zu) (zu zr )]T = Cx + DFdwhere

C = A(3, :)1 1 0 0 0 0

0 1 0 0 0 1

, D =

Bm(3, :)0

0

and A(3, :) and Bm(3, :) denote the third row of A and Bm ,respectively.

To deal with these three performance requirements simul-taneously, a quadratic performance index is defined as

J = limT

T0

yT y + F2d rdt (7)

where contains the weighting factors among sprung massacceleration, suspension deflection, and tire deflection, and r isa weighting factor to be determined by the designer.

The performance index (7) can be rewritten as

J = limT

T0

x T Qx + 2x T N Fd + FTd RFd dt (8)

where Q = CT C , N = CT D and R = DT D + r .

III. SUBOPTIMAL CONTROLLER DESIGNIf ignoring the constraint on the semiactive damping, the

aforementioned problem is a linear quadratic regulation prob-lem and an optimal controller can be derived as

Fd = R1(BT P + NT )x (9)

where P is the solution of the following algebraic Riccatiequation (ARE):

AP + P A (P B + N)R1(BT P + NT ) + Q = 0. (10)As shown in [21], the optimal solution for a quarter car

semiactive suspension involves time-varying Riccati equationsand no analytical solution can be obtained. However,suboptimal control laws based on, such as the clipped optimalmethod and the steepest gradient method, are fine engineeringapproximations in practice, which are derived, respectively,in the following.

Proposition 1: The COC law is

Fd =

Fdmin , Fd < FdminFd , Fdmin Fd FdmaxFdmax , Fd > Fdmax

(11)

where

Fdmin =(cmin cv0)|zs zu |, Fdmax = (cmax cv0)|zs zu |.As shown in [21], the performance index (8) is related to

the optimal passive suspension as

Jsemi = Jpassive +

0

(2x T N Fd + FTd RFd

)dt

where P is the solution of the Lyapunov equation

AT P + P A = Q. (12)The SGC law is derived as follows.

Proposition 2: The SGC law can be derived using (11)where the P matrix in (9) is derived by (12) insteadof (10).

Note that the COC law and SGC law share the samestructure, where the only difference is the procedure todetermine P . The COC law is equivalent to making the semi-active suspension approximate the active suspension, whereasthe SGC law is equivalent to increasing the improvement ofthe semiactive suspension compared with the passive one [21].Since both of these two control laws heavily depend on thevalues of the weight coefficients, it is hard to say, whichis better for a general semiactive suspension system [22].In this brief, both of them are employed to demonstrate thebenefits of the inerter for the considered semiactive suspensionsystems.

IV. QUARTER CAR MODEL SIMULATIONThe vehicle parameters are taken from [2] with

ms = 250 kg, mu = 35 kg, kt = 150 kN/m, and thesuspension static stiffness ks is selected from 10 to 120 kN/m.The vehicle forward speed V is assumed to be 30 m/s andthe nominal value of the semiactive damper is chosen ascv0 = 1500 Ns/m. In addition, r in (7) is chosen as 0 andthe bounds of semiactive damping coefficient are chosen ascmin = 0 Ns/m and cmax = 3000 Ns/m.



A. Quantitative Performance IndexesTo quantitatively compare the performances among these

configurations shown in Fig. 3, the performance measures ,acc, sus, and rhd representing the overall, ride comfort,suspension deflection, and road holding performances,respectively, are defined as the root mean square (rms) valuesof y, zs , zs zu , and zu zr , respectively.

The weighting factor is chosen as

= diag{

1normacc

2 ,1

normsus2 ,

2

normrhd2

}(13)

where normacc , normsus , and normrhd are the normalized valuesof acc, sus, and rhd, respectively. In this brief, basedon the open-loop system (passive system) with the nom-inal damping coefficients cv0 = 1500 Ns/m, one obtainsnormacc = 0.9962,normsus = 0.0058, and normrhd = 0.0028. Notethat for the above choice of , one has

2 = 2acc

normacc2 + 1

2susnormsus

2 + 22rhd

normrhd2

where 1 and 2 are weighting factors between ride comfort,suspension deflection, and road holding.

In this brief, three groups of 1 and 2 values are selected as

1 = 0.5, 2 = 0.5 :Preference of ride comfort over road holding

1 = 0.5, 2 = 1 :Equal preference of ride comfort and road holding

1 = 0.5, 2 = 1.5 :Preference of road holding over ride comfort.

B. Optimization of the Passive Part ParametersThe NelderMead simplex method is employed with various

starting points for each configuration to obtain the optimalvalues of the component coefficients (a similar procedure canbe found in [2] and [4]), and a set of optimal parametersare obtained with respect to different static stiffness anddifferent choices of 1 and 2. The detailed values of theseparameters at ks = 80 kN/m are shown in Table III. SinceC4 and C7 reduce to C3 and C6, respectively, they are notshown in Table III. Note that the nominal damping cv0 hasbeen considered in optimizing the passive part, which meansthat a parallel damper with damping coefficient cv0 is addedto each configuration when acting as passive suspension toobtain the optimal parameters.

C. Simulation ResultsNumerical simulations under both the COC and SGC are

conducted, where the simulation time is chosen as 20 s andthe average road profile (Class C road) is employed. Notethat similar results are obtained from these two suboptimalcontrol laws. For brevity, only the results under the COCare shown in Figs. 49. Since the relaxation spring kb inC4 and C7 provides no improvement for ride comfort [2], [5],C4 and C7 reduce to C3 and C6, respectively, when 1 = 0.5and 2 = 0.5.

TABLE IIIPARAMETERS IN PASSIVE PART WHEN ks = 80 kN/m FOR QUARTER CAR

MODEL (kS ARE IN kN/m, cS ARE IN kNs/m, bS ARE IN kg)

Fig. 4. Comparison of the overall performance when 1 = 0.5,2 = 0.5 under the COC. Left: quantitative values. Right: percentageimprovement over C1.

Fig. 5. Comparison of the ride comfort, suspension deflection, and roadholding when 1 = 0.5, 2 = 0.5 under the COC.

Fig. 6. Comparison of the overall performance when 1 = 0.5,2 = 1 under the COC. Left: quantitative values. Right: percentage improve-ment over C1.

From Figs. 4, 6, and 8, it is observed that for all theselected 1 and 2, the configurations with inerter performbetter than the conventional semiactive strut C1, where over10% improvements have been obtained. It is also shown inFigs. 4, 6, and 8 that the series arrangements of inerter



Fig. 7. Comparison of the ride comfort, suspension deflection, and roadholding when 1 = 0.5, 2 = 1 under the COC.

Fig. 8. Comparison of the overall performance when 1 = 0.5,2 = 1.5 under the COC. Left: quantitative values. Right: percentageimprovement over C1.

Fig. 9. Comparison of the ride comfort, suspension deflection, and roadholding when 1 = 0.5, 2 = 1.5 under the COC.

(C3C7) provide more improvement than the parallel arrange-ment C2. From Figs. 5, 7, and 9, it is seen that both ridecomfort and road holding are simultaneously improved by theseries arrangements C3C7. Meanwhile, suspension deflectionis degraded, which is consistent with the results in [5] onthe analysis of passive suspensions with inerter. However, theimprovement of the ride comfort by the parallel arrangementC2 is obtained by sacrificing both suspension deflection androad holding, which is a potential drawback of the parallelarrangement of inerter.

A comparison between the COC and SGC is also conductedand the percentages of improvement of the SGC over theCOC are shown in Fig. 10, where we see that for the selected1 and 2, these two suboptimal control laws do not differfrom each other by too much (only gaps of at most 1.5% areshown in Fig. 10). It is also observed that for the consideredvehicle model, the COC tends to perform better than theSGC in the low static stiffness range (1030 kN/m), whereasthe SGC tends to do better in the high static stiffness range(70120 kN/m).

Fig. 10. Percentages of improvement of the SGC over the COC. Positivesign means improvement, whereas negative sign means degradation.

Fig. 11. Semiactive full car model.

V. APPLICATION TO A FULL CAR MODEL

In this section, we proceed to applying the proposed semiac-tive suspensions with inerter to a seven degree of freedom fullcar model, as shown in Fig. 11, whereas a detailed descriptionof this model can be found in [2]. The following subscriptsare used: 1) f r ; 2) f l; 3) rr ; and 4) rl denote the front-right,front-left, rear-right, and rear-left, respectively. The proposedsemiactive suspensions are equipped at each corner of the fullcar model, and it is assumed that the passive part at each cornerof the vehicle possesses the same structure, but the values ofthe components coefficients can be different, which will bedetermined by numerical optimization.

By assuming that the angles of the sprung mass in thepitch and roll directions are small enough, and using themethod of modeling the passive parts in the quarter car case,a state-space model integrating the full car model, the passiveparts, and a first-order road model can be obtained as follows:

x = Ax + B Fd + Bww (14)where x = [xTs , x Tu , x Tp , x Tr ]T , xs = [zs, , ]T , xu = [zu f r ,zu f l , zurr , zurl ]T , x p, and xr are states of the passive partmodel and road model, respectively. Fd contains the forcesgenerated by the semiactive damper at each corner.

The road input considered in this full car case is a paralleltrack, which can be obtained by passing different white noiseprocesses wl(t) and wr (t) through the same linear filter (5)and ignoring the correlation between the left and right roadexcitations. The Pad approximation method is employed toapproximate the time delay between the front wheels and therear wheels.



The controlled outputs are defined as

y = [x Ts (x1 xu)T (xu Zr )T ]Twhere xs , (x1 xu), and (xu Zr ) represent the sprungmass accelerations, suspension deflections, and tire deflections,respectively, and

x1 = [zs f r , zs f l , zsrr , zsrl ]T= ET xs

Zr = [zr f r , zr f l , zrrr , zrrl ]T

E = 1 1 1 1l f l f lr lr

t f t f tr tr

.

Similarly, a performance index is defined as

J = limT

T0

yT y + FTd Fd dt

= limT

T0

x T Qx + 2x N Fd + FTd RFd dt (15)

where Q = CT C , N = CT D, and R = DT D+, where

is a weighting factor determined by the designer, and theweighting factor is chosen as = diag{1,2,3}, with

1 = diag{

1normacc

2 ,1

normacc2 ,

1normacc

2

}

2 = 1 diag{

1normsus

2 ,1

normsus2 ,

1normsus

2 ,1

normsus2

}

3 = 2 diag{

1normrhd

2 ,1

normrhd2 ,

1normrhd

2 ,1

normrhd2

}

where normacc , normsus , and normrhd are the normalized values ofacc, sus, and rhd, respectively. acc, sus, and rhd are therms values of xs , (x1 xu) and xu zr , respectively, whichquantitatively represent ride comfort, suspension deflection,and road holding, respectively. In this way, one has

2 = 2acc

normacc2 + 1

2susnormsus

2 + 22rhd

normrhd2

where 1 and 2 are weighting factors between ride comfort,suspension deflection, and road holding.

A. Suboptimal Control LawsSimilar to the quarter car case, two suboptimal control laws

can be derived for the considered full car model.Proposition 3: The COC law is

Fdi =

Fdi_ min, Fdi < Fdi_ minFdi , Fdi_min Fdi Fdi_ maxFdi_ max, Fdi > Fdi_ max

(16)

where i denotes f r , f l, rr , and rl, separatelyFdi_ min = (cmin cv0i)|x1i xui |Fdi_ max = (cmax cv0i)|x1i xui |

Fd = R1(BT P + NT )x (17)and P is the solution of the following ARE:

AP + P A (P B + N)R1(BT P + NT ) + Q = 0.

TABLE IVPARAMETERS IN PASSIVE PART WHEN k f = kr = 80 kN/m FOR THE FULL

CAR MODEL (kS ARE IN kN/m, cS ARE IN kNs/m, bS ARE IN kg)

Fig. 12. Comparison of the overall performance when 1 = 0.5,2 = 1 under the COC. Left: quantitative values. Right: percentage improve-ment over C1.

Proposition 4: The SGC law can be derived using (16)but the P matrix in (17) is derived by solving the followingLyapunov function AT P + P A = Q.B. Full Car Model Simulation

The following parameters taken from [2] are employed:1) ms = 1600 kg; 2) I = 1000 kgm2; 3) I = 450 kgm2;4) t f = 0.75 m; 5) tr = 0.75 m; 6) l f = 1.15 m;7) lr = 1.35 m; 8) m f = 50 kg; 9) mr = 50 kg; 10) kt f =250 kN/m; and 11) ktr = 250 kN/m. The vehicle forwardspeed V is assumed to be 30 m/s and the nominal values of thesemiactive damper are chosen as cv0i = 1500 Ns/m, i denotesf r , f l, rr , and rl, separately, and the bounds of semiactivedamping coefficient are chosen as cmin = 0 Ns/m andcmax = 3000 Ns/m. Since similar results are obtained fordifferent choices of 1 and 2 in the quarter car case,1 = 0.5 and 2 = 1 are selected in the full car modelsimulation. The average road (Class C road) is employedand the simulation time is 20 s. in (15) is cho-sen as diag{1e6, 1e6, 1e6, 1e6}. The normalized val-ues are obtained from the open-loop simulation asnormacc = 1.3837,normsus = 0.0137, and normrhd = 0.0060.

Note that in the quarter car simulation, C7 always reducesto C6 for most of static stiffness, and C6 performs slightlybetter than C5 due to the employment of an extracenteringspring. Hence, in the full car case, the parallel arrangement C2and the series arrangement C6 are considered, which arecompared with the conventional strut C1. First of all, theoptimal parameters of the passive parts at each corner areobtained using the NelderMead simplex method. Similarto the quarter car case, a set of optimal parameters withrespect to different static stiffness are obtained and the detailedparameters at 80 kN/m are summarized in Table IV.

Then, a semiactive damper is equipped at each corner,which is controlled by the proposed suboptimal control laws.



Fig. 13. Comparison of the ride comfort, suspension deflection, and roadholding when 1 = 0.5, 2 = 1 under the COC.

Fig. 14. Comparison between the SGC and the COC. Positive number meansSGC better than COC, whereas negative number means COC better than SGC.

As shown in Figs. 12 and 13, the simulation results confirmthe findings in the quarter car simulation that the proposedsemiactive suspensions with inerter indeed improve the over-all performance compared with the conventional semiactivesuspension C1, where almost 7% improvement is shownin Fig. 12, and both ride comfort and road holding can beimproved by the series arrangement C6. However, as shownin Fig. 14, for the considered full car model, the COCconsistently performs better than the SGC except C2 at highstatic stiffness, which is different from the quarter car case.This confirms the conclusion that the performance of thesesuboptimal control laws heavily depends on the selectedparameters [22].

VI. CONCLUSIONIn this brief, performance benefits of using inerter in semi-

active suspensions have been demonstrated by proposing anovel structure for semiactive suspension with inerter, whichconsists of a passive part and a semiactive part. Six differentsemiactive suspension struts, each of which employs an inerterin the passive part and a semiactive damper in the semiactivepart, were introduced. Two suboptimal control laws, that is,the COC and the SGC laws, were derived to control the semi-active damping coefficients. An augmented state-space modelintegrating the vehicle model, the suspension model, and afirst-order road model was established for both a quartercar model and a full car model. Extensive simulationswere conducted, showing that the proposed semiactive strutscan significantly improve the overall suspension perfor-mance, including ride comfort, suspension deflection, and roadholding. For example, >10% improvements over the conven-tional semiactive suspension without inerter were observed forthe quarter car model. Moreover, comparisons between thesetwo suboptimal control laws and between these semiactivesuspension struts, were conducted to facilitate the practicalimplementation of the proposed semiactive suspension struts,

which is expected to find some applications in the nearfuture.

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Performance Benefits of Using Inerter in Semiactive Suspensions

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