Journal of SystemDesign andDynamics

Vol.1, No.2, 2007

An Active Suspension Controller Achievingthe Best Ride Comfort

at Any Specified Location on A Vehicle∗

Masahiro OYA∗∗, Hiroshi HARADA∗∗ and Yoshiaki ARAKI∗∗∗∗∗ Kyushu Institute of Technology

1–1 Sensui, Tobata-ku, Kitakyushu 804–8550, Japan

E-mail: oya@cntl.kyutech.ac.jp∗∗∗ University of East Asia

2–1 Ichinomiyagakuen, Shimonoseki 751–8503, Japan

AbstractIn this paper, a new active suspension control scheme is developed so that ride com-fort becomes best at any specified location on vehicle body. To achieve this end, twoideal vehicles are designed in which ride comfort becomes best at each different lo-cation. Then, linearly combining the two ideal vehicles, a combined ideal vehicle isconstructed. It should be noted that we can easily force ride comfort at a specifiedlocation become best in the proposed combined ideal vehicle by setting only one de-sign parameter. To achieve the good property stated above in actual vehicles, a robusttracking controller is proposed. It is shown by carrying out numerical simulations thatride comfort at a specified location can be easily improved in the closed loop systemusing the proposed combined ideal vehicle.

Key words : Automobile, Ride Comfort, Ideal Model, Robust Tracking Control

1. Introduction

Recently, to achieve good ride and good handling qualities, a large amount of controlstrategies using active automotive suspensions have been proposed(1) – (16). In reference pa-pers (1) – (9), control strategies have been developed based on quarter-car model, and controlstrategies have been developed based on half-car model or full-car model in reference papers(10) – (16). For early papers, refer to the survey paper (17). Let’s consider the case whenthe acceleration of vehicle body is controlled so as to be zero. Then, the best ride comfort isachieved at every location on vehicle body. However, some serious problems arise. For exam-ple, the suspension stroke may over an admitted range and the handling qualities may becomeworse because of large variation of tire deflection. Therefore, in conventional strategies whichis proposed to improve the ride comfort, active suspension controllers are designed by usinga trial and error method so that the suspension stroke lies within an admitted range and thehandling qualities don’t become worse. Using the conventional controllers, the ride comfortisn’t same at every location on vehicle body and becomes best at only one location on vehiclebody. Each every time the best location for the ride comfort has to be changed, we have toredesign another different controller by using a trial and error method. Namely, plenty of timeis required to move best location for ride comfort.

In this paper, to overcome the problem stated above, we propose a new strategy to controlactive suspensions. In the closed loop system using the proposed active suspension controller,the ride comfort at a specified location becomes best in some sense. We call it ’semi-best’hereafter. At first, two ideal vehicles are designed in which the ride comfort becomes best ateach of two different locations. Then, the combined ideal vehicle is proposed in which the twoideal vehicles are combined linearly. The combined ideal vehicle has the good property that

∗Received 15 Jan., 2007 (No. T-04-0667)Japanese Original: Trans. Jpn. Soc.Mech. Eng., Vol.71, No.701, C (2005),pp.137–144 (Received 9 June, 2004)[DOI: 10.1299/jsdd.1.245]

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Fig. 1 Two wheels model.

the location where the ride comfort becomes semi-best can be easily moved without redesignof two ideal vehicles. Finally, an active suspension controller is proposed so that the motionof an actual vehicle tracks the motion of the proposed combined ideal vehicle.

2. Dynamic equation

Let’s consider the two wheels model shown in Fig. 1. The explanations of parameters areshown in Table 1. It is assumed that pitching angle θ(t) is small, and then, dynamic equationof vehicles is given as follows. The symbol I2 denotes 2 × 2 unit matrix.

xz(t) = M−1d(t) − H−1w(t), xu(t) = −Kuxu(t) − M−1u (HT )−1d(t) − w(t)

xz(t) = H−1[z f (t) − w f (t), zr(t) − wr(t)]T

xu(t) = [zu f (t) − w f (t), zur(t) − wr(t)]T

d(t) = HT (−C xs(t) − Kxs(t) + u(t))

(1)

xs(t) = Hxz(t) − xu(t), w(t) = [w f (t), wr(t)]T , u(t) = [uf (t), ur(t)]T

M = (T Th )−1diag[m, ic]T−1

h , Mu = diag[mu f , mur], K = diag[k f , kr]C = diag[c f , cr], Ku = M−1

u diag[ku f , kur]

H =

1 a1 − a

, Th = I2 − Dh, D =

0 10 0

(2)

The control objective here is to develop an active suspension controller so that the verticalacceleration at any specified location on vehicle body can be reduced to small value. To meetthe objective, the following assumptions are made for actual vehicles considered here.A1 Vertical acceleration q(t)T = [z f (t), zr(t)] is measured.A2 Suspension displacement xs(t) and its velocity xs(t) are measured.A3 Tire deflection Hxz(t) and its velocity H xz(t) are measured.A4 The values of vehicle parameters are known. Additionally, the values of vehicle systemparameters don’t vary except for sprung mass m, moment of inertia ic of vehicle body and thecenter of gravity.

3. Robust tracking controller

In the explanation below, 0n, In denote n × n zero matrix and n × n unit matrix. Todistinguish between the matrix M using values of actual vehicle parameters and the matrix Musing values of nominal vehicle parameters, the matrix using nominal values is denoted byM. Additionally, we use the relation that the acceleration y(, t) = [z(t), θ(t)]T at a location on vehicle body can be described as y(, t) = (I2 + D)M−1d(t) where z denotes verticalacceleration at the location .

3.1. Preparation to design a combined ideal vehicleFor the help of understanding the explanation in the following subsection, at first, it is

explained how to design an ideal vehicle achieving good ride comfort.

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Table 1 Notation of two wheels modelSymbol MeaningC,CG center and center of gravity of vehicle bodyzcg, θ vertical displacement at CG and pitchingz f , zr vertical displacement of vehicle body at positions on front and rear wheel axlezu f , zur vertical displacement of front and rear unsprung massw f , wr vertical displacement of road disturbance added to front and rear wheelv longitudinal speed of vehiclem, ic sprung mass and moment of inertia of vehicle bodya half of vehicle body lengthh, distances from C to CG and from C to Pmu f ,mur front and rear unsprung massk f , kr front and rear suspension stiffnessc f , cr front and rear suspension damping rateku f , kur front and rear tire spring stiffnessu f , ur front and rear wheel active control force

Table 2 Nominal values of parametersm 781 kg ic 990 kgm2

h 0.04 m a 1.38 mk f 27160 N/m kr 29420 N/mc f 4000 Ns/m cr 2500 Ns/m

mu f 69 kg mur 96 kgku f 229000 N/m kur 255000 N/m

Let’s consider the new state xT (t) = [xTz (t), xT

u (t), xTz (t), xT

u (t), dT (t)] including thesignal d(t) relating to the vertical acceleration of vehicle body. Then, the dynamic equation isdescribed as

x(t) = Ax(t) + Bux(t) − Dww(t), ux(t) = u(t)+u(t)A = Γ + BF, DT

w = [02, 02, (H−1)T , I2, 02], B = [02, 02, 02, 02, H]T

, (3)

Γ =

02 02 I2 02 02

02 02 02 I2 02

02 02 02 02 M−1

02 −Ku 02 02 −M−1u (HT )−1

02 02 02 02 02

F = [−KH, − CKu + K, − (C+K)H, C+K, F5]F5 = −C(HM−1 + M−1

u (HT )−1) − (HT )−1

. (4)

Defining the nominal state xN(t)T = [xNz(t)T , xNu(t)T , xNz(t)T , xNu(t)T , dN(t)] and thenominal input uNx(t) = uN(t) + uN(t), it follows from Eqs. (3), (4) that the dynamic equationof the nominal vehicle with nominal values shown in Table 2 is given as follows.

xN(t) = AxN(t) + BuNx(t) − Dww(t), A = Γ + BF (5)

Where the symbol • denotes a matrix in which the matrix M is replaced with M.In the case of utilizing LQ optimal control for the nominal vehicle, the disturbance w(t)

is recognized as an impulse disturbance in the nominal vehicle Eq. (5), and then, a controlleris designed(18) so that a quadratic criterion is minimized. Here, let’s consider the followingcriterion

J =∫ ∞

0

(qNm (t)T EqqNm (t) + xNz(t)

T EzxNz(t) + xNs(t)T EsxNs(t)

+xNu(t)T EuxNu(t) + uNx(t)T RuNx(t)

)dt (6)

where Eq, Ez, Es, Eu are weighting matrixes, qNm (t) = [zNm (t), θN(t)]T is the state containingthe vertical acceleration zNm (t) at the location = m on the nominal vehicle body and the

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Fig. 2 Frequency responses in closed loop system using the controller Eq. (8).

pitching acceleration θN(t), and xNs(t) = HxNz(t) − xNu(t) denotes the suspension displace-ment. Using the positive definite solution of the Riccati equation

AT

P + PA − PBR−1BT P = −Q

Q = diag[Q1, 04, Q2], Q1 =

HT EsH + Ez − HT Es

−EsH Es + Eu

Q2 = ((MTm)−1)T Eq(MTm)−1, Tm = I2 − Dm

, (7)

an active suspension controller to minimize the criterion Eq. (6) is given by(18)

uNx(t) = −R−1BT PxN(t). (8)

Based on the nominal vehicle Eq. (5), ideal vehicles generating ideal motion for theactual vehicle are designed in the next subsection. To make it easy to understand the designstrategy for ideal vehicles proposed in the next subsection, numerical simulation results areshown in the case where a controller Eq. (8) is designed for the nominal vehicle whosenominal parameters are shown in Table 2. The longitudinal velocity of vehicle is v = 100 ×1000/3600 m/s.

Fig. 2 shows the frequency responses in the closed loop system using the controllerEq. (8). Figs. 2(a), (b) show gain diagrams of transfer functions from the derivative ofroad disturbance w f (t) = wr(t − 2a/v) to the vertical acceleration zNm (t) at the location =m = −1.5 m on the nominal vehicle body and the pitching acceleration θN(t). Figs. 2(c),(d) show gain diagrams of transfer functions from the road disturbance w f (t) = wr(t − 2a/v)to the front and rear tire deflections xNu(t) = [xNu f (t), xNur(t)]T . Figs. 2(e), (f) show gaindiagrams of transfer functions from the road disturbance w f (t) = wr(t − 2a/v) to the frontand rear suspension displacements xNs(t) = [xNs f (t), xNsr(t)]T . In Fig. 2, thick lines showfrequency responses in the passive vehicle (nominal vehicle without the controller Eq. (8))and thin lines show frequency responses in the closed loop system using the controller Eq.(8). The controller is designed so that the vertical acceleration at the neighborhood of thelocation = −1.5m on the nominal vehicle body is reduced to a small value. To achieve asmall acceleration at = −1.5m, the weighting matrixes used in the controller design are set

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Fig. 3 Maximum vertical acceleration with respect to location .

as

Eq = (T−1q )T diag[5, eq]Tq

−1, eq = 1, Es = esI2, es = 0, Eu = euI2

eu = 105, Ez = 1.75 × 104 × ezI2, ez = 1, R = 10−7I2, Tq = I2 − Dq

. (9)

The weighting matrixes are chosen so that the vertical acceleration at the location = m =−1.5m is suppressed as small as possible on the condition that the increase of the gain relatingto the suspension displacement in the controlled vehicle from that in the passive vehicle iswithin 10 dB and the increase of the gain relating to the tire deflection in the controlled vehiclefrom that in the passive vehicle is as small as possible.

To investigate the gain characteristics relating to the vertical acceleration at the otherlocation on the controlled vehicle body, the maximum gain with respect to the road distur-bances of 3Hz or less is investigated at every location . As a result, Fig. 3 is obtained. InFig. 3(a), thick line shows the maximum gain curve relating to the vertical acceleration onthe passive vehicle body and thin line shows the maximum gain curve relating to the verticalacceleration on the controlled vehicle body. In Figs. 3 (b)-(d), the variation of the maximumgain curve is shown in the case when the weighting matrixes are changed. It can be seen fromFigs. 3 (b) that in the case of decreasing the value of the parameter q, the location where themaximum gain curve becomes minimum is translated parallel to transverse direction, and itcan be seen from Fig. 3 (c) that in the case of decreasing the value of the parameter eq, thegradient of the maximum gain curve with respect to the location becomes small. It is alsoseen from Fig. 3 (d) that in the case of increasing the value of the parameter eu, the maximumgains become small at every location. Based on the properties, the weighting matrixes aredetermined by using a trial and error method, and then, an ideal vehicle can be designed.

3.2. Design of a combined ideal vehicleAn ideal vehicle is developed based on the properties stated in the previous subsection.

That is, a controller Eq. (8) is designed by using a trial error method so that the verticalacceleration can be reduced to a small value at a specified location = m on the nominalvehicle body . In the designed controller Eq. (8), let’s substitute R1, P1 for positive definitematrixes R, P. Then, we propose the following ideal vehicle Eq. (10) generating the idealtrajectory. The state xd1(t) = [xzd1(t)T , xT

ud1(t), xzd1(t)T , xTud1(t), dd1(t)T ] denotes the ideal state

for the state x(t) of the actual vehicle Eq. (3).

xd1(t) = ηd1(t) + DwH xz(t)ηd1(t) = (A − BR−1

1 BT P1)xd1(t) − DwHM−1d(t)

(10)

The ideal vehicle is constructed so that the following differential equation is satisfied.

xd1(t) = (A − BR−11 BT P1)xd1(t) − Dww(t). (11)

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When the actual vehicle track the ideal vehicle Eq. (10), ride comfort becomes best atonly one location = m. To develop an ideal vehicle so that ride comfort becomes semi-bestat any specified location, let’s consider the following ideal vehicle Eqs. (13), (14) transformedby using the functions Ω(y1, y2), T (y1, y2) with respect to real numbers y1, y2.

xm1(t) = Ω(p, m)xd1(t) = [xTzm1(t), xT

um1(t), xTzm1(t), xT

um1(t), dTm1(t)]T

Ω(y1, y2) = diag[T (y1, y2), I2, T (y1, y2), I2, MT (y1, y2)M

−1]T (y1, y2) = I2 − D(y1 − y2)

(12)

In the Eq. (12) p is a design parameter introduced to specify the location where the verticalacceleration of vehicle body must be small.

xm1(t) = Γxm1(t) + BFm1xm1(t) − DmDm1dm1(t) −Ω(p, m)Dww(t) (13)

Fm1 = (HT )−1MT (p, m)M−1

HT (F − R−11 BT P1)Ω(p, m)−1

Dm1 = M−1u (HT )−1(p − m)MDM

−1, Dm = [02, 02, 02, I2, 02]T

(14)

The vertical acceleration and the pitching acceleration (it is denoted briefly as the vehicleacceleration hereafter) at a location on the each ideal vehicle body Eq. (11) and Eq. (13)are given as yd1(, t) = (I2 + D)M

−1dd1(t), ym1(, t) = (I2 + D)M

−1dm1(t). Then, since it

follows from Eq. (12) that the relation dm1(t) = MT (p, m)M−1

dd1(t) holds, the relationym1(, t) = yd1( − p + m, t) can be derived. From the fact it is seen that the maximum gaincurve relating to the vertical acceleration in the ideal vehicle Eq. (13) is the same curve as thetranslated curve from the maximum gain curve in the ideal vehicle Eq. (11) (see Fig. 3(a)).Namely, the maximum gain curve relating to the vertical acceleration becomes minimum atthe neighborhood of the specified location = p on the ideal vehicle Eq. (13). From thefact it can be concluded that the location where ride comfort become semi-best can be easilymoved to a specified location by setting values of p. However, deriving error equation for thetracking error xm1(t) = x(t) − xm1(t) from Eq. (3) and Eq. (13), it is seen that it is difficult toforce the actual vehicle Eq. (3) track the ideal vehicle even in the case of M = M due to theinfluence of dm1(t) and w(t).

To resolve the problem stated above, let’s consider to use another ideal vehicle. Namely,using the same design procedure stated above, the second ideal vehicle is designed so that thevertical acceleration becomes small at the neighborhood of another location = −m. Thesecond ideal model is represented by replacing the subscript ’1’ with ’2’ in the first idealvehicle Eq. (10). Using the state xd2(t), the second ideal vehicle can be described by

xd2(t) = (A − BR−12 BT P2)xd2(t) − Dww(t). (15)

Let’s consider the following ideal vehicle translated by using the state transformation xm2(t) =Ω(p,−m) ×xd2(t) = [xT

zm2(t), xTum2(t), xT

zm2(t), xTum2(t), dT

m2(t)]T .

xm2(t) = Γxm2(t) + BFm2xm2(t) − DmDm2dm2(t) −Ω(p,−m)Dww(t) (16)

Fm2 = (HT )−1MT (p,−m)M−1

(F − R−12 BT P2)Ω(p,−m)−1

Dm2 = M−1u (HT )−1(p+m)MDM

−1

(17)

It is seen that the maximum gain curve with respect to the location in the ideal vehicleEq. (16) is the same curve as the translated curve from the maximum gain curve in the idealvehicle Eq. (15). Namely, the vertical acceleration becomes minimum at the neighborhood ofthe specified location = p, and it is expected that dm1(t) becomes the same as dm2(t). Usingthe facts, let’s design a combined ideal vehicle.

To combined the two ideal vehicles linearly, let’s define γ1 = (p + m)/(2m) and γ2 =

−(p − m)/(2m) satisfying the relation

γ1 + γ2 = 1, (p − m)γ1 + (p + m)γ2 = 0. (18)

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Then, we propose the new state xm(t) for the combined ideal vehicle given by xm(t)=γ1 xm1(t)+γ2xm2(t). Defining dm(t) = dm1(t)−dm2(t), the combined ideal vehicle xm(t) = [xT

zm(t), xTum(t),

xTzm(t), xT

um(t), dTm(t)]T is described as

xm(t) = Γxm(t) + B(γ1Fm1xm1(t) + γ2Fm2xm2(t)) − Dww(t) − Dm∆(t), (19)

∆(t) = M−1u (HT )−1

2p − 2m2m

MDM−1

dm(t). (20)

If the norm of the signal dm(t) is small and can be ignored, and if the matrix M is equal toM, it is easy to design an active suspension controller so that the tracking error between theactual vehicle Eq. (3) and the combined ideal vehicle Eq. (19) becomes asymptotically stable.This is the main reason for using a combined ideal vehicle containing two ideal vehicles. Inthe proposed combined ideal vehicle, it is expected that ride comfort becomes semi-best at thespecified location p. This fact is examined by using numerical simulations in section 4.

3.3. Trajectory tracking controllerIn general, the variation of M = M − M may become large due to the variation of

vehicle weight. From this reason, in the case of using only a constant gain feedback withx(t) = x(t) − xm(t), it is difficult to design an active suspension controller achieving goodrobust performance. Therefore, in order to guarantee good robust performance also in thecase of M 0, a trajectory tracking control strategy proposed in Ref. (8) is used.

The tracking error between the actual vehicle and the combined ideal vehicle Eq. (19) isdefined as

xz(t) = xz(t) − xzm(t), xu(t) = xu(t) − xum(t), d(t) = d(t) − MM−1

dm(t). (21)

If the signal d(t) becomes zero, the vehicle acceleration y(, t) at the location on the actualvehicle body is given by y(, t) = (I2 + D)M−1d(t) = (I2 + D)M

−1dm(t). From this, it can

be seen that the vehicle acceleration y(, t) at the location tracks the vehicle accelerationym(, t) = (I2 + D)M

−1dm(t) at the location on the combined ideal vehicle body even if

the matrix M varies. Moreover, if the signals xz(t) and xu(t) become zero, the suspensiondisplacement and the tire deflection in the actual vehicle track that in the combined idealvehicle. In the following, paying the attention to the tracking error defined in Eq. (21), anactive suspension controller is developed to force the actual vehicle track the combined idealvehicle.

Using the same manner proposed in Ref. (8), the following new signals µ(t) and ξ0(t)are defined by using the positive definite and diagonal matrixes Λα = diag[α1, α2] and Λβ =diag[β1, β2]. The design parameters αi and βi are positive. In the below, symbols s, L, L−1

denote Laplace operator, Laplace transform and inverse Laplace transform, respectively.

µ(t) = d(t) − HT MuL−1[s(sI2 + Λα)−1ΛβL[ ˙xu(t)]

]ξ0(t) = xz(t) − M−1HT MuL−1

[(sI2 + Λα)−1ΛβL[xu(t)]

] (22)

Using the signals Eq. (22) and the polynomial Au(s) = (sI2 + Λα)(s2I2 + Ku) + Λβs2, thetracking error xu(t) can be described by

L[xu(t)] = Au(s)−1(sI2 + Λα)L[∆u(t)] − Au(s)−1(sI2 + Λα)M−1u (HT )−1µ(t),(23)

∆u(t) = ∆(t) − M−1u (HT )−1MM

−1dm(t). (24)

It can be ascertained that all real part of roots of det[Au(s)] = 0 become negative for any pos-itive design parameters αi and βi. According to Hurwitz’s stability criterion, if the controllerachieving µ(t) = 0 can be designed, it is seen from Eq. (23) that the tracking errors xu(t) and˙xu(t) become stable. Moreover, in the case when the signals ξ0(t) and ξ0(t) are zero, from Eq.(22) it follows that the tracking errors d(t), xz(t) and ˙xz(t) are also stable. Namely, when the

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Fig. 4 Configuration of the proposed control system.

state ξ(t) = [(Mξ0(t))T , (Mξ0(t))T , µ(t)T ]T is asymptotically stable, the stability of all signalsin the closed loop system can be guaranteed. Especially in the case of M = M, since it isexpected that the signal ‖dm(t)‖ becomes small and the signal ‖∆u(t)‖ becomes also small, thenorms of the tracking errors defined in Eq. (21) become small, and then, the actual vehicletrack the combined ideal vehicle. Control performance in the case of M M is confirmed bycarrying out numerical simulations later.

In the below, a method to design an active suspension controller achieving asymptoticallystability of the state ξ(t) is shown. The state equation using the state ξ(t) is given as

ξ(t) = Aξξ(t) + Bξ(ux(t) + ∆ξ(t)), (25)

∆ξ(t) = Fx(t) − (HT )−1MM−1

dm(t) + MuΛβ

(Ku xu(t)

+ M−1u (HT )−1 d(t) − ∆u(t) + L−1

[s(sI2 + Λα)−1ΛαL[ ˙xu(t)]

])

Aξ =

02 I2 02

02 02 I2

02 02 02

, Bξ =

02

02

HT

. (26)

Using the positive definite solution Pξ of the Riccati equation

ATξ Pξ + PξAξ − PξBξB

Tξ Pξ = −I6, (27)

and constructing a controller as

ux(t) = −BTξ Pξξ(t) − ∆ξ(t), (28)

it can be seen that the state ξ(t) becomes asymptotically stable. Although the signal dm(t) iscontained in the signal ∆ξ(t), using the fact that the relation (HT )−1BT Dw = 02 holds in Eq.(10) and the relations obtained by replacing ’1’ with ’2’ in Eq. (10), it can be concluded thatthe signal dm(t) can be generated by available signals xd1(t), xd2(t) and d(t). In order to makeit easy to understand the whole configuration of the closed loop system using the proposedcontroller, the configuration of the proposed closed loop system is shown in Fig. 4.

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Fig. 5 Characteristics of combined ideal vehicle.

Fig. 6 Frequency responses in combined ideal vehicle.

4. Numerical simulation results

Numerical simulation results are shown below to demonstrate effectiveness of the pro-posed active suspension controller. Nominal values of the actual vehicle parameters are set asshown in Table 2 and longitudinal velocity is set to be v = 100 × 1000/3600m/s.

At first, the combined ideal vehicle is designed in which ride comfort can be improvedwithin the range of ±1m from the center on the combined ideal vehicle body. Namely, thetwo ideal vehicles Eqs. (13) and (16) are designed so that ride comfort of each ideal vehiclebecomes best at the each neighborhood of the specified two different locations = m = −1.5mand = −m = 1.5m. In the ideal two vehicles, the increase of the gain relating to thesuspension displacement from that in the passive vehicle is within 10 dB and the increase ofthe gain relating to the tire deflection is suppressed as much as possible. The ideal vehicleEq. (13) is designed by using the weighting matrixes Eq. (9) and the ideal vehicle Eq. (16) isdesigned by using the following weighting matrixes

Eq2= (T−1q2 )T diag[5, eq2]Tq2

−1, eq2=10, Es2=es2I2, es2=0, Eu2=eu2I2

eu2=105, Ez2=1.75 × 104 × ez2I2, ez2=1, R2=10−7I2, Tq2= I2 − Dq2

. (29)

The frequency characteristics of the two designed ideal vehicles are shown in Fig. 5 (a). InFig. 5 (a), the maximum value of the gain diagram relating to the vertical acceleration at eachlocation on each of the two ideal vehicles is plotted with respect to the location . As shownin Fig. 5 (a), the maximum gain curves become minimum at the neighborhood of the eachlocation = −1.5m and = 1.5m, and ride comfort becomes best at the neighborhood of the

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Fig. 7 Vertical acceleration in the closed loop system using the proposed activesuspension controller Eq. (28).

each locations = −1.5m and = 1.5m. In Fig. 5 (b), the maximum gain curves relating tothe vertical acceleration on the combined ideal vehicle is plotted in the case when the locationsto be specified by designers are set as p = −0.8m, p = 0m and p = 0.8m. As shown in Fig.5 (b), the location where ride comfort becomes semi-best can be easily changed by setting theonly one design parameter p. However, there is the problem that the semi-best location inFig. 5 (b) is different from the specified location p = ±0.8m. To investigate the fact in detail,the relation between the semi-best location and the specified location p is investigated. Fig.5 (c) shows the relation between the semi-best location and the specified location p. In thecase of changing the value of the design parameter p, the minimum values of the maximumgain curves are plotted with respect to the semi-best location in Fig. 5 (d).

In Fig. 5 (c), for example, a polynomial approximation p = r0 + r1 + r22 + · · · can be

obtained by using a least squares method and so on. It is easy from the polynomial approx-imation to obtain the value of p corresponding to the location where ride comfort must besemi-best. Moreover, as shown in Fig. 5 (d), in the case when the semi-best location on thecombined ideal vehicle is set within the range of = ±1.5m, it can be seen that the minimumgain at the semi-best location is designed so as to be about 19.5dB. As stated above, it can beconcluded that ride comfort becomes semi-best at the specified location by setting only onedesign parameter p in the proposed combined ideal vehicle.

In the case of p = −0.8,−0.3, 0, 0.3, 0.8, the gain diagrams relating to tire deflection andsuspension displacement at the location = 0m from the derivative of the road disturbancew f (t) = wr(t − 2a/v) in the combined ideal vehicle are shown in Fig. 6. The thick lines showthe gain diagrams in the passive vehicle with nominal values shown in Table 2 and thin linesshow the gain diagrams in the combined ideal vehicle. As for tire deflection, it is seen fromFigs. 6 (a), (b) that the maximum values of the gain diagrams in the passive vehicle and thecombined ideal vehicle are of the same. As for suspension displacement, it can be seen formFigs. 6 (c), (d) that the increases of the gain from the gain in the passive vehicle becomes lassthan 10 dB.

Fig. 7 shows the maximum gain curves relating to the vertical acceleration. The thicklines show the maximum gain curves in the combined ideal vehicle and thin lines show the

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Fig. 8 Frequency responses in closed loop system using the proposed active suspensioncontroller Eq. (28).

maximum gain curves in the closed loop system (CLS) using the proposed controller Eq. (28).In the case of setting p to be p = 0m, the maximum gain curves are shown in Fig. 7 (a). Inthe case of setting p to be p = −0.8m and p = 0.8m, the maximum gain curves are shownin Fig. 7 (b). Moreover, in the case of adding weights (madd = 0, 60, 120kg) at the center ofgravity on the actual vehicle body, the variations of the maximum gain curves in the closedloop system are shown in Figs. 7 (c), (d). Figs. 7 (e), (f) show the variations of the maximumgain curve in the closed loop system in the case of adding a weight of 60kg at the center ofgravity and adding a weight of 60kg at the locations = −1.5, 1.5m. Fig. 8 shows the gaindiagrams relating to tire deflection and suspension displacement at the location = 0m in thecase of p = 0m and adding weights (madd = 0, 60, 120kg) at the center of gravity. The thickline show the gain diagram in the combined ideal vehicle and thin lines show the gain diagramin the closed loop system. Although some difference are arisen because of the influence of thesignal ∆u(t) in Eq. (24), it is seen from Fig. 7 and Fig. 8 that the gain characteristic betweenthe combined ideal vehicle and the closed loop system are mostly same.

It is concluded from the above results that using the proposed active suspension con-troller, the specified location becomes semi-best by setting only one design parameter p with-out applying a trail and error method even if the weight of the actual vehicle body changesand the center of the gravity moves.

5. Conclusion

The new method to control active suspensions is proposed. Using the proposed method,the specified location on the actual vehicle body can be easily set to be semi-best. The follow-ing properties are shown by carrying out numerical simulations. In the proposed combinedideal vehicle, any specified location on the combined ideal vehicle body become semi-best bysetting only one design parameter p. Then, it is shown that using a proposed model trackingcontroller, the gain characteristics between the actual vehicle and the combined ideal vehiclebecome almost same. Moreover, it is shown that tracking performance hardly varies even ifthe weight of the vehicle body changes and the center of gravity moves.

References

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