active suspension design

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International Journal of Automation and Computing 7(4), November 2010, 419-427DOI: 10.1007/s11633-010-0523-7

Robust Active Suspension Design Subject toVehicle Inertial Parameter Variations

Hai-Ping Du 1 Nong Zhang 21 School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong NSW 2522, Australia

2 Mechatronics and Intelligent Systems, Faculty of Engineering, University of Technology, Broadway NSW 2007, Australia

Abstract: This paper presents an approach in designing a robust controller for vehicle suspensions considering changes in vehicleinertial properties. A four-degree-of-freedom half-car model with active suspension is studied in this paper, and three main performancerequirements are considered. Among these requirements, the ride comfort performance is optimized by minimizing the H norm of thetransfer function from the road disturbance to the sprung mass acceleration, while the road holding performance and the suspensiondeection limitation are guaranteed by constraining the generalized H 2 (GH 2 ) norms of the transfer functions from the road disturbanceto the dynamic tyre load and the suspension deection to be less than their hard limits, respectively. At the same time, the controllersaturation problem is considered by constraining its peak response output to be less than a given limit using the GH 2 norm as well.By solving the nite number of linear matrix inequalities (LMIs) with the minimization optimization procedure, the controller gains,which are dependent on the time-varying inertial parameters, can be obtained. Numerical simulations on both frequency and bumpresponses show that the designed parameter-dependent controller can achieve better active suspension performance compared with thepassive suspension in spite of the variations of inertial parameters.

Keywords: Active suspension, half-car suspension model, parameter-dependent control, linear matrix inequalities (LMIs).

1 Introduction

The use of active suspensions has been considered formany years and various approaches have been proposed toimprove the active suspension performance [1 3] . There aremany different performance requirements being consideredby automotive makers for an advanced vehicle suspensionsystem. These requirements include ride comfort, handlingor road holding capability, and suspension deection lim-itation, etc. To meet these conicting demands, multiobjective functional control of vehicle suspensions [4 9] has at-tracted much attention recently because it can reduce theconservativeness of the approach that minimizes differentperformance requirements with one single objective func-tional. In the multiobjective active suspensions, the H 2or H norm of the transfer function from the road dis-turbance to the sprung mass acceleration can be used tospecify the ride comfort performance, while the generalizedH 2 (GH 2 ) norm can be used to constrain the suspensiondeection and the H norm can be used to specify theroad holding performance, etc. The combination of the op-

timization procedure can emerge as, for example, minimiz-ing 1 H 2 + 2 GH 2 subject to H < , where 1 and 2are positive weighting coefficients, > 0 is a performanceindex [5] ; minimizing H or H 2 subject to hard constrains(e.g., suspension deection, tyre deection, actuator satu-ration, etc.) [7 9] ; minimizing H 2 subject to H < [6] , etc.

On the other hand, it has been widely documented thatchanges in vehicle inertial properties can have a direct ef-fect on vehicle comfort, handling, and braking performance.Changes in inertial properties are most pronounced in ve-hicles where ratio of passengers/cargo to the vehicle sprung

Manuscript received August 29, 2008; revised September 8, 2009This work was supported by the Australian Research Council

(No.ARC LP0560077) and the University of Technology, Sydney,

Australia.

mass can have a large variation. Vehicles that fall intothis category are sport utility vehicles (SUVs), militaryand commercial vehicles, and also small and light vehicleswhere a load of four passengers represents a large percent-age change. To achieve more stringent levels of safety, com-fort and fuel efficiency, the accurate estimates of the in-ertial properties are necessary. Hence, both on-line andoff-line estimation methods have been developed to identifythe inertial parameters, e.g., vehicle mass, vehicle centre of gravity, and vehicle pitch, roll, and yaw moments of iner-tia about the vehicle centre of gravity, based on the mea-surement available signals [10 13] . In fact, with the develop-ment of on-line estimation of the inertial parameters [13] , theparameter-dependent control technique [14] can be appliedto realize robust control of the vehicle suspension despitethe changes of vehicle inertial parameters.

In this paper, the parameter-dependent control strategyis applied to design a robust vehicle suspension consider-ing vehicle inertial parameter variations. A four-degree-of-freedom half-car model is used to study the performanceof a vehicle suspension system in terms of the heave and

pitch motions, the suspension deection, and the tyre de-ection performance features. Three main performance re-quirements (ride comfort, road holding capability, and sus-pension deection limitation) and the changes in inertialproperties are considered by constructing an appropriateparameter-dependent state feedback controller to providethe trade-off between these requirements. Among the con-icting requirements, the ride comfort performance is opti-mized by minimizing the H norm of the transfer functionfrom the road disturbance to the sprung mass accelerations(both the heave and pitch), while the road holding perfor-mance and the suspension deection limitation are guaran-teed by constraining the generalized H 2 (GH 2 ) norms of the transfer functions from the road disturbance to the dy-


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420 International Journal of Automation and Computing 7(4), November 2010

namic tyre load and the suspension deection to be lessthan their hard limits, respectively. At the same time, theactuator saturation problem is considered by constrainingthe peak values of the output control forces to be less thangiven limits. The feasible solution for such a controller isobtained by solving the nite number of linear matrix in-

equalities (LMIs). Numerical simulations on both frequencyand bump responses show that the designed parameter-dependent controller can achieve good active suspensionperformance in spite of the variations of inertial parame-ters.

The rest of this paper is organized as follows. Section 2presents the problem formulation for the robust activeH /GH 2 control for a half-car active suspension model.The design approach for the parameter-dependent con-troller based on the solvability of LMIs is presented inSection 3. Section 4 presents the design result and per-formance evaluation. Finally, we conclude our ndings inSection 5.

Notation: R n denotes the n -dimensional Euclidean spaceand R n m the set of all n m real matrices. For a realsymmetric matrix W, the notation of W > 0 (W < 0) isused to denote its positive- (negative-) deniteness. refers to either the Euclidean vector norm or the matrixspectral norm. Also, I is used to denote the identity matrixof appropriate dimensions. To simplify notation, is usedto represent a block matrix which is readily inferred bysymmetry.

2 Problem formulation for robust ac-tive suspension control

This paper takes the general half-car suspension model,

which is shown in Fig.1, as an example. The half-car sus-pension model is represented by a linear four degree-of-freedom (DOF) system. It consists of a single sprung mass(car body) connected to two unsprung masses (front andrear wheels) at each end. The sprung mass is free to heaveand pitch, while the unsprung masses are free to bouncevertically with respect to the sprung mass. The passivesuspensions between the sprung mass and unsprung massesare modeled as linear viscous dampers and spring elements,while the tyres are modeled as simple springs without damp-ing components. In parallel with the passive suspensions,two hydraulic actuators are installed between the sprungmass and unsprung masses to provide active forces. Re-gardless of actuator dynamics, we consider active forces ascontrol inputs.

In Fig. 1, m s is the mass of the car body, m uf and m urare the unsprung masses on the front and rear wheels, re-spectively. I is the pitch moment of inertia about the cen-tre of mass, is pitch angle. z c is the displacement of thecentre of mass. z sf is the front body displacement and z sris the rear body displacement. z uf and z ur are the frontand rear unsprung mass displacements, respectively. z rf and z rr are the front and rear terrain height displacements,respectively. l1 is the distance between the front axle andthe centre of mass, and l2 is the distance between the rearaxle and the centre of mass. ksf , ksr , and csf , csr denotethe stiffness and damping coefficient of passive suspension

elements for the front and rear wheels. Similarly, ktf and

ktr denote the front and rear tyre stiffness, and u f and u rrepresent the front and rear actuator force inputs.

Fig.1 Four POF half-car suspension model

By applying Newton s second law of motion and usingthe static equilibrium position as the origin for both thedisplacement of the centre of mass and the angular displace-ment of the car body, the motion equations for the half-carsuspension model can be formulated (since these equationscan be derived straightforwardly, they are omitted here forsimplicity). By dening the state variables of the half-carmodel as

x1 (t ) = z sf (t ) z uf (t): Front suspension deectionx2 (t ) = z sf (t ) : Front body vertical velocity

x3 (t ) = z sr (t) z ur (t): Rear suspension deectionx4 (t ) = z sr (t) : Rear body vertical velocityx5 (t ) = z uf (t) z rf (t ) : Front tyre deectionx6 (t ) = z uf (t) : Front wheel vertical velocityx7 (t ) = z ur (t) z rr (t) : Rear tyre deectionx8 (t ) = z ur (t) : Rear wheel vertical velocity. (1)

and dening x(t ) = [ x1 (t) x2 (t) x3 (t) x4 (t) x5 (t) x6 (t)x7 (t) x8 (t)]T the motion equations can be further expressedas

x(t) = Ax (t) + B 1 w(t) + B 2 u(t) (2)

where

A =

0 1 0 0

ksf a1 csf a1 ksr a2 csr a20 0 0 1

ksf a2 csf a2 ksr a3 csr a30 0 0 0

ksf m uf

csf m uf

0 0

0 0 0 0

0 0 ksr

m urcsr

m ur


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H. P. Du and N. Zhang / Robust Active Suspension Design Subject to Vehicle Inertial Parameter Variations 421

0 1 0 00 csf a1 0 csr a20 0 0 10 csf a2 0 csr a30 1 0 0

ktf

m uf csf

m uf 0 00 0 0 1

0 0 ktr

m ur csr

m ur

B 1 =0 0 0 0 1 0 0 00 0 0 0 0 0 1 0

T

B 2 =0 a1 0 a2 0

1m uf

0 0

0 a2 0 a3 0 0 0 1

m ur

T

u(t ) = u f (t)

u r (t), w(t) = z rf

(t)z rr (t)

a1 =1

m s +l21I

a2 =1

m s l1 l2I

a3 =1

m s+ l

22

I .

In this paper, the following performance aspects of a half-car suspension system are taken into account:

1) Ride comfortRide comfort can be quantied by the body acceleration.

In this paper, both the heave and pitch accelerations arechosen as the control output, i.e.,

z 1 (t) =q 1 z c (t)

q 2 (t)(3)

where q 1 and q 2 are weights, and normally, q 1 is chosen as1 and q 2 = q 1 l1 l2 [1] . z c (t) is derived from (2) as

z c (t) =1

m s {ksf [z sf (t) z uf (t)]csf [z sf (t) z uf (t)] ksr [z sr (t) z ur (t)]csr [z sr (t ) z ur (t)] + u f (t) + u r (t)} (4)

and (t) can be obtained from motion equations. In or-der to design an active suspension to perform adequatelyin a wide range of shock and vibration environments, theH norm is chosen as the performance measure since H norm of a linear time-invariant (LTI) system is equal to theenergy-to-energy gain and its value actually gives an up-per bound on the root-mean-square (RMS) gain. Hence,our goal is to minimize the H norm of the transfer func-tion T z 1 w from the disturbance w(t) to the control outputz 1 (t) (optimal control), where the H norm of the transferfunction T z 1 w is dened as

T z 1 w = supRe( s ) > 0

T z 1 w (s) (5)

to improve the ride comfort performance.2) Suspension deection limitationIn order to avoid damaging vehicle components and gen-

erating more passenger discomfort, the active suspension

controllers must be capable of preventing the suspensionfrom hitting its travel limits. Therefore, we need to guar-antee the suspension deection

|z sf (t) z uf (t)| z f max|z sr (t ) z ur (t)| z r max (6)

where z f max and z r max are the maximum suspension de-ection hard limits, under any road disturbance input andvehicle running conditions. The suspension travel spacedoes not need to be minimal but its peak value need to beconstrained. Since the L norm of a mathematical func-tion in time-domain actually denes the peak value of thefunction [15 , 16] , i.e.,

z

supt z T (t)z (t ) (7)

we will optimize the L norm of the suspension deectionoutput under the energy-bounded road disturbance input,that is,

w 2

0wT (t )w(t)d t < (8)

i.e., wL2 [0, ) , to realize the hard requirement for thesuspension deection. This is the generalized H 2 (GH 2 ) orenergy-to-peak optimization problem [15] .

3) Road holding abilityIn order to ensure a rm uninterrupted contact of wheels

to road, the dynamic tyre load should not exceed the staticones for both the front and rear tyres [17] , i.e.,

ktf (z uf (t ) z rf (t)) < F f , ktr (z ur (t ) z rr (t)) < F r (9)where F f and F r are static tyre loads which can be calcu-lated by

F f + F r = ( m s + m uf + m ur )gF r (l1 + l2 ) = m s gl1 + m ur g(l1 + l2 ). (10)

This is also a peak value optimization problem which canbe dealt with by the same way used on the suspension de-ection.

4) Actuator saturation effectIn terms of the limited power provided by the actuator,

the active force for the suspension system should be con-ned to a certain range, i.e.,

|u f (t )| u f max , |u r (t)| u r max . (11)This peak value optimization problem can be dealt with

by the same way as mentioned above.Hence, we dene the hard constraints on the suspensiondeections, tyre loads, and active forces as control output

z 2 (t) =

z sf (t) z uf (t)z f max

z sr (t) z ur (t)z r max

ktf (z uf (t) z rf (t))F f

ktr (z uf (t) z rr (t))F r

u f (t)u f maxu r (t )

u r max

. (12)


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In summary, the suspension system is described as

x(t) = Ax (t ) + B 1 w(t ) + B 2 u(t)z 1 (t) = C 1 x(t) + D 12 u(t)z 2 (t) = C 2 x(t) + D 22 u(t) (13)

where x(t ), w(t), A, B 1 , and B 2 are dened as in (2); u(t)is the control input; z 1 (t) and z 2 (t) are the control outputs;and

C 1 =q 1 00 q 2

ksf m s

csf m s

ksrm s

csrm s

l1 ksf I

l1 csf I

l2 ksrI

l2 csrI

0 csf m s

0 csrm s

0 l1 csf

I 0 l2 csr

I

D 12 = q 1 00 q 2

1m s

1m s

l1I

l2I

C 2 =

1z f max

0 0 0 0 0 0 0

0 0 1z r max

0 0 0 0 0

0 0 0 0 ktf F f

0 0 0

0 0 0 0 0 0 ktrF r

0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

D 22 =

0 00 00 00 01

u f max0

0 1u r max

.

In practice, the inertial properties of vehicles, such asvehicle sprung mass and pitch moment inertia, are oftenaltered due to the load variations. When the time-varyingparameters are considered in the model (13), the vehiclemodel becomes a parameter-varying model and this modelis expressed as

x(t) = A( (t))x(t) + B 1 ( (t)) w(t) + B 2 ( (t))u(t)z 1 (t) = C 1 ( (t))x(t) + D 12 ( (t ))u(t)z 2 (t) = C 2 ( (t))x(t) + D 22 ( (t ))u(t) (14)

where the matrices A( (t)) , B1 ( (t)) , B2 ( (t)) , C 1 ( (t)) ,C 2 ( (t)) , D12 ( (t)), and D 22 ( (t)) are continuous func-tions of (t ) which is the time-varying parameter vectorand can be measured or estimated in real-time. Assumematrices A( (t )) , B1 ( (t)) , B2 ( (t)) , C 1 ( (t)) , C 2 ( (t)) ,

D 12 ( (t )), and D 22 ( (t)) are constrained to the polytope

P given by

P =

(A, B 1 , B 2 , C 1 , C 2 , D 12 , D 22 ) ( (t)) :(A, B 1 , B 2 , C 1 , C 2 , D 12 , D 22 ) ( (t) =

N

i =1i (t) (A, B 1 , B 2 , C 1 , C 2 , D 12 , D 22 ) i ,

N

i =1i (t) = 1 , i (t ) 0, i = 1 , , N.

(15)

where (t) does not necessarily represent the actual time-varying parameter (t ) of the dynamical system, but thereexists a linear relationship between (t) and (t) that canbe easily determined from the physical model whenever (t)affects affinely the linear system. It is clear that the knowl-edge of the value of (t) denes a precisely known systeminside the polytope P described by the convex combinationof its N vertices. Throughout the paper, the vertices of thepolytope P are referred as Ai , B 1 i , B 2 i , C 1 i , C 2 i , D 12 i , D 22 i ,i = 1 , , N. Also, it is assumed that the parameter (t)can vary arbitrarily fast, that is, there is no bound on (t).This is truly suitable to the case when the passengers/cargoget in or get off the vehicle.

In this paper, the aim of the robust active suspension de-sign is to nd a parameter-dependent state feedback controllaw

u(t) = K ( (t ))x(t) =N

i =1

i (t)K i x(t)

N

i =1

i (t) = 1 , i (t) 0, i = 1 , , N (16)

such that the closed-loop system given by

x(t) = A( (t)) x(t) + B 1 ( (t ))w(t) +B2 ( (t))K ( (t))x(t)

z 1 (t) = C 1 ( (t))x(t) + D 12 ( (t))K ( (t)) x(t )z 2 (t) = C 2 ( (t))x(t) + D 22 ( (t))K ( (t)) x(t )

(17)

has the following properties: 1) the closed-loop system isquadratically stable; 2) performance T z 1 w < isminimized subject to z 2 < 2 w 2 for all nonzerow L2 [0, ) and the prescribed constant 2 > 0, whereT z 1 w denotes the closed-loop transfer function from the roaddisturbance w(t) to the control output z 1 (t).

3 Parameter-dependent controller de-sign

By referring to [18], the quadratic stability of closed-loopsystem (17) with T z 1 w < and z 2 < 2 w 2 isequivalent to the existence of a symmetric matrix P > 0such that

P A ( (t)) T + A( (t))P +W ( (t)) T B 2 ( (t)) T + B 2 ( (t)) W ( (t ))

B 1 ( (t))

I


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P C 1 ( (t)) T + W ( (t)) T D 12 ( (t)) T

0

2 I < 0 (18)

and

P P C 2 ( (t ))T

+ W ( (t))T

D 22 ( (t))T

22 I

> 0 (19)

are feasible, where W ( (t)) = K ( (t ))P = N i =1 i (t )W i ,N i =1 i (t) = 1 , i (t) 0, i = 1 , , N.Since N i =1 i (t ) = 1 , the entry P A ( (t))

T + A( (t ))P +W ( (t )) T B 2 ( (t )) T + B 2 ( (t ))W ( (t)) in (18) can be re-written as

P A ( (t)) T + A( (t))P +W ( (t)) T B 2 ( (t)) T + B 2 ( (t ))W ( (t)) =

P N

i =1

i (t)ATi +N

i =1

i (t )Ai P +

N

i =1

i (t)W TiN

i =1

i (t)B T2 i +

N

i =1

i (t)B 2 iN

i =1

i (t )W i =

N

i =1

2i (t)Q i +N 1

i =1

N

j = i +1

i (t)j (t)Q ij

where Q i = P A Ti + Ai P + W Ti B T2 i + B 2 i W i , Q ij = P (A i +Aj )T + ( Ai + Aj )P + W Ti B T2 j + B 2 j W i + W Tj B T2 i + B2 i W j .Similarly, the other entries in (18) can be re-written in the

same way, and thus, the left hand of inequality (18) can beexpressed as

P A ( (t)) T + A( (t ))P +W ( (t)) T B 2 ( (t)) T +

B 2 ( (t))W ( (t))B 1 ( (t))

I

P C 1 ( (t)) T + W ( (t)) T D 12 ( (t)) T

0

2 I =

N

i =1

2i (t )R i +N 1

i =1

N

j = i +1

i (t)j (t)R ij (20)

where

R i =P A Ti + A i P + W Ti B T2 i + B 2 i W i

B 1 i P C T1 i + W Ti D T12 i

I 0 2 I

(21)

and

R ij =

P (Ai + Aj )T + ( Ai + A j )P +W Tj B T2 i + B 2 i W j + W Ti B T2 j + B 2 j W i

B 1 i + B 1 jP (C 1 i + C 1 j )+

W Ti D T12 j + W Tj D T12 i

2I 0 2 2 I

. (22)

To ensure (18), it is equivalent to simultaneously guar-anteeing R i < 0 for i = 1 , , N, and R ij < 0 fori = 1 , , N 1, and j = i + 1 , , N. Similarly, to guar-antee (19), it is equivalent to ensuring

P P C T2 i + W Ti D T22 i

22 I

> 0, i = 1 , , N (23)

and

2P P (C 2 i + C 2 j )T + W Ti D T22 j + W Tj D T22 i

2 22 I

> 0

i = 1 , , N 1, j = i + 1 , , N. (24)In summary, if there exist matrix P > 0, matrices W i ,

i = 1 , , N, and scalars > 0, 2 > 0 such that LMIs(21), (22), (23), and (24) are satised simultaneously, thenthe parameter-dependent state feedback control gain K =

N i =1 i (t)K i , with K i = W i P

1 , N i =1 i (t ) = 1 , i (t) 0,i = 1 ,

, N, results in the stable closed-loop system with

the performance T z 1 w < and z 2 < 2 w 2 .If the H performance is minimized, the optimal con-troller can be found.

4 Design example

In this section, we will apply the proposed approach todesign the parameter-dependent state feedback controllerfor the half-car suspension model described in Section 3.The parameters of the half-car suspension model selectedfor this study are listed in Table 1.

Table 1 System parameter values used in the half-carsuspension model

Parameters Valuesm s 575 kg

m uf 60kgk sf 16812N/mk tf 190000N/mc sf 1000N s/ml 1 1.38mI 769 kg m

2

m ur 60kgk sr 16 812kN/mk tr 190000N/mc sr 1000N s/ml 2 1.36m


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The design constraints and the design constants for ourcontrollers are listed in Table 2.

Table 2 The values of design constants chosen in thecontroller design procedure

Parameters Values

z f max 0.08 m

z r max 0.08 m

F f 4014.5N

F r 3580 .5 N

u f max 1500N

u r max 1500N

q1 1

q2 1

2 24

Here, we assume only the vehicle sprung mass m s andthe pitch moment of inertia I are varied due to the vehicle

load variation and both m s and I can uctuate aroundtheir nominal values by 30 %. To express the parametervariations in linear form, we dene the time-varying pa-rameters as

1 (t) =1

m s (t), 2 (t) =

1I (t)

(25)

and dene the four vertices of the polytope P as[ 1 min 2 min ] , [ 1 max 2 min ] ,[ 1 min 2 max ] , [ 1 max 2 max ] . (26)

Accordingly, the convex coordinates i (t), i = 1 , 2, 3, 4, aredened as

1 (t) = (t )(t)2 (t) = (1 (t))(t)3 (t) = (t )(1 (t))4 (t) = (1 (t))(1 (t )) (27)

where (t)( 1 max 1 (t )) / ( 1 max 1 min ), (t ) =( 2 (t) 2 min )/ ( 2 max 2 min ).If the changes in more inertial properties are considered,for example, the location of vehicle centre of gravity l1 ,we could dene more time-varying parameters as 3 (t ) =l1 (t)/I (t), 4 (t) = l21 (t )/I (t). For brevity, these contentsare not further discussed in this paper.

With the above dened parameter-varying model, theparameter-dependent controller gain can be obtained bysolving the LMIs (21), (22), (23), and (24) for matricesP > 0 and W i , i = 1 , 2, 3, 4, with the minimization proce-dure for . Then, the performance of the designed con-troller is evaluated in the following sections.

4.1 Frequency response

When the sprung mass and pitch moment of inertia arenot varied, i.e., the nominal case, the frequency responses of heave and pitch accelerations from the ground disturbancefor the designed active suspensions are shown in Fig. 2. As acomparison, the frequency response for the passive suspen-sion is plotted as well. The frequency responses of suspen-

sion deection and tyre deection of the front wheel from

the ground disturbance are also plotted in Fig. 2. For clar-ity, the frequency responses of suspension deection andtyre deection of the rear wheel from the ground distur-bance, which are similar to the front wheel, are not plottedin the gure. Generally speaking, a vehicle s ride comfort isalways conicted with its road holding capability and sus-

pension deection. In the proposed approach, the trade-off between different performances can be achieved by tuningthe value of parameter 2 . To show this clearly in the g-ure, when the value of 2 is varied from 6 to 36, we plotthe frequency response of the active suspension in Fig. 2,where 2 is used as 6 , 12, 20, 28, and 36, respectively. Sincethe human body is more sensitive to vibrations of 48 Hz inthe vertical direction and of 12 Hz in the horizontal direc-tion according to ISO 2361, the ride comfort is frequencysensitive. It can be seen from this gure that when 2 is in-creased, the better ride comfort is obtained, where a signif-icant improvement on ride comfort in the frequency rangebetween 1 Hz and 8 Hz is achieved, while the suspensiondeections and the tyre deections around the wheel hopfrequency are worsened. Hence, we can choose an appropri-ate 2 to manage the conicting requirements. Accordingto the analysis from Fig. 2, we choose 2 = 24 in our designprocess as given in Table 2.

(a) Heave acceleration

(b) Pitch acceleration


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(c) Suspension deection (front)

(d) Tyre deection (front)

Fig.2 Frequency responses of suspensions with nominal inertialparameters for different 2 , where dot line is used for passivesuspension, and solid line is used for active suspension

When the sprung mass and pitch moment of inertia arealtered, we plot the frequency responses of heave and pitchaccelerations, suspension deection, and tyre deection of the front wheel, from the ground disturbance in Fig. 3. Forclarity, only the nominal case and two other vertex cases,where Case 1 corresponds to increasing both sprung massand pitch moment of inertia by 30% of their nominal values,and Case 2 corresponds to decreasing both sprung mass andpitch moment of inertia by 30 % of their nominal values, areplotted in the gure. It can be seen from this gure thatno matter the variations of inertial properties, the activesuspension always shows better frequency responses thanthe passive suspension.

4.2 Bump response

Consider the case of an isolated bump in an otherwisesmooth road surface, the corresponding ground displace-ment for the front wheel is given by [7]

z rf (t) =

a2

(1 cos(2v 0

lt)) , if 0 t l

v00, if t > l

v0

(28)

where a and l are the height and the length of the bump.

We choose a = 0 .1 m, l = 5 m and the vehicle forward ve-locity as v0 = 45 km/h. In this paper, the road conditionfor the rear wheel z rr (t) is assumed to be the same with thefront wheel but with a time delay of ( l1 + l2 )/v 0 . The cor-responding ground velocities for the front and rear wheelsare plotted in Fig. 4.

The bump responses of the passive suspension and theactive suspension are compared in Fig. 5, where bump re-sponses of the heave acceleration, pitch acceleration, sus-pension deections, tyre loads, and active forces are plotted

(a) Heave acceleration

(b) Pitch acceleration

(c) Suspension deection (front)


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(d) Tyre deection (front)Fig.3 Frequency responses for suspensions with varied inertialparameters, where dot line is used for passive suspension, andsolid line is used for active suspension

Fig. 4 Ground velocity

Fig.5 Bump responses for suspensions with nominal inertial pa-rameters and varied inertial parameters: passive suspension (dotline); active suspension (solid line); bounds (ne dot line)

for the nominal and the above mentioned two vertex cases.

It can be seen from Fig. 5 that the lower bump response

quantities of the heave and pitch accelerations for activesuspension compared with the passive suspension systemare obtained. Compared with the passive suspension, thesuspension deections (front and rear) and the dynamic tyreloads (front and rear) of active suspension are all guaranteedto be less than their hard limits in spite of the large bump

energy. On the contrary, the passive suspension bump re-sponses violate these limits. Similarly, the active forces arewithin the saturation limits as well for active suspension. Itis conrmed that the designed robust active suspension canrealize the good suspension performance when driving overa pronounced bump road in spite of the inertial parametervariations.

5 Conclusions

This paper presents a parameter-dependent controller de-sign approach for vehicle suspension with the considerationof changes in vehicle inertial parameters. The control objective is expressed as minimizing the ride comfort perfor-mance ( H norm) subject to the tyre load ( GH 2 norm), thesuspension deection ( GH 2 norm), and the actuator force(GH 2 norm) being constrained to given limitations in spiteof the variations of inertial parameters. The designed sus-pension is applied to a half-car suspension model with largechanges in sprung mass and pitch moment of inertia. Nu-merical simulation validates that the vehicle suspension per-formance is improved with the designed controller in spiteof the inertial parameter variations.

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Hai-Ping Du received the Ph. D. de-gree in mechanical design and theory fromShanghai Jiao Tong University, Shanghai,PRC in 2002. He was awarded the excellentPh. D. Thesis Prize by Shanghai ProvincialGovernment in 2004. He is currently a lec-turer in the School of Electrical, Computerand Telecommunications Engineering, Fac-ulty of Informatics, University of Wollon-gong, Australia. Previously, he worked as a

research fellow in the Faculty of Engineering, University of Tech-nology, Sydney from 2006 to 2008, as a post-doctoral researchassociate in the University of Hong Kong and Imperial CollegeLondon from 2002 to 2003 and 2004 to 2005, respectively.

His research interests include robust control theory and engi-neering applications, soft computing, dynamic system modelling,model and controller reduction, and smart materials and struc-tures.

E-mail: [email protected] (Corresponding author)

Nong Zhang received the Ph. D. degreein 1989 from the University of Tokyo,Japan. In the same year, as a research as-sistant professor, he joined the Faculty of Engineering of the University. In 1992, asa research fellow, he joined the Engineer-ing Faculty of the University of Melbourne,Australia. In 1995, he joined the Facultyof Engineering of the University of Tech-nology, Sydney, Australia. He is a member

of ASME and a fellow of the Society of Automotive Engineers,Australasia.

His research interests include experimental modal analysis,rotor dynamics, vehicle powertrain dynamics, and recently hy-draulically interconnected suspension and vehicle dynamics.

E-mail: [email protected]

active suspension design

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