The Missing Mechanical Circuit Element

8/21/2019 The Missing Mechanical Circuit Element

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10 IEEE CIRCUITS AND SYSTEMS MAGAZINE 1531-636X/09/$25.00©2009 IEEE FIRST QUARTER 2009

Feature

The Missing

MechanicalCircuit ElementMichael Z.Q. Chen,

Christos Papageorgiou,

Frank Scheibe, Fu-Cheng Wang,

and Malcolm C. Smith

Abstract

In 2008, two articles in Autosport revealeddetails of a new mechanical suspension

component with the name “J-damper”which had entered Formula One Racing

and which was delivering significant per-formance gains in handling and grip. From

its first mention in the 2007 Formula One“spy scandal” there was much specula-

tion about what the J-damper actuallywas. The Autosport articles revealed that

the J-damper was in fact an “inerter” and

that its origin lay in academic work onmechanical and electrical circuits at Cam-

bridge University. This article aims to pro-vide an overview of the background and

origin of the inerter, its application, andits intimate connection with the classical

theory of network synthesis.

© LAT PHOTOGRAPHIC


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FIRST QUARTER 2009 IEEE CIRCUITS AND SYSTEMS MAGAZINE 11

1. Introduction

The standard analogies between mechanical and

electrical networks are universally familiar to stu-

dents and engineers alike. The basic modelling

elements have the following correspondences:

spring 4 inductor

damper 4 resistor

mass 4 capacitor,

where force relates to current and velocity to voltage. It

is known that the correspondence is perfect in the case

of the spring and damper. A fact which is also known,

but frequently glossed over, is that there is a restric-

tion in the case of the mass. All the above elements

except the mass have two “terminals” (for a mechani-

cal element the terminals are the attachment pointswhich should be freely and independently movable in

space). In contrast, the mass element has only one such

terminal—the centre of mass. It turns out that the mass

element is analogous to a grounded electrical capacitor

(see Sidebar I).

The above correspondence is so familiar that one does

not think to question it. However, a careful examination

M.Z.Q. Chen is with the Department of Engineering, University of Leicester, Leicester LE1 7RH, U.K. C. Papageorgiou is with Red Bull Technology Ltd.,

Milton Keynes MK7 8BJ, U.K. F. Scheibe is with the BMW Group, 80788 Munich, Germany. F.-C . Wang is with the Depar tment of Mechanical Engineer- ing, National Taiwan University, Taipei 10617, Taiwan. M.C. Smith is with the Department of Engineering, University of Cambridge, Cambridge, CB2

1PZ, U.K.

One of the principal motivations for the introduction of

the inerter in [38] is the synthesis of passive mechanical

networks. The fact that the mass element, together with the

spring and damper, is insufficient to realize the totality ofpassive mechanical impedances can be seen using the force-

current analogy between mechanical and electrical circuits.

In this analogy, force and current are the ‘‘through-variables’’

and velocity and voltage are the

‘‘across-variables’’ [35]. More-

over, the terminals of mechani-

cal and electrical elements are

in one-to-one correspondence.

For the mechanical elements

the spring and damper havetwo independently movable

terminals, whereas the termi-

nals of the mass are its centre

of mass and a fixed point in

an inertial frame (mechanical

ground). The mass is therefore

analogous to a grounded ca-

pacitor. In contrast, the inerter

is a two-terminal device, analo-

gous to an ungrounded capaci-

tor, with both terminals freely

and independently movable.

Fig. 1 shows a table of ele-

ment correspondences in the

force-current analogy with the inerter replacing the mass ele-

ment. The admittance Y (s ) is the ratio of through to across

quantities, where s is the standard Laplace transform variable.

For mechanical networks in rotational form the through andacross variables are torque and angular velocity, respectively.

For further background on network analogies see [23], [35],

and [38].

ELECTRICAL AND MECHANICAL NETWORK ANALOGIES

ElectricalMechanical

Spring

Inerter

Damper

Inductor

Capacitor

Resistor

i

i

i i

i

i

v 2 v 1

v 2 v 1

v 2 v 1

v 2 v 1

v 2 v 1

v 2 v 1

F F

F

F

F

F Y (s ) = bs

Y (s ) = c

Y (s ) = Cs

dF

dt = k (v 2 − v 1)

F = b d (v 2 − v 1)

dt i = C

d (v 2 − v 1)

dt

F = c (v 2 − v 1)

Y (s ) = k s

Y (s ) = 1Ls

Y (s ) = 1R

di dt = 1

L(v 2 − v 1)

(v 2 − v 1)1R

i =

Figure 1. Electrical and mechanical circuit symbols and correspondences. In theforce-current analogy forces substitute for currents and velocities substitute for

voltages. The admittance Y (s) maps velocity and voltage into force and current, re-spectively. (The symbol s is the standard Laplace transform variable.)

Digital Object Ident ifier 10.1109/MCAS.2008.931738


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12 IEEE CIRCUITS AND SYSTEMS MAGAZINE FIRST QUARTER 2009

of the classical theory of electrical networks suggestsotherwise. The famous result of Bott and Duffin [3] says

that an arbitrary passive driving-point impedance can

be realized as a two-terminal network comprising resis-

tors, capacitors and inductors only. Since the mapping

to mechanical circuits is power-preserving it is natural

to expect that arbitrary passive mechanical impedances

can be similarly realized. But there is a snag. A circuit in

which neither terminal of a capacitor is grounded will not

have a mechanical analogue. In applications where both

mechanical terminals are movable (such as a vehicle sus-

pension system) the restriction is a very real one.To bypass the snag a new mechanical modelling ele-

ment was proposed by Smith [38]. The element has two

terminals, and has the property that the applied force at

the terminals is proportional to the relative acceleration

between them. It was shown that such devices can be built

in a relatively simple manner [37], [38]. A new word “inert-

er” was coined to describe such a device. As well as offer-

ing new possibilities for “passive mechanical control” in a

variety of applications, the inerter brought out strong con-

nections with the classical theory of electrical circuit syn-

thesis, reviving old questions and suggesting new ones.

Since the birth of the inerter in the Engineering De-

partment at Cambridge University a number of applica-

tions have been proposed and investigated. Alongside

the successful application in Formula One racing (see

Sidebar II) the general applicability to vibration absorp-

tion and automotive suspensions has been considered

[29], [38], [40]. The use of the inerter in mechanical

steering compensators of high-performance motor-

cycles was studied in [14], [15]; by replacing the con-

ventional steering damper with a serial inerter-damper

layout, it was shown that two significant instabilities,

“wobble” and “weave”, can be stabilized simultaneous-ly. Further research saw the inerter proposed for train

suspension systems [44], [46], in which the inerter was

located in both the body-bogie and bogie-wheel connec-

tions. Recently, the inerter has been studied for building

suspension control [43], where three building models

being used to analyse the suspension performance.

In all cases, the introduction of the inerter device has

been shown to offer performance advantages over con-

ventional passive solutions.

This article describes the background to the inerter,

the connections with classical electrical circuit theory,and its applications. The rest of this article is organised

as follows. Section II presents the physical construc-tions of the inerter. Section III reviews passive network

synthesis, considers the suspension synthesis solution

of restricted complexity, and presents a new test for

positive-realness. Section IV presents positive-real syn-

thesis using matrix inequalities and the analytical solu-

tions for optimal ride comfort and tyre grip. In Section V

the development of a simulation-based methodology is

presented for the analysis and optimal design of nonlin-

ear passive vehicle suspensions. Section VI presents a

behavioural approach to play in mechanical networks.

Conclusions are given in Section VII.

2. The Inerter and its Physical Embodiments

Let us focus attention first on the five familiar two-ter-

minal modelling elements: resistor, capacitor, inductor,

spring, and damper. Each is an ideal modelling element,

with a precise mathematical definition. At the same

time, each is a model for physical devices whose behav-

iour is an approximation to the ideal. The same is true

for the inerter.

As an ideal modelling element, the inerter is defined

to be a two-terminal mechanical device such that the

applied force at the terminals is proportional to the rela-

tive acceleration between them. The constant of pro-

portionality is called the inertance and has the units

of kilograms. For this to be a useful definition, realistic

embodiments are needed. The meaning of “realistic”

was elaborated in [38]. It was argued that the inerter de-

vice should have a small mass relative to the inertance b

which should be adjustable independently of the mass.

Also, the device should function properly in any spatial

orientation, it should support adequate linear travel

and it should have reasonable overall dimensions. In-

erters with these features can be mechanically realizedin various ways. In [38], a rack-and-pinion inerter (see

Fig. 3(a)) was proposed using a flywheel that is driven

by a rack and pinion, and gears. Other methods of con-

struction are described in [37], e.g. using hydraulics

or screw mechanisms. Fig. 3(b) shows a schematic of

a ball-screw inerter and an example of such a device

is pictured in Fig. 4. For such devices the value of the

inertance b is easy to compute [37], [38]. In general, if

the device gives rise to a flywheel rotation of a radians

per meter of relative displacement between the termi-

nals, then the inertance of the device is given by b 5 J a where J is the flywheel’s moment of inertia.

As an ideal modelling element, the inerter is defined to be a two-terminal

mechanical device such that the applied force at the terminals is

proportional to the relative acceleration between them.


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Like other modelling elements, the deviation of in-

erter embodiments from ideal behaviour should be keptin mind. Typical effects which have been observed and

quantified include backlash, friction and elastic effects

[20], [26], [27], [28], [45]. Backlash (mechanical play) in

a physical inerter is a particularly interesting issue, theo-

retically and practically, which is discussed in Section VI.

3. Passive Network Synthesis

The literature on passive electrical network synthesis is

both rich and vast. Excellent introductions to the field

can be found in [1], [2], [17], [24], [42]. The concept of

passivity can be translated over directly to mechanicalnetworks as follows. Suppose that 1 F , v 2 represents the

force-velocity pair associated with a two-terminal me-

chanical network, then passivity requires:

32`

F 1 t 2v 1 t 2dt $ 0for all admissible time functions F 1 t 2 , v 1 t 2 and all T . If Z 1s 2 is the real rational impedance or admittance function of

a linear time-invariant two-terminal network, it is well-

known that the network is passive if and only if Z 1s 2 ispositive-real [1], [24]. Let Z 1s 2 be a real-rational function.Then Z 1s 2 is defined to be positive-real if Re 3 Z 1s 2 4 $ 0in the open right half plane (ORHP), i.e. for all s with

Re 3s 4 . 0 . The following is a well-known equivalent con-dition for positive-realness.

After the initial ‘‘discovery’’ of the inerter, Professor Smith

did some calculations which indicated a potential perfor-

mance advantage for vehicle suspensions which might be large

enough to interest a Formula One team. Cambridge Universityfiled a patent on the device [37] and then approached McLaren

Racing in confidence. McLaren was interested to try out the

idea and signed an agreement with the University for exclusive

rights in Formula One for a limited period. After a rapid devel-

opment process the

inerter was raced for

the first time by Kimi

Raikkonen at the

2005 Spanish Grand

Prix, who achieved a

victory for McLaren

(see Fig. 2).

During devel-

opment McLaren

invented a decoy

name for the inerter

(the ‘‘J-damper’’) to

keep the technology

secret from its com-

petitors for as long

as possible. The ‘‘J’’

has no actual mean-

ing, and of course

the device is not a damper. The idea behind the decoy name was

to make it difficult for personnel, who might leave McLaren to

join another Formula One team, to transfer information about

the device, and in particular to make a connection with the

technical literature on the inerter which Professor Smith and

his group were continuing to publish. This strategy succeeded in

spectacular fashion during the 2007 Formula One ‘‘spy scandal’’when a drawing of the McLaren J-damper came into the hands

of the Renault engineering team. This incident was reported to

the FIA World Motor Sport Council who convened to consider

the matter in Monaco on 6th December 2007. A full transcript

of the proceedings is

available on the FIA

official website [16].

During the De-

cember hearing,

neither the World

Motor Sport Council

nor McLaren made

pub l ic what the

J-damper was. After-

wards speculation

increased on inter-

net sites and blogs

about the function

and purpose of the

device and there

were many amus-

ing and erroneous

guesses. Finally,

the truth was discovered by the Autosport magazine. Two ar-

ticles appeared in May 2008 which revealed the Cambridge

connection and that the J-damper was an inerter [19], [31].

FROM THOUGHT-EXPERIMENT TO FORMULA ONE RACING

Figure 2. Kimi Raikkonen at the Spanish Grand Prix 2005 driving

the McLaren MP4-20 to victory on the first racing deployment of the

inerter. Photo courtesy of LAT Photographic.


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Theorem 1: [1], [24]: Z 1s 2 is positive-real if andonly if

Z 1s 21. is analytic in Re 3s 4 . 0;Re 3 Z 1 j v 2 4 $ 02. for all v with v not a pole of Z 1s 2 ;

poles on the imaginary axis and infinity are simple3.

and have non-negative residues.

An alternative necessary and sufficient condition for

positive-realness is as follows.

Theorem 2: [48], [49]: Let Z

1s

2 5 p

1s

2 / q

1s

2 , where p

1s

2

and q 1s 2 are coprime polynomials. Then Z 1s 2 is positive-real if and only if p 1s 2 1 q 1s 21. is Hurwitz;Re 3 Z 1 j v 2 4 $ 02. for all v with v not a pole of Z 1s 2 .

In [3] Bott and Duffin showed that any rational positive-

real function can be realized as the driving-point imped-

ance of a two-terminal network comprising resistors,

inductors and capacitors only. Making use of the force-

current analogy (see Sidebar I) and the new modelling

element (inerter) it can be seen that, given any positive-

real function Z 1s 2 , there exists a passive two-terminalmechanical network whose impedance equals Z 1s 2 ,which consists of a finite interconnection of springs,dampers and inerters. The ability to synthesise the mostgeneral positive-real impedance allows the designer to

achieve the optimal performance among passive me-

chanical networks. Fig. 5 shows

a specific mechanical network to-

gether with a physical realization

constructed at Cambridge Univer-

sity Engineering Department.

Efficiency of realization, as de-

fined by the number of elements

used, is much more important for

mechanical networks than electri-

cal networks. In this section, we

consider the class of realizations

in which the number of dampers

and inerters is restricted to one in

each case while allowing an arbi-

trary number of springs (which is

the easiest element to realize prac-

tically). Some examples of this

class have been given in Figs. 10

and 12 (Section IV). This problemis analogous to restricting the

number of resistors and capaci-

tors, but not inductors, in electri-

cal circuit synthesis [10]. Such

questions involving restrictions

on both resistive and one type

of reactive element have never

been considered. This contrasts

with the problems of minimal resistive and minimal re-

active synthesis which have well-known solutions when

transformers are allowed ([13], [50], see also [1]). In ourproblem, we impose the condition that no transformers

Figure 3. Schematics of two embodiments of the inerter.

(a) Rack and pinion inerter, (b) ballscrew inerter.

Gear

Rack

Terminal 2 Terminal 1

Pinions

Flywheel

(a)

Terminal 2 Terminal 1ScrewNut Flywheel

(b)

Figure 4. Ballscrew inerter made at Cambridge University Engineering Depart-

ment; Mass


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are employed, due to the fact that large lever ratios can

give rise to practical problems. Such a case can occur if

there is a specification on available “travel” between two

terminals of a network, as in a car suspension. A large le-

ver ratio may necessitate a large travel between internal

nodes of a network, which then conflicts with packaging

requirements.We show that the problem considered here is closely

related to the problem of one-element-kind multi-port

synthesis. We then review the definition of paramountcy

and its connection to transformerless synthesis. Five

circuit realizations are then presented to cover the gen-

eral class under consideration.

We consider a mechanical one-port network Q con-

sisting of an arbitrary number of springs, one damper

and one inerter. We can arrange the network in the form

of Fig. 6 where X is a three-port network containing all

the springs. The impedance matrix of X defined by

£v ^ 1v ^ 2v ^ 3

§ 5 s £ L1 L4 L5 L4 L2 L6 L5 L6 L3

§ £ F ^ 1 F ^ 2 F ^ 3

§ 5: sL £ F ^ 1 F ^ 2 F ^ 3

§ ,where L is a non-negative definite matrix and ^ denotes

Laplace transform. And the admittance of Q is

F ^ 1

v ^ 15

a3s3 1 a2s

2 1 a1s 1 a0

b4s4 1 b3s

3 1 b2s2 1 b1s

, (1)

where a3 5 bc 1 L2 L3 2 L62 2 , a2 5 bL3, a1 5 cL2, a0 5 1,b4 5 bc det 1 L 2 , b3 5 b 1 L1 L3 2 L52 2 , b2 5 c 1 L1 L2 2 L42 2 andb1 5 L1.

The admittance (1) effectively has only six parame-

ters which can be adjusted among the seven coefficients.

To see this note that b and c can be set to be equal to 1

and the following scalings carried out: L1 S R1, cL2 S R2,

bL3 S R3, " cL4 S R4, " bL5 S R5, " bcL6 S R6, to leave(1) invariant. The resulting admittance is Y 1s 2 5 1

R2 R3 2 R62

2s3 1 R3s

2 1 R2s 1 1

s 1det Rs3 1 1 R1 R3 2 R52 2s2 1 1 R1 R2 2 R42 2s 1 R1 2 (2)and

R J £ R1 R4 R5 R4 R2 R6 R5 R6 R3

§ 5 T £ L1 L4 L5 L4 L2 L6 L5 L6 L3

§T ,where

T 5 £1 0 00 " c 00 0

" b

§ and R is non-negative definite.

We will now consider the conditions on L or R that

will ensure that X corresponds to a network of springs

only (and no transformers). To this end we introduce

the following definition.

Definition 1: A matrix is defined to be paramount if its

principal minors, of all orders, are greater than or equal

to the absolute value of any minor built from the same

rows [6], [36].

It has been shown that paramountcy is a necessary

condition for the realisability of an n-port resistive

Figure 5. Inerter in series with damper with centring springs.

(a) Circuit diagram and (b) mechanical realization.

k 1

k 2c

b

(b)(a)

Figure 6. General one-port containing one damper and

one inerter.

v 1

F 1

X

F 2

F 3

v 2

v 3

c

b


7/17


network without transformers [6], [36]. In general,

paramountcy is not a sufficient condition for the re-

alisability of a transformerless resistive network anda counter-example for n 5 4 was given in [7], [47].

In [41, pp. 166–168], however, it was proven that

paramountcy is necessary and sufficient for the real-

isability of a resistive network without transformers

with order less than or equal to three ( n # 3). The

construction of [41] for the n 5 3 case makes use of

the network containing six resistors with judicious re-

labelling of terminals and changes of polarity.We now state a theorem from [8], [9], [12] which pro-

vides specific realizations for the Y 1s 2 in the form of Fig. 6for any X that contains springs only and no transform-

ers. The realizations are more efficient than would be

obtained by directly using the construction of Tellegen

in that only four springs are needed. This is due to the

fact that Theorem 3 exploits the additional freedom in the

parameters b and c to realize the admittance (2). Alterna-

tive realizations can also be found which are of similar

complexity (see [8]).

Theorem 3: [8], [9], [12] Given Y 1s 2 in the form ofFig. 6 where X contains only springs. Then Y 1s 2 can berealized with one damper, one inerter, and at most four

springs in the form of Fig. 7(a)–7(e).

If we take a closer look at Eq. (2), it is a bi-cubic func-

tion multiplied by 1 / s . It appears difficult to determine

necessary and sufficient condition for positive-realness

of this class using existing results (Theorems 1 and 2).

The convenient test provided by Theorem 2 is then no

longer applicable and detailed checking of the residue

conditions in Theorem 1 is still needed. This motivated

the search for the improved test of Theorem 4.

Theorem 4: [8], [11] Let Z 1s 2 5 p 1s 2 / q 1s 2 , where p 1s 2 and q 1s 2 have no common roots in the ORHP. Then Z 1s 2 is positive-real if and only if

p 1s 2 1 q 1s 21. has no roots in the ORHP;Re 3 Z 1 j v 2 4 $ 02. for all v with j v not a pole of Z 1s 2 .

When p 1s 2 and q 1s 2 are coprime, the “only if” impli-cation is stronger in Theorem 2 than Theorem 4 while

the reverse is the case for the “if” implication. The latter

fact means that Theorem 4 is more powerful for testing

the positive-realness of a given function. Although The-orem 4 appears only subtly different from Theorem 2 it

gives a significant advantage, as seen in testing some

classes of low-order positive-real functions [8], [11].

4. Vehicle Suspension

In general, a good suspension should provide a com-

fortable ride and good handling for a reasonable range

of suspension deflections. The specific criteria used

depend on the purpose of the vehicle. From a system

design point of view, there are two main categories of

disturbances on a vehicle, namely road and load dis-turbances (the latter being a simple approximation to

c

b

k 1

k 1

k 1

k 1

k 1

k 2

k 2

k 2 k 2

k 2

k 3

k 3

k 3 k 3

k 3

k 4

k 4

k 4

k 4

k 4

(a)

(b) (c)

(d) (e)

c b

c

b

c

b

c

b

Figure 7. Network realizations of Theorem 3 [8], [9],and [12]. (a) Case (i), (b) Case (ii), (c) Case (iii), (d) Case

(iv), and (e) Case (v).


8/17


driver inputs in elementary vehicle models). Standard

spectra are available to model stochastic road profile

inputs. Load disturbances can be used to model forces

induced by driver inputs such as accelerating, brak-

ing and cornering. In this way, suspension design can

be thought of as a problem of disturbance rejection to

selected performance outputs (e.g., vertical body ac-celeration, body pitch deflection, tyre deflection and

suspension travel).

Passive suspensions contain elements such as

springs, dampers, inerters and possibly levers, which

can only store or dissipate energy, i.e. there is no en-

ergy source in the system. They therefore provide a

simpler and cheaper means of suspension design and

construction at the expense of performance limita-

tions than active suspensions (with energy sources).

Generally a suspension needs to be “soft” to insu-

late against road disturbances and “hard” to insulateagainst load disturbances. It is well-known that these

objectives cannot be independently achieved with a

passive suspension [21], [39]. However, the use of in-

erters in addition to springs and dampers can alleviate

the necessary compromises between these two goals

[29], [40].

In the next section, we show how suspension net-

works can be designed using a linear matrix inequal-

ity (LMI) approach (Section A). We also present some

results on global optima which can be derived as a

function of the quarter-car model parameters for some

specific networks (Section B).

A. Design of Optimal Passive

Suspension Networks

We summarize the approach of [29] where the sus-

pension design problem was formulated as an opti-

mal control problem over positive real admittances.

The solution of the optimization problem made use

of matrix inequalities and required the application of

a local, iterative scheme due to the non-convexity of

the problem. Even so, the design method was able to

come up with new network topologies involving inert-ers that resulted in considerable improvement in the

individual performance measures. It was also pos-

sible to formulate and solve multi-objective optimiza-

tion problems.

1) The quarter-car model: The quarter-car model pre-

sented in Fig. 8 is the simplest model to consider for sus-

pension design. It consists of the sprung mass ms, the

unsprung mass mu and a tyre with spring stiffness kt .

The suspension strut provides an equal and opposite

force on the sprung and unsprung masses by means of

the positive-real admittance function Y 1s 2 which relatesthe suspension force to the strut velocity. In this section

we will assume further that Y 1s 2 5 K 1s 2 1 ks/s, where K 1s 2 is positive-real and has no pole at s 5 0 and ks is fixed at the desired static stiffness. Here we fix the

parameters of the quarter-car model as: ms 5 250 kg,

mu 5 35 kg, and kt 5 150 kN/m.

2) The control synthesis paradigm: In order to syn-

thesise admittances over the whole class of positive-

real functions, we use a control synthesis paradigm

along with a state-space characterisation of positive-

realness. The search for positive-real admittances

is formulated as a search for positive-real “control-

lers” K 1s 2 as shown in Fig. 9 where w represents theexogenous disturbances (e.g. z r and F s ) and z rep-

resents outputs to be controlled, e.g. sprung mass

acceleration, tyre force, etc. The characterisation of

positive-realness of the controller is achieved with

the following result.

F s

Y (s )

z s

z u

z r

m u

k t

m s

Figure 8. Quarter-car vehicle model.

G (s )

K (s )

F

z w

v 2 − v 1

Figure 9. The control synthesis paradigm applied for the

synthesis of a positive-real admittance K(s).


9/17


Lemma 2 (Positive real lemma [4]): Given that,

K 1s 2 5 c Ak BkC k Dk

d2 5 C k 1sI 2 Ak 221 Bk 1 Dk , (3)then K

1s

2 is positive-real if and only if there exists P k . 0

that satisfies the Linear Matrix Inequality (LMI)

c Ak P k 1 P k Ak P k Bk 2 C k Bk

T P k 2 C k 2 DkT 2 Dk

d # 0.

3) Generalized plant for the optimization of tyre grip:

In this section we will focus on a single aspect of perfor-

mance, namely the tyre grip which is related to the tyre

normal loads. We will use the r.m.s dynamic tyre load

parameter J 3 [40] for a standard stochastic road profile

given by

J 3 5 2p" V k 7s21T z r Skt 1 z u2 z r 2 1 j v 2 72, (4)where k is a road roughness parameter and V the vehicle

velocity.

We now calculate the generalized plant, G J 3 1s 2 ,corresponding to the block diagram of Fig. 9 and the

performance measure J 3 . The performance output

corresponding to J 3 is given by z 5 kt 1 z u 2 z r 2 and theexcitation input is the road disturbance signal w 5 z r .

The measurement signal for the controller is the rela-

tive velocity of the suspension, z

#

s 2 z

#

u and the control-ler output is the suspension force F . It was shown in

[29] that,

G J 3 1s 2 5 F D0 2ksms

0 ksms1 0 0 0

0 ksmu 0 2ks 1 kt

mu

0 0 1 0

Tc 2 mskt 0 2 mukt 0

1 0 2 1 0d

D 0 2 1ms0 0kt mu

1mu

0 0

T c0 0

0 0d

V .

Given a controller K 1s 2 of order nk , with state-space rep-resentation as in (3), let the state-space representation

of the closed-loop system resulting from the intercon-

nection of the generalized plant G J 3 1s 2 and the controllerbe given by:

£ x #

x #

k

e z u 2 z r

§ 5 cA cl BclCcl 0

d £ xxkzr

§ .

Theorem 5: There exists a strictly positive-real control-

ler K 1s 2 of order nk such that J 3 , 2p" 1V k 2kt n and Acl is stable, if and only if the following matrix inequality

problem is feasible for some X cl . 0, X k . 0, Q, n and

Ak, Bk, C k, Dk of compatible dimensions:

c Acl T X cl 1 X cl Acl X cl Bcl Bcl

T X cl 2 I d , 0, c X cl C cl T

C cl Q d . 0,

tr 1Q 2 , n2, c A

k

T X k

1 X k

Ak

X k

Bk

2 C k

T

BkT X k 2 C k 2 Dk

T 2 Dk d , 0.

K c

(a)

K c b

(b)

K

k

c

(c)

K

c

b

(d)

b

K

c

c

k 1

k 1

(e)

b

k b

K

k 1

k 1

(f)

Figure 10. Passive suspension networks incorporating

springs, dampers and inerters. Here, ks 5K. (a) S1, (b) S3,

(c) S2, (d) S4, (e) S5, and (f) S7.


10/17


The first three LMIs are neces-

sary and sufficient conditions

for the existence of a stabilising

controller that achieves an upper

bound of n on the closed-loop

H2-norm [34]. The fourth LMI

further restricts the control-ler to be strictly positive-real.

Without the positive-real con-

straint the H2-synthesis prob-

lem can be formulated as an

LMI problem as shown in [34].

With the positive-real constraint

it is not obvious how to do so,

hence an iterative optimization

method is employed to solve

the Bilinear Matrix Inequal-

ity (BMI) problem locally. Themethod, which is described in

[18], is to linearise the BMI us-

ing a first-order perturbation

approximation, and then itera-

tively compute a perturbation

that ‘slightly’ improves the con-

troller performance by solving

an LMI problem. The proposed

scheme was implemented in YALMIP [22], which is a

MATLAB toolbox for rapid prototyping of optimiza-

tion problems. A feasible starting point must be given

to the algorithm.

4) Tyre grip optimization results: The optimization of

the J 3 measure was attempted in [40] over various fixed

structure suspensions (see Fig. 10). In contrast, the it-

erative algorithm implemented in YALMIP was used to

optimize J 3 over general second-order admittances K 1s 2 in order to investigate whether 3 can be improved fur-

ther. The optimization was performed for ks ranging from

1 3 104 N/m to 12 3 104 N/m in steps of 2000 N/m. The

comparison of the optimization results obtained with

YALMIP with those obtained by fixed-structure optimiza-

tion are presented in Fig. 11.The optimization results obtained with YALMIP are

presented as three distinct curves suggesting that the

structure of the suspension changes as the static stiffness

varies. At low and high stiffness the YALMIP second-order

admittance can do better than both the second-order

S5 layout and the third-order S7 layout. An encouraging

feature of the optimization algorithm is that it allows the

change in the structure of the admittance as the static

stiffness varies in order to obtain the minimum value of

3 . In the intermediate range K

1s

2 turns out to be the net-

work S10 shown in Fig. 12 consisting of an inerter, damperand spring in series [29].

B. Analytical Solutions for Optimal

Ride Comfort and Tyre Grip

The approaches of [29], [40] both require extensive nu-

merical optimizations. The question whether the solu-

tions obtained are global optima is not rigorously settled.Also, if a new set of vehicle parameters is chosen, the

1 2 3 4 5 6 7 8 9 10 11 12

x 104

350

400

450

500

550

600

650

700

Static Stiffness in N/m

J 3

Optimization Results for J 3 for Quarter-Car Model

S2 (Damper with Relax. Spring)S3 (Damper, Inerter in Parallel)S4 (Damper, Inerter in Series)S5S7YALMIP 1E4 < ks < 1.8E4 N/m

YALMIP 2E4 < ks < 6.5E4 N/m

YALMIP 6.6E4 < ks < 12E4 N/m

S1 (Damper)

Figure 11. Comparison of YALMIP optimization results with fixed-structure optimi-

sation results for J3. (See Figure 10 for the configurations.)

Figure 12. Additional passive suspension networks incor-

porating springs, dampers, and inerters (a) S9 and (b) S10.

K

k c

b

(a)

K

k

c

b

(b)


11/17


numerical optimizations must be repeated. In [33] both of

these issues are addressed for ride comfort and tyre gripperformance measures in a quarter-car vehicle model.

Six suspension networks of fixed structure are selected:

S1–S4 in Fig. 10 and S9–S10 in Fig. 12. Global optima are

derived as a function of the quarter-car model parame-

ters. The optima are also parameterised in terms of sus-

pension static stiffness, which can therefore be adjusted

to approximately take account of other performance re-

quirements, such as suspension deflection and handling.

1) The quarter-car model and suspension networks:

We consider again the quarter-car model described in

Fig. 8, where Y 1s 2 is the admittance of one of the candi-date suspension networks.Network S1 models a conventional parallel spring-damper suspension and S2 contains a “relaxation

spring” in series with the damper. S3, S4, S9 and S10

show extensions incorporating an inerter and possibly

one “centring spring” (cf. [40]) across the damper. The

mechanical admittance Y 1s 2 for three of these layouts(S3, S9, S10) is now given for illustration:

Y 3 5 K

s 1 c 1 sb,

Y 9

5 K

s 1

as

k 1 sc1

1

sbb21

,

Y 10 5 K

s 1 a s

k1

1

c 1

1

sbb21.

2) Performance measures and analytical expression: In

addition to the r.m.s. dynamic tyre load parameter J 3

defined in (4) we also consider a ride comfort measure.

This is the r.m.s. body acceleration in response to a

standard stochastic road profile and is equal to

J 1 5 2p" V k

7s21

T z r S

$

z s 1 jw 2 72.See [40] for detailed derivations of the performancemeasures.

An analytical expression of the H2-norm of the (sta-

ble) transfer function G 1s 2 can be computed from a mini-mal state-space realization as

7G 72 5 7C 1sI 2 A 221 B 72 5 1CLC 21 / ,where the matrix L is the unique solution of the

Lyapunov equation

AL 1 LAT 1 BBT 5 0. (5)

The matrix L is then determined

from the linear equations in (5)

and the performance measures

are then given by

J i 5 2p" V k H i , Sj ,where H 5 CLC T and i indicates the

performance measure index and j the suspension network number.

3) Optimal solutions for mixed

performance of J 1 and 3: Optimal

performance solutions for 1 and J 3

individually and for suspension net-

works S1–S4, S9 and S10 have been

computed in [33]. Furthermore, it

is also important to consider com-

bined optimal vehicle performance

across different measures. Here we

present the results for a mixed J 1 and 3 measure:

m u = 35 kg

m u = 20 kg

K = 15 kNm−1

K = 55 kNm−1

K = 35 kNm−1

J

3 ( × 1 0 2 )

J 1

3.5

0.8 1 1.2 1.4 1.6 1.8

4

4.5

5

5.5

6

6.5 S1

S2

S3

S4

S9

S10

S4

S9

Figure 13. Analytical solutions for global optimum of mixed performance J1 and J3 for net-works S1–S4, S9 and S10 for three static stiffness values with quarter-car parameter valuesms 5 250 kg, kt 5 150 kNm

21, mu 5 35 kg (for K5 15,35,55 kNm21

2 and mu 5 20 kg

(for K5 15 kNm21 2 . Smaller magnitudes in J1 and J3 are beneficial.

Passive suspensions provide a simpler and cheaper means of suspension design and

construction at the expense of performance limitations than active suspensions.


12/17


H 1,3: Sj 5

11 2 a

2ms H 1, Sj 1 a H 3, Sj , (6)

where a [ 30,1 4 is a weighting between 1 and 3. Thescaling factor ms is inserted to approximately norma-

lise the measures and simplify the resulting formulae.

Eq. (6) can be optimized with respect to the suspen-

sion parameters [33]. The resulting optimal solutions

are drawn for a particular mu, ms and kt in Fig. 13. In

general it can be seen that networks involving inert-

ers (especially S9 and S10) offer performance advan-

tages over conventional networks for both 1 and 3

combined. The results also show that ride comfort( 1 ) deteriorates as suspension static stif fness in-

creases, and that tyre grip improves as unsprung

mass is decreased, for all suspension networks.

5. Simulation-Based, Optimal Design of Passive

Vehicle Suspensions Involving Inerters

In this section we will present the development of a

simulation-based methodology for the analysis and op-

timal design of nonlinear passive vehicle suspensions.

The methodology makes use of a nonlinear vehicle

model which is constructed in the Matlab/Simulink

toolbox SimMechanics. The vehicle model is in a 4-post

rig configuration and it allows the detailed representa-

tion of the suspension geometry and the nonlinearities

of the suspension elements. Several aspects of suspen-

sion performance are considered such as ride comfort,

tyre grip and handling. For each aspect of performance

we will propose time-domain performance measures

that are evaluated after a simulation run. For the ride

comfort and tyre grip performance we define appropri-

ate road disturbance inputs and for the handling per-

formance we define appropriate torque disturbances

acting on the sprung mass. The results demonstratethe performance improvements which can be achieved

using inerters over a conventional arrangement using

nonlinear dampers.

A. Nonlinear Vehicle Model

The nonlinear vehicle model considered in this study

is typical of a high-performance sports car with a fairly

accurate description of the suspension geometry and

the characteristics of the suspension elements. The ap-

proximate parameters used for the vehicle model are

given by its sprung mass ms 5 1500 kg and its momentsof inertia about its roll, pitch and yaw axes respectively

( I x 5 400 kgm2, I y 5 2300 kgm

2, I z 5 2500 kgm2 ), the front

unsprung masses each with a mass of muf 5 50 kg,

and the rear unsprung masses each with a mass of

mur 5 55 kg. Both the front and rear suspensions are

of a double wishbone arrangement with a front static

stiffness of 55 kN/m and a rear static stiffness of

50 kN/m. The tyres are modelled as vertical springs of

stiffness 350 kN/m (rear) and 320 kN/m (front). Both

the front and rear suspensions are a parallel arrange-

ment of a spring with a nonlinear damper. The non-

linear dampers have a dual rate characteristic with a

smooth transition between the hard and soft settings.

Such a dual-rate damper characteristic has been found

to provide better combined performance in ride com-

fort and handling than a linear damper [30]. A static

view of the animation of the vehicle model is shown in

Fig. 14 in its nominal state, i.e. with no external distur-

bances applied to it.

B. Definition of Disturbances

For the evaluation of the ride comfort and tyre grip we

use a kerbstrike road profile. The kerbstrike has height

h0, length 1 m, and transition ramps of unity slope. Let

v be the speed of travel of the vehicle and y the height of

the kerb. Then we have:

y 1 t 2 5 vt , 0 , t # h0v

,

y 1 t 2 5 h0, h0v , t # 1 2 h0v ,

Figure 14. A static view of the animation of the SimMechanics

vehicle model. The viewer is at the rear and elevated withrespect to the model.

Front LeftUnsprung

Mass

Rear RightUnsprung

MassRear Left

Road Disturbance

Front RightSuspension

StrutSprung Mass

Efficiency of realization, as defined by the number of elements used, is much

more important for mechanical networks than electrical networks.


13/17


y 1 t 2 5 1 2 vt , 1 2 h0v , t # 1v. The kerbstrike initially appears at the front left wheel

and subsequently at the rear left wheel delayed by L/v

seconds, where L is the wheelbase of the vehicle.

The load disturbances used for the assessment

of handling are pitch and roll step signals applied

on the sprung mass. Due to the left-right symmetry

of the vehicle model the roll disturbance is a step

about the roll axis that results in a negative roll an -

gle of the sprung mass for some fixed time and thenthe step is removed so that the sprung mass recov-

ers zero roll angle. The disturbance about the pitch

axis is chosen such that it results in both pitching-

up and pitching-down of the sprung mass since

there is no front-rear symmetry. Again with the

removal of the pitch disturbance the vehicle pitch

angle is restored to zero. The actual magnitude of

the pitch and roll disturbances is specified accord-

ingly by tak ing into account the relative importance

of the handling performance over the ride comfort

and tyre gr ip performance.

C. Definition of Performance Measures

The performance measure for the ride comfort consid-

ers the weighted accelerations of the sprung mass,

namely the heave ( z $ ), pitch ( u

$

) and roll ( w$ ) accelera-

tions. The acceleration weights are taken from [5] andrepresent discomfort felt by humans due to mechanical

vibrations. The performance measure for tyre grip con-

siders the tyre forces at the four wheel stations. The

time-domain measures for ride comfort and tyre grip

are defined as:

J 8t 5" trace 1z$wz$wT 2 y r 5kerbstrike , J 9t 5" trace 1Ft FtT 2 y r 5kerbstrike ,

where the signal z

$

w 5 3 z $

,u

$

,w

$

4 denotes the weighted ac-celeration responses of the sprung mass, r denotes theroad elevations at the four wheel stations and F t denotesthe tyre forces. It is easy to see that

J 8t 5" z $ T z $ 1 u$ T u$ 1 w$ T w$ so it represents the square root of the sum of the ener-

gies squared of the relevant signals. In the case of the

kerbstrike disturbance the resulting signals are finite

energy signals.

In order to define the time-domain handling mea-

sures we assume that we know the desired handling

responses of the vehicle in the pitch and roll channels,

both in bump and rebound in case they are different.

The energy of the error (possibly weighted) between

the actual and the desired response can then be used as

a time-domain handling measure.

If the energy of the error is small

then the handling of the vehicle is

close to the desired handling per-

formance. The time-domain han-

dling measure is defined as:

H t 5" erollT eroll 1 epitchT epitch , (7)where eroll is the error signal due

to the application of the roll dis-

turbance and epitch is the error

signal due to the application of

the pitch disturbance.

D. Optimal Design of Nonlinear

Suspensions

In this section we use the non-linear simulation model and the

Figure 15. The new suspension network and the admittance function of the linear

series connection of the spring, damper and inerter.

k s

c n

k

c

b 10−1 100 101 102

10−1 100 101 102

10110

2

103

104Admittance of (c + k + b ) Network

Frequency in Hz

G a i n i n N s / m

−100

−50

0

50

100

Frequency in Hz

P h a s e i n °

As well as offering new possibilities for “passive mechanical control” in a variety of

applications, the inerter brought out strong connections with the classical theory

of electrical circuit synthesis, reviving old questions and suggesting new ones.


14/17


defined performance measures to design a suspensionnetwork involving nonlinear dampers and inerters in or-

der to improve the ride comfort, tyre grip and handling

when compared to the performance achieved with the

default nonlinear damper characteristic. The approach

was to use the default network topology of the parallel

combination of the spring and the nonlinear damper

with an extra parallel network consisting of a series con-

nection of a spring, a damper and an inerter as shown in

Fig. 15. The cost function

J 51

2 J 8t

J 8t 01

1

2 J 9t

J 9t 0

was optimized over the front/rear soft settings of the

nonlinear dampers and the front/rear parameters of the

series network, where the subscript “0” denotes the per-

formance of the default suspension. The following val-

ues were obtained after optimization:

J 8t

J 8t 05 0.98,

J 9t

J 9t 05 0.945,

H t

H t 05 1.003.

The above results indicate that the tyre grip is im-

proved by 5% without deteriorating the ride comfort

and handling performances. It is expected that in-

cluding the hard settings of the nonlinear dampers

as decision variables in the optimization and also us-

ing a cost function that includes all aspects of per-

formance will also result in an improvement of the

handling performance.

6. Play in mechanical networks with inerters

A physical inerter as shown in Fig. 3 contains me-

chanical play or backlash in e.g. the rack and pinionmechanism which may affect the performance of the

device, its closed-loop stability and its mechanical

durability. This section addresses the mathematical

modelling of passive mechanical networks includ-

ing play and their physical accuracy. The results are

based on [32] and have shown that the treatment of

play as an input-output operator in mechanical net-

works leads to unsatisfactory solutions from a physi-

cal point of v iew. In contrast, a behavioural definition

of play (ideal play) does not suffer from these objec-

tions and appears more reasonable from a physicalpoint of view.

A. The Play Operator A number of different play definitions have been proposed

in the literature: the dead-zone (Fig. 16(a)) and hyster-

esis model (Fig. 16(b)) with the latter commonly used as

a basis for a formal mathematical approach to play. Both

definitions aim to describe an apparently well-defined

phenomena and give rise to two different mathematical

descriptions. This raises the question of which model, or

indeed either, is more satisfactory?

The behaviour of the play operator in Fig. 16(b) can

be expressed as a condition of three hybrid states.

Here, the position of the piston ( z 1 ) and cylinder ( z 2 )are considered to be the input and output (follower),

respectively.

(engagement—extension): z 2 5 z 1 1 P, z 1#

5 z 2#

# 0,

(engagement—compression): z 2 5 z 1 2 P, z 1#

5 z 2#

$ 0,

(disengagement): | z 1 2 z 2| , e, z #

2 5 0.

For a simple mechanical network incorporating the play

operator ( H ) in series with a damper (Fig. 17) several

properties can be identified that are unsatisfactory from

a physical point of view, [32]:

During disengagement the force through the play1.

element is not necessarily zero.

Figure 17. Damped harmonic oscillator network. The letter

i indicates the input and f the follower.

z y u

f i

m 1 m 2

c

k

H

Given any positive-real function Z (s), there exists a passive two-terminal

mechanical network whose impedance equals Z (s), which consists

of a finite interconnection of springs, dampers and inerters.

(a) (b)

x

y

− ∋ ∋ x

y

− ∋ ∋

Figure 16. (a) Graph of dead-zone play model. (b) Graph ofhysteresis play model.


15/17


The solutions of the network equations depend2.

on the choice of inertial frame, namely, the addi-

tion of a constant velocity to all states may changeswitching times or eliminate them altogether.

During engagement the force through the play ele-3.

ment is not restricted in sign.The behaviour of the network is not invariant to a4.

switch of terminals of the play operator.

B. Ideal Play

Following the shortcomings of the above play operator,

a behavioural definition for ideal play was proposed in

[32] which does not suffer from this criticism. Consider

a physical representation of play as shown in Fig. 18(a)

where z 1, z 2 are the terminal positions and F is the equal

and opposite force applied at the terminals. The ideal play

is defined to be completely characterised by the followingthree states:

(engagement—extension): z 2 2 z 1 5 e, F # 0,

(engagement—compression): z 2 2 z 1 5 2 e, F $ 0,

(disengagement): | z 2 2 z 1| , e, F 5 0.

Note that the definition is invariant to terminal reversal

and by definition always admits a force through the de-

vice of appropriate sign (see Fig. 18(b) for the modelling

symbol). Also, we note that this definition allows the me-

chanical network to maintain invariance to the choice of

inertial frame, since the three states only depend on the

difference between z 1 and z 2 .

However, since the ideal play does not admit an

input-output graph, mathematical properties like well-

posedness and the exclusion of limit points of switch-

ing are arrived at by analysing individual transition

scenarios, [32]. By means of the network example shown

in Fig. 19, one can show that at engagement of play im-

pulsive forces P may occur and multiple solutions are

obtained. Energy is dissipated when 2 P 0 , P # P 0, where

P 0 is the impulse strength required for play to coalesce

at engagement, Fig. 20.In order to regain well-posedness and capture the range

of solutions indicated in Fig. 20, the network in Fig. 19 was

systematically extended by the addition of compliance

springs and dampers. This highlights a connection with

the work of Nordin et al . [25] who proposed a model for

backlash which is equivalent to the semi-ideal model in

Fig. 21. This model was shown to be effective in modelling

the practical behaviour of inerter with play [26].

7. Conclusions

This paper has described the background and ap-plication of a newly introduced mechanical circuit

F F

z 1 z 2 z 3k 1

c 1P

Figure 21. Semi-ideal play model with displacementszd 5 z1 2 z3, zn 5 z1 2 z2, and z p 5 z2 2 z3.

Figure 19. Harmonic oscillator network with an inerter andideal play.

z y u

m 1 m 2

b

k

P

Figure 20. Change in kinetic energy due to an impulse of

strength P at t = t0. Energy is dissipated when 2P0 , P #P0 and energy increases when P , 2P0.

P 02P 0

0P

E ( t 0) − E ( t 0)−+

The inerter is defined to be a two-terminal mechanical device such that the applied

force at the terminals is proportional to the relative acceleration between them.

Figure 18. (a) Physical representation of play. (b) Terminalmodelling symbol for play.

F F

z 1 z 2z 1 z 2

P Q 2

(a) (b)

∋

P


16/17


element, the inerter, from its origin in modelling and

circuit synthesis through to a high-profile applica-

tion in Formula One racing. The role of the inerter to

make the analogy between electrical and mechanical

circuits exact has been emphasised. From a practical

point of view, the inerter allows the most general pas-

sive mechanical impedances to be synthesised, whichis not possible using the traditional analogy in which

the mass element is used. From a theoretical point

of view, the subject of transformerless synthesis of

one-port networks is reopened with some interesting

new twists. Several application areas for the inerter

have been highlighted. The paper has given a detailed

treatment of the application of the inerter to vehicle

suspensions and discussed the deviation from ideal

behaviour of practical devices.

Michael Z.Q. Chen was born in Shang-hai. He graduated from Nanyang Tech-

nological University, Singapore, in 2003

with a B.Eng. degree in Electrical and

Electronic Engineering, and from Cam-

bridge University in 2007 with a Ph.D.

degree in Control Engineering. He is

currently a Lecturer in the Department of Engineering

at the University of Leicester, England. He is a Fellow of

the Cambridge Philosophical Society, a Life Fellow of the

Cambridge Overseas Trust, and a member of the IEEE.

Since 2008, he has been an Associate Editor of the IES

Journal B–Intelligent Devices & Systems and a reviewer of

the IEEE Transactions on Circuits & Systems, Automatica,

the International Journal of Adaptive Control & Signal Pro-

cessing, and the Journal of Sound & Vibration, amongst

others. His research interests include: passive network

synthesis, vehicle suspensions, complex networks, and

statistical mechanics.

Christos Papageorgiou was born in Li-

massol, Cyprus. He graduated from the

University of Cambridge, UK, in 1999

with an M.Eng./B.A degree in Electricaland Information Sciences and in 2004

with a Ph.D. degree in Control Engineer-

ing. He held positions as a research as-

sociate in the Control Group at Cambridge University,

as a researcher in the Electrical and Computer Engi-

neering Department of the University of Cyprus and as

a research assistant in the Automatic Control Group

of Linköping University. His research interests include

vehicle suspension control, flight control design and

clearance, modelling and identification of mechanical

devices and the application of convex optimization incontroller design and analysis.

Frank Scheibe was born in Bremen, Ger-

many. He received the M.Eng. degree in

Electrical and Electronic Engineering

from Imperial College London in 2003,

and the Ph.D. degree in Control Engi-

neering from Cambridge University in

2008. In 2005 he worked for McLarenRacing Ltd and in 2007/08 was a Vehicle Dynamics En-

gineer with McLaren Automotive Ltd, Woking, England.

He is currently a Research and Development Engineer

with the BMW Group, Munich, Germany. His research

interests include nonlinear mechanical systems, vehi-

cle suspensions, and hybrid engine control.

Fu-Cheng Wang was born in Taipei,

Taiwan, in 1968. He received the B.S.

and M.S. degrees in mechanical engi-

neering from National Taiwan Univer-sity, Taipei, Taiwan, in 1990 and 1992,

respectively, and the Ph.D. degree in

control engineering from Cambridge

University, Cambridge, U.K., in 2002. From 2001 to

2003 he worked as a Research Associate in the Con-

trol Group of the Engineering Department, University

of Cambridge, U.K. Since 2003 he has been with the

Control Group of Mechanical Engineering Department

at National Taiwan University, in which he is now an

Associate Professor. His research interests include ro-

bust control, inerter research, suspension control, fuel

cell control, medical engineering and fuzzy systems.

Malcolm C. Smith received the B.A.

degree in mathematics, the M.Phil.

degree in control engineering and op-

erational research, and the Ph.D. de-

gree in control engineering from the

University of Cambridge, Cambridge,

U.K., in 1978, 1979, and 1982, respec-

tively. He was subsequently a Research Fellow at the

German Aerospace Center, DLR, Oberpfaffenhofen,

Germany, a Visiting Assistant Professor and ResearchFellow with the Department of Electrical Engineering

at McGill University, Montreal, Canada, and an As-

sistant Professor with the Department of Electrical

Engineering at the Ohio State University, Columbus,

OH. He returned to Cambridge University as a Lec-

turer in the Department of Engineering in 1990, be-

came a Reader in 1997, and Professor in 2002. He is a

Fellow of Gonville and Caius College, Cambridge, U.K.

His research interests include control system design,

frequency response methods, H-infinity optimiza-

tion, nonlinear systems, active suspension, and me-chanical systems. Prof. Smith was a corecipient of the


17/17

George Axelby Outstanding Paper Award in the IEEE

Transactions on Automatic Control in 1992 and 1999,

both times for joint work with T. T. Georgiou. He is a

Fellow of the IEEE.

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Documents

The Missing Mechanical Circuit Element