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

Feature

The Missing Mechanical Circuit ElementMichael Z.Q. Chen, Christos Papageorgiou, Frank Scheibe, Fu-Cheng Wang, and Malcolm C. Smith

AbstractIn 2008, two articles in Autosport revealed details 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 actually was. The Autosport articles revealed that the J-damper was in fact an “inerter” and that its origin lay in academic work on mechanical 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, and its intimate connection with the classical theory of network synthesis.

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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 inductordamper 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 points which 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 Department 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 of

passive 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 have

two 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 and

across 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

ii

i

i

v2 v1

v2 v1

v2 v1

v2 v1

v2 v1

v2 v1

FF

F

F

F

F Y(s) = bs

Y(s) = c

Y(s) = Cs

dFdt = k(v2 − v1)

F = bd(v2 − v1)

dti = C

d(v2 − v1)dt

F = c(v2 − v1)

Y(s) = ks Y(s) = 1

Ls

Y(s) = 1R

didt = 1

L (v2 − v1)

(v2 − v1)1R

i =

Figure 1. Electrical and mechanical circuit symbols and correspondences. In the force-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 Identifier 10.1109/MCAS.2008.931738


12 IEEE CIRCUITS AND SYSTEMS MAGAZINE FIRST QUARTER 2009

of the classical theory of electrical networks suggests otherwise. 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 realized in 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 Ja2 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.



Like other modelling elements, the deviation of in-erter embodiments from ideal behaviour should be kept in 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 mechanical networks as follows. Suppose that 1F, v 2 represents the

force-velocity pair associated with a two-terminal me-chanical network, then passivity requires:

3T

2`

F 1 t 2v 1 t 2dt $ 0

for 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 is positive-real [1], [24]. Let Z 1s 2 be a real-rational function. Then Z 1s 2 is defined to be positive-real if Re 3Z 1s 2 4 $ 0 in 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 University

filed 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).

Dur ing 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 hear ing,

neither the World

Motor Sport Council

nor McLaren made

p u b l i c w h a t th e

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. F inal ly,

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.



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

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

poles on the imaginary axis and infi nity are simple 3. 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 3Z 1 jv 2 4 $ 02. for all v with jv 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-terminal mechanical network whose impedance equals Z 1s 2 , which consists of a finite interconnection of springs, dampers and inerters. The ability to synthesise the most general 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 problem is 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 our problem, 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<1 kg, Inertance (adjustable) = 60–240 kg. (a) Complete with outer case, (b) ballscrew, nut and flywheel, (c) flywheel removed, (d) thrust bearing.

(a) (b)

(c) (d)



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

£ v̂1

v̂2

v̂3

§ 5 s £L1 L4 L5

L4 L2 L6

L5 L6 L3

§ £ F̂1

F̂2

F̂3

§ 5: sL £ F̂1

F̂2

F̂3

§ ,

where L is a non-negative definite matrix and ^ denotes Laplace transform. And the admittance of Q is

F̂1

v̂15

a3s3 1 a2s

2 1 a1s 1 a0

b4s4 1 b3s

3 1 b2s2 1 b1s

, (1)

where a3 5 bc 1L2L3 2 L62 2 , a2 5 bL3, a1 5 cL2, a0 5 1,

b4 5 bc det 1L 2 , b3 5 b 1L1L3 2 L52 2 , b2 5 c 1L1L2 2 L4

2 2 and b1 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

1R2R3 2 R6

2 2s3 1 R3s2 1 R2s 1 1

s 1det Rs3 1 1R1R3 2 R52 2s2 1 1R1R2 2 R4

2 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.

k1

k2c

b

(b)(a)

Figure 6. General one-port containing one damper and one inerter.

v1

F1

X

F2

F3

v2

v3

c

b



network without transformers [6], [36]. In general, paramountcy is not a sufficient condition for the re-alisability of a transformerless resistive network and a 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. 6 for 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 of Fig. 6 where X contains only springs. Then Y 1s 2 can be realized 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 3Z 1 jv 2 4 $ 02. for all v with jv 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

k1

k1

k1

k1

k1

k2

k2

k2 k2

k2

k3

k3

k3 k3

k3

k4

k4

k4

k4

k4

(a)

(b) (c)

(d) (e)

cb

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).



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 insulate against 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 NetworksWe 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 relates the 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 the exogenous disturbances (e.g. zr and Fs ) 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.

Fs

Y (s)

zs

zu

zr

mu

kt

ms

Figure 8. Quarter-car vehicle model.

G (s)

K (s)

F

z w

v2 − v1

Figure 9. The control synthesis paradigm applied for the synthesis of a positive-real admittance K(s).



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

K 1s 2 5 cAk Bk

Ck Dkd2 5 Ck 1sI 2 Ak 221Bk 1 Dk

, (3)

then K 1s 2 is positive-real if and only if there exists Pk . 0 that satisfies the Linear Matrix Inequality (LMI)

cAkTPk 1 PkAk PkBk 2 Ck

T

BkTPk 2 Ck 2 Dk

T 2 Dkd # 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 J3 [40] for a standard stochastic road profile given by

J3 5 2p"Vk 7s21TzrSkt1zu2zr2 1 jv 2 72, (4)

where k is a road roughness parameter and V the vehicle velocity.

We now calculate the generalized plant, GJ31s 2 ,

corresponding to the block diagram of Fig. 9 and the performance measure J3 . The performance output corresponding to J3 is given by z 5 ekt 1zu 2 zr 2 and the excitation input is the road disturbance signal w 5 zr. 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,

GJ31s 2 5 F D

0 2 ksms

0 ksms

1 0 0 00 ks

mu0 2 ks 1 kt

mu

0 0 1 0

Tc 2 ms

kt0 2 mu

kt0

1 0 2 1 0d

D 0 2 1ms

0 0ktmu

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 GJ3

1s 2 and the controller be given by:

£ x#

x#k

ezu 2 zr

§ 5 cAcl Bcl

Ccl 0d £ x

xk

zr

§ .

Theorem 5: There exists a strictly positive-real control-ler K 1s 2 of order nk such that J3 , 2p"1Vk 2kt

n and Acl is stable, if and only if the following matrix inequality problem is feasible for some Xcl . 0, Xk . 0, Q, n2 and Ak, Bk, Ck, Dk of compatible dimensions:

cAclTXcl 1 XclAcl XclBcl

BclTXcl 2 I

d , 0, cXcl CclT

Ccl Qd . 0,

tr 1Q 2 , n2, cAkTXk 1 XkAk XkBk 2 Ck

T

BkTXk 2 Ck 2 Dk

T 2 Dkd , 0.

K c

(a)

K c b

(b)

K

k

c

(c)

K

c

b

(d)

b

K

c

c

k1

k1

(e)

b

kb

K

k1

k1

(f)

Figure 10. Passive suspension networks incorporating springs, dampers and inerters. Here, ks 5 K. (a) S1, (b) S3, (c) S2, (d) S4, (e) S5, and (f) S7.



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. The method, 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 pro blems. A feasible starting point must be given to the algorithm.

4) Tyre grip optimization results: The optimization of the J3 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 J3 over general second-order admittances K 1s 2 in order to investigate whether J3 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 J3 . In the intermediate range K 1s 2 turns out to be the net-work S10 shown in Fig. 12 consisting of an inerter, damper and spring in series [29].

B. Analytical Solutions for Optimal Ride Comfort and Tyre GripThe 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 J3 for Quarter-Car Model

S2 (Damper with Relax. Spring)S3 (Damper, Inerter in Parallel)S4 (Damper, Inerter in Series)S5 S7YALMIP 1E4 < ks < 1.8E4 N/mYALMIP 2E4 < ks < 6.5E4 N/mYALMIP 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)



numerical optimizations must be repeated. In [33] both of these issues are addressed for ride comfort and tyre grip performance 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:

Y3 5Ks

1 c 1 sb,

Y9 5Ks

1 a sk 1 sc

11sbb21

,

Y10 5Ks

1 a sk

11c

11sbb21

.

2) Performance measures and analytical expression: In addition to the r.m.s. dynamic tyre load parameter J3 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

J1 5 2p"Vk 7s21TzrS$zs1 jw 2 72.

See [40] for detailed derivations of the performance measures.

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 221B 72 5 1CLCT 21/2,

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

Ji 5 2p"VkHi,Sj,

where H 5 CLCT and i indicates the performance measure index and j the suspension network number.

3) Optimal solutions for mixed performance of J1 and J3: Optimal performance solutions for J1 and J3 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 J1 and J3 measure:

mu = 35 kg

mu = 20 kg

K = 15 kNm−1

K = 55 kNm−1

K = 35 kNm−1

J 3 (×

102 )

J1

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 values ms 5 250 kg, kt 5 150 kNm21, mu 5 35 kg (for K 5 15,35,55 kNm21 2 and mu 5 20 kg (for K 5 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.



H1,3:Sj 5 11 2 a 2ms2H1,Sj 1 aH3,Sj, (6)

where a [ 30,1 4 is a weighting between J1 and J3. The scaling factor ms

2 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 J1 and J3 combined. The results also show that ride comfort (J1) deteriorates as suspension static stiffness 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 demonstrate the performance improvements which can be achieved using inerters over a conventional arrangement using nonlinear dampers.

A. Nonlinear Vehicle ModelThe 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 moments of inertia about its roll, pitch and yaw axes respectively

(Ix 5 400 kgm2, Iy 5 2300 kgm2, Iz 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. Defi nition of DisturbancesFor 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 #h0

v,

y 1 t 2 5 h0, h0

v, t #

1 2 h0

v,

Figure 14. A static view of the animation of the SimMechanics vehicle model. The viewer is at the rear and elevated with respect 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 muchmore important for mechanical networks than electrical networks.



y 1 t 2 5 1 2 vt, 1 2 h0

v, 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 then the 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 taking into account the relative importance of the handling performance over the ride comfort and tyre grip performance.

C. Defi nition of Performance MeasuresThe 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] and represent 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:

J8t 5"trace 1z$wz$

wT 2 yr5kerbstrike

,

J9t 5"trace 1Ft FtT 2 yr5kerbstrike

,

where the signal z$

w 5 3z$,u$,w$ 4 denotes the weighted ac-

celeration responses of the sprung mass, yr denotes the road elevations at the four wheel stations and Ft denotes the tyre forces. It is easy to see that

J8t 5"z$ Tz$

1 u$ T

u$

1 w$ Tw$

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:

Ht 5"erollT eroll 1 epitch

T 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 SuspensionsIn 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.

ks

cn

k

c

b 10−1 100 101 102

10−1 100 101 102

101

102

103

104Admittance of (c + k + b) Network

Frequency in Hz

Gai

n in

Ns/

m

−100

−50

0

50

100

Frequency in Hz

Pha

se in

°

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.



defined performance measures to design a suspension network 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 512

J8t

J8t0

112

J9t

J9t0

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:

J8t

J8t0

5 0.98, J9t

J9t0

5 0.945, Ht

Ht0

5 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 pinion mechanism 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 view. In contrast, a behavioural definition of play (ideal play) does not suffer from these objec-tions and appears more reasonable from a physical point of view.

A. The Play OperatorA 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 (z1) and cylinder (z2) are considered to be the input and output (follower), respectively.

(engagement—extension): z2 5 z1 1 P, z1#

5 z2#

# 0,

(engagement—compression): z2 5 z1 2 P, z1#

5 z2#

$ 0,

(disengagement): |z1 2 z2| , 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 play 1. element is not necessarily zero.

Figure 17. Damped harmonic oscillator network. The letter i indicates the input and f the follower.

z yu

f i

m1 m2

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 of hysteresis play model.



The solutions of the network equations depend 2. on the choice of inertial frame, namely, the addi-tion of a constant velocity to all states may change switching 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 a 4. switch of terminals of the play operator.

B. Ideal PlayFollowing 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 z1, z2 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 following three states:

(engagement—extension): z2 2 z1 5 e, F # 0,

(engagement—compression): z2 2 z1 5 2 e, F $ 0,

(disengagement): |z2 2 z1| , 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 z1 and z2 .

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 2P0 , P # P0, where P0 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

z1 z2 z3k1

c1P

Figure 21. Semi-ideal play model with displacements zd 5 z1 2 z3, zn 5 z1 2 z2, and zp 5 z2 2 z3.

Figure 19. Harmonic oscillator network with an inerter and ideal play.

z yu

m1 m2

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.

P02P0

0 P

E( t0) − E( t0)−+

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) Terminal modelling symbol for play.

FF

z1 z2z1 z2

PQ2

(a) (b)

∋

P



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, which is 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 Electrical and 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 in controller 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 McLaren

Racing 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 engineer ing 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 Research Fellow 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



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.

References[1] B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthe-sis. Englewood Cliffs, NJ: Prentice-Hall, 1973.[2] V. Belevitch, “Summary of the history of circuit theory,” Proc. IRE, vol. 50, no. 5, pp. 848–855, 1962.[3] R. Bott and R. J. Duffi n, “Impedance synthesis without use of trans-formers,” J. Appl. Phys., vol. 20, pp. 816, 1949.[4] S. Boyd, L. El Ghaoui, E. Feron, and B. Balakrishnan, “Linear matrix inequalities in system and control theory,” SIAM, 1994.[5] British Standard Guide to Measurement and Evaluation of Human Exposure to Whole-Body Mechanical Vibration and Repeated Shock, BS 6841, 1987.[6] I. Cederbaum, “Conditions for the impedance and admittance ma-trices of n-ports without ideal transformers,” Proc. IEE, vol. 105, pp. 245–251, 1958.[7] I. Cederbaum, “Topological considerations in the realization of re-sistive n-port networks,” IRE Trans. Circuit Theory, vol. CT-8, no. 3, pp. 324–329, 1961.[8] M. Z. Q. Chen, “Passive network synthesis of restricted complexity,” Ph.D. thesis, Cambridge Univ. Eng. Dept., U.K., Aug. 2007.[9] M. Z. Q. Chen and M. C. Smith, “Mechanical networks comprising one damper and one inerter,” in Proc. 2008 European Control Conf., 2007, Kos, Greece, July 2007, pp. 4917–4924.[10] M. Z. Q. Chen and M. C. Smith, “Electrical and mechanical passive network synthesis,” in Recent Advances in Learning and Control (Lec-ture Notes in Control and Information Sciences), vol. 371. New York: Springer-Verlag, 2008, pp. 35–50.[11] M. Z. Q. Chen and M. C. Smith, “A note on tests for positive-real functions,” IEEE Trans. Automat. Control, to be published.[12] M. Z. Q. Chen and M. C. Smith, “Restricted complexity network real-isations for passive mechanical control,” IEEE Trans. Automat. Control, to be published.[13] S. Darlington, “Synthesis of reactance 4-poles which produce prescribed insertion loss characteristics,” J. Math. Phys., vol. 18, pp. 257–353, 1939.[14] S. Evangelou, D. J. N. Limebeer, R. S. Sharp, and M. C. Smith, “Con-trol of motorcycle steering instabilities—passive mechanical compen-sators incorporating inerters,” IEEE Control Syst. Mag., pp. 78–88, Oct. 2006.[15] S. Evangelou, D. J. N. Limebeer, R. S. Sharp, and M. C. Smith, “Steer-ing compensation for high-performance motorcycles,” Trans. ASME, J. Appl. Mech., vol. 74, no. 2, pp. 332–346, 2007.[16] FIA’s decision on the “spy scandal”. [Online]. Available: http://www.fia.com/mediacentre/Press_Releases/FIA_Sport/2007/December/071207-01.html, 2007.[17] E. A. Guillemin, Synthesis of Passive Networks. New York: Wiley, 1957.[18] A. Hassibi, J. How, and S. Boyd, “A path-following method for solv-ing BMI problems in control,” in Proc. American Control Conf., San Diego, CA, 1999, pp. 1385–1389.[19] M. Hughes, “A genius idea, and why McLaren hasn’t tried to stop others using it,” Autosport J., 2008.[20] J. Z. Jiang, M. C. Smith, and N. E. Houghton, “Experimental test-ing and modelling of a mechanical steering compensator,” in Proc. 3rd IEEE Int. Symp. Communications, Control and Signal Processing (ISCCSP 2008), 2008, pp. 249–254.[21] D. Karnopp, “Theoretical limitations in active vehicle suspensions,” Vehicle Syst. Dyn., vol. 15, pp. 41–54, 1986.[22] J. Löfberg. (2004). YALMIP 3. [Online]. Available: http://control.ee.ethz.ch/~joloef/yalmip.msql[23] A. G. J. MacFarlane, Dynamical System Models. London: Harrap, 1970.[24] R. W. Newcomb, Linear Multiport Synthesis. New York: McGraw-Hill, 1966.

[25] M. Nordin, J. Galic, and P.-O. Gutman, “New models for backlash and gear play,” Int. J. Adaptive Control Signal Process., vol. 11, pp. 49–63, 1997.[26] C. Papageorgiou, N. E. Houghton, and M. C. Smith, “Experimental testing and analysis of inerter devices,” ASME J. Dyn. Syst., Measure. Control, to be published.[27] C. Papageorgiou, O. G. Lockwood, N. E. Houghton, and M. C. Smith, “Experimental testing and modelling of a passive mechani-cal steering compensator for high-performance motorcycles,” pre-sented at the Proc. European Control Conf., Kos, Greece, July 2–5, 2007.[28] C. Papageorgiou and M. C. Smith, “Laboratory experimental test-ing of inerters,” in Proc. 44th IEEE Conf. Decision and Control and 2005 European Control Conf., Dec.12–15, 2005, pp. 3351–3356.[29] C. Papageorgiou and M. C. Smith, “Positive real synthesis using matrix inequalities for mechanical networks: application to vehicle suspension,” IEEE Trans. Control Syst. Technol., vol. 14, pp. 423–435, 2006.[30] C. Papageorgiou and M. C. Smith, “Simulation-based analysis and optimal design of passive vehicle suspensions,” Tech. Rep. CUED/F-INFENG/TR 599, Apr. 2008.[31] C. Scarborough, “Technical insight: Renault’s J-damper,” Autosport J., 2008.[32] F. Scheibe and M. C. Smith, “A behavioural approach to play in mechanical networks,” SIAM J. Control Optim., vol. 47, pp. 2967–2990, 2009.[33] F. Scheibe and M. C. Smith, “Analytical solutions for optimal ride comfort and tyre grip for passive vehicle suspensions,” Vehicle Syst. Dyn., to be published.[34] C. Scherer, P. Gahinet, and M. Chilali, “Multi-objective output-feed-back control via LMI optimisation,” IEEE Trans. Automat. Control, vol. 42, no. 7, pp. 896–911, 1997.[35] J. L. Shearer, A. T. Murphy, and H. H. Richardson, Introduction to System Dynamics. Reading, MA: Addison-Wesley, 1967.[36] P. Slepian and L. Weinberg, “Synthesis applications of paramount and dominant matrices,” in Proc. Nat. Electron. Conf., 1958, vol. 14, pp. 611–630.[37] M. C. Smith, “Force-controlling mechanical device,” U.S. Patent 7 316 303, Jan. 8, 2008.[38] M. C. Smith, “Synthesis of mechanical networks: the inerter,” IEEE Trans. Automat. Control, vol. 47, no. 10, pp. 1648–1662, 2002.[39] M. C. Smith and G. W. Walker, “Performance limitations and con-straints for active and passive suspension: A mechanical multi-port approach,” Vehicle Syst. Dyn., vol. 33, pp. 137–168, 2000.[40] M. C. Smith and F.-C. Wang, “Performance benefi ts in passive ve-hicle suspensions employing inerters,” Vehicle Syst. Dyn., vol. 42, pp. 235–257, 2004.[41] B. D. H. Tellegen, Theorie der Wisselstromen. Gronigen, The Nether-lands: Noordhoff, 1952.[42] M. E. Van Valkenburg, Introduction to Modern Network Synthesis. New York: Wiley, 1960.[43] F.-C. Wang, C.-W. Chen, M.-K. Liao, and M.-F. Hong, “Performance anal-yses of building suspension control with inerters,” in Proc. 46th IEEE Conf. Decision and Control, New Orleans, LA, Dec. 2007, pp. 3786–3791.[44] F.-C. Wang, M.-K. Liao, B.-H. Liao, W.-J. Su, and H.-A. Chan, “The performance improvements of train suspension systems with mechanical networks employing inerters,” Vehicle Syst. Dyn., to be published.[45] F.-C. Wang and W.-J. Su, “The impact of inerter nonlinearities on ve-hicle suspension control,” Vehicle Syst. Dyn., vol. 46, no. 7, pp. 575–595, 2008.[46] F.-C. Wang, C.-.H. Yu, M.-L. Chang, and M. Hsu, “The performance improvements of train suspension systems with inerters,” in Proc. 45th IEEE Conf. Decision and Control, San Diego, CA, Dec. 2006, pp. 1472–1477.[47] L. Weinberg, “Report on circuit theory,” Tech. Rep., XIII URSI As-sembly, London, U.K., Sept. 1960.[48] L. Weinberg and P. Slepian, “Realizability condition on n-port net-works,” IRE Trans. Circuit Theory, vol. 5, pp. 217–221, 1958.[49] L. Weinberg and P. Slepian, “Positive real matrices,” Indiana Univ. Math. J., vol. 9, pp. 71–83, 1960.[50] D. C. Youla and P. Tissi, “N-port synthesis via reactance extraction, Part I,” in IEEE Int. Conv. Rec., 1966, pp. 183–205.


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