Feature
The Missing
Mechanical
Circuit Element
Michael Z.Q. Chen,
Christos Papageorgiou,
Frank Scheibe, Fu-Cheng Wang,
and Malcolm C. Smith
Abstract
In 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 performance gains in handling and grip. From
its first mention in the 2007 Formula One
“spy scandal” there was much speculation 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 Cambridge University. This article aims to provide an overview of the background and
origin of the inerter, its application, and
its intimate connection with the classical
theory of network synthesis.
© LAT PHOTOGRAPHIC
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ELECTRICAL AND MECHANICAL NETWORK ANALOGIES
O
ne of the principal motivations for the introduction of
force-current analogy with the inerter replacing the mass ele-
the inerter in [38] is the synthesis of passive mechanical
ment. The admittance Y (s) is the ratio of through to across
networks. The fact that the mass element, together with the
quantities, where s is the standard Laplace transform variable.
spring and damper, is insufficient to realize the totality of
For mechanical networks in rotational form the through and
passive mechanical impedances can be seen using the force-
across variables are torque and angular velocity, respectively.
current analogy between mechanical and electrical circuits.
For further background on network analogies see [23], [35],
In this analogy, force and current are the ‘‘through-variables’’
and [38].
and velocity and voltage are the
‘‘across-variables’’ [35]. Moreover, the terminals of mechani-
Mechanical
cal and electrical elements are
in one-to-one correspondence.
F
F
For the mechanical elements
v2
dF
= k(v2 − v1)
dt
the spring and damper have
two independently movable
terminals, whereas the termiof mass and a fixed point in
v2
an inertial frame (mechanical
F=b
d(v2 − v1)
dt
Y(s) = ks
v1
Spring
F
F
nals of the mass are its centre
Electrical
Y(s) = bs
i
i
v2
di
dt
= L1 (v2 − v1)
i
v2
i= C
Y(s) = 1
Ls
Inductor
i
v1
Inerter
v1
v1
d(v2 − v1)
dt
Y(s) = Cs
Capacitor
ground). The mass is therefore
F
analogous to a grounded capacitor. In contrast, the inerter
F
v2
is a two-terminal device, analo-
i
i
Damper
i = 1 (v2 − v1)
R
v2
v1
Y(s) = 1
R
v1
F = c(v2 − v1)
gous to an ungrounded capacitor, with both terminals freely
Y(s) = c
Resistor
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, respectively. (The symbol s is the standard Laplace transform variable.)
and independently movable.
Fig. 1 shows a table of element correspondences in the
1. Introduction
he standard analogies between mechanical and
electrical networks are universally familiar to students and engineers alike. The basic modelling
elements have the following correspondences:
T
spring
damper
mass
4
4
4
inductor
resistor
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 restriction in the case of the mass. All the above elements
except the mass have two “terminals” (for a mechanical 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
Digital Object Identifier 10.1109/MCAS.2008.931738
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 Engineering, National Taiwan University, Taipei 10617, Taiwan. M.C. Smith is with the Department of Engineering, University of Cambridge, Cambridge, CB2
1PZ, U.K.
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11
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.
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 resistors, 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 suspension system) the restriction is a very real one.
To bypass the snag a new mechanical modelling element 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 “inerter” was coined to describe such a device. 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.
Since the birth of the inerter in the Engineering Department at Cambridge University a number of applications have been proposed and investigated. Alongside
the successful application in Formula One racing (see
Sidebar II) the general applicability to vibration absorption and automotive suspensions has been considered
[29], [38], [40]. The use of the inerter in mechanical
steering compensators of high-performance motorcycles was studied in [14], [15]; by replacing the conventional steering damper with a serial inerter-damper
layout, it was shown that two significant instabilities,
“wobble” and “weave”, can be stabilized simultaneously. 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 connections. 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 conventional 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
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as follows. Section II presents the physical constructions 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 synthesis using matrix inequalities and the analytical solutions 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 nonlinear 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-terminal 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 behaviour 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 relative acceleration between them. The constant of proportionality 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 device 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. Inerters 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 construction 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 terminals, then the inertance of the device is given by b 5 Ja 2
where J is the flywheel’s moment of inertia.
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FROM THOUGHT-EXPERIMENT TO FORMULA ONE RACING
fter the initial ‘‘discovery’’ of the inerter, Professor Smith
A
the device, and in particular to make a connection with the
did some calculations which indicated a potential perfor-
technical literature on the inerter which Professor Smith and
mance advantage for vehicle suspensions which might be large
his group were continuing to publish. This strategy succeeded in
enough to interest a Formula One team. Cambridge University
spectacular fashion during the 2007 Formula One ‘‘spy scandal’’
filed a patent on the device [37] and then approached McLaren
when a drawing of the McLaren J-damper came into the hands
Racing in confidence. McLaren was interested to try out the
of the Renault engineering team. This incident was reported to
idea and signed an agreement with the University for exclusive
the FIA World Motor Sport Council who convened to consider
rights in Formula One for a limited period. After a rapid devel-
the matter in Monaco on 6th December 2007. A full transcript
opment process the
of the proceedings is
inerter was raced for
available on the FIA
the first time by Kimi
official website [16].
Raikkonen at the
During the De-
2005 Spanish Grand
cember hearing,
Prix, who achieved a
neither the World
victory for McLaren
Motor Sport Council
(see Fig. 2).
nor McLaren made
During devel-
public what the
opment McL aren
J-damper was. After-
invented a decoy
wards speculation
name for the inerter
increased on inter-
(the ‘‘J-damper’’) to
net sites and blogs
keep the technology
about the function
secret from its com-
and purpose of the
petitors for as long
as possible. The ‘‘J’’
has no actual mean-
device and there
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.
ing, and of course
were many amusing and erroneous
guesses. Finally,
the device is not a damper. The idea behind the decoy name was
the truth was discovered by the Autosport magazine. Two ar-
to make it difficult for personnel, who might leave McLaren to
ticles appeared in May 2008 which revealed the Cambridge
join another Formula One team, to transfer information about
connection and that the J-damper was an inerter [19], [31].
Like other modelling elements, the deviation of inerter 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, theoretically 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 1 F, v 2 represents the
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force-velocity pair associated with a two-terminal mechanical network, then passivity requires:
T
3
2`
F 1 t 2 v 1 t 2 dt $ 0
for all admissible time functions F 1 t 2 , v 1 t 2 and all T . If Z 1 s 2
is the real rational impedance or admittance function of
a linear time-invariant two-terminal network, it is wellknown that the network is passive if and only if Z 1 s 2 is
positive-real [1], [24]. Let Z 1 s 2 be a real-rational function.
Then Z 1 s 2 is defined to be positive-real if Re 3 Z 1 s 2 4 $ 0
in the open right half plane (ORHP), i.e. for all s with
Re 3 s 4 . 0 . The following is a well-known equivalent condition for positive-realness.
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13
Rack
Pinions
3. poles on the imaginary axis and infinity are simple
and have non-negative residues.
An alternative necessary and sufficient condition for
positive-realness is as follows.
Theorem 2: [48], [49]: Let Z 1 s 2 5 p 1 s 2 /q 1 s 2 , where p 1 s 2
and q 1 s 2 are coprime polynomials. Then Z 1 s 2 is positivereal if and only if
1. p 1 s 2 1 q 1 s 2 is Hurwitz;
Terminal 2
Gear
Flywheel Terminal 1
2. Re 3 Z 1 jv 24 $ 0 for all v with jv not a pole of Z 1 s 2 .
(a)
In [3] Bott and Duffin showed that any rational positivereal function can be realized as the driving-point impedance of a two-terminal network comprising resistors,
inductors and capacitors only. Making use of the forcecurrent analogy (see Sidebar I) and the new modelling
element (inerter) it can be seen that, given any positivereal function Z 1 s 2 , there exists a passive two-terminal
Terminal 2
Nut Flywheel Screw
Terminal 1
mechanical network whose impedance equals Z 1 s 2 ,
which consists of a finite interconnection of springs,
(b)
dampers and inerters. The ability to synthesise the most
Figure 3. Schematics of two embodiments of the inerter.
general positive-real impedance allows the designer to
(a) Rack and pinion inerter, (b) ballscrew inerter.
achieve the optimal performance among passive mechanical networks. Fig. 5 shows
a specific mechanical network together with a physical realization
constructed at Cambridge University Engineering Department.
Efficiency of realization, as defined by the number of elements
used, is much more important for
mechanical networks than electrical networks. In this section, we
consider the class of realizations
(a)
(b)
in which the number of dampers
and inerters is restricted to one in
each case while allowing an arbitrary number of springs (which is
the easiest element to realize practically). 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 capacitors, but not inductors, in electri(c)
(d)
cal circuit synthesis [10]. Such
Figure 4. Ballscrew inerter made at Cambridge University Engineering Departquestions involving restrictions
ment; Mass<1 kg, Inertance (adjustable) = 60–240 kg. (a) Complete with outer
on both resistive and one type
case, (b) ballscrew, nut and flywheel, (c) flywheel removed, (d) thrust bearing.
of reactive element have never
been considered. This contrasts
Theorem 1: [1], [24]: Z 1 s 2 is positive-real if and with the problems of minimal resistive and minimal reonly if
active synthesis which have well-known solutions when
1. Z 1 s 2 is analytic in Re 3 s 4 . 0;
transformers are allowed ([13], [50], see also [1]). In our
2. Re 3 Z 1 jv 2 4 $ 0 for all v with jv not a pole of Z 1 s 2 ;
problem, we impose the condition that no transformers
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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 lever 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 general class under consideration.
We consider a mechanical one-port network Q consisting 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^ 1
L1
^
£ v2 § 5 s £ L4
v^ 3
L5
L4
L2
L6
L5 F^ 1
F^ 1
^
L6 § £ F2 § 5: sL £ F^ 2 § ,
L3 F^ 3
F^ 3
where a3 5 bc 1 L2L3 2 L26 2 , a2 5 bL3, a1 5 cL2, a0 5 1,
b4 5 bc det 1 L 2 , b3 5 b 1 L1L3 2 L25 2 , b2 5 c 1 L1L2 2 L24 2 and
b1 5 L1.
The admittance (1) effectively has only six parameters 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 1 s 2 5
1 R2R3 2 R 26 2 s 3 1 R3s 2 1 R2s 1 1
s 1 det Rs 1 1 R1R3 2 R 25 2 s 2 1 1 R1R2 2 R 24 2 s 1 R1 2
k2
c
(b)
Figure 5. Inerter in series with damper with centring springs.
(a) Circuit diagram and (b) mechanical realization.
(1)
3
b
(a)
where L is a non-negative definite matrix and ^ denotes
Laplace transform. And the admittance of Q is
a3s 3 1 a2s 2 1 a1s 1 a0
F^ 1
5
,
v^ 1 b4s 4 1 b3s 3 1 b2s 2 1 b1s
k1
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
F2
(2)
F1
v2
c
v3
b
and
R1
R J £ R4
R5
R4
R2
R6
R5
L1
R6 § 5 T £ L4
R3
L5
L4
L2
L6
L5
L6 § T,
L3
v1
X
F3
where
1
T 5 £0
0
0
"c
0
and R is non-negative definite.
FIRST QUARTER 2009
0
0 §
"b
Figure 6. General one-port containing one damper and
one inerter.
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In [41, pp. 166–168], however, it was proven that
paramountcy is necessary and sufficient for the realisability 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 relabelling of terminals and changes of polarity.
We now state a theorem from [8], [9], [12] which provides specific realizations for the Y 1 s 2 in the form of Fig. 6
for any X that contains springs only and no transformers. 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). Alternative realizations can also be found which are of similar
complexity (see [8]).
k2
k1
k3
c
k4
b
(a)
k1
k2
k3
k1
c
k4
k3
k2
c
b
k4
b
(c)
(b)
k1
c
k3
k3
b
k4
k1
k2
k4
k2
c
b
(d)
(e)
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).
network without transformers [6], [36]. In general,
paramountcy is not a sufficient condition for the realisability of a transformerless resistive network and
a counter-example for n 5 4 was given in [7], [47].
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Theorem 3: [8], [9], [12] Given Y 1 s 2 in the form of
Fig. 6 where X contains only springs. Then Y 1 s 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 function 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 1 s 2 5 p 1 s 2 /q 1 s 2 , where p 1 s 2
and q 1 s 2 have no common roots in the ORHP. Then Z 1 s 2
is positive-real if and only if
1. p 1 s 2 1 q 1 s 2 has no roots in the ORHP;
2. Re 3 Z 1 jv 2 4 $ 0 for all v with jv not a pole of Z 1 s 2 .
When p 1 s 2 and q 1 s 2 are coprime, the “only if” implication 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 Theorem 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 comfortable 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 disturbances (the latter being a simple approximation to
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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, braking 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 acceleration, 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 energy source in the system. They therefore provide a
simpler and cheaper means of suspension design and
construction at the expense of performance limitations than active suspensions (with energy sources).
Generally a suspension needs to be “soft” to insulate 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 inerters 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 networks can be designed using a linear matrix inequality (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 suspension design problem was formulated as an optimal 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 inerters that resulted in considerable improvement in the
individual performance measures. It was also possible to formulate and solve multi-objective optimization problems.
1) The quarter-car model: The quarter-car model presented in Fig. 8 is the simplest model to consider for suspension 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 1 s 2 which relates
the suspension force to the strut velocity. In this section
FIRST QUARTER 2009
Fs
ms
zs
Y (s)
mu
zu
kt
zr
Figure 8. Quarter-car vehicle model.
we will assume further that Y 1 s 2 5 K 1 s 2 1 ks/s, where
K 1 s 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 synthesise admittances over the whole class of positivereal functions, we use a control synthesis paradigm
along with a state-space characterisation of positiverealness. The search for positive-real admittances
is formulated as a search for positive-real “controllers” K 1 s 2 as shown in Fig. 9 where w represents the
exogenous disturbances (e.g. zr and Fs ) and z represents 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.
z
w
G (s)
v2 − v1
F
K (s)
Figure 9. The control synthesis paradigm applied for the
synthesis of a positive-real admittance K(s).
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Lemma 2 (Positive real lemma [4]): Given that,
K1s2 5 c
Ak Bk
2
d 5 Ck 1 sI 2 Ak 2 21Bk 1 Dk ,
Ck Dk
(3)
then K 1 s 2 is positive-real if and only if there exists Pk . 0
that satisfies the Linear Matrix Inequality (LMI)
K
ATk Pk 1 PkAk
c T
B k Pk 2 C k
PkBk 2
d # 0.
2 D Tk 2 Dk
c
K
C Tk
c
b
3) Generalized plant for the optimization of tyre grip:
In this section we will focus on a single aspect of performance, 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 7s 21Tzr S kt1zu 2zr2 1 jv 2 7 2,
(4)
where k is a road roughness parameter and V the vehicle
velocity.
We now calculate the generalized plant, GJ3 1 s 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 e kt 1 zu 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, zs 2 zu and the controller output is the suspension force F . It was shown in
[29] that,
(b)
(a)
k
0
1
D
0
F
GJ3 1 s 2 5
0
c
K
c
K
2 mkss
0
0
0
0
1
ks
mu
0
2 mkts
1
0
0
ks
ms
0
T
2 ks m1u kt
0
2 mktu
21
0
d
0
0
0
2 m1s
0
mu
1
mu
0
0
c
0 0
d
0 0
D kt
T
V
.
c
Given a controller K 1 s 2 of order nk , with state-space representation as in (3), let the state-space representation
of the closed-loop system resulting from the interconnection of the generalized plant GJ3 1 s 2 and the controller
be given by:
b
(c)
(d)
kb
k1
£
e zu 2 zr
c
K
k1
c
K
b
k1
b
k1
(e)
(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.
18
IEEE CIRCUITS AND SYSTEMS MAGAZINE
#
x
#
xk
§ 5 c
Acl
Ccl
x
Bcl
d £ xk § .
0
zr
Theorem 5: There exists a strictly positive-real controller K 1 s 2 of order nk such that J3 , 2p" 1 Vk 2 kt n and Acl
is stable, if and only if the following matrix inequality
problem is feasible for some Xcl . 0, Xk . 0, Q, n 2 and
Ak, Bk, Ck, Dk of compatible dimensions:
c
ATclXcl 1 XclAcl
B TclXcl
tr 1 Q 2 , n 2, c
XclBcl
Xcl
d , 0, c
2I
Ccl
ATk Xk 1 XkAk
B Tk Xk 2 Ck
C Tcl
d . 0,
Q
XkBk 2 C Tk
d , 0.
2 D Tk 2 Dk
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J3
The first three LMIs are necesOptimization Results for J3 for Quarter-Car Model
sary and sufficient conditions
700
for the existence of a stabilising
S1 (Damper)
controller that achieves an upper
S2 (Damper with Relax. Spring)
bound of n on the closed-loop
650
S3 (Damper, Inerter in Parallel)
S4 (Damper, Inerter in Series)
H2-norm [34]. The fourth LMI
S5
further restricts the controlS7
600
YALMIP 1E4 < ks < 1.8E4 N/m
ler to be strictly positive-real.
YALMIP 2E4 < ks < 6.5E4 N/m
Without the positive-real conYALMIP 6.6E4 < ks < 12E4 N/m
550
straint the H2-synthesis problem can be formulated as an
LMI problem as shown in [34].
500
With the positive-real constraint
it is not obvious how to do so,
450
hence an iterative optimization
method is employed to solve
400
the Bilinear Matrix Inequality (BMI) problem locally. The
method, which is described in
350
1
2
3
4
5
6
7
8
9
10
11
12
[18], is to linearise the BMI using a first-order perturbation
x 104
Static Stiffness in N/m
approximation, and then iteraFigure 11. Comparison of YALMIP optimization results with fixed-structure optimitively compute a perturbation
sation results for J3. (See Figure 10 for the configurations.)
that ‘slightly’ improves the controller performance by solving
an LMI problem. The proposed
scheme was implemented in YALMIP [22], which is a
MATLAB toolbox for rapid prototyping of optimization problems. A feasible starting point must be given
to the algorithm.
4) Tyre grip optimization results: The optimization of
k
the J3 measure was attempted in [40] over various fixed
structure suspensions (see Fig. 10). In contrast, the itc
k
erative algorithm implemented in YALMIP was used to
optimize J3 over general second-order admittances K 1 s 2
K
in order to investigate whether J3 can be improved furK
c
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
b
b
YALMIP with those obtained by fixed-structure optimization are presented in Fig. 11.
The optimization results obtained with YALMIP are
(b)
(a)
presented as three distinct curves suggesting that the
structure of the suspension changes as the static stiffness
Figure 12. Additional passive suspension networks incorvaries. At low and high stiffness the YALMIP second-order
porating springs, dampers, and inerters (a) S9 and (b) S10.
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 B. Analytical Solutions for Optimal
change in the structure of the admittance as the static Ride Comfort and Tyre Grip
stiffness varies in order to obtain the minimum value of The approaches of [29], [40] both require extensive nuJ3 . In the intermediate range K 1 s 2 turns out to be the net- merical optimizations. The question whether the soluwork S10 shown in Fig. 12 consisting of an inerter, damper tions obtained are global optima is not rigorously settled.
and spring in series [29].
Also, if a new set of vehicle parameters is chosen, the
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19
Passive suspensions provide a simpler and cheaper means of suspension design and
construction at the expense of performance limitations than active suspensions.
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 parameters. The optima are also parameterised in terms of suspension static stiffness, which can therefore be adjusted
to approximately take account of other performance requirements, 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 1 s 2 is the admittance of one of the candidate suspension networks.
Network S1 models a conventional parallel springdamper 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 1 s 2 for three of these layouts
(S3, S9, S10) is now given for illustration:
Y3 5
Y9 5
s
1 21
K
1a
1 b ,
s
k 1 sc sb
Y10 5
s 1
1 21
K
1a 1 1 b .
s
k c sb
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 7s 21Tzr S z$s 1 jw 2 7 2.
See [40] for detailed derivations of the performance
measures.
An analytical expression of the H2-norm of the (stable) transfer function G 1 s 2 can be computed from a minimal state-space realization as
7G 7 2 5 7C 1 sI 2 A 2 21B 7 2 5 1 CLC T 2 1/2,
where the matrix L is the unique solution of the
Lyapunov equation
K
1 c 1 sb,
s
AL 1 LAT 1 BB T 5 0.
The matrix L is then determined
from the linear equations in (5)
and the performance measures
are then given by
S1
6.5
K = 15 kNm−1
S2
S3
6
S4
J3 (×102)
5.5
S9
K = 55 kNm−1
Ji 5 2p"VkHi,Sj,
S10
5
S4
mu = 35 kg
4.5
4
S9
K = 35 kNm−1
mu = 20 kg
3.5
0.8
1
1.2
1.4
1.6
1.8
J1
Figure 13. Analytical solutions for global optimum of mixed performance J1 and J3 for networks S1–S4, S9 and S10 for three static stiffness values with quarter-car parameter values
ms 5 250 kg, kt 5 150 kNm 21, mu 5 35 kg (for K 5 15,35,55 kNm 21 2 and mu 5 20 kg
(for K 5 15 kNm 21 2 . Smaller magnitudes in J1 and J3 are beneficial.
20
IEEE CIRCUITS AND SYSTEMS MAGAZINE
(5)
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 J1 and J3: Optimal
performance solutions for J1 and J3
individually and for suspension networks S1–S4, S9 and S10 have been
computed in [33]. Furthermore, it
is also important to consider combined optimal vehicle performance
across different measures. Here we
present the results for a mixed J1
and J3 measure:
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Efficiency of realization, as defined by the number of elements used, is much
more important for mechanical networks than electrical networks.
H1,3:Sj 5 1 1 2 a 2 m 2s H1,Sj 1 aH3,Sj,
(6)
where a [ 3 0,1 4 is a weighting between J1 and J3. The
scaling factor m 2s is inserted to approximately normalise the measures and simplify the resulting formulae.
Eq. (6) can be optimized with respect to the suspension 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 inerters (especially S9 and S10) offer performance advantages over conventional networks for both J1 and J3
combined. The results also show that ride comfort
(J1) deteriorates as suspension static stiffness increases, 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 optimal 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 representation of the suspension geometry and the nonlinearities
of the suspension elements. Several aspects of suspension 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 appropriate road disturbance inputs and for the handling performance 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 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 approximate 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
FIRST QUARTER 2009
Front Left
Unsprung
Mass
Sprung Mass
Front Right
Suspension
Strut
Rear Right
Unsprung
Mass
Rear Left
Road Disturbance
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.
(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 arrangement of a spring with a nonlinear damper. The nonlinear 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 comfort 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 disturbances 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 #
h0
,
v
h0
1 2 h0
y 1 t 2 5 h0,
,t#
,
v
v
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21
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.
y 1 t 2 5 1 2 vt,
1 2 h0
1
,t# .
v
v
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 angle of the sprung mass for some fixed time and then
the step is removed so that the sprung mass recovers zero roll angle. The disturbance about the pitch
axis is chosen such that it results in both pitchingup 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 accordingly by taking into account the relative importance
of the handling performance over the ride comfort
and tyre grip performance.
tions. The acceleration weights are taken from [5] and
represent discomfort felt by humans due to mechanical
vibrations. The performance measure for tyre grip considers the tyre forces at the four wheel stations. The
time-domain measures for ride comfort and tyre grip
are defined as:
$ $
J8t 5 "trace 1 zwzTw 2 yr5kerbstrike ,
J9t 5 "trace 1 Ft FTt 2 yr5kerbstrike ,
$$ $
$
where the signal zw 5 3 z,u ,w 4 denotes the weighted acceleration 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
Gain in Ns/m
so it represents the square root of the sum of the energies 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 measures we assume that we know the desired handling
responses of the vehicle in the pitch and roll channels,
C. Definition of Performance Measures
The performance measure for the ride comfort consid- both in bump and rebound in case they are different.
ers the weighted accelerations of the sprung mass, The energy of the error (possibly weighted) between
$
$
$
namely the heave ( z ), pitch ( u ) and roll ( w ) accelera- 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
Admittance of (c + k + b) Network
104
close to the desired handling performance. The time-domain han3
10
k
dling measure is defined as:
ks
b
cn
102
101
10−1
100
101
Frequency in Hz
102
100
101
Frequency in Hz
102
Phase in °
50
0
−50
−100
10−1
Figure 15. The new suspension network and the admittance function of the linear
series connection of the spring, damper and inerter.
22
IEEE CIRCUITS AND SYSTEMS MAGAZINE
(7)
where eroll is the error signal due
to the application of the roll disturbance and epitch is the error
signal due to the application of
the pitch disturbance.
100
c
Ht 5 "e Trolleroll 1 e Tpitchepitch ,
D. Optimal Design of Nonlinear
Suspensions
In this section we use the nonlinear simulation model and the
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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.
J9t
Ht
J8t
5 0.98,
5 0.945,
5 1.003.
J8t0
J9t0
Ht0
The above results indicate that the tyre grip is improved by 5% without deteriorating the ride comfort
and handling performances. It is expected that including the hard settings of the nonlinear dampers
as decision variables in the optimization and also using a cost function that includes all aspects of performance 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 mechanical 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 including 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 networks leads to unsatisfactory solutions from a physical point of view. In contrast, a behavioural definition
of play (ideal play) does not suffer from these objections and appears more reasonable from a physical
point of view.
FIRST QUARTER 2009
#
#
(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, z2 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]:
1. During disengagement the force through the play
element is not necessarily zero.
y
y
∋
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 performance of the default suspension. The following values were obtained after optimization:
∋
1 J8t 1 J9t
1
2 J8t0 2 J9t0
−
x
∋
J5
A. The Play Operator
A number of different play definitions have been proposed
in the literature: the dead-zone (Fig. 16(a)) and hysteresis 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.
x
∋
defined performance measures to design a suspension
network involving nonlinear dampers and inerters in order 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 connection of a spring, a damper and an inerter as shown in
Fig. 15. The cost function
−
(a)
(b)
Figure 16. (a) Graph of dead-zone play model. (b) Graph of
hysteresis play model.
k
m1
m2
f
H
i
c
z
u
y
Figure 17. Damped harmonic oscillator network. The letter
i indicates the input and f the follower.
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23
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.
2∋
Q
F
P
F
P
z1
z2
z1
z2
(a)
(b)
Figure 18. (a) Physical representation of play. (b) Terminal
modelling symbol for play.
k
m1
m2
P
z
u
b
y
Figure 19. Harmonic oscillator network with an inerter and
ideal play.
E(t 0+ ) − E(t 0− )
P0
2P0
0
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.
c1
F
z1
k1
P
z2
F
z3
Figure 21. Semi-ideal play model with displacements
zd 5 z1 2 z3, zn 5 z1 2 z2, and zp 5 z2 2 z3.
2. The solutions of the network equations depend
on the choice of inertial frame, namely, the addition of a constant velocity to all states may change
switching times or eliminate them altogether.
24
3. During engagement the force through the play element is not restricted in sign.
4. The behaviour of the network is not invariant to a
switch of terminals of the play operator.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
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 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 device of appropriate sign (see Fig. 18(b) for the modelling
symbol). Also, we note that this definition allows the mechanical 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 wellposedness and the exclusion of limit points of switching 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 impulsive 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 application of a newly introduced mechanical circuit
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element, the inerter, from its origin in modelling and
circuit synthesis through to a high-profile application 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 passive 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 Shanghai. He graduated from Nanyang Technological University, Singapore, in 2003
with a B.Eng. degree in Electrical and
Electronic Engineering, and from Cambridge 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 Processing, 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 Limassol, 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 Engineering. He held positions as a research associate in the Control Group at Cambridge University,
as a researcher in the Electrical and Computer Engineering 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.
FIRST QUARTER 2009
Frank Scheibe was born in Bremen, Germany. He received the M.Eng. degree in
Electrical and Electronic Engineering
from Imperial College London in 2003,
and the Ph.D. degree in Control Engineering from Cambridge University in
2008. In 2005 he worked for McLaren
Racing Ltd and in 2007/08 was a Vehicle Dynamics Engineer 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, vehicle 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 engineering from National Taiwan University, 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 Control 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 robust 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 operational research, and the Ph.D. degree in control engineer ing from the
University of Cambridge, Cambridge,
U.K., in 1978, 1979, and 1982, respectively. 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 Assistant Professor with the Department of Electrical
Engineering at the Ohio State University, Columbus,
OH. He returned to Cambridge University as a Lecturer in the Department of Engineering in 1990, became 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 optimization, nonlinear systems, active suspension, and mechanical systems. Prof. Smith was a corecipient of the
IEEE CIRCUITS AND SYSTEMS MAGAZINE
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25
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
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IEEE CIRCUITS AND SYSTEMS MAGAZINE
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J., 2008.
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Dyn., to be published.
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FIRST QUARTER 2009
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