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Passive Network Synthesis Revisited
Malcolm C. Smith
Department of Engineering
University of Cambridge
U.K.
Mathematical Theory of Networks and Systems (MTNS)
17th International Symposium
Kyoto International Conference Hall
Kyoto, Japan, July 24-28, 2006
Semi-Plenary Lecture
1
Passive Network Synthesis Revisited Malcolm C. Smith
Outline of Talk
1. Motivating example (vehicle suspension).
2. A new mechanical element.
3. Positive-real functions and Brune synthesis.
4. Bott-Duffin method.
5. Darlington synthesis.
6. Minimum reactance synthesis.
7. Synthesis of resistive n-ports.
8. Vehicle suspension.
9. Synthesis with restricted complexity.
10. Motorcycle steering instabilities.
MTNS 2006, Kyoto, 24 July, 2006 2
Passive Network Synthesis Revisited Malcolm C. Smith
Motivating Example – Vehicle Suspension
Performance objectives
1. Control vehicle body in the face of variable loads.
2. Insulate effect of road undulations (ride).
3. Minimise roll, pitch under braking, acceleration and
cornering (handling).
MTNS 2006, Kyoto, 24 July, 2006 3
Passive Network Synthesis Revisited Malcolm C. Smith
Quarter-car Vehicle Model (conventional suspension)
PSfrag replacements
tyre
load
disturbances
road
disturbances
mu
ms
damperspring
kt
MTNS 2006, Kyoto, 24 July, 2006 4
Passive Network Synthesis Revisited Malcolm C. Smith
The Most General Passive Vehicle Suspension
PSfrag replacements
mu
Z(s)
ms
zrkt
Replace the spring and damper with a
general positive-real impedance Z(s).
But is Z(s) physically realisable?
MTNS 2006, Kyoto, 24 July, 2006 5
Passive Network Synthesis Revisited Malcolm C. Smith
Electrical-Mechanical Analogies
1. Force-Voltage Analogy.
voltage ↔ force
current ↔ velocity
Oldest analogy historically, cf. electromotive force.
2. Force-Current Analogy.
current ↔ force
voltage ↔ velocity
electrical ground ↔ mechanical ground
Independently proposed by: Darrieus (1929), Hahnle (1932), Firestone (1933).
Respects circuit “topology”, e.g. terminals, through- and across-variables.
MTNS 2006, Kyoto, 24 July, 2006 6
Passive Network Synthesis Revisited Malcolm C. Smith
Standard Element Correspondences (Force-Current Analogy)
v = Ri resistor ↔ damper cv = F
v = L didt inductor ↔ spring kv = dFdt
C dvdt = i capacitor ↔ mass m dv
dt = F
PSfrag replacements v1
v1
v2
v2
ii
FF
Electrical
Mechanical
What are the terminals of the mass element?
MTNS 2006, Kyoto, 24 July, 2006 7
Passive Network Synthesis Revisited Malcolm C. Smith
The Exceptional Nature of the Mass Element
Newton’s Second Law gives the following network interpretation of the mass
element:
• One terminal is the centre of mass,
• Other terminal is a fixed point in the inertial frame.
Hence, the mass element is analogous to a grounded capacitor.
Standard network symbol
for the mass element:PSfrag replacements
v1 = 0v2
F
MTNS 2006, Kyoto, 24 July, 2006 8
Passive Network Synthesis Revisited Malcolm C. Smith
Table of usual correspondences
PSfrag replacements
•
ElectricalMechanical
spring
mass
damper
inductor
capacitor
resistor
i
i
ii
i
i
v1
v1
v1
v1
v1
v1 = 0v2
v2
v2
v2
v2
v2
F
FF
FF
MTNS 2006, Kyoto, 24 July, 2006 9
Passive Network Synthesis Revisited Malcolm C. Smith
Consequences for network synthesis
Two major problems with the use of the mass element for synthesis of
“black-box” mechanical impedances:
• An electrical circuit with ungrounded capacitors will not have a direct
mechanical analogue,
• Possibility of unreasonably large masses being required.
Question
Is it possible to construct a physical device such that
the relative acceleration between its endpoints is pro-
portional to the applied force?
MTNS 2006, Kyoto, 24 July, 2006 10
Passive Network Synthesis Revisited Malcolm C. Smith
One method of realisation
PSfrag replacements
terminal 2 terminal 1gear
rack pinions
flywheel
Suppose the flywheel of mass m rotates by α radians per meter of relative
displacement between the terminals. Then:
F = (mα2) (v2 − v1)
(Assumes mass of gears, housing etc is negligible.)
MTNS 2006, Kyoto, 24 July, 2006 11
Passive Network Synthesis Revisited Malcolm C. Smith
The Ideal Inerter
We define the Ideal Inerter to be a mechanical one-port device such
that the equal and opposite force applied at the nodes is
proportional to the relative acceleration between the nodes, i.e.
F = b(v2 − v1).
We call the constant b the inertance and its units are kilograms.
The ideal inerter can be approximated in the same sense that real springs,
dampers, inductors, etc approximate their mathematical ideals.
We can assume its mass is small.
M.C. Smith, Synthesis of Mechanical Networks: The Inerter,
IEEE Trans. on Automat. Contr., 47 (2002), pp. 1648–1662.
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Passive Network Synthesis Revisited Malcolm C. Smith
A new correspondence for network synthesis
PSfrag replacements
••
ElectricalMechanical
spring
inerter
damper
inductor
capacitor
resistor
i
i
ii
i
i
v1
v1
v1
v1
v1
v1
v2
v2
v2
v2
v2
v2
Y (s) = ks
dFdt = k(v2 − v1)
Y (s) = bs
F = bd(v2−v1)dt
Y (s) = c
F = c(v2 − v1)
Y (s) = 1Ls
didt = 1
L (v2 − v1)
Y (s) = Cs
i = C d(v2−v1)dt
Y (s) = 1R
i = 1R (v2 − v1)
FF
FF
FF
Y (s) = admittance =1
impedance
MTNS 2006, Kyoto, 24 July, 2006 13
Passive Network Synthesis Revisited Malcolm C. Smith
Rack and pinion inerter
made at
Cambridge University
Engineering Department
mass ≈ 3.5 kg
inertance ≈ 725 kg
stroke ≈ 80 mm
MTNS 2006, Kyoto, 24 July, 2006 14
Passive Network Synthesis Revisited Malcolm C. Smith
Damper-inerter series arrangement
with centring springs
PSfrag replacements
MTNS 2006, Kyoto, 24 July, 2006 15
Passive Network Synthesis Revisited Malcolm C. Smith
Alternative Realisation of the Inerter
PSfrag replacements
screw nut flywheel
MTNS 2006, Kyoto, 24 July, 2006 16
Passive Network Synthesis Revisited Malcolm C. Smith
Ballscrew inerter made at Cambridge University Engineering Department
Mass ≈ 1 kg, Inertance (adjustable) = 60–180 kg
MTNS 2006, Kyoto, 24 July, 2006 17
Passive Network Synthesis Revisited Malcolm C. Smith
Electrical equivalent of quarter car model
PSfrag replacements
mu
Fs
Y (s)
ms
zs
zu
zr
kt
PSfrag replacements
zr
mu
Y (s)
ms Fs
k−1t
+−
Y (s) = Admittance =Force
Velocity=
Current
Voltage
MTNS 2006, Kyoto, 24 July, 2006 18
Passive Network Synthesis Revisited Malcolm C. Smith
Positive-real functions
Definition. A function Z(s) is defined to be positive-real if one of the
following two equivalent conditions is satisfied:
1. Z(s) is analytic and Z(s) + Z(s)∗ ≥ 0 in Re(s) > 0.
2. Z(s) is analytic in Re(s) > 0, Z(jω) + Z(jω)∗ ≥ 0 for all ω at which
Z(jω) is finite, and any poles of Z(s) on the imaginary axis or at infinity
are simple and have a positive residue.
MTNS 2006, Kyoto, 24 July, 2006 19
Passive Network Synthesis Revisited Malcolm C. Smith
Passivity Defined
Definition. A network is passive if for all admissible v, i which are square
integrable on (−∞, T ], ∫ T
−∞v(t)i(t) dt ≥ 0.
Proposition. Consider a one-port electrical network for which the impedance
Z(s) exists and is real-rational. The network is passive if and only if Z(s) is
positive-real.
R.W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966.
B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, 1973.
MTNS 2006, Kyoto, 24 July, 2006 20
Passive Network Synthesis Revisited Malcolm C. Smith
O. Brune showed that any (ratio-
nal) positive-real function could
be realised as the impedance
or admittance of a network
comprising resistors, capacitors,
inductors and transformers.
(1931)
MTNS 2006, Kyoto, 24 July, 2006 21
Passive Network Synthesis Revisited Malcolm C. Smith
Minimum functions
A minimum function Z(s) is a positive-real function with no poles or zeros
on jR ∪ ∞ and with the real part of Z(jω) equal to 0 at one or more
frequencies.PSfrag replacements
ReZ(jω)
0 ωω1 ω2
MTNS 2006, Kyoto, 24 July, 2006 22
Passive Network Synthesis Revisited Malcolm C. Smith
Foster preamble for a positive-real Z(s)
Removal of poles/zeros on jR ∪ ∞. e.g.
s2 + s+ 1
s+ 1= s+
1
s+ 1↑
lossless
↓s2 + 1
s2 + 2s+ 1=
(2s
s2 + 1+ 1
)−1
Can always reduce a positive-real Z(s) to a minimum function.
MTNS 2006, Kyoto, 24 July, 2006 23
Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle
Let Z(s) be a minimum function with Z(jω1) = jX1 (ω1 > 0).
Write L1 = X1/ω1 and Z1(s) = Z(s)− L1s.
Case 1. (L1 < 0)
PSfrag replacements
Z(s) Z1(s)
L1 < 0
(negative inductor!)
Z1(s) is positive-real. Let Y1(s) = 1/Z1(s). Therefore, we can write
Y2(s) = Y1(s)− 2K1s
s2 + ω21
for some K1 > 0.
MTNS 2006, Kyoto, 24 July, 2006 24
Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle (cont.)
Then:
PSfrag replacements
Z(s) Z2(s)
L1 < 0
L2 > 0
C2 > 0
where L2 = 1/2K1, C2 = 2K1/ω21 and Z2 = 1/Y2.
Straightforward calculation shows that
Z2(s) = sL3+ Z3(s)↑
proper
where L3 = −L1/(1 + 2K1L1). Since Z2(s) is positive-real, L3 > 0 and Z3(s)
is positive-real.
MTNS 2006, Kyoto, 24 July, 2006 25
Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle (cont.)
Then:
PSfrag replacements
Z(s) Z3(s)
L1 < 0
L2 > 0
L3 > 0
C2 > 0
To remove negative inductor:
PSfrag replacements
L1 L3M
Lp L2Ls
Lp = L1 + L2
Ls = L2 + L3
M = L2
Some algebra shows that: Lp, Ls > 0 and M2
LpLs= 1 (unity coupling
coefficient).
MTNS 2006, Kyoto, 24 July, 2006 26
Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle (cont.)
Realisation for completed cycle:
PSfrag replacements
M
Lp LsZ(s) Z3(s)
C2
MTNS 2006, Kyoto, 24 July, 2006 27
Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle (cont.)
Case 2. (L1 > 0). As before Z1(s) = Z(s)− L1s
PSfrag replacements
Z(s) Z1(s)
L1 > 0
(no need for
negative inductor!)
Problem: Z1(s) is not positive-real!
Let’s press on and hope for the best!!
As before let Y1 = 1/Z1 and write
Y2(s) = Y1(s)− 2K1s
s2 + ω21
.
Despite the fact that Y1 is not positive-real we can show that K1 > 0.
MTNS 2006, Kyoto, 24 July, 2006 28
Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle (cont.)
Hence:
PSfrag replacements
Z(s) Z2(s)
L1 > 0
L2 > 0
C2 > 0
But still Z2(s) is not positive-real. Again we can check that
Z2(s) = sL3+ Z3(s)↑
proper
where L3 = −L1/(1 + 2K1L1).
This time L3 < 0 and Z3(s) is positive-real.
MTNS 2006, Kyoto, 24 July, 2006 29
Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle (cont.)
So:
PSfrag replacements
Z(s) Z3(s)
L1 > 0
L2 > 0
L3 < 0
C2 > 0
As before we can transform to:
PSfrag replacements
M
Lp LsZ(s) Z3(s)
C2
where Lp, Ls > 0 and M2
LpLs= 1 (unity coupling coefficient).
MTNS 2006, Kyoto, 24 July, 2006 30
Passive Network Synthesis Revisited Malcolm C. Smith
R. Bott and R.J. Duffin showed
that transformers were un-
necessary in the synthesis of
positive-real functions. (1949)
MTNS 2006, Kyoto, 24 July, 2006 31
Passive Network Synthesis Revisited Malcolm C. Smith
Richards’s transformation
Theorem. If Z(s) is positive-real then
R(s) =kZ(s)− sZ(k)
kZ(k)− sZ(s)
is positive-real for any k > 0.
Proof.
Z(s) is p.r. ⇒ Y (s) =Z(s)− Z(k)
Z(s) + Z(k)is b.r. and Y (k) = 0
⇒ Y ′(s) =k + s
k − sY (s) is b.r.
⇒ Z ′(s) =1 + Y ′(s)1− Y ′(s) is p.r.
R(s) = Z ′(s) after simplification.
MTNS 2006, Kyoto, 24 July, 2006 32
Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.)
Idea: use Richards’s transformation to eliminate transformers from Brune
cycle.
As before, let Z(s) be a minimum function with Z(jω1) = jX1 (ω1 > 0).
Write L1 = X1/ω1.
Case 1. (L1 > 0)
Since Z(s) is a minimum function we can always find a k s.t. L1 = Z(k)/k.
Therefore:
R(s) =kZ(s)− sZ(k)
kZ(k)− sZ(s)
has a zero at s = jω1.
MTNS 2006, Kyoto, 24 July, 2006 33
Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.)
We now write:
Z(s) =kZ(k)R(s) + Z(k)s
k + sR(s)
=kZ(k)R(s)
k + sR(s)+
Z(k)s
k + sR(s)
=1
1Z(k)R(s) + s
kZ(k)
+1
kZ(k)s + R(s)
Z(k)
.
MTNS 2006, Kyoto, 24 July, 2006 34
Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.)
Z(s) =1
1Z(k)R(s) + s
kZ(k)
+1
kZ(k)s + R(s)
Z(k)
PSfrag replacements
Z(k)k
Z(s)
kZ(k) Z(k)R(s)
Z(k)R(s)
MTNS 2006, Kyoto, 24 July, 2006 35
Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.)
We can write: 1Z(k)R(s) = const× s
s2+ω21
+ 1R1(s) etc.
PSfrag replacements
R1
R2
Z(s)
MTNS 2006, Kyoto, 24 July, 2006 36
Passive Network Synthesis Revisited Malcolm C. Smith
Example — restricted degree
Proposition. Consider the real-rational function
Yb(s) = ka0s
2 + a1s+ 1
s(d0s2 + d1s+ 1)
where d0, d1 ≥ 0 and k > 0. Then Yb(s) is positive real if only if the following
three inequalities hold:
β1 := a0d1 − a1d0 ≥ 0,
β2 := a0 − d0 ≥ 0,
β3 := a1 − d1 ≥ 0.
MTNS 2006, Kyoto, 24 July, 2006 37
Passive Network Synthesis Revisited Malcolm C. Smith
Brune Realisation Procedure for Yb(s)
Foster preamble always sufficient to complete the realisation if β1, β2 > 0.
(No Brune or Bott-Duffin cycle is required).
A continued fraction expansion is obtained:
Yb(s) = ka0s
2 + a1s+ 1
s(d0s2 + d1s+ 1)
=k
s+
1s
kb+
1
c3 +1
1
c4+
1
b2s
where kb =kβ2
d0, c3 = kβ3, c4 =
kβ4
β1, b2 =
kβ4
β2and
β4 := β22 − β1β3.
PSfrag replacementsk
c3
b2
kb
c4
MTNS 2006, Kyoto, 24 July, 2006 38
Passive Network Synthesis Revisited Malcolm C. Smith
Darlington Synthesis
Realisation in Darlington form:
a lossless two-port terminated
in a single resistor.
PSfrag replacements
I1 I2
V1 V2
R ΩZ1(s) Lossless
network
For a lossless two-port with impedance:
Z =
Z11 Z12
Z12 Z22
we find
Z1(s) = Z11R−1(Z11Z22 − Z2
12)/Z11 + 1
R−1Z22 + 1.
MTNS 2006, Kyoto, 24 July, 2006 39
Passive Network Synthesis Revisited Malcolm C. Smith
Writing
Z1 =m1 + n1
m2 + n2=
n1
m2
m1/n1 + 1
n2/m2 + 1,
where m1, m2 are polynomials of even powers of s and n1, n2 are polynomials
of odd powers of s, suggests the identification:
Z11 =n1
m2, Z22 = R
n2
m2, Z12 =
√R
√n1n2 −m1m2
m2.
Augmentation factors are necessary to ensure a rational square root.
Once Z(s) has been found, we then write:
Z(s) = sC1 +s
s2 + α2C2 + · · ·
where C1 and C2 are non-negative definite constant matrices.
MTNS 2006, Kyoto, 24 July, 2006 40
Passive Network Synthesis Revisited Malcolm C. Smith
Darlington Synthesis (cont.)
Each term in the sum is realised in the form of a T-circuit and a series
connection of all the elementary two-ports is then made:
PSfrag replacements
X1(s) X2(s)
X3(s)
1 : n
Ideal
Network 1Network 2
I = 0
PSfrag replacements
X1(s)
X2(s)
X3(s)
1 : n
Ideal
Network 1
Network 2
I = 0
MTNS 2006, Kyoto, 24 July, 2006 41
Passive Network Synthesis Revisited Malcolm C. Smith
Electrical and mechanical realisations of the admittance Yb(s)
PSfrag replacements
k2
k3
λ1
λ2
k1
b
c
k−12 H
k−13 H
1: − ρ
k−11 H b F
R1 Ω
PSfrag replacements
k2k3
λ1λ2
k1
b c
k−12 H
k−13 H
1: − ρk−1
1 H
b F
R1 Ω
MTNS 2006, Kyoto, 24 July, 2006 42
Passive Network Synthesis Revisited Malcolm C. Smith
Minimum reactance synthesis
PSfrag replacementsZ(s)
Nondynamic
Network
X
sL1
1sC1
Let L1 = · · · = C1 = · · · = 1. If
M =
M11 M12
M21 M22
is the hybrid matrix of X, i.e. v1
i2
= M
i1
v2
,
then
Z(s) = M11−M12(sI+M22)−1M21.
D.C. Youla and P. Tissi, “N-Port Synthesis via Reactance Extraction, Part I”, IEEEInternational Convention Record, 183–205, 1966.
MTNS 2006, Kyoto, 24 July, 2006 43
Passive Network Synthesis Revisited Malcolm C. Smith
Minimum reactance synthesis
Conversely, if we can find a state-space realisation Z(s) = C(sI −A)−1B +D
such that the constant matrix
M =
D −CB −A
has the properties
M +M ′ ≥ 0,
diagI,ΣM = MdiagI,Σ
where Σ is a diagonal matrix with diagonal entries +1 or −1. Then M is the
hybrid matrix of the nondynamic network terminated with inductors or
capacitors, which realises Z(s).
A construction is possible using the positive-real lemma and matrix
factorisations.
B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, 1973.
MTNS 2006, Kyoto, 24 July, 2006 44
Passive Network Synthesis Revisited Malcolm C. Smith
Synthesis of resistive n-ports
Let R be a symmetric n× n matrix.
A necessary and sufficient condition for R to be realisable as the driving-point
impedance of a network comprising resistors and transformers only is that it
is non-negative definite.
No necessary and sufficient condition is known in the case that transformers
are not available.
A general necessary condition is known: that the matrix is paramount.1
A matrix is defined to be the paramount if each principal minor of the matrix
is not less than the absolute value of any minor built from the same rows.
It is also known that paramountcy is sufficient for the case of n ≤ 3.2
1I. Cederbaum, “Conditions for the Impedance and Admittance Matrices of n-ports without
Ideal Transformers”, IEE Monograph No. 276R, 245–251, 1958.2P. Slepian and L. Weinberg, ”Synthesis applications of paramount and dominant matrices”,
Proc. National Electron. Conf., vol. 14, Chicago, Illinois, Oct. 611-630, 1958.
MTNS 2006, Kyoto, 24 July, 2006 45
Passive Network Synthesis Revisited Malcolm C. Smith
Simple Suspension Struts
PSfrag replacements
b
kb
k c
k1
k2
PSfrag replacements
bkb
k c
k1
k2
PSfrag replacements
b
kb k
c
k1
k2
PSfrag replacements
b
kb k
ck1
k2
Layout S1 Layout S2 Layout S3 Layout S4
parallel series
MTNS 2006, Kyoto, 24 July, 2006 46
Passive Network Synthesis Revisited Malcolm C. Smith
Performance Measures
Assume:
Road Profile Spectrum = κ|n|−2 (m3/cycle)
where κ = 5× 10−7 m3cycle−1 = road roughness parameter. Define:
J1 = E[z2s(t)
]ride comfort
= r.m.s. body vertical acceleration
MTNS 2006, Kyoto, 24 July, 2006 47
Passive Network Synthesis Revisited Malcolm C. Smith
Optimisation of J1 (ride comfort)
1 2 3 4 5 6 7 8 9 10 11 120.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
PSfrag replacements
static stiffness
J1
% J11 2 3 4 5 6 7 8 9 10 11 12
0
2
4
6
8
10
12
14
16
18
PSfrag replacements
static stiffness
J1
%J
1
(a) Optimal J1 (b) Percentage improvement in J1
Key: layout S1 (bold), layout S2 (dashed), layout S3 (dot-dashed), and
layout S4 (solid).
M.C. Smith and F-C. Wang, 2004, Performance Benefits in Passive Vehicle Sus-
pensions Employing Inerters, Vehicle System Dynamics, 42, 235–257.
MTNS 2006, Kyoto, 24 July, 2006 48
Passive Network Synthesis Revisited Malcolm C. Smith
Control synthesis formulation
PSfrag replacements
F
F
Fs
K(s)
zr
zs
zumu
kt
ms
ks
PSfrag replacements
F
FsK(s)
zr
zs
zu
mu
kt
ms
ks zwG(s)
K(s)
F zs − zu
Ride comfort: w = zr, z = zs
Performance measure: J1 = const.× ‖Tzr→szs‖2
MTNS 2006, Kyoto, 24 July, 2006 49
Passive Network Synthesis Revisited Malcolm C. Smith
Bilinear Matrix Inequality (BMI) formulation
Let K(s) = Ck(sI − Ak)−1Bk +Dk and Tzr→szs = Ccl(sI −Acl)−1Bcl.
Theorem. There exists a positive real controller K(s) such that
‖Tzr→szs‖2 < ν and Acl is stable, if and only if the following problem is
feasible for some Xcl > 0, Xk > 0, Q, ν2 and Ak, Bk, Ck, Dk of compatible
dimensions:A
TclXcl +XclAcl XclBcl
BTclXcl −I
< 0,
Xcl CTcl
Ccl Q
> 0, tr(Q) < ν2,
A
TkXk +XkAk XkBk − CTkBTk Xk − Ck −DT
k −Dk
< 0.
C. Papageorgiou and M.C. Smith, 2006, Positive real synthesis using matrix inequalities for mechanical
networks: application to vehicle suspension, IEEE Trans. on Contr. Syst. Tech., 14, 423–435.
MTNS 2006, Kyoto, 24 July, 2006 50
Passive Network Synthesis Revisited Malcolm C. Smith
A special problem
What class of positive-real functions Z(s) can be realised using one damper,
one inerter, any number of springs and no transformers?
PSfrag replacements
Z(s) X
c
b
Leads to the question: when can X be realised as a network of springs?
MTNS 2006, Kyoto, 24 July, 2006 51
Passive Network Synthesis Revisited Malcolm C. Smith
Theorem. Let
Y (s) =(R2R3 −R2
6) s3 +R3 s2 +R2 s+ 1
s(detRs3 + (R1R3 −R25) s2 + (R1R2 −R2
4) s+R1), (1)
where R :=
R1 R4 R5
R4 R2 R6
R5 R6 R3
is non-negative definite.
A positive-real function Y (s) can be realised as the driving-point admittance
of a network comprising one damper, one inerter, any number of springs and
no transformers if and only if Y (s) can be written in the form of (1) and there
exists an invertible diagonal matrix D = diag1, x, y such that DRD is
paramount.
An explicit set of inequalities can be found which are necessary and sufficient
for the existence of x and y.
M.Z.Q. Chen and M.C. Smith, Mechanical networks comprising one damper and one inerter, in preparation.
MTNS 2006, Kyoto, 24 July, 2006 52
Passive Network Synthesis Revisited Malcolm C. Smith
Collaboration with Imperial College
Application to motorcycle stability.
At high speed motorcycles can experience significant steering instabilities.
Observe: Paul Orritt at the 1999 Manx Grand Prix
MTNS 2006, Kyoto, 24 July, 2006 53
Passive Network Synthesis Revisited Malcolm C. Smith
Weave and Wobble oscillations
Steering dampers improve wobble (6–9 Hz) and worsen weave (2–4 Hz).
Simulations show that steering inerters have, roughly, the opposite effect to
the damper. Root-loci (with speed the varied parameter):
Steering damper root-locus Steering inerter root-locus
−18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 40
10
20
30
40
50
60
PSfrag replacements
Real Axis
Imag
inar
yA
xis
wobble
weave
6
−15 −10 −5 0 50
10
20
30
40
50
60
PSfrag replacements
Real AxisIm
agin
ary
Axi
s
wobble
weave
6
Can the advantages be combined?
S. Evangelou, D.J.N. Limebeer, R.S. Sharp and M.C. Smith, 2006, Steering compensation for high-
performance motorcycles, Transactions of ASME, J. of Applied Mechanics, 73, to appear.
MTNS 2006, Kyoto, 24 July, 2006 54
Passive Network Synthesis Revisited Malcolm C. Smith
Solution — a steering compensator
... consisting of a network of dampers, inerters and springs.
Needs to behave like an inerter at weave frequencies and like a damper at
wobble frequencies.
PSfrag replacements
damper
inerter
realimaginary
forbidden
PSfrag replacements
damper
inerter
realimaginary
forbidden
Prototype designed by N.E. Houghton and manufactured in the Cambridge University
Engineering Department.
MTNS 2006, Kyoto, 24 July, 2006 55
Passive Network Synthesis Revisited Malcolm C. Smith
Conclusion
• A new mechanical element called the “inerter” was introduced which is
the true network dual of the spring.
• The inerter allows classical electrical network synthesis to be mapped
exactly onto mechanical networks.
• Applications of the inerter: vehicle suspension, motorcycle steering and
vibration absorption.
• Economy of realisation is an important problem for mechanical network
synthesis.
• The problem of minimal realisation of positive-real functions remains
unsolved.
MTNS 2006, Kyoto, 24 July, 2006 56
Passive Network Synthesis Revisited Malcolm C. Smith
Acknowledgements
Research students Michael Chen, Frank Scheibe
Design Engineer Neil Houghton
Technicians Alistair Ross, Barry Puddifoot, John Beavis
Collaborations Fu-Cheng Wang (National Taiwan University)
Christos Papageorgiou (University of Cyprus)
David Limebeer, Simos Evangelou, Robin Sharp
(Imperial College)
MTNS 2006, Kyoto, 24 July, 2006 57