Download Passive Network Synthesis Revisited Malcolm C. Smith Department

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
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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
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Passive Network Synthesis Revisited
Malcolm C. Smith
Quarter-car Vehicle Model (conventional suspension)
load
PSfrag replacements
disturbances
ms
spring
damper
mu
kt
tyre
road
disturbances
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
The Most General Passive Vehicle Suspension
ms
Z(s)
Replace the spring and damper with a
general positive-real impedance Z(s).
ag replacements
mu
zr
But is Z(s) physically realisable?
kt
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Electrical-Mechanical Analogies
1. Force-Voltage Analogy.
voltage
current
↔
↔
force
velocity
Oldest analogy historically, cf. electromotive force.
2. Force-Current Analogy.
current
voltage
electrical ground
↔
↔
↔
force
velocity
mechanical ground
Independently proposed by: Darrieus (1929), Hähnle (1932), Firestone (1933).
Respects circuit “topology”, e.g. terminals, through- and across-variables.
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Standard Element Correspondences (Force-Current Analogy)
v = Ri
resistor
di
v = L dt
inductor
C dv
dt = i
capacitor
↔
damper
spring
kv =
↔
mass
m dv
dt = F
↔
i
PSfrag replacements
cv = F
dF
dt
i
v2
v1
Electrical
F
F
v2 Mechanical
v1
What are the terminals of the mass element?
MTNS 2006, Kyoto, 24 July, 2006
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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
PSfrag replacements
for the mass element:
F
v2
MTNS 2006, Kyoto, 24 July, 2006
v1 = 0
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Passive Network Synthesis Revisited
Malcolm C. Smith
Table of usual correspondences
Electrical
Mechanical
F
i
F
i
v2
v2
F
•
v2
v1
spring
inductor
i
v1 = 0
F
i
v2
capacitor
i
i
v2
MTNS 2006, Kyoto, 24 July, 2006
v1
v1
mass
F
v2
v1
damper
v1
resistor
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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 proportional to the applied force?
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
One method of realisation
rack
pinions
PSfrag replacements
terminal 2
gear
flywheel
terminal 1
Suppose the flywheel of mass m rotates by α radians per meter of relative
displacement between the terminals. Then:
F = (mα2 ) (v̇2 − v̇1 )
(Assumes mass of gears, housing etc is negligible.)
MTNS 2006, Kyoto, 24 July, 2006
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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(v̇2 − v̇1 ).
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.
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
A new correspondence for network synthesis
Electrical
Mechanical
F
v2
dF
dt
F
k
s
i
spring
di
dt
•F Y (s) = bs
i
F
v2
F =
i
v2
v1
Y (s) =
1
Ls
v1
= k(v2 − v1 )
•
Y (s) =
v1
d(v2 −v1 )
b dt
F
F
v2
inerter
Y (s) = c
=
− v1 ) inductor
i
v2
v1
i = C d(v2dt−v1 )
i
i
v1
v2
Y (s) = Cs
capacitor
Y (s) =
1
R
v1
F = c(v2 − v1 )
damper
i=
Y (s) = admittance =
MTNS 2006, Kyoto, 24 July, 2006
1
L (v2
1
R (v2
− v1 )
resistor
1
impedance
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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
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Passive Network Synthesis Revisited
Malcolm C. Smith
Damper-inerter series arrangement
with centring springs
PSfrag replacements
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Passive Network Synthesis Revisited
Malcolm C. Smith
Alternative Realisation of the Inerter
PSfrag replacements
screw
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nut flywheel
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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
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Passive Network Synthesis Revisited
Malcolm C. Smith
Electrical equivalent of quarter car model
Fs
g replacements PSfrag replacements
ms
Y (s)
zs
zu
kt−1
żr +
−
mu
Y (s)
mu
ms
Fs
kt
zr
Y (s) = Admittance =
MTNS 2006, Kyoto, 24 July, 2006
Force
Velocity
=
Current
Voltage
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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
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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 ],
Z 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
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Passive Network Synthesis Revisited
Malcolm C. Smith
O. Brune showed that any (rational) positive-real function could
be realised as the impedance
or admittance of a network
comprising resistors, capacitors,
inductors and transformers.
(1931)
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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
MTNS 2006, Kyoto, 24 July, 2006
ω1
ω2
ω
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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
+1
2
s +1
−1
Can always reduce a positive-real Z(s) to a minimum function.
MTNS 2006, Kyoto, 24 July, 2006
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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) − L1 s.
Case 1. (L1 < 0)
L1 < 0
placements
Z(s)
Z1 (s)
(negative inductor!)
Z1 (s) is positive-real. Let Y1 (s) = 1/Z1 (s). Therefore, we can write
2K1 s
Y2 (s) = Y1 (s) − 2
s + ω12
for some K1 > 0.
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
PSfrag
replacements
The Brune
cycle (cont.)
Then:
L1 < 0
L2 > 0
Z(s)
Z2 (s)
C2 > 0
where L2 = 1/2K1 , C2 = 2K1 /ω12 and Z2 = 1/Y2 .
Straightforward calculation shows that
Z2 (s) = sL3 +
Z3 (s)
↑
proper
where L3 = −L1 /(1 + 2K1 L1 ). Since Z2 (s) is positive-real, L3 > 0 and Z3 (s)
is positive-real.
MTNS 2006, Kyoto, 24 July, 2006
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PSfrag
replacements
Passive Network
Synthesis Revisited
Malcolm C. Smith
The Brune cycle (cont.)
Then:
L1 < 0
L3 > 0
L2 > 0
Z(s)
Z3 (s)
C2 > 0
placements
To remove negative inductor:
L1
M
Lp
Ls
L3
L2
Some algebra shows that: Lp , Ls > 0 and
coefficient).
MTNS 2006, Kyoto, 24 July, 2006
Lp = L 1 + L 2
Ls = L 2 + L 3
M = L2
M2
Lp Ls
= 1 (unity coupling
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Passive Network Synthesis Revisited
Malcolm C. Smith
PSfrag replacements
The Brune cycle (cont.)
Realisation for completed cycle:
M
Z(s)
Lp
Ls
Z3 (s)
C2
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
The Brune cycle (cont.)
Case 2. (L1 > 0). As before Z1 (s) = Z(s) − L1 s
g replacements
L1 > 0
Z(s)
Z1 (s)
(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) −
2K1 s
.
2
2
s + ω1
Despite the fact that Y1 is not positive-real we can show that K1 > 0.
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
PSfrag
replacements
The Brune
cycle (cont.)
Hence:
L1 > 0
L2 > 0
Z(s)
Z2 (s)
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 + 2K1 L1 ).
This time L3 < 0 and Z3 (s) is positive-real.
MTNS 2006, Kyoto, 24 July, 2006
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PSfrag
replacements
Passive Network
Synthesis Revisited
Malcolm C. Smith
The Brune cycle (cont.)
So:
L1 > 0
L3 < 0
L2 > 0
Z(s)
Z3 (s)
C2 > 0
PSfrag replacements
As before we can transform to:
M
Lp
Z(s)
Ls
Z3 (s)
C2
where Lp , Ls > 0 and
M2
Lp Ls
MTNS 2006, Kyoto, 24 July, 2006
= 1 (unity coupling coefficient).
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Passive Network Synthesis Revisited
Malcolm C. Smith
R. Bott and R.J. Duffin showed
that transformers were unnecessary in the synthesis of
positive-real functions. (1949)
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Richards’s transformation
Theorem. If Z(s) is positive-real then
kZ(s) − sZ(k)
R(s) =
kZ(k) − sZ(s)
is positive-real for any k > 0.
Proof.
Z(s) − Z(k)
is b.r. and Y (k) = 0
Z(s) is p.r. ⇒ Y (s) =
Z(s) + Z(k)
k+s
0
Y (s) is b.r.
⇒ Y (s) =
k−s
0
1
+
Y
(s)
0
is p.r.
⇒ Z (s) =
1 − Y 0 (s)
R(s) = Z 0 (s) after simplification.
MTNS 2006, Kyoto, 24 July, 2006
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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:
has a zero at s = jω1 .
MTNS 2006, Kyoto, 24 July, 2006
kZ(s) − sZ(k)
R(s) =
kZ(k) − sZ(s)
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Passive Network Synthesis Revisited
Malcolm C. Smith
Bott-Duffin construction (cont.)
We now write:
Z(s) =
=
=
MTNS 2006, Kyoto, 24 July, 2006
kZ(k)R(s) + Z(k)s
k + sR(s)
Z(k)s
kZ(k)R(s)
+
k + sR(s)
k + sR(s)
1
1
+
1
s
k
Z(k)R(s) + kZ(k)
Z(k)s +
R(s)
Z(k)
.
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Passive Network Synthesis Revisited
Malcolm C. Smith
Bott-Duffin construction (cont.)
Z(s) =
1
1
Z(k)R(s) +
kZ(k)
s
kZ(k)
+
1
k
Z(k)s
+
R(s)
Z(k)
Z(k)R(s)
PSfrag replacements
Z(s)
Z(k)
k
MTNS 2006, Kyoto, 24 July, 2006
Z(k)
R(s)
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Passive Network Synthesis Revisited
Malcolm C. Smith
Bott-Duffin construction (cont.)
We can write:
1
Z(k)R(s)
= const ×
s
s2 +ω12
+
1
R1 (s)
etc.
R1
Z(s)
PSfrag replacements
R2
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Example — restricted degree
Proposition. Consider the real-rational function
a0 s2 + a 1 s + 1
Yb (s) = k
s(d0 s2 + d1 s + 1)
where d0 , d1 ≥ 0 and k > 0. Then Yb (s) is positive real if only if the following
three inequalities hold:
β1
β2
β3
MTNS 2006, Kyoto, 24 July, 2006
:= a0 d1 − a1 d0 ≥ 0,
:= a0 − d0 ≥ 0,
:= a1 − d1 ≥ 0.
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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:
a0 s2 + a 1 s + 1
Yb (s) = k
s(d0 s2 + d1 s + 1)
1
k
+
=
1
s
s
k
+
PSfrag
replacements
1
kb
c3 +
1
1
+
c4
b2 s
kβ2
kβ4
kβ4
, c3 = kβ3 , c4 =
, b2 =
and
d0
β1
β2
2
β4 := β2 − β1 β3 .
where kb =
MTNS 2006, Kyoto, 24 July, 2006
kb
c4
c3
b2
38
PSfrag replacements
Passive Network Synthesis Revisited
Malcolm C. Smith
Darlington Synthesis
I1
Realisation in Darlington form:
a lossless two-port terminated
in a single resistor.
Z1 (s) V1
For a lossless two-port with impedance:

Z11
Z=
Z12
we find
I2
Z12
Z22
Lossless
network
V2
RΩ


2
R−1 (Z11 Z22 − Z12
)/Z11 + 1
.
Z1 (s) = Z11
R−1 Z22 + 1
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Writing
m 1 + n1
n1 m1 /n1 + 1
=
,
Z1 =
m 2 + n2
m2 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:
√
√
n1
n2
n 1 n2 − m 1 m 2
, Z22 = R
, Z12 = R
.
Z11 =
m2
m2
m2
Augmentation factors are necessary to ensure a rational square root.
Once Z(s) has been found, we then write:
s
C2 + · · ·
Z(s) = sC1 + 2
s + α2
where C1 and C2 are non-negative definite constant matrices.
MTNS 2006, Kyoto, 24 July, 2006
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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:
lacements
PSfrag replacements
X1 (s)
X2 (s)
1: n
I=0
Ideal
etwork 1
etwork 2
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Network 1
1: n
X3 (s)
I=0
X1 (s)
X2 (s)
X3 (s)
Network 2
Ideal
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Passive Network Synthesis Revisited
ag replacements
Malcolm C. Smith
PSfrag replacements
k2
k3
Electrical
and mechanical realisations of the admittance Yb (s)
λ1
λ2
k2−1 H
1: − ρ
k1
b
k3−1c
k1−1 H
MTNS 2006, Kyoto, 24 July, 2006
k3
k2
H
bF
k2−1 H
−1
k
R1 Ω 3 H
1: − ρ
k1−1 Hk1
λ2
λ1
b
c
bF
R1 Ω
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Passive Network Synthesis Revisited
Malcolm C. Smith
Minimum reactance synthesis
sL1
eplacements
Z(s)
Nondynamic
Network
X
1
sC1
Let L1 = · · · = C1 = · · · = 1. If


M11 M12


M=
M21 M22
is the hybrid matrix of X, i.e.




i
v
 1  = M  1 ,
i2
v2
then
Z(s) = M11 −M12 (sI+M22 )−1 M21 .
D.C. Youla and P. Tissi, “N-Port Synthesis via Reactance Extraction, Part I”, IEEE
International Convention Record, 183–205, 1966.
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Minimum reactance synthesis
Conversely, if we can find a state-space realisation Z(s) = C(sI − A)−1 B + D
such that the constant matrix


D −C


M=
B −A
has the properties
M + M0
diag{I, Σ}M
≥ 0,
= M diag{I, 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
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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
1
I. Cederbaum, “Conditions for the Impedance and Admittance Matrices of n-ports without
Ideal Transformers”, IEE Monograph No. 276R, 245–251, 1958.
2
P. 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
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Passive Network Synthesis Revisited
Malcolm C. Smith
Simple Suspension Struts
cementsPSfrag replacements
b
kb
PSfrag replacements
PSfrag replacements
c
kb
k
c
k
c
b
kb k
k1
kb k
b
k2
k1
k2
k1
k2
Layout S1
b
k1
k2
Layout S2
parallel
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c
Layout S3
Layout S4
series
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Passive Network Synthesis Revisited
Malcolm C. Smith
Performance Measures
Assume:
Road Profile Spectrum = κ|n|−2
(m3 /cycle)
where κ = 5 × 10−7 m3 cycle−1 = road roughness parameter. Define:
J1
2 = E z̈s (t)
ride comfort
= r.m.s. body vertical acceleration
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Optimisation of J1 (ride comfort)
2.6
18
2.4
16
2.2
14
2
12
replacements
% J1
J1
1.8
1.6
PSfrag replacements
1.4
4
1
J1
0.8
0.6
8
6
1.2
% J1
10
1
2
3
4
5
6
7
8
9
static stiffness
(a) Optimal J1
10
11
12
2
0
1
2
3
4
5
6
7
8
9
10
11
12
static stiffness
(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 Suspensions Employing Inerters, Vehicle System Dynamics, 42, 235–257.
MTNS 2006, Kyoto, 24 July, 2006
48
Fs
Passive Network Synthesis Revisited
Malcolm C. Smith
K(s)
Control synthesis formulation zr
zs
zu
Fs
PSfrag replacements
mu
kt
ms
zs m s
F
ks w
K(s)
ks
z
G(s)
F
mu
zu
żs − żu
F
K(s)
kt
zr
Ride comfort: w = zr , z = żs
Performance measure: J1 = const. × kTẑr →sẑs k2
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Bilinear Matrix Inequality (BMI) formulation
Let K(s) = Ck (sI − Ak )−1 Bk + Dk and Tẑr →sẑs = Ccl (sI − Acl )−1 Bcl .
Theorem. There exists a positive real controller K(s) such that
kTẑr →sẑs k2 < ν and Acl is stable, if and only if the following problem is
feasible for some Xcl > 0, Xk > 0, Q, ν 2 and Ak , Bk , Ck , Dk of compatible
dimensions:




T
ATcl Xcl + Xcl Acl Xcl Bcl
Xcl Ccl
< 0, 
> 0, tr(Q) < ν 2 ,

T
Bcl
Xcl
−I
Ccl Q


ATk Xk + Xk Ak Xk Bk − CkT
 < 0.

BkT Xk − Ck
−DkT − Dk
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
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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
c
Z(s)
X
b
Leads to the question: when can X be realised as a network of springs?
MTNS 2006, Kyoto, 24 July, 2006
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Passive Network Synthesis Revisited
Malcolm C. Smith
Theorem. Let
(R2 R3 − R62 ) s3 + R3 s2 + R2 s + 1
,
Y (s) =
s(det R s3 + (R1 R3 − R52 ) s2 + (R1 R2 − R42 ) s + R1 )


R1 R4 R5



where R :=  R4 R2 R6 
 is non-negative definite.
R5 R6 R3
(1)
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 = diag{1, 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
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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
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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
60
Steering inerter root-locus
60
6
6
wobble
50
50
40
40
Imaginary Axis
Imaginary Axis
wobble
30
weave
20
0
−18
weave
20
10
PSfrag replacements
30
10
PSfrag replacements
−16
−14
−12
−10
−8
−6
Real Axis
−4
−2
0
2
4
0
−15
−10
−5
Real Axis
0
5
Can the advantages be combined?
S. Evangelou, D.J.N. Limebeer, R.S. Sharp and M.C. Smith, 2006, Steering compensation for highperformance motorcycles, Transactions of ASME, J. of Applied Mechanics, 73, to appear.
MTNS 2006, Kyoto, 24 July, 2006
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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
PSfrag replacements
damper
damper
real
imaginary
inerter
real
inerter imaginary
forbidden
forbidden
Prototype designed by N.E. Houghton and manufactured in the Cambridge University
Engineering Department.
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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.
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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)
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