realization of a special class of admittances with one ...web.hku.hk/~mzqchen/cdc2012.pdf ·...

Realization of a Special Class of Admittanceswith One Damper and One Inerter

Michael Z. Q. Chen

Department of Mechanical EngineeringThe University of Hong Kong

December 11, 2012

Outline

1 Introduction

2 Problem Formulation

3 Main Results

4 Conclusion

Passive Networks

DefinitionA network is passive if the behaviors of its terminals satisfies:∫ T

−∞I(t)V (t)dt ≥ 0,

for all the physically acceptable current I(t), voltage V (t), and allT .

Classical Electrical Circuit Theory

Passivity is characterised in terms of terminal behaviour (impedanceor admittance functions).

i

i

vElectricalNetwork

Admittance = Y (s) = i(s)/v(s). Impedance = Z (s) = Y (s)−1.

14/46

B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory Approach,Prentice Hall, 1973.

Positive-Real Functions

DefinitionZ (s) is defined to be a positive-real function if Z (s) is analyticand Re Z (s) ≥ 0 in Re [s] > 0.

Vehicle dynamics, engineering thought-experiments and Formula One racing M.C. Smith

Positive-real functions

real

imaginary

forbidden

The admittance (or impedance) of a passive network must be positive-real, i.e.Re(Y(jω))≥ 0 for all ω . (Not too difficult to prove.)

The University of Hong Kong, 13 January 2011 23

A Necessary Condition of Passive Networks

Defining admittance as Y (s) = I(s)/V (s), and impedance asZ (s) = V (s)/I(s), we can obtain the following conclusion.

TheoremThe impedance (admittance) of any passive network must be apositive-real function.

What about the converse?

R.W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966.

Brune’s WorkPassive Network Synthesis without Transformers M.C. Smith

i

iv

PassiveNetwork

forbidden

O. Brune (1931): the driving-point impedance of a linearpassive two-terminal network ispositive-real. Conversely:

Any (rational) positive-real func-tion can be realised using resis-tors, capacitors, inductors andtransformers.

YY Fest 2010 3

In 1931, O. Brune establisheda procedure that can realizeany positive-real function withpassive resistors, inductors,capacitors, and transformers.

O. Brune, “Synthesis of a finite two-terminal networkwhose driving-point impedance is a prescribed functionof frequency,” J. Math. Phys., vol. 10, pp. 191–236,1931.

Minimum Functions

DefinitionA minimum function Z (s) is a positive-real function with no polesand zeros on jR ∪ {∞} and the real part of Z (jω) equal to 0 atone or more frequencies.

Minimum Functions

A minimum function Z (s) is a positive-real function with no poles orzeros on jR∪{∞} and with the real part of Z (jω) equal to 0 at one ormore frequencies.

ReZ (jω)

0 ωω1 ω2

17/46

Foster Preamble for a Positive-Real Function

Removal of poles on jR ∪ {∞}

Z = sL + Z1, L > 0.

Passive Network Synthesis without Transformers M.C. Smith

Foster preamble for a positive-real Z(s)

Removal of poles on jR∪{∞}

Z = sL+Z1, (Z1 proper)L

Z1

Removal of zeros on jR∪{∞}

Z =(

Ass2+ω2 +Y1

)

−1

CL

Y1

Subtract minimum real part

Z = R+Z2R

Z2

Not necessarily a unique process

YY Fest 2010 5

Removal of zeros on jR ∪ {∞}

Z =

(2Kis

s2 + ω2i+ Y1

)−1

, Ki > 0.





Z1


Z =(

Ass2+ω2 +Y1

)

−1

CL

Y1


Z = R+Z2R

Z2


YY Fest 2010 5


Z = R + Z2, R > 0.





Z1


Z =(

Ass2+ω2 +Y1

)

−1

CL

Y1


Z = R+Z2R

Z2


YY Fest 2010 5

Foster Preamble can yield any positive-real function to aminimum one.

The Brune Cycle

Hence for any minimum function Z (s), we havePassive 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

It can be proven that Z3(s) is positive-real, δ[Z3(s)] < δ[Z (s)]and L1L3 < 0. However, the negative inductor can be absorbedin transformers.

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

Lp = L1 + L2 > 0,Ls = L2 + L3 > 0,M = L2 > 0.

Brune Synthesis

Since δ[Z3(s)] < δ[Z (s)] and Z3(s) is positive-real after a Brunecycle. Therefore, we conclude that

Brune Synthesis = Foster Preambles + Brune Cycles

can realize any positive-real function with passive resistors, in-ductors, capacitors and transformers.

The Theorems of Brune (1931) and Bott-Duffin (1949):

Any (rational) positive-real function can be realised as the admittance(or impedance) of a network comprising resistors, capacitors,inductors only.

PositiveReal

Function

CircuitRealisation

constructive procedure

16/46

Bott-Duffin’s WorkPassive 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

In 1949, R. Bott and R. J. Duf-fin established a new realiza-tion procedure for any positive-real function without trans-formers.

R. Bott and R. J. Duffin, “Impedance synthesis withoutuse of transformers,” Journal of Applied Physics, vol. 20,no. 8, pp. 816–816, 1949.

Bott-Duffin Cycle

Theorem (Richards’s Transformation)If Z (s) is positive-real, then R(s) = (kZ (s) − sZ (k))/(kZ (k) −sZ (s)) is positive-real for any k > 0, and δ[R(s)] ≤ δ[Z (s)].


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

δ[R1] = δ[R2] < δ[Z ], whereZ (s) is a minimum positive-real function.

Redundant elements Exist

Bott-Duffin Synthesis

Bott-Duffin Synthesis = Foster Preambles + Bott-Duffin Cyclescan realize any positive-real function using finite number of re-sistors, capacitors, and inductors.


The three linear, passive two-terminal electricalelements

resistor capacitor inductor

Why three?

YY Fest 2010 2

We call the resistor, capacitor, and inductor as the three basictwo-terminal electrical elements.

Passive Mechanical Systems

Force-Current Analogy:


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

Passivity: ∫ T

−∞F (t)v(t)dt ≥ 0

Mechanical Immittances:

Z (s) = v/F , Y (s) = F/v

Passive Mechanical SystemsPassive 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

Z (s) must be positive-real.

But is Z (s) always realizablephysically?

Analogy of Basic Elements (Force-Current Method)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


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


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

The Inerter

DefinitionThe (ideal) inerter is a two-terminal mechanical device with theproperty that the equal and opposite force applied at the termi-nals is proportional to the relative acceleration between them,that is, F = bv where v = v1 − v2.

Chapter 1: Introduction 3

terminal 2 terminal 1gear

rack pinions

flywheel

Figure 1.2: Schematic of a mechanical model of an inerter.

k1

k2c

b

(a) Circuit diagram (b) Mechanical realisation

Figure 1.3: Inerter in series with damper with centring springs.

M. C. Smith, “Synthesis of mechanical network: the inerter,” IEEE Trans. Automat. Control, vol. 47, no. 10, pp. 1648–1662, 2002.

Applications of the Inerter

1 Vehicle suspension systems control

2 Motorcycle steering control

3 Building systems control

4 Train suspension systems control

5 ... ... ... (Future)

Three Basic Passive Mechanical Elements

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

Feature

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

AbstractIn 2008, two articles in Autosport revealed details of a new mechanical suspension component with the name “J-damper” which had entered Formula One Racing and which was delivering significant per-formance gains in handling and grip. From its first mention in the 2007 Formula One “spy scandal” there was much specula-tion about what the J-damper actually was. The Autosport articles revealed that the J-damper was in fact an “inerter” and that its origin lay in academic work on mechanical and electrical circuits at Cam-bridge University. This article aims to pro-vide an overview of the background and origin of the inerter, its application, and its intimate connection with the classical theory of network synthesis.

© LAT PHOTOGRAPHIC

Authorized licensed use limited to: University of Leicester. Downloaded on March 13, 2009 at 06:47 from IEEE Xplore. Restrictions apply.

E lectricalMechanical

spring

inerter

damper

inductor

capacitor

resistor

i

i

ii

i

i

v1v1

v1

v1

v1

v1

v2

v2

v2

v2

v2

v2

FF

FF

FF 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 =

1L (v2 − v1)

Y(s) = C s

i = C d(v2−v1)dt

Y(s) = 1R

i = 1R (v2 − v1)

The theory of passive electricalnetworks can be completely trans-planted into mechanical networks.

M. Z. Q. Chen, C. Papageorgiou, F. Scheibe, F.-C. Wang, and M. C. Smith, “The missing mechanical circuit element,”IEEE Circuits Syst. Mag., vol. 9, no. 1, pp. 10–26, 2009.

Mechanical Networks Synthesis and ExistingProblems

Theory of passive network synthesis

Passive mechanical systems design

The interest of the investigation is

renewed

Redundantelements limit its

practical use

Outline

1 Introduction


3 Main Results

4 Conclusion

A Class of Functions to be Investigated

Consider a class of admittance functions in the form of

Y (s) = ka0s2 + a1s + 1

s(d0s2 + d1s + 1), (1)

where a0, a1, d0, d1 ≥ 0, and k > 0. The mechanical admit-tances of many suspension struts are always of this form.

TheoremY (s) in the form of (1) where a0, a1, d0, d1 ≥ 0, and k > 0 ispositive-real, if and only if

a0d1 − a1d0 ≥ 0, a0 − d0 ≥ 0, a1 − d1 ≥ 0.

[1] M.C. Smith, “Synthesis of mechanical networks: the inerter,” IEEE Trans. Automat. Control, vol. 47, no. 10,pp. 1648–1662, 2002.

[2] M. Z. Q. Chen and M. C. Smith, “A note on tests for positive-real functions,” IEEE Trans. Automat. Control, vol. 54,no. 2, pp. 390–393, 2009.

Realization by Brune

It is firstly pointed out in [1] that any positive-real Y (s) in thisform is realizable asSMITH: SYNTHESIS OF MECHANICAL NETWORKS: THE INERTER 1657

Fig. 13. First realization of the admittance (13).

Fig. 14. Second realization of the admittance (13).

. This corresponds to the simplest suspension strutof a spring in parallel with a damper with admittance

where the damper constant is given by kNsm .Fig. 16 shows the step response fromto with rather lightdamping in evidence. This highlights one of the difficulties withconventional suspension struts which are very stiffly sprung.

2) Design 2: SD Admittance With Multiple Dampers:Forthe same optimization problem, but with as in(12) and , direct searches using the Nelder–Mead sim-plex method led to no improvement on the value of

obtained in Design 1 with one damper. This situation ap-pears to persist for a higher number of dampers as in Fig. 12.Further direct searches in the parameter space converged to-ward a set of pole-zero cancellations, and consequent reductionin McMillan degree in , leaving only the solution obtainedin Design 1. These claims are backed only by computationalevidence, with a formal proof being lacking.

Fig. 15. Plot of damping ratio� versusT andT in Design 1.

Fig. 16. Response ofz to a unit step atz : Design 1 (–) and Design 3 (–�).

3) Design 3: Degree 3 Positive-Real Admittance:For thesame optimization problem, but with the admittance

as in equation (13), and no SD restriction, directsearches using the Nelder–Mead simplex method led to clearimprovements. The following parameters

(27)

gave a value of . The improved damping is demon-strated in Fig. 16 compared to the case of Design 1. The posi-tive-real nature of is illustrated in the Bode plot of Fig. 17,which also clearly shows that this solution is employing phaseadvance. The latter fact proves that is not an SD admit-tance, as is also seen by . The valuesfor the constants in the realizations of Figs. 13 and 14 are givenby

kNsm kNsm

kg kNsm

kNsm kg

These values appear to be within the bounds of practicality.

by Brune Synthesis (Foster Preamble).[1] M. C. Smith, “Synthesis of mechanical networks: the inerter,” IEEE Trans. Automat. Control, vol. 47, no. 10,pp. 1648–1662, 2002.

Realization with One Damper and One InerterChapter 4: Restricted Complexity Synthesis and Paramountcy 41

v1

F1

X

F2

F3

v2

v3

c

b

Figure 4.1: General one-port containing one damper and one inerter.

where

T =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0

0√

c 0

0 0√

b

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(4.4)

and R is non-negative definite. From the expression det(R) = R1R2R3 − R1R26 − R2R2

5 − R3R24 +

2R4R5R6, we note that (4.3) depends on sign(R4R5R6) but not on the individual signs of R4, R5 and

R6 and sign(R4R5R6) will be used to divide the class of Y(s) realisable in the form of Fig. 4.1 into

subclasses in Chapter 5.

If we now suppose a Y(s) in the form of (4.3) is given with R non-negative definite, then

it is a standard fact [12, Chapter 4, pages 173–179] that Y(s) can be realised in the form of Fig. 4.1

with X consisting of springs and transformers (levers), and with b = c = 1. This chapter and

Chapter 5 address the following question: suppose a Y(s) in the form of (4.3) is given, what addi-

tional conditions are necessary for Y(s) to be realisable with one damper, one inerter, any number of

springs but no transformers (levers)? Before further treatment to address this question in Chapter 5,

we will discuss a closely related concept paramountcy and its association with transformerless real-

isation and then present a necessary and sufficient condition for a third-order non-negative definite

matrix to be reducible to a paramount matrix using a diagonal transformation in Section 4.2.

ProblemConsider any positive-real admittance function Y (s) in the form:

Y (s) = ka0s2 + a1s + 1

s(d0s2 + d1s + 1),

where a0, a1, d0, d1 ≥ 0, and k > 0. What is the necessary andsufficient condition for Y (s) to be realizable with one damper,one inerter, and an arbitrary number of springs? What about thecovering configurations?

Outline

1 Introduction


3 Main Results

4 Conclusion

Review of Its Positive-Realness

TheoremConsider a class of admittance functions in the form of

Y (s) = ka0s2 + a1s + 1

s(d0s2 + d1s + 1),

where a0, a1, d0, d1 ≥ 0, and k > 0. Y (s) is positive-real if andonly if

a0d1 − a1d0 ≥ 0,a0 − d0 ≥ 0,a1 − d1 ≥ 0.

A Degenerate Case

Let p(s) = a0s2 + a1s + 1 and q(s) = d0s2 + d1s + 1, thenit is known from [1] that the resultant of p(s) and q(s) can becalculated as

Rk := (a0−d0)2−(a0d1−a1d0)(a1−d1).

k1

1 1

1

cb

1

k2

If Rk = 0, then Y (s) reduces to

Y (s) = kas + 1

s(ds + 1)=

ks+

1ds

k(a−d) +1

k(a−d)

,

where a ≥ d ≥ 0. Hence Y (s) can be realized by a networkconsisting of at most one damper and two springs.

[1] I. Gohberg, P. Lancaster, and L. Rodman, “Positive real matrices,” Indiana University Mathematics Journal, vol. 9,

pp. 71–83, 1960.

Review of a General Conclusion

TheoremA positive-real function Y (s) can be realized as the driving-pointadmittance of a network comprising one damper, one inerter,and arbitrary number of springs but no levers if and only if Y (s)can be written in the form of

Y (s) =(R2R3 − R2

6)s3 + R3s2 + R2s + 1

s(det Rs3 + (R1R3 − R2

5)s2 + (R1R2 − R24)s + R1

) ,where R in the form of

R =

R1 R4 R5R4 R2 R6R5 R6 R3

(2)

is non-negative definite, and satisfies either Condition A or Con-dition B.

Review of a General Conclusion

Condition AThere exists a first- or second-order minors of R being zero.

Condition B1. R4R5R6 < 0;2. R4R5R6 > 0, R1 > (R4R5/R6), R2 > (R4R6/R5) and R3 >

(R5R6/R4);3. R4R5R6 > 0, R3 < (R5R6/R4), and R1R2R3 + R4R5R6−R1R2

6 − R2R25 ≥ 0;

4. R4R5R6 > 0, R2 < (R4R6/R5), and R1R2R3 + R4R5R6−R1R2

6 − R3R24 ≥ 0;

5. R4R5R6 > 0, R1 < (R4R5/R6), and R1R2R3 + R4R5R6−R3R2

4 − R2R25 ≥ 0.

M. Z. Q. Chen and M. C. Smith, “Restricted complexity network realizations for passive mechanical control,” IEEETrans. Automat. Control, vol. 54, no. 10, pp. 2290–2301, 2009.

Covering Configurations2294 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 10, OCTOBER 2009

where as defined in (6) is non-negative definite and there ex-ists an invertible diagonal matrix such that

is paramount.Proof: (Only if.) As in Section III, we can write the

impedance of in the form of (2). Since is realized usingsprings only (no transformers), we claim that the matrix in(4) is paramount. This follows since the results of Section IV onresistive networks carry through in an exactly analogous wayto one-element-kind networks (e.g., comprising capacitors onlyor inductors only without transformers or mutual inductances)[21]. The transformation to (4) as in Section III now providesthe required matrix with the property that isparamount where and .

(If.) If we define and , then isparamount. Using the construction of Tellegen (see Section IV,Fig. 6), we can find a network consisting of six springs and notransformers with impedance matrix equal to . Using this net-work in place of in Fig. 5 provides a driving-point admittancegiven by (3) which is equal to (15) after the same transformationof Section III.

We now combine Lemmas 3, 4, and 5 to obtain the followingmain theorem.

Theorem 6: A positive-real function can be realized asthe driving-point admittance of a network in the form of Fig. 5,where has a well-defined impedance and is realizable withsprings only and , if and only if can be written inthe form of (15) and satisfies the conditions of either Lemma3 or Lemma 4.

In Theorem 7, we provide specific realizations for thein Theorem 6 for all cases where satisfies the conditions ofLemma 4. The realizations are more efficient than would be ob-tained by directly using the construction of Tellegen (see Sec-tion IV, Fig. 6) in that only four springs are needed. This is dueto the fact that Theorem 7 exploits the additional freedom in theparameters and to realize the admittance (15). Alternative re-alizations can also be found which are of similar complexity (see[6], [8]). The singular cases satisfying the conditions of Lemma3 are treated from first principles in Theorem 8 where it is shownthat at most three springs are needed. In comparison, the con-struction of Tellegen does not treat the singular cases explicitly.

Theorem 7: Given in the form of (15) where as de-fined in (6) is non-negative definite and satisfies the conditionsin Lemma 4. Then can be realized with one damper, oneinerter and four springs as one of the following constructions:

i) , can be realized in the form of Fig. 7with

Fig. 7. Network realization of Theorem 7, case (i).

Fig. 8. Network realization of Theorem 7, case (ii).

ii) , ,and , can be realized in the formof Fig. 8 with

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2294 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 10, OCTOBER 2009

where as defined in (6) is non-negative definite and there ex-ists an invertible diagonal matrix such that

is paramount.Proof: (Only if.) As in Section III, we can write the

impedance of in the form of (2). Since is realized usingsprings only (no transformers), we claim that the matrix in(4) is paramount. This follows since the results of Section IV onresistive networks carry through in an exactly analogous wayto one-element-kind networks (e.g., comprising capacitors onlyor inductors only without transformers or mutual inductances)[21]. The transformation to (4) as in Section III now providesthe required matrix with the property that isparamount where and .

(If.) If we define and , then isparamount. Using the construction of Tellegen (see Section IV,Fig. 6), we can find a network consisting of six springs and notransformers with impedance matrix equal to . Using this net-work in place of in Fig. 5 provides a driving-point admittancegiven by (3) which is equal to (15) after the same transformationof Section III.

We now combine Lemmas 3, 4, and 5 to obtain the followingmain theorem.

Theorem 6: A positive-real function can be realized asthe driving-point admittance of a network in the form of Fig. 5,where has a well-defined impedance and is realizable withsprings only and , if and only if can be written inthe form of (15) and satisfies the conditions of either Lemma3 or Lemma 4.

In Theorem 7, we provide specific realizations for thein Theorem 6 for all cases where satisfies the conditions ofLemma 4. The realizations are more efficient than would be ob-tained by directly using the construction of Tellegen (see Sec-tion IV, Fig. 6) in that only four springs are needed. This is dueto the fact that Theorem 7 exploits the additional freedom in theparameters and to realize the admittance (15). Alternative re-alizations can also be found which are of similar complexity (see[6], [8]). The singular cases satisfying the conditions of Lemma3 are treated from first principles in Theorem 8 where it is shownthat at most three springs are needed. In comparison, the con-struction of Tellegen does not treat the singular cases explicitly.

Theorem 7: Given in the form of (15) where as de-fined in (6) is non-negative definite and satisfies the conditionsin Lemma 4. Then can be realized with one damper, oneinerter and four springs as one of the following constructions:

i) , can be realized in the form of Fig. 7with

Fig. 7. Network realization of Theorem 7, case (i).

Fig. 8. Network realization of Theorem 7, case (ii).

ii) , ,and , can be realized in the formof Fig. 8 with


CHEN AND SMITH: RESTRICTED COMPLEXITY NETWORK REALIZATIONS 2295

Fig. 9. Network realization of Theorem 7, case (iii).

iii) , and, can be realized in

the form of Fig. 9 with

iv) , and, can be realized in


Fig. 10. Network realization of Theorem 7, case (iv).

v) , and, can be realized in


Proof: By direct substitution it can be checked that the ad-mittance for each of Figs. 7–11 is equal to (15). These five real-izations cover all the cases in Lemma 4. A direct derivation ofthe formulae is given in [6], [8].

Theorem 8: Given in the form of (15) where as de-fined in (6) is non-negative definite. If one or more of the first-


CHEN AND SMITH: RESTRICTED COMPLEXITY NETWORK REALIZATIONS 2295












2296 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 10, OCTOBER 2009

Fig. 11. Network realization of Theorem 7, case (v).

or second-order minors of is zero, then can be realizedwith at most one damper, one inerter and three springs.

Proof: See Appendix D.

VI. EXAMPLE OF NON-REALIZABILITY

In [18, Theorem 4], it was shown that a real-rational function

(16)

where , and , is positive real if only if thefollowing three inequalities hold:

(17)

(18)

(19)

The classical synthesis procedures of Brune and Darlingtonwere applied to the admittance (16), and the realizations shownin Fig. 12 and Fig. 13 were obtained [18]. It will be observedthat two dampers and one inerter are needed in Fig. 12, while inFig. 13, a transformer (lever) is required. It is interesting to askif the class of admittances defined by (16) can be realized usingone damper, one inerter and no transformers. We will show inthe example below that this is not always possible. First of all,we need to establish the following result.

Theorem 9: Let be any positive-real admittance of theform of (16) with . Then such an admittance canbe written in the form of (4) with non-negative definite and

if and only if

Fig. 12. Brune realization of equation (16).

Fig. 13. Darlington realization of equation (16).

with

Proof: If (16) equals (4) there must be a degree drop (i.e., ) or at least one pole-zero cancel-

lation occurs in (4). Suppose we have a common factor in theform of . By comparing the coefficients, the followingidentities hold:

(20)


More Explicit Form

To make better use of the conclusion, we will convert the generaltheorem to an equivalent but more explicit one, i.e., in the formof

Y (s) =α3s3 + α2s2 + α1s + 1β4s4 + β3s3 + β2s2 + β1s

,

and the conclusions are in terms of αi , βi . Hence the followingrelations are needed.

α3 = R2R3 − R26 , α2 = R3, α1 = R2, β4 = det(R),

β3 = R1R3 − R25 , β2 = R1R2 − R2

4 , β1 = R1.

First Lemma

Lemma 1The matrix R defined as

R :=

R1 R4 R5R4 R2 R6R5 R6 R3

with entries satisfying

α3 = R2R3 − R26 , α2 = R3, α1 = R2, β4 = det(R),

β3 = R1R3 − R25 , β2 = R1R2 − R2

4 , β1 = R1.

is non-negative definite if and only if α1, α2, α3, β1, β2, β3, β4≥ 0.

Second Lemma

Lemma 2The form

Y (s) =(R2R3 − R2

6)s3 + R3s2 + R2s + 1

s(det Rs3 + (R1R3 − R2

5)s2 + (R1R2 − R24)s + R1

)with R non-negative definite and


,

with α1, α2, α3, β1, β2, β3, β4 ≥ 0 can be expressed by eachother with the corresponding coefficients equal if and only if

α3 ≤ α1α2, β3 ≤ α2β1, β2 ≤ α1β1,

(β4 + 2α1α2β1 − β1α3 − α1β3 − α2β2)2

= 4(α1β1 − β2)(α2β1 − β3)(α1α2 − α3).

Third Lemma

Lemma 3Consider a non-negative definite matrix R in the form of (2), andthe variables α1, α2, α3, β1, β2, β3, and β4 satisfy

α3 = R2R3 − R26 , α2 = R3, α1 = R2, β4 = det(R),

β3 = R1R3 − R25 , β2 = R1R2 − R2

4 , β1 = R1.(3)

Let W := 2α1α2β1 + β4−α1β3−α2β2−α3β1, W1 := α1α2 −α3,W2 := α2β1 − β3, and W3 := α1β1 − β2. Then there exists atleast one of the first- or second-order minors of R being zero ifand only if at least one of the variables α1, α2, α3, β1, β2, β3,W1, W2, W3, β1−W/(2W1), α1−W/(2W2) and α2−W/(2W3)is zero.

Fourth Lemma

Lemma 4Consider a non-negative definite matrix R as defined in (2),whose all first- or second-order minors are non-zero, and thevariables α1, α2, α3, β1, β2, β3, and β4 satisfy (3). LetW := 2α1α2β1 + β4 − α1β3 − α2β2 − α3β1, W1 := α1α2 − α3,W2 := α2β1 − β3, and W3 := α1β1 − β2. Then there exists an in-vertible D = diag{1, x , y} such that DRD is a paramount matrixif and only if one of the following conditions holds:

1. W < 0;2. W > 0, β1 > (W/(2W1)), α1 > (W/(2W2)), α2 >

(W/(2W3));3. W > 0, α2 < (W/(2W3)) and β4 +α1β3 +α3β1−α2β2 ≥ 0;4. W > 0, α1 < (W/(2W2)) and β4 +α2β2 +α3β1−α1β3 ≥ 0;5. W > 0, β1 < (W/(2W1)) and β4 +α1β3 +α2β2−α3β1 ≥ 0.

The equivalent General Condition

Theorem 1A positive-real function Y (s) can be realized as the driving-pointadmittance of a network with one inerter, one damper, and ar-bitrary number of springs with the assumption that X contains awell-defined impedance, if and only if Y (s) can be written in theform of


,

where the coefficients α1, α2, α3, β1, β2, β3, β4 ≥ 0, and furthersatisfy the conditions of Lemma 2, and the conditions of eitherLemma 3 or Lemma 4.

The Possible Cases

To express any positive-real function

Y (s) = ka0s2 + a1s + 1

s(d0s2 + d1s + 1),

where a0,a1,d0,d1 ≥ 0, k > 0, and Rk 6= 0 in the form of


,

there are only two possible cases.

The First Case

For the first case,

Y (s) = ka0s2 + a1s + 1

s(d0s2 + d1s + 1)=

α3s3 + α2s2 + α1s + 1β4s4 + β3s3 + β2s2 + β1s

,

the coefficients can be regarded as follows:

α3 = 0, α2 = a0, α1 = a1, β4 = 0, β3 =d0

k, β2 =

d1

k, β1 =

1k,

which are all non-negative. It’s obvious that the conditions ofLemma 3 always hold. After calculations, the conditions of Lemma2 can be equivalent to

a0d1 = a1d0.

The Second Case

For the second case, by

Y (s) = k(Ts + 1)(a0s2 + a1s + 1)s(Ts + 1)(d0s2 + d1s + 1)

=α3s3 + α2s2 + α1s + 1β4s4 + β3s3 + β2s2 + β1s

,

the coefficients can be regarded as follows:

α3 = a0T , α2 = a0 + a1T , α1 = a1 + T , β4 = d0T/k ,β3 = (d0 + d1T )/k , β2 = (d1 + T )/k , β1 = 1/k , T > 0.

Substitute the above coefficients into the conditions of Lemma2, Lemma 3, and Lemma 4. After a series of calculations, theconditions can be equivalent to

d20

(a0d1 − a1d0)(a1 − d1)≥ 1,

and T must be expressed as

T =

√a0d1 − a1d0

a1 − d1. (4)

The Realizability Condition

Theorem 2Consider a positive-real function

Y (s) = ka0s2 + a1s + 1

s(d0s2 + d1s + 1),

where a0, a1, d0, d1 ≥ 0, k > 0 and Rk 6= 0. It can be realizedas the driving-point admittance of a network with one damper,one inerter, and arbitrary number of springs but no levers withassumption that X has a well-defined impedance, if and only if

d20

(a0d1 − a1d0)(a1 − d1)≥ 1

ora0d1 − a1d0 = 0.

The Covering Realizations

When a0d1 − a1d0 = 0, it can be indicated that a0 > d0 sinceRk 6= 0 (a0 ≥ d0 for positive-realness).If a0d1 = a1d0 and a0 > d0 = 0, then d1 = 0. Therefore,

Y (s) = ka0s + ka1 +ks,

with k1 = k > 0, b = ka0 > 0, and c = ka1 ≥ 0.If a0d1 = a1d0 and a0 > d0 > 0, then we have

Y (s) =ks+ k

(a0 − d0)s + (a1 − d1)

d0s2 + d1s + 1

=ks+

1d0

k(a0−d0)s + 1

k(a0−d0)s+k(a1−d1)

,

with k1 = k > 0, k2 = k(a0 − d0)/d0 > 0, b = k(a0 − d0) > 0,and c = k(a1 − d1) ≥ 0.


Whend2

0(a0d1 − a1d0)(a1 − d1)

≥ 1,

Y (s) can be realized by the network as follows.CHEN AND SMITH: RESTRICTED COMPLEXITY NETWORK REALIZATIONS 2295













The values can be expressed as

k1 =ka0d1(a0d1 − a1d0)[(a1 − d1)T + (a0 − d0)]

d0(a1 − d1)(d1T + d0)[(a0 − d0)T + (a0d1 − a1d0)],

k2 =ka0a1d1T [(a1 − d1)T + (a0 − d0)]

2

(a1 − d1)(d1T + d0)[(a0 − d0)T + (a0d1 − a1d0)]2,

k3 =ka0T [(a1 − d1)T + (a0 − d0)]

(d1T + d0)[(a0 − d0)T + (a0d1 − a1d0)],

k4 =ka0T [d0T + (a1d0 − a0d1)][(a1 − d1)T + (a0 − d0)]

(d1T + d0)[(a0 − d0)T + (a0d1 − a1d0)]2,

b =ka2

0(a0d1 − a1d0)[(a1 − d1)T + (a0 − d0)]

(a1 − d1)[(a0 − d0)T + (a0d1 − a1d0)]2,

c =ka2

0d21 (a0d1 − a1d0)[(a1 − d1)T + (a0 − d0)]

2

(a1 − d1)2(d1T + d0)2[(a0 − d0)T + (a0d1 − a1d0)]2,

where T is defined in (4). It is noted that k1, k2, k3,b, c > 0, andk4 ≥ 0.

Outline

1 Introduction


3 Main Results

4 Conclusion

Conclusion

1 Introduction to passive network synthesis.

2 Introduction to the inerter.

3 An equivalent and more explicit condition for any positive-real admittance to be realizable with one damper, one in-erter, and an arbitrary number of springs is established.

4 Making use the equivalent condition, a necessary and suf-ficient condition for the realizability of the class of positive-real functions under investigation is derived.

5 Canonical configurations to cover the condition are presented.

The End

Thank You Very Much!

realization of a special class of admittances with one ...web.hku.hk/~mzqchen/cdc2012.pdf ·...

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