05256285 Restricted Complexity Network Realizations

2290 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 10, OCTOBER 2009

Restricted Complexity Network Realizationsfor Passive Mechanical Control

Michael Z. Q. Chen, Member, IEEE, and Malcolm C. Smith, Fellow, IEEE

Abstract—This paper considers a realization problem of re-stricted complexity arising in an approach to passive control ofmechanical systems. This approach is based on synthesizing apositive-real admittance or impedance function using springs,dampers and inerters. This paper solves the following problem:what is the most general class of mechanical admittances whichcan be realized if the number of dampers and inerters is re-stricted to one in each case, while allowing an arbitrary number ofsprings and no transformers (levers)? The solution uses elementextraction of the damper and inerter followed by the derivationof a necessary and sufficient condition for the one-element-kind(transformerless) realization of an associated three-port network.This involves the derivation of a necessary and sufficient conditionfor a third-order non-negative definite matrix to be reducible toa paramount matrix using a diagonal transformation. It is shownthat the relevant class of mechanical admittances can be parame-trized in terms of five circuit arrangements each containing foursprings.

Index Terms—Electric circuits, mechanical networks, networksynthesis, passivity.

I. INTRODUCTION

T HE purpose of this paper is to pose and solve a realizationproblem of restricted complexity which arises in an ap-

proach to mechanical control based on the idea of synthesizinggeneral passive networks. This approach was made possible bythe introduction of a new circuit element, coined the inerter [18],which provides a means to realize an arbitrary positive-real me-chanical admittance or impedance with networks comprisingsprings, dampers and inerters. Applications of the method to ve-hicle suspension [15], [20], control of motorcycle steering in-stabilities [12], [13] and vibration absorption [18] have beenidentified which give performance advantages over more con-ventional passive solutions. (See [7] for more details.) Althoughfully active control will always have the greatest potential (e.g.,in terms of adaptation or ease of programmability), this needs tobe balanced against increased implementation complexity, costand power requirements.

The proposed approach to mechanical control allows resultsof classical electrical circuit synthesis to be directly exploited.

Manuscript received February 27, 2007; revised December 12, 2007. Firstpublished September 22, 2009; current version published October 07, 2009.Recommended by Associate Editor K. Fujimoto.

M. Z. Q. Chen is with the Department of Automation, Nanjing Universityof Science and Technology, Nanjing 210094, China (e-mail: [email protected]; [email protected]).

M. C. Smith is with the Department of Engineering, University of Cambridge,Cambridge CB2 1PZ, U.K. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2009.2028953

However, efficiency of realization, as defined by the number ofelements used, is much more important for mechanical networksthan electrical networks. The only general method for trans-formerless electrical synthesis—the method of Bott and Duffin[2] and its variants [23]—appears to be highly non-minimal. Theassociated minimum complexity realization problem is far frombeing solved in general.

This paper considers the class of realizations in which thenumber of dampers and inerters is restricted to one in eachcase while allowing an arbitrary number of springs (whichis the easiest element to realize practically). This problem isanalogous to restricting the number of resistors and capacitors,but not inductors, in electrical circuit synthesis (see also [9]).Such questions involving restrictions on both resistive andone type of reactive element have never been considered. Thiscontrasts with the problems of minimal resistive and minimalreactive synthesis which have well-known solutions whentransformers are allowed [1], [11], [26]. In our problem, weimpose the condition that no transformers are employed, dueto the fact that large lever ratios can give rise to practicalproblems. Such a case can occur if there is a specification onavailable “travel” between two terminals of a network, as in acar suspension. A large lever ratio may necessitate a large travelbetween internal nodes of a network, which then conflicts withpackaging requirements.

We show that the problem considered in this paper is closelyrelated to the problem of one-element-kind multi-port synthesis.In particular, we will show that a necessary and sufficient condi-tion of Tellegen [22] for the realizability of resistive three-portsallows us to deduce that at most six springs are needed to re-alize the required class. Our approach goes further to prove thatat most four springs are needed. In addition, an explicit con-struction is given comprising five different circuit arrangementsto cover all cases.

This paper is organized as follows. Section II reviews the def-inition and properties of the inerter. Some background on elec-trical-mechanical circuit analogies, passive network synthesisand positive-real functions is briefly reviewed. In Section III theproblem of synthesizing passive mechanical networks with onedamper and one inerter is formulated. Section IV reviews thedefinition of paramountcy and provides a necessary and suf-ficient condition for a third-order non-negative definite matrixto be reducible to a paramount matrix using a diagonal trans-formation. In Section V, the explicit solution of the synthesisproblem is presented. Section VI shows that a simple class oflow-order positive-real admittances considered in [18] cannotalways be realized using one damper, one inerter and no trans-formers. Conclusions are given in Section VII.

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CHEN AND SMITH: RESTRICTED COMPLEXITY NETWORK REALIZATIONS 2291

Fig. 1. Free-body diagram of a two-terminal mechanical element with force-velocity pair �� where � � � � � .

Fig. 2. Schematic of a mechanical model of an inerter.

II. NETWORK SYNTHESIS

A. The Inerter

An ideal two-terminal mechanical element called the inerterwas introduced in [18] with the property that the (equal and op-posite) force applied at the terminals is proportional to the rel-ative acceleration between them. In the notation of Fig. 1, theinerter obeys the force-velocity law , wherethe constant of proportionality is called the inertance and hasthe units of kilograms and , are the velocities of the twoterminals. In order to be practically useful, the device shouldhave a small mass (relative to ) and its inertance should beadjustable independently of the mass. Also, the device shouldfunction properly in any spatial orientation, it should supportadequate linear travel and it should have reasonable overall di-mensions. Such a device can be constructed using a flywheelthat is driven by a rack and pinion, and gears (see Fig. 2). Thevalue of the inertance is easy to compute in terms of the var-ious gear ratios and the flywheel’s moment of inertia [18]. Ingeneral, if the device gives rise to a flywheel rotation of ra-dians per meter of relative displacement between the terminals,then the inertance of the device is given by whereis the flywheel’s moment of inertia. Other methods of construc-tion are described in [17].

B. Passive Network Synthesis

One of the principal motivations for the introduction of the in-erter in [18] was the synthesis of passive mechanical networks.It was pointed out that the standard form of the electrical-me-chanical analogy (where the spring, mass and damper are anal-ogous to the inductor, capacitor and resistor) was restrictive forthis purpose, because the mass element effectively has one ter-minal connected to ground. In order that the full power of elec-trical circuit synthesis theory be translated over to mechanicalnetworks, it is necessary to replace the mass element by a gen-uine two-terminal element—the inerter. Fig. 3 shows the new

Fig. 3. Circuit symbols and correspondences with defining equations and ad-mittance � ��.

Fig. 4. Inerter in series with damper with centring springs. (a) Circuit diagram;(b) mechanical realization.

table of element correspondences in the force-current analogywhere force and current are the ‘through’ variables and velocityand voltage are the ‘across’ variables. The admittance isthe ratio of through to across quantities, where is the standardLaplace transform variable. Fig. 4 shows a particular circuit to-gether with a mechanical realization constructed at CambridgeUniversity Engineering Department.

The theory of passive network synthesis has been widelystudied in the electrical engineering literature [1], [14]. Theconcept of passivity can be translated over directly to mechan-ical networks as follows. Suppose that represents theforce-velocity pair associated with a two-terminal mechanicalnetwork, then passivity requires

(1)


Fig. 5. General one-port containing one damper and one inerter.

for all admissible time functions , and all . Ifis the real rational impedance or admittance function of a lineartime-invariant two-terminal network, it is well-known that thenetwork is passive if and only if is positive-real [1], [10],[14]. Furthermore, given any positive-real function , thereexists a two-terminal mechanical network whose impedanceequals , which consists of a finite interconnection ofsprings, dampers and inerters. See [18] for details.

III. PROBLEM FORMULATION

We consider a mechanical one-port network consisting ofan arbitrary number of springs and levers (transformers), onedamper and one inerter. We can arrange the network in the formof Fig. 5 where is a three-port network containing all thesprings and levers. We bring in mild assumptions that the net-work has a well-defined admittance and the network hasa well-defined impedance (see Appendix A). As in the proofof [19, Theorem 8.1/2] we can derive an explicit form for theimpedance matrix. This is defined by

(2)

where is a non-negative definite matrix. ( denotes Laplacetransform.) Setting and and elimi-nating and gives the following expression for the admit-tance of :

(3)

where , , , ,, ,

and . Note that requires that fornon-negative definiteness which means that the admittance doesnot exist. Thus the assumption of existence of the admittancerequires that . (See Appendix A for further discussionon the assumptions in the problem formulation.)

The admittance (3) effectively has only six parameters whichcan be adjusted among the seven coefficients. To see this notethat and can be set to be equal to 1 and the following scalings

carried out: , , ,, , , to leave (3) invariant. The

resulting admittance is

(4)

and

where

(5)

and is non-negative definite. From the expression, we note that

(4) depends on but not on the individual signsof , and .

If we now suppose a in the form of (4) is given withnon-negative definite, then it is a standard fact [3, Chapter 4,

pages 173–179] that can be realized in the form of Fig. 5with consisting of springs and transformers (levers), and with

. This paper addresses the following question: sup-pose a in the form of (4) is given, what additional condi-tions are necessary for to be realizable with one damper,one inerter, any number of springs but no transformers (levers)?

IV. TRANSFORMERLESS REALIZATION AND PARAMOUNTCY

This section reviews the concept of paramountcy and its rolein transformerless synthesis. New results are also established fora third-order non-negative matrix to be reducible to a paramountmatrix by a diagonal transformation.

Definition 1: A matrix is defined to be paramount if its prin-cipal minors, of all orders, are greater than or equal to the abso-lute value of any minor built from the same rows [4], [16].

It has been shown that paramountcy is a necessary conditionfor the realizability of an -port resistive network without trans-formers [4], [16]. In general, paramountcy is not a sufficientcondition for the realizability of a transformerless resistive net-work and a counter-example for was given in [5], [24].However, in [22, pp.166–168], it was proven that paramountcyis necessary and sufficient for the realizability of a resistive net-work without transformers with order less than or equal to three

. The construction of [22] for the case makesuse of the network containing six resistors shown in Fig. 6 withjudicious relabelling of terminals and changes of polarity. A re-working (in English) of Tellegen’s proof is given in [6], [8].

In the next two lemmas we establish a necessary and sufficientcondition for a third-order non-negative definite matrix

(6)

to be reducible to a paramount matrix using a transformation ofthe form of (5).


Fig. 6. Tellegen’s circuit for the construction of resistive 3-ports without trans-formers.

Lemma 2: Let be non-negative definite and suppose thatall first- and second-order minors are non-zero.1 Then isa paramount matrix for an invertible if andonly if the following inequalities are satisfied by and :

(7)

(8)

(9)

Furthermore, is a paramount matrix, for some, if and only if the following inequali-

ties are satisfied:

(10)

(11)

(12)

1For a non-negative definite matrix �, it is possible to have non-zero first-

and second-order minors and yet be singular, e.g.

� � �

� � ��

� ��

.

(13)

(14)

Proof: See Appendix B.Lemma 3: Let be non-negative definite. If any first- or

second-order minor of is zero, then there exists an invertiblesuch that is a paramount matrix.

Proof: See [6], [8] for details.We now establish an equivalent necessary and sufficient con-

dition to Lemma 2 which will be used to establish our main re-sult in Section V.

Lemma 4: Let be non-negative definite and suppose thatall first- and second-order minors are non-zero. Then satis-fies conditions (10)–(14) of Lemma 2 if and only if one of thefollowing holds:

i) ;ii) , ,

and ;iii) , and

;iv) , and

;v) , and

.Proof: See Appendix C.

Remark: It is shown in the proof of Lemma 4 that, under theassumption that is non-negative definite and ,at most one of the three expressions

is negative. This fact is needed later to see that certain parametervalues in the realization theorems of Section V are positive.

V. MAIN RESULTS

This section derives a necessary and sufficient condition forthe realizability of an admittance function using one damper,one inerter, an arbitrary number of springs and no levers (trans-formers) (Theorem 6). The proof relies on the results of Sec-tion IV and the construction of [22]. A stronger version of thesufficiency part of this result, which shows that at most foursprings are needed, is given in Theorem 7 with explicit circuitconstructions. Singular cases are treated in Theorem 8.

Lemma 5: A positive-real function can be realized as thedriving-point admittance of a network in the form of Fig. 5, where

has a well-defined impedance and is realizable with springsonly and , if and only if can be written in the form of

(15)


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


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-


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)


(21)

The expressions for follow by equating powers offor all but the highest power in (20) and (21). The latter equation,after substitution for in , reduces to

(22)

Squaring both sides and factorising leads to

The negative solution for is not admissible since. Taking the positive solution for then solves (22)

and gives the required formulae. The degree drop (i.e., ) corresponds to the limiting case where

.Conversely, with as defined, (20) and (21)

hold. From the same equations it follows immediately that allprincipal minors of are non-negative.

Remark: If Theorem 9 holds, it is easily observed thatand , which corresponds to Case

(iii) in Lemma 4.Example: Consider the admittance function

It is observed that , , . Therefore, ispositive-real. However,

, so that the second condition of Case (iii) in Lemma4 fails. Therefore, cannot be realized in the form of Fig. 5.

VII. CONCLUSION

This paper has studied a realization problem arising in anapproach to passive mechanical control. The problem requiresthe number of dampers and inerters to be restricted to one ineach case while no transformers are allowed but the number ofsprings is in principle unrestricted. The solution made use of anelement extraction procedure of mixed type in which one of theelements extracted is resistive (the damper) and the other is re-active (the inerter). This approach establishes a connection withthe problem of single-element-kind multi-port synthesis. It hasbeen shown that at most four springs are needed to realize anymechanical admittance (or impedance) within the class consid-ered. Specific realizations were presented to cover all cases. Partof the proof relies on the derivation of a necessary and sufficientcondition for a third-order non-negative definite matrix to be re-ducible to a paramount matrix using a diagonal transformation.An example was presented of a positive-real admittance whichcan be realized with at most one damper and one inerter only iftransformers are allowed.

APPENDIX AASSUMPTIONS IN PROBLEM FORMULATION

The expression (3) was derived under the assumption that thenetwork in Fig. 5 has a well-defined impedance. We nowexplain why this is only a very mild assumption.

It has already been pointed out in Section III that there are ef-fectively only six parameters which can be adjusted among theseven coefficients other than . If a similar expression to (3) isderived under the assumption that has a well-defined admit-tance, then it turns out that and there are again only sixadjustable parameters among the remaining seven coefficients.In both cases, this dependence among the coefficients can be ex-pressed in the following equation:

(23)

If is not assumed to have either an impedance or admit-tance, then it turns out that also takes the form of (3)in which (23) holds but with no coefficients constrained to benonzero (when , ). This can be argued as follows.

Following [14], [25], given an arbitrary -port network whichis a finite interconnection of resistors, there should be constantmatrices and such that

with invertible, where and are vectors of voltagesand currents respectively.

Now consider an arbitrary -port network consisting ofsprings. By similar reasoning, there exist constant matricesand such that

with invertible, where and are vectors of displace-ments and forces respectively. This implies the existence of aconstant symmetric matrix satisfying(where ) with the property that

( being the identity matrix). Then, with ,the admittance takes the form of (3) and each , is equalto with (23) being satisfied. (Weremark that is not the scattering matrix since it is definedin terms of force-displacement pairs rather than force-velocitypairs.)

We conclude that the only cases that are excluded from theachievable admittances under the assumption that has a well-defined impedance are limiting cases where some coefficientsin (3) go to 0 or . These can be approached arbitrarily byrealizations in the assumed class.

APPENDIX BPROOF OF LEMMA 2

Writing down all the inequalities for the first order minorsof according to the definition of paramountcy and rear-ranging gives

(24)

(25)

(26)


Notice that the inequalities (24)–(26) can be satisfied individu-ally for some choices of and since

(27)

(28)

(29)

Moreover, the set of and satisfying (24)–(26) is non-emptyif and only if the following two intervals have a non-emptyintersection:

It can be shown that this intersection is non-empty if and onlyif the following two inequalities hold:

These inequalities are always satisfied since it is easily shownfrom (27)–(29) that for any non-negativedefinite .

Similarly, writing down all the inequalities for thesecond order minors of according to the definitionof paramountcy and rearranging gives

(30)

(31)

(32)

Again the inequalities (30)–(32) can be satisfied individually forsome choices of and since

(33)

(34)

(35)

Moreover, the set of and satisfying (30)–(32) is non-emptyif and only if the following two intervals have a non-empty in-tersection:

It can be shown that this intersection is non-empty if and onlyif the following two inequalities hold:

In fact, these inequalities are always satisfied since it can beeasily shown from (33)–(35) that

for any non-negative definite matrix .Combining (24)–(26) and (30)–(32) together gives inequali-

ties (7)–(9).To show the equivalence with (10)–(14) we note that the ex-

istence of and satisfying (7)–(9) is equivalent to each ofthe intervals in (7)–(9) being non-empty, i.e., (10)–(12), and anon-empty intersection between the interval in (9) and the al-lowed range for implied by (7) and (8), i.e., (13), (14).

APPENDIX CPROOF OF LEMMA 4

(If.) We first show that both (i) and (ii) imply that the condi-tions of Lemma 2 are satisfied. We then provide a sequence ofsteps to show that (iii) conditions of Lemma 2. The proofsthat both (iv) and (v) imply the conditions of Lemma 2 aresimilar.Step 1) (i) conditions of Lemma 2.

It is easy to see that given , (7)–(9)reduce to

(36)

(37)

(38)

As pointed out in Appendix B the set of satis-fying (36)–(38) is always non-empty. Therefore, inthe case of , the conditions of Lemma2 are always satisfied.

Step 2) (ii) conditions of Lemma 2.Note that

(39)

and

(40)

Since (39) 0 and (40) 0, (7) reduces to

(41)

Similarly, (8) reduces to

(42)

and (9) reduces to

(43)


It has been shown in Appendix B that the set ofsatisfying (41), (42) and (43) is always non-

empty. Therefore, (ii) Lemma 2.Step 3) (iii) conditions of Lemma 2.

We claim that

cannot both be negative. To see this, assume tothe contrary that both are negative, as well as

. Then

which contradicts the non-negative definiteness of. Similarly, we can show that

cannot both be negative and also

cannot both be negative. Therefore, at most one ofthe three expressions

is negative.Since , it is implied that

and . As in Step 2we can check that (8), (9) of Lemma 2 reduce to

(44)

(45)

Our next step is to show that the interval defined in (7) isnon-empty. Since for any non-negative definite

the following inequalities hold:

(46)

(47)

By assumption

(48)

Adding (48) to (46) and (47), respectively, gives

which imply that

which ensures that the interval in (7) is non-empty.The set of satisfying (7), (44), (45) is non-empty iff the

intersection of the following two intervals is non-empty:

and

This holds iff the following two inequalities hold:

This is equivalent to the following four inequalities being satis-fied:

(49)

(50)

(51)

(52)

We will now verify these individually.Rewriting (48) as

together with

shows that

which is equivalent to (49). Similarly

which is equivalent to (51).


We now observe from the assumptions that the following in-equality holds:

This is equivalent to

which implies (50). Similarly, (52) holds.This completes the proof that (iii) Lemma 2. Similarly it

can be shown that (iv) Lemma 2 and (v) Lemma 2.(Only if.) Obviously Lemma 2 implies either

or . As shown in Step 3 of the (If.) part of theproof, at most one of the three expressions

(53)

is negative. Hence (ii) holds or exactly one of the expressions in(53) is negative. To complete the proof we need to show that thefinal inequality in (iii) is implied by the first two [and similarlyfor (iv) and (v)]. Since , it is implied that

and . Under this conditionit has been shown that (7)–(9) of Lemma 2 reduce to (7), (44),(45). Since Lemma 2 holds, it is necessary that the intervals in(44) and (45) are non-empty. Each of these inequalities implies

Similarly we can show the cases for (iv) and (v).

APPENDIX DPROOF OF THEOREM 8

We selectively describe five typical cases when only one ofthe first- or second-order minors of is zero. All other caseswhen one minor vanishes are similar. When more than one ofthe first- or second-order minors of is zero the realizationssimplify even further.

1) . The admittance fails to exist in this case.2) . Then

with , and, which is a spring in parallel with a series connec-

tion of a spring and an inerter.3) . Then

with , ,and , which is realized using one damper, oneinerter and two springs.

4) . Then

with , ,,

and , which is realizedusing one damper, one inerter and three springs.

5) . Then

with , ,, and ,

which is realized using one damper, one inerter and threesprings.

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[16] P. Slepian and L. Weinberg, “Synthesis applications of paramountand dominant matrices,” in Proc. Nat. Elect. Conf., 1958, vol. 14, pp.611–630.

[17] M. C. Smith, “Force-Controlling Mechanical Device,” U.S. PatentPCT/GB02/03056, Jul. 4, 2001.

[18] M. C. Smith, “Synthesis of mechanical networks: The inerter,” IEEETrans. Automat. Control, vol. 47, no. 10, pp. 1648–1662, Oct. 2002.


[19] M. C. Smith and G. W. Walker, “A mechanical network approach toperformance capabilities of passive suspensions,” in Proc. WorkshopModelling Control Mech. Syst., London, U.K., Jun. 17–20, 1997, pp.103–117.

[20] M. C. Smith and F.-C. Wang, “Performance benefits in passive ve-hicle suspensions employing inerters,” Vehicle Syst. Dyn., vol. 42, pp.235–257, 2004.

[21] A. Talbot, “Some fundamental properties of networks without mutualinductance,” in Proc. Inst. Elect. Eng., 1955, pp. 168–175.

[22] B. D. H. Tellegen, Theorie der Wisselstromen. Groningen, TheNetherlands: P. Noordhoff, 1952.

[23] M. E. Van Valkenburg, Introduction to Modern Network Synthesis.New York: Wiley, 1960.

[24] L. Weinberg, Report on Circuit Theory XIII URSI Assembly, London,U.K., Tech. Rep., Sep. 1960.

[25] D. C. Youla, L. J. Castriota, and H. J. Carlin, “Bounded real scatteringmatrices and the foundations of linear passive network theory,” IRETrans. Circuit Theory, vol. 6, no. 1, pp. 102–124, 1959.

[26] D. C. Youla and P. Tissi, “ � -port synthesis via reactance extraction,part I,” in Proc. IEEE Int. Convention, 1966, pp. 183–205.

Michael Z. Q. Chen (M’08) was born in Shanghai,China. He received the B.Eng. degree in electricaland electronic engineering from Nanyang Techno-logical University, Singapore, in 2003 and a Ph.D.degree in control engineering from CambridgeUniversity, Cambridge, U.K., in 2007.

From 2007 to 2009, he was a Lecturer in the De-partment of Engineering, University of Leicester, Le-icester, U.K. Since 2009, he has been a Professor inthe Department of Automation, Nanjing Universityof Science and Technology, Nanjing, China. Since

2008, he has been a Reviewer for Automatica, Physica A, the InternationalJournal of Adaptive Control & Signal Processing, and the Journal of Sound &Vibration. He is also an external Reviewer for the Research Grants Council ofHong Kong. His research interests include: passive network synthesis, vehiclesuspensions, complex networks, statistical mechanics, and systems biology.

Dr. Chen is a Fellow of the Cambridge Philosophical Society and a Reviewerof the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS.

Malcolm C. Smith (M’90–SM’00–F’02) receivedthe B.A. (M.A.) degree in mathematics, the M.Phil.degree in control engineering and operational re-search, and the Ph.D. degree in control engineering,from Cambridge University, Cambridge, U.K., in1978, 1979, and 1982, respectively.

He was subsequently a Research Fellow at theGerman Aerospace Center, Oberpfaffenhofen,German, a Visiting Assistant Professor and Re-search Fellow with the Department of ElectricalEngineering at McGill University, Montreal, QC,

Canada, and an Assistant Professor with the Department of Electrical Engi-neering, Ohio State University, Columbus. In 1990, he joined the EngineeringDepartment, University of Cambridge, where he is currently a Professor. Hisresearch interests are in the areas of robust control, nonlinear systems, electricaland mechanical networks, and automotive applications.

Dr. Smith received the 1992 and 1999 George Axelby Best Paper Awards, inthe IEEE TRANSACTIONS ON AUTOMATIC CONTROL.

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