» LECTURE NOTES

Transmission Matrices for Physical-System Modeling

KHALED F. ALJANAIDEH and DENNIS S. BERNSTEIN

ransfer functions model input–output relationships [7, pp. 10-35–10-37], [8, pp. 132–134], [9, pp. 16–32] [10, pp. 49–57], in linear systems. For example, viewing as an [11, Ch. 3], [12, pp. 208–214], [13, Ch. 12], [14, Ch. 11], [15, Ch. 11], Tinput, can be seen as an output. Likewise, [16, pp. 24–27]. Reference [17] includes detailed definitions from a battery gives rise to current flow. The gain, and further references for these matrices. Transformations poles, and zeros of these transfer functions provide insight among six versions of these parameters are given in [6, into the response to inputs. p. 317], where the b parameters are most closely related to Transfer functions can be derived by applying the laws power transmission matrices for electrical circuits as they are of dynamics and circuits, such as Newton’s and Kirch- defined in the present article. hoff’s laws. An alternative approach to obtaining them is PTMs and their variants have been used in diverse to develop models of mechanical and electrical compo- mechanical and electrical applications. PTMs are employed nents and connect them by accounting for power (the flow in [11] to derive equations of motion for lumped mechani- of ) by means of power-conjugate variables (variables cal systems as well as continuum structures, such as shafts whose product is power). In mechanical systems, force and and beams. For mechanical applications, the use of PTMs velocity are power-conjugate variables; in electrical sys- circumvents the necessity of free-body analysis and elimi- tems, voltage and current are power-conjugate variables. nates the need to determine reaction and . Input–output models can be used to relate effort and flow On the electrical side, PTMs are used to model distrib- variables, which are power conjugate. In structural model- uted networks, including transmission lines [3], [5, Ch. 1, ing, power-conjugate variables are force and velocity in pp. 200–218], [12, pp. 212–214], [13, pp. 361–362]. translational motion and moment and angular velocity in As described in “Summary,” a benefit of power trans- rotational motion; in electrical systems, they are voltage and mission matrices is that analogous models can be derived current, respectively; in fluid and acoustic systems, they are pressure and volume velocity; and in thermodynamic sys- tems, they are temperature difference and entropy flow rate. Summary Since power-conjugate variables occur in pairs, they are ower transmission matrices (PTMs) for structures are connected by power transmission matrices (PTMs), which are P 22# matrices that relate force and velocity at one ter- 22# transfer functions. PTMs have a variety of forms and minal to force and velocity at another terminal. This tech- names arising from the way the variables are defined and nique provides an elegant approach to deriving transfer ordered. They are called four-terminal structures, four-terminal functions for structures consisting of , springs, and networks, quadripole, transfer matrices, and two-port networks. dashpots, that is, dampers. PTMs were developed during Depending on the choice of input and output signals, the the 1970s and 1980s, but they are rarely mentioned today entries of the matrices are known as ABCD parameters, except briefly in textbooks. The first goal of this tutorial is to cascade parameters, chain parameters, four-pole param- highlight the utility of this modeling technique. A related aim eters, impedance parameters, admittance parameters, is to bring inerters into this framework. The article revisits hybrid parameters, inverse hybrid parameters, scatter- the classical topic of analogies by deriving PTMs for two- ing parameters, scattering transfer parameters, and trans- port networks. The across-through analogy is used to con- mission parameters [1, pp. 28–31], [2, pp. 69, 70, 315–317], struct circuits whose transfer functions match those of the [3, pp. 56–58], [4, pp. 243–257], [5, pp. 3–15], [6, pp. 312–330], analogous structure. This article is intended for all students and practitioners of control systems who may benefit from

Digital Object Identifier 10.1109/MCS.2019.2925257 awareness of this modeling technique. Date of publication: 16 September 2019

88 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 1066-033X/19©2019IEEE across domains of application. The study of analogies terminals rather than the inertial acceleration of a between mechanical and electrical systems has a long as explained in “What Is an Inerter?” Mechanical net- history. The classic book [18], which remains the unique works can be built from four components (masses, inert- reference devoted to the topic, encompasses mechanical, ers, springs, and dashpots), whereas electrical networks electrical, and acoustic systems. Circuit models for struc- can be constructed from only three (, , tures are discussed within the context of analog computa- and ). Some brief remarks are provided on the tion in [19, Ch. 9]. A recent survey [20] provides a detailed history of the development of electromechanical analogies. In an electrical circuit, potential can be defined relative to a constant potential level, such as an earth ground. Analo- What Is an Inerter? gously, the velocity of a point in a structure can be defined he relationship between the acceleration of a mass and relative to a point that has constant inertial velocity, for exam- T the force applied to the mass is given by Newton’s second ple, a point on an inertially nonrotating massive body [21]. law. The acceleration vector is the second derivative of a posi- Such a point provides an inertial ground. Voltage and velocity tion vector, where the vector is defined relative to an are relative (across) variables, whereas force and current are unforced particle, and the derivative is taken with respect to an absolute (through) variables [22, pp. 18–22], [23, p. 20], [24, inertial frame. In the absence of a force, an unforced particle pp. 45–47], [25, p. 21], [26, pp. 39–58]. In electrical applica- moves with constant speed along a straight line with respect tions, an earth ground provides an approximately infinite to an inertial frame; consequently, any unforced particle can volume that can transfer electrons without losing its charge be used as a reference point for defining inertial acceleration. neutrality. Analogously, in structural applications, an iner- It is possible to consider acceleration relative to an arbi- tially nonrotating massive body has approximately infinite trary point and with respect to a noninertial frame. An inert- mass and does not accelerate translationally or rotationally er is a device that has two physical terminals with the prop- in response to reaction forces and torques. A physical ter- erty that the difference between the reaction forces applied minal is a physical point of attachment, such as the wires to the physical terminals is proportional to the acceleration emanating from a or the ends of a , while a of one of the physical terminals relative to the other, which reference terminal is an inertial or earth ground [27], [28]. also serves as the reference terminal [36]–[38]. Unlike the There are two competing analogies between electrical case of a mass shown in Figure 2, the reference terminal and mechanical systems, denoted in this article by fE-vI and for an inerter need not have constant inertial velocity; its fI-vE, each of which (for historical reasons) has at least four motion is simply the motion of one of the physical terminals. names. For example, fE-vI is called the Maxwell analogy, As discussed in [37], an inerter can be realized using a direct analogy, , and effort-flow analogy, ball screw or a rack and pinion. Figure S1(a) shows a ball- and it associates force with potential and velocity with cur- screw inerter where the forces applied to the physical ter- rent. On the other hand, fI-vE is known as the Firestone anal- minals are converted to a , which is applied to the nut ogy, inverse analogy, , and across-through flywheel. The push-top toy in Figure S1(b) operates essen- analogy, and it pairs force with current and velocity with tially by the same principle as the ball-screw inerter. potential. Early papers [29]–[31] discuss various aspect of these analogies. With voltage and current taken as analo- gous to velocity and force, respectively, we note that mass Thrust Bearing m, c, and stiffness k are related to

C, resistance R, and L by mC= , cR= 1/, and T2 T1 kL= 1/. Observe that capacitors and masses store energy provided by the across variables voltage and velocity, respec- tively, whereas inductors and springs store energy provided Nut Flywheel by the through variables current and force, respectively. Ball Screw (a) (b) Bond graph modeling is based on the effort-and-flow analogy fE-vI [2, pp. 343–418], [32, pp. 260–270], [33], [34], FIGURE S1 (a) A ball-screw inerter. Physical terminal T is con- [35, p. 123–167], [25, pp. 113–129]. Bond graphs are related to 1 nected to a ball screw, and physical terminal T2 is connected transmission matrices that are based on fE-vI in [2, pp. 69, to the housing. The difference Tf = f2 - f1 between the forces 70, 315–317]. In contrast, the present article focuses on fI-vE; applied to the physical terminals is proportional to the torque x the reason for this choice is the analogy between inertial applied to the nut flywheel; the torque x is proportional to the angular acceleration of the nut flywheel; and the angular ac- and earth grounds as well as the preservation of series and celeration of the nut flywheel is proportional to the acceleration parallel connections across domains. Ta of T2 relative to T1. Consequently, Tf is proportional to Ta. This article encompasses the inerter as a distinct (b) A push-top toy operates essentially by the same principle mechanical component [36]–[38], in which the reaction force as the ball-screw inerter. is proportional to the relative acceleration of the physical

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 89 memristor and memdashpot, which are hysteretic compo- ance, and inertance are improper transfer functions, that nents that relate charge to flux density and to is, the degree of the numerator is greater than that of the position, respectively. denominator. This terminology is summarized in Table 1. This article provides a tutorial on power transmission A transmissibility is a relationship between two vari- matrices. It derives power transmission matrices for the ables of the same type [40], [41], [42, 30 pp. 2-31–2-36]. Force- basic mechanical components and shows how they can be to-force, velocity-to-velocity, and position-to-position combined by series and parallel connections to obtain trans- relationships constitute a force transmissibility, velocity fer functions for specified inputs and outputs. Power trans- transmissibility, and position transmissibility, respectively. mission matrices are formulated in terms of the differential As shown in [43], transmissibilities can be represented in operator p rather than the Laplace s or its harmonic steady- terms of behaviors [27], [44]. state specialization .~; this setting offers -domain To represent dynamics in the time domain, the transfer equations that account for arbitrary initial conditions [39]. functions in this article are expressed in terms of the differ- 3 ential operator p = d/dt rather than the complex Laplace MODELING MECHANICAL SYSTEMS USING variable s. To clarify this distinction, note that the mass- TRANSFER FUNCTIONS spring system In mechanical systems, the transfer functions from force to position, force to velocity, and force to acceleration are called mqp +=kq f (1) compliance, admittance, and accelerance, respectively. Their reciprocals are called stiffnance, impedance, and inertance, can be written as respectively. Thus, for example, impedance is the transfer function from velocity to force. Note that stiffnance, imped- mqp2 +=kq f, (2)

whose solution can be represented as TABLE 1 The names of the transfer functions for structures. Admittance is the transfer function from force to velocity; q = 1 f. (3) impedance is its reciprocal. mkp2 +

Force to Motion Motion to Force Note that (3) is a representation of the time-domain solu- Position Compliance Stiffnance tion of (1), which includes the free response due to q()0 and Velocity Admittance Impedance qo ()0 as well as the forced response due to f. In contrast, the Acceleration Accelerance Inertance Laplace-domain equation

qst()= 1 tfs() (4) ms2 + k

f2 T2 T1 f1 Structure accounts for the forced response of (1) but assumes that q()0 v2 v1 and qo ()0 are zero. In fact, an additional term is needed to T0 (a) represent the free response in the Laplace domain. Con- sequently, although G()p in (3) and Gs() in (4) have the f2 f1 same form, their meaning is different. Modeling based on P p is used in behaviors as discussed in [44]. Of particular v2 v 1 relevance to this article, a similar approach is used in [8] (b) within the context of power transmission matrices. Further discussion about the distinction between the Laplace s and

FIGURE 1 (a) A structure relating force f1 and velocity v1 to force f2 the operator p can be found in [39]. and velocity v2. The forces f1 and f2, which are absolute (through) variables, are applied to physical terminals T1 and T2, respective- POWER TRANSMISSION MATRICES FOR ly. The v1 and v2 of T1 and T2, respectively, are relative (across) variables that are defined relative to the reference terminal MECHANICAL SYSTEMS # T0. Since the axis direction is to the left, positive values of f1 and f2 A power transmission matrix of a structure is a 22 matrix indicate pushing or pulling to the left. Specifically, f1 is the reaction that relates the force and velocity at one physical terminal force applied by the structure to the structure on its right, whereas to the force and velocity at another physical terminal. Fig- f2 is the reaction force applied to the structure by the structure on ure 1 shows a structure with force f1 at physical terminal T1 its left. By Newton’s third law, these reaction forces are equal and opposite in relation to the adjacent structures. (b) The power trans- and force f2 at physical terminal T2. Force f1 is the reaction mission matrix P of the structure relates the force and velocity at force applied by the structure to the structure on its right, one terminal to the force and velocity at the other terminal. whereas f2 is the reaction force applied to the structure by

90 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 the structure on its left. This convention allows all positive ft21()=+ft() mvp 1()t , (8) forces to point to the left and avoids an excessive number vt21()= vt(). (9) of minus signs. Unlike the usual convention, the positive real axis points to the left, and positive values of f1 and f2 The elementary power transmission matrix of a mass with indicate that these forces are pushing to the left. The choice m is thus given by of axis direction has the pleasant feature that series connec- tions constructed from right to left correspond to products of PTMs constructed from right to left. Figure 1 also shows Admittance and Impedance Matrices velocities v1 and v2 of T1 and T2 relative to the reference power transmission matrix maps the force and velocity terminal, T0, respectively. at one terminal to the force and velocity at another ter- The power transmission matrix P, depicted in Figure 1, is A minal. A nice feature of this formulation is that series con- the 22# matrix that relates (,fv11) to (,fv22) according to nections can be modeled simply by forming the product of matrices of this type. Variations of these matrices map forc- ft2() ft1() = P()p . (5) es to velocities and velocities to forces. Specifically, by as- ==vt2()GGvt1() suming that p12 ! 0 and rearranging (5) and (6) we derive

v1 - p11 1 f1 Since the entries of P are functions of the differential oper- = 1 , (S1) ator p, (5) is a time-domain equation that fully accounts for ;;v2EEp12 -1 p22 ;f2E the initial conditions of all variables. Writing where the coefficient matrix is an admittance matrix. As- ! p11 p12 suming that p21 0, the inverse relation is given by P = , (6) ;p21 p22E f1 p22 -1 v1 =- 1 , (S2) ;;f2EEp21 1 - p11 ;v2E it follows that p11 is a force transmissibility (force/force), p12 is an impedance (force/velocity), p21 is an admittance where the coefficient matrix is an impedance matrix. (velocity/force), and p22 is a velocity transmissibility (veloc- As an example, the impedance matrix for a spring satis- ity/velocity). The entries of a power transmission matrix fies consist of two transmissibilities, one impedance and one f1 1 -1 v1 admittance. Power transmission matrices are closely related = k . (S3) to admittance and impedance matrices. For details, see ;;f2EEp 1 -1 ;v2E “Admittance and Impedance Matrices.” Since this matrix is singular, it follows that the spring does not have an admittance matrix. Similarly, the admittance THE ELEMENTARY POWER TRANSMISSION matrix for a mass satisfies MATRICES v1 - 1 1 f1 In this section, the power transmission matrices are derived = 1 . (S4) for the mass, inerter, spring, and dashpot; the correspond- ;;v2EEmp - 1 1 ;f2E ing transmission matrices are the elementary power trans- Since this matrix is singular, it follows that the mass does mission matrices. All forces and motion are assumed to not have an impedance matrix. occur along a line, with the positive direction in all figures Under fvI- E, the impedance matrix, which is the analog taken to be toward the left. This convention leads to block of the mechanical admittance, can be written as diagrams that are consistent with matrix multiplication. E1 Z11 Z12 I1 Consider the mass m, shown in Figure 2, whose physi- = . (S5) ;;E2EEZ21 Z22 ;I2E cal terminals T1 and T2 are fixed attachment points on the body. The reference terminal T0 coincides with a point with Now, assume that EZ22= L I , where ZL is the load imped- an inertial velocity that is constant and corresponds to the ance from I2 to E2 . The resulting input impedance, Zin, motion of an unforced particle [21], which may be embed- from I1 to E1 is given by ded in an inertially nonrotating massive body; in analogy with circuits, an inertially nonrotating massive body can ZZ12 21 ZZin =+11 . (S6) be viewed as an inertial ground. It follows from Newton’s ZZL - 22 third law that Alternatively, assume that EZ11= S I , where ZS is the

source impedance from I1 to E1. The corresponding output ma121()tf=-()tf()t , (7) impedance, Zout, from I2 to E2 results from where -ft1() is the reaction force on the structure due to the ZZ12 21 ZZout =+22 . (S7) ZZS - 11 body on its right, and a1()t is the inertial acceleration of the mass relative to T0. It thus follows that

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 91 1 0 =3 1 f2 T m T f2 T b T Pbin(,p).1 (14) 2 1 f1 2 1 f1 >bp H v2 v1 v2 v1

(a) (b) Next, for the spring with stiffness k shown in Figure 2, k c note that f2 f2 T T1 f T2 T1 f1 2 1 v v v 2 v1 2 1 ft21()= ft(), (15) (c) (d) ft12()=-kq((tq)(1 t)), (16)

FIGURE 2 (a) A mass m with forces f1 and f2 acting on its physical where q1 and q2 are the positions of T1 and T2 relative to terminals T1 and T2, which are rigidly attached to the body at its center of mass. The dynamics of a mass require that the refer- T0, respectively. It thus follows that ence terminal T0 have constant inertial velocity. (b) An inerter b with forces f1 and f2 acting on its physical terminals T1 and T2. The ppft12()=-kq((tq)(p 12tk)) =-((vt)(vt1 )), (17) physical terminals translate relative to each other, with velocities v and v relative to the reference terminal T . For all t $ 0, f (t) = 1 2 0 1 which implies that f2(t). (c) A spring with forces f1 and f2 acting on its physical termi- nals and velocities v1 and v2 relative to the reference terminal T0. p For all t $ 0, f (t) = f (t). (d) A dashpot with forces f and f acting on vt21()=+ft() vt1(). (18) 1 2 1 2 k its physical terminals and velocities v1 and v2 relative to the refer- $ ence terminal T0. For all t 0, f1(t) = f2(t). The elementary power transmission matrix of a spring with stiffness k is thus given by

10 =3 p c Pks(,p).1 (19) f k f > k H 2 f1 2 f1 v T T v T T1 2 2 1 v1 2 2 v1 Next, for the dashpot with viscosity c shown in Fig- (a) (b) ure 2, it follows that

FIGURE 3 (a) A shunted spring. (b) A shunted dashpot. For both ft21()= ft(), (20) structures, T1 and T2 are rigidly connected to each other. ft12()=-cv((tv)(1 t)). (21)

3 1 mp Therefore, Pmm(,p).= (10) ;01E 1 vt21()=+ft() vt1 (), (22) c A mass can be connected to other structures using one or both of its terminals. In the former case, it is a sprung and thus the elementary power transmission matrix of a mass, and the force on its free terminal is zero; otherwise, dashpot with viscosity c is produced by it is unsprung. 1 0 Next, consider the inerter shown in Figure 2 that has 3 Pcd()= 1 1 . (23) physical terminals T1 and T2 and reference terminal T0. = c G Unlike a mass, the reference terminal of an inerter need not be an inertial ground. The relative inertia b can be realized Figure 3(a) shows a shunted spring k, which is a spring as an inerter (see “The Power Transmission Matrix of an connected to an inertial ground. The corresponding ele- Inerter”) and satisfies mentary power transmission matrix is calculated by

k ba((21ta)(-=tf)) 12()tf= ()t , (11) 3 1 Pks,sh(,p).= p (24) >01H where a1 and a2 are the accelerations of T1 and T2 relative to T0, respectively. Therefore, Similarly, Figure 3(b) shows a shunted dashpot c, which is a dashpot connected to an inertial ground. The corresponding ft21()= ft(), (12) elementary power transmission matrix is given by

ft22()=-bvp((tv)(1 t)), (13) 3 1 c Pcd,sh()= . (25) ;01E and thus the elementary power transmission matrix of an inerter with relative inertia b is given by Note that

92 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 det Pmms(,pp)(==detdPbin ,) et Pk(,p)(= det Pcd )

==det Pksd,sh,(,p)(det Pcsh ).= 1 (26) Power Transmission Matrices for Levers and

It will be shown that all power transmission matrices onsider a lever of length l with endpoints e1 and e2, formed from series and parallel connections of masses, Cwhose distances from the fulcrum are l1 and ll21=-l , inerters, springs, dashpots, shunted springs, and shunted respectively. Letting ~()t denote the angular velocity of dashpots share this property. Additional mechanical com- the lever around the fulcrum, it follows that vt11()= ~()tl ponents are the lever and . The power transmis- is the velocity of e1 along a circular arc with radius l1, and sion matrix for the lever as well as its electrical analog, the vt22()= ~()tl is the velocity of e2 along a circular arc with 3 , is derived in “Power Transmission Matrices radius l2 . Hence, vt21()= (/1 m)(vt), where m = ll12/ is the for Levers and Transformers.” A brief description of the velocity ratio. Assume that forces -f1 and f2 are applied gyrator is given in “What Is a Gyrator?” orthogonally to the lever at e1 and e2, respectively, and ~()t is constant. The total moment on the lever is zero, and

RECIPROCITY thus -+ft11()lf22()tl = 0. Hence, ft21()= mft(), where m is Consider the structure modeled by the power trans- the mechanical advantage. The power transmission matrix mission matrix in Figure 1. Suppose that terminal T2 is for the lever is given by fixed to the reference terminal T0. Therefore, v2 = 0, and m 0 (5) becomes 3 Pl ()m = 0 1 . (S8) > m H fp211 p12 f1 = . (27) The circuit analog of the lever is the transformer, which ;;0EEp21 p22 ;v1E consists of primary and secondary windings that transfer It follows that electrical energy. Let I1 and E1 denote the current through the primary windings and the potential drop across the pri- p22 mary windings, respectively; similarly, let I2 and E2 denote f1 =- v1 (28) p21 the current through the secondary windings and the poten- and tial drop across the secondary windings, respectively. Then,

fp21=+11fp12 v1 (29) It21()= aI ()t , (S9) 1 pp12 21 - pp11 22 Et21()= Et(), (S10) = v1 . (30) a p21 where the turns ratio a is the ratio of the number of turns Hence, the transfer function G12 from f2 to v1 is given by in the primary winding to the number of turns in the sec-

p21 ondary winding. For a > 1, the transformer is step-down; G12 = . (31) pp12 21 - pp11 22 for a < 1, it is step-up. The power transmission matrix of a transformer is produced by Alternatively, suppose that terminal T1 is fixed to refer- = a 0 ence terminal T0. In that case, v1 0, and (5) becomes 3 Pat ()= 0 1 . (S11) = aG f2 p11 p12 f1 = . (32) ;;v2EEp21 p22 ;0E

Therefore, What Is a Gyrator? vp22= 11f , (33) gyrator is an idealized electrical component whose A power transmission matrix is given by and the transfer function G21 from -f1 to v2 results from 1 3 0 =- Pag ()= a . (S12) Gp21 21, (34) >a 0H where -f1 is the force on the structure. Since only the off-diagonal entries of Pg are nonzero, it fol-

The structure is reciprocal if transfer functions G12 and lows that the gyrator converts currents to potential and vice

G21 are equal. Therefore, the structure is reciprocal if and versa. In effect, a gyrator can be used to make a capaci- only if tor emulate the signal processing ability of an and vice versa. In practice, a gyrator can be approximately real-

p21 ized using resistors and operational amplifiers. =-p21 . (35) pp12 21 - pp11 22

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 93 3 1 0 Hence, the structure is reciprocal if and only if Next, multiplying (39) by D = , it follows that ;0 -1E

det P = 1. (36) f1 f2 = DPrev D . (40) ;;v1EEv2 The following result follows from properties of series and parallel connections discussed in subsequent sections. On the other hand, inverting P yields

f1 f2 Proposition 1 = P-1 . (41) ;;v1EEv2 All structures constructed from series and parallel connec- -1 tions of masses, inerters, springs, dashpots, and transform- Hence, DPrev DP= . The structure is symmetric if and ers are reciprocal. only if PD-1 = PD and the diagonal entries of P are iden- Since det P = 1, inverting (5) yields tical. It follows from the symmetry condition PD-1 = PD that ;;det P = 1. By reciprocity, the stronger condition ft1 () ft2 () = P()p -1 , (37) det P = 1 holds. vt1 ()GGvt2 () == Intuitively, the mass, inerter, spring, and dashpot are where symmetric structures, since the effect of applying a force and velocity at one terminal on the other terminal does not change p22 -p12 -1 -1 P = . (38) if the terminals are relabeled. In fact, Pmm (,pp)(= DPm mD,), ;-p21 p11 E -1 -1 Pbin (,pp)(= DPin bD,), Pkss(,pp)(= DP kD,), and Pcd (, -1 Consequently, p11 and p22 can be viewed as force and pp)(= DPd cD,). However, many structures are not sym- velocity transmissibilities. metric, as shown by Example 1.

SYMMETRY POWER TRANSMISSION MATRICES Let P be the power transmission matrix of the structure S FOR SERIES CONNECTIONS with inputs (,f11v ) and outputs (,fv22) diagrammed in Fig- The series connection of a pair of structures requires ure 1 and modeled by (5). Now, let Srev be the structure, with that the neighboring physical terminals be rigidly joined.

T1 relabeled as T2 and vice versa. This relabeling is equivalent Because of the convention that left-side arrows point left to reversing the direction of the positive axis, which replaces and right-side arrows point right, it follows from Newton’s v1 and v2 with -v2 and -v1, respectively. Furthermore, f1 third law that adjacent forces at the common physical ter- and f2 become -f2 and - f1, respectively. The interpretation minal must be equal, that is, have the same sign and mag- of these forces is the opposite of the convention for defining nitude. A single common reference terminal is assumed forces along the positive axis; under that convention, -f2 and for the entire structure. The reference terminal must have

-f1 are equivalent to f2 and f1, respectively. Therefore, the constant inertial velocity in the case where at least one power transmission matrix Prev of Srev satisfies structure in the series connection is a mass; otherwise, it may be arbitrary. f1 f2 = Prev . (39) Consider the series connection of the structures shown -v1 -v2 ;;EE in Figure 4, where, for all in=-11,,f ,

Prev is the reverse power transmission matrix of S, and the structure is symmetric if PPrev = . ffii,,21= + 1, (42)

vvii,,21= + 1, (43)

where fi,1 and fi,2 and vi,1 and vi,2 are the forces and veloci- = f f T T ties of the ith structure in the figure. For all in1,,, let Pi 2 2 Structure Structure Structure 1 f1 n 1 be the power transmission matrix of the ith structure, that is, v2 2 v1 fi,2 fi,1 (a) = Pi , (44) ;;vi,2EEvi,1 f2 f1 Pn P2 P1 where v2 v1 9 pi,11 pi,12 (b) Pi = . (45) ;pi,21 pi,22E FIGURE 4 A series connection of n structures. (a) The structures Note that, for convenience, the arguments of Pi are omitted. are connected in series by rigidly joining their physical terminals, Henceforth, the time argument will also be excluded. where T1 and T2 are the physical terminals of the first and last structure, respectively. (b) A block diagram of the series connec- To find the equivalent transmission matrix that relates tion in terms of transmission matrices. (,fv11) to ()f22, v in Figure 4, note that

94 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 f2 fn,1 PPser()pp= sm(,kP)(m,)p = Pn ;;v2EEvn,1 10 1 mp 1 mp 2 fn-11, ==p p mp , (50) = PPnn-1 > 1H ;01E + 1 ;vn-11, E k > kk H f1 = PPnn-11gP . (46) ;v1E which is not symmetric. Finally, the power transmission matrix of the series connection of a spring with stiffness k Therefore, the power transmission matrix that results from and an inerter with relative inertia b in Figure 5(e) is calcu- connecting P1,,f Pn in series is given by lated by

3 PPser = nnPP-11g . (47) PPseri()pp= s(,kP)(n b,)p 10 1 0 1 0 Note that the ordering of the factors in (47) depends on the ==p 1 p 1 , (51) >>1HH1 + 1 left-to-right ordering, from (,fv11) to ()f22, v , of the struc- kkbp > bp H tures with the power transmission matrices PP1,,f n, respectively, depicted in Figure 4. Since transmission which is symmetric. matrices may not commute, the product (47) depends on the physical ordering of the structures in the series con- POWER TRANSMISSION MATRICES nection. Observe that, if Pser is the product of elementary FOR PARALLEL CONNECTIONS power transmission matrices, det Pser = 1. The parallel connection of two structures requires that the physical terminals on each side of the structures be rigidly Example 1: Series Connections joined together. Consequently, on each side of a parallel Consider the series connection of two springs with connection, the velocities of the physical terminals relative stiffnesses k1 and k2 illustrated in Figure 5(a). The to the reference terminal are equal. power transmission matrix of this series connection is given by k k2 1 f2 T T 2 1 f1 PPser()pp= ss(,kP21)(k ,)p

10 10 1 0 v2 v1 ==pp 11 1 1 + p 1 >>kk21HH>c kk12m H (a) c R 1 0V f T 2 c1 S W 2 2 T1 f1 S p W = -1 1 , S 11 W (48) v2 v S + W 1 c kk12m T X (b) k c f T T which shows that the equivalent stiffness is 2 2 1 f1 -1 ((1 k1)+ ()1 k2 ) . The equivalent viscosity of the v2 v1 series connection of two dashpots with viscosities c1 -1 (c) and c2 shown in Figure 5(b) is ((11cc12)(+ )) . The power transmission matrix of the series connection of a k f2 T2 m T1 f dashpot with viscosity c and a spring with stiffness k in 1 v Figure 5(c) results from 2 v1 (d) ser pp= sdp PP() (,kP)(c,) k 10 1 0 f2 T2 b T 1 0 1 f1 ==pp1 1 , (49) 1 1 + 1 v >>kkHHc > c H 2 v1 (e) which shows that the admittance of a spring and dashpot in series is given by ()p kc1 + ()1 . Note that these three FIGURE 5 (a) A series connection of two springs with stiffnesses k1 series connections are symmetric structures. The power and k2. (b) A series connection of two dashpots with viscosities c1 and c2. (c) A series connection of a dashpot with viscosity c and transmission matrix of the series connection of a spring a spring with stiffness k. (d) A series connection of a mass with with stiffness k and a mass with inertia m in Figure 5(d) inertia m and a spring with stiffness k. (e) A series connection of is produced by an inerter with relative inertia b and a spring with stiffness k.

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 95 Consider the parallel connection of the structures illus- be the power transmission matrix that represents the ith trated in Figure 6, where, for all ij,{d 1,,f n}, structure in the parallel connection in Figure 6.

If two masses with m1 and m2 are connected vvij,,11= , (52) in parallel, the resulting structure is a mass with inertia vvij,,22= , (53) mm12+ . If a mass is connected in parallel with a spring, and it constrains the motion of the physical terminals, and the n spring cannot be compressed. The same situation occurs = ff11/ i, , (54) where a mass is connected in parallel with a dashpot. The i = 1 n instance where one of the structures in parallel is a mass is ff22= / i, , (55) not considered. However, each structure in a parallel con- i = 1 nection may be a series connection of masses and at least where fi,1 and fi,2 are the forces at the common physical one inerter, spring, or dashpot in series. terminals, and vi,1 and vi,2 are the velocities of the common As shown in “The Derivation of Ppar,” the power trans- physical terminals. Let mission matrix Ppar that relates (,fv11) to (,fv22) in Figure 6 is calculated by 3 pi,11 pi,12 Pi = (56) 2 ;pi,21 pi,22E 1 aacb- Ppar = , (57) b =1 c G where

n n n Structure 33pi,11 3 pi,22 ab==1 c = 1 //,,/ . (58) i ==1 pi,21 i 1 ppi,,21 i = 1 i 21

Note that division by zero would occur if one of the struc- tures were a mass; however, as discussed above, this case is Structure f T T f 2 2 2 1 1 not allowed. The expressions of (58) are given in [45], [46], and [7, pp. 10–36]. Related expressions are given in [13, pp. v2 v1 342, 343] and [47].

Example 2: Parallel Connection Consider the parallel connection of two structures, where one Structure structure is the series connection of a mass with inertia m1 n and a spring with stiffness k, and the other one is the series

connection of a mass with inertia m2 and a dashpot with vis- FIGURE 6 The parallel connection of n structures. The structures cosity c, shown in Figure 7. The power transmission matrix of are connected in parallel by rigidly joining their right physical ter- the series connection of the mass and the spring is given by minals to the physical terminal T1 and their left physical terminals to the physical terminal T2. 10 1 m1 p 1 m1 p 2 P1()p ==p p m1 p , (59) 1 ;01E + 1 > k H > kk H k m1 while the power transmission matrix of the series connec- f T T 2 2 1 f1 tion of the dashpot and the mass is given by

v2 v m2 c 1 R m2 p V 10 S + 1 m2 pW 1 m2 p S c W P1()p ==1 . (60) 01 1 S 1 W FIGURE 7 The series connection of a mass and spring is linked in ; E > c H S 1 W parallel with the series connection of a dashpot and mass. T c X Applying (57) and (58) to (59) and (60) yields

f m f2 + f T2 T1 f1 2 3 2 1 mc2 pp++km12mmpp++()12mc()+ kp Ppar ()p = 2 , v2 v1 ckp + = p mc1 pp++k G m (61) (a) (b) which is the power transmission matrix of the parallel con- FIGURE 8 (a) A mass with an external force, f, acting on it. (b) The nection in Figure 7. It can be seen that det Ppar = 1; however, external force f is applied to the terminal to the left of m. the structure is not symmetric.

96 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 MODELING EXTERNAL FORCES ft2 ()+ ft() 1 mfp 1 ()t Consider mass m pictured in Figure 8, where f is an exter- = . (64) = vt2 () G ;01E =vt1 ()G nal force acting on it. Using Newton’s second law, it follows from Figure 8 that MODELING STRUCTURES WITH BOUNDARY mvp 12()tf=+()tf()tf- 1 ()t . (62) CONDITIONS

Since vv21= , it follows that (,fv11) and ()f22, v are related by Example 3: The Single-Degree-of-Freedom Structure Consider the structure depicted in Figure 9(a), where exter- ft2 () 1 mfp 1 ()t - ft() nal force f is applied to the mass. The inerter, spring, and =+ (63) =vt2 ()G ;01E =vt1 ()G ; 0 E dashpot are connected in parallel, and the mass is con- nected in series with the inerter, spring, and dashpot. or, equivalently, ­Figure 9(b) shows the physical terminals of the structure. It

The Derivation of Ppar e consider the parallel connection of n = 2 power trans- Now, substituting (S22) into (S20) results in Wmission matrices; the case of n power transmission ma- a ac trices follows by induction. First, define pf11,,1112+=pf11 21 ()ff11 ++21 - d v1 . (S23) b cmb

p11, 1 p11, 2 P1 = (S13) Next, summing ;p12, 1 p12, 2E

p21, 1 p21, 2 fp12 =+11,,11fp1112 v1 (S24) P2 = (S14) ;p22, 1 p22, 2E fp22 =+21,,12fp1212 v1 (S25)

so that yields

f12 f11 f22 f21 ff12 +=22 pf11,,1112++pf11 21 ()pp11,,22+ 12 v1 . (S26) ==P1 ,,P2 (S15) ;;v2EEv1 ;;v2EEv1 Substituting (S23) into (S26) gives

where v2 is the common velocity of the physical output termi- a ac ff12 +=22 ()ff11 ++21 ++pp11,,2212 - d v1 nals of P1 and P2. It follows that b c b m 1 =+[(aaff11 21)(++cb[]pp11,,22+-12 d )]v1 b vp21=+,,21 fp11 1221v , (S16) 1 det P1 det P2 =+aa()ff11 21 +-cb + v1 . (S27) vp22=+,,21 fp21 2221v , (S17) b ;;c p12,,1 p221 EEm

P1 P2 which imply that Now consider the case where and represent series connections of masses, springs, and dashpots. It follows that p11, 1 pp11,,1122 == pf11, 111 =-v2 v1, (S18) detdPP12et 1, and thus (S27) implies that p12, 1 p12, 1

p21, 1 pp21,,1222 1 2 pf21, 121 =-v2 v1 . (S19) ff12 +=22 [(aaff11 ++21)(cb- )]v1 . (S28) p22, 1 p22, 1 b

Adding (S18) and (S19) yields Combining (S22) and (S28) yields (57). This proves (57) in the

case where P1 and P2 are each the power transmission ma- pf11,,1112+=pf11 21 advv21- , (S20) trices of the series connections of masses, springs, and dash-

pots. Note that det Ppar = 1. where Consider the case where P1 and P2 are each either the power

9 pp11,,1122 pp21,,1222 transmission matrices of the series connections of masses, springs, d =+. (S21) p12, 1 p22, 1 and dashpots (as in the previous instance) or the power transmis- sion matrices of the series connections of parallel connections of Next, dividing (S16) by P12, 1 and (S17) by P22, 1, adding the re- masses, springs, and dashpots. In this case, detdPP12==et 1, sulting equations, and dividing by b yields and again (57) holds. Continuing in a similar fashion, it follows that, 1 c for all structures constructed from series and parallel connections vf21=+()12fv11+ . (S22) bb of masses, springs, and dashpots, (57) holds.

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 97 follows from (57) that the power transmission matrix P of with T1, it follows that v1 = 0. Viewing the left edge of the the parallel inerter-spring-dashpot connection is given by mass as physical terminal T2, let v2 represent the velocity of the mass. Since no force is applied to the left edge of the 10 mass, it is clear that f2 = 0. Using the boundary conditions p 1 = f2 = P()p = 1 . (65) v 0 and 0, it follows from (64) that >bcpp2 ++k H 10 ft() 1 mp ft1() The wall is assumed to be an inertially nonrotating mas- = p vt2 () 01 1 0 = G ; E >bcpp2 ++k H ; E sive body, and it serves as physical terminal T1 and refer- R 2 V ence terminal T0. Let f1 represent the force applied to the S mp W ft1()+ ft1() S 2 W wall by the spring and dashpot, and since T0 is colocated bcpp++k = S W. (66) S p W ft1() S bcpp2 ++k W f T X

b It thus follows that k 2 m ()mb++ppck+ ft()= ft1 (), (67) c bcpp2 ++k p vt2 ()= 2 ft1 (). (68) (a) bcpp++k b Combining (67) and (68) yields T m T f2 + f = f 2 k 1 f1 p vt2 ()= 2 ft(). (69) v ()mb++ppck+ 2 c v1 = 0

Alternatively, solving (68) for f1 gives (b) bcpp2 ++k FIGURE 9 (a) An inerter, spring, and dashpot are connected in paral- ft1 ()= vt2 (), (70) p lel; this substructure is connected in series with a mass. The external force f is applied to the mass, which is sprung. (b) The physical termi- which expresses the unknown force f1 applied to the wall nal T1 is attached to the wall and thus has zero velocity. The physical by the structure in terms of v2. Substituting f1 given by (70) terminal T2 is attached to the left side of the mass and is free. into (67) and solving for v2 yields (69). Although this differ- ent derivation of (69) involves more steps, it utilizes (68), which can be viewed as a feedback connection, as illus- f trated in Figure 10 [where H denotes the transfer function f f = 0 f 2 1 in (68)]. This feedback connection specifies the unknown P P v2 2 1 v1 = 0 reaction force f1 applied by the wall in terms of the veloc- ity, v2, of the mass. (a) f Example 4: The Two-Degrees-of-Freedom Structure Consider the structure pictured in Figure 11(a), where the f f = 0 f 2 1 external force f is applied to the first mass. Note that the

v2 P2 P1 v1 = 0 springs and masses are connected in series. Figure 11(b) shows the physical terminals of the structure. Using the boundary

conditions v1 = 0 and f2 = 0, it follows from (63) that

H 0 ft1 () -ft() =+Pmms(,22pp)(Pk,)Pmms(,11pp)(Pk,) ;;vt2 ()EEcm00; E (b) (71)

FIGURE 10 (a) A block diagram of the single-degree-of-freedom R 1 4 2 V S [(mm12pp++()km21km22++km12 kk12)(ft1 )W structure in Figure 9 represented as the product of two elementary kk12 S 2 W power transmission matrices. The external force f is added to f2, = S -+()km12p kk12ft()] W, which is zero since the mass is sprung. The goal is to determine S 1 3 W S [(mk1 pp++12kfpp)(11tk)(- ft)] W the transfer function from the external force f to the velocity v2 of kk12 T X m2. (b) A modification of the block diagram in (a). The transfer (72) function H given by (70) specifies the force f1 on the wall in terms of v2. By relating v2 to f1, H closes a feedback loop. as illustrated in Figure 12(a). Therefore,

98 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 4 2 0 =+[(mm12ppkm12++km21 km22)]+ kk12ft1 () entries of a transmission matrix consist of two transmis- 2 sibilities, one stiffnance and one compliance. -+km12()p kf2 ()t , (73) Relating energy transmission matrices to power trans- 1 3 vt2 ()=+[(mk1 pp12+-kfpp)(11tk)(ft)]. (74) mission matrices using vq11= p and vq22= p yields kk12

Solving (73) for ft1 () yields f

k2 k1 2 km12()p + k2 m2 m1 ft1 ()= 4 2 ft(). (75) mm12pp++()km12 km21++km22 kk12

(a) Finally, substituting (75) into (74) gives k k m 2 f m 1 T T2 2 1 1 f = 0 f k2 p 2 1 vt2 ()= 2 2 2 ft(). (76) v2 v1 = 0 ()mk1 pp++12()mk22+ km2 p (b)

FIGURE 11 (a) A structure with two masses and two springs. (b) The Alternatively, solving (73) for ft() produces springs k1 and k2 and masses m1 and m2 are connected in series, and the external force f is applied to the terminal to the left of m1. 4 2 mm12pp++()km12 km21++km22 kk12 ft()= 2 ft1 (). (77) km12()p + k2 f f = 0 Next, substituting ft() given by (77) into (74) results in 2 f1 v2 4 2 P4 P3 P2 P1 v1 = 0 km21pp((mk2 ++12mk21mk++22mk))p 12k vt2 ()= 2 2 2 2 ft1 (). km12()pp++km21[( km12)( pp++kk22)]m2 (a)

(78) f f = 0 Therefore, 2 f1 v2 P P P P v = 0 2 2 2 2 4 3 2 1 1 km12()pp++km21[( km12)( pp++kk22)]m2 ft1 ()= 4 2 vt2 (), km21pp((mk2 ++12mk21mk++22mk))p 12k (79) H which expresses the unknown force f1 applied by the wall (b) in terms of v2. Substituting ft1 () given by (79) into (74) and FIGURE 12 (a) A block diagram of the two-degrees-of-freedom struc- solving for vt2 () yields (76). While this derivation of (76) ture in Figure 11 represented as the product of four elementary power is more tedious, it utilizes (79), which can be viewed as a transmission matrices. The external force f is added to the output of feedback connection, as depicted in Figure 12(b), where H m2. The goal is to determine the transfer function from the external denotes the transfer function in (79). This feedback con- force f to the velocity v2 of m2. (b) A modification of the block diagram in (a). The transfer function H given by (79) specifies the force f on nection specifies the unknown reaction force f1 applied 1 the wall in terms of v . By relating v to f , H closes a feedback loop. by the wall. 2 2 1

ENERGY TRANSMISSION MATRICES

FOR STRUCTURES f2 T2 T1 f1 Structure f An energy transmission matrix, E, of a structure with phys- 2 f1 q E # 1 ical terminals T1 and T2 is a 22 matrix that describes q2 q1 q2 the relationship between the forces f1 and f2 and the posi- (a) (b) tions q1 and q2 of T1 and T2, as shown in Figure 13(a). As depicted in Figure 13(b), E satisfies FIGURE 13 (a) A structure relating force f1 and position q1 to force f2 ft2 () ft1 () and position q2. The forces f1 and f2, which are absolute (through) = E()p , (80) variables, are applied to the physical terminals T and T , respec- ==qt2 ()GGqt1 () 1 2 tively. The positions q1 and q2 of T1 and T2, respectively, are rela- where tive (across) variables defined relative to the reference terminal, T . (b) The input–output representation of (5) showing the input e11 e12 0 E = . (81) signals f1 and q1 and output signals f2 and q2. The energy trans- ;e21 e22E mission matrix E(p) of the structure represents the relationship be- Note that e11 is a force transmissibility, e12 is a stiffnance, e21 tween the force and position of one physical terminal to the force is a compliance, and e22 is a position transmissibility. The and position of another physical terminal.

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 99 p11 pp12 The energy transmission matrices for series and parallel 1 0 1 0 EP==0 1 p21 . (82) connections satisfy the rules given for power transmis- > H ;0 pE > p22 H p p sion matrices.

The elementary energy transmission matrices for the mass, Example 5: Energy Analysis of the inerter, spring, and dashpot are given by Single-Degree-of-Freedom Structure Consider the structure in Figure 9(a), where external force f 2 3 1 mp is applied to the mass. The inerter, spring, and dashpot are Emm (,p),= (83) ;01E connected in parallel, and the mass is connected in series 10 with the inerter, spring, and dashpot. Figure 14 shows the 3 1 Ebin (,p),= 1 (84) physical terminals of the structure. It follows from (57) that >bp2 H the energy transmission matrix E of the parallel inerter- 1 0 3 spring-dashpot connection is given by Eks (,p),= 1 1 (85) > k H 1 0 10 3 E()p = 1 . (88) 1 2 1 Ecd (,p).= 1 (86) >bcpp++k H >cp H

As in the case of the elementary power transmission matrices, Let f1 represent the force applied to the wall by the spring

and dashpot, and since T0 is colocated with T1, it follows

det Emmi(,pp)(= det Ebn ,) that q1 = 0. Viewing the left edge of the mass as physical ter-

==detdEksd(,pp)(et Ec,)= 1. (87) minal T2, let q2 represent the position of the mass from the

reference terminal, T0. Because no force is applied to the left

edge of the mass, it follows that f2 = 0. Using the boundary b conditions q1 = 0 and f2 = 0, it follows from (64) that f + f = fT m 2 2 k T1 f1 2 10 c ft() 1 mp ft1() = 1 1 =qt2()G ;01E >bcpp2 ++k H ; 0 E q1 = 0 R 2 V S mp W q2 ft1()+ ft1() S bcpp2 ++k W = S W. (89) FIGURE 14 The physical terminal T is attached to the wall and thus S 1 W 1 2 ft1 () has zero position. The physical terminal T is attached to the left S bcpp++k W 2 T X side of the mass and is free. It thus follows that f f ++2 + f2 = 0 f1 ()mbppck ft()= ft1 (), (90) bcpp2 ++k q2 E2 E1 q1 = 0 1 qt2 ()= ft1 (). (91) (a) bcpp2 ++k f f f2 = 0 f1 Combining (90) and (91) results in

q2 E2 E1 q1 = 0 1 qt2 ()= ft(). (92) ()mb++pp2 ck+ H Finally, as in the case of power transmission matrices, (b) the feedback connection is shown in Figure 15, where H FIGURE 15 (a) A block diagram of the single-degree-of-freedom denotes the transfer function that satisfies structure in Figure 9 represented as the product of two elementary ft12()= Hq()p ()t , (93) energy transmission matrices. The external force f is added to f2, which is zero since the mass is sprung. The goal is to determine that is, the transfer function from the external force f to the position q2 of m2. (b) A modification of the block diagram in (a). The transfer Hb()pp=+2 ckp + . (94) function H given by (94) specifies the forcef 1 on the wall in terms of q2. By relating q2 to f1, H closes a feedback loop.

100 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 POWER TRANSMISSION MATRICES level). The latter is the natural choice in electrical systems. FOR ELECTRICAL SYSTEMS This point is discussed in [29]. In electrical systems, power is the product of potential E A longstanding difficulty in applying fI-vE concerns and current I. The unit of potential is the volt, where 1 V = 1 the ability to realize capacitors with masses. A capaci- joule/coulomb (J/C). The unit of current is the ampere, tor that has one terminal grounded can be realized by a where 1 A = 1 C/s. Current is the derivative of charge Q, mass. The mechanical analogy of an ungrounded capaci- that is, IQ= o . The product EI has the dimensions of tor is less obvious. This point is mentioned in [22, p. 20], power, whose unit is the watt, where 1 W = (1 V)(1 A) = which states, “Nongrounded capacitors have no mechani- (1 J/C)(1 C/s) = 1 J/s. cal analog.” One solution to this problem is discussed As with structures, power transmission matrices can be in [24, p. 234], which uses a transformer to produce an defined for circuits. This can be done by expressing an anal- “ungrounded” mass. The present article takes advantage ogy between mechanical systems and electrical systems of the inerter. and recasting the results obtained for mechanical systems. Another distinction between fE-vI and fI-vE is the type Taking advantage of the fact that the product of current and of external source that arises in an analogy to external potential has the dimensions of power, this can be done by force. In fE-vI, the external source is potential, whereas in 1) associating force ( f ) with potential (E) and velocity (v) fI-vE, it is current. Converting a structure to a circuit by with current (I) (the fE-vI analogy) or 2) associating force (f) means of fE-vI yields a circuit with an applied potential, with current (I), and velocity (v) with potential (E) (the fI-vE while the conversion through fI-vE produces a circuit with analogy). These analogies, which are shown in Table 2, have an applied current. various traditional names. For instance, fE-vI has names Two additional mechanical/electrical analogies can be including the Maxwell analogy, direct analogy, impedance defined for power transmission matrices. Replacing E and analogy, and effort-flow analogy, whereas fI-vE is termed IQ= o in fE-vI with Q and Eo , respectively, yields fQ-vEo . the Firestone analogy, inverse analogy, mobility analogy, The reference terminal for power transmission matrices is and across-through analogy. defined in terms of a potential rate. Although stiffness k One argument in favor of fE-vI is that the units of is analogous to capacitance, neither mass nor viscosity has potential [joule per coulombs (volts)] and the units of cur- analogs in terms of R, L, or C. rent [coulombs per second (amperes)] are closely aligned, respectively, with the units of force [joules per meter (newtons)] and the units of velocity (meters per second). TABLE 2 The mechanical-electrical analogies. In analogy Hence, fE-vI merely entails replacing meters with cou- fE-vI, the external source is potential, and the position lombs, and transforming mechanical systems to electrical and velocity reference terminals are charge and current systems involves replacing m, c, and k with L, R, and 1/C, levels, respectively. In analogy fQ − vE˙, the external source is respectively. Under this analogy, the reference terminal for current, and the position and velocity reference terminals are power transmission matrices is defined in terms of a cur- potential and potential-rate levels, respectively. In analogy fl-vE, the position and velocity reference terminals are rent level. magnetic-flux and potential levels, respectively. In analogy In fI-vE, force units (newtons) are analogous to cur- f U − vI˙, the position and velocity reference terminals are rent units [coulombs per second (amperes)], and veloc- current and current-rate levels, respectively. Note that, in ity units (meters per second) are analogous to potential analogies fQ − vE˙ and f U − vI˙, m and c have no analogs units [joule/coulombs (volts)]. The reference terminal in terms of R, L, and C. A may represent a mass or an inerter, depending on whether or not it is shunted to for power transmission matrices is defined in terms of ground. potential E. Transforming mechanical systems to electri- cal systems necessitates replacing m, c, and k with C, 1/R, Mechanical and 1/L, respectively. Variables f q v a m, b c k There are two reasons why fE-vI may seem to be more Electrical E Q = # I QIo = QIp = o L R 1/C natural than fI-vE as an analogy to mechanical systems. As analogy noted, replacing meters with coulombs transforms force fE-vI to potential and velocity to current. The analogy between Electrical Q = # I E Eo Ep C mass and inductance reflects the fact that a mass stores analogy o kinetic energy due to velocity v while an inductor stores fQ-vE “kinetic” energy due to the flow of charge (current). Simi- Electrical I U = # E Uo = E Up = Eo C 1/R 1/L larly, the analogy between stiffness and the reciprocal of analogy capacitance reflects the storage of potential energy. These fI-vE o p energy analogies are less clear for fI-vE. On the other hand, Electrical U = # E I I I L the reference terminal for fE-vI is a current level, in contrast analogy fU-vIo with the reference terminal for fI-vE (which is a potential

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 101 Alternatively, replacing I and E in fI-vE by U = # E and case of the earth ground, whose neutrality is unaffected by Io in fI-vE, respectively, yields fvU- Io, where U is magnetic accumulating or shedding charge, the inertial velocity of an flux with units of webers and 1 Wb = 1 V-s. The refer- inertially nonrotating massive body is unaffected by forces. ence terminal for power transmission matrices is defined Consequently, under fI-vE, Newton’s second law f = mpv is in terms of a current rate. While stiffness k is analogous precisely I = CpE, where an electrical ground with infinite to inductance, neither mass nor viscosity have analogs in capacity plays the role of an inertial frame. terms of R, L, or C. Although the choice has pros and cons, we adopt the It is interesting to note that none of the analogies in Table 2 analogy fI-vE. Under it, is the transfer explicitly requires a reference terminal that is analogous to function from current to potential. The analogous trans- the unforced particle specified by Newton’s second law for fer function is from force to velocity, which, according to force and inertia [21]. In fI-vE, only potential difference is rel- Table 1, is mechanical admittance. Under fI-vE, electrical evant, and there is no need to define an inertial ground. Mass impedance and admittance are inconsistent with mechani- is analogous to capacitance, and the idealized model of a cal impedance and admittance. Within the circuit setting, capacitance assumes the ability to send and receive electrons a fourth element can be considered, the memristor, whose to and from ground without affecting the ground’s poten- mechanical analog is the memdashpot. For details, see tial. The electrical ground in a circuit is implicitly assumed “What Is a Memristor?” to have infinite capacity. Specifically, the earth ground pro- vides an approximately infinite volume that can absorb or POWER TRANSMISSION MATRICES FOR CIRCUITS lose electrons without losing its charge neutrality. Analo- Consider the circuit with physical terminals T1 and T2 gously, in structural applications, an inertially nonrotating in Figure 16. Under fI-vE, power transmission matrices massive body provides a reference point for defining the transform current and potential inputs to current and inertial acceleration of a particle subject to forcing. As in the potential outputs, that is,

What Is a Memristor? n the analogy fI-vE, force f and velocity v are analogous to cur- the relationship between charge and magnetic flux. It might be Irent I and potential E, respectively. The integral of force (mo- expected that the integration operators would cancel and that the mentum h) and the integral of velocity (position q) are analo- remaining edge would be equivalent to resistance. The idealized gous to charge Q and magnetic flux U, respectively. These four element of the remaining edge turns out to be a hysteretic opera- mechanical quantities and their electrical analogs can be used tor; this is the memristor [49]. In the mechanical instance, three to label the four vertices of two squares displayed in Figure S2. of the edges correspond to masses, springs, and dashpots. The In the electrical case, three of the edges correspond to resistors, remaining edge is a hysteretic dashpot called a memdashpot [50], capacitors, and inductors. The remaining edge can be viewed as [51], which is the mechanical analog of the memristor.

. Capacitor . Mass E = Q = I v = q h = f 1/C 1 1/m 1 c R 1/ Resistor Dashpot Memristor Memdashpot

. Inductor . I = Q = E Spring q = v 1 f = h 1 L 1/k

(a) (b)

FIGURE S2 A memristor and memdashpot. (a) The four vertices of the square correspond to the four electrical variables that are analogous to force and velocity and their integrals in fI-vE. The four edges of the square correspond to resistors, capacitors, and inductors; the remaining edge is the memristor, which is an idealized hysteretic device. (b) The mechanical analog of the memris- tor; that is, the memdashpot.

102 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 I2 I1 = P . (95) ;;E2EEE1 I2 T2 T1 I1 Circuit When two circuits are joined, the common physical termi- nals constitute a node through which current passes. The E2 E1 arrows for I1 and I2 indicate the direction of current flow in the case where E1 and E2 are constant, EE21> , and II21= ; in this instance, II12= > 0. In the case where E1 and E2 (a) are constant, EE21< , and II21= , then II12= < 0, and the current flows opposite to the indicated direction. This sign I2 I1 P convention, which is consistent with the one for forces and E2 E1 structures, avoids the need for the minus signs that appear (b) in the definition of ABCD matrices (as explained in “Power Transmission Matrices and ABCD Parameters”). FIGURE 16 (a) A circuit relating current I and potential E to current I As a special case, consider the capacitor with physical ter- 1 1 2 and potential E2. The currents I1 and I2, which are absolute (through) 1 2 0 minals T and T and reference terminal T that appears in variables, flow through the physical terminalsT 1 and T2, respectively, Figure 17(a). For a capacitor with capacitance C, it follows that which are nodes. The arrows for I1 and I2 indicate the direction of cur- rent flow whenE 2 > E1 and I2 = I1; in this case, I1 = I2 > 0. When E2 < E1 11 and I = I , I = I < 0, and the current flows opposite to the indicated Et21()-=Et() It12()= It(). (96) 2 1 1 2 CppC direction. The potentials E1 and E2 of T1 and T2, respectively, are rela- tive (across) variables defined in relation to the reference terminal T0 Thus the elementary power transmission matrix is given by (ground). (b) The power transmission matrix P of the circuit relates

the current and potential at T1 to the current and potential at T2. 10 3 PCcap(,p).= 1 1 (97) >Cp H

For the resistor with resistance R shown in Figure 17(b), Power Transmission Matrices it follows that and ABCD Parameters related representation of power transmission matrices is given by ABCD parameters, which have the form Et2()-=Et112() RI ()tR= It() (98) A

E2 E1 = PABCD , (S29) and thus ;;I2 EE-I1 where Et21()=+RI ()tE1()t . (99) A B PABCD = . (S30) Hence, the elementary power transmission matrix is found by ;C DE

3 10 Consequently, P can be written in terms of ABCD param- PRres(,p).= (100) ;R 1E eters as C -D For the inductor with inductance L shown in Figure 17(c), P = . (S31) ;A -BE it follows that

The minus sign preceding I1 reflects the convention that

Et2 ()-=Et112() LIpp()tL= It(). (101) the current is positive in the case where it is flowing into either physical terminal [14, p. 11.6]. Hence, the elementary power transmission matrix is given by

T C I = I T R T T L T I I2 = I1 2 T1 I1 2 1 2 1 I1 I2 = I1 2 1 1

E2 E1 E2 E1 E2 E1

(a) (b) (c)

FIGURE 17 (a) A capacitor with capacitance C and current I1 = I2 flowing through it with potentials E1 and E2 at its physical terminals T1 and T2, respectively. (b) A resistor with resistance R and current I1 = I2 flowing through it with potentialsE 1 and E2 at its physical terminals T1 and T2, respectively. (c) An inductor with inductance L and current I1 = I2 flowing through it with potentialsE 1 and E2 at its physical terminals T1 and T2, respectively.

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 103 3 10 Under fI-vE, power transmission matrices transform PLind(,p).= (102) ;Lp 1E current and potential inputs into current and potential outputs, according to (95). Energy transmission matrices Note that power transmission matrices (97), (100), and transform current and magnetic-flux inputs into current (102) for the capacitor, resistor, and inductor, respectively, and magnetic-flux outputs, that is, are analogous to power transmission matrices (14), (23), I2 I1 and (19) for the inerter, dashpot, and spring, respectively. = E()p . (107) ;;UU2EE1 To obtain an analog for the mass, we consider the shunted capacitor in Figure 18(a). Letting I3 denote the current The elementary energy transmission matrices for the capac- through the capacitor, it follows that II21=+I3 . Since itor, resistor, and inductor are given by

EE12= is the potential drop across the capacitor, 1 0 3 1 II21=+CpE1 . (103) ECcap (,p),= 1 (108) >Cp2 H The elementary power transmission matrix for the shunted 10 capacitor is produced by 3 ERres(,p),= R 1 (109) > p H 3 1 Cp PCcap,sh (,p).= (104) 3 10 ;01E ELind ()= . (110) ;L 1E Likewise, the elementary power transmission matrices for the shunted resistor and shunted inductor are given by RECIPROCITY AND SYMMETRY FOR CIRCUITS 1 Consider the circuit modeled by the power transmission 3 1 Pres,sh ()R = R , (105) matrix in Figure 16. Shorting terminal T2 to ground yields = G 0 1 E2 = 0, and (95) becomes 1 3 1 Ip211 p12 I1 PLind,sh (,p).= Lp (106) = . (111) >01H ;;0EEp21 p22 ;E1E

Note that the shunted capacitor is analogous to the mass, and The transfer function G12 from I2 to E1 is given by the shunted resistor and shunted inductor are analogous to the p21 shunted dashpot and shunted spring, respectively. Observe G12 = . (112) pp12 21 - pp11 22 that none of the circuits in Figure 17 is connected to ground, and T0 serves as an arbitrary potential reference level; in effect, Alternatively, shorting T1 to ground yields E1 = 0, and

T0 may be a floating ground. However, in Figure 18, all three (95) becomes circuits are connected to ground, and T0 is assumed to be an I2 p11 p12 I1 earth ground, whose potential is constant. = . (113) ;;E2EEp21 p22 ;0E

ENERGY TRANSMISSION MATRICES The transfer function G21 from –I1 to E2 is given by FOR ELECTRICAL SYSTEMS Gp21 =- 21 . (114) For energy transmission matrices, the analogies between mechanical and electrical systems require a variable that corre- The circuit is reciprocal if the transfer functions G12 and G21 sponds to position, where the electrical analog of position is the are equal, that is, if and only if det P = 1, as in the case of integral of the electrical analog of velocity. For fE-vI, the electri- structures. This property is demonstrated in [14, p. 11.3]. cal analog of position is charge; for fQ-vEo , it is potential; for A circuit is symmetric if reversal of its physical ter- fI-vE, it is magnetic flux; and for fvU- Io, it is current. minals yields the same power transmission matrix.

I T T1 I1 T 2 2 I2 T2 1 I1 I2 T2 T1 I1

E C E E E E L 2 1 2 R 1 2 E1

(a) (b) (c)

FIGURE 18 (a) A shunted capacitor with capacitance C and currents I1 and I2 flowing through its physical terminals T1 and T2 with poten- tials E1 and E2, respectively. (b) A shunted resistor with resistance R and currents I1 and I2 flowing through its physical terminals T1 and T2 with potentials E1 and E2, respectively. (c) A shunted inductor with inductance L with currents I1 and I2 flowing through its physical terminals T1 and T2 with potentials E1 and E2, respectively.

104 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 As in structures, a circuit is symmetric if and only if The circuits in Figure 19(a) and (b) can be connected in the diagonal entries of its power transmission matrix series, and the corresponding power transmission matrices are equal, that is, if and only if the current and potential are provided by the rules for series connections of struc- transmissibilities are equal. Symmetric power transmis- tures. Consequently, the power transmission matrix for the sion matrices play a role in the analysis of transmission circuit in Figure 19(c) is calculated by lines as described in “Power Transmission Matrices and R 1 V Transmission Lines.” S 1 W S Z2 ()p W PP12()pp()= , (117) S Z1 ()p W SERIES AND PARALLEL CONNECTION OF CIRCUITS SZ1 ()p 1+ W Z2 ()p The power transmission matrices produced by (97)–(106) are T X special cases of the circuits shown in Figure 19. The circuit whereas the power transmission matrix for the circuit in with impedance Z1 in Figure 19(a) is not grounded. There- Figure 19(d) results from fore, II21= , and the power transmission matrix is given by Z1 ()p 1 10 1+ PP21()pp()= Z2 ()p Z2 ()p . (118) P1 ()p = . (115) ;Z1 ()p 1E > Z1 ()p 1 H Likewise, the power transmission matrix for the shunt circuit with impedance Z2 in Figure 19(b) is found by TRANSFORMING STRUCTURES INTO CIRCUITS AND VICE VERSA One of the benefits of analogies is the ability to convert a 1 1 P2 ()p = Z2 ()p . (116) structure into a dynamically equivalent circuit and vice >01H versa. As depicted in [20], this can be done under fE-vI by

Power Transmission Matrices and Transmission Lines ne of the main applications of power transmission matrices it follows that [12, p. 213] is to approximate the modeling of transmission lines [5, Ch. 1, O A B cosh a Za0 sinh pp. 200–218], [12, pp. 212–214], [13, pp. 361, 362]. Although = sinh a (S35) C A cosh a ; E > Z0 H a transmission is a distributed , an approxi- mate model can be constructed by viewing the transmission and thus [52, p. 32] line as the series connection of n identical power transmission A B n cosh an Za0 sinh n matrices. Assuming that each circuit in the series connection = sinh an . (S36) C A cosh an ; E > Z0 H is symmetric, it follows that

If the is terminated with the impedance Z0, n In A B I1 the input impedance Zin of the transmission line is produced = . (S32) ;;EnEEC A ;E1E by [12, p. 214]

Eanncosh nZ+ 0 Iasinh n Defining Zin = . (S37) (/EZnn0)sinh an + Iacosh n

3 -1 aA= cosh (S33) These expressions can be used to analyze wave propagation 3 ZB0 = /,C (S34) in transmission lines [5, Ch. 1, pp. 200–218], [12, pp. 214–217].

Z I = I 1 I T Z1 I T Z1 2 1 T2 T1 I1 2 2 T1 I1 I2 T2 T1 I1 2 2 T1 I1

E E Z E Z E Z2 E E2 1 2 2 1 E2 2 1 E2 1

(a) (b) (c) (d)

FIGURE 19 (a) An ungrounded circuit with impedance Z1. (b) A shunted circuit with impedance Z2. (c) An ungrounded circuit with imped- ance Z1 in series with a shunted circuit with impedance Z2. (d) An ungrounded circuit with impedance Z1 in series with a shunted circuit with impedance Z2, where the order of the circuits is reversed relative to (c).

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 105 replacing series connections with parallel connections and R 1 0V S W parallel connections with series connections. Under fI-vE, the p P()p = S 1W. (119) S 2 p 1 W conversion is more direct, with series connections of struc- S Cp ++ W tures replaced by series connections of circuits and likewise T RLX for parallel connections. To demonstrate that feature, the cir- Note that the parallel connection of the capacitor, induc- cuit analogs of Examples 3 and 4 are constructed. In these tor, and resistor is linked in series with capacitor C0 (which examples, external forces are replaced by current sources. is shunted to ground), as shown in Figure 19. Using the

boundary conditions E1 = 0 and II2 = (where I2 denotes

Example 6: Circuit Analog of the Single- the current flowing through T2), it follows from (118) that Degree-of-Freedom Structure R 1 0V Under fI-vE, Figure 20 presents the circuit analog of the S W It() 1 C0 p p It1 () single-degree-of-freedom structure from Figure 9(a). The = S 1W Et2 () 01S 2 p 1 W 0 inerter b, spring k, and dashpot c in parallel in Figure 9(a) ;;EES Cp ++ W; E T RLX are replaced by the capacitor C, inductor L, and resistor R 2 V S C0 p W R, respectively. The RLC components are connected in It1 ()+ It1 () S 2 p 1 W series with the capacitor C0, which is shunted to ground in S Cp ++ W = S RL W (120) ­analogy to the mass m. Furthermore, the external force f is p , S It1 () W replaced by the I. Therefore, S p W Cp2 ++1 S RL W T X C where I1 denotes the current flowing through T1 . Next,

T2 L 2 p 1 ()CC0 ++p + RL It()= It1 (), (121) R p Cp2 ++1 RL

E2 C0 p Et2 ()= It1 (). (122) I2 = I 2 p 1 T1 Cp ++ I1 RL E1 = 0 Combining (121) and (122) yields FIGURE 20 A circuit analog of the single-degree-of-freedom struc- p ture shown in Figure 9. The capacitors C and C0, the inductor L, Et2 ()= It(). (123) 2 p 1 and the resistor R in parallel are analogous to the inerter, mass, ()CC0 ++p + spring, and dashpot in parallel. The current source plays the role RL of the external force. Example 7: Circuit Analog of a Single- Degree-of-Freedom Structure Consider the single-degree-of-freedom structure detailed in f Figure 21(a), where the mass and inerter are in series, with a k spring and dashpot in parallel. The transmission matrix of the parallel spring–dashpot connection results from m b c 10 = p P()p 1 . (124) >ckp + H (a) L Moreover, the parallel spring–dashpot connection is linked C T2 in series with the inerter and the mass, and thus, R R 2 V S mbpp++cm()bk++()mb W mp ft() S bcp + k W ft1 () E2 C0 () = S 2 W . (125) I2 = I =vt()G bcpp++k ; 0 E I T1 S 1 W 1 S bcpp()+ k W E1 = 0 T X

(b) It follows from (125) that FIGURE 21 (a) A single-degree-of freedom structure with a mass and inerter in series with a spring and dashpot in parallel. (b) The mbpp2 ++cm()bk++()mb ft()= ft1 (), (126) analogous circuit. bc()p + k

106 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 bcpp2 ++k Example 8: Circuit Analog of a Two-Degrees- vt()= ft1 (), (127) bcpp()+ k of-Freedom Structure Under fI-vE, Figure 22 displays the circuit analog of the and thus of the two-degrees-of-freedom structure in Figure 11(a). bcpp2 ++k Masses m1 and m2 and springs k1 and k2 are replaced by vt()= ft(). (128) pp((bm 2 ++bm)(ckp ++bm)) shunted capacitors C1 and C2 and inductors L1 and L2, respectively, which are connected in series. The external Under fI-vE, Figure 21(b) shows the circuit analog force f is replaced by the current source I, which is added of the single-degree-of-freedom structure displayed in between C1 and L2. In analogy to (72), where m1, m2, k1,

Figure 21(a). The parallel connection of the spring k and and k2 correspond to C1, C2, 1/L1, and 1/L2, respectively, dashpot c, which are connected in series with the inerter we have (134), shown at the bottom of the page, and thus, b in Figure 21(a), is replaced by the ­parallel connection of the inductor L and resistor R, which are connected I2 = 0 in series with the capacitor C. The RLC components are T2 connected in parallel with capacitor C0, which is shunted to ground in analogy to the mass m. The external force L C C L1 f is replaced by the current source I. In analogy to (65), E2 2 2 I 1 where b, k, and c are analogous to C, 1/L, and 1/R, respec- I1 T1 tively, the power transmission matrix P of the capacitor- E1 = 0 inductor-resistor connection results from FIGURE 22 The circuit analog of the two-degrees-of-freedom struc- ture in Figure 11. The capacitors C and C and inductors L and L 10 1 2 1 2 in parallel are analogous to the masses m and m and the springs P()p = CLRLpp2 ++R . (129) 1 2 1 k1 and k2, respectively. The current source I applied between > CRpp()+ L H capacitor C1 and inductor L2 is analogous to the external force f applied between mass m1 and spring k2. Note that the capacitor-inductor-resistor connection is in series to capacitor C0 (which is shunted to ground), as in T2 Figure 19. Using the boundary conditions E1 = 0 and II2 = , it follows from (118) that I2

+ C R R 2 V E2 = E E S CC0 pp()LR ++LRp W – 1 + C0 p S W T1 It() CRpp()+ L It1 () I1 = S 2 W . (130) E = 0 ;;Et2 ()EECLRLpp++R 0 1 S 1 W S CRpp()+ L W T X (a)

It thus follows that I1 = 0 T 2 T 2 1 CC0 LRpp++()CC00LC++()CR It()= It1 (), (131) I CR()+ Lp 2 2 E = E + C R E CLRLpp++R 2 – E 1 Et2 ()= It1 (). (132) CRpp()+ L

Substituting I1 from (132) into (131) produces (b) Cpp2 ++11 RL FIGURE 23 A resistor-capacitator circuit with a voltage source. In Et2 ()= It(). (133) 2 11 (a), only the capacitor is modeled as shunted; in (b), both the resis- pp((CC0 ++CC00)(p ++CC)) R L tor and capacitor are modeled as shunted.

2 2 0 CL22p ++1 CC21ppL1 1 C1 p It1 () -It() = + ;Et2()E = L2 p 1 GGeo= L1 p 1 ;;00EE 2 4 2 2 2 -+()CL22p 11It()++()CCLL1 212pppCL11 ++CL22 CL21p + It1() = 2 , (134) = -+LI2 pp()CL112LLp ++12LI1()t G

OCTOBER 2019 « IEEE CONTROL SYSTEMS MAGAZINE 107 2 4 2 01=-()CL22pp++It() (CCLL1 212 + CL11p is relative. This article presents a self-contained treatment 2 2 of the application of power and energy transmission matri- ++CL22ppCL21 + 1)(It1 ) (135) ces to modeling mechanical systems composed of masses, =- ++2 + Et22() Lt()ppIC()112LLp LL12It1 (). (136) inerters, springs, and dashpots. The examples illustrate the role of feedback connections in deriving transfer func-

Solving (135) for I1 yields tions. The across-through analogy between structures and circuits is used to obtain analogous results for electrical 2 CL22p + 1 systems. Through further analogies, this technique can be It1 ()= 4 2 2 2 It(). CCLL1 212p ++++CL11pppCL21 CL22 1 applied to acoustic and thermal systems. (137) We believe that power and energy transmission matri- Substituting (137) in (136) produces ces provide an elegant approach to modeling structures and circuits. Although this technique is at least 80 years 1 p old, it has gained very little traction in absolute terms and L2 Et2 ()= It(), (138) relative to the literature on bond graphs. One reason for this 2 112 C2 2 C1 pp++C2 + p ccL1 mmL2 L2 deficit is that the development of transmission matrices in structures and circuits has largely occurred independently, which is analogous to (76). and a complete treatment is not available. Another possible reason, as we have found, is that care is needed in working Example 9: Circuit With a Voltage Source with transmission matrices to understand the roles of iner- Consider the resistor-capacitator circuit in Figure 23, where tial and electrical grounds, include the inerter, and formu- the goal is to determine the transfer function from the late conventions for obtaining the correct signs that respect input voltage E to current I2 . Only the capacitor is mod- Newton’s third law and the direction of current flow. eled as shunted with the boundary conditions E1 = 0 and It is interesting that circuits are composed of three ele-

EE2 = , as seen in Figure 23(a). Using the corresponding ments (resistors, inductors, and capacitors), whereas struc- power transmission matrices yields tures are composed of four elements (masses, inerters, springs, and dashpots), with the restriction that a mass can- I211 Cp 10 I = . (139) not be placed in parallel with an inerter, spring, or dashpot. ;;EEE01;;R 10EE This discrepancy can be traced to the fact that ball-screw and Therefore, rack-and-pinion inerters take advantage of coupling between translational and rotational motion, which has no counter- It21()=+()1 RCp It(), (140) part in electrical components. Under the analogy fI-vE, a

Et()= RI1 ()t . (141) capacitor may represent a mass or an inerter; likewise, under the analogy fE-vI, an inductor may represent those compo-

Substituting I1 from (141) into (140) yields nents. The analogies between structures and circuits possess subtleties and surprises that warrant future investigation. 1 It2 ()=+CEp ()t . (142) In view of the analogy between structures and circuits (not ` R j to mention acoustics and thermodynamics), it is reasonable to Alternatively, the capacitor and resistor are both mod- expect that modeling techniques applicable in one domain eled as shunted with the boundary conditions EE2 = and are applicable in other ones. For example, Kirchhoff’s current

I1 = 0, shown in Figure 23(b). Using the corresponding power law has the mechanical interpretation that the summation transmission matrices produces of forces acting on a particle with zero mass is zero. Under the analogy fI-vE, Newton’s second law is equivalent to the 1 o I2 1 Cp 1 0 capacitor equation IC= E. Along the same lines, Lagrang- = R . (143) ;;EEE01=0 1 G ;E1E ian dynamics can be used to model circuits [48]. These and other ramifications of dynamical analogies remain to be fully Therefore, explored to facilitate control-system applications. 1 It21()=+CEp ()t , (144) ` R j ACKNOWLEDGMENTS We thank Gary H. Bernstein for discussions on earth Et()= Et1 (). (145) grounds and Kent Lundberg for help with constructing cir-

Substituting E1 from (145) into (144) yields (142). cuit diagrams. Our interest in this topic was motivated by [27] and [28]. It is hoped that the present article will encour- CONCLUSIONS age interested readers to revisit the use of power trans- Transmission matrices relate pairs of power-conjugate vari- mission matrices for physical-system modeling and take ables, where one variable is absolute and the other variable advantage of this method for control-system applications.

108 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2019 AUTHOR INFORMATION [21] D. S. Bernstein, “Newton’s frames,” IEEE Contr. Syst. Mag., vol. 28, Khaled F. Aljanaideh ([email protected]) received the pp. 17–18, Feb. 2008. [22] I. J. Busch-Vishniac, Electromechanical Sensors and Actuators. New York: B.S. degree in mechanical engineering from Jordan Univer- Springer-Verlag, 1999. sity of Science and Technology, Irbid, Jordan, in 2009. He re- [23] P. P. L . Regtien, Sensors for Mechatronics, 3rd ed. Amsterdam, The Neth- ceived the M.S.E. and M.S. degrees in aerospace engineering erlands: Elsevier, 2012. [24] J. L. Shearer, A. T. Murphy, and, H. H. Richardson, Introduction to System and applied mathematics and the Ph.D. degree in aerospace Dynamics. Reading, MA: Addison-Wesley, 1967. engineering from the University of Michigan, Ann Arbor, in [25] C. W. De Silva, Mechatronics: An Integrated Approach. Boca Raton, FL: 2011, 2014, and 2015, respectively. He is an assistant profes- CRC, 2005. [26] E. Skudrzyk, Simple and Complex Vibratory Systems. Philadelphia, PA: sor in the Department of Aeronautical Engineering, Jordan Univ. of Pennsylvania Press, 1968. University of Science and Technology. He was a postdoctoral [27] J. C. Willems, “The behavioral approach to open and interconnected research fellow in the Department of Aerospace Engineering systems,” IEEE Contr. Syst. Mag., vol. 27, no. 6, pp. 46–99, 2007. [28] J. C. Willems, “Terminals and ports,” IEEE Circ. Syst. Mag., vol. 10, no. 4, at the University of Michigan from 2015 to 2016, where he also pp. 8–26, 2010. was a part-time visiting assistant professor from 2017 to 2019. [29] J. Miles, “Applications and limitations of mechanical-electrical analo- His research interests include system identification and fault gies, new and old,” J. Acoust. Soc. Amer., vol. 14, no. 3, pp. 183–192, 1943. [30] S. Rubin, “Mechanical immittance- and transmission-matrix con- detection for aerospace and mechanical systems. cepts,” J. 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