derivation of input-state-output port-hamiltonian systems from bond graphs

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Simulation Modelling Practice and Theory 17 (2009) 137–151

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Derivation of Input-State-Output Port-HamiltonianSystems from bond graphs

Alejandro Donaire a,b,*, Sergio Junco a

a Departamento de Electronica, Facultad de Ingenierıa, Universidad Nacional de Rosario, Riobamba 245 bis, S2000EKE Rosario, Argentinab Consejo Nacional de Investigaciones Cientıficas y Tecnicas – CONICET, 27 de febrero 210 bis, S2000EZP Rosario, Argentina

Received 23 April 2007; received in revised form 21 November 2007; accepted 16 February 2008Available online 23 February 2008

Abstract

This paper presents methods to obtain models in the form of Input-State-Output Port-Hamiltonian Systems from cau-sal nonlinear bond graph models. This is done first establishing equivalences among key variables in both domains throughthe comparison of the expressions of the stored system energy in both formalisms. Later, with the help of the general field-representation of bond graphs and its associated standard implicit form, the functions characterizing this class ofPort-Hamiltonian Systems, i.e., interconnection, dissipation and input/output matrices, as well as their properties, areimmediately expressed in terms of bond graphs parameters. Under suitable assumptions, the method supports the directderivation of Input-State-Output Port-Hamiltonian Systems – which is an explicit type of PHS – even from bond graphshaving causally coupled dissipators and storages in derivative causality, which are known to imply algebraic and implicitdifferential equations. The methods are illustrated with some application examples covering different causal situations.Besides its intrinsic interest as a technique for model conversion, the contribution is seen as a useful step towards imple-menting Port-Hamiltonian based control system design methods with the support of BG techniques.� 2008 Elsevier B.V. All rights reserved.

Keywords: Bond graphs; State-Input-Output Port-Hamiltonian Systems; Model equivalences

1. Introduction

Stemming from the well-known Hamiltonian formalism rooted in classical mechanics, Port-HamiltonianSystems (PHS) constitute an important class of models for (possibly) nonlinear physical systems. Finite-dimensional PHS result from the network modeling of lumped-parameters physical systems with independentstorages and dissipative elements that exchange energy with the environment through power ports [20,21]. AsPHS clearly express the phenomena of storage, dissipation and exchange of energy in the system, they havebeen successfully used – in their particular form of Port-Controlled Hamiltonian Systems with dissipation,

1569-190X/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.simpat.2008.02.007

* Corresponding author. Address: Departamento de Electronica, Facultad de Ingenierıa, Universidad Nacional de Rosario, Riobamba245 bis, S2000EKE Rosario, Argentina. Tel.: +54 341 4808543.

E-mail addresses: [email protected] (A. Donaire), [email protected] (S. Junco).

mailto:[email protected]

mailto:[email protected]

138 A. Donaire, S. Junco / Simulation Modelling Practice and Theory 17 (2009) 137–151

or PCHD – for controller design of physical system based on passivity theory, such as energy shaping, inter-connection and damping assignment [17].

The bond graph (BG) formalism is a powerful modeling technique for physical systems [15] whose graph-ical description directly reveals the management of energy in the system (storage, dissipation, etc.) as well asthe interconnection structure through which internal and external power exchange occurs. Simultaneously, thegraphical augmentation of BGs with the causal relationships among the system variables allows the formula-tion of different types of mathematical models. The simultaneous exhibition of physical and mathematicalinformation makes of BG a suitable tool to support engineers in many stages of system design and analysis,involving modeling, simulation, choice of instrumentation/actuator architecture, controller synthesis andsupervision [4,11,12,15,18,19]. There are several research results implementing linear and nonlinear formalmethods for control system analysis and synthesis in the BG domain, like in [1,4,9,13,19,24]. The graphical,network-oriented nature of BGs models, its closeness to engineering understanding, and the concurrenceon them of both, the physical and the mathematical system description, make appealing and possible toexploit the empirical, physical or engineering knowledge that the designer has about the system. In this sense,heuristic controllers leading to energy and power shaping methods have been presented in [14].

The common physical foundations of PHS and BG modeling, and the interest of performing PCHD-basedcontrol synthesis methods with the support of BG techniques, motivated the research leading to the resultspresented in this paper. While some results in this direction are presented in [7] (an improved version of[6], a conference paper), a companion paper appearing also in this journal, this paper concentrates in the der-ivation of PHS from BG. Here, nonlinear BGs are considered which are causally augmented following thesequential causality assignment procedure (SCAP-procedure, [15]). This procedure entails the best effort tomaximize the number of BG-storages in integral causality, but depending on the interconnection structureit is possible that some of them remain in derivative causality, and that some causally coupled resistors appear,yielding a BG that here will be called a (causally) constrained BG. Both situations, associated with algebraicconstraints, lead to mathematical models in the form of implicit DAE-systems [3].

Despite this fact, under certain solvability assumptions related to properties of the BG elements, themethod provided in this paper directly yields explicit dynamic models pertaining to the important PHS classknown as Input-State-Output (ISO) PHS, even for constrained BG. Starting with the total energy as commonsystem function in both formalisms, and equating the corresponding expressions of the energy stored in thesystem, the equivalence among key variables in both formalisms is first established. These are the state vari-ables, their derivatives and the energy gradient in the PHS formulation, and – basically, for the simplest causalsituation in the BG domain – the energy variables, and the input and output of the storages on the BG side.Then, with the help of the general field-representation of bond graphs and its associated standard implicitform, the functions characterizing the Input-State-Output PHS, i.e., the interconnection, dissipation andinput/output matrices, as well as their properties, are immediately expressed in terms of bond graphs param-eters. Closely related results are presented in [10], where a geometric approach to the dynamic system definedby a BG is followed. It is first shown that a Dirac structure can be associated to the (power-conserving) inter-connection structure of an acausal BG. It is then proved that the equations describing the BG correspond to aPHS, which are in general implicit PHS, as Dirac structures allow handling systems with algebraic constraints[21]. Further, after a discussion of well-posedness of the models, simulatable representations are provided,whose differential indexes are analyzed in view of numerical simulation.

Founded on the acausal description of the BG interconnection by the Dirac structure, [10] provides a con-cise formulation that, at the same time, can be regarded as a mathematical theory of BG (valid, in fact, forgeneral network-type representations of physical systems [21]). The causal approach followed in this paper– which uses standard BG techniques familiar to the engineering BG community – dictates not a theoretical,but a procedural, equational approach. Though the ISO PHS – the final format of the models derived in thispaper, as already discussed – is a special case of the implicit PHS in [10], the methods in this paper allow tohandle BG with algebraic constraints. What makes possible the immediate formulation of the explicit modelISO PHS from a constrained BG is the use of the causality rules directly in the BG domain.

The remaining of the paper is organized as follows: in Section 2 the basic features of PCHD and BG modelsare summarized, and the equivalence of variables is given. Section 3 derives PCHD models from BG via its

A. Donaire, S. Junco / Simulation Modelling Practice and Theory 17 (2009) 137–151 139

state-space implicit form. Three application examples are presented in Section 4. Finally, Section 5 reportssome conclusions.

2. Input-State-Output Port-Hamiltonian Systems and bond graphs

2.1. ISO PH systems

The standard Hamiltonian equations of motion of autonomous conservative systems, formulated in clas-sical mechanics, has been extended to the explicit equation system (1) with the introduction of the Rayleighdissipation function Rð _qÞ and the vector of generalized forces T. There, H(q,p) is the total energy of the systemexpressed as a function of the (vectors of) configuration variables q and generalized momenta p. This kind ofsystems has shown successful to represent many classes of physical systems common in engineering problems[16,22]. The attention given by system theoreticians to this important formalism motivated its generalizationto the form (2), known as ISO PHS [21] and frequently referred to as Port-Controlled Hamiltonian Systemswith dissipation [17,20] in the context of control system design.1 As usual in control theory, an output equa-tion has been added to the first equation:

_q ¼ oHðq; pÞop

_p ¼ � oHðq; pÞoq

� oRð _qÞo _q

þ T

ð1Þ

_x ¼ ½JðxÞ � RðxÞ� oHoxðxÞ þ gðxÞu

y ¼ gTðxÞ oHoxðxÞ

ð2Þ

Summarizing, in the PCHD formalism (2), the state variables x 2 Rn are the energy variables, the smoothfunction HðxÞ : Rn ! R represents the total energy stored in the system, and u; y 2 Rm are the input- and out-put-port power variables, respectively. Inputs and outputs are conjugate variables so that their product rep-resents the power exchanged between the system and the environment. The input vector u is modulated by then � m matrix g(x) which also defines the output vector y. The n � n skew-symmetric matrix J(x) = �JT(x) re-veals the power-conserving interconnection structure in the model while the dissipation structure is capturedby the symmetric matrix R = RT P 0; both matrices depend smoothly on x.

There are systems where the control acts through the interconnection structure (e.g., power electronicdevices when the switched system behavior can be approximated by a smooth system). For this kind of sys-tems, the model (2) can be further generalized by (3) [17], where J(x,u) = �JT(x,u):

_x ¼ ½Jðx; uÞ � RðxÞ� oHoxðxÞ þ gðxÞu ð3Þ

A most important property of PCHD systems is its I–O passivity, with the energy H of the system being thestorage function and s = yTu the supply rate [20]. The power balance Eq. (4) shows that the stored energy isdissipated by the action of R(x) because of its properties of symmetry and positive definiteness.

_HðxÞ ¼ oHox

� �T

_x ¼ oHox

� �T

JðxÞ � RðxÞ½ � oHoxþ oH

ox

� �T

gðxÞu ¼ � oHox

� �T

RðxÞ oHoxþ yTu ð4Þ

Integrating (4) over the time-interval [t0 , t1] results in the well-known dissipative inequality below, which ascer-tain the passive properties of PCHD systems [23].

Hðxðt1ÞÞ 6 Hðxðt0ÞÞ þZ t1

t0

yTudt

1 In the sequel the acronym PCHD will be retained for the designation of these systems.


2.2. General field representation of bond graphs

Almost any BG can be represented at the highest level of abstraction by the generic model of Fig. 1. There,the BG-components are assembled together in different fields conveniently defined according to the energy- orpower-processing features of the components. Four groups can be distinguished: the dissipative- or R-field,collecting the components in which power is lost from the system; the storage-field, with the energy-conserva-tive elements C and I; the source-field, with the sources Se and Sf that supply power into the system; and thegeneral junction structure or JS-field, putting together the 0, 1, (M)TF and (M)GY, which capture the power-conservative interconnection of the system components.

Most convenient to establish a link between the BG and the PCHD formalism is to uncover the mathemat-ical structure of the BG. As it is well known, this is done assigning input-output causality to all BG-compo-nents. This computational causality assignment to the BG can also be represented at the highest abstract levelby a general structure, as shown in Fig. 2 [2], where the storage-field has been partitioned in two fields accord-ing to the integral or derivative causality assignment (ICA or DCA) of its components. This structure definesthe set of implicit differential-algebraic equations given in (5) and Table 1, which from here on will be referredto as the BG-Standard Implicit Form (BG-SIF). In (5), the outputs of the JS-field are related to its inputs, whileTable 1 conveys the causal form of the (linear and nonlinear) element constitutive laws (CLs) considered inthis paper.

The notation is as follows:

– Xi = [xi,1,xi,2, . . . ,xi,n]T, the state vector, contains the energy variables corresponding to the I- or C-ele-ments in ICA, momentum (p) on I-elements and displacement (q) on C-elements. It has been assumed thatthere are n storage elements in ICA.– Xd = [xd,1,xd,2, . . . ,xd,l�n]T contains the energy variables corresponding to the I- or C-elements in DCA, pon I-elements and q on C-elements. It has been considered that there are l � n storages in DCA.– Zi = [zi,1,zi,2, . . . ,zi,n]T and Zd = [zd,1,zd,2, . . . ,zd,l�n]T contain the co-energy variables associated to Xi andXd, flows f on I-elements and efforts e on C-elements.– Di and Do contain the efforts and flows variables entering and exiting from resistances, respectively. It hasbeen assumed that there are p resistors.– U contains the efforts and flows variables imposed by the sources.

Junction Structure0, 1, (M)TF, (M)GY

Se, Sf

RI, C

Source

Fig. 1. General structure of an acausal bond graph.

JunctionStructure

Se, Sf

R

C

C

I

I

zi

zd

zi

zd

U

DoDi

xd.

xi.

xd.

xi.

Fig. 2. General structure of a causal bond graph.

Table 1Constitutive laws

Non linear Linear

Zi = f(Xi) Zi = FXi

Do = l(Xi)Di multiplicative modulated resistors Do = LDi

Xd = g(Zd) Xd = GZd


_Xi

Zd

Di

264375 ¼ S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

264375

Zi

_Xd

Do

U

2666437775 ð5Þ

Some matrix properties are: S11 and S33 skew-symmetric matrices, S31 ¼ �ST13 and S21 ¼ �ST

12 , along this pa-per it is assumed that S22 = 0, S23 = 0, S32 = 0 and dS21/dt = 0 [2].

Along this paper, the vector Zd is considered to be a function of Xd, written Zd = g�1(Xd). As the energymust be conserved, the matrices of/oXi, og/oZd and og�1/oXd, as well as their linear versions F, G and G�1,satisfy the Maxwell’s reciprocal relations, which means that they are symmetrical matrices [15].

The resistors are considered to be truly dissipative, resulting the matrix l(Xi) positive definite; particularly,the constitutive matrix of the linear resistor field satisfies L > 0.

2.3. Equivalence between PCHD and BG variables

The series RLC circuit of the Fig. 3 is first considered as motivation. The Hamiltonian function of the cir-cuit is defined in (7) by the inductor flux pL and the capacitor charge qC. The PCHD model (6) is obtainedusing the Kirchhoff’s laws, the Hamiltonian gradient vector and the element constitutive laws.

Computing the Hamiltonian gradient respect to the states results that oH/opL is the inductor current f andoH/oqC is the capacitor voltage e, corresponding to the output variables of the I- and C-elements, respectively.Choosing, as usual, the BG energy variables as state vector, implies that their time derivatives are the input vari-ables of the storages. Then, the states in the PCHD and BG models are identical, that is x = Xi. This yields dx/dt = dXi/dt, i.e., the time derivative of the Hamiltonian state vector equals the input vector of the storage field.Further, oH/ox = Zi, because the entries of the Hamiltonian gradient are the outputs of the storages (the induc-tor current and the capacitor voltage). A simple inspection of the BG, the dynamic model (6) and the variableequivalence, shows that the entries of J are closely related with the direct causal paths between the capacitor andthe inductor, and the entries of R are related to the interconnection between the storages and the resistor. As thevoltage and the current associated to the input power port are the input and the output of (6), respectively, theinput matrix g is defined by the interconnection structure linking the power source and the storages.

_pL

_qC

� �¼

0 �1

1 0

� ��

R 0

0 0

� �� oH=opL

oH=oqC

� �þ

1

0

� �U

y ¼ 1 0½ �oH=opL

oH=oqC

� � ð6Þ

1Se:U C:CL

U

I:L

R:R

C

R

VoltageSource

Capacitor

Inductance

Resistor

SeriesConnection

PowerPorts

JuntionStructure

I-Field

C-Field

R-Field

SourceField

PowerBonds

a b

Fig. 3. (a) Series RLC circuit and (b) bond graph model in ICA.


where

HðpL; qCÞ ¼p2

L

2Lþ q2

C

2Cð7Þ

The previous introductory example shows a variable equivalence among the PCHD and BG models, and sug-gests that J, R and g are related to causal paths in the BG. Along the paper, the equivalence between bothformalisms will be formally established for more general cases. As starting point, the PCHD state vector ischosen to be equal to the state vector of the BG-SIF, i.e. x = Xi. Then, an equation relating the BG co-energyvariables and the gradient of the energy function is derived.

General Case. NonlinearR’s, nonlinear storages in ICA and DCA, and no DCA-storages causally determined

by sources.This case covers the most general BG models considered in this paper. Multiport nonlinear storag-es, including IC-elements, satisfying the Maxwell’s reciprocal relations are allowed to be part of the model.The vector-valued functions f and g of Table 1 can be written as follows:

f ðXiÞ ¼

f1ðXiÞf2ðXiÞ

..

.

fnðXiÞ

266664377775; gðZdÞ ¼

g1ðZdÞg2ðZdÞ

..

.

gl�nðZdÞ

266664377775

It can be seen that the CL of the elements in ICA only depend on the energy variables of elements in ICA, andthe CL of the elements in DCA only depend on the co-energy variables of the elements in DCA. Then, mul-tiport elements with mixed causality are not considered in this case.

Computing the energy function E in the BG via integration of the power P, the product of the input andoutput variables of each storages, yields

EðXi;XdÞ ¼Z

ZTi

_Xi dt þZ

ZTd

_Xd dt ¼Z

f TðXiÞdXi þZðg�1ðXdÞÞT dXd ð8Þ

The energy gradient oE/oX can be split-off in the two subgradients oE(X)/oXi = f(Xi) = Zi and oE(X)/oXd=g�1(Xd) = Zd. In this case the energy functions of the BG and PCHD formulations, E(Xi,Xd) and H(x), respec-tively, are different, because they have different arguments (yet, when evaluated on the same system state-Xi = x for our choice of the state variables- they take identical values for they represent the energy storedin the system). Thus, the BG energy function needs to be recalculated as function of only Xi. Writing the en-ergy itself as a function of only the state vector Xi yields an expression eEðXiÞ of the kind (9), with eE ¼ H andXi = x. Applying the chain rule to (9) results in (10), which is the equation looked for.

eEðXiÞ ¼ EðXi;XdÞ ¼ EðXi; gðZdÞÞ ¼ EðXi; gðS21ZiÞÞ ¼ EðXi; gðS21f ðXiÞÞÞ ¼ HðXiÞ ð9Þ

oHoXi

¼ I� of ðXiÞoXi

T

S12

ogðZdÞoZd

S21

" #Zi ð10Þ

The variable equivalence given by (10) becomes simpler for particular cases. Two cases are distinguished.

Case 1. Nonlinear BG with all storages in ICA. BG models considered in this case has not storages in DCA,which implies that S12 = S21 = 0. Then, the previous Eq. (10) reduces to the identity (11), which confirms thevariable equivalences suggested in the introductory example of the RLC circuit (recall that this example BGhas all its storages in ICA).

Zi ¼oEðXiÞ

oXi

¼ oHðxÞox

ð11Þ

Table 2Equivalence of variables

PCHD model BG model

State vector x � Xi Vector of energy variables � state vectorDerivative of the state vector _x � _Xi Input vector of the storage field in ICAEnergy gradient respect to the state vector oH

ox $ Zi Output vector of the storage field in ICA(equivalents according to (10)–(12))


Case 2. LinearR’s, linear storages in ICA and DCA, without DCA-storages causally determined by sources.

Given the hypothesis of linear storages, (10) is reduced to (12) by setting of/oXi = F and og/oZd = G.

oH
oXi
¼ ½I� FTS12GS21�Zi ð12Þ

Eqs. (10)–(12) show that the co-energy variable vector Zi has an equivalent vector in model (3). Table 2summarizes the correspondences among the PCHD and BG variables.

3. Formulating PCHD models from BG

This section addresses the derivation of PCHD models from BG, the main result of the paper. The equiv-alence of variables summarized in Table 2 suggests that the PCHD model in terms of BG variables must havethe form given in (13). With (13) as objective, the BG-SIF is manipulated to explicitly determine the matricesSs and Sg. After that, through direct comparison of this model with the PCHD state equations, a BG inter-pretation of the PCHD parameters, i.e., J, R and g, is found.

_Xi ¼ SsðXi;UÞZi þ SgðXiÞU ð13ÞIn subsection 3.1 the explicit differential equations for general nonlinear causally constrained BG are derived.In addition, the expressions for particular nonconstrained and linear cases are presented. In subsection 3.2,Theorem 1 presents formulae for J, R and g, and characterize them vis-a-vis some BG properties. Finally, sup-ported by the results of this section, subsection 3.3 presents a procedure to build PCHD models, as an alter-native to the traditional derivation of Hamiltonians of physical systems.

3.1. Explicit models from causally constrained bond graphs

As already stated in the Introduction, despite the fact that causally constrained BGs usually yield implicitDAE-systems, an explicit differential equation of the form (13) can be derived from the BG-SIF under suitablesolvability assumptions. In the General Case and Case 2 of the previous section, the PCHD model themselvesdo not follow immediately from (13) because the Zi vectors do not equal the energy gradient oH/ox, but theyare related by (10) and (12), respectively. Case 1 is simpler because of the identity Zi = oH/ox.

General Case. Nonlinear R’s, nonlinear storages in ICA and DCA, and no DCA-storages causally determined by

sources. The algebraic manipulations of the BG-SIF lead to the particular form of (13) which yields the PCHD(14) when using (10). For this to be equivalent to (3) the assumption dS21/dt = 0 was made. This assumptionallows to find an explicit dynamic model from the BG-SIF that matches the explicit PCHD form. In the BG lan-guage, it restricts the interconnection structure between the storages subfields in integral and derivative causalityto be non modulated. When the interconnection matrix S21 is not a constant matrix, i.e., it depends on the statesXi, or explicitly on time, the model gets the form of an implicit Port-Hamiltonian system [10,20,21] or a time-varying PCHD system [8], respectively, mismatching the explicit ISO PHS form looked for.

_Xi ¼ I� of ðXiÞoXi

S12

ogðZdÞoZd

S21

� ��T

S11 þ S13lðXiÞ½I� S33lðXiÞ��1S31

h iI� of ðXiÞ

oXi

S12

ogðZdÞoZd

S21

� ��1oHoXi

þ I� of ðXiÞoXi

S12ogðZdÞoZd

S21

� ��T

½S14 þ S13lðXiÞ½I� S33lðXiÞ��1S34�U ð14Þ

with Zd = S21 f(Xi)


Case 1. Nonlinear BG with all storages in ICA. The particular expression of the general model (15) for this caseis obtained by setting S12 = S21 = 0 in (14).

_Xi ¼ fS11 þ S13lðXiÞ½I� S33lðXiÞ��1S31g

oHoXi

þ fS14 þ S13lðXiÞ½I� S33lðXiÞ��1S34gU ð15Þ

Case 2. LinearR’s, linear storages in ICA and DCA, without DCA-storages causally determined by sources.
Under these hypotheses, the PCHD model (16) results from (14), replacing of/ oXi , og/ oZd and l(Xi) by F, G
and L, respectively.

_Xi ¼ ½I� S12GS21F��1 ½S11 þ S13LðI� S33LÞ�1S31�½I� FS12GS21��1 oH

oXi

þ ½S14 þ S13LðI� S33LÞ�1S34�U

� �ð16Þ

In order to compact the notation of (14), which will be useful for the next section, the symmetric and the skew-symmetric component of the matrix W = S13 l(Xi) [I � S33 l(Xi)]

�1 S31 , as well as the matrix Kd and its trans-pose are defined as in (17)–(19), respectively.

W sy ¼S13lðXiÞ½I� S33lðXiÞ��1

S31 þ ½S13lðXiÞ I � S33lðXiÞ��1S31

h iT

2ð17Þ

W sk ¼S13lðXiÞ½I� S33lðXiÞ��1

S31 � ½S13lðXiÞ I � S33lðXiÞ��1S31

h iT

2ð18Þ

Kd ¼ I� of ðXiÞoXi

S12

ogðZdÞoZd

S21

� ��1

() KTd ¼ I� S12

ogðZdÞoZd

S21

of ðXiÞoXi

� ��1

ð19Þ

3.2. Matrix and function equivalences

In this section the expressions of J, R and g are found in terms of the matrices of the BG-SIF, and thedependence of the skew-symmetry of J and the symmetry and positive (semi-)definiteness of R in terms ofthe BG properties are investigated. The next theorem covers the General Case.

Theorem 1. Let be a nonlinear BG with the associated constitutive laws of Table 1 and the BG-SIF (5). Storages

in ICA and DCA, and coupled dissipative elements (S33 – 0) are admitted but no sources causalizing storages in

DCA (S24 = 0). Then, the following identities and properties hold:

ð1Þ Jðx; uÞ ¼ KTd S11Kd þ KT

d WskKd ¼ S11;d þWsk;d ð20Þð2Þ RðxÞ ¼ �KT

d WsyKd ¼ �Wsy;d ð21Þð3Þ gðxÞ ¼ KT

d ½S14 þ S13lðXiÞ½I� S33lðXiÞ��1S34� ð22Þ

where Wsy and Wsk are the matrices presented in (17) and (18), and Wsy,d and Wsk,d are the symmetric and theskew-symmetric parts, respectively, of the matrix Wd ¼ KT

d W Kd, such that Wd = Wsy,d + Wsk,d. The matrixKd is defined in (19).

(4) The symmetric component Wsy,d of Wd is nonzero iff S13–0; S31–0 and l(Xi) – 0.(5) The skew-symmetric component Wsk,d of Wd is nonzero iff S13–0; S31–0, l(Xi) – 0, S33 – 0 and rank

ðS13Þ > 1.(6) And, finally, provided that l(Xi) > 0

(a) R(x) = �Wsy,d > 0 iff rank ðS31Þ ¼ n(b) R(x) = �Wsy,d P 0 iff rank ðS31Þ < n

where S31 ¼ S31Kd and S13 ¼ KTd S13.


Proof. See Appendix A.In the particular case that all the storages admit ICA (Case 1), the matrix Kd results the identity

matrix, S31 ¼ S31; S13 ¼ S13, and the expression of J, R and g of the theorem above can be reduced to(23)–(25).

Jðx; uÞ ¼ S11 þ S13

½lðXiÞ½I� S33lðXiÞ��1 � ½Iþ lðXiÞS33��1lðXiÞ�

2S31 ð23Þ

RðxÞ ¼ �S13

½lðXiÞ½I� S33lðXiÞ��1 þ ½Iþ lðXiÞS33��1lðXiÞ�

2S31 ð24Þ

gðxÞ ¼ S14 þ S13lðXiÞ½I� S33lðXiÞ��1S34 ð25Þ

Remark. The results of the theorem can be interpreted as follows: Properties #2 and #4 mean that the PCHD-dissipation matrix R is entirely determined by the R-field of the BG and its interaction with the storage-field.This is an intuitively obvious result. Properties #1 and #5 mean that the presence of causally coupled resistorsin the BG determines a contribution of its R-field to the PCHD-structure matrix J. This is not immediate fromintuition. Property #6 means that the PCHD-dissipation matrix R is positive definite iff all the storages imposecausality to the BG R-field. Otherwise it is positive semidefinite.

The results for the linear BG with storages in ICA and DCA (Case 2) is obtained, as a particular case ofTheorem 1, substituting of/oXi = F, og/oZd = G and l(Xi) = L in (20)–(22).

3.3. Formulating PCHD models of physical systems over bond graphs

The previous results provide an alternative to the classic methods used to obtain a Hamiltonian System of aphysical system. Summarizing, the following procedure should be applied. Construct a BG of the system andcausally augment it applying the SCAP algorithm. The BG energy variables constitute the PCHD state vector.Then, write-down the JS-matrices (Sij) associated to the BG-SIF; this is done computing loop gains of BGcausal paths following standard causality-based rules [3]. Finally, apply the appropriate formulae of Theorem

1 to get the desired PCHD model.This procedure avoids some difficult stages in classical methods, like selecting the most suitable generalized

coordinates and deciding the choice of the independent variables among them.

Remark. The previous procedure implies using (20)–(22) of Theorem 1, but matrix manipulations can beavoided in some cases where it is possible to work directly on the BG, as shown in the application section.

4. Examples and applications

The next examples illustrate the links between PCHD and BG models just established. The first one, a Per-manent Magnet Synchronous Motor (PMSM), is a Source-Controlled BG with multiplicative nonlinearities(state modulation of MGYs). The second system is a linear RLC-circuit with coupled dissipative elements(algebraic loop) where the contribution of the R-elements to J(x) and R(x) is evidenced. The third systemis a DC–DC boost converter averaged model where the control acts through the junction structure, i.e., itis a Structure-Controlled BG.

4.1. Permanent magnet synchronous motor

Fig. 4 presents the BG of the PMSM [1]. All the elements – up to the nonlinear state-modulated gyra-tors –, have linear constitutive laws, and all storages are in ICA. Because this model belongs to the classof system referred to as Case 1, simple particular expressions are used to simplify the derivation of thePCHD model.


The energy variables of the storages I:Ld, I:Lq and I:J, are the components of the state vector Xi = [x1, x2,x3]T = [pd, pq, px]T; the outputs of the storages compose the vector Zi = [Id, Iq, x]T, and the input vector isdefined by the sources, U = [Vd, Vq, TL].

The constitutive laws for the PMSM are given by the matrices F = diag{1/Ld, 1/Lq, 1/J} and L = diag{R,R, b}, with R, b > 0, which implies that the R’s are truly dissipative. The system energy results form (8) asHðXiÞ ¼ XT

i FXi=2.The entries of S11, S13, S31, S14 and S34 given in (26) are computed following the causal paths in Fig. 4 after

well known rules [3]. For example, the 1-row 3-column entry of S11 is obtained reading the gain of the causalpath which connects the output of the mechanical storage I:J with the input of the electrical storage I:Ld asshown in Fig. 4.

S11 ¼0 0 npx2

0 0 �npðx1 þ /eÞ�npx2 npðx1 þ /eÞ 0

264375; S13 ¼

�1 0 0

0 �1 0

0 0 �1

264375; S31 ¼

1 0 0

0 1 0

0 0 1

264375;

� S14 ¼1 0 0

0 1 0

0 0 �1

264375; S34 ¼ 0 ð26Þ

The matrices defining the PCHD model are obtained according to (15), but reading causal paths instead ofperforming matrix calculations: as there are no coupled resistors, S33 = 0, which implies Wsk,d = 0 and, thus,J = S11 and R is given by the dissipative field and its interconnection with the storages. Then, the PCHD dy-namic and output Eq. (27) can be directly derived using the expressions of Mason’s rules on BGs. The matrixmanipulations (20)–(22), necessary in more complicated cases, would yield -of course- the same results:J(x) = S11, R(x) = � S13LS31 = L and g(x) = S14.

_Xi ¼ ½S11JðxÞ� L

RðxÞ�Zi þ S14

gðxÞU

Y ¼ ST14

gTðxÞZi

ð27Þ

The interconnection structural matrix J(x) is modulated by the states, as it is shown in (26). The dissipationmatrix is positive definite, which can be checked using the fact that all the R-elements are truly-dissipative andrank (S31) = 3, or applying the Sylvester’s criteria.

MGY

TF

MGY

1

1

1 1

R:R

I:Lq

I:Ld

R:R

np

R:b

I:J

Vq:Se

Vd:Se

Se:TL

:

::

LqIq

LdId+φeIq

Id

ω

Fig. 4. BG of the PMSM.


4.2. RLC circuit

The circuit of Fig. 5a has the BG representation shown in Fig. 5b. There are three elements in ICA andthree resistors, two of them statically coupled. This coupling will show how the R-elements contribute tothe structure with a skew-symmetric component which is power-conserving. Because all the storages are inICA, this is a Case 1-type system.

The Junction Structural Matrices Sij are read from the BG (see [2] for details).

S11 ¼0 �1 �1

1 0 0

1 0 0

264375; S13 ¼

0 �1 0

1 0 0

1 0 �1

264375; S31 ¼

0 �1 �1

1 0 0

0 0 1

264375; S33 ¼

0 �1 0

1 0 0

0 0 0

264375 ð28Þ

The constitutive laws are defined by the matrices F = diag{1/L1, 1/C2, 1/C3} and L = diag{1/R4, R5, 1/R6},where L1, C2, C3, R4, R5 and R6 > 0.

Recall (23) and (24), rewritten below as (29), with the replacement l(Xi) = L. Clearly, S11 contributes to J(x)because it is skew-symmetric. Because the presence of coupled resistors, Theorem 1 anticipates that W, the sec-ond right-hand side term of (29), generated by the R-elements, has a symmetric and a skew-symmetric com-ponent, Wsy and Wsk, respectively.

JðxÞ �RðxÞ ¼ S11 þ S13L½I� S33L��1S31 ð29Þ

W ¼ �S13LðI� S33LÞ�1S31 ¼Wsy þWsk ¼

1

R4 þ R5

R4R5 0 0

0 1 1

0 1 ðR4 þ R5 þ R6Þ=R6

264375þ 0 R5 R5

�R5 0 0

�R5 0 0

264375

8><>:9>=>;

ð30Þ

Finally, the structural interconnection and dissipative matrices are computed from (29) and (30) in (31) and(32).

JðxÞ ¼ S11 �Wsk ¼

0 �1� R5

R4þR5�1� R5

R4þR5

1þ R5

R4þR50 0

1þ R5

R4þR50 0

2666437775 ð31Þ

RðxÞ ¼Wsy ¼1

R4 þ R5

R4R5 0 0

0 1 1

0 1 ðR4 þ R5 þ R6ÞR6

26643775 > 0 ð32Þ

The positive definiteness of R(x) is assured by the fact that all the R’s are truly-dissipative and rank

(S31) = n = 3.

R5

R6R4

C2C3L1

0

0

1

R:R4

R:R5

R:R6

C2:C

L1:I

C3:C

a b

Fig. 5. (a) Circuit diagram and (b) BG model.

RCL

E

μ=1

μ=0

01 R:RMTF

C:CI:L

E:Se :

μ

a b

Fig. 6. (a) Circuit diagram and (b) BG average model.


The algebraic loop shown in Fig. 5b relates R4 and R5, which are the R-elements appearing in Wsk. Thiscoupled resistors generate the skew-symmetric components in (31), which do not dissipate energy but contrib-ute to the interconnection structural matrix J(x).

4.3. Boost converter

The boost converter average model has the particularity that the control input u = l, the PWM (pulsewidth modulation) duty ratio, acts on the interconnection structure [17], as shows the MGY in Fig. 6b [5].

The state vector, the storage output vector and the input vector are Xi=[pL, qC]T, Zi = [IL, VC]T and U = E,respectively; the CLs are defined by the matrices L = 1/R1 and F = diag{1/L1, 1/C1}; and the structural matri-ces obtained from the BG model are given in (33).

S11 ¼0 �l

l 0

� �; S13 ¼

0

�1

� �; S31 ¼ 0 1½ �; S14 ¼

1

0

� �ð33Þ

The skew-symmetric matrix S11 results from the causal interaction of the I- and C-elements through the MTFof gain l. The dynamic model presented in (34) is equivalent to the boost converter model given in [17]. Asseen in J(x, l) = J(l) = S11, the interconnection structure depends on l, i.e., it is modulated by an input,and the dissipative matrix R(x) is positive semi-definite, which can be inferred by the facts that the rank

(S31) = 1 < n and the resistor R1 > 0.

_Xi ¼0 �l

l 0

� ��

0 0

0 1=R1

� �� Zi þ

1

0

� �E ð34Þ

5. Conclusion

A method to derive Input-State-Output PHS or Port-Controlled Hamiltonian Systems with dissipationfrom bond graphs has been presented. The dependency of the parameters (functions and matrices) of theHamiltonian form on the BG properties has been analyzed. The Standard Implicit form and energy propertiesof BG models were used to obtain the results. As the matrices of the BG Standard Implicit form can beobtained algorithmically, the Hamiltonian parameters may be automatically computed by some adequatesoftware.

It has been shown that the presence of coupled R-elements on the BG determines the existence of symmetricand skew-symmetric components in the matrix contributed by the coupled R-field of the BG. It means thatadding damping to the system could modify the interconnection matrix J(x) when an algebraic loop isproduced.

The method can be seen as the enabling step of a procedure for the construction of PHS models through theBG technique. This is worth from an engineering point of view because, on the one hand, as a network-typerepresentation technique, the BG method honors the usual interconnection topology of technical systems andprovides an object-oriented modeling tool, and, on the other hand, avoids employing classical analytical meth-ods that, in some cases, may show formulation difficulties.

Current research focuses on control system design in the BG domain using the theoretical support alreadyavailable for PCHD, but also taking advantage of the physical information intuitively provided by BG.


Appendix A

Lemma 1. The following auxiliary result will be used later. It means that positive definiteness of the coupled

R-field holds iff the R-field itself is positive definite. It follows from the skew-symmetry of S33 and the fact that theinverse of a positive matrix is also positive. Indeed, l(Xi) [I � S33 l(Xi)]

�1 > 0, [I � S33 l(Xi)]

l(Xi)�1 > 0, l(Xi)

�1 � S33 > 0, l(Xi)�1 > 0, l(Xi) > 0.

Proof of Theorem 1.Proof of Properties #1–3. A direct comparison between (3) and (14) leads to (35) and (36), with Zd = S21

f(Xi). The latter equation directly proves (22).

Jðx; uÞ � RðxÞ ¼ I� S12

ogðZdÞoZd

S21

of ðXiÞoXi

� ��1

½S11 þ S13lðXiÞ½I� S33lðXiÞ��1S31�

I � of ðXiÞoXi

S12

ogðZdÞoZd

S21

� ��1ð35Þ

gðxÞ ¼ I� S12

ogðZdÞoZd

S21

of ðXiÞoXi

� ��1

½S14 þ S13lðXiÞ½I� S33lðXiÞ��1S34� ð36Þ

Let S be the right hand-side of (35). Then, J and R can be computed as Ssk = (S-ST)/2 and Ssy = (S + ST)/2,the skew-symmetric and symmetric parts of S, respectively. Defining the matrices Wsy, Wsk and Kd as in (17)–(19), respectively, yields immediately (37) from (35) and (36), which proves properties #1, #2 and #3 ofTheorem 1.

Jðx; uÞ ¼ KTd S11Kd þ KT

d WskKd ¼ S11;d þWsk;d

RðxÞ ¼ �KTd WsyKd ¼ �Wsy;d

gðxÞ ¼ KTd ½S14 þ S13lðXiÞ½I� S33lðXiÞ��1

S34�ð37Þ

Defining S31 ¼ S31Kd and S13 ¼ KTd S13, with S13 ¼ �ST

31, and writing Wsk,d and Wsy,d as in (38) and (39),respectively, allows to proves of the remaining properties.

Proof of Properties #4: Wsy,d is nonzero iff S13–0; S31–0 and l(Xi) – 0. From the definition of Wsy,d asgiven in (38), where S13 ¼ �ET; S31 ¼ E, and � D < 0 because l(Xi) > 0,D > 0, which results from Lemma1 (the scalar 2 has been built-in into D). It can also be proved that l(Xi) = 0,D = 0.

Wsy;d ¼S13 lðXiÞ½I� S33lðXiÞ��1 þ ½Iþ lðXiÞS33��1

lðXiÞh i

S31

2¼ �ETDE ð38Þ

The main diagonal elements of ETDE are Wsy;d;ii ¼ �eTi Dei, with ei being the ith column of E, i = 1, . . . ,n. The

negative definiteness of �D yields Wsy,d,ii – 0 with ei – 0, implying that Wsy,d – 0. Conversely, it is trivial thatS13 ¼ ST

31 ¼ 0 _ lðX iÞ ¼ 0()Wsy;d ¼ 0

Proof of Properties #5: Wsk,d is nonzero iff S13–0; S31–0; lðX iÞ–0;S33–0 and rank ðS13Þ > 1.

Wsk;d ¼S13 lðXiÞ½I� S33lðXiÞ��1 � ½Iþ lðXiÞS33��1

lðXiÞh i

S31

2¼ �ETCE ð39Þ

where S13 ¼ �ET; S31 ¼ E;C is a skew-symmetric matrix, and S33 = 0, C = 0. Because of the skew-symme-try of Wsk,d, its main diagonal is zero. The remaining entries Wsk;d;ij ¼ �eT

i Cej with i – j are nonzero if thecolumns of E are linearly independent, otherwise Wsy,d,ij = 0. But the entries cannot be all zero because itwould imply that the rank of ET ¼ �S13 ¼ ST

31 is 1, and by hypothesis it is greater than one. Thus, theskew-symmetric matrix Wsk,d is nonzero.

It is trivial that S33 = 0)Wsk,d = 0, S13 ¼ ST31 ¼ 0)Wsk;d ¼ 0 and l(Xi) = 0)Wsk,d = 0. The first state-

ment has the interpretation that without coupled R’s there are no contribution to J(x) matrix from the dissi-pative field. The same result is obtained if there are no connection between dissipative and storage fields orwhen there are not R’s in the model.


Proof of Properties #6: The positive definiteness of l(Xi) assures at least the positive semi-definiteness ofR(x). The assumption on l(Xi) is satisfied by any truly-dissipative 1-port R-element (meaning that the e–f char-acteristics goes though the origin and is completely contained in the first and third quadrants of the e–f plane).If in addition the rank of S31 ¼ n, then the matrix R(x) is positive definite. If rank ðS31Þ < n, then R(x) is onlypositive semi-definite.

The positive definiteness R(x) > 0 holds when for any v 2 Pn, v – 0, the following is satisfied:RðxÞ > 0() � vTW sy;dv > 0() � vT½Wd þWT

d �v > 0:The last term of the previous equivalence holds iff �Wd is positive definite. This condition is analyzed next,

where the property S13 ¼ �ST31 is used: �vTWdv > 0() � vTS13lðX iÞ½I� S33lðX iÞ��1

S31v > 0() ðS31vÞTlðX iÞ½I� S33lðX iÞ��1

S31v > 0:Let be z ¼ S31v 2 P p. If rank ðS31Þ ¼ n, then z – 0"v – 0 and z = 0 for v = 0. Thus, from the fact that

l(Xi) > 0, l(Xi) [I-S33 l(Xi)]�1 > 0, it results that ðS31vÞTlðX iÞ½I� S33lðX iÞ��1

S31v ¼ zTlðX iÞ½I� S33lðX iÞ��1z >

08v–0ð) z–0Þ, what proves that �Wsy,d = R(x) > 0 is positive definite.If rank ðS31Þ < n, then R(x) is only positive semi-definite because ðS31vÞTlðX iÞ½I� S33lðX iÞ��1

S31v P08v; and ðS31vÞTlðX iÞ½I� S33lðX iÞ��1

S31v ¼ 0 for any v – 0. Consequently, z = 0 for any v – 0 sincedim(kerðS31ÞÞ > 0. It is the case when the number of dissipative elements is less than the number of storages,i.e. p < n. h

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