intelligent powertrain design

Jimmy C. Mathews

Advisors: Dr. Joseph Picone Dr. David Gao

Powertrain Design Tools Project

The BOND GRAPH Methodology for Modeling of Continuous Dynamic Systems

INTELLIGENT POWERTRAIN DESIGN

of 43Intelligent Powertrain Design

Outline

• Dynamic Systems and Modeling

• Bond Graph Modeling Concepts

• Sample Applications of Bond Graph Modeling

• The Generic Modeling Environment (GME) and Bond Graph Modeling

• Future Concepts


• Dynamic Systems Related sets of processes and reservoirs (forms in which matter or energy exists) through which material or energy flows, characterized by continual change.

• Common Dynamic Systems electrical, mechanical, hydraulic, thermal among numerous others.

• Real-time Examples moving automobiles, miniature electric circuits, satellite positioning systems

• Physical systems Interact, store energy, transport or dissipate energy among subsystems

• Ideal Physical Model (IPM) The starting point of modeling a physical system is mostly the IPM.

• To perform simulations, the IPM must first be transformed into mathematical descriptions, either using Block diagrams or Equation descriptions

• Downsides – laborious procedure, complete derivation of the mathematical description has to be repeated in case of any modification to the IPM [3].

Dynamic Systems and Modeling


Computer Aided Modeling and Design of Dynamic Systems

• Basic Concepts

Physical System

Schematic Model

Classical Methods, Block Diagrams OR Bond

Graphs

Simulation and Analysis Software

Output

Data Tables & Graphs

STEP 1: Develop an ‘engineering model’

STEP 2: Write differential equations

STEP 3: Determine a solution

STEP 4: Write a program

The Big Question??

Differential Equations

GME + Matlab/Simulink

Fig 1. Modeling Dynamic Systems [1]


Bond Graphs vs. Block Diagrams [5, 8]

• Block Diagrams Early attempt to deal with heterogeneity, closely related to the emergence of automatic control, nice example of information hiding, very successful and good environments like Simulink, Easy V, and VisSim available presently.

• Familiar and versatile graphical notation to represent Signal Flow.

• Downsides i. Do not provide a suitable notation for depicting physical system models because not all block diagrams represent physical processes. ii. Energetic Coupling between elements/systems - - - energy exchange implies interaction, i.e. a bilateral, two-way influence of each system on the other. Block diagrams fundamentally depict a unilateral influence of one system on another. Hence, to describe energetic interaction of two systems/elements in terms of signal flow, the output of one should be the input of another and vice versa.

Fig 2. Block Diagram of Energetic Interaction [8]

iii. When two systems interact energetically, we must have the block representation as in figure 2 (or its converse). In contrast, the block diagrams shown below might represent possible operations on signals or information, but neither represents any possible energetic interaction.


• Bond Graphs Close correspondence between the bond graph and the physical system it represents.

• Conserves the physical structural information as well as the nature of sub-systems which are often lost in a block diagram.

• Can be directly derived from the IPM. When the IPM is changed, only the corresponding part of a bond graph has to be changed. Advantage of making the model very amenable to modification for ‘model development’ and ‘what if?’ situations.

• Account for all the energy in physical systems and provide a common link among various engineering systems. Use analogous power and energy variables in all domains, but allow the special features of the separate fields to be represented.

• The only physical variables required to represent all energetic systems are power variables [effort (e) & flow (f)] and energy variables [momentum ε(t) and displacement F(t)].

• The dynamics of physical systems are derived by the application of instant-by-instant energy conservation. Actual inputs are exposed.

• Linear and non-linear elements are represented with the same symbols; non-linear kinematics equations can also be shown.

• Provision for active bonds. Physical information involving information transfer, accompanied by negligible amounts of energy transfer are modeled as active bonds.

Some more advantages will be discussed after dealing with the concept of causality.

Bond Graphs vs. Block Diagrams (contd..)


Bond Graph Methodology

• Invented by Henry Paynter in the 1961, later elaborated by his students Dean C. Karnopp and Ronald C. Rosenberg

• A Bond Graph is an abstract representation of a system where a collection of components interact with each other through energy ports and are placed in a system where energy is exchanged [2]. • A bond-graph model consists of subsystems which can

either describe idealized elementary processes or non-idealized processes. Non-idealized processes can either be non-linear equation models or bond graph sub models [3].

• Subsystems can have two type of ports: power ports and signal ports.

• Power ports specify both an effort variable and flow variable. Signal ports specify only one variable, a flow or an effort or a mathematical variable.

• Two types of knots in bond graphs, 0 junctions and 1 junctions; represent domain-independent generalizations of Kirchoff’s laws.

• Connects are called bonds, indicate power between various subsystems. The half arrows indicates positive power flow orientation. The full arrows indicate signal flows.

• Bond is characterized by the value of an instantaneous power, computed as the product of effort and flow variables (e.g. voltage and current in the electrical domain).

Fig 3. Subsystems of a bond graph [3]


The Bond Graph Modeling Formalism

• Bond Graph’s Reach?

Electrical

Mechanical Translation

Mechanical Rotation

Hydraulic/Pneumatic

Chemical/Process Engineering

Thermal

Magnetic Fig 4. Multi-Energy Systems Modeling using Bond Graphs


Fig. 5 Electric elements with power ports [4]

The Bond Graph Modeling Formalism (contd..)

• Two different physical domains are considered: the Electrical and the Mechanical domains.

• Electrical Domain

To facilitate conversion of electrical circuits to bond graphs, represent different elements (Voltage Source, Resistor, Capacitor, Inductor) with visible ports (figure 5).

To these ports, we connect power bonds denoting energy exchange between elements.

• Mechanical Domain

Mechanical elements like Force, Spring, Mass, Damper are similarly dealt with.


The R – L - C circuitThe power being exchanged by a port with the rest of the system is a product of the voltage and the current: P = u * i

The equations for the resistor, capacitor and inductor are: u_R = i * R

u_C = 1/C * (∫idt)u_L = L * (di/dt); or i = 1/L * (∫u_L dt)


Fig 6. The RLC Circuit [4]

1


The Spring-Mass-Damper SystemPort variables on the bond graph elements are force on the element port and velocity of the element port. P = F * v

The equations for the damper (damping coefficient, α), spring (coefficient, KS) and mass are: F_d = α * v

F_s = KS * (∫v dt) = 1/CS * (∫ vdt)F_m = m * (dv/dt); or v = 1/m * (∫F_m dt); Also, F_a = force


Fig 7. The Spring Mass Damper System [4]


Analogies!

Lets compare! We see the following analogies between the mechanical and electrical elements:

• The Damper is analogous to the Resistor.• The Spring is analogous to the Capacitor, the mechanical compliance corresponds with the

electrical capacity.• The Mass is analogous to the Inductor.• The Force source is analogous to the Voltage source.• The common Velocity is analogous to the loop Current.

Notice that the bond graphs of both the RLC circuit and the Spring-mass-damper system are identical. Still wondering how??

• Various physical domains are distinguished that each is characterized by a particular conserved quantity. Table 1 illustrates these domains with corresponding flow (f), effort (e), generalized displacement (q), and generalized momentum (p).

• Note that power = effort x flow in each case.• Also note, the bond graph modeling language is domain-independent.



f

flow

e

effort

q = ∫f dt

generalized displacement

p = ∫e dt

generalized momentum

Electromagnetic i

current

u

voltage

q = ∫i dt

charge

λ = ∫u dt

magnetic flux linkage

Mechanical Translation

v

velocity

f

force

x = ∫v dt

displacement

p = ∫f dt

momentum

Mechanical Rotation ω

angular velocity

T

torque

θ = ∫ω dt

angular displacement

b = ∫T dt

angular momentum

Hydraulic / Pneumatic

φ

volume flow

P

pressure

V = ∫φ dt

volume

τ = ∫P dt

momentum of a flow tube

Thermal T

temperature

FS

entropy flow

S = ∫fS dt

entropy

Chemical μ

chemical potential

FN

molar flow

N = ∫fN dt

number of moles

Table 1. Domains with corresponding flow, effort, generalized displacement and generalized momentum




• Foundations of Bond Graphs

Based on the assumptions that satisfy basic principles of physics; a. Law of Energy Conservation is applicable b. Positive Entropy production c. Power Continuity

• Closer look at Bonds and Ports

Power port or port: The contact point of a sub model where an ideal connection will be connected.

Power bond or bond: The connection between two sub models; drawn by a single line (Fig. 8)

Bond denotes ideal energy flow between two sub models; the energy entering the bond on one side immediately leaves the bond at the other side (power continuity).

Energy flow along the bond has the physical dimension of power, being the product of two variables; called power-conjugated variables.

Fig. 8 Energy flow between two sub models represented by a bond [4]

A Bef

(directed bond from A to B)



• Two views of Interpretation of Power Bond

1. As an interaction of energy; connected subsystems for a load to each other by their energy exchange; embodies an exchange of a physical quantity. 2. As a bilateral signal flow; interpreted as effort and flow flowing in opposite direction, thus determining the computational direction of the bond variables; w.r.t. one of the sub models, effort is the input and flow is the output and vice versa for the other sub model.

• Determining the direction of Effort and Flow

During modeling it need not be decided what the computational direction of the bond variables is, however it is necessary to derive the mathematical model (set of differential equations) from the graph. Process of determining the computational direction of the bond variables is called causal analysis; indicated in the graph by the so-called causal stroke, (indicating the direction of the effort), called the causality of the bond (figure 9).

Fig. 9 Why is the Power direction not shown? [4]



• Bond Graph Elements

Bond graph elements are drawn as letter combinations (mnemonic codes) indicating the type of element. The bond graph elements are the following:

C storage element for a q-type variable, e.g. capacitor (stores charge), spring (stores displacement)

L storage element for a p-type variable, e.g. inductor (stores flux linkage), mass (stores momentum)

R resistor dissipating free energy, e.g. electric resistor, mechanical friction

Se, Sf sources, e.g. electric mains (voltage source), gravity (force source), pump (flow source)

TF transformer, e.g. an electric transformer, toothed wheels, lever

GY gyrator, e.g. electromotor, centrifugal pump

0, 1 0 and 1–junctions, for ideal connecting two or more sub models



• Storage Elements

Two types; C – elements & I – elements; q–type and p–type variables are conserved quantities and are the result of an accumulation (or integration) process; they are the state variables of the system.

C – element (capacitor, spring, etc.)

q is the conserved quantity, stored by accumulating the net flow, f to the storage element.

resulting balance equation dq/dt = f

Fig. 10 Examples of C - elements [4]

Equations for linear capacitor and linear spring:

dq/dt = i, u = (1/C) * q

dx/dt = v, F = k * x = (1/C) * x

For a capacitor, C [F] is the capacitance and for a spring, K [N/m] is the stiffness and C [m/N] the compliance.



I – element (inductor, mass, etc.)

p is the conserved quantity, stored by accumulating the net effort, e to the storage element.

resulting balance equation dp/dt = f

Fig. 11 Examples of I - elements [4]

Equations for linear inductor and linear mass:

dλ/dt = u, i = (1/L) * λ

dp/dt = F, V = (1/m) * p

For an inductor, L [H] is the inductance and for a mass, m [kg] is the mass. For all other domains, an I – element can be defined.



R – element (electric resistors, dampers, frictions, etc.)

R – elements dissipate free energy and energy flow towards the resistor is always positive.

Algebraic relation between effort and flow, lies principally in 1st or 3rd quadrant.e = r (f)

Fig. 12 Examples of Resistors [4]

Electrical resistance value [Ω] given by Ohm’s law; u = R * I

If the resistance value can be controlled by an external signal, the resistor is a modulated resistor, with mnemonic MR. E.g. hydraulic tap: the position of the tap is controlled from the outside, and it determines the value of the resistance parameter.

In the thermal domain, the dissipator irreversibly produces thermal energy, the thermal port is drawn as a kind of source of thermal energy. The R becomes an RS.



Fig. 13 Examples of Sources [4]

Sources (voltage sources, current sources, external forces, ideal motors, etc.)

Sources represent the system-interaction with its environment. Depending on the type of the imposed variable, these elements are drawn as Se or Sf.

Source elements are used to give a variable a fixed value, for example, in case of a point in a mechanical system with a fixed position, a Sf with value 0 is used (fixed position means velocity zero).

Fig. 14 Example of Modulated Voltage Source [4]

When a system part needs to be excited, often a known signal form is needed, which can be modeled by a modulated source driven by some signal form (figure 14).



Transformers (toothed wheel, electromotor, etc.)

An ideal transformer is represented by TF and is power continuous (i.e. no power is stored or dissipated). The transformations can be within the same domain (toothed wheel, lever) or between different domains (electromotor, winch).

e1 = n * e2 & f2 = n * f1

Efforts are transduced to efforts and flows to flows; n is the transformer ratio. Only one dimensionless parameter n is required to describe effort transduction and flow transduction. n is a defined as follows: e1 and f1 belong to the bond pointing towards TF.

Fig. 15 Examples of Transformers [4]

If n is not constant, it becomes an input signal to the modulated transformer, MTF.



Gyrators (electromotor, pump, turbine)

An ideal gyrator is represented by GY and is power continuous (i.e. no power is stored or dissipated). Real-life realizations of gyrators are mostly transducers representing a domain-transformation.

e1 = r * f2& e2 = r * f1

r is the gyrator ratio and is the only parameter required to describe both equations. R has a physical dimension (same as R-element), since r is the relation between effort and flow.

Fig. 16 Examples of Gyrators [4]

Gyrator is defined by one bond pointing towards and other bond pointing away.

If r is not constant, the gyrator is a modulated gyrator, a MGY.



Junctions

Junctions couple two or more elements in a power continuous way; there is no storage or dissipation at a junction.

0 – junction

Represents a node at which all efforts of the connecting bonds are equal. E.g. a parallel connection in an electrical circuit.

The sum of flows of the connecting bonds is zero, considering the sign. The power direction determines the sign of flows: all inward pointing bonds get a plus and all outward pointing bonds get a minus.

0 – junction can be interpreted as the generalized Kirchoff’s Current Law.

Additionally, equality of efforts (like electrical voltage) at a parallel connection.

Fig. 17 Example of a 0-Junction [4]



1 – junction

Is the dual form of the 0-junction (roles of effort and flow are exchanged).

Represents a node at which all flows of the connecting bonds are equal. E.g. a series connection in an electrical circuit.

The efforts of the connecting bonds sum to zero. Again, the power direction determines the sign of flows: all inward pointing bonds get a plus and all outward pointing bonds get a minus.

1- junction can be interpreted as the generalized Kirchoff’s Voltage Law.

In the mechanical domain, 1-junction represents a force-balance, and is a generalization of Newton’ third law.

Additionally, equality of flows (like electrical current) through a series connection.

Fig. 18 Example of a 1-Junction [4]



Some Miscellaneous Stuff!

Power Direction: The power is positive in the direction of the power bond. A port that has incoming power bond consumes power. E.g. R, C. If power is negative, it flows in the opposite direction of the half-arrow.

R, C, and I elements have an incoming bond (half arrow towards the element) as standard, which results in positive parameters when modeling real–life components.

For source elements, the standard is outgoing, as sources mostly deliver power to the rest of the system.

For TF– and GY–elements (transformers and gyrators), the standard is to have one bond incoming and one bond outgoing, to show the ‘natural’ flow of energy.

These are constraints on the model!

Duality:

The role of effort and flow in the storage elements (C, I) are interchanged. They are each other’s dual form.

A gyrator can be used to decompose an I-element to a GY and C element and vice versa.



• Causal Analysis

Causal analysis is the determination of the signal direction of the bonds. The energetic connection (bond) is now interpreted as a bi-directional signal flow. The result is a causal bond graph, which can be seen as a compact block diagram.

Causal analysis covered by modeling and simulation software packages that support bond graphs; Enport, MS1, CAMP-G, 20 SIM

Four different types of constraints need to be discussed prior to following a systematic procedure for bond graph formation and causal analysis.]

• Causal Constraints

Fixed Causality (Se, Sf)

Fixed causality is the case when equations allow only one of the two port variables to be the outgoing variable. An effort source (Se) has by definition always its effort variable as signal output, and has the causal stroke outwards. This causality is called effort-out causality or effort causality. A flow source (Sf) clearly has a flow-out causality or flow causality.

May occur at non-linear elements, where the equations for that port cannot be inverted (e.g. division by zero). e

Sef

e

Sff

Se ef

Sf ef


Constrained Causality (TF, GY, 0-junction, 1-junction)

Constrained causality is defined when a relations exist between the causalities of the different ports of the element. At a TF, one of the ports has effort-out causality and the other has flow-out causality.

OR

Similarly, at a GY, both ports have either effort-out causality or flow-out causality.

At a 0–junction, where all efforts are the same, exactly one bond must bring in the effort. This implies that 0–junctions always have exactly one causal stroke at the side of the junction.

The causal condition at a 1–junction is the dual form of the 0-junction. All flows are equal, thus exactly one bond will bring in the flow, implying that exactly one bond has the causal stroke away from the 1–junction.

Preferred Causality (C, I)

Causality determines whether an integration or differentiation w.r.t time is adopted in storage elements. Integration has a preference over differentiation because:

1. At integrating form, initial condition must be specified.


TF

e1 e2

f2f1n

TF

e1 e2

f2f1n


2. Integration w.r.t. time can be realized physically; Numerical differentiation is not physically realizable, since information at future time points is needed.

3. Another drawback of differentiation: When the input contains a step function, the output will then become infinite.

Therefore, integrating causality is the preferred causality. C-element will have effort-out causality and I-element will have flow-out causality. (figures 10 & 11).

Indifferent causality (Linear R)

Indifferent causality is used, when there are no causal constraints! At a linear R, it does not matter which of the port variables is the output.

There is no difference choosing the current as incoming variable and the voltage as outgoing variable, or the other way around.



• Electrical Circuit # 1 (R-L-C) and its Bond Graph model

STEP 1: Determine which physical domains exist in the system and identify all basic elements like C, I, R, Se, Sf, TF, GY. Give each element a unique name.

STEP 2: Indicate a reference effort for each domain in the Ideal Physical Model (reference velocity with positive direction for the mechanical domains). Note that references in the mechanical domain have a direction.

Generation of the connection / junction structure.

STEP 3: Identify all other efforts (mechanical domains: velocities) and give them unique names.

STEP 4: Draw these efforts (mechanical: velocities), and not the references, graphically by 0–junctions (mechanical: 1–junctions). Keep if possible, the same layout as the IPM.

+-

U0

U1U2

U3

Examples


STEP 5: Identify all effort differences (mechanical velocity(=flow) differences) needed to connect the ports of all elements enumerated in Step 1. Differences have a unique name.

STEP 6: Construct the effort differences using a 1–junction (mechanical: flow differences with a 0–junction) and draw them as such in the graph.

STEP 4: 0 0 0

STEP 5, 6: 0 1 0 1 0

STEP 7: The junction structure is now ready and the elements can be connected. Connect the port of all elements found at step 1 with the 0–junctions of the corresponding efforts or effort differences (mechanical: 1–junctions of the corresponding flows or flow differences).

STEP 8: Simplify the resulting graph by applying the following simplification rules: 1. A junction between two bonds can be left out, if the bonds have a ‘through’ power direction (one bond incoming, the other outgoing).

2. A bond between two the same junctions can be left out, and the junctions can join into one junction.

3. Two separately constructed identical effort or flow differences can join into one effort or flow difference.

Examples (contd..)

U1 U2 U3

U1 U2 U3

0: U12 0: U23


STEP 7:

STEP 8:

0: U12

0 1 0 1 0Se : UC : C

0: U23

R : R I : L

U1 U3U2

Examples (contd..)

1Se : U

C : C

R : R

I : L


Examples (contd..)

The Causality Assignment Algorithm:

STEP 1a. Chose a fixed causality of a source element, assign its causality, and propagate this assignment through the graph using the causal constraints. Go on until all sources have their causalities assigned.

STEP 1b. Chose a not yet causal port with fixed causality (non-invertible equations), assign its causality, and propagate this assignment through the graph using the causal constraints. Go on until all ports with fixed causality have their causalities assigned.

STEP 2: Chose a not yet causal port with preferred causality (storage elements), assign its causality, and propagate this assignment through the graph using the causal constraints. Go on until all ports with preferred causality have their causalities assigned.

1Se : U

C : C

R : R

I : L

1a.

2.

1Se : U

C : C

R : R

I : L


STEP 3: Chose a not yet causal port with indifferent causality, assign its causality, and propagate this assignment through the graph using the causal constraints. Go on until all ports with indifferent causality have their causalities assigned.

Examples (contd..)

3.

1Se : U

C : C

R : R

I : L

• Electrical Circuit # 2 and its Bond Graph model

R1

R2R3

C1

C2

L1

R1 R2 R3

C2C1L1


Examples (contd..)

• A DC Motor and its Bond Graph model

Km

Se1:Ua Se2:τ


Generation of Equations from Bond Graphs

1Se : U

C : C

R : R

I : L1

2

4

3

Fig. 19 Bond Graph of a series RLC circuit

• A causal bond graph contains all information to derive the set of state equations.

• Either a set of Ordinary first-order Differential Equations (ODE) or a set of Differential and Algebraic Equations (DAE).

• Write the set of mixed differential and algebraic equations.• For a bond graph with n bonds, 2n equations can be

formed, n equations each to compute effort and flow or their derivatives.

• Then, the algebraic equations are eliminated, to get final equations in state-variable form.

For the given RLC circuit, Se = e1= U;

e2 = R * f2;

(de3/dt) = (1/C) * f3;

(df4/dt) = (1/L) * e4;

f1 = f4; f2 = f4; f3 = f4;

e4 = e1 - e2 - e3

Hence, e1 - e2 - e3 = U – (R * f2) – e3 = U – (R * f4) – e3

(df4/dt) = (1/L) * (U – (R * f4) – e3) - - - - - - - (i)


Also, (de3/dt) = (1/C) * f3 = (1/C) * f4 - - - - - - - - (ii)

In matrix form, (dx/dt) = Ax + Bu

(de3/dt) 0 1/C e3 0

= + U

(df4/dt) -1/L -R/L f4 1/L

Generation of Equations from Bond Graphs (contd..)


Some Points to Note:

One of the most important features of bond graphs is ‘easy determination of causality’.

For computer algorithms to solve equations, representing the physics of real systems, it is essential that proper input and output causality be maintained.

State variables and computational problems are known completely after assigning causality, even before the modeler derives a single equation.

Modeling in terms of bond graphs helps one focus on modeling the physical effects without bothering about the computational issues such as generation of a consistent system of equations.

B.G. on one hand relate closely to the structure of the system being modeled, while on the other hand, they contain enough information to derive other system representations like state-space equations.

B.G can be drawn or a B.G. description of the system can be created before causality is considered. In contrast, causality has to be considered before a block diagram can be drawn. E.g. the decision as to whether a resistor has a voltage or current as output has to be made before a block diagram can be constructed. In B.G., causality can be automatically assigned after the system has been described.

Generation of Equations from Bond Graphs (contd..)


The Bond Graph Metamodeling Environment in GME


Applications in GME Metamodeling Environment

• RLC Circuit


Applications in GME Metamodeling Environment (contd..)

DC Motor model


Future Concepts

• Defining the GME Approach for analysis of Bond Graphs [1]

Conventional Approach Probable GME / Matlab Approach

1. Determination of Physical System and specifications from the requirements.

2. Draw a functional Block Diagram.

3. Transform the physical system into a schematic.

4. Use Schematic and obtain a mathematical model, a block diagram or a state representation.

5. Reduce the block diagram to a close loop system.

6. Analyze, design and test.

1. Identify the physical system elements and represent a word Bond Graph.

2. Represent a bond graph model in GME.

3. GME interpreters generate equations in a suitable form (e.g. state space variable matrix form) suitable for analysis in Matlab.

4. Use Matlab, to analyze, design and test.


Future Concepts (contd..)

Fig. The Simulation Generation Process [7]

Bond Graph Interpreters in GME ??

• Creating Bond Graph Interpreters


• Advanced Bond Graph Techniques

Expansion of Bond Graphs to Block Diagrams

Bond Graph Modeling of Switching Devices

Bond Graphs as Object-oriented physical-systems modeling

Hierarchical modeling using Bond Graphs

Use of port-based approach for Co-simulation

Future Concepts (contd..)


References

1. Granda J. J, “Computer Aided Design of Dynamic Systems” http://gaia.csus.edu/~grandajj/

2. Wong Y. K., Rad A. B., “Bond Graph Simulations of Electrical Systems,” The Hong Kong

Polytechnic University, 1998

3. http://www.ce.utwente.nl/bnk/bondgraphs/bond.htm

4. Broenink J. F., "Introduction to Physical Systems Modeling with Bond Graphs,"

University of Twente, Dept. EE, Netherlands.

5. Granda J., Reus J., "New developments in Bond Graph Modeling Software Tools: The Computer

Aided Modeling Program CAMP-G and MATLAB," California State

University, Sacramento

6. http://www.bondgraphs.com/about2.html

7. Vashishtha D., “Modeling And Simulation of Large Scale Real Time Embedded Systems,” M.S.

Thesis, Vanderbilt University, May 2004

8. Hogan N. "Bond Graph notation for Physical System models," Integrated

Modeling of Physical System Dynamics

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