INC 693, 481 Dynamics System and Modelling:
Linear Graph Modeling I
Dr.-Ing. Sudchai Boonto
Assistant Professor
Department of Control System and Instrumentation Engineering
King Mongkut’s Unniversity of Technology Thonburi Thailand
Introduction to Linear Graph Models
node
effort
variable
flow variable
branch
graph is constructed from:
ˆ A set of branches that each represent an energy port associated
with a passive of source system element.
ˆ A set of nodes that represent the points of interconnection of the
lumped elements.
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Introduction to Linear Graph Models
C
C
111
000
Capacitance
L
Inductance
R
Resistance
graph is constructed from:
ˆ Capacitance elements (except electrical capacitance) must have
their effort variable defined with respect to a constant reference
value:
ˆ velocity on mass with respect to a constant velocity inertial
reference frame
ˆ in fluid system the pressure is measured with respect to the reference
pressure.
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Introduction to Linear Graph Models
ˆ The arrow on the graph element of each type is drawn in the
direction for which:
ˆ e the effort variable associated with the branch.
ˆ the flow variable f is defined as having a positive value.
ˆ For source elements the arrow associated with the branch
designates the sign associated with the source variable:
ˆ for a flow source the arrow designates the direction defined for
positive flow variable flow, and
ˆ for an effort source the arrow designates the direction defined for the
effort variable drop.
ˆ the arrow on an effort variable source branch is commonly drawn
toward the reference node.
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Introduction to Linear Graph Models
11
00
k
00
11
00
11
b
m
00
11
00
11
00
1111111111
00000000
00
11
v
v
Fs (t)
Figure: a simple mechanical system
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
b
k
m
1111
0000
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Element Interconnection Laws
Compatibility
ˆ Compatibility law represents a set of constraints on effort
variables in the graph that may be related to physical laws. The
sum of the effort variable drops on the branches around any closed
loop in linear graph is identically zeros, or:
N
∑
ei = 0 for any N forming a closed loop
i=1
1
B
2
Loop
A
C
4
D
1111
0000
3
4
∑
ei = e1 − e2 + e3 − e4
i=1
= (eA − eB ) − (eC − eB )
+ (eC − eD ) − (eA − eD ) = 0
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Element Interconnection Laws
Compatibility
The physical interpretation of the compatibility law in the various energy
domains is:
ˆ Mechanical systems: the velocity drops across all elements sum
to zero around any closed path path in a linear graph.
ˆ Electrical systems: the compatibility law is identical to KVL
which states that the summation of all voltage drops around any
closed loop in an electrical circuit is identically zero.
ˆ Fluid systems: Pressure is a scalar potential which must sum to
zero around any closed path in a fluid system.
ˆ Thermal systems: Themperature is a scalar potential which must
sum to zero around any closed path in a thermal system.
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Element Interconnection Laws
Continuity
ˆ Continuity law specifies constraints on the flow variables in a
linear graph that may be related to physical laws. The sum of flow
variables flowing into any closed contour drawn on a linear graph is
zeros, or
N
∑
fi = 0 for any N branches that intersect a closed contour
i=1
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
J 8/20 I }
Element Interconnection Laws
Continuity
1
1
4
A
2
5
C
B
3
closed
contour
3
6
closed
contour
2
for the right hand figure:
f1 − f4 + f5 = 0,
f2 − f5 − f6 = 0,
−f3 + f4 + f6 = 0
for the closed contour:
f1 + f2 − f3 = (f4 − f5 ) + (f5 + f6 ) − (f4 + f6 ) = 0
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
J 9/20 I }
Element Interconnection Laws
Continuity
The principle of continuity corresponds to the following physical
constraints:
ˆ Mechanical systems: In a translational (or rotatinal) mechanical
system continuity at a node arises as a direct expression of
Newton’s laws of motion.
ˆ Electrical systems: the continuity at an electrical node is KCL
which states that the sum of currents flowing into any node in a
circuit must be zero.
ˆ Fluid system: the sum of volume flow rates into the junction must
be zero.
ˆ Thermal systms: the continuity of heat flow rate ensures that
there is no accumulation of heat at any junction between elements.
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Sign Conventions on One-Port System Elements
Electrical Circuit
A
i=
iR
Vs (t)
+
−
R
A
R
Vs (t)
i = − R1 Vs
1
R Vs
Vs (t)
R
(a) electrical system
the compatibility equation for the middle graph is
−Vs + vR = 0
then
iR =
1
Vs
R
the compatibility equation for the right graph is
Vs + vR = 0
then
1
iR = − Vs .
R
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
J 11/20 I }
Sign Conventions on One-Port System Elements
Mechanical System
v
1
0
F = F (t)
0
1
0
1
m
m
F
(t)
0
1
0
1
0
1
1 1111
1111110
000000
0000
v
F (t)
F (t)
v
v
11
00
F = −F (t)
00
11
00
11
m
m
F (t)
00
11
00
11
00
11
00 1111
11
11111
00000
0000
m
F (t)
F = −F (t)
01
0
1
m
m
1010 F (t)
10 111
000000
111111
000
00000010 1111
111111
0000
000
111
m
m
F = F (t)
11
00
00
11
m
m
F (t)
00
11
00
11
00
11
00000
11111
000
111
00 1111
11
00000
11111
0000
000
111
m
F (t)
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Procedure of System Graph
General Procedure
1. defined the system: inputs, outputs, energy and required elements
2. for a schematic,or pictorial, model of the physical system
3. determine the system sources, energy storage and dissipation
elements
4. Identify the effort variables and draw a set of nodes
5. determine the appropriate nodes for each lumped element, and
inert each element into the graph
6. select a set of sign conventions for the passive elements and draw
the arrows on the graph
7. select the sign conventions for the system source elements
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Mechanical Translational System Models
Fs (t)
m
v
Physical system
1
0
0
1
0
1
0
1
0
1
0
1
0
01
1
0
1
0
1
0
01
1
0
1
0
1
0
01
1
0
1
0
1
0
01
1
0
01
1
0
1
compatibility:
Fs (t)
0110
1010
m
v
10
1010
k
10
1010
1111111111
0000000000
Schematic representation
v
Fs (t)
k
000 v
111
m
ref
=0
Linear graph
continuity
vm = vk
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
Fs − Fk − Fm = 0
J 14/20 I }
Mechanical Rotational System Models
b2
ωs (t)
111111
000000
111111
000000
Motor
Flywheel J
ωJ (t)
Bearing b1
b2
11
00
00
11
00
11
00
11
111
000
A
B
A
ωs (t)
B
Physical system
b1
J
1111
0000
0000
1111
Linear graph
ˆ We have an angular velocity source ωs (t), a rotary inertia J, a
rotary damper b1 and a rotary damper b2
ˆ the angular velocity ωJ of flywheel must be defined relative to the
fixed reference node.
ˆ the inner bearing rotates at the same angular velocity as the
flywheel then b1 is parallel with the flywheel.
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Electrical Models
L1
A
Vs (t)
L2
B
+
−
C
C1
C2
RL
G
(a) Electrical model
L1
B
L2
A
C
C1
C2
Vs (t)
111
000
000 G
111
RL
(b) Linear graph
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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Fluid System Models
Qs (t)
A
Patm
Reservoir
R1
Tank
C1
A
R2
C2
B
Qout
Qs (t)
R1
B
R2
000
111
11 P
00
atm
ˆ the pressure drop across valve R1 is PA − PB and so it is inserted
between the two nodes A and B
ˆ the outlet valve R2 discharges between the storage tank pressure
PA and the reference pressure Patm , and so it is connected in
parallel with C2 .
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
J 17/20 I }
Fluid System Models
Long pipe effect
Rp
Tank Cf
B
pipe Rp , Ip
Ip
C
B
A
Rf
Rf
Ps (t)
A
Patm
Cf
111 P
000
atm
the pipe is assumed to:
ˆ dissipate energy through frictional losses at the walls
ˆ to store energy associated with the motion of the fluid within the
pipe
ˆ they can be approximated by a combination of a single lumped
resistance Rp and a fluid inductnace Ip .
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
J 18/20 I }
Fluid System Models
Long pipe effect
The two elements have a common flow Q and are described by the
elemental equation:
PRP = RP Q
PIP = IP
dQ
dt
for the resistance
for the iductance
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
J 19/20 I }
Reference
1. Wellstead, P. E. Introduction to Physical System Modelling,
Electronically published by:
www.control-systems-principles.co.uk, 2000
2. Banerjee, S., Dynamics for Engineers, John Wiley & Sons, Ltd.,
2005
3. Rojas, C. , Modeling of Dynamical Systems, Automatic Control,
School of Electrical Engineering, KTH Royal Institute of
Technology, Sweeden
4. Fabien, B., Analytical System Dynamics: Modeling and Simulation
Springer, 2009
INC 693, 481 Dynamics System and Modelling: , Linear Graph Modeling I
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