system analysis through bond graph modeling robert mcbride may 3, 2005
TRANSCRIPT
System Analysis through Bond Graph Modeling
Robert McBride
May 3, 2005
Overview
• Modeling– Bond Graph Basics– Bond Graph Construction
• Simulation• System Analysis
– Efficiency Definition and Analysis– Optimal Control– System Parameter Variation
• Conclusions• References
Modeling: Bond Graph Basics
• Bond graphs provide a systematic method for obtaining dynamic equations.– Based on the 1st law of thermodynamics.
– Map the power flow through a system.
– Especially suited for systems that cross multiple engineering domains by using a set of generic variables.
– For an nth order system, bond-graphs naturally produce n, 1st-order, coupled equations.
– This method easily identifies structural singularities in the model. Algebraic loops can also be identified.
Modeling: Bond Graph Basic Elements• The Power Bond
The most basic bond graph element is the power arrow or bond.
There are two generic variables associated with every power bond, e=effort, f=flow.
e*f = power.
e
fA B
Power moves from system A to system B
Modeling: Bond Graph Basics
• effort/flow definitions in different engineering domains
Effort e Flow f
Electrical Voltage [V] Current [A]
Translational Force [N] Velocity [m/s]
Rotational Torque [N*m] Angular Velocity[rad/sec]
Hydraulic Pressure [N/m2] Volumetric Flow
[m3/sec]
Chemical Chemical Potential[J/mole]
Molar Flow[mole/sec]
Thermodynamic Temperature[K]
Entropy FlowdS/dt [W/K]
Modeling: Bond Graph Basic Elements• Power Bonds Connect at Junctions.
• There are two types of junctions, 0 and 1.
0 11
2
3
45
11
12
13
Efforts are equal
e1 = e2 = e3 = e4 = e5
Flows sum to zerof1+ f2 = f3 + f4 + f5
Flows are equal
f11 = f12 = f13
Efforts sum to zeroe11+ e12 = e13
• I for elect. inductance, or mech. Mass
• C for elect. capacitance, or mech. compliance
• R for elect. resistance, or mech. viscous friction
• TF represents a transformer
• GY represents a gyrator
• SE represents an effort source.
• SF represents a flow source.
Modeling: Bond Graph Basic ElementsI
C
R
TFm
e1
f1
e2
f2
e2 = 1/m*e1f1 = 1/m*f2
GYe1
f1
e2
f2d
f2 = 1/d*e1f1 = 1/d*e2
SE
SF
Modeling: Bond Graph Construction
SE 1
R:R1
0
C:C1
1
R:R2
I:L1
SineVoltage1
This bond graph is a-causal
• Causality determines the SIGNAL direction of both the effort and flow on a power bond.
• The causal mark is independent of the power-flow direction.
Modeling: Bond Graph ConstructionCausality
e
f
f
e
Modeling: Bond Graph ConstructionIntegral Causality
e
fI
e
f sI1
f
e Cf
e
sC1
Integral causality is preferred when given a choice.
Modeling: Bond Graph ConstructionNecessary Causality
0 11
2
3
45
11
12
13
eEfforts are equal
f
Flows are equale1 = e3 = e4 = e5 ≡ e2 f11 = f13 ≡ f12
Modeling: Bond Graph Construction
SE 1
R:R1
0
C:C1
1
R:R2
I:L1
SineVoltage1
This bond graph is Causal
Modeling: Bond Graph Construction From the System Lagrangian
• Power flow through systems of complex geometry is often difficult to visualize.
• Force balancing methods may also be awkward due to the complexity of internal reaction forces.
• It is common to model these systems using an energy balance approach, e.g. a Lagrangian approach.
Question: Is there a method for mapping the Lagrangian of a system to a bond graph representation?
0 VTL0
dt
d
ii qq
LL
Modeling: Lagrangian Bond Graph Construction
1. Assume that the system is conservative.
2. Note the flow terms in the Lagrangian. The kinetic energy terms in the Lagrangian will have the form ½ I * f 2 where I is an inertia term and f is a flow term.
3. Assign bond graph 1-junctions for each distinct flow term in the Lagrangian found in step 2.
4. Note the generalized momentum terms.
5. For each generalized momentum equation solve for the generalized velocity.
ii q
p
L
iq
Modeling: Lagrangian Bond Graph Construction (cont.)
6. Note the equations derived from the Lagrangian show the balance of efforts around each 1-junction.
7. If needed, develop the Hamiltonian for the conservative system.
8. Add non-conservative elements where needed on the bond graph structure.
9. Add external forces where needed as bond graph sources.
10. Use bond graph methods to simplify if desired.
Modeling: Lagrangian Bond Graph, Gyroscope Example
2222
222
coscos
sin2
1
CCC
BAAATL
Modeling: Lagrangian Bond Graph, Gyroscope Example
1. The system is already conservative.
2. Rewrite the Lagrangian to note the flow terms.
3. Form 1-junctions for θ, ψ, and φ.
4. Generalized momentums are
cos2
1
2
1
cossin2
1
22
222
CCAA
CCCBAL
. . .
coscossin 22 CCCCBA
Lp
AA
Lp
cos
CCL
p
Modeling: Lagrangian Bond Graph, Gyroscope Example
5. Solve for the generalized velocities.
CCCBA
Cp
22 cossin
cos
AA
p
C
Cp cos
Modeling: Lagrangian Bond Graph, Gyroscope Example
6. Complete Lagrange Equations
. .
0sincos
cossin2cossin2
cossin 22
CC
CCBA
CCCBAL
dt
dp
0sincos
CCCL
dt
dp
0sincossin
cossin
2
2
CCC
BAAAL
dt
dp
Note P*f Cross Terms
Modeling: Lagrangian Bond Graph, Gyroscope Example
. .
Overview
• Modeling– Bond Graph Basics– Bond Graph Construction
• Simulation• System Analysis
– Efficiency Definition and Analysis– Optimal Control– System Parameter Variation
• Conclusions• References
Common Bond Graph Simulation Flow Chart
Bond Graph Construction
Equation Formulation
Simulation Code Development
Model Analysis through Simulation
Simulation Environment
Question: Does Such a Simulation Environment Exist?
The Dymola Simulation Environment
• Dymola/Modelica provides an object-oriented simulation environment.
• Dymola is very capable of handling algebraic loops and structural singularities.
• Dymola does not have any knowledge of bond graph modeling. A bond graph library is needed within the framework of Dymola.
The Dymola Bond Graph Library
• The bond graph library consists of a Dymola model for each of the basic bond graph elements.
• These elements are used in an object-oriented manner to create bond graphs.
The Dymola Bond Graph Library: Bonds
The Dymola Bond Graph Library: Junctions
The Dymola Bond Graph Library: Passive Elements
The Dymola Gyroscope Bond Graph Model
The Dymola Gyroscope Bond Graph Model
Gyroscopically Stabilized Platform
Gyroscopically Stabilized Platform with Mounted Camera
Overview
• Modeling– Bond Graph Basics– Bond Graph Construction
• Simulation• System Analysis
– Efficiency Definition and Analysis– Optimal Control– System Parameter Variation
• Conclusions• References
System Analysis: Servo-Positioning System
System Analysis: Motor Dynamics
System Analysis: Fin Dynamics
System Analysis: Backlash Model
System Analysis: Servo Controllers
1
999985.0*095.0)(
z
zZG
@ 6000 Hz
453.0
688.0*172.0)(2
z
zzY
@ 1200 Hz
Control Scheme 1 Control Scheme 2
System Analysis: Servo Step Response
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.284.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3F
in P
ositi
on (
deg)
Time (sec)
5 (deg) Step: Hinge Moment = -6 (N*m/deg)
NL1NL2
System Analysis:Controller Efficiency Definition
• By monitoring the output power and normalizing by the input power an efficiency calculations is defined as
• Bond graph modeling naturally provides the means for this analysis.
tf
o tf
InIn
tf
OutOuttf
o tf
tf
controller dtdtFlowEffort
dtFlowEffortdt
dtInputPower
dtrOutputPowe
0
0
0
0
*
*
System Analysis: Servo Step Response Efficiency
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
-6In
teg(
|Fin
Ene
rgy|
)
Time (sec)
5 (deg) Step: Hinge Moment = -6 (N*m/deg)
NL1NL2
System Analysis:Controller Efficiency
• The power flow through a bond graph model of the plant can be used to compare the effectiveness of different control schemes regardless of the architecture of the controller design, and without limiting the analysis to linear systems.
Question: Can the controller efficiency be used to measure optimality of controller gain selection?
System Analysis: Missile System
System Analysis: Missile System Bond Graph
System Analysis: Missile System 3-Loop Autopilot
1
1
1
System Analysis: Missile System Dymola Model
Missile System Analysis: Performance Index Minimization
dttf
PwYA
AYC
AwZA
AZC
AwPI 0
2 3
2 2
2 1
dB3MarginGain
20Margin Phase
%30Undershoot
%20Overshoot
Linear Constraints
0 0
Missile System Analysis: Performance Index Minimization
αδ
θ = q.
Sample Optimal Control Gains and Response
Variable Set 1 Set 2 Set 4KA 0.07836 0.10696 0.11625KR 0.30587 0.23254 0.21848WI 36.11504 24.68886 21.86830
KDC 1.13686 1.10027 1.09226PI 0.06014 0.06105 0.06224
Gain Marg. 3.465 dB 3.000 dB 3.000 dBPhase Marg. 180˚ 61.535˚ 45.0055˚Overshoot 0% 0.92% 6%
Undershoot 30.00% 25.60% 24.05%Pole1 51.51505 (-1 + i) 32.68964 (-1 + i) -26.845 + 28.459iPole2 51.51505 (-1 - i) 32.68964 (-1 - i) -26.845 - 28.459iPole3 -17.17168 -29.27869 -36.70808
0 0.05 0.1 0.15 0.2 0.25-14
-12
-10
-8
-6
-4
-2
0
2
4
6
Mis
sile
Acc
eler
atio
n (G
's)
Time (sec)
Complete System: -12.92 G step response
Set 1Set 2Set 430% Undershoot
Sample Optimal Gain Efficiency
0 0.05 0.1 0.15 0.2 0.250
0.5
1
1.5
2
2.5
3
3.5x 10
-6
Act
uato
r E
ffec
ienc
y
Time (sec)
Complete System: -12.92 G step response
Set 1Set 2Set 4
System Analysis:Controller Efficiency
• The efficiency signal can be used as a benchmark when comparing efficiencies of different gain selections.
• Constraint violation is assumed when the efficiency signal is more proficient than the benchmark.
Question: How do the efficiency signals compare against an optimal control autopilot such as an SDRE design?
System Analysis: Missile System Dymola Model
System Analysis:Autopilot Response Comparison
0 0.05 0.1 0.15 0.2 0.25-15
-12
-9
-6
-3
0
3
6
Mis
sile
Acc
eler
atio
n (G
's)
Time (sec)
Complete System: -12.92 G step response
Set 2Set 4SDRE30% Undershoot
0 0.05 0.1 0.15 0.2 0.250
0.5
1
1.5
2
2.5x 10
-6
Act
uato
r E
ffec
ienc
y
Time (sec)
Complete System: -12.92 G step response
Set 2Set 4SDRE
System Analysis: Varying Mass Parameter Efficiency
• Often a system’s mass parameters change as parts replacements are made.
• The autopilot gain selection, chosen with the original mass parameters, may no longer be valid for the changed system.
• The efficiency signal can be used to determine if a controller gain redesign is necessary.
System Analysis:Mass Parameter Variations
0 0.05 0.1 0.15 0.2 0.25-20
-15
-10
-5
0
5
Mis
sile
Acc
eler
atio
n (G
's)
Time (sec)
Complete System: -12.92 G step response
Xcg = 8.5Xcg = 8.875Xcg = 9.25Xcg = 9.625Xcg = 10 (Nominal)Xcg = 10.375Xcg = 10.7530% Undershoot20% Overshoot
0.04 0.06 0.08 0.1 0.12 0.14 0.160.06
0.061
0.062
0.063
0.064
0.065
0.066
0.067
Per
form
ance
Ind
ex
Time (sec)
Complete System: 1 G step response
Xcg = 8.5Xcg = 8.875Xcg = 9.25Xcg = 9.625Xcg = 10 (Nominal)Xcg = 10.375Xcg = 10.75
System Analysis:Mass Parameter Variations
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
-6
Act
uato
r E
ffec
ienc
y
Time (sec)
Complete System: -12.92 G step response
Xcg = 8.5Xcg = 8.875Xcg = 9.25Xcg = 9.625Xcg = 10 (Nominal)Xcg = 10.375Xcg = 10.75
Conclusions
• A method for creating a bond graph from the system Lagrangian was provided.
• A Dymola Bond Graph Library was constructed to allow system analysis directly from a bond graph model.
• A controller efficiency measurement was defined.• The controller efficiency measurement was used to
compare controllers with different control structures and gain sets to better determine a proper gain set/control structure.
• The efficiency signal is also useful for determining the need for gain re-optimization when a system undergoes changes in its design.
References• Cellier, F. E., McBride, R. T., Object-Oriented Modeling of Complex
Physical Systems Using the Dymola Bond-Graph Library. Proceedings, International Conference of Bond Graph Modeling, Orlando, Florida, 2003, pp. 157-162.
• McBride, R. T., Cellier, F. E., Optimal Controller Gain Selection Using the Power Flow Information of Bond Graph Modeling. Proceedings, International Conference of Bond Graph Modeling, New Orleans, Louisiana, 2005, pp. 228-232.
• McBride, R. T., Quality Metric for Controller Design. Raytheon Missile Systems, Tucson AZ 85734, 2005.
• McBride, R. T., Cellier, F. E., System Efficiency Measurement through Bond Graph Modeling. Proceedings, International Conference of Bond Graph Modeling, New Orleans, Louisiana, 2005. pp. 221-227.
• McBride, R. T., Cellier, F. E., Object-Oriented Bond-Graph Modeling of a Gyroscopically Stabilized Camera Platform. Proceedings, International Conference of Bond Graph Modeling, Orlando, Florida, 2003, pp. 157-223.
• McBride, R. T., Cellier, F. E., A Bond Graph Representation of a Two-Gimbal Gyroscope. Proceedings, International Conference of Bond Graph Modeling, Phoenix, Arizona, 2001, pp. 305-312.
Backups
Modeling: Lagrangian Bond Graph, Ball Joint Table
cos2
1
2
1cos
2
1sin
2
1 23
2222
221 mglIIIIL
Modeling: Lagrangian Bond Graph, Ball Joint Table
System Analysis:Linear Autopilot Power IO
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2-4
-3
-2
-1
0
1
2
3
4
Out
put
Pow
er (
N*m
/s)
Time (sec)
5 (deg) Step: Hinge Moment = 0 (N*m/deg)
PID1PID2PID3
0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.1450
500
1000
1500
2000
2500
3000
3500
4000
4500
Inpu
t P
ower
(N
*m/s
)
Time (sec)
5 (deg) Step: Hinge Moment = 0 (N*m/deg)
PID1PID2PID3
System Analysis:Linear Autopilot Energy IO
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
50
100
150
200
250
300
350
400
Inpu
t E
nerg
y (N
*m)
Time (sec)
5 (deg) Step: Hinge Moment = 0 (N*m/deg)
PID1PID2PID3
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Out
put
Ene
rgy
(N*m
)
Time (sec)
5 (deg) Step: Hinge Moment = 0 (N*m/deg)
PID1PID2PID3
System Analysis:Linear Autopilot Normalized Energy
and Integral (|Normalized Energy|)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
1
2
3
4
5
6x 10
-6
Inte
g(|F
in E
nerg
y|)
Time (sec)
5 (deg) Step: Hinge Moment = 0 (N*m/deg)
PID1PID2PID3
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
1
2
3
4
5
x 10-4
|Fin
Ene
rgy|
Time (sec)
5 (deg) Step: Hinge Moment = 0 (N*m/deg)
PID1PID2PID3
System Analysis:Linear Autopilot Efficiency Comparison
0 1 2 3 4 5 6 7 8 9 104
4.2
4.4
4.6
4.8
5
5.2
Fin
Pos
ition
(de
g)
Time (sec)
5 (deg) Step: Hinge Moment = -6 (N*m/deg)
PID1PID2PID3
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7x 10
-8
Inte
g(|F
in E
nerg
y|)
Time (sec)
5 (deg) Step: Hinge Moment = -6 (N*m/deg)
PID1PID2PID3
System Analysis:Missile Parameters