System Analysis through Bond Graph Modeling

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System Analysis through Bond Graph Modeling

• Robert McBride
• May 3, 2005

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Overview

• Modeling
• Bond Graph Basics
• Bond Graph Construction
• Simulation
• System Analysis
• Efficiency Definition and Analysis
• Optimal Control
• System Parameter Variation
• Conclusions
• References

3
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.

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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, eeffort, fflow.
• ef power.

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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 Nm Angular Velocityrad/sec
Hydraulic Pressure N/m2 Volumetric Flow m3/sec
Chemical Chemical PotentialJ/mole Molar Flowmole/sec
Thermodynamic TemperatureK Entropy FlowdS/dt W/K
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Modeling Bond Graph Basic Elements

• Power Bonds Connect at Junctions.
• There are two types of junctions, 0 and 1.

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Modeling Bond Graph Basic Elements

• 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.

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Modeling Bond Graph Construction
This bond graph is a-causal
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Modeling Bond Graph ConstructionCausality

• 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.

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Modeling Bond Graph ConstructionIntegral
Causality
Integral causality is preferred when given a
choice.
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Modeling Bond Graph ConstructionNecessary
Causality
5
4
0
1
1
11
3
13
2
12
f
e
Efforts are equal
Flows are equal
e1 e3 e4 e5 e2
f11 f13 f12
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Modeling Bond Graph Construction
RR1
RR2
SE
0
1
1
SineVoltage1
CC1
IL1
This bond graph is Causal
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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?
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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.

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Modeling Lagrangian Bond Graph Construction
(cont.)

1. Note the equations derived from the Lagrangian
   show the balance of efforts around each
   1-junction.
2. If needed, develop the Hamiltonian for the
   conservative system.
3. Add non-conservative elements where needed on the
   bond graph structure.
4. Add external forces where needed as bond graph
   sources.
5. Use bond graph methods to simplify if desired.

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Modeling Lagrangian Bond Graph, Gyroscope
Example
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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 ?, ?, and f.
4. Generalized momentums are

.
.
.
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Modeling Lagrangian Bond Graph, Gyroscope
Example

1. Solve for the generalized velocities.

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Modeling Lagrangian Bond Graph, Gyroscope
Example

1. Complete Lagrange Equations

.
.
Note Pf Cross Terms
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Modeling Lagrangian Bond Graph, Gyroscope
Example
.
.
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Overview

• Modeling
• Bond Graph Basics
• Bond Graph Construction
• Simulation
• System Analysis
• Efficiency Definition and Analysis
• Optimal Control
• System Parameter Variation
• Conclusions
• References

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Common Bond Graph Simulation Flow Chart
Question Does Such a Simulation Environment
Exist?
Bond Graph Construction
Equation Formulation
Simulation Environment
Simulation Code Development
Model Analysis through Simulation
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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.

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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.

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The Dymola Bond Graph Library Bonds
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The Dymola Bond Graph Library Junctions
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The Dymola Bond Graph Library Passive Elements
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The Dymola Gyroscope Bond Graph Model
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The Dymola Gyroscope Bond Graph Model
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Gyroscopically Stabilized Platform
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Gyroscopically Stabilized Platform with Mounted
Camera
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Overview

• Modeling
• Bond Graph Basics
• Bond Graph Construction
• Simulation
• System Analysis
• Efficiency Definition and Analysis
• Optimal Control
• System Parameter Variation
• Conclusions
• References

33
System Analysis Servo-Positioning System
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System Analysis Motor Dynamics
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System Analysis Fin Dynamics
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System Analysis Backlash Model
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System Analysis Servo Controllers
Control Scheme 1
Control Scheme 2
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System Analysis Servo Step Response
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System AnalysisController 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.

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System Analysis Servo Step Response Efficiency
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System AnalysisController 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?
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System Analysis Missile System
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System Analysis Missile System Bond Graph
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System Analysis Missile System 3-Loop Autopilot
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System Analysis Missile System Dymola Model
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Missile System Analysis Performance Index
Minimization
Linear Constraints
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Missile System Analysis Performance Index
Minimization
a
d
.
? q
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Sample Optimal Control Gains and Response
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Sample Optimal Gain Efficiency
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System AnalysisController 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?
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System Analysis Missile System Dymola Model
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System AnalysisAutopilot Response Comparison
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System Analysis Varying Mass Parameter
Efficiency

• Often a systems 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.

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System AnalysisMass Parameter Variations
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System AnalysisMass Parameter Variations
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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.

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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.

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• Backups

59
Modeling Lagrangian Bond Graph, Ball Joint Table
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Modeling Lagrangian Bond Graph, Ball Joint Table
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System AnalysisLinear Autopilot Power IO
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System AnalysisLinear Autopilot Energy IO
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System AnalysisLinear Autopilot Normalized
Energy and Integral (Normalized Energy)
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System AnalysisLinear Autopilot Efficiency
Comparison
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System AnalysisMissile Parameters