system analysis through bond graph modeling robert mcbride may 3, 2005

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