System Analysis through Bond Graph Modeling Robert Mc. Bride 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 1 st 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, 1 st -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 Ø The most basic bond graph element is the power arrow or bond. Ø There are two generic variables associated with every power bond, e=effort, f=flow. Ø e*f = power. A e f Power moves from system A to system B B
Modeling: Bond Graph Basics • effort/flow definitions in different engineering domains Effort e Flow f Voltage [V] Current [A] Force [N] Velocity [m/s] Rotational Torque [N*m] Angular Velocity [rad/sec] Hydraulic Pressure [N/m 2] Volumetric Flow [m 3/sec] Chemical Potential [J/mole] Molar Flow [mole/sec] Temperature [K] Entropy Flow d. S/dt [W/K] Electrical Translational Thermodynamic
Modeling: Bond Graph Basic Elements • Power Bonds Connect at Junctions. • There are two types of junctions, 0 and 1. 5 1 4 0 3 2 11 1 13 12 Efforts are equal Flows are equal e 1 = e 2 = e 3 = e 4 = e 5 f 11 = f 12 = f 13 Flows sum to zero Efforts sum to zero f 1+ f 2 = f 3 + f 4 + f 5 e 11+ e 12 = e 13
Modeling: Bond Graph Basic Elements • I for elect. inductance, or mech. Mass I • C for elect. capacitance, or mech. compliance C • R for elect. resistance, or mech. viscous friction R • TF represents a transformer e 1 f 1 TF • GY represents a gyrator e 1 f 1 GY d • SE represents an effort source. SE • SF represents a flow source. SF m e 2 f 2 e 2 = 1/m*e 1 f 1 = 1/m*f 2 = 1/d*e 1 f 1 = 1/d*e 2
Modeling: Bond Graph Construction R: R 1 SE 1 R: R 2 0 1 Sine. Voltage 1 C: C 1 This bond graph is a-causal I: L 1
Modeling: Bond Graph Construction Causality • 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. e f
Modeling: Bond Graph Construction Integral Causality e f I C e f 1 s. I 1 s. C Integral causality is preferred when given a choice.
Modeling: Bond Graph Construction Necessary Causality 5 1 4 0 3 1 11 2 e Efforts are equal e 1 = e 3 = e 4 = e 5 ≡ e 2 13 12 f Flows are equal f 11 = f 13 ≡ f 12
Modeling: Bond Graph Construction R: R 1 SE 1 R: R 2 0 1 Sine. Voltage 1 C: C 1 This bond graph is Causal I: L 1
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?
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.
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
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
Modeling: Lagrangian Bond Graph, Gyroscope Example 5. Solve for the generalized velocities.
Modeling: Lagrangian Bond Graph, Gyroscope Example 6. Complete Lagrange Equations . . 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 Question: Does Such a Simulation Environment Exist? Bond Graph Construction Equation Formulation Simulation Environment Simulation Code Development Model Analysis through Simulation
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 Control Scheme 1 @ 6000 Hz Control Scheme 2 @ 1200 Hz
System Analysis: Servo Step Response
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.
System Analysis: Servo Step Response Efficiency
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 0 Linear Constraints 0
Missile System Analysis: Performance Index Minimization δ. θ=q α
Sample Optimal Control Gains and Response
Sample Optimal Gain Efficiency
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
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
System Analysis: Mass Parameter Variations
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. , Mc. Bride, 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. • Mc. Bride, 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. • Mc. Bride, R. T. , Quality Metric for Controller Design. Raytheon Missile Systems, Tucson AZ 85734, 2005. • Mc. Bride, 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. • Mc. Bride, 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. • Mc. Bride, 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
Modeling: Lagrangian Bond Graph, Ball Joint Table
System Analysis: Linear Autopilot Power IO
System Analysis: Linear Autopilot Energy IO
System Analysis: Linear Autopilot Normalized Energy and Integral (|Normalized Energy|)
System Analysis: Linear Autopilot Efficiency Comparison
System Analysis: Missile Parameters
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