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 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 n th 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  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 f A B Power moves from system A to system B

Modeling: Bond Graph Basics effort/flow definitions in different engineering domains Effort eFlow f ElectricalVoltage [V]Current [A] TranslationalForce [N]Velocity [m/s] RotationalTorque [N*m]Angular Velocity [rad/sec] HydraulicPressure [N/m 2 ]Volumetric Flow [m 3 /sec] ChemicalChemical Potential [J/mole] Molar Flow [mole/sec] ThermodynamicTemperature [K] Entropy Flow dS/dt [W/K]

Modeling: Bond Graph Basic Elements Power Bonds Connect at Junctions. There are two types of junctions, 0 and Efforts are equal e1 = e2 = e3 = e4 = e5 Flows sum to zero f1+ f2 = f3 + f4 + f5 Flows are equal f11 = f12 = f13 Efforts sum to zero e11+ 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 Elements I C R TF m e1 f1 e2 f2 e2 = 1/m*e1 f1 = 1/m*f2 GY e1 f1 e2 f2 d f2 = 1/d*e1 f1 = 1/d*e2 SESF

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 Construction Causality e f f e

Modeling: Bond Graph Construction Integral Causality e f I e f sIsI 1 f e C f e sCsC 1 Integral causality is preferred when given a choice.

Modeling: Bond Graph Construction Necessary Causality e Efforts are equal f Flows are equal e1 = e3 = e4 = e5 ≡ e2f11 = 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?

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

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 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

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 Hz Control Scheme 1Control Scheme 2

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 Linear Constraints 00

Missile System Analysis: Performance Index Minimization αδ θ = q.

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

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 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 McBride, R. T., Quality Metric for Controller Design. Raytheon Missile Systems, Tucson AZ 85734, McBride, R. T., Cellier, F. E., System Efficiency Measurement through Bond Graph Modeling. Proceedings, International Conference of Bond Graph Modeling, New Orleans, Louisiana, pp 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 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

Backups

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