Design of Disturbance Rejection Controllers for a Magnetic Suspension System

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Design of Disturbance Rejection Controllers for a
Magnetic Suspension System

• By Jon Dunlap
• Advisor Dr. Winfred K.N. Anakwa
• Bradley University
• April 27, 2006

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Outline Of Presentation

• Goal
• System Information
• Previous Lab Work
• Preliminary Lab Work
• Internal Model Principle
• Design Process
• Results
• Conclusion

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Goal

• Multiple Controllers for Multiple Disturbances
• Digital Controllers
• Created In Simulink
• xPC Target Box Serving as Controller Container
• Minimize Steady-State Error, Overshoot and
  Setting Time
• Act As A Stepping Stone From Previous Work
• Practical Use in Antenna Stabilization

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Method

• Method of Choice
• Internal Model Principle
• B.A. Francis W.M. Wonham
• The Internal Model Principle of Control Theory,
  1976
• Chi-Tsong Chen
• Linear System Theory and Design, 3rd, 1999
• Analogous to an Umbrella

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Functional Description

• Host PC using Simulink and xPC software
• xPC Target Box with Controllers
• Magnetic Suspension System
• Feedback Incorporated 33-210

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Block Diagram
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Previous Lab Work

• Using Classical Controller
• Will It Reject Disturbances?

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Previous Lab Work

• Results of Classical Controller With Disturbance
• Rejected Step Disturbance

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Preliminary Lab Work
Disturbance Laplace Equation
kCos(aT)
kSin(aT)
k-unit step
k-unit ramp

• Laplace Transfer Functions Found
• Later Converted To Discrete Using Zero-order Hold

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Internal Model Principle

• Uses a Model to Cancel Unstable Poles of
  Reference and Disturbance Inputs to Provide
  Asymptotic Tracking and Disturbance Rejection
• Model Is Least Common Multiple of Unstable or
  Zero Continuous Denominator Poles
• Ramp Disturbance Input 0,0
• Step Reference Input 0
• Model P 0,0

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Design Approach

• Disturbance Removed Model Inserted
• Must Stabilize Plant at all Times
• Disturbance Should Never Affect Plant Output
• 3 Known, 2 Unknown
• A(z)D(z)P(z) B(z)N(z)

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Diophantine Equation

• A(z)D(z)P(z) B(z)N(z) F(z)
• Want
• Choose Poles to Form F Polynomial
• Discrete, Close to 1
• Need
• Order of Controller
• Order of D(z)P(z) 1 Order of Controller
• Order of F
• 2(Order of D(z)P(z))-1 Order of F

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Keep In Mind - Account for Model when
Implementing Controller

• Controller Order Assumes Denominator without
  Model
• Adding Model Increases Order Beyond Designed
  Value
• Ex. If D(z)P(z)4, then Controller3
• But Model2 so Controller Denominator really
  should be 1

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Pole Placement

• Ideal Situation
• Tsettle 60ms
• O.S. 18
• .479
• Wn
• Wn 139.1788
• Wnlt

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Pole Placement Problems and Solution

• Complex Poles Give Oscillations
• Wn lt Polesgt.92
• All Poles Close To 1 Is Too Slow
• Speed Up With Poles Closer To Origin
• Iterative Design Approach Required
• Working Poles For Ramp Rejection At
• .9947, .9716, .9275, .9, .01
• F(z)

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Diophantine Solution

• A(z)D(z)P(z) B(z)N(z) F(z)
• Combine D(z)P(z) to Equal D(z)
• For Each X(z)
• System of Equations To Be Solved Simultaneously
• A0D0B0N0F0AnDnBnNnFn

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Diophantine Solution

• Using Previous Example

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Actual Values For Ramp Controller

• N(z)
• D(z)
• P(z)
• F(z)
• B(z)
• A(z)
• A(z)P(z)

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xPC Simulink Implementation

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Results Stability

• 2V Step Disturbance at 2.00V Set Point
• 5V/s Ramp Disturbance at 2.00V Set Point

2.042
2.042
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Results Tracking

1. 5V/s Ramp Disturbance

6Hz .5V Peak-Peak Sine Wave Input
5.98Hz .7V Peak-Peak Sine Wave Output
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Conclusion

• Problems
• Simulation Does Not Match Plant
• Pole Locations are Hard to Find
• lt300mV Error at Start Up
• Future
• Continue to Implement Sinusoidal and Square Wave
• Fine Tune Ramp With Better Poles

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Questions?
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Magnetic Suspension System

• Control and Disturbance Signal Create Current
• Current Induces Magnetic Field
• Field Suspends Ball
• Sensor Translates Location into Voltage

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xPC Target Box and Host PC

• Using 10 V ADC and DAC
• Host Uploads Controller and Commands
• Process Position Data and Passes Control

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Modeling Hybrid Systems

• Simulink treats any model that has both
  continuous and discrete sample times as a hybrid
  model, presuming that the model has both
  continuous and discrete states. Solving such a
  model entails choosing a step size that satisfies
  both the precision constraint on the continuous
  state integration and the sample time hit
  constraint on the discrete states. Simulink meets
  this requirement by passing the next sample time
  hit, as determined by the discrete solver, as an
  additional constraint on the continuous solver.
  The continuous solver must choose a step size
  that advances the simulation up to but not beyond
  the time of the next sample time hit. The
  continuous solver can take a time step short of
  the next sample time hit to meet its accuracy
  constraint but it cannot take a step beyond the
  next sample time hit even if its accuracy
  constraint allows it to.

http//www.mathworks.com/access/helpdesk/help/tool
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