State Space Control of a Magnetic Suspension System

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State Space Control of a Magnetic Suspension
System

• Margaret Glavin
• Supervisor Prof. Gerard Hurley

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Introduction

• Proportional and Derivative Control
• PWM Control
• State Space Control
• Applications of the Suspension System

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State Space Background

• Developed since 1960s
• Modern control theory
• State variable method of describing differential
  equations
• Not one unique set of state variables to describe
  the state space of the system

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State Equations

• dx/dt Ax Bu
• y Cx Du
• A State Matrix
• B Input Matrix
• C Output Matrix
• D Direct transmission Matrix

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Block Diagram
y

D

u

B
1/s
C
x
x

A
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Steps for State Space Design

• State Matrices
• Controllability and feedback gain
• Observability and observer gain
• Combine both
• Introducing reference input

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Equations

• Differential equation for system
• Transfer function

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Methods to Calculate Space State Matrices from
Transfer Functions

• Canonical forms
• Controllable canonical form
• Observable canonical form
• Jordan canonical form
• Modal canonical form
• Diagonal canonical form
• MatLab

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State Space Matrices
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Controllability

• Controllability matrix
• Matrix rank is n or n linearly independent column
  vectors
• If determinant is non zero system is controllable

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Feedback Gain Matrix

• Used to place the poles
• If controllable poles placed at any location
• Methods to calculate matrix
• Direct substitution method
• Transformation matrix
• Ackermanns formula

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Reference Input

• K matrix calculated with input set to zero
• Kc input gain
• Overcomes steady state error
• Kc(1/(C(-1/(A-Bk))B)

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Observer

• State variables not always available
• Observer designed to estimate the state variables
• Full state observer
• Reduced state observer

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Observability

• Observability matrix
• Matrix rank is n or has
• n linearly independent
• column vectors
• Determinant is a non
• zero value

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Observer Gain

• Used to place the observer poles
• Poles two to five times faster than controller
  poles
• Same methods of calculation used as for feedback
  gain matrix

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Simulink

• Part of the Matlab Program
• Used to draw and simulate block diagrams
• Graphs at different points in the system can be
  plotted

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PSpice

• MicroSim Corporation
• Designing and simulating circuits
• Schematic capture or netlist
• Libraries
• Modelling transfer function
• Saves time and money

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Hardware

• Building circuit
• Testing circuit
• Fault finding
• Part of circuit already built

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Applications

• MagLev train
• Floats above guide way
• Two types
• Reach speeds of 310 mph (500 kph)
• Frictionless bearings

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Questions