State Space Control of a Magnetic Suspension

System

Margaret GlavinSupervisor: Prof. Gerard

Hurley

Introduction

Proportional and Derivative Control

PWM Control

State Space Control

Applications of the Suspension

System

State Space BackgroundDeveloped since 1960’s

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

State Equationsdx/dt = Ax + Buy = Cx + Du

A – State Matrix B – Input Matrix C – Output Matrix D – Direct transmission Matrix

Block Diagram

B 1/s C

D

A

u

+

+

+

+x’ x

y

Steps for State Space Design

State Matrices

Controllability and feedback gain

Observability and observer gain

Combine both

Introducing reference input

EquationsDifferential equation for system

Transfer function

0''2

' 2

2

22

2

2

ia

ILNx

a

ILN

dt

xdM dd

22

2

)(

)(

nwsIg

sY

sX

Methods to Calculate Space State Matrices from Transfer

FunctionsCanonical forms

Controllable canonical form

Observable canonical form

Jordan canonical form

Modal canonical form

Diagonal canonical form

MatLab

State Space Matrices

01

0 2nwA

0

1B

I

gC

20 0D

ControllabilityControllability matrix

Matrix rank is n or n linearly independent column vectors

If determinant is non zero system is controllable

BAABAM nc

1

Feedback Gain MatrixUsed to place the polesIf controllable poles placed at any locationMethods to calculate matrix

Direct substitution methodTransformation matrixAckermann’s formula

Reference Input

K matrix calculated with input set

to zero

Kc input gain

Overcomes steady state error

Kc=(1/(C*(-1/(A-Bk))*B)

Observer

State variables not always

available

Observer designed to estimate the

state variables

Full state observer

Reduced state observer

ObservabilityObservability matrixMatrix rank is n or has

n linearly independent column vectors

Determinant is a non zero value

1

2

n

o

CA

CA

CA

C

M

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

Simulink

Part of the Matlab Program

Used to draw and simulate block

diagrams

Graphs at different points in the

system can be plotted

Vsum3

vsum3

t

time

refInput

ref i/p

-K-

l2 (-32.3)

-K-

l1 (-2581)

int2

int2

int1

int11

b1 (1)

1

a21 (1)

K*u

a12 (1473)

Vsum5

Vsum5

Vsum4

Vsum4

Vsum2

Vsum2

Vsum1

Vsum1

-9.288

s +-14732

Transfer FcnStep (1V)

Output

Plant output

-K-

Kc (-2.153)

-K-

K2 (3973)

-K-

K1 (100)

1s

Integrator1

1s

Integrator

69

Gs (69)

Clock

-K-

C2 (-640)

PSpiceMicroSim Corporation

Designing and simulating circuits

Schematic capture or netlist

Libraries

Modelling transfer function

Saves time and money

Title

Size Document Number Rev

Date: Sheet of

<Doc> <RevCode>

<Title>

A

1 1Wednesday, March 23, 2005

+

-

OUT

U1

OPAMPR1

1k

Vsum3

int2

int1

C2

R3

2.153k

Vsum5

Vsum4

0

Position sensor

l1

b1

a12

Plant o/p

Vsum2

ref i/p

Vsum1

kc

l2

a21

V1

TD = 0

TF = 0PW = 0.5sPER = 1s

V1 = 0v

TR = 0

V2 = 1V

0 00

+

-

OUT

U4

OPAMP

+

-

OUT

U5

OPAMP

R7

1.7523k

R8

2581.2kR9

1k

R10

2581.2k

0

R11

10k

R12

1000k

C1

100u

0

0

0

+

-

OUT

U6

OPAMP

+

-

OUT

U7

OPAMPR13

10k

R14

1000k

R15

32.3k

R16

1k

C2

100u

R17

32.3k

0

0

+

-

OUT

U8

OPAMP

+

-

OUT

U9

OPAMP

0

R18

1k

R19

39.73k

R20

3973k

+

-

OUT

U20

OPAMPR44

10k

R45

10k

+

-

OUT

U10

OPAMP

R46

10k

R21

1k

R22

1k

R47 10k

0

R23

2.153k

+

-

OUT

U19

OPAMP

0

R34

9.288

C3

2.8m

+

-

OUT

U15

OPAMP

R35

-38.38

R36

1 C4

-26m

0

+

-

OUT

U16 OPAMP

R37

1k

R38

1k

+

-

OUT

U17OPAMP

R39

38.38

+

-

OUT

U18

OPAMPR40

1k

R41

69k

Feedback gain

Plant

R24

1k

R25

640k

0

R42

1kR43

1k

Hardware

Building circuit

Testing circuit

Fault finding

Part of circuit already built

Applications

MagLev train

Floats above guide way

Two types

Reach speeds of 310 mph (500 kph)

Frictionless bearings

Questions