State Space Control of a Magnetic Suspension System Margaret Glavin Supervisor: Prof. Gerard Hurley.
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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
top related