nonlinear fuzzy pid control phase plane analysis standard surfaces performance
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Nonlinear Fuzzy PID Control
•Phase plane analysis•Standard surfaces•Performance
Phase Plane
x
x
O
2221212
2121111
xaxax
xaxax
Trajectory
Axx
ectorvelocity v
ectorposition v
x
x
212111
222121
1
2
xaxa
xaxa
x
x
Slope
Ax0mEquilibriu
Equilibrium Points
x1
x2
Stable node
x1
Time [s] x1
x2
Unstable node
x1
Time [s]
x1
x2
Stable focus
x1
Time [s] x1
x2
Unstable focus
x1
Time [s]
x1
x2
Center point
x1
Time [s] x1
x2
Saddle point
x1
Time [s]
000Re
0Re0Re
21
RC
CC
RR
iii
iiii
ii
Closed Loop (1/s2)
BuAxx
eG
eG
c
c
BAxx
u
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
e
ce
Example: 1/s2
pc
pc
ppp
p
KB
KA
RefKxKxRefKx
xx
eK
0,
0
10
112
21
BAxBuAxx
pKA
point saddle0
pointcentre0
p
p
K
K
Example: Stopping a Car
-20 0 20
-20
-10
0
10
20
Position [m]
Spe
ed [
m/s
]
0 2 4-25
-20
-15
-10
-5
0
Pos
ition
[m
]Time [s]
-20 0 20
-20
-10
0
10
20
Position [m]
Spe
ed [
m/s
]
0 2 4-25
-20
-15
-10
-5
0
Pos
ition
[m
]
Time [s]
Open loop
Closed loop
Phase Plane
0 5 100
0.5
1
1.5C
ontr
olle
d ou
tput
y
0 5 10-2
0
2
4
6
Con
trol
sig
nal U
Seconds
-100 0 100-100
-50
0
50
100
CE
E
-1000
100
-1000
100-200
0
200
ECE
u
Rule Base With 4 Rules
1. If error is Neg and change in error is Neg then control is NB3. If error is Neg and change in error is Pos then control is Zero7. If error is Pos and change in error is Neg then control is Zero9. If error is Pos and change in error is Pos then control is PB
-100 0 100-100
-50
0
50
100
CE
E
1
3
7
9
Surfaces: Linear and Saturation
-1000
100
-1000
100-200
0
200
ECE
u
-100 -50 0 50 1000
0.5
1
input family
mem
bers
hip
-1000
100
-1000
100-200
0
200
ECE
u
-100 -50 0 50 1000
0.5
1
input familym
embe
rshi
p
Linear
Saturation
Surfaces: Deadzone and Quantizer
Deadzone
Quantizer
-1000
100
-1000
100-200
0
200
ECE
u
-100 -50 0 50 1000
0.5
1
input family
mem
bers
hip
-1000
100
-1000
100-200
0
200
ECE
u
-100 -50 0 50 1000
0.5
1
input familym
embe
rshi
p
Example: FPD Control of 1/s2
-1 0 1-1
-0.5
0
0.5
1
e
ce
0 5 10 15-2
-1
0
1
2
Time [s]
x1
-1 0 1-1
-0.5
0
0.5
1
ece
0 5 10 15-2
-1
0
1
2
Time [s]
x1-100
0100
-1000
100-200
0
200
ECE
u-100 0 1000
0.5
1
input familym
embe
rshi
p
Example: FPD+I Control of 1/s2
-1 0 1-1
-0.5
0
0.5
1
e
ce
0 20 40-2
-1
0
1
2
Time [s]
x1
-1 0 1-1
-0.5
0
0.5
1
ece
0 20 40-2
-1
0
1
2
Time [s]
x1-100
0100
-1000
100-200
0
200
ECE
u-100 0 1000
0.5
1
Input family
Mem
bers
hip
Hand-Tuning
1. Adjust GE (or GCE) to exploit universe
2. Set GIE = GCE = 0; tune GU
3. Increase GU, then increase GCE
4. Increase GIE to remove final offset
5. Repeat from 3) until GU is large as possible
Limit Cycle
-1 0 1-1
0
1
e
ce
0 20 40-2
0
2
Time [s]
x1
-1 0 1-1
0
1
e
ce
0 20 40-2
0
2
Time [s]
x1
-1000
100
-1000
100-200
0
200
ECE
u-100 0 1000
0.5
1
input familym
embe
rshi
p
Input Universe Saturation
-1 0 1-1
0
1
e
ce
0 20 40-2
0
2
Time [s]
x1
-1 0 1-1
0
1
e
ce
0 20 40-2
0
2
Time [s]
x1
-1000
100
-1000
100-200
0
200
ECE
u-100 0 1000
0.5
1
Input familyM
embe
rshi
p
Design Procedure*
• Build and tune a conventional PID controller first.• Replace it with an equivalent linear fuzzy controller.• Make the fuzzy controller nonlinear.• Fine-tune the fuzzy controller.
*) Relevant whenever PID control is possible, or already implemented
Bode Plot: Linear FPD
0
20
40
Ma
gn
itud
e [
dB
]
10-1
100
101
0
50
100
Ph
ase
[d
eg
]
Frequency [rad]
Bode Plot: Nonlinear FPD
0.5
1
1.5
2
Ma
gn
itud
e
saturation
quantizerlinear
deadzone
10-1
100
101
0
20
40
60
Ph
ase
[d
eg
]
Frequency [rad]
saturation
quantizerlinear
deadzone
Nyquist: Nonlinear FPD+I of 1/(s+1)3
-2 0 2-2
-1
0
1
2Kp = 4.8, Ti = 15/8, Td = 15/32
quantizer
saturationdeadzone
linear
Nyquist: Nonlinear FPD+I of 1/s2
-2 0 2-2
-1
0
1
2Kp = 0.5, Ki = 0, Td = 1
quantizer
saturationdeadzone
linear
Nyquist: Nonlinear FPD+I ofe-2s/(s+1)
-2 0 2-2
-1
0
1
2Kp = 4.8, 1/Ti = 1, Td = 0.46875
quantizer
saturationdeadzone
linear
Nyquist: Nonlinear FPD+I of 25/(s+1)(s2+25)
-2 0 2-2
-1
0
1
2Kp = -0.25, 1/Ti = -1, Td = 0
quantizer
saturationdeadzone
linear
Fuzzy + PID Configurations
ProcessPID
Fuzzy
ProcessPID
Fuzzy
ProcessPID
Fuzzy
ProcessPID
(a) (b)
(c) (d)
Summary
• Phase plane analysis
• Standard surfaces
• Performance