A Brief Review

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Nov 15, 2013 (3 years and 9 months ago)

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I.D.Landau :Digital Control/System Identification
1
Robust R-S-T Digital Control
and
Open Loop System Identification
A Brief Review
I.D. Landau
Laboratoire d’Automatique de Grenoble(INPG/CNRS), France
(landau@lag.ensieg.inpg.fr)
ADAPTECH, 4 Rue du Tour de l’Eau, St. Martin d’Hères(38), France
(info@adaptech.com)
IEEE
Advanced Process Control
Workshop, Vancouver, April 29-May 1, 2002
I.D.Landau :Digital Control/System Identification
2
Applications of R-S-T Controllers
Peugeot (PSA)
Sollac (Florange)
Hot Dip Galvanizing
Double Twist Machine
(Pourtier)
360° Flexible Arm (LAG)
I.D.Landau :Digital Control/System Identification
3
Implementation of R-S-T Digital Controllers
PLC Leroy implements
R-S-T digital controllers and
Data acquisition modules
ALSPA 320
ALSPA 320 implements
R-S-T digital controllers and
Data acquisition modules
I.D.Landau :Digital Control/System Identification
4
u
yRef.
+
+
DESIGN
METHOD
MODEL(S)
Performance
specs.
PLANT
IDENTIFICATION
Robustness
specs.
CONTROLLER
1) Identification of the dynamic model
2) Performance and robustness specifications
3) Compatible controller design method
4) Controller implementation
5) Real-time controller validation
(and on site re-tuning)
6) Controller maintenance (same as 5)
Controller Design and Validation
I.D.Landau :Digital Control/System Identification
5
Outline
Robust digital control
-The R-S-T digital controller
-Basic design
-Robustness issues
-An example
Open loop system identification
-Data acquisition
-Model complexity
-Parameter estimation
-Validation
I.D.Landau :Digital Control/System Identification
6
Robust Digital Control
I.D.Landau :Digital Control/System Identification
7
Computer
(controller)
D/A
+
ZOH
PLANT
A/D
CLOCK
Discretized Plant
r(t)
u(t)
y(t)
The R-S-T Digital Controller
r(t)
m
m
A
B
T
S
1
A
Bq
d
R
u(t)
y(t)
Controller
Plant
Model
+
-
)1()(
1



t
y
t
y
q
I.D.Landau :Digital Control/System Identification
8
r(t)
m
m
A
B
T
S
1
A
Bq
d

R
u(t)
y(t)
Controller
Plant
Model
+
-
The R-S-T Digital Controller
Plant Model:
)(
)(*
)(
)(
)(
1
11
1
1
1









qA
qBq
qA
qBq
qG
dd
A
A
n
n
qaqaqA





...1)(
1
1
1
)(*...)(
111
1
1 




qBqqbqbqB
B
B
n
n
R-S-T Controller:
)()()1(*)()()(
111
tyqRdtyqTtuqS








)()()()()(
11111








qRqBqqSqAqP
d
I.D.Landau :Digital Control/System Identification
9
r
A
B
y
mm
)/(*

+
-
R
1
q
-d
B
A
S
T
A
B
m
m
r(t)
y (t+d+1)
*
u(t)
y(t)
q
-(d+1)
P(q
-1
)
q
-(d+1)
B*(q )
-1
B*(q )
-1
B(1)
q
-(d+1)
B
m
(q )
B*(q )
-1
-1
A
m
(q ) B(1)
-1
Pole Placement with R-S-T Controller
';'
S
H
S
R
H
R
SR


Controller :
:,
SR
H
H
fixed parts
Regulation: R’ and S’ solutions of:
dominant
poles
auxiliary
poles
Tracking :
FDR
d
S
PPPRBHqSAH




''
)
1
(
/
B
P
T

Reference trajectory: y*
computer file
I.D.Landau :Digital Control/System Identification
10
Connections with other Control Strategies
- Digital PID :
1
1;2





qHnn
SSR
-Tracking and regulation with independent objectives(MRC):
F
D
P
P
B
P
*

(Hyp.: B* has stable damped zeros)
- Minimum variance tracking and regulation (MVC):
C
B
P
*

noise model
(Hyp.: B* has stable damped zeros)
- Internal Model Control (IMC):
F
AP
P

A has stable damped poles)
I.D.Landau :Digital Control/System Identification
11
+
-
R
1
q
-d
B
A
S
T
r(t) u(t)
y(t)
Plant
Model
p(t)
b(t)
+
+
+ +
(disturbance)
(measurement noise)
The Sensitivity Functions
S
yp
(q
-1
) =
AS
AS + q
-d
BR
=
AS
P
Output sensitivity function (p -> y)
S
up
(q
-1
) = -
AR
AS + q
-d
BR
= -
AR
P
Input sensitivity function (p -> u)
S
yb
(q
-1
) = -
q
-d
BR
AS + q
-d
BR
= -
q
-d
BR
P
Noise sensitivity function (b -> y)
S
yp
- S
yb
= 1
I.D.Landau :Digital Control/System Identification
12
-1





1
crossover
frequency
Re H
Im H
G
|H
OL
|=1

CR
1

CR
2

CR
3
Robustness Margins
z =
e
j
> 29°M 0.5 G 2 ; 



M
0.5 (-6dB),
 > T
s

M =
1+H
OL
(z
-1
)
min
=
S
yp
(z
-1
)
max
-1
=
S
yp
-1
(
)




= min
i


i

CR
i
Modulus Margin:
Delay Margin:
Typical values:
The inverse is not necessarily true!
I.D.Landau :Digital Control/System Identification
13
Im G
Re G

G (e )
-j

uncertainty
disk

W

Robust Stability
Family of plant models:
),,('
xy
W
G
F
G


G – nominal model;
1
)(
1



z

)(
1

z
W
xy
- size of uncertainty
Robust stability condition:
a related sensitivity
function
a type of uncertainty
1


xyxy
WS

1

xyxy
WS
defines the size of the
tolerated uncertainty
defines an upper template
for the modulus of the
sensitivity function
There also lower templates (because of the relationship between various sensitivity fct.)
I.D.Landau :Digital Control/System Identification
14
S
yp
max
= - M
S
yp
dB
0.5f
s
0
delay
margin
nominal
perform.
( G’ = G + W
a
)
S
up
dB
actuator effort
size of the tolerated additive uncertainty W
a
0
0.5f
s
S
up
-1
Templates for the Sensitivity Functions
Output Sensitivity
Function
Input Sensitivity
Function
I.D.Landau :Digital Control/System Identification
15
Robust Controller Design
Pole placement with sensitivity functions shaping
F
D
P
P
P

R
H
R
R
'

S
H
S
S
'


SRD
H
and
H
of
part
and
P
Allow to shape the sensitivity functions
-Iterative
Choosing and using band stop filters
FjSjFiRi
P
H
P
H
/,/
F
P
Several approaches to design :
-Convex optimization
(see Langer, Landau,Automatica, June99,Optreg (Adaptech) )
I.D.Landau :Digital Control/System Identification
16
MIRROR
DETECTOR
RIGID FRAMES
LIGHT
SOURCE
POT.
ENCODER
SERVO.
LOCAL
POSITION
COMPUTER
TACH.
ALUMINIUM
360° Flexible Arm
I.D.Landau :Digital Control/System Identification
17
Frequency characteristics
Poles-Zeros
360° Flexible Arm
(Identified Model)
I.D.Landau :Digital Control/System Identification
18
1
2
3
4
5
6
7
8
9
-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
Frequence [Hz]
Module [dB]
Syp - Sensibilité perturbation-sortie
A
B
D
C
gabarit
1
2
3
4
5
6
7
8
9
-30
-20
-10
0
10
20
30
40
50
60
Frequence [Hz]
Module [dB]
Sup - Sensibilité perturbation-entrée
A
D
B
C
gabarit
A- without auxiliary poles
B- with auxiliary poles
C- with stop band filter
D- with stop band filter
11
/
FS
PH
2
2
/
F
R
PH
Shaping the Sensitivity Functions
Output Sensitivity Function - S
yp
Input Sensitivity Function - S
up
I.D.Landau :Digital Control/System Identification
19
Open Loop System Identification
I.D.Landau :Digital Control/System Identification
20
I/0 Data Acquisition
under anExperimental Protocol
Model Complexity Estimation
(
or Selection
)
Choice of the Noise Model
Parameter Estimation
Model Validation
Yes
No
Control
Design
System Identification Methodology
I.D.Landau :Digital Control/System Identification
21
I/O Data Acqusition
Signal : a P.R.B.S sequence
Magnitude : few % of the input operating point
Clock frequency :
Length :
Largest pulse :
3,2,1;)/1(


p
f
p
f
sclock
)(
frequency
sampling
f
s

sss
N
fTcellsofnumberNpT/1,;)12(
1




s
NpT
Length : < allowed duration of the experiment
Largest pulse : (rise time)
R
t

P.R.B.S
.
NpT
s
t
r
I.D.Landau :Digital Control/System Identification
22
power
amplifier
filter
u(t)
y(t)
inertia
d.c.motor.
tacho
generator
M
TG
An I/O File
I.D.Landau :Digital Control/System Identification
23
Complexity Estimation from I/O Data
Objective :
To get a good estimation of the model complexity
directly from noisy data
),,(
d
n
n
B
A
complexity
penalty term
error term
(should be unbiased)
S(n,N)
0
n
CJ
(n)
J
(n
)
minimum
n
opt


),
ˆ
()
ˆ
(minmin
ˆ
ˆˆ
N
n
S
n
J
CJ
n
nn
opt



),max(
d
n
n
n
B
A


To get a good order estimation, J should tend to the value for
noisy free data when (use of instrumental variables)


N
I.D.Landau :Digital Control/System Identification
24
u(t)
y(t)
Plant
ADC
Discretized plant
+
-
Parameter
adaptation
algorithm
y(t)

(t)
Estimated
model
parameters
)(
ˆ
t

Adjustable
discrete-time
model
DAC +
ZOH
Parameter Estimation
I.D.Landau :Digital Control/System Identification
25
+
+u(t) y(t)
1
A
q
-d
B
A
e(t)
)()()()()(:1
1
1
tetuqBqtyqAS
d





(Recursive) Least Squares
+
+
u(t) y(t)
q
-d
B
A
w(t)
)()()()()()(:2
1
1
1
twqAtuqBqtyqAS
d






Ouput Error(O.E.)
Instrumental Variable…
+
+
u(t) y(t)
1
q
-d
B
A
e(t)
CA


)()(/1)()()()(:4
1
1
1
teqCtuqBqtyqAS
d






Generalized Least Squares
+
+
u(t) y(t)
A
q
-d
B
A
e(t)
C
)()()()()()(:3
111
teqCtuqBqtyqAS
d 


Extended Least Squares
O.E. with Extended Prediction Model
(Recursive) Maximum Likelihood
«Plant + Noise » Models
~ 33%
~ 64%
~ 1%
~ 2%
I.D.Landau :Digital Control/System Identification
26
Parameter Estimation Methods
)()()(*)()(*)1(
11
tdtuqBtyqAty
T










vector
t
measuremen
vector
parameter




;
Plant Model
)()(
ˆ
)1(
ˆ
ttty
T

Estimated model
vectornobservatiovectorparameterestimated  

;
ˆ
)1(
ˆ
)1()()(
ˆ
)1()1(
00
 tytytttyt
T

Prediction error
Parameter adaptation algorithm
2)(0;1)(0
)()()()()()1(
)1()()1()(
ˆ
)1(
ˆ
2
1
2
1
1
1
0




tt
ttttFttF
tttFtt
T






)()(
t
f
t



I.D.Landau :Digital Control/System Identification
27
Parameter Estimation Methods
I- Based on the asymptotic whitening of the prediction error
(Recursive Least Squares, Extended Least Squares, Recursive
Max. Likelihood, O.E. with Extended Prediciton Model )
II- Based on the asymptotic decorrelation between the prediction
error and the observation vector
(Output Error, Instrumental Variable)
I.D.Landau :Digital Control/System Identification
28
1;
17.2
)(  i
N
iRN
normalized
crosscorelation
number
of data
1
N
2.17
N
97%
136.0)(256



iRNN
15.0)(

iRN
pratical value :
+
-
u
y
y

Plant
Model
q
-1
q
-1
ARMAX (ARARX) predictor
Whiteness
Test
+
-
u
y
y

Plant
Model
q
-1
Output Error Predictor
Decorrelation
Test
Validation of Identified Models
Statistical Validation
I.D.Landau :Digital Control/System Identification
29
Software Tools for Implementing the Methodology
System Identification
-Winpim (Adaptech)
identification in open loop and closed loop operation
-CLID (Adaptech)
identification in closed loop (Matlab Toolbox)
Controller Design
-Winreg (Adaptech)
design and optimisation of R-S-T digital controllers
-Optreg (Adaptech)
automated design of robust digital controllers (under Matlab)
Real-time implementation
-Wintrac (Adaptech): cascade digital control
I.D.Landau :Digital Control/System Identification
30
« Personal » References
Landau.I.D.,(1990) System Identification and Control Design, Prentice Hall, N.J.,USA
Landau I.D., (1995): « Robust digital control of systems with time delay
(the Smith predictor revisited) », Int. J. of Control, vol. 62, pp.325-347.
Landau I.D.,Lozano R.,M’Saad M., (1997):Adaptive Control,Springer, London,U.K.
Landau I.D., (1993) Identification et Commande des Systèmes, 2nd edition,
Hermes, Paris (June)
Landau I.D., (2002) Commande des Systèmes – Conception, identification
et mise en œuvre,Hermes, Paris (June)