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I.D.Landau :Digital Control/System Identification1
Robust R-S-T Digital Controland
Open Loop System IdentificationA Brief Review
I.D. LandauLaboratoire d’Automatique de Grenoble(INPG/CNRS), France
([email protected] )ADAPTECH, 4 Rue du Tour de l’Eau, St. Martin d’Hères(38), France
([email protected] )
IEEE Advanced Process Control
Workshop, Vancouver, April 29-May 1, 2002
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I.D.Landau :Digital Control/System Identification2
Applications of R-S-T Controllers
Peugeot (PSA)
Sollac (Florange)Hot Dip Galvanizing
Double Twist Machine(Pourtier)
360° Flexible Arm (LAG)
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Implementation of R-S-T Digital Controllers
PLC Leroy implementsR-S-T digital controllers andData acquisition modules
ALSPA 320ALSPA 320 implements R-S-T digital controllers andData acquisition modules
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I.D.Landau :Digital Control/System Identification4
u yRef.
++
DESIGNMETHOD
MODEL(S)
Performancespecs.
PLANT
IDENTIFICATION
Robustnessspecs.
CONTROLLER
1) Identification of the dynamic model2) Performance and robustness specifications3) Compatible controller design method4) Controller implementation5) Real-time controller validation
(and on site re-tuning)6) Controller maintenance (same as 5)
Controller Design and Validation
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I.D.Landau :Digital Control/System Identification5
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
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I.D.Landau :Digital Control/System Identification6
Robust Digital Control
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Computer(controller)
D/A+
ZOHPLANT A/D
CLOCK
Discretized Plant
r(t) u(t)y(t)
The R-S-T Digital Controller
r(t)
m
m
AB
TS1
ABq d−
R
u(t) y(t)
Controller
PlantModel
+
-
)1()(1 −=− tytyq
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I.D.Landau :Digital Control/System Identification8
r(t)
m
m
AB
TS1
ABq d−
R
u(t) y(t)
Controller
PlantModel
+
-
The R-S-T Digital Controller
Plant Model:)(
)(*)(
)()(1
11
1
11
−
−−−
−
−−− ==
qAqBq
qAqBqqG
dd
A
A
nn qaqaqA −−− +++= ...1)( 1
11 )(*...)( 111
11 −−−−− =++= qBqqbqbqB B
B
nn
R-S-T Controller: )()()1(*)()()( 111 tyqRdtyqTtuqS −−− −++=
Characteristic polynomial (closed loop poles):
)()()()()( 11111 −−−−−− += qRqBqqSqAqP d
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I.D.Landau :Digital Control/System Identification9
rABy mm )/(* =
+
-
R
1 q-d
B
AST
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
';' SHSRHR SR ==Controller : :, SR HH fixed parts
Regulation: R’ and S’ solutions of:
dominantpoles
auxiliarypolesTracking :
FDRd
S PPPRBHqSAH ==+ − ''
)1(/ BPT =
Reference trajectory: y*computer file
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Connections with other Control Strategies
- Digital PID : 11;2 −−=== qHnn SSR
-Tracking and regulation with independent objectives(MRC):
FD PPBP *= (Hyp.: B* has stable damped zeros)
- Minimum variance tracking and regulation (MVC):
CBP *=noise model
(Hyp.: B* has stable damped zeros)
- Internal Model Control (IMC):
FAPP = (Hyp.: A has stable damped poles)
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I.D.Landau :Digital Control/System Identification11
+
-
R
1 q-d
BAS
Tr(t) u(t) y(t)
PlantModel
p(t)
b(t)
+
+
+ +
(disturbance)
(measurement noise)
The Sensitivity Functions
Syp(q-1) = ASAS + q-dBR
= ASP
Output sensitivity function (p -> y)
Sup(q-1) = - ARAS + q-dBR
= - ARP
Input sensitivity function (p -> u)
Syb(q-1) = - q-dBRAS + q-dBR
= - q-dBRP
Noise sensitivity function (b -> y)
Syp - Syb = 1
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I.D.Landau :Digital Control/System Identification12
-1
∆Φ1
∆1
∆Μ
1
crossoverfrequency
Re H
Im H
G
|HOL|=1ω CR
1
ω CR2
ω CR3
Robustness Margins
z = ejω
> 29°∆M 0.5 ∆G 2 ; ∆Φ≥ ≥⇒
∆M 0.5 (-6dB), ∆τ > Ts≥
∆M = 1+HOL(z-1) min = Syp(z-1) max-1 = Syp
-1( ) ∞( )
∆τ = mini ∆Φi
ωCRi
Modulus Margin:
Delay Margin:
Typical values:
The inverse is not necessarily true!
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Im G
Re Gω = π
G (e )-j ω
uncertaintydisk
δ W
ω = 0
Robust Stability
Family of plant models:
),,(' xyWGFG δ∈
G – nominal model; 1)( 1 ≤∞
−zδ
)( 1−zWxy- size of uncertainty
Robust stability condition:a related sensitivity
functiona type of uncertainty
1<∞xyxyWS
⇓1−
< xyxy WSdefines the size of thetolerated uncertainty
defines an upper templatefor the modulus of the
sensitivity function
There also lower templates (because of the relationship between various sensitivity fct.)
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Sypmax= - M∆Syp dB
0.5fs0
delay marginnominal
perform.
( G ’ = G + δWa )
Sup dB
actuator effort
size of the tolerated additive uncertainty Wa
00.5f s
Sup-1
Templates for the Sensitivity Functions
Output SensitivityFunction
Input SensitivityFunction
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Robust Controller Design
Pole placement with sensitivity functions shaping
FD PPP =
RHRR '=
SHSS '=
Nominal performance: SRD HandHofpartandP
Allow to shape the sensitivity functions
-Iterative Choosing and using band stop filters FjSjFiRi PHPH /,/FP
Several approaches to design :
-Convex optimization(see Langer, Landau, Automatica, June99, Optreg (Adaptech) )
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MIRROR
DETECTOR
RIGID FRAMES
LIGHTSOURCE
POT.
ENCODER
SERVO.
LOCALPOSITION
COMPUTERTACH.
ALUMINIUM
360° Flexible Arm
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Frequency characteristics Poles-Zeros
360° Flexible Arm
(Identified Model)
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1 2 3 4 5 6 7 8 9-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
Frequence [Hz]
Mod
ule
[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]
Mod
ule
[dB
]
Sup - Sensibilité perturbation-entrée
A
D
B
C
gabarit
A- without auxiliary polesB- with auxiliary polesC- with stop band filterD- with stop band filter
11 / FS PH22 / FR PH
Shaping the Sensitivity Functions
Output Sensitivity Function - Syp Input Sensitivity Function - Sup
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Open Loop System Identification
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I/0 Data Acquisition under anExperimental Protocol
Model Complexity Estimation(or Selection)
Choice of the Noise Model
Parameter Estimation
Model Validation
YesNo ControlDesign
System Identification Methodology
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I/O Data Acqusition
Signal : a P.R.B.S sequenceMagnitude : few % of the input operating pointClock frequency : Length :Largest pulse :
3,2,1;)/1( == pfpf sclock )( frequencysamplingf s =sss
N fTcellsofnumberNpT /1,;)12( 1 ==−−
sNpT
Length : < allowed duration of the experimentLargest pulse : (rise time)Rt≥
P.R.B.S.
NpTs
tr
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poweramplifier
filter
u(t)
y(t)
inertia
d.c. motor .
tachogenerator
M
TG
An I/O File
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Complexity Estimation from I/O Data
Objective :To get a good estimation of the model complexitydirectly from noisy data
),,( dnn BA
complexitypenalty term
error term(should be unbiased)
S(n,N)
0 n
CJ(n)
J (n )
minimum
nopt
[ ]),ˆ()ˆ(minminˆˆˆ
NnSnJCJnnnopt +==),max( dnnn BA +=
To get a good order estimation, J should tend to the value fornoisy free data when (use of instrumental variables) ∞→N
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u(t) y(t)Plant ADC
Discretized plant
+
-
Parameteradaptationalgorithm
y(t)
ε (t)
Estimated modelparameters
)(ˆ tθ
Adjustablediscrete-time
model
DAC +ZOH
Parameter Estimation
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+
+u(t) y(t)
1A
q-d B
A
e(t)
)()()()()(:1 11 tetuqBqtyqAS d += −−−
(Recursive) Least Squares
+
+u(t) y(t)q-d B
A
w(t)
)()()()()()(:2 111 twqAtuqBqtyqAS d −−−− +=
Ouput Error(O.E.)Instrumental Variable…
+
+u(t) y(t)
1
q-d B
A
e(t)
CA
[ ] )()(/1)()()()(:4 111 teqCtuqBqtyqAS d −−−− +=
Generalized Least Squares
+
+u(t) y(t)A
q-d B
A
e(t)
C
)()()()()()(:3 111 teqCtuqBqtyqAS d −−−− +=
Extended Least SquaresO.E. with Extended Prediction Model(Recursive) Maximum Likelihood
«Plant + Noise » Models
~ 33%
~ 64%
~ 1%
~ 2%
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Parameter Estimation Methods
)()()(*)()(*)1( 11 tdtuqBtyqAty Tψθ=−+−=+ −−
vectortmeasuremenvectorparameter −− ψθ ;
Plant Model
)()(ˆ)1(ˆ ttty T φθ=+°Estimated model
vectornobservatiovectorparameterestimated −− φθ ;ˆ
)1(ˆ)1()()(ˆ)1()1( 00 +−+=Φ−+=+ tytytttyt TθεPrediction error
Parameter adaptation algorithm
2)(0;1)(0
)()()()()()1(
)1()()1()(ˆ)1(ˆ
21
21
11
0
<≤≤<ΦΦ+=+
+Φ++=+−−
tt
ttttFttF
tttFttT
λλλλ
εθθ
[ ])()( tft φ=Φ
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Parameter Estimation Methods
I- Based on the asymptotic whitening of the prediction error(Recursive Least Squares, Extended Least Squares, RecursiveMax. Likelihood, O.E. with Extended Prediciton Model )
II- Based on the asymptotic decorrelation between the predictionerror and the observation vector(Output Error, Instrumental Variable)
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1;17.2)( ≥≤ iN
iRN
normalizedcrosscorelation
number of data
1N2.17
N
97%
136.0)(256 ≤→= iRNN
15.0)( ≤iRNpratical value :
+
-
u
y
y
εPlant
Model
q-1
q-1
ARMAX (ARARX) predictor
WhitenessTest
+
-
u
y
y
ε
Plant
Model
q-1
Output Error Predictor
DecorrelationTest
Validation of Identified Models
Statistical Validation
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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
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« 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, identificationet mise en œuvre, Hermes, Paris (June)