I.D.Landau : From Robust Control to Adaptive Control 1 I.D. Landau Laboratoire d’Automatique de Grenoble, (INPG/CNRS), France Mayo 2003 Del Control Robusto al Control Adaptable
I.D.Landau : From Robust Control to Adaptive Control1
I.D. LandauLaboratoire dAutomatique de Grenoble, (INPG/CNRS), France
Mayo 2003
Del Control Robusto al Control Adaptable
I.D.Landau : From Robust Control to Adaptive Control2
I.D. LandauLaboratoire dAutomatique de Grenoble, (INPG/CNRS), France
May 2003
From Robust Control to Adaptive Control
I.D.Landau : From Robust Control to Adaptive Control3
Robust Control
uncertaintiesstructured (parameter variations)
unstructured (often in high frequencies)
Performance may be limited (for large plant uncertainties)
Adaptive Control
-Well suited for handling parameter variations- Should work correctly in the presence of unstructureduncertainties (parasitics)
- Problems for large and abrupt changes in plant parameters
I.D.Landau : From Robust Control to Adaptive Control4
Robust Control plays an important role in Adaptive Control(directly or indirectly)
Adaptive Control can improve the performances of aRobust Controller
Identification in Closed Loop allows to establish linksbetween Robust Control and Adaptive Control
I.D.Landau : From Robust Control to Adaptive Control5
Outline
- Introduction
- Identification in closed loop
- Experimental results (flexible transmission)
- Adaptive control strategies
- Robust control design for adaptive control
- Parameter estimators
- Adaptive control with multiple models
- Experimental results (flexible transmission)
- Adaptive rejection of unknown disturbances
- Experimental results (active suspension)
- Concluding remarks
I.D.Landau : From Robust Control to Adaptive Control6
Plant Identification in Closed Loop
There are systems where open loop operationis not suitable ( instability, drift, .. )
A controller may already exist ( ex . : PID )
Iterative identification and controller redesign
Re-tuning of the controllera) to improve achieved performancesb) controller maintenance
Why ?
Cannot be dissociated from thecontroller and robustness issues
May provide better design models ! !
I.D.Landau : From Robust Control to Adaptive Control7
m
axismotor
d.c.motor
Positiontransducer
axisposition
ref
load
Controlleru(t)
y(t)
ADC
R-S-Tcontroller
DAC+
+
Identification in Closed Loop
The flexible transmission
I.D.Landau : From Robust Control to Adaptive Control8
What is the good model ?
closed loop identified model open loop identified model
fs = 20Hz 0% load
I.D.Landau : From Robust Control to Adaptive Control9
output
control
real system
controller design using theopen loop identified model
output
control
real system
computed polecl. loop syst. pole
Benefits of identification in closed loop (1)
The pattern of identified closed loop poles is different fromthe pattern of computed closed loop poles
I.D.Landau : From Robust Control to Adaptive Control10
real system
output
control
controller computed using theclosed loop identified model
output
control
real system
computed polecl. loop syst. pole
Benefits of identification in closed loop (2)
The computed and the identified closed loop poles are very close
I.D.Landau : From Robust Control to Adaptive Control11
Notations
K G
v(t) p(t)y
v(t)r u
-
)q(A)q(Bq)q(G 1
1d1
=
)q(S)q(R)q(K 1
11
=
Sensitivity functions :
KG11)z(S 1yp +
=
KG1K)z(S 1up +
=
KG1KG)z(S 1yr +
=
KG1G)z(S 1yv +
=
)z(R)z(Bz)z(S)z(A)z(P 11d111 +=
; ; ;
Closed loop poles :
True closed loop system :(K,G), P, SxyNominal simulated(estimated) closed loop : xyS,P),G,K(
I.D.Landau : From Robust Control to Adaptive Control12
-1
1
1
1
crossover
frequency
Re H
Im H
G
|HOL|=1 CR
1
CR2
CR3
Robustness Margins
z = ej
> 29M 0.5 G 2 ;
M 0.5 (-6dB), > Ts
M = 1+HOL(z-1) min = Syp(z-1) max-1 = Syp -1( )
( )
= mini iCR
i
Modulus Margin:
Delay Margin:
Typical values:
The inverse is not necessarily true!
Critical frequency region for control
I.D.Landau : From Robust Control to Adaptive Control13
Syp max= - MSyp dB
0.5fs0
delaymarginnominalperform.
( G = G + Wa )
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
Critical frequency region for control
Critical frequency region for control
I.D.Landau : From Robust Control to Adaptive Control14
noise
q-dBA
w
Plant++
T 1/S
R
+
-r
u y
ru
+
+
q-dBA
w
ASPru
yu +
+
ARP
+
+Syp
-Sup
Identification in Closed Loop
Open loopinterpetation
- take advantage of the improved input spectrum- are insensitive to noise in closed loop operation
Objective : development of algorithms which:
I.D.Landau : From Robust Control to Adaptive Control15
CLOE Algorithms
Objective of the Identification in Closed Loop
(identification for control)
Find the plant model which minimizes the discrepancybetween the real closed loop system and the simulated closed loop system.
noise
Controller Plant
Controller Model
r u y
y
+-
++
ParametricAdaptationAlgorithm
Simulated System
Closed Loop Output Error
I.D.Landau : From Robust Control to Adaptive Control16
Closed Loop Output Error Identification Algorithms(CLOE)
K
P.A.A.
G
r
u
u y
y
CL
+
+
+
K
G
Excitation added to controller output
K G
G
ru
uy
y+
+
+
P.A.A.
CL
K
+
+p
Excitation added to reference signal
Same algorithm but different properties of the estimated model
I.D.Landau : From Robust Control to Adaptive Control17
Iterative Identification in Closed Loop and Controller Re-Design
Repeat 1, 2, 1, 2, 1, 2,CL
CL
Step 1 : Identification in Closed Loop-Keep controller constant-Identify a new model such that
Step 2 : Controller Re Design- Compute a new controller such that
w1/S
RPlant
R
1/S
Model
++
+
+
+
-
-
-CL
r u y
u y
T q-d B/A
q-d B/A
I.D.Landau : From Robust Control to Adaptive Control18
Adaptive Control Basic Schemes
- Indirect adaptive control- Direct adaptive control (the controller is directly estimated)
Performancespecifications
AdjustableController Plant+
-
ControllerDesign
Plant Model
Estimation
Adaptation loop
AdjustableController Plant+
-
Performancespecifications
ControllerEstimation
Adaptation loop
Indirect Adaptive Control Direct Adaptive Control
I.D.Landau : From Robust Control to Adaptive Control19
time
Parameter Estimation+
Controller Computationt t+1
time
Fixed (or time varying)Controller computed at( t)
+Parameter Estimation
t t+N
Controllercomputedat (t +N)
Iterative Identification and Controller Redesign versus (Indirect) Adaptive Control
N = 1 : Adaptive Control
The iterative procedure introduces a time scale separation between identification / control design
N = SmallAdaptive ControlN = LargeIterative Identification in C.L.And Controller Re-design
Plant Identification in C.L. +Controller Re-design
N
I.D.Landau : From Robust Control to Adaptive Control20
The flexible transmission
m
axismotor
d.c.motor
Positiontransducer
axisposition
ref
load
Controlleru(t)
y(t)
ADC
R-S-Tcontroller
DAC
Adaptive Control of a Flexible Transmission
I.D.Landau : From Robust Control to Adaptive Control21
Adaptive Control of a Flexible Transmission
Frequency characteristics for various load
Rem.: the main vibration mode varies by 100%
I.D.Landau : From Robust Control to Adaptive Control22
Robust Control Design for Adaptive Control
parameter variations(low frequency) Adaptation Robust Design
unstructureduncertainities(high frequency)
Basic rule : The input sensitivity function (Sup) should be small inmedium and high frequencies
Pole Placement :- Opening the loop in high frequncies (at 0.5fs)- Placing auxiliary closed loop poles near the high frequency polesof the plant model
Generalized Predictive Control :- Appropriate weighting filter on the control term in the criterion
How to achieve this ?
I.D.Landau : From Robust Control to Adaptive Control23
Robust Control Design for Adaptive Control(Flexible Transmission)
a) Standard pole placement (1 pair dominant poles + h.f. aperiodic poles)b) Opening the loop at 0.5fs (HR = 1 + q-1)c) Auxiliary closed loop poles near high frequency plant poles
I.D.Landau : From Robust Control to Adaptive Control24
Parameter Estimators for Adaptive Control
I.D.Landau : From Robust Control to Adaptive Control25
Classical Indirect Adaptive Control
PLANT
disturb.u y
+
++
q- dBA
CL
q-dBA
-
P.A.A.
y
Reference AdjustableController
q -1Filter Filter
Adaptationmechanism
(design)
- Uses R.L.S. type estimator (equation error)- Sensitive to output disturbances- Requires adaptation freezing in the absence of persistent excitation- The threshhold for adaptation freezing is problem dependent
I.D.Landau : From Robust Control to Adaptive Control26
Closed Loop Output Error Parameter Estimatorfor Adaptive Control
PLANT
disturb.u y+
+ +CL
q-dBA
-1S
R
P.A.A.
y
Reference AdjustableController
AdjustableController
q-1
Adaptationmechanism
(design)
- Insensitive to output disturbances- Remove the need for adaptation freezing in the absence ofpersistent excitation
- CLOE requires stability of the closed loop- Well suited for adaptive control with multiple models
q- dBA
+
I.D.Landau : From Robust Control to Adaptive Control27
Adaptive Control Effect of Disturbances
Classical parameter estimator(filtered RLS) CLOE parameter estimator
Disturbances destabilize the adaptive system when using RLS parameter estimator(in the absence of a variable reference signal)
I.D.Landau : From Robust Control to Adaptive Control28
Adaptive Control with Multiple Models
I.D.Landau : From Robust Control to Adaptive Control29
Supervisory Control
PLANT
MODELS
CONTROLLERS
SUPERVISOR
G1
G2
Gn
Kn
K2
K1
+
+
+
-
-
- 1
2
n...
.. y
Performance criterion:
nijettJ it
j
jtii ...2,1,0,0;)()()(
2
0
)(2=+=
=
)(min tJiiSwitching rule:
Rem. : stability requires the use of hysteresis or time delay in switching
I.D.Landau : From Robust Control to Adaptive Control30
Adaptive Control with Multiple Models
n is small (for the flexible transmission n = 3)Multiple models : improvement of the adaptation transientsCLOE Estimator : reduction of the false swithchings, performance improvement
SUPERVISOR
G1
G2
Gn
+
+
-
-
- 1
2
n..
yPLANT
+
+
-G
0-
+
CL
u
u y
Controller
Controller
r
P.A.A.
GAdaptive model
Fixed models
I.D.Landau : From Robust Control to Adaptive Control31
Adaptive Control versus Robust Control
Load variations : 0% 100% (in several steps)Rem : The robust controller used is the winner of an international
benchmark test for robust control of the flexible transmission (EJC, no.2., 1995)
I.D.Landau : From Robust Control to Adaptive Control32
Adaptation Transients
Adaptive Control with Multiple Models Classical Adaptive Control (simulation)
0 = adaptive ; 1= 0% ; 2 = 50% ; 3 = 100%
Load variations : 100% 0% (in two steps at 19s and 29s)
I.D.Landau : From Robust Control to Adaptive Control33
Adaptive Control with Multiple Models
The plant models are not in the model set
0 = adaptive ; 1= 0% ; 2 = 50% ; 3 = 100%
Load variations : 75% 25%
I.D.Landau : From Robust Control to Adaptive Control34
Adaptive rejection of unknown disturbancesApplication to active suspension
I.D.Landau : From Robust Control to Adaptive Control35
Rejection of unknown disturbancesRejection of unknown disturbances
Problem: Attenuation of unknown and/or variable stationary disturbanceswithout using an additional measurement
Solution: Direct adaptive feedback control Methodology: Based on the
Internal model principle Sensitivity function Q - parametrization Direct adaptive control algorithm
Objective: Computation of a controller with an adaptive internal model of thedisturbance
Rem: Stationary disturbances models have poles on the unit circle
Hypothesis: Plant model parameters are constant and known
I.D.Landau : From Robust Control to Adaptive Control36
).()(')();()(')(
:Controller
111
111
=
=
qHqSqSqHqRqR
S
R
Internal model principle: HS(z-1)=Dp(z-1)
Closed loop system. NotationsClosed loop system. Notations
)()(qD
1)P(q
)(q)(q)S'(q)HA(qy(t) 1-
p1-
-1-1-1S
-1t
N p =
++
A / Bq-d S / Ru(t)
Controller Plant )(p1 t
)(t
pp DN / Dirac circle;unit on the poles
edisturbanc ticdeterminis : )()()(
)( 11
1
=
=
(t) D
tqDqN
tp
p
p
p
)()()()()( :poles CL
)()()()(
)()()( :Output
11111
11
11
11
+=
==
qRqBzqSqAqP
tpqStpqP
qSqAty
d
yp
I.D.Landau : From Robust Control to Adaptive Control37
Central contr: [R0(q-1),S0(q-1)].
Bezout: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1).
Control: S0(q-1) u(t) = -R0 (q-1) y(t)
Q-parameterization :R(z1)=R0(q-1)+A(q-1)Q(q-1);S(q-1)=S0(z-1)-q-dB(q-1)Q(q-1).Q(q-1) computed such as [R(q-1),S(q-1)]
contain the internal model of the disturb.
Direct adaptative control (Direct adaptative control (QQ--parameterizationparameterization))
pd MDBQqS = 0
Control: S0(q-1) u(t) = -R0 (q-1) y(t) - Q (q-1) w(t),
where w(t) = A (q-1) y(t) - q-dB (q-1) u(t).
CL poles: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1).
The internal modelcan be tuned with Q
I.D.Landau : From Robust Control to Adaptive Control38
Goal: minimize y(t) (according to a certain criterion).
).()()(
)()()()()( 11
1
1
10 twqQ
qPqBqtw
qPqSt
d
=
)( of valueestimatedan be Let 11 qQ)(t,qQ -
Direct Adaptive Control (unknown Dp)
)0 termedisturbanc )1((
)1()()(
)()],1()([)1(
thatshowcan We
1
111
=+
+++=+
tv
tvtwqP
qBqqtQqQtd
Hypothesis: Identified (known) plant model (A,B,d).
[ ] [ ] )()P(q
))Q(q(qq-)(qS)()(qD)(q
)P(q))Q(q(qq-)(qS)A(qy(t)
e.disturbanc ticdeterminis : )()()(
)(Consider
1-
1-1-d-1-0
1-p
1-
1-
1-1-d-1-0
1-
1
1
1
twBtNB
tqDqN
tp
p
p
p
==
=
(Based on an ideea of Y. Z. Tsypkin)
Leads to a directadaptive control
I.D.Landau : From Robust Control to Adaptive Control39
Plant
ModelModel
^Adaptationalgorithm
Direct adaptive rejection of unknown disturbances
The order of the Q polynomial depends upon the order of the disturbance modeldenominator (DP) and not upon the complexity of the plant model
Less parameters to estimate than for the identification of the disturbance model Much simpler than indirect adaptive control
I.D.Landau : From Robust Control to Adaptive Control40
The Active Suspension
Residualforce
(acceleration)measurement
Activesuspension
Primary force(acceleration)(the shaker)
I.D.Landau : From Robust Control to Adaptive Control41
The Active Suspension SystemThe Active Suspension System
controller
residual acceleration (force)
primary acceleration / force (disturbance)
1
23
4
machine
support
elastomere cone
inertia chamber
pistonmainchamber
hole
motor
actuator(pistonposition)
sTs 00125.0=++
A / Bq-d S / R
D / Cq 1-d
u(t)
ce)(disturban
(t)up
Controller
force) (residualy(t)
Plant )(p1 t
Two paths :PrimarySecondary (double differentiator)
I.D.Landau : From Robust Control to Adaptive Control42
Active Suspension
Primary path
Frequency Characteristics of the Identified ModelsSecondary path
0;16;14 === dnn BA
I.D.Landau : From Robust Control to Adaptive Control43
Adaptive disturbance rejection
Closedloop
Open loop
Disturbance : Chirp
Initialization of theadaptive controller
25 Hz47 Hz
I.D.Landau : From Robust Control to Adaptive Control44
Concluding Remarks
- Identification in closed loop establishes a bridge betweenrobustness and adaptation
- Iterative identification in closed loop and controller re-designis a two times scales adaptive control
- Robust linear design in high frequency is needed for adaptivecontrol schemes
- The multiple models approach to adaptive control improvessignificantly the adaptation transients
- Robust control gives hints for adaptive rejection of unknowndisturbances
- High speed simple adaptive direct control scheme for rejectionof unknown disturbances has been proposed and tested.
I.D.Landau : From Robust Control to Adaptive Control45
References
Morse A.S. (1995) Control using logic based switching in Trends inControl (A. Isidori, ed.) Springer Verlag, London, U.K.
Narendra K.S., Balakrishnan (1997) Adaptive control using multiple models IEEE Tr. on Aut. Control, AC-42, pp. 171-187
Karimi A., Landau I.D.(2000) Robust adaptive control of a flexible transmission system using multiple models , IEEE Tr. on Contr.Syst.Technology.March
Landau I.D., Lozano R., MSaad M., (1997) : Adaptive Control, Springer, London,U.K.
I.D.Landau I.D(1999) From robust control to adaptive control Control Eng.Practice,vol 7,no10, pp1113-1124
Landau I.D., (2001) : Identification in closed loop : a powerful design tool(better models, simple controllers) , Control Engineering Practice, vol. 9, no. 1, pp. 51- 65.
A. Constantinescu, I.D. Landau (2002) Adaptive narrow band disturbance rejection in active vibration control, Proceedings of IFAC World Congress 02, Barcelona, Spain
See also: http://www-lag.ensieg.inpg.fr/landau/bookIC
I.D.Landau : From Robust Control to Adaptive Control46
web site :
Commande des systmesconception,identification et mise en oeuvre
http://www-lag.ensieg.inpg.fr/landau/bookIC