Événement - date MEA’10, December 09 th, 2010 1/30 Performances analysis of controlled interval...
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Transcript of Événement - date MEA’10, December 09 th, 2010 1/30 Performances analysis of controlled interval...
Événement - date
MEA’10, December 09th, 2010 1/30
Performances analysis of
controlled interval systems
using H∞ approach
Événement - date
MEA’10, December 09th, 2010 2/30
• Introduction
• Robust control approaches for (parametric) uncertain systems
• A posteriori performances analysis
• Example with the control of piezoelectric cantilevers
• Conclusion
Outline
Événement - date
MEA’10, December 09th, 2010 3/30
• Introduction
• Robust control approaches for (parametric) uncertain systems
• A posteriori performances analysis
• Example with the control of piezoelectric cantilevers
• Conclusion
Événement - date
MEA’10, December 09th, 2010 4/30
1- Introduction
Control design requirements
Good knowledge on the system
Good knowledge on its environment
accurate model
Sys
tèm
e p
hys
iqu
e
?
Événement - date
MEA’10, December 09th, 2010 5/30
1- Introduction
Characteristics of physical systems
Complexity
nonlinearities (hysteresis, creep: ”piezo systems”)
time varying parameters
etc
vibration, variation of the ambient temperature, etc
disturbance effects and imprecise measurements
etc
Sensitivity to the environmental conditions
Modeling & control
difficulties
Non-perfect model
Sim
plifi
catio
ns
Model subject to parametric uncertainty
Événement - date
MEA’10, December 09th, 2010 6/30
Reference OutputController Model
U+-
uncertainties
Control of parametric uncertain systems
1- Introduction
Achieve robust performances
Closed-loop control
robust controller+
Événement - date
MEA’10, December 09th, 2010 7/30
Robust controller: a controller that takes into account the uncertainties during the synthesis and ensures the stability and the performances for the closed-loop.
Definitions
Robust stability: the stability of the closed-loop obtained by a robust controller despite of uncertainties.
Robust performances: the performances of the closed-loop ensured by a robust controller in presence of uncertainties.
1- Introduction
Événement - date
MEA’10, December 09th, 2010 8/30
• Introduction
• Robust control approaches for (parametric) uncertain systems
• A posteriori performances analysis
• Example with the control of piezoelectric cantilevers
• Conclusion
Événement - date
MEA’10, December 09th, 2010 9/30
2- Robust control approaches for parametric uncertain systems
H∞ approach
robust controller
K(s)
Outputsystem
Input
(s)
uncertainties matrix
- Implementation difficulties
- Complex controllers
µ-synthesis
H∞-loopshaping
H∞-LPV
H∞ standard
Événement - date
MEA’10, December 09th, 2010 10/30
Interval analysis replacing real numbers by intervals[Moore, 1966]
Robust control using interval analysis
2- Robust control approaches for parametric uncertain systems
Guaranteed stability & robust performances
[Bondia. CDC03] [Tchen. CCE00][Khadraoui. CDC10] Principle:
Theory of control Interval analysis
Robust control law
Parametric uncertainty Modeling
Required specification
Accounts
- Reduced order controllers
- Natural modeling of parametric uncertainties
Événement - date
MEA’10, December 09th, 2010 11/30
• Introduction
• Robust control approaches for (parametric) uncertain systems
• A posteriori performances analysis
• Example with the control of piezoelectric cantilevers
• Conclusion
Événement - date
MEA’10, December 09th, 2010 12/30
3- A posteriori performances analysis
Objective
Based on H∞ standard, verify if the controller C(s) ensures required specifications.
y eC(s) [G](s,[a],[b])
+
- u
- : an interval system,[ ]( ,[ ],[ ])G s a b
- C(s): a given controller computed with interval control method.
Specifications:
- Settling time tr ≤
- Static error ≤
- No (or small) overshoot
- etc…
?Does the controller ensures the required specifications.
Événement - date
MEA’10, December 09th, 2010 13/30
Reminding H∞ principle
+
-
x2
T1(s)
T2(s)+
+
x1
e1
e2
y1
y2
1- Stability: [Zames. 1966]
Stability ensured
1)()( 21
sTsT
Small gain theorem
3- A posteriori performances analysis
The H∞ norm of a SISO system G(s) is defined by: )(sup)( jwGsGw
Événement - date
MEA’10, December 09th, 2010 14/30
Reminding H∞ principle
2- H∞ Standard: [Doyle and Glover. 1989]
zS
W1 z1
z2
C.S
W2
W3 d
yC(s) G(s)r
u+
-
++
b
Specifications:
- Settling time/bandwidth
- Static error/static gain
- No (or small) overshoot
- Control moderation
- Disturbances rejection
- Stability
3- A posteriori performances analysis
1.(1 . ) .( )u C C G r b
)()1( 1 brGC
Événement - date
MEA’10, December 09th, 2010 15/30
Choice of the Weighting functions
bandwidth (min)
Overshoot (max)
static error (max)
1
11
( ) p
z
sw
W ss
w
2
3
11
zw
tr
21 %
1p z
Dw w
3- A posteriori performances analysis
Weighting W1(s)
)(
1
1 sW
( )S s
Événement - date
MEA’10, December 09th, 2010 16/30
Choice of the Weighting functions
3- A posteriori performances analysis
Weighting W2(s)
( ). ( )C s S s
2
1
( )W s
1
k
3 ( )W s
Weighting W3(s)
2
1( ) .
1
sW s k
s
maxin
s
Vk
y
1 , 1
Événement - date
MEA’10, December 09th, 2010 17/30
Fl(P(s),C(s))
H∞ standard problem: Compute the controller C(s) such as:
( ( ), ( ))lF P s C s
( ) 1
- Stability
-
3- A posteriori performances analysis
W3 d
yC(s) G(s)
r u+
-
++
b
S
W1
C.S
W2z
z1
z2
P(s): augmented system
C(s)
P(s)
u
H∞ standard formalism
d
re
2
1
z
zz
Événement - date
MEA’10, December 09th, 2010 18/30
Interval closed-loop performances analysis via H∞ -standard approach
Specifications:
- Settling time tr ≤
- Static error ≤
- No (or small) overshoot
Given a controller C(s), If the controller satisfies:
then, the required specifications are ensured defined in terms of weighting functions
1( )[ ]( ) 1W s S s
1
1[ ]( )
( )S s
W s
3- A posteriori performances analysis
1([ ]( ,[ ],[ ]), ( )) ( )[ ]( )lF P s a b C s W s S s
y eC(s) [G](s,[a],[b])
+
- u
zW1(s)
Événement - date
MEA’10, December 09th, 2010 19/30
H∞ norm of the sensitivity of an interval system
Given an interval system, the maximal H∞ norm of its sensitivity function is achieved
at twelve (out of sixteen) Kharitonov vertices. [Long Wang, 2002]
[ ]( )[ ]( )
[ ]( ) [ ]( )
f sS s
f s g s
0 1
0 1
[ ]( ) [ ] [ ] ... [ ]
[ ]( ) [ ] [ ] ... [ ]
nn
mm
f s a a s a s
g s b b s b s
Let consider a sensitivity function of an open-loop interval system :[ ]( )
[ ]( )[ ]( )
g sT s
f s
3- A posteriori performances analysis
2 3 4 52 311 0 1 4 5
2 3 4 51 2 512 0 3 4
2 3 4 50 3 421 1 2 5
2 3 4 50 1 4 522 2 3
( ) ...,
( ) ...,
( ) ...,
( ) ...,
f s a a s a s a s a s a s
f s a a s a s a s a s a s
f s a a s a s a s a s a s
f s a a s a s a s a s a s
The four Kharitonov polynomials corresponding to and : [ ]( )f s [ ]( )g s
2 3 4 52 311 0 1 4 5
2 3 4 51 2 512 0 3 4
2 3 4 50 3 421 1 2 5
2 3 4 50 1 4 522 2 3
( ) ...,
( ) ...,
( ) ...,
( ) ...,
g s b b s b s b s b s b s
g s b b s b s b s b s b s
g s b b s b s b s b s b s
g s b b s b s b s b s b s
Événement - date
MEA’10, December 09th, 2010 20/30
1 1 2 2( ) (1111), (1212), (2222), (2121), (1112), (1222),i j i j
(2221), (2111), (1211), (2212), (2122), (1121).
With:
2 2
1 1 2 22 2 1 1
( )( ) max
( ) ( )i j
i j i ji j i j
f sS s
f s g s
The maximal H∞ norm of the sensitivity function is defined by:
3- A posteriori performances analysis
Événement - date
MEA’10, December 09th, 2010 21/30
• Introduction
• Robust control approaches for (parametric) uncertain systems
• A posteriori performances analysis
• Example with the control of piezoelectric cantilevers
• Conclusion
Événement - date
MEA’10, December 09th, 2010 22/30
4- Example with the control of piezoelectric cantilevers
Fig. 1: a) Manipulation of micro-object; b) Piezocantilever subjected to an electrical excitation
(a)
(b)
Événement - date
MEA’10, December 09th, 2010 23/30
Interval model
22 1
22 1 0
[ ] [ ] 1[ ]( ,[ ],[ ])
[ ] [ ] [ ]
b s b sG s a b
a s a s a
0[ ] [1.283,1.448]a 6
1[ ] [5.234,5.37] 10a 8
2[ ] [8.753,9.844] 10a
41[ ] [1.807,1.809] 10b
82[ ] [6.992,8.08] 10b
4- Example with the control of piezoelectric cantilevers
Specifications:
- Settling time tr ≤ 30ms
- Static error ≤ 1%
- No overshoot 0.2 400( )
sC s
s
Interval
Control
[Khadraoui, SWIM’10]
Based on H∞ standard, prove the robustness of the PI controller to ensure the required specification
Événement - date
MEA’10, December 09th, 2010 24/30
1
11
( ) p
z
sw
W ss
w
From the required specification:
2
3
11
zw
tr
21 %
1p z
Dw w
Robustness study of the controller C(s)
4- Example with the control of piezoelectric cantilevers
After numerical application:
1
1 1
( ) 100
s
W s s
1)(
1
1
sW
H∞ Norm of )(
1
1 sW
Événement - date
MEA’10, December 09th, 2010 25/30
1 [ ]( )[ ]( )
1 ( ) [ ]( ,[ ],[ ]) [ ]( ) [ ]( )
f sS s
C s G s a b f s g s
[ ]( ) 0.9646S s
Using the method proposed in [Long Wang, 2002], we obtain:
1
[ ]( ) 0.9646
11
( )
S s
W s
1
1[ ]( )
( )S s
W s
Specifications hold
4- Example with the control of piezoelectric cantilevers
Robustness study of the controller C(s)
H∞ norm of the sensitivity function [S](s)
Événement - date
MEA’10, December 09th, 2010 26/30
The controller has played its role and the required specifications are satisfied
Fig. 1: magnitudes of the sensitivity function compared to the magnitude of the weighting function
[S](s) = (1+C(s).[G](s,[a],[b]))-1
W1(s)1
4- Example with the control of piezoelectric cantilevers
Événement - date
MEA’10, December 09th, 2010 27/30
Fig. 1: Experimental results using PI controller for two piezocantilevers compared with the desired behaviors
4- Example with the control of piezoelectric cantilevers
Événement - date
MEA’10, December 09th, 2010 28/30
• Introduction
• Robust control approaches for (parametric) uncertain systems
• A posteriori performances analysis
• Example with the control of piezoelectric cantilevers
• Conclusion
Événement - date
MEA’10, December 09th, 2010 29/30
Experimental results with piezocantilevers prove the efficiency of the proposed method,
H∞ standard approach confirms the robustness of the computed controller,
Performances analysis for a controlled system with uncertainties,
The interval analysis represents a guaranteed tools for modeling and control system with uncertainties.
5- Conclusion
Événement - date
MEA’10, December 09th, 2010 30/30
Thank You For Your
Attention