Événement - date

MEA’10, December 09th, 2010 1/30

Performances analysis of

controlled interval systems

using H∞ approach

sofiane.khadraoui@femto-st.fr

mrakoton@femto-st.fr

plutz@femto-st.fr

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