PARAMETER DEPENDENT LYAPUNOV FUNCTIONS FOR STABILITY OF LINEAR PARAMETER VARYING SYSTEMS Nedia...

PARAMETER DEPENDENT LYAPUNOV FUNCTIONS FOR

STABILITY OF LINEAR PARAMETER VARYING SYSTEMS

Nedia Aouani, Salah Salhi, Germain Garcia, Mekki Ksouri

Research Unit of System Analysis and Control ACS, National Engineering School of Tunis

University of Toulouse, LAAS-CNRS

OUTLINE

Motivations Problem formulation New representation of the time derivative of the

parameter New LMI based conditions for stability analysis of

LPV polytopic systems Numerical example Conclusion

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MOTIVATIONS• Over the last two decades, LPV systems has undergone a

wealth of practical and theoretical developments [CAO&al, 2004], [Geromel&al, 2006], [Montagner&al, 2009]. All these works treat the problem of stability and establish conditions for analysis purposes.

• As to the uncertain parameters, they can be modeled under different structures : affine , polytopic or rational dependence. One difficulty remains how to represent the time derivative of the uncertainty in the case it is assumed to vary in a polytopic domain with bounded rates.

• Parameter Dependent Lyapunov Functions (PDLF) are ivestigated the last ten years [Peaucelle&al, 2000], and the LMIs became a powerful skill to deal with such problems.


MOTIVATIONS

• As to Parameter Dependent Lyapunov Functions, some specific forms have been investigated all along the littérature: the class of Polynomial PDLFs [Chesi, 2003, 2004, 2005, 2007], [Oliveira,2005], the class of rational ones [Scorletti, 1995], [Lu, 1996] and the class of affine ones [Feron, 1996], [Gahinet, 1996], [Peaucelle, 2001].

Main Idea: Use of PDLFs of particular forms that have been used for LTI systems by [Ebihara, 2005] , for the case of LPV systems.

Reduction of conservatism in the proposed conditions


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PROBLEM FORMULATION

( ) ( ( )) ( ) ( )x t A t x t 1

1( ( )) ( )

N

i ii

A t t A

(2)

1( ) , : : 1, 0 , number of vertices,

NN

N N i ii

t where N

(3)

( ) ; 0i t r r (4)

N. AOUANI 5

Linear Parameter Varying system

The system matrices

The time varying parameter varies in a vertex such that

The parameter’s rate of variation

Objectives Assess robust stability of the system

NEW REPRESENTATION OF THE TIME DERIVATIVE OF THE PARAMETER

( )i j kr

1

( ) ; : : 1, 0MM

M M j jj

t

Previous representations of the parameter’s rate of variation

, is a constanti i i i • [Cao & al, 2004]

1 1,( ) ( )

( ) ( )

( ) : 1, 0 ,

1.. ,

MM Mj j M j j

j j

Nj

t t h

t satifies t

t

h for all j M are given vectors

r

•

[Geromel & al, 2006]

• [Xie & al, 2005] 1 1

( ) ( ) : ( ) 0, ( ) 1, 2N N r

v k k k kk k

t t t t N

1

( ) : : 1, 0;KK

K K k kk

t

Lemma1


Assumptions:• Stability condition:

With:

• The time derivative of Lyapunov matrix is given by:

Where and


( ) ( ) ( ( )) ( ( ))P P r P t P t

( ) Nt ( ) Nt

( )( ( )) 0, ( ( )) 0, : 0, 0

( )2nx t

V x t V x t y R y A I yx t

( ( )) ( ( )) , ( ( )) 0T n nV x t x P t x P t

NEW LMI BASED CONDITIONS FOR STABILITY ANALYSIS OF LPV POLYTOPIC SYSTEMS

The system (1) is asymptotically stable if there exist positive definite symmetric matrices , matrices ,

and such that the following LMI holds:


Proposition

1

2

( )0

0j k i

ii

r P P P FHe A I

FP

1,.., ; 1,.., ; 1,..,i N j N k N

iP jP

kP ( 1, 2)n nlF l


TheoremThe system (1) is asymptotically stable if there exist positive definite symmetric matrices , matrices and such that the following LMI holds:

And


iP ,j kP P ( 1, 2,3, 1, 2)n mlmF l m

11 12

2 2 21 22

31 32

( ) 0 00

0 0 0 ( ) 00

0 0 0

1,.., ; 1,.., ; 1,..,

j kT i

c ii

r P P F FA I

W W He F FA I

F F

i N j N k N

Where 2 cW and are given by

,222 21 22 21 22

,2 2

0 0 1, , ,

0 1 0

T n nnc

n n n

IW W W W W

I

22

Ti i

i n

i

P AP

A I


We consider the following LPV system [Geromel & al, 2006]

Such that the matrices Ai are taken:

The uncertain parameter

The time derivative of the uncertain parameter is bounded such that ;Purpose:Delimit the region of the plane (,) with and Such that the global asymptotic stability is preserved.

N. AOUANI 10

( ) ( ( )) ( )x t A t x t

1 22 2 2 2

0 1 0 1;

( ) ( )A A

w w

1; 0.05;w

r ri 2

r

NUMERICAL EXAMPLE

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0 4 0 1

1: : 1, 0

NN

N i ii

0 1

NUMERICAL EXAMPLE


0 1 2 3 40

0.5

1

1.5

2

Alpha

Omega

Theorem1 (Present paper)

Theorem1 (Geromel et al)

NEUMERICAL EXAMPLE

If we consider the system (1) with N=3:

And

We numerically verify the feasibility of the point

1 2 32 2 2 2

0 1 0 1;

( 3 ) ( )A A A

w w

4r

203, 1.5


CONCLUSION

• We have proposed in this paper a new stability condition formulated in terms of LMI constraints, for an LPV continuous system under polytopic uncertainty structure.

• Further analysis conditions can be deduced following the same ideas and increasing redundancy


THANK YOU FOR YOUR ATTENTION


PARAMETER DEPENDENT LYAPUNOV FUNCTIONS FOR STABILITY OF LINEAR PARAMETER VARYING SYSTEMS Nedia...

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