Performances of unknown input observers for chaotic LPV maps in a stochastic context

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Performances of unknown input observers for chaotic LPV maps in a stochastic context

Gilles Millérioux, Jamal Daafouz

To cite this version:

Gilles Millérioux, Jamal Daafouz. Performances of unknown input observers for chaotic LPV maps in a stochastic context. Nonlinear Dynamics, Springer Verlag, 2006, 44, pp.205-212. �10.1007/s11071- 006-1970-0�. �hal-00114846�

PERFORMANCES OF UNKNOWN INPUT OBSERVERS FOR CHAOTIC LPV MAPS IN A STOCHASTIC CONTEXT

G. Mill´erioux and J. Daafouz

Centre de Recherche en Automatique de Nancy (CRAN),

Universit´e Henri Poincar´e Nancy 1, Institut National de Polytechnique de Lorraine

e-mail:millerioux@esstin.uhp-nancy.fr, jdaafouz@ensem.inpl-nancy.fr WWW:http://www.cran.uhp-nancy.fr

Abstract

Jointly modeling chaotic maps as LPV systems and using Unknown Input Observers for retrieving the infor- mation in a secure communication scheme has previously been motivated in a deterministic context [1]. In this paper, some new theoretical results from a control theory point of view, concerning the design in a stochastic and so more realistic context of Unknown Input Observers for chaotic LPV systems is provided. The design of such observers is expressed in terms of the resolution of a finite set of Matrix Inequalities constraints and guarantees some pre- scribed performances on the state reconstruction error.

I. INTRODUCTION

The well-known practical interest of chaos syn- chronization lies in the potential applications in com- munications and more specifically in the possibilities of encoding or masking messages by embedding the information into the dynamics of a chaotic system.

The information to be masked plays the role of an external input for the dynamical system and is not transmitted to the receiver. Hence, the receiver sys- tem must be designed such that the information can be unmasked, given the only available output data consisting of a function of the state vector. In [1], a brief survey of the main approaches proposed in the literature is given. Then, a novel approach based on Unknown Input Observer (UIO) for a noise-free con- text is proposed.

In a deterministic context, Unknown Input Observers have been largely investigated for linear systems [2][3][4][5]. On the other hand, in a stochastic con- text, there does not exist a lot of results. For time- invariant systems, we can mention the works of [6]

while for linear time-varying systems, the reader can refer to [7], but none of those classes of systems can exhibit chaotic behaviors and so have no interest for chaos-based communications purposes.

The aim of this paper is to state some new results concerning the design of Unknown Input Observers

for Linear Parameter Varying Systems in a stochastic context. The interest of LPV systems lies in the fact that a large amount of chaotic systems enter this class.

Furthermore, the UIO design guaranteeing some pre- scribed performances can be carried out in a tractable way by solving Matrix Inequalities.

Notation :

stands for the symmetric block of a positive definite matrix, and stand for the zero and the identity matrix of appropriate size.

,

and

!

"

$#%

&

.

II. UNKNOWN INPUT OBSERVERS FORLPV

SYSTEMS

Consider the general state space realization of LPV discrete-time systems in a noisy context.

'

&$(*)+ ,-/.01& 2 3-4 2 576

8 9:& 2 ;<6 (1)

where

&>=@?BA

, ⁸

C=D?FE

,

,G=D?HAIJA

,

3K=@?BAIJL

,

9M=N? E:IJA

,

;O=P? E7IJL

.

4Q=P? L

is the input,

6RQ=

?HS

is the disturbance acting on the dynamics through

5

and acting on the measurement through

;

.

,

is of class

9 )

with respect to the entries of a^T -dimensional time-varying parameter vector

.D+/.

)

JU$V$V$VWU .0X

. In [8], it has been shown that a lot of chaotic maps can be modeled by LPV discrete-time systems with

.J

being a function of the state vector

Y

. Since

&

evolves chaotically,

.

is bounded in a hypercube ^Z . As a result,

,

lies in a compact set which can always be embedded in a polytope, that is :

,[/.JH \

]^

#*)%_

^

/.J`,

^

(2) where the

, ^

’s correspond to the vertices of the convex hull ^acb%d

, )

U$V$V$VWU

,

\fe

. The

_

’s belong to the compact set ^g

dih

= ? \ U h

h ) U$V$V$VU

h \ U h ^

kj!lDm&n

and

" \^

#*)

h ^ po

and

they can always be expressed as functions of class

9

with respect to the

.

’s. The advantage of such a de- composition lies in the fact that the design problems turn into the resolution of a finite set of constraints in- volving only the vertices of the convex hull.

For secure communication purposes,

4

plays the role of the information to be masked and acts as an un- known input. ⁸

is the signal transmitted to the re- ceiver. The structure of the required Unknown Input Observers for the recovering of

4

is reminded from [1].

&$(*)p7,-/.0R

T

/.J 9

& 2 T /.01

8 2

8

$(*)

(3) with

A

9

and ^T

/.0 " \^

#*)

_ ^ /.J

T ^

. The gains

and ^T

^

’s (ⁿ

o

U$V$V$VWU

) are unknown matrices to be computed.

From (1) and (3), it is straightforward to show that the state reconstruction error

&

is governed by :

$(*) /.0

Y273-4&2 /.0 6f; 61$(*)

(4) with

/.0< " \

^

#*)

_ ^ 7,

^ T ^ 9

and

/.0

" \^

#*)

_ ^

75

T ^;

.

Before dealing with the performances on the state re- construction when disturbances act on the system, it is necessary to remind how the global stability of the null solution of (4) can be guaranteed when

60Q

l . Some details can be found in [1].

Theorem 1. The global stability of the null solution of (4) with

6R:

l is ensured if

i)

9f3 B

3 B

, ii) there exist symmetric matrices

^

, matrices

^

and

^

such that,^{m n U}

= d o

U$V$V$V U

e ! d o U$V$V$VWU

e

, the following set of Linear Matrix Inequalities is feasible.

" ^

^ , ^ ^9 ^ 2 ^

#%$'&(

l (5)

The time-varying gain is given by ^T

/.

" \^

#*)

_ ^ T ^

with^T

^

*)

)

^ ^

.

Proof: On one hand, according to the definition of

, the equality

3 l

entails that

must be subject to

3p+c9f3

(6) and i) ensures the existence of the solution

of (6).

Its general expression is :

3Q9f3 -, 2/.

E

@9f3 9f3 -,

(7) with

.

an arbitrary matrix. Then, whenever

satis- fies (7),

73M

l and so (4) turns into an input inde- pendent dynamics :

$(*) /.0 20:/.J610f; 61$(*)

(8) On the other hand, the proof follows a reasoning sim- ilar to the one carried out in [9]. All the relations are valid^{m n U}

= d o

U$V$V$V U

e ! d o U$V$V$VWU

e

. a) Since

2$

is strictly positive, one has :

^ ) )

$ ^ j ^ 2 ^

32$

b) Substitute

^

by

^T ^

in (5) and take into account the inequality above yields :

" ^

^ 7,

^ T ^ 9 ^ ) )

$ ^

&(

l (9)

which is equivalent to

465

" ^

%$J7,

^ T ^ 9 %$

&

574

(10) with

4 "

^ ) )

$ &

and so to

" ^

%$J7,

^ T ^9 %$ & ( l

(11) since

^

and

2$

are full rank matrices.

c) For each ⁿ

o

U$V$V$VWU

, multiply the correspond- ing

o

U$V$V$VWU

inequalities (11) by

_ $

$(*) and sum.

Then, multiply the resultingⁿ

o

U$V$V$VWU

inequalities by

_ ^

and sum again. We obtain :

" 8

8

$(*)9

8

$(*) & ( l

(12) with

8

N " \

^

#*)

_ ^ ^

and

8

$(*) " \^

#*)

_ ^

$(*) ^

. Applying the Schur complement formula gives :

8

$(*)-

8

;:

l m (13)

It is shown in [9] that ^<>=

?A0? ? (

, a function de- fined by^<

@

U _

1H@

8 A@W

with

8

N " \^

#*)

_ ^ ^

and

_

<=

g acts as a Lyapunov function for (4) when

6 l

and ensures the poly-quadratic stability of (4) which is sufficient to global asymptotical stability.

This completes the proof.

In the forthcoming sections, the case

6J/B l

, that is the stochastic context, is considered and constitutes the main result of the paper.

III. OBSERVER DESIGN WITH PRESCRIBED PERFORMANCES

A. Bounded gain

We define the upper bound denoted of the gain as a scalar verifying :

#%

@%

6 : (14)

where

@:9

,

6:6 6

$(*)

.

Theorem 2. The gain corresponding to (4) with

6B

l is less than if

i)

9f3 B

3 B

, ii) there exist symmetric matrices

^

, matrices

^

and

^

such that^{m n U}

f=

d o U$V$V$VU

e ! d o U$V$V$V U

e

, the set (21) of Linear Matrix Inequalities is feasible.

The time-varying gain is given by ^T

/.

" \^

#*)

_ ^ T ^

with^T

^ ) )

^ ^

.

Proof : For the same reason motivated in the proof of Theorem 1, condition i) ensures the existence of a matrix

such that

73 l

holds and turns (4) into an input independent dynamics. Besides, define the matrices

^ ^

B

B

U

^ ^

B

B

and

^

7,

^ T ^9 ^

75

T ^; 1f;

) ) 9

(21) can be rewritten :

" ^

^ ^ ^ 2 ^

$& ( (15)

Following the same three steps a) to c) as in the proof of Theorem 1, feasibility of (15) implies that

$(*)

:

l (16)

with

7

8

B

B

U

$(*) 8

$(*)

B

B

and

f;

) )

Equation (16) can be rewritten like (22). Then, multi- ply left and right respectively by

6

and its trans- pose gives :

<

$(*)

U _ (*) < U _

2 ) ) 9

9

1 6

6*:

l

(17) Consider (17) from

l

to and sum leads to:

< \ (*)

U _ \

(*) 2 ) ) \

]

$#%

9 1

9 \

] #%

6

6*:

l

(18) Yet,^<

\

(*)

U _ \

(*)

\ (*)

8 \

(*)

( l

. Hence :

) ) \

]

$#%

9 9 : \

]

$#%

6

61

(19) When tends toward infinity, this relation is equiva- lent to (14). This completes the proof.

B. Peak-to-peak gain

Let be the upper bound of the peak-to-peak gain defined as the ratio between

and

6

!

"

#6 : (20)

$$$$$$ ^

B

B

^ 7,

^ ^ 9 ^

75 ^; ^

f; ^ 2 ^ $

9 B

B

&%

%%%%%

( l

(21)

8

$(*)- 8 2 ) ) ' '

H

8

$(*)-

8 (*)

B

f;Q 8 (*)9 cf;Q 8

$(*) f;Q 8 (*)W7; ( l

(22)

Theorem 3. The peak-to-peak gain corresponding to (4) with

6RB

l is less than ^{( l} if

i)

9f3 B

3 B

,

ii) there exist symmetric positive definite matrices

^

, matrices

^

, scalars

= l U o

, ^{h ( l} such that,

m n U P=

d o U$V$V$VWU

e ! d o U$V$V$VWU

e

, the Matrix In- equalities (29) are fulfilled.

The time-varying gain is given by ^T

/.

" \^

#*)

_ ^ T ^

with^T

^ ) )

^ ^

.

Proof: For the same reason motivated in the proof of Theorem 1, condition i) turns (4) into an input in- dependent dynamics. On one hand, again, consider- ing the first inequality of (29) and following the same steps from a) to c) as in the proof of Theorem 1 yields :

$$ o 5 8

hH

hH

8

$(*)9

8

$(*) 8 (*) 7;

8

$(*)

&%

%

( l

(23) Applying the Schur complement formula and somes basic manipulations yields (30). Then, multiply (30) left and right respectively by

6

and its transpose, entails that :

<

$(*)

U _

$(*) : o < U _

Y2

h

6

m

(24) Applying the Gronwall-lemma in the discrete case gives :

< U _

1 : h

#6*

m (25)

On the other hand, multiplying the second inequality of (29) by

_ ^

and sum fromⁿ

o

to gives :

" 8

1

)

h

&(

l (26)

Besides, multiply (26) left and right respectively by

61

and its transpose leads to :

o

:

%< U _

2 h

W 61

m (27)

Finally, combining (25) and (27) and taking into ac- count that

h ( l

from the second inequality of (29) leads to :

:

6

(28) And yet, (28) is equivalent to (20).

Remark

Note that the Matrix Inequalities coresponding to the peak-to-peak performances are not linear unlike the ones related to the gain. They involve a product of two unknowns, say and

^

, and a nonlinear depen- dence on which prevents the convexity.

REFERENCES

[1] G. Millerioux and J. Daafouz. Unknown input observers for message-embedded chaos synchronization of discrete-time systems. International Journal of Bifurcation and Chaos, 14(4), April 2004.

[2] M. Darouach, M. Zazadinski, and S. J. Xu. Full-order ob- servers for linear systems with unknown inputs. IEEE Trans.

on Automatic Control, 39(3):606–609, March 1994.

[3] S-K. Chang, W-T. You, and P-L. Hsu. Design of general structured observers for linear systems with unknown inputs.

J. Franklin Inst., 334(2):213–232, 1997.

[4] F. Yang and R. W. Wilde. Observers for linear systems with unknown inputs. IEEE Trans. on Automatic Control, 33(7):677–681, July 1988.

[5] Y. Guan and M. Saif. A novel approach to the design of un- known input observers. IEEE Trans. on Automatic Control, 36(5):632–635, May 1991.

[6] Nikoukhah R. Innovations generation in the presence of un- known inputs: application to robust failure detection. Auto- matica, 30:1851–1867, 1994.

[7] M. Darouach, M. Zazadinski, and M. Boutayeb. Extension to minimum variance estimation for systems with unknown inputs. Automatica, 39:867–876, 2003.

[8] G. Millerioux and J. Daafouz. Polytopic observer for global synchronization of systems with output measurable nonlin- earities. International Journal of Bifurcation and Chaos, 13(3):703–712, March 2003.

[9] J. Daafouz and J. Bernussou. Parameter dependent lyapunov functions for discrete time systems with time varying para- metric uncertainties. Systems and Control Letters, 43:355–

359, 2001.

$$ o 5 ^

hH

hH

^ 7,

^ ^9 ^

75

^; ^

f; ^ 2 ^

3%$

&%

%

( l U " ^ )

h & ( l

(29)

8

$(*)9@ o 8

H

8

$(*)-

hH 2/

8

$(*)

f;Q

8 (*)9 7; 8

$(*)

hH

2 f;Q

8

$(*) f;

: l

(30)