Stability of Linear Dynamical Systems with Fuzzy Parameters

Le Hung Lan Department of Cybernetics, Faculty of Electrical –Electronic Engineering

University of Transport and Communications Hanoi, Vietnam

E-mail: lehunglan@fpt.vn

Abstract

In the paper the problem of stability analysis of

uncertain linear systems with parameters described by fuzzy functions is studied. Each coefficient is considered as a variable interval parametrized by the degree of confidence in validity of the corresponding model. Then the system stability can be solved by using some extended robust stability criteriums for nonsymmetric intervals. 1. Introduction

It is well-known that it is not possible to obtain an accurate model of system, some uncertainty is always present. A system has parametric uncertainty if there exists a mathematical model of it, but the values of some of its parameters are not exactly known. Many times, an estimation of the uncertain parameter values are given by an expert. In this case, the uncertain parameter can be represented by a fuzzy number

iq~ ,

with membership function ]1,0[)( ∈= iqμα . The membership value α can be interpreted as the confidence degree in that the value of the parameter equals its nominal value. Thus, a value 1=α indicates precise knowledge ( )ker(

~

ii qq ∈ ) whereas 01=α represents maximum

uncertainty ( ∈iq supp(iq

~ ). By using α -cuts concept

the coefficient iq~

can be represented as varying interval.

[ ])(),()( iiii

def

i qqq ααα +−= where iα is the confidence degree in that coefficient

(.)−iq and (.)+

iq are strictly increasing and decreasing functions, respectively, so that

++−− == iiii qqqq )0(,)0( . (1)

Let us consider an uncertain linear dynamical continuous system with single input and single output described by the differential equation.

)(...)()(...)()( 0~

)(~

0~

)1(1

~)( tubtubtyatyaty m

mn

nn ++=+++ −

− (2)

where ji ba~~

, are fuzzy numbers. This linear system with fuzzy parametric

uncertainty can be represented by the transfer function

0~

11

~0

~1

1~~

~

~~~~

...

...

),(

),(),,(asas

bsbsb

bsU

asYabsGn

nn

mm

mm

+++

++==−

−

−− (3)

Given a confidence degrees bj

ai αα , for the

parameters ji ba~~

, , the confidence degree in the model can be defined as the confidences vector

),...,,...,,( 110bm

ban

a ααααα −= . System (2) can then be represented by the family of transfer functions

α

αα

α⎥⎦⎤

⎢⎣⎡++⎥⎦

⎤⎢⎣⎡+

⎥⎦⎤

⎢⎣⎡++⎥⎦

⎤⎢⎣⎡

=⎥⎦⎤

⎢⎣⎡

−− 0

~1

1~

0~~

~~

...

...),(

asas

bsbqsG

nn

n

mm

(4)

where [ ] [ ])(),(,)(),(~~

bjj

bjjj

aii

aiii bbbaaa αααα

αα

+−+− =⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡ ,

i=1, …, n, j=0, … , m are interval components, corresponding to the α -cuts of the uncertain

parameters ji ba~~

, . For a given confidence degree, the transfer function (3) corresponds to a model with interval parameters. Thus, a fuzzy parametric uncertain system is a generalization of an interval system [1,2,5,6] and the robust stability theory can be usefull for study of this system class. Firstly, we extend some robust stability criteriums and next show that how can apply them for fuzzy dynamical system stability analysis.

Fifth International Conference on Fuzzy Systems and Knowledge Discovery

978-0-7695-3305-6/08 $25.00 © 2008 IEEE

DOI 10.1109/FSKD.2008.164

115

Fifth International Conference on Fuzzy Systems and Knowledge Discovery

978-0-7695-3305-6/08 $25.00 © 2008 IEEE

DOI 10.1109/FSKD.2008.164

115

2. Extended robust stability criteriums of interval systems 2.1. Interval characteristic polynomial

Tsypkin and Polyak [5] introduced a simple graphical robust stability criterium for a family of interval polynomials, where there is symmetric parametric perturbation

01

1 ...)( asasasp nn

nn +++= −

− (5)

iii aaa Δ≤− γ0 . (6)

The extended Tsypkin-Polyak involves the case of nonsymmetric parametric perturbation

+− Δ≤−≤Δ− iiii aaaa γγ 0 (7) Denote the nominal polynomial

00

101

00 ...)( asasasp nn

nn +++= −

− (8) and

},...,0,),({)( 0 niaaaajp iiii =Δ≤−≤Δ−=Π +− γγωωγ (9) a set on complex plan, ∞<≤ ω0 .

The set )(ωγΠ here is a rectangle:

...,)(

...,)(

...,)(

...,)(

),())()(Im()(

),())()(Re()(

67

45

231

67

45

231

66

44

220

66

44

220

0

0

+Δ+Δ+Δ+Δ=

+Δ+Δ+Δ+Δ=

+Δ+Δ+Δ+Δ=

+Δ+Δ+Δ+Δ=

≤−≤−

≤−≤−

−+−+−

+−+−−

−+−+−

+−+−−

+−

+−

ωωωωωωωωωωωωωωωω

ωγωωωγωγωωωγ

aaaaS

aaaaT

aaaaS

aaaaS

TjpjpT

SjpjpS

(10)

hence the condition )(0 ωγΠ∉ means that the inequalities

)()(Im)(),()(Re)( 00 ωγωωγωγωωγ +−+− ≤≤−≤≤− TjpTSjpS (11) can not be satisfied simultaneously for any ∞<≤ ω0 . If we introduce

[ ]{ }[ ]{ }

)(/)()(),(/)()(

,)()()()()(5.0)(

,)()()()()(5.0)(

...)(Im)(

...)(Re)(

00

0

0

405

203

01

010

404

202

00

00

ωωωωωωωωωωωωωωωωωω

ωωωωωωωωω

TVySUx

TTsignVTTT

SSsignUSSS

aaajpV

aaajpU

==

−++=

−++=

−+−==

−+−==

+−+−

+−+−

−

(12)

then the condition (13) can be written as { } γωω >)(,)(max yx (for 0>ω ). As )(ωx and

)(ωy have the same signs as )(),( 00 ωω VU , the plot )()()( ωωω jyxz += goes through the same quadrants

as the plot )(0 ωjp . If we apply Mikhailov’s test for

stability of a fixed polynomial )(0 sp with regard to

conditions 0>na (it give γ>Δ −nn aa /0 ) and for

0=ω , we arrive to the following Extended Tsypkin-Polyak criterion [6].

Theorem 1. The following conditions are necessary and sufficient for robust stability of the family (7), (9):

a) The plot )()()( ωωω jyxz += , ∞<≤ ω0 goes through n quadrants in counterclockwise direction.

b) It does not intersect γ -square (i.e. the set γγ ≤≤ yx , ).

c) Boundary conditions hold: γγγ >∞>∞> )(,)(,)0( yxx .

2.2. Closed-loop systems with interval plant

Let us consider the closed-loop control system with fixed controller

)()()(

sCsDsR = , (13)

and interval plant

nmsasaasbsbb

asAbsBbasP n

n

mm <

++++++== ,

......

),(),(),,(

10

10 (14)

,...,,1,0,,...,,1,0, 00 niabbmibbb iiiiii =Δ≤−=Δ≤− γγ (15)

where 0,0 ≥≥ ii βα are ranges of bounded coefficients perturbations, 0≥γ - margin of

perturbations, 00 , ii ab - nominal coefficients. In [7] the author proposed the simple graphical

criterium for system robust stability test and simultaneously computing stability margin maxγ .

Note that the coefficient interval perturbations given by (15) are symmetric, next we extend the criterium for the case of nonsymmetric interval perturbations. It means that

,...,,0,,...,,0, 00 niaaaamibbbb iiiiiiii =Δ≤−≤Δ=Δ≤−≤Δ +−+− γγγγ (16)

where −− ΔΔ ii ab , - left range of perturbations, ++ ΔΔ ii ab , - right range of perturbations.

Let the nominal system with plant

nn

mm

sasaasbsbb

sAsBs 00

100

001

00

0

00 ...

...)()()(P

++++++==

is stable. If denote

116116

...

...,

...,

...,

,)(

67

45

231

67

45

231

66

44

220

66

44

220

000

+Δ+Δ+Δ+Δ=Δ

+Δ+Δ+Δ+Δ=Δ

+Δ+Δ+Δ+Δ=Δ

+Δ+Δ+Δ+Δ=Δ

+=

−+−++

+−+−−

−+−++

+−+−−

ωωωωωωωωωωωω

ωω

aaaaa

aaaaa

aaaaa

aaaaa

vjujA

odd

odd

even

even

AA

(17)

then the set ),( ajA ω is rectangle in complex plan

with center )(0 ωjA , left width −Δ evenaγ , right width +Δ evenaγ , above height +Δ oddaγω , and under heght

−Δ oddaγω . Similarly we have the values

,,),(0−+ ΔΔ eveneven bbjB γγω −+ ΔΔ oddodd bb γωγω , for the

rectangle ),(B bjω .

Denote RRj vjuejR ωρω ϕ +==)(

( ) ( )RA

RA

A

RA

RA

A

vuuvvvvuuu 002002

0201;1 −−=+−=

−−

ρω

ρ,

( ) ( )RB

RB

B

RB

RB

B

uvvuvvvuuu 00002

00 ; +−=−−=−−

ω , (18)

,,,, 00000000

ABB

ABB

BAA

BAA uucvvdvvduuc

−−−−−=−=−=−=

( ) ( ) ( )[ ]

( )( ) ( )

( )( ) ( )

( ) ( ) ( )[ ],5,0

},)]([

)](){[(5,0

},)]([

)](){[(5,0

,5,0

22222

2

2

2

2

1

AAAAAA

RRoddoddoddodd

RRevenevenevenevenA

RRoddoddoddodd

RRevenevenevenevenA

Aeveneveneveneven

A

csignccccc

vvsignbbbb

uusignbbbbc

vvsignbbbb

uusignbbbbc

csignaaaac

−+−+

+−+−

+−+−−

+−+−

+−+−+

−+−+

−++=

Δ−Δ+Δ+Δ+

+Δ−Δ−Δ+Δ=

Δ−Δ−Δ+Δ+

+Δ−Δ+Δ+Δ=

Δ−Δ−Δ+Δ=

ω

ρ

ω

( ) ( ) ( )[ ]

( )( )( )

( ) },)]()[(

)](){[(5,0

,})]()[(

)](){[(5,0

,5,0

2

2

1

RRoddoddoddodd

RRevenevenevenevenA

RRoddoddoddodd

RRevenevenevenevenA

Aoddoddoddodd

A

uusignbbbb

vvsignbbbbd

uusignbbbb

vvsignbbbbd

dsignaaaad

+−+−

+−+−−

+−+−

+−+−+

−+−+

Δ−Δ−Δ+Δ+

+Δ−Δ−Δ+Δ=

Δ−Δ+Δ+Δ+

+Δ−Δ+Δ+Δ=

Δ−Δ−Δ+Δ=

ρ

( ) ( ) ( )[ ]( ) ( ) ( )[ ]

( )( ) ( )

( )( ) ( )

( ) ( ) ( )[ ],5,0

,/})]([

)](){[(5,0

,/})]([

)](){[(5,0

,5,0

,5,0

22222

22

2

22

2

1

22222

BBBBBB

RRoddoddoddodd

RRevenevenevenevenB

RRoddoddoddodd

RRevenevenevenevenB

Beveneveneveneven

B

AAAAAA

csignccccc

vvsignaaaa

uusignaaaac

vvsignaaaa

uusignaaaac

csignbbbbc

dsignddddd

−+−+

+−+−

+−+−−

+−+−

+−+−+

−+−+

−+−+

−−+=

Δ−Δ−Δ+Δ+

+Δ−Δ−Δ+Δ=

Δ−Δ+Δ+Δ+

+Δ−Δ+Δ+Δ=

Δ−Δ+Δ+Δ=

−++=

ρω

ρω

( ) ( ) ( )[ ],5,01B

oddoddoddoddB dsignbbbbd −+−+ Δ−Δ+Δ+Δ=

( )

( ) ,/})]()[(

)](){[(5,02

2

ρRRoddoddoddodd

RRevenevenevenevenB

uusignaaaa

vvsignaaaad+−+−

+−+−+

Δ−Δ+Δ+Δ+

+Δ−Δ−Δ+Δ=

( )( )

( ) ( ) ( )[ ]BBBBBB

RRoddoddoddodd

RRevenevenevenevenB

dsignddddd

uusignaaaa

vvsignaaaad

−+−+

+−+−

+−+−−

−−+=

Δ−Δ−Δ+Δ+

+Δ−Δ+Δ+Δ=

22222

2

2

5,0

,/})]()[(

)](){[(5,0

ρ

BB

B

LBB

B

LAA

A

LAA

A

L dddv

cccu

dddv

cccu

21

2

21

2

21

1

21

1 ,,,+

=+

=+

=+

=

Construct plot

( ) ∞<≤−−+++

+−−++=

ω

ω

0,)()()(

)()()()(212121

212121

LLLLLL

LLLLLL

vvsignvvvvj

uusignuuuuL (19)

Theorem 2. Necessary and sufficient conditions for robust stability of the interval system (13),(14),(16) are as follows:

(a) Nominal system is stable, (b) The plot )(ωL does not intersect the γ -square

(i.e the square with vertices ),( γγ ±± . Proof.

The closed-loop system characteristic polynomial corresponds to the set of polynomials with interval parameters

),()(),()(),,( asAsCbsBsDabsH += . (20) Following the Zero Exclusion Principle the entire

set is stable if and only if a) nominal polynomial is stable and b) ),,(0 abjH ω∉ . But ),,( abjH ω is the sum of two rectangles: ),()(1 bjBjD ωω=Π and

),()(2 ajAjC ωω=Π . So the condition ),,(0 abjH ω∉ means that the rectangles 12 , Π−Π do not intersect. The condition b) in theorem is implied from the simple note: two rectangles do not intersect iff all vertices of each rectangle do not belong to another one [7].

Remark. The main advantage of the two above robust stability criteriums is that by constructing plot

)(ωz or )(ωL it is not only possible to verify the system stability with given value 0≥γ , but also find maximal value maxγ , where maxγ - dimension of the bigest square touched them. The system stability remains preserved for all maxγγ < , in other word, maxγ is system stability margin. 3. Stability of systems with fuzzy coefficients

117117

Consider fuzzy system described by fuzzy

characteristic polynomial

,...)( 0~

11

~~~asasasp n

nn

n +++= −− (21)

or closed-loop system with fixed controller (13) and fuzzy plant

nmsasaa

sbsbb

asA

bsBbasPn

n

mm <

+++

+++== ,...

...

),(

),(),,( ~

1

~

0

~

~1

~~

~

~~~~

0 , (22)

where the coefficients are fuzzy numbers with triangular or trapezoidal membership functions. The α -cut represents in a compact way the family of characteristic polynomials

)(...)()()( 001

11 αααα asasasp nnn

nnn

def+++= −

−− , (23)

[ ] niaaa iiiiii ...,,0,)(),()( =∈ +− ααα , or transfer functions

)(...)()()(...)(

001

11

00ana

nnna

nn

bmbmm

asasabsbP

αααααα

+++++= −

−−

, (24)

[ ] niaaa aii

aii

aii ...,,0,)(),()( =∈ +− ααα ,

[ ] mibbb bii

bii

bii ...,,0,)(),()( =∈ +− ααα ,

parameterized by the confidence degree in the model. Note that their coefficient are nonsymmetrical

functional intervals with centers ioi aa

~ker)1( ∈ or

ioi bb

~ker)1( ∈ . So the desired theorems give suitable

tools for system stability analysis. Two main problems can be raised in system

stability analysis here: firstly, given different confidence levels iα (or b

iai αα , ), the answer of

system stability can be obtained by examining plot )(ωz or )(ωL with 1=γ ; furthmore, in the case of

common confidence level α and triangular membership functions, using those graphical criteriums we can find stability margin minα (the system stability remain preserved for all minαα > ), where

maxmin 1 γα −= . (25) It is not so simple for the case of trapezoidal

membership functions. Here for fuzzy number iq~

we must define

.)0(),0(

,1)1()0(

)0()1()0(

)0(

,)0()0()1()1(

)0()1()0()1(

00

00

0

iiiiii

ii

ii

ii

iii

iiii

iiiii

qqqqqq

qqqq

qqqqm

qqqqqqqqq

−=Δ−=Δ

≥−

−=−

−=

−+−−=

++−−

++

+

−−

−

+−−+

+−−+

(26)

Then we can apply the above two theorems to compute maxγ . Besides this case is different with the above triangular case: each coefficient has own stability margin

),1( maxmin γα −= ii m (27)

The maximal value of them can be considered as the overall system stability margin i

minmin max αα = . 4. Conclusion

We have considered above the problem of fuzzy system stability that can be solved effectively by the robust stability approach. The proposed methods are based on extended robust stability criteriums for the case of nonsymmetric interval and α -cuts concepts. They can be used for linear continuous dynamical system with coefficients being described by triangular or trapezoidal fuzzy functions. 5. References [1] Bondia J., Pico J. “Analysis of systems with variable parametric uncertainty using fuzzy functions”, Proc. European control conference ECC’99, Karlsruhe, Germany, 1999. [2] Bondia J., Pico J. Analysis of linear systems with fuzzy parametric uncertainty, Fuzzy Sets and Systems. 2003. V.135. p. 81-121. [3] Argoun M.B. “Frequency domain conditions for the stability of perturbed polynomials”, IEEE Trans. Automat. Control. 1987. AC-32. V.10. p.913-916. [4] Husek P., Pytelkova R., “Analysis of systems with parametric uncertainty described by fuzzy functions”, Proc. 10th Mediterranean Conference of Control and Automation – MED2002. Lisbon. Portugal, July 9-12, 2002. [5] Tsypkin Ya.Z., Polyak B.T, Frequency domain criteria for lp-robust stability of continuous linear systems, IEEE Trans. Aut. Control, 1991. N.1. 1464-1469. [6] Le Hung Lan, “Analiz robastnoj ustoichivosti system s nechetkimi parametrami. Avtomatika I Telemechanhika.” N.4, 98-109. /in Russian/ = “Robust stability of fuzzy-parameter systems. Automation and Remote Control,” 2005, N.4, 645-661 /in English/. [7] Le Hung Lan, “Modifisirovanui chastotnui criterii robastnoi ustoichivosti zamknutux system. Avtomatika I Telemexhanika”. N.8.119-130. 1993. /in Russian/

118118