Observer Based H infinity Controller for Nonlinear Singular Networked Control Systems

) 8 1 0 2 E C E I( g n ir e e n i g n E n o it a c i n u m m o C d n a c i n o rt c e l E , n o it a m r o f n I n o e c n e r e f n o C l a n o it a n r e t n I 8 1 0 2 8 7 9 : N B S

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, y g o l o n h c e T n o it a m r o f n I f o e g e ll o C u s g n a i J , g n ir e e n i g n E l a c ir t c e l E d n a l a c i n a h c e M f o l o o h c S a n i h C , 3 5 1 4 1 2 i x u W : s d r o w y e

K Observer, H-inifntiyconrtoller, Netwo -rk induceddelay, Nonilnea rsingula rsystem.

.t c a r t s b

A Basedonobserve,rf o raclas so fnonilnea rconitnuou ssingula rnetworkedconrtols ystems , e m it e h t g n ir e d i s n o

c -varying network-induced delay ,a wn e conrtolle rand observe rc -o designing e u q i n h c e

t a redevelopedwtihH-in ifntiyperformancenorm ,asufifcien tcondiitonf ort heexistenceo f e

h

t invesitgated conrtolle ri sdeirved ,and a systemaitc design approach is addressed by using v o n u p a y

L -Krasovskii funciton and ilnea rmatirx inequailites (LMIs )fo rthe conrto lle rdesigning . , y ll a n i

F anexamplei sprovidedt odemonsrtatethevaildtiyoft heproposedmethod.

n o it c u d o r t n I e h

T singula rsystemha saprofoundbackgroundi nengineeirngpracitce ,whicharealsor eferredt o d e z il a r e n e g r o s m e t s y s r o t p ir c s e d s

a state-spaces ystem ,st heyareoft heoreitcali nteres tandalsohave y

n a

m appilcaitonsini ndusrty,s ucha selectirca lnetworkanalysis ,powers ystem ,seconomics ystems , e t a g it s e v n i o t e d a m e r e w s tr o f f e t a e r G . n o o s d n a , l o rt n o c s s e c o r p l a c i m e h

c singular ys stemt heory

t s a p e h t g n ir u d s n o it a c il p p a d n

a 30 years ,see [1 6~ ]and the reference stherein. Stabiilzaiton o r

y l e d i w n e e b e v a h n g i s e d r e v r e s b

o used i n con rto landsigna lprocessingi nt hel astf ewdecades( see [7 1~ 2 )] R . ecenlty ,many approache shave been developed to design esitmators ro observer sfo r

r a l u g n i

s systems,s ee [13~ 81 ]andt her eferencest herein. I 1n 5] [ and[ 16],l oca lasymptoitcobserver s e

r e

w obtainedf o rgenera lnonilnea rdescirptors ystems ,thef -u llordera ndr educed-orde robserversf o r p

i

L schtizdescirpto rsystem swa spresentedi n[ 17]. Thedesignprocedurei ssimple ,bu t tii susually t n e d n e p e d s i h c i h w n o it i d n o c t n e i c if f u s e h t y f s it a s o t d r a

h ont hechoiceo fcoordinate rtansformaiton

. x ir t a m t a h t n w o n k l l e w s i

tI inanNCS,t hemosts igniifcan tfeatureist henetworki nduceddelays ,which d e s u a c y ll a u s u e r

a by ilmtiedb tisr ateoft hecommunicaiton channels ,orbysigna lprocessingand n o it a g a p o r

p ,these generally b irng snegaitve effect son NCS stablitiy and performance .Atlhough r

e h

t eweres omei mpo trantr esutl sachievedf ors ingula rnetworkeds ystems[ 1 ] t 8 , he observaitonand g n il l o rt n o

c fu trherenhancest hei mpo tranceoft hes tudyonnon ilnears ingulars ystems hw ti network n o i s s i m s n a

rt it -medelay . tIi snecessary to develop anovelobserve randconrtolle rco-designingfo r r a e n il n o

n singularnetworkedsystem .s : s w o ll o f s a d e z i n a g r o s i r e p a p s i h

T Frislty, weinrtoducet heproblem to bestudied, then a new

o c r e v r e s b o d n a r e ll o rt n o

c -designingt echniqueha sbeenpresented wtiht heLMIst echnique .Laslty , .s tl u s e r d e s o p o r p r u o f o s s e n e v it c e f f e e h t e t a rt s n o m e d o t n e v i g s i e l p m a x e n a n o it a t o

N :TheRn _{na dR}n*m_{denotet hen-dimensiona lEucildean spaceandt hese to fal ln×m r ea l} ; y l e v it c e p s e r , s e c ir t a

m I i sthe identtiy matirx wtih approp irate dimension ;The superscirpt T ; x ir t a m a f o e s o p s n a rt e h t s t n e s e r p e

r X referst oEucildeannormoft hevectorX .ThenotaitonP> 0

t a h t s n a e

m Pisr eals ymmetircandposiitvedeifntie.I ns ymmetircblockmat irces ,weuseanasteirsk * to represen ta term tha ti sinduced by symmerty, diag{...}stand sfo ra block-diagona lmatirx ,

d n

a He(M)stand sf orM+M.

0

,) ( ) , , ( ) ( ) ( ) (

) 0 ( ,) (

t D u x t f t u B t x A t x E

x x t x C y

ω + +

+ = 

 _{= =}



 _{)( 1}

e r e h

w _x(_t)_∈Rnis t hestatevector ,_y∈Rrist hemeasureoutpu,t _u(t)∈Rmi ssystem i nput,_A ,_E∈Rn×n , m

n

B_∈R × ,_C∈Rr×n dan _D∈Rr×mareconstantmat irces, _ω(k)∈Rqist hedisturbancei npu twhichbelong s

o

t L2[0,∞) , x(0)=x0 i s the compaitble iniita l condiiton , assume tha t 0<rank(E)=s<n .

) , ,

(_{t x u} p

f

f = ∈R i s a nonilnea r vector-valued funciton wtih itme-varying , f(0,0,u)=0 , to any )

, ,

(t xu a nd (_t,_x,_u)_∈R×Rn_×Rm _{,whichs aitsifest hef o llowinggloba lLipsch tizcondiiton :}

) ( ) , ,

(t xu Fxt

f ≤ , f(t,x,u)−f(t,x,u) ≤ F(x(t)−x(t)) , ( 2)

e r e h

w _F∈Rn×p i saknownconstan tmatirx..

Int hi spaper ,tii sassumedt hatt hemeasureddatai srtansmittedt oobservert hroughnetwork. The d

e r e d i s n o c e h t f o e r u t c u rt

s SNCSi ss howni nFigure1

. 1 e r u g i

F Thes rtuctureo f SN CS

e h

T followingassumpitonsf ort he SN eCSa r neededthroughout ht si paper:

1 n o it p m u s s

A eT h it -med irven sensor ssample the system output y(t) wtih samp ilng Tpeirod, n

e h

t thes ampleddata y(kT) i spackeitzed and s entt ot heobserve ratt he kT (k∈N )instan tovert he

. k r o w t e n

2 n o it p m u s s

A T heevent-d irvenobserverr eceivest hemeasureddatapacket satt he kT+dk instant ,

n e h

t complete s tis aciton ,dk i sthe it -mevarying network- induced delay a t itme s tep k. Every

e h t d n a d e p m a t s e m it s i a t a d d e l p m a

s observe ralway susest hemostr ecen tdatapacke.t

o c y l n o e w , t u o p o r d a t a d k r o w t e n t u o h ti w r e d n u , r e p a p s i h t n

I nside rtha tthe network-induced

.l e n n a h c r e v r e s b o o t s r o s n e s e h t n i s t s i x e s y a l e d

( s n o it p m u s s a e h t r e d n u , ) 1 ( m e t s y s e h t g n ir e d i s n o

C A1 )and( A2 ,)l et

) ( ) (t Kx t

u = , )( 3

e r e h

w _x(_t)_∈Rn _{i san esitmaiton o fthe system state vecto rx(t) , a f -ull orde rnonilnea robserve r}

o

c nsrtuctedast hef ollowingf orm:

[

]

[

1

]

,) ( ) ( ) ( ) , , ( ) ( ) ( ) (

) 1 (

, k

k

t D T k x C T k y L u x t f t x K B A t x E

d T k d T k t

ω

+

 = + + + − +

 

+ + + ∈ ∀ 



( 4)

e r e h

w _K∈Rm×n _,L∈Rn×r _{ea r theconrtolle randobserver ng s ai tobedesigned,r especitvely.}

Deifne d(t)=t−kT ,∀t∈

[

kT+dk,(k+1)T+dk+₁

]

,then( 4 )canbew irttenas

) , , ( ) ( ) ( )

(t A BK x t f t x u x

E = + + +L

[

y(t−d(t))−Cx(t−d(t))

]

+Dω(t) , )( 5

h ti w

) (

0≤d t ≤dM ,0<d(t)<µ <1 . ( 6)

a dn dM ist heuppe rbounddelay.

t e

L e(t)=x(t)−x(t) ,from( 1)t o( 4) ,wehaveerro rdynamics ystem

[

( ()) ( ())

]

)

( )

(t Aet L Cxt d t Cx t d t e

E = − − − − +[f(t,x,u)−f(t,x,u)]+Dω(t) , ( 7)

r o t c e v d e t n e m g u a e n if e

D _ξ(_t)_=xT(_t) _eT(_t)_T



) ( ) , , ( ) ) ( ( ) ( )

(t A t Ad t d t f t xu D t

Eξ = ξ + ξ − + + ω , ( 8)

e r e h w

0 0

E E=_{ E}_



 , 0

K B K B A

A=_ + −_A _ 

 ,

0 0

F F=_{ F}_



 ,

0 0 0 d

A =_{ LC}_ 

 ,

D D=_D_



 ,

) , , (

) , , ( ) , , (

u x t f

f =_{f t xu f t x u} _

− _

 .

s tl u s e R n i a M

o f ll a h s e w , n o it c e s s i h t n

I cu sont hes tudyo fr obust conrto lle rbasedonobserve rco-designing f o r .

) 8 ( m e t s y s e h

t Before presenitng the main resutls ,we fris tinrtoduce deifniiton and lemma sa s ,

s w o ll o

f whichwli lbeessenitalf o rou rdeirvaito . n

1 n o it i n if e

D [5]

T )

1 hesingular system (8) iss aidt oberegularandi mpulsef ree ,t ha ti ,s thepai r(E,A) isr egular

. e e r f e s l u p m i d n

a T hesingula rsystem (8) si admissible, t hat i s ,when f ≡0 ,system (8)i sregular ,

d n a e e r f e s l u p m

i masy pto itcallystable.

2 )Fo rgivens calar sdMand γ>0,t hes ystem (8) wtih u(t)=0iss aidt ober obusltyadmissiblewtih

H∞performanceγ,s ucht hat z(t)₂≤γ ω(t)₂ fo ranynonzerodisturbanceω(t)∈L2[0,∞). a

m m e

L 1 .Fo rgivenf ul lcolumnmatirx _B∈Rn×m _{thes ingula rvaluedecomposing}

0 VT

U B= _∆_



 ,and

y l n

o X1∈Rn×n, 1 11

2 2

0

0 T

X

U U

X = _{ X} _ 

 ,then there exisitng matirx

m n

X_∈R× ,

1B BX

X = ,where_U∈Rn×n,

m m

V_∈R × are o trhodox matirces , the nonzero singula r value o f _B i s

{

}

1,..,. m

g a i

d σ σ

=

∆ , and

) ( ) ( 2

2 n m n m

X _∈R − × − _,

1

1 mm

X _∈R × _.

.r e p a p s i h t f o s tl u s e r n i a m e h t e d i v o r p w o n e W

m e r o e h

T 1 .Fo rprescirbeds calarγ >0andd_M ,>0 fit hereexis tposiitvedeifn tiematirces P ,Q,R

d n

a _S∈R2n×2n _{,and scalarsλ >0,} 1 0

λ > ,λ >2 0 saitsfying t hefollowing matirce sinequaltiy then there

t s i x

e s aconrto ller( 3 )basedonanonilnea robserverfors ystem(8) ,and thesystem i sadmissiblewtih H-in ifntiyperformance.

0 T T_{P P E}

E = ≥ _,

4 1 3 1 2 1 1 1

4 2 2

2

3 3

4 4

2 0 0

0 0 0

0

D P

I

γ 

Ω Ω Ω Ω



 _{∗ Ω Ω}

 



 <

=

Ω ∗ ∗ Ω

 

Ω ∗ ∗

∗ _





 _{∗ ∗ ∗ ∗ − }



, 9( )

e r e h

w : 1

1

1 =Q−He(XA)−dM−ETSE−λXFFTXT

Ω , ₁₂ T T

d

A X Y A − − =

Ω , 1

3

1 =dM−ETSE

Ω , Ω14=PT+X−ATZT ,

1 2

2 =−He(YAd)−λYFFTYT

Ω , ₂₄ T T

dZ

A Y− =

Ω , 1

3

3 =−Q−dM− ETSE

Ω , ₄₄ ( ) ₂ T T

MR He Z ZFF Z

d + −λ −

=

Ω .

: f o o r

P Deifning the fo llowing Lyapunov-Krasovskii funciton fo rsystem (8 )wtih ω(t)=0 sa

s w o ll o

f :

3 2

1() () ()

)

(t V t V t V t

V = + + , (10)

r o

F simpilctiy , let ξ =t: ξ(t) , ξt−d(t):=ξ(t−d(t)) , f := f(t,x,u) , wtih V1(t)=ξtTETPξt ,

2() ₍₎ d

t _T t t t d

t Q t

t

V ξ ξ

−

=

∫

,

0

3() ( ( )) ( ( ))d d

M

t _T

t

d E S E

t

V _β ξ α ξ α α β

+ −

=

∫

∫

  , thenalongt he rtajectoryofs ystem( 8) ,tii sdeirvedt ha t

1(t) tTPT[E t] [E t]TP t

V =ξ ξ + ξ ξ T T[ ] [ ]T T T T t t t

t t

t P Eξ Eξ Pξ f f ξ F Fξ

ξ + − +

≤   , (11)



V₂(t)=ξ_tT_Qξ

t−(1−µ)ξt−d(t)

T _Qξ

3() ( ) ( ) ( ) ( )d M

t _T

T

t t t

t

M E S E _{t d} E S E

d t

V ξ ξ ξ ξ α

−

−

=  

∫

 

 _{, (13)}

r o

F V3(t) ,therehas

3(t) dM(E t)TS(E t)

V ≤ ξ ξ () ₁1 1₁ ()

) ( )

(

T _{T T}

M M

T T

M

M M M

t t d E SE d E SE

d t d

t d E SE d E SE

ξ ξ

ξ ξ

− −

− −



−  

 

−_{    }

−

− _ − _{ }

   , (14)

n i p u o r g e r ,) 4 1 ( ~ ) 1 1 ( g n i n i b m o

C gt het erm soft heml eadst o

3 2

1() () ()

)

(t V t V t V t

V ≤  + +  2 T T ₍₎ ( )T t t d t

t X ξ Y Eξ Z f

ξ − 

 + + +

+ _  _ ×Eξ_t−Aξ_t−A_dξ_t−_d(t)−f ≤ξ ΩtT ξt , (15)

e r e h w

)

( M ( )

T T T

T T T

t d t t d t t

t ξ ξ ξ Eξ

ξ = − −   ,

4 1 3 1 2 1 1 1

4 2 2

2 3 3

4 4

0

0 0 Ω Ω Ω

Ω 





 _{∗ Ω Ω}



 _<

= Ω



 ∗ ∗ Ω



 _{∗ ∗ ∗ Ω}

 

, da n X,Y,Zaref -reeweighitngmat irce s .

L te λmin(−Ω)=λ0 then λ >0 0 ,BySchu rcomplement( 9b) ,whichcanbeobtain Ω<0 ,therefore

2

0 0

)

( T

t t

t

t

V ≤ξ Ωξ ≤−λ ξ < _{. (16)}

When ω(t)≠0 ,le tu snowprovet hatt hes ystemha sH-in ifntiyperformance γ , sa )

(t V ≤ T

t t ξ

ξ Ω +_ξT(_t)_PT_Dω(_t)_+ωT(_t)_DT_Pξ(_t) , da n

2 _{() ()} )

( )

( T T T

T

t t

t t t

x C C t

x −γ ω ω +ξΩξ _+ξT(_t)_PT_Dω(_t)_+ωT(_t)_DT_Pξ(_t)= T ₀ t t ς

ς Ω < (17)

w eh er () ( ) ()

T T T

T T

t d t t d t t

t ξ ξ ξ M Eξ ω t

ς = − −  .

Thisi mpiles y(t)₂≤γ ω(t)₂,t hi scompletest heproof of Theorem1.

x e n e h t n

I tpatr ,basedonTheorem1,a s ufifcientc ondiitonf ort hee xistenceo f a sutiableconrtolle r r

e v r e s b o d n

a si presentedasf o llow .s

2 m e r o e h

T .Fo rprescirbed scalardM >0a nd γ >0 ,and scalars λ>0 ,λ >1 0 ,λ >2 0, fi t here exis t

s e c ir t a m e v it i s o

p 1 2

3

P P P

P

 

=_{ ∗ }



 ,

2 1

3

Q Q Q

Q

 

=_{ ∗ }



 ,

2 1

3

R R R

R

 

=_{ ∗ }



 ,

2 1

3

S S S

S

 

=_{ ∗ }



 , da n X1,X2,K,L , such tha t

e h

t inequailites(17)aref easible

0 T T_{P P E}

E = ≥ ,

6 1 5 1 4 1 3 1 2 1 1 1

5 2 4 2 2

2

3 3

5 4 4 4

5 5

2 0 0

0 0

0 ₀

0 0

I

γ 

Ω Ω Ω Ω Ω Ω



 _{∗ Ω Ω Ω}

 



 ∗ ∗ Ω

< =

Ω  

Ω Ω ∗ ∗

∗ _





 _{∗ ∗ ∗ ∗ Ω}

 

− ∗ ∗ ∗ ∗

∗ 

 _



 _{, ( ) 18}

e r e h w

1 1

1 1 1

1

2

0

T M

K B X K B X A

X _{d E SE}

e H Q

A X

−



 + − ₋

− =

Ω _{ }



 ,

1 1

1 1 2 1

2

0 0 0

0 T T

K B X K B X A X

L C A

X

λ  + − − 

=

Ω    ₋ 

 



 ,Ω22=−He(YAd) Ω13=Ω13 , 16

D D

  =

Ω _{ }



 ,

1 5 1

2 0 0

F X

F X

 

− =

Ω _{ }



 ,

1 1

1 2 4

1

2 2

0 0

0 T

T T T T T

T T T

T

X X

B K X A P

X X

A B

K

λ  + + 

− =

Ω _{   }

−   

 ,Ω33=Ω33,

1 1 2

4 2

2 0 0

0

0

0 T T

X X L

C λ

λ  +  

− =

Ω _{   }

− _

   ,

1 1 5 2

2 0 0

F X

F X

λ 

−

Ω _{ }



 ,

1 1 2 4

4

2 2

0 0

T

M T

X X R d

X X

λ  + 

+ − =

Ω _{ }

+ _

1 1 2 4

4

2 2

0 0

T

M T

X X R d

X X

λ  + 

+ − =

Ω _{ }

+ _

 ,

1 2 5 4

2 0 0

F X

F X

λ  

−

Ω _{ }



 , 55

0 0

I I

λ 

− =

Ω _{ }

 

h ti w n e s o h c e b n a c h c i h w , ) 4 ( r e v r e s b o d n a ) 3 ( r e ll o rt n o c d e ri s e d e v a h ) 8 ( s S C N S e h t n e h T

s a s r e t e m a r a

p 1

1 1 VTK

X V

K_{= ∆}− _{∆ ,} 1 2 L

X L₌ − _.

: f o o r

P By Schurcomplemen,ti nequaltiy( 9)i sequivalentt o( 19)

4 1 3 1 2 1 1 1

4 2 2

2 3 3

4 4

5 5

2

0 0

0 0

0 ₀

0 0

D F X

F Y

F Z

I

γ 

Ω Ω Ω Ω −



 _{∗ Ω Ω −}

 



 _{∗ ∗ Ω}

< =

Ω  

− Ω ∗ ∗

∗ _





 _{∗ ∗ ∗ ∗ Ω}

 

− ∗ ∗ ∗ ∗

∗ 

 _



   

 

 

 

, (19)

e r e h w

1 1

1 =Q−He(XA)−dM− ETSE

Ω , Ω₁₂ =Ω₁₂ , Ω₁₃=Ω₁₃ , Ω₁₄=Ω₁₄ , Ω₂₂=Ω₂₂ , Ω₂₄=Ω₂₄ , Ω₃₃=Ω₃₃ ,

2 2 5

5 =−λIn× n

Ω ,

4

4 =−dMR+He(Z)

Ω .

e n if e

D 1

2 0 0

X X

X

 

=_{ }



 ,andl et

1 1

2 0 0

X Y

X

λ  

= _{ }



 ,

1 2

2 0 0

X Z

X

λ  

= _{ }



 ,accordingt olemma1 ,t hereexist s

Rsaitsfying X1B=BR ,and l et L=X2L ,K=RK ,thensubsttiuitng X ,Y ,Z ,K ,L into( 19 ,) again

g n i y l p p

a Schurcomplemen tonehas(18b) ,a s

1 1

1 1

2 2 0

, ,

,

0 0

T

T X U X U XB BRRK K

V U B

X

∆  

 _{= = =}

= _{   }



   , (20)

s t e g e n o , ) 0 2 ( m o r

f 1

1 1

T_K

V X V

K_{= ∆}− _∆ , 1

2 L

X

L= − . Thi scompletest heproo .f

e l p m a x E

f o s s a l c a r e d i s n o

C nonilnear singulars ystems( 1 ,)t hemainparameter sborrowedf rom Example2 1

[ e c n e r e f e r f

o 7],

0 1 0

1 0 0

0 0 0

E

 

 

=_{ }



 _



, 10 01 12

1 2 1

A

− 





 ₋

=_{ }



 _



, 21 , 00

1 2

D B

   

  

_{− =}

=_{   }

  

 _{  }



,

( )

3 0 1

0 1

0 )

, , ( , 1 1 2

n i s 8 5 3 3 . 0

u x t f C

x

 



 ₌ 

=_{   }



 _{ }

 

,

e t a t s l a it i n i m e t s y s e h

t _xT(_t)₌[0,1,1]T ,the observe riniita lstate is _xT_(t)=[0,0,0]T, assuming the

d o ir e p g n il p m a s m e t s y

s T=0.01s ,dM=0.02s ,and t he it -mevaryingnetwork-induced delayi sr andom n

i d(t)∈

[

0,0.02

]

, tle γ=0.5 ,λ₁=0.,1λ₂=0.2 ,usingMaltabLM Iconrto lToo lboxt os olvet heLMIsi n

: s a s r e t e m a r a p h ti w r e ll o rt n o c d n a r e v r e s b o e h t e v a h e w , ) 7 1 (

0 0 0

0 1 , 2 8 5 7 . 0

5 9 8 . 0 0 3

0 0 . 2

T

L K

 

 

 



 ₌

=_{   }

 



 _{  }



.

n o i s u l c n o C

r e v r e s b o d n a r e ll o rt n o c w e n a , k r o w t e n n o d e s a b , e c n a b r u t s i d r a e n il n o n h ti w s m e t s y s r a l u g n i s r o F

o

c -designingt echniquehavebeenpresentedbyconsrtucitnganaugmentedplan twtiht hedisturbance k

r o w t e n d n

d e n i a t b o e b n a c s m e t s y s d e k r o w t e n r a l u g n i

s simutlaneouslywtiht heLMIst echnique, t heproposed

l a s u a c s i r e v r e s b

o ,andt hes ystemi sr obusltys tablewtihH-in ifntiyperformancenorm .

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