1 (Introduction)

[1][2][3]

[4][5][6][7]

Lyapunov [8-13]

(60274009)(20020145007)

[10]Lyapunov [11]

[12]

[13]Lyapunov

Lyapunov

Lyapunov

Brunowsky

Lyapunov LyapunovLyapunov

2 (Problem Formulation

1 1 1

1. , 110004 E-mail: mengbo_422@126.com

: LyapunovLyapunov Lyapunov

ε

: , , , , ,

State Feedback Controller Design for a Kind of Nonlinear Singularly Perturbed System

MENG Bo1, JING Yuanwei1, Chao Shen1

1. College of Information Science and Engineering, Northeastern University, Shenyang 110004 E-mail: mengbo_422@126.com

Abstract: The feedback stabilization of MIMO nonlinear singularly perturbed systems is considered. A Lyapunov function for the overall system is established through the Lyapunov function of the linear part and that of the zero dynamic and boundary layer. The upper bound expression of ε is given to obtain the condition of asymptotically stability for the system. The simulation results show the effectiveness and feasibility of the controller.

Key Words: Nonlinear System, MIMO, Singularly Perturbed, Boundary Layer, Two-time-scale, Stabilization

5100

978-1-4244-1734-6/08/$25.00 c© 2008 IEEE

1 1 1 1

2 2 2 2

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

( )

x F x z u f x Q x z g x uz F x z u f x Q x z g x uy h xε

= = + += = + +

= (1)

nxx B∈ ⊂ p

zz B∈ ⊂

xB zBmu∈

my∈ ε 1( , , )F x z u ,

2 ( , , )F x z u x zB B× 1(0,0,0)F =

2 (0,0,0) 0F = 1( )f x 2 ( )f x 1( )g x 2 ( )g x( )h x 1( )Q x 2 ( )Q x

()

(1)

2 ( )Q x xx B∈0ε = (1)

1 1 1( ) ( ) ( )sx f x Q x z g x u= + + (2)

2 2 20 ( ) ( ) ( )sf x Q x z g x u= + + (3) sz z 2 ( )Q x (3)

[ ] [ ]12 2 2( ) ( ) ( )sz Q x f x g x u−= − + (4)

(4) (2)( ) ( )

( )s

x F x G x uy h x

= +

= (5)

[ ][ ]

11 2 1 2

11 2 1 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

F x f x Q x Q x f x

G x g x Q x Q x g x

−

−

= −

= − (6)

(5)(1)

tτε

= (7)

τ (1)

[ ]1 1 1

2 2 2

d( ) ( ) ( )

dd

( ) ( ) ( )d

x f x Q x z g x u

z f x Q x z g x u

ετ

τ

= + +

= + + (8)

0ε =

2 2 2d

( ) ( ) ( )dz f x Q x z g x uτ

= + + (9)

0t = x(0)x (9)

1 [ ]2 2( ), ( )Q x g x( )K x 2 2( ) ( ) ( )Q x g x K x+ xx B∈

(Hurwitz uniformly in xx B∈ )

3 (Main Results)

3.1 (Design of Feedback Controller)

( )u u K x z= + (10)

( )K x 2 2( ) ( ) ( )Q x g x K x+( 1 )(10) (1)

1 1 1d

( ) ( ) ( ) ( )dx f x g x K x z g x ut

= + + (11)

[ ]2 2 2 2d

( ) ( ) ( ) ( ) ( )dz f x Q x g x K x z g x ut

ε = + + + (12)

[ ]2 2 2 2d

( ) ( ) ( ) ( ) ( )dz f x Q x g x K x z g x uτ

= + + + (13)

sy z z= −

[ ] [ ]12 2 2 2( ) ( ) ( ) ( ) ( )sz Q x g x K x f x g x u−= − + +

y

[ ]2 2d

( ) ( ) ( )dy Q x g x K x yτ

= + (14)

2 2( ) ( ) ( )Q x g x K x+sz

szsy

d( ) ( )

d( )s

x F x G x uty h x

= +

= (15)

[ ]1 1 1( ) ( ) ( ) ( ) ( )F x f x Q x g x K x= − +

[ ] 12 2 2( ) ( ) ( ) ( )Q x g x K x f x−+

[ ]1 1 1( ) ( ) ( ) ( ) ( )G x g x Q x g x K x= − +

[ ] 12 2 2( ) ( ) ( ) ( )Q x g x K x g x−+

2 (15) { }1, , mr r⋅ ⋅ ⋅ 1 2r r r= + +

mr n⋅ ⋅ ⋅ + < 1 i≤ , j m≤ 1ik r≤ −

xx B∈ ( ) 0kiFGL L h x = { }( )G span G x=

m m× 1( ) ( ( )) ( ( ))irij iFGA x a x L L h x−= =

xx B∈

2008 Chinese Control and Decision Conference (CCDC 2008) 5101

11

11 1

1 11

1

1

11

( )

( )

( )( )

( )( )

( )

m

m

rr F

mm

rmmr F

n rn r

h x

L h x

h xx

L h xx

x

ξ

ξ

ξ ξη

ξϕη

ϕη

−

−

−−

= = Φ =

(15) Brunowsky

[ ]( , )

( , ) ( , )s

QA B F G u

y Cx

η η ξξ ξ η ξ η ξ

=

= + +

=

{ }1

0 1 0

, , ,0 0 10 0 0

i i

m i

r r

A diag A A A

×

= =

{ } [ ]1 1, , , 0 0 1

im i r

B diag B B B Τ

×= =

{ } [ ]1 1, , , 1 0 0

im i r

C diag C C C×

= =

[ ] [ ]1( ) ( ) ( )u A x b x c x−= − +

1 111( ) ( ), ( )mrr

mF Fb x L h x L h xΤ−−=

[ ]1( ) ( ), , ( )mc x c x c x Τ=

11 1 1 1

11 1

( ), , ( )mrr

k kk k mF F

k kL h x L h xβ β

Τ− −

= =

=

ikβ ( )p s =

11 2 0i i

i

r ri i irs s sβ β β −+ + + =

( , )QA

η η ξξ ξ

=

=

{ }1

1 2

0 1 0

, , ,0 0 1

i i i

m i

i i ir r r

A diag A A A

β β β×

= =

− − −

3.2 (Stability Analysis)

3 (15) ( ,0)Qη η=0η =

( )p s 3 P( )L η P Lyapunov

A P PA IΤ + = −( )L η η

21( ,0)

L Q η α ηη

∂ ≤ −∂

2L α ηη

∂ ≤∂

1 2,α α( , ) ( ) ( )V L kNη ξ η ξ= +

Lyapunov ( )N Pξ ξ ξΤ=

max2N Pλ ξξ

∂ ≤ ⋅∂

max ( )Pλ PLyapunov

( )P x Lyapunov

[ ]2 2( ) ( ) ( ) ( )Q x g x K x P xΤ+

[ ]2 2( ) ( ) ( ) ( )P x Q x g x K x I+ + = −

( , ) ( )W x y y P x yΤ=

[ ] 22 2( ) ( ) ( )

W Q x g x K x y yy

∂ + ≤ −∂

2W b yx

∂ ≤∂

bLyapunov

( , ) ( ) ( ) ( , )v x y L kN W x yη ξ= + +k

[ ]2 2

( , )( , )

( ) ( ) ( )

QL N Wv x y kA

yQ x g x K x y

η ξξ

η ξ∂ ∂ ∂=∂ ∂ ∂

+

( ) ( )

( )

1 1

1

, , ( , ) , , ( , )

, , ( , )

s sV F x z u x z F x z u x zxW F x z u x zx

∂+ −∂∂+∂

5102 2008 Chinese Control and Decision Conference (CCDC 2008)

1 ( ) ( )1 1, , ( , ) , , ( , )s sV F x z u x z F x z u x zx

∂ −∂

(11) (15) ( )1 , , ( , )W F x z u x zx

∂∂

x

[ ]

( ) ( )

( )

[ ]

( ) ( )

( )

2

211 1

1

2 2 211

1 1

1

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

, , ( , ) , , ( , )

, , ( , )

( , ) ( ,0)

, , ( , ) , , ( , )

, , ( , )

s s

s s

L Lv x y Q Q Q k

Vy F x z u x z F x z u x zx

W F x z u x zx

k y

L Q Q

V F x z u x z F x z u x zxW F x z u x zx

η η ξ η ξη η

ε

α η ξ ε

η ξ ηη

−

−

∂ ∂= + − −∂ ∂

∂− + −∂

∂+∂

≤ − − −

∂+ −∂∂+ −∂∂+∂

1( , , )F x z u x zB B× ( , )Q η ξ

1(0,0,0) 0F = (0,0) 0Q =

1 2 3, , 0k k k > ( , ) x zx z B B∀ ∈ ×

( ) ( )1 1, , ( , ) , , ( , )s sF x z u x z F x z u x z−

1 1sk z z k y≤ − =

2( , ) ( ,0)Q Q kη ξ η ξ− ≤

( )1 3, , ( , )F x z u x z k≤

2 2 211 2 2

22 1 max 1 3

( , )

2 ( )

v x y k y k

k y P k y bk y

y y

α η ξ ε α η ξ

α η λ ξ

η ξ η ξ

−

Τ

≤ − − − +

+ + +

= − Λ

1 1 2

1 31

2 3 4

kα λ λλ λλ λ ε λ−

− −Λ = − −

− − −

2 2 2 11 2 3 max 1 4 3, , ( ) ,

2 2k k P k bkα αλ λ λ λ λ= = = =

21 1

2 21 4 2 1 1 4 2

,( )

kα λεα λ λ α ε α λ λ∗ ∗= =

+ − +

0 ε ε∗< < k 0 k k∗< <Λ (1)

4 (Simulations)

1 1 3 2 1 1 2

23 3 1 3 2

4 5 1 4

1 2 1

1 1 4 2 2 1 5

, 2

(2 ),

2( ) , ( )

x x z x x x u ux x u x ux z x x xz x x z uy h x x y h x x xε

= + = + +

= − + + += − = − −= − + += = = = −

{ } { }5 1 , 1x zB x R x B z R z= ∈ ≤ = ∈ ≤

1 2u z= −ddy yτ

= −

1 21 1

Aξ ξη ξ ξ η

== − − −

[ ]1 2A diag A A=

1 2

0 1 0 1,

2 1 3 1A A= =

− − − −

2 21 3

2 1( )

2 1 3A x

x x− −

=− − − −

1 22 31 2 1 1 4 3

2( )

x xb x

x x x x x x+

=− + + +

4 1 22 3

1 5 1 2 1 1 4 3

4 2( )

3( )x x x

c xx x x x x x x x

+ −=

− + − + + +

3 2 2 21 1 2 1 2 2 3 3 4

3 22 1 1 2 1 2 4 5

6 9 6 3 2

4 8 3 4 2

u x x x x x x x x zu x x x x x x x

= + − + − −

= − + − + + −

1 2 2

1 3

1 2 3 4

1,4, 64.396, 0.10.5, 2, 2.4072, 6.44

0.250.0958,

1 (6.44 4)

kk k b

k

α α

λ λ λ λ

εε∗ ∗

= = == = == = = =

= =− +

1 [0 1 0.5 0.5 1]x= − −1z = 0.01ε =

2z

/tτ ε=

2008 Chinese Control and Decision Conference (CCDC 2008) 5103

5 (Conclusions)

Lapunovε

0 2 4 6 8 10 12 14-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x

t/s

0 2 4 6 8 10 12 14-0.4

-0.2

0

0.2

0.4

0.6

0.8

1z

t/s

1:

-1-0.5

00.5

1

-1-0.5

0

0.51-1

-0.5

0

0.5

1

x1x2

z

2 1 2, ,x x z( 1 22z x x= − )

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[7] , , , , .H2 , , Vol.22, No.1,

86-91, 2005. [8] Panagiotis D Christofides. Output feedback control of

Nonlinear two-time-scale systems. Proc. of the IEEE ACC, Albuquerque, New Mexico, 1729-1733, 1997.

[9] Panagiotis D Christofides. Robust output feedback control of nonlinear singularly perturbed systems, Automatica, vol.36, 45-52, 2000.

[10] Saberi A, Khalil H. Quadratic-type Lyapunov functions for singularly perturbed systems, IEEE Trans on Automatic Control, Vol.29, No.6, 542-550, 1984.

[11] C. C. chen. Global exponential stabilization for nonlinear singularly perturbed systems, IEE Proc-control Theory Appl, Vol.145, No.4, 377-382, 1999.

[12] Yong S, Jong T. Control of Nonlinear Singularly Perturbed Systems Using Gain Scheduling, IEICE Trans Fundamentals, Vol.E85-A, No.9, 2175-2179, 2002.

[13] J. W. Son, J. T. Lim. Robust stability of nonlinear singularly perturbed system with uncertainties, IEE Proc-control Theory Appl, Vol.153, No.1, 104-11, 2006.

5104 2008 Chinese Control and Decision Conference (CCDC 2008)