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May 12, 2004 - for linear descriptor systems with delay in state. Fan YANG. 1 ,2. , Qingling ZHANG. 1. (1. Institute of Systems Science , Northeastern University ...
Journal of Control Theory and Applications 1 ( 2005) 76 - 84

Delay2dep e nde nt H- infinity c ontrol f or linear descrip t or syste ms wit h delay in state 1 ,2

Fan YAN G

, Qingling ZHAN G

1

( 1 . Institute of Systems Science , Northeastern University ,Shenyang Liaoning 110004 , China ;

2 . Department of Mathematics , Tonghua Teachers’College , Tonghua Jilin 134002 , China)

  A bs t ract : A delay2dependent H- infinity control for descriptor systems with a state2delay is investigated. The purpose of the problem is to design a linear memoryless state2feedback controller such that the resulting closed2loop system is regular ,impulse free and stable with an H- infinity norm bound. Firstly , a delay2dependent bounded real lemma (B RL ) of the time2delay descriptor systems is presented in terms of linear matrix inequalities (LMIs) by using a descriptor model transformation of the system and by taking a new Lyapunov2 Krasovsii functional. The introduced functional does not require bounding for cross terms , so it has less conservation. Secondly , with the help of the obtained bounded real lemma , a sufficient condition for the existence of a new delay2dependent H- infinity state2feedback controller is shown in terms of nonlinear matrix inequalities and the solvability of the problem can be obtained by using an iterative algorithm involving convex optimization. Finally , numerical examples are given to demonstrate the effectiveness of the new method presented. Ke ywor ds : Delay descriptor systems ; Delay2dependent ; H- infinity control ; Bounded real lemma (B RL ) ; Linear matrix in2 equalities (LMIs)

aircraft stabilization , chemical engineering systems , lossless

1  I nt r o d uctio n

transition

lines ,

power

systems ,

robotic

systems

 Since the late 1980s , the H ∞control theory has been

etc [ 11~15 ] . To the best of our knowledge , there are only

developed because of its both practical and theoretical

few papers on the delay2dependent

significance [ 1~3 ] . A great deal of effort has been devoted

descriptor systems with delay. Recently , E. Fridman &

to the

investigation

systems [ 4~7 ] . The

of main

H ∞ control for reason is

that

time2delay

time2delay

phenomenon is often encountered in control systems either

H ∞ control for

U . Shaked[ 7 ] reduced the conservatism of delay2dependent conditions by applying the descriptor model transformation and by taking a new Lyapunov2Krasovskii functional [ 8 ] .

in the state , the control input , or the measurements and the

The results are based on the so2called Park ’ s inequality for

time2delay systems often caused instability and performance

bounding cross terms.

degradation of systems [ 8~10 ] .

 The main objective of the delay2dependent H ∞ control

 The existing results for H ∞control of time2delay systems in this paper is to design a memoryless state feedback deal with delay2independent and delay dependent . The controller such that the resulting closed2loop system is delay2independent type is generally conservative , especially regular ,impulse free ,stable ,as well as satisfying a prescribed when a delay is short . In order to reduce the conservatism of

delay2dependent

type , there

have

been

H ∞ norm bound constraint , which it allows a maximum

various

delay size for a fixed H ∞performance bound or achieves a

approaches and some of the methods for nondescriptor

minimum H ∞ performance bound for a fixed delay size . By

systems with state2delay have been extended to descriptor

using a descriptor model transformation of the system and

system with delay.

by taking a new Lyapunov2Krasovskii functional according

 Descriptor time2delay systems describe a broad class of

to Park[ 4 ] , which does not require bounding for cross

systems which are not only of theoretical interest but also

terms , a new delay2dependent bounded real lemma for

have great practical significance . We have found that they

systems with a state2delay is derived in terms of LMIs. Based

often appear in various engineering systems , including

on the bounded real lemma obtained ,a sufficient condition

 Received 12 May 2004 ; Revised 24 January 2005 . This work was supported by the Technology Foundation of Liaoning Province ( No. 200140104) .

F. YANG et al . / Journal of Control Theory and Applications 1 ( 2005) 76 - 84

77

for the existence of H ∞ control is given in terms of

control input , ω( t )

nonlinear matrix inequality. An iterative algorithm involving

disturbance signal and z ( t ) ∈R is the controlled output .

convex optimization was developed to solve the nonlinear

The matrix E ∈R may be singular , we shall assume that rank ( E) = r < n. A 0 , A 1 , B 0 , B 1 , C0 , C1 , D are known

matrix inequality , which is similar to Moon et al [ 16 ] .

 This paper is organized as follows : the delay2dependent

q

∈ L 2 [ 0 , ∞) is the exogenous p

n ×n

real constant matrices with appropriate dimensions. The

H ∞ control problem for descriptor systems which we deal

time delay h > 0 is assumed to be unknown and with

with is stated precisely in Section 2 ; A new bounded real

known bound g h.

lemma is obtained in Section 3 ; In Section 4 , based on the

 For the sake of convenience , we assume that A i1 A i2 Ir 0

obtained bounded real lemma ,a sufficient condition for the existence of a delay2dependent H ∞state2feedback control is

E =

0

given in terms of nonlinear matrix inequality and the Section ; Examples are given to illustrate the effectiveness we proposed about the delay2dependent H ∞ control in Section

5 ; Conclusion is given in the last Section of this paper .

 N ot a ti ons  Throughout the paper , the superscript W T stands for the transpose of any matrix W ; R denotes the set n of real numbers ; R denotes the n- dimensional Euclidean ) space ; ・ stands for the induced matrix 22norm ; λmin ( ・ ) λ ( and max ・ denote the minimum and the maximum n ×m eigenvalue of the corresponding matrix , respectively ; R is the set of n × m real matrices ; Cn [ a , b ] denotes the

B i1

Bi =

procedure to solve the problem is also detailed in the same

0

, Ai =

A i3

,

A i4

( 2)

, Ci = [ Ci1  Ci2 ] , ( i = 0 , 1) .

B i2

For a prescribed scalar γ > 0 , define the performance index J (ω) =



∫( z ( s) z ( s) - γω ( s) ω( s) ) d s . (3) T

2

T

0

The singular delay system of (1a) with u ( t ) = 0 ,ω( t ) = 0 can be written as Eg x ( t ) = A 0 x ( t ) + A 1 x ( t - h) .

( 4)

 D ef i niti on 1 [ 10 ]  1) The singular delay system ( 4) is said to be regular and impulse free if the pair ( E , A ) is regular ( i . e . det ( s E - A ) is not identically zero ) and impulse free ( i . e . deg ( det ( s E - A ) ) = rank ( E) ) .

Banach space of continuous vector functions mapping the

 2) The singular delay system ( 4) is said to be stable if for any ε > 0 there exists a scalar δ(ε) > 0 such that ,for q are square integral over [ 0 , ∞) is denoted by L 2 [ 0 , ∞) ; any compatible initial conditions < ( t ) satisfying x t (θ) = x ( t + θ) (θ ∈[ - g h ,0 ] , g h is a known positive sup < ( t ) ≤δ(ε) , the solution x ( t ) of system ( 4)

interval [ a , b ] into R n ; the space of functions in R q that

- h ≤t ≤0

delay ; I denotes

an

identity

matrix

of

appropriate

dimension and 3 represents the elements below the main diagonal of a symmetric block matrix ; the notation X > 0 ( respectively , X ≥ 0) , for X ∈ R n ×n means that the matrix X is real symmetric positive definite ( respectively , positive semi2definite ) ; matrices ,if not explicitly stated ,are assumed to have compatible dimensions.

 Consider the following class of descriptor system with

( 1a) C0 x ( t ) ( 1b)

,

( 1c) x ( t ) = 0 , t ≤0 , T T n T ( ) ( ) ( ) where x t = [ x1 t  x2 t ] ∈R is the system state vector , x1 ( t ) ∈ R , x2 ( t ) ∈ R r

n-

t →∞

then the solution x ( t ) of ( 4) is said to be asymptotically stable .  3) The singular delay system ( 4) is said to be admissible if it is regular ,stable and impulse free . The objective of this paper

is

to

design

a

state2feedback

memoryless

u ( t ) = Kx ( t ) , K ∈R

state delay : Eg x ( t ) = A 0 x ( t ) + A 1 x ( t - h ) + B 0 u ( t ) + B 1ω( t ) ,

Du ( t ) C1 x ( t - h)

≤εfor t ≥0. Furthermore , lim x ( t ) = 0 ,

x ( t)

delay2dependent H ∞2controller

2  D esc rip tio n a n d p r eli mi na ries

z ( t) =

satisfies

l ×n

( 5)

,

such that for a given real number γ > 0 , the closed2loop system ( 1) with controller ( 5) is admissible (ω( t ) = 0) and for all nonzero ω( t ) inequality holds : Gzw ( s ) ∞ < γ i . e.

q

∈ L 2 [ 0 , ∞) , the following z ( t)

2

< γ ω( t )

2

,

i . e. J (ω) < 0 , where J (ω) has the structure ( 3) .

 Pr op ositi on 1 [ 7 ]  Assume that A 04 is nonsingular. For

ω( t ) ∈ L 2q [ 0 , ∞) , the solution to ( 1a) with u ( t ) = 0 r l , u ( t ) ∈ R is the and ( 1c ) exists and is unique on [ 0 , t 1 ] for all t 1 > 0 .  L e m m a 1 [ 12 ]  If there exist matrix U = U T > 0 and

F. YANG et al . / Journal of Control Theory and Applications 1 ( 2005) 76 - 84

78

0

matrix Pf that satisfy the following LMI : PfT A 04

+

T A 04

PfT A 14

Pf + U

3

A g1 =

< 0,

- U

then A 04 is nonsingular and the difference operator D : Cn - r [ - h , 0 ] → R given by n

0

A 11 , A g2 = A 13

0

A 12 , B g1 = A 14

B 11 . B 12

( 8)

Taking a Lyapunov2Krasovskii functional candidate V ( t ) for system ( 7) : 5

-1

D ( x2 t ) = x 2 ( t ) + A 04 A 14 x 2 ( t - h ) is stable for delay h ∈ [ 0 , g h ] ( i . e . the difference equation D ( x 2 t ) = 0 is asymptotically stable for delay h ∈[ 0 , g h ]) . Furthermore , under additional assumption that Pf > 0 the

“fast system ” x 2 ( t ) = x 2 ( t ) + A 14 x 2 ( t - h) g

where , T V 1 ( t ) = xg ( t ) g E Pg x ( t) , t

∫x ( s) S x ( s) d s , = ∫x ( s) S x ( s) d s , = ∫∫ y ( s) S y ( s) d s dθ, t- h t

is asymptotically stable for h ∈ [ 0 , g h ]. -1  By Lemma 1 ,it follows that A 04 A 14 < 1 and from A 04 is nonsingular , we know the pair ( E , A ) is regular and

V3 ( t)

impulse free . According to Definition 1 , the delay singular system ( 4) is regular and impulse free .

V5 ( t)

 L e m m a 2 [ 12 ]  Suppose A 04 A 14 < 1 . If there exist positive numbers α,β,γ and a continuous function V : Cn [ - h , 0 ] → R such that β 0 and the inequality ( 17) holds , we have A 04 is

T

T

( 16)

< 0,

0

t θ

T T P32 A 04 + A 04 P32 + S 2

   + hy ( t ) S 3 y ( t ) -

0 Y1

3 0 Y2 +

+

=

t

T

y (θ) y ( s)

0 θ- h

Y2T ,

Thus

T

x1 (θ)

t θ

T Ψ22 = - P31 - P31 + g hS 3 + g h X22 .

t ) S 1 x1 ( t ) -

T

P31 A 12

T

T T T Ψ13 = P21 A 02 + P22 A 04 + A 03 P32 ,

x 1T (

T

T

T Y1 ,

P31 + g h X12 +

T A 01

C02

P21 A 12 + P22 A 14

P31 A 11 - Y2

T

+ (

T

P21 A 11 + P22 A 13 - Y1

T

T P21 A 01

( 15)

.

0

Ψ22

T P22 A 03

0 T C02

 Pr oof  For u = 0 ,ω = 0 , in order to prove system ( 1) is regular and impulse free , we substitute ( 11) into ( 13) and ( 13) can be changed into the following LMI

3

T

T

C01 C02

0

C01

Ψ13

T P21 A 01

T

0

Ψ12

Ψ12 = P1 -

 =

0 T C02

Ψ11

+ S1 + g h X11 + Y1 +

x1T (

0

T

where

Ψ11 =

C01 C01 C g C g =

X22 0 ,

3

S2 T

( 8) and X12 0

79

t

- x 1 ( t - h) ) +

∫y ( s) S y ( s) d s t- h

T

3

T T =g x ( t) ( g hX g+ g Y1 ) g x ( t) - 2 g x ( t) g Y2 x 1 ( t - h )

F. YANG et al . / Journal of Control Theory and Applications 1 ( 2005) 76 - 84

80 t

+

∫ t- h

where

Therefore ,in the case of u = 0 ,ω = 0 , we obtain T T V g( x t ) ≤g x ( t) ( P A g0 + Ag0T P + Sg) g x ( t)

T T T T T ξ= [g x ( t )  ω ( t )  x 1 ( t - h )  x 2 ( t - h) ] .

T T + 2g x ( t) P ( A g1 x 1 ( t - h) + Ag2 x2 ( t - h) )

- h) S 2 x 2 ( t - h) -

∫ 0

∫y ( s) S y ( s) d s T

Now we integrate equality ( 18) in t from 0 to ∞, because V ( 0) = 0 , V ( x ∞) ≥0 and ∞

T

- x1 ( t - h ) S 1 x1 ( t - h ) - x2T ( t t

   + z T ( t ) z ( t ) ,

( 19)

T y ( s) S 3 y ( s) d s .

∫x ( t) C C x ( t) d t + ∫x ( t - h) C C x ( t - h) d t ,

+ g x ( t) ( g hX g+g Y1 ) g x ( t) - 2 g x ( t) g Y2 x 1 ( t T

t

T y ( s) S 3 y ( s) d s



+ 2g x T ( t ) PTA g2 x 2 ( t - h) - x1T ( t

T

2

t - h) S 2 x2 ( t - h)

d

T

T

2

T

0

T

1

0



∫[ z ( t) z ( t) - γω ( t) ω( t) ]d t   ≤ ξ Ω ξd t < - μ x ( t ) , μ > 0 , ∫

( 18)

T

2



 x 1T (

 x 2T (

t - h)

T

0

where T

t - h) ] ,

T

0

2

1

that is to say J (ω) < 0 , if ( 13) holds.

and T

P

T Ag0 + Ag0

∞)

By using a Schur complement argument and ( 13) , we have

T

ζ= [g x ( t)

1

t



=ζ Ω2ζ,

T

T 1

∫[ z ( t) z ( t) - γω ( t ) ω( t) + d tV ( x ) ]d t  = ∫[ z ( t) z ( t) - γω ( t) ω( t) ]d t + V ( x   = ξ Ω ξd t . ∫ ∞

T T + g Y1 ) g x ( t) + 2 g x ( t) ( P A g1 - g Y2 ) x 1 ( t - h )

- h) S 1 x 1 ( t - h) -

T

0

T T =g x ( t) ( P A g0 + Ag0T P + Sg + g hX g

T x2 (

0

thus

∫ t- h

T 0

0

T

- h) +

T

0



3

t- h



T z ( s) z ( s) d s =

P+ Sg + g hX g + gY1

Ω2 =

P Ag1 - g Y2

P Ag2

3

- S1

0

3

3

- S2

T

T



4   D ela y2dep e n de nt s t a t e2f e e d bac k

.

H∞

co nt r ol

 Theorem 1 will play an important role in solving the H ∞ complement lemma , it is easy to see that Ω2 < 0 , control problem. Now we apply Theorem 1 to the H ∞ therefore , V g( x t ) < 0 and there exist two scalars α > 0 ,β control problem of system ( 1) .  In order to look for the state2feedback gain matrix K , > 0 such that Lyapunov2Krasovsii functional ( 9) satisfies

Suppose ( 13) holds , then from ( 13) and using the Schur

2

α x1 ( t )

≤ V ( x t ) ≤β x ( t )

2

we assume

,

u ( t ) = Kx ( t ) , K = [ K1  K2 ] ,

where

then A g0 and C g C g in ( 13) be replaced by

α =λmin ( P1 ) , β =λmax ( P1 ) + g hλmax ( S 1 ) + g hλmax ( S 2 )

2 + (t - g h ) λmax

X12

Y1

3

X22

Y2

3

3

S3

By Lemma 1 and Lemma 2 , system

( 1)

0

I

0

A 01 + B 01 K1

- I

A 02 + B 01 K2

A 03 + B 02 K1

0

A 04 + B 02 K2



A0 =

2 + (t - g h ) λmax ( S 3 )

X11





C C=C g C g + K D DK T

T

is internally =

d T 2 T V ( x t ) + z ( t ) z ( t ) - γ ω ( t ) ω( t ) dt P

  ≤ξT

T Ag0 +Ag0 P+ Sg+ ghXg + gY1

P Bg1 T

P Ag1 - gY2 T

P Ag2

3 3

- S1

0 0

3

3

3

- S2

0

T

+

ξ

T

T C01 C01

0

3

0

3

3 T

T

-γI 3 2

, ( 21)

and .

asymptotically stable and when ω( t ) ≠0

T

( 20)

T

T

T C01 C02

0 T C02

C02 T

T

K1 D D K1

0

3

0

0

3

3

K2T D T D K2

K1 D D K2 ( 22)

respectively. Applying the above2obtained B RL to ( 21) and ( 22) , results in a nonlinear matrix inequality

F. YANG et al . / Journal of Control Theory and Applications 1 ( 2005) 76 - 84









P A 0 + A 0 P + Sg + g hX g+ g Y1 + C C

P B g1 T

P A g1 - g Y2

3

2 - γI

0

3

3

3

3

T

T

T

T C11

C11

3

T

- S 2 + C12 C12

0

Q31

= P

-1

,

( 24)

g h Z12 g h Z22 3

3 3

Q SgQ = T

0

0

, Z12 =

Z22 =

T Q31 S 3 Q31

,

Multiply ( 23) by diag{ Q , I , Q1 , Q1 } and its transpose on

N 11 =

T Q22 S 2 Q22

, N 13 = Q22 S 2 Q32 ,

the left and on the right , respectively , we have

N 33 = Q32 S 2 Q32 ,

0 < Q1 =

Q32

∈R

r ×r

.

T

Φ

Bg1

3

-γ I

3

3

3

3

Ag1 Q1 - Q g Y2 Q1

2

0 -

T Q1 ( S1 -

M 11 < 0,

T T Q1 C11 C12 Q1

Q1

3

g hQ X gQ = g h T

0

T C11 C11)





T

3 3

M12

0

M22

0 ,

3

0 T

T

T

M12 = Q1 X12 Q31 + Q21 X22 Q31 , T



T Φ = A 0 Q + Q T A 0T + Q T ( Sg + g hX g+g Y1 + C C) Q.

M22 = Q31 X22 Q31 ,

Substituting ( 8) , ( 14) , ( 15) , ( 21) , ( 22) and ( 24) into ( 25) we obtain

Q g Y1 Q =

T

T

N1 + N1

N2

0

3 3

0

0 , 0

T



3

A0 Q

- N1 Q21

A g1 Q1 - Q g Y 2 Q1 = T

0

Q31

  = - Q21 +A01 Q1 +A02 Q22 + B01 W1 - Q31 A02 Q32 + B01 W2 , A03 Q1 + A04 Q22 + B 02 W1 0 A04 Q32 + B 02 W2 T

≈ ≈

T

Q C C Q= Q

T

T

C01 C01 + K1 D D K1

0

3

0

3

3

T

A 13 Q1 Q S iQ = U i  ( i = 1 , 2) , T

T

N 1 = Q1 Y1 Q1 + Q21 Y2 Q1 , N 2 = Q31 Y2 Q1 ,

T

T

T

C01 C02 + K1 D D K2

0 T C02

C02 +

Q T T K2 D D K2

( C01 Q1 + C02 Q22) ( C01 Q1 + C02 Q22 ) + W1T D T DW1

0

3

0

3

3

T

=

A 11 Q1 - N 2 ,

T

( 26)

W 1 = K1 Q1 + K2 Q22 , W 2 = K2 Q32 .

T

T

M11 = Q1 X11 Q1 + Q1 X12 Q21 + Q21 X12 Q1 + Q21 X22 Q21 ,

T T - Q1 ( S2 - C12 C12) Q1

( 25) ≈

,

T

Ag2 Q1

T

,

N 33

Z11 =

0 T Q1

T Q21 S 3 Q31

N 13

T Q21 S 3 Q21

Q22

( 23)

< 0.

C12

U1 + g h Z11 + N 11

0

Q21

Q =

T

0 T C11

In order to obtain an LMI , define Q1

P A g2

T

- S1 +

81

( C01 Q1 + C02 Q22 ) T C02 Q32 + W1T D T DW2

0 T Q32

T C02

.

C02 Q32 + W 2T D T DW 2

Applying the Schur complements lemma to ( 25) and obtain the following LMI :