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 :
