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H∞ tracking of uncertain stochastic time-delay systems: Memory state-feedback controller design
Applied Mathematics and Computation 249 (2014) 356–370
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
H1 tracking of uncertain stochastic time-delay systems: Memory state-feedback controller design Jianwei Xia a,b, Ju H. Park b,⇑, Tae H. Lee b,⇑, Baoyong Zhang c a
School of Mathematics Science, Liaocheng University, Shandong 252000, China Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 712-749, Republic of Korea c School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China b
a r t i c l e
i n f o
Keywords: Stochastic systems Robust H1 tracking control Stochastic stability Time delay
a b s t r a c t The problem of state robust H1 tracking control for uncertain stochastic systems with interval time-varying delay is considered in this paper. The target is to design a memory state feedback controller such that the resulting closed-loop augmented system is robustly stochastically asymptotically stable and the states of original systems can follow the reference signal with given H1 performance. Firstly, by constructing delay-partitioningdependent Lyapunov–Krasovskii functional with reciprocally convex approach, a delaydependent condition guaranteeing the robust H1 tracking performance is proposed in a set of matrix inequalities. Secondly, new stability criteria for stochastic systems with time-varying delay are given with less conservativeness than some recent results. Thirdly, memory state feedback controller is designed via a set of linear matrix inequalities (LMIs) which can be solved by Matlab LMI toolbox. At last, numerical simulations are addressed to show the effectiveness of the proposed methods. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In the past several decades, the study for stochastic systems has drawn much attention due to their significant rolesin many branches of engineering and science applications, such as missile autopilot control, satellites attitude control and chemical process control, etc [1,2]. It is well known that time delay is usually inevitable encountered and often the main reason for instability and poor performance for almost all practical models [3–13]. Meanwhile, there has been a growing interest in the stability analysis for delayed system in recent years. For the reason that the delay-dependent results are usually less conservative than the delay-independent ones,much more consideration has been focused on the technique design to achieve delay-dependent stability conditions[14–26]. Naturally, a great deal of results related to stability analysis and synthesis for stochastic systems with time delays have been reported in the recent literature. For example, the delay-independent and delay-dependent conditions of exponential stability, robust stabilization, robust H1 control, H1 filtering and L2 L1 filtering were proposed in [27–35], respectively. Especially, the authors in [30] provided a novel fault-tolerant control method for Markovian jump stochastic systems and the results can be well applied to the real application systems. On the other hand, state/output tracking is an important issue of the system and control theory, which has been extensively applied in robot control, flight control, signal processing and other practical fields [36,37]. The target of tracking
⇑ Corresponding authors. E-mail addresses: [email protected] (J. Xia), [email protected] (J.H. Park), [email protected] (T.H. Lee), [email protected] (B. Zhang). http://dx.doi.org/10.1016/j.amc.2014.10.029 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
control problem is to design a controllers such that the state of the closed-loop system follows a prescribed reference signal. And some results of this topic have been made in this field in recent years. For example, the problem of state H1 tracking control for continuous and discrete linear systems were considered in [38,39]. When time delay was considered, delayindependent and delay-dependent sufficient conditions of state H1 tracking problem were obtained for time-delay Markovian jump systems and switched systems in [40–42], respectively. When the stochastic terms were considered in the systems, the output feedback tracking control and adaptive neural tracking control for stochastic systems were considered in [43,44]. However, to the best of authors’ knowledge, there is few effort to study the state H1 tracking control problem for uncertain stochastic systems with time-varying delays, which motivates our present study. In this paper, we address the problem of memory state robust H1 tracking control for uncertain stochastic systems interval time-varying delays. By a novel delay-partitioning-dependent Lyapunov–Krasovskii functional with reciprocally convex approach, a delay-dependent condition guaranteeing the robust H1 tracking performance is obtained. Meanwhile, improved stability criteria for stochastic systems with time-varying delay are proposed with less conservativeness. Then, existence conditions and design methods of the desired controller are constructed by solving a set of linear matrix inequalities. Some numerical examples will show the verification of our design methods. Notation:Throughout the paper, for symmetric matrices X and Y, the notation X P Y (respectively, X > Y) means that the matrix X Y is positive semi-definite (respectively, positive definite); I is the identity matrix with appropriate dimension; M T represents the transpose of the matrix M; Rn denotes the ndimensional Euclidean space; 0mn represents a zero matrix with m n dimensions; jj jj denotes the Euclidean norm for vector or the spectral norm of matrices; symðAÞ denotes A þ AT ; ðX; F; P) is a probability space, where X is the sample space, F is the
r-algebra of subsets of the sample space, and
P is the probability measure on F; and L2F 0 ð½h; 0; Rn Þ denotes the family of all F 0 measurable Cð½h; 0; Rn Þ-valued random variables n ¼ fnðhÞ : h 6 h 6 0g such that suph6h60 EjnðhÞj2 g < 1, where Efg stands for the expectation operator with respect to some probability measure P. The notations X > 0(P 0) is used to denote a symmetric positive-definite (positive-semidefinite) matrix. In symmetric block matrices or complex matrix expressions, we use an asterisk to represent a term that is induced by symmetry, and diagfg stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. System description and preliminaries Consider the following uncertain stochastic system with time delay
dxðtÞ ¼ ½AðtÞxðtÞ þ Ad ðtÞxðt dðtÞÞ þ Bu ðtÞuðtÞ þ Br ðtÞrðtÞdt þ ½CðtÞxðtÞ þ DðtÞxðt dðtÞÞdxðtÞ;
ð1Þ
xðtÞ ¼ /ðtÞ; t 2 ½2d2 ; 0;
ð2Þ
where x 2 Rn is state vector; uðtÞ 2 Rp is control input vector; rðtÞ 2 Rq is bounded disturbance; dðtÞ is time differential func_ 6 l, where d1 ; d2 and l are known constant scalars; tion representing time-varying delay and satisfies 0< d1 6 dðtÞ 6 d2 ; dðtÞ wðtÞ is a one-dimension Brownian motion satisfying EfdwðtÞg ¼ 0 and EfdwðtÞ2 g ¼ dt; /ðtÞ is a vector-valued initial continuous function defined on the interval ½2d2 ; 0; the uncertain matrices AðtÞ; Ad ðtÞ; Bu ðtÞ; Br ðtÞ; CðtÞ and DðtÞ are assumed to
½ AðtÞ Ad ðtÞ Bu ðtÞ Br ðtÞ CðtÞ DðtÞ ¼ ½ A Ad
Bu
Br
C
D þ MJðtÞ½ Na
Nad
Nbu
N br
Nc
Nd ;
ð3Þ
where A; Ad ; Bu ; Br ; C; D; M; N a N ad ; N bu ; N br ; N c , and N d are real constant matrices. JðtÞ is an unknown norm-bounded time-varying matrix satisfying
J T ðtÞJðtÞ 6 I:
ð4Þ
The reference signal is assumed to be generated by the following deterministic system
_ yðtÞ ¼ EyðtÞ þ Ed yðt dðtÞÞ þ v ðtÞ;
ð5Þ
yðtÞ ¼ wðtÞ; 8t 2 ½2d2 ; 0;
ð6Þ
where yðtÞ 2 Rn is reference signal state vector; v ðtÞ is a bounded reference input; and E and Ed are known real matrices with appropriate dimensions. Without loss of generality, it is assumed that this reference system is stable. Design memory state feedback controller in the following form
uðtÞ ¼ K 1 xðtÞ þ K 2 xðt dðtÞÞ þ F 1 yðtÞ þ F 2 yðt dðtÞÞ;
ð7Þ
where K 1 ; K 2 ; F 1 and F 2 are the controller gains to be determined. Remark 1. It is well known that, because more information is considered, memory controllers usually lead to the better performances than memoryless ones for the designed systems. In this paper, we discuss a synthesis problem of memory controller which guarantee the closed-loop augmented system is robustly stochastically asymptotically stable and the original states can follow the reference signal with given H1 performance. Applying the controller (7) into the system (1), we obtain an augmented closed-loop system as following
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
dnðtÞ ¼ ½ðAðtÞ þ Bu ðtÞK1 ÞnðtÞ þ ðAd ðtÞ þ Bu ðtÞK2 Þnðt dðtÞÞ þ Br ðtÞrðtÞdt þ ½CðtÞnðtÞ þ DðtÞnðt dðtÞÞdwðtÞ;
uðtÞ ¼
/ðtÞ
wðtÞ
; 8t 2 ½2d2 ; 0;
ð8Þ ð9Þ
where
" nðtÞ ¼
xðtÞ yðtÞ
#
"
;
rðtÞ ¼
rðtÞ
# ;
v ðtÞ "
A 0
#
K1 ¼ ½ K 1 "
M
F 1 ;
K2 ¼ ½ K 2
F 2 ;
#
þ JðtÞ½ Na 0 ; 0 E 0 " # " # Ad 0 M Ad ðtÞ ¼ Ad þ MJðtÞN ad ¼ þ JðtÞ½ Nad 0 ; 0 Ed 0 " # " # Bu M Bu ðtÞ ¼ Bu þ MJðtÞN bu ¼ þ JðtÞN bu ; 0 0 " # " # Br 0 M þ JðtÞ½ Nbr 0 ; Br ðtÞ ¼ Br þ MJðtÞN br ¼ 0 I 0 " # " # C 0 M CðtÞ ¼ C þ MJðtÞN c ¼ þ JðtÞ½ Nc 0 ; 0 0 0 " # " # D 0 M þ JðtÞ½ Nd 0 ; DðtÞ ¼ D þ MJðtÞN d ¼ 0 0 0 AðtÞ ¼ A þ MJðtÞN a ¼
ð10Þ
and uðtÞ denotes the initial condition of system (8). Define state tracking error as
eðtÞ ¼ xðtÞ yðtÞ;
ð11Þ
and denote the tracking performance constraint as
E
Z
tf
eðtÞT WeðtÞdt
6 c2
0
Z
tf
rðtÞT rðtÞdt þ d;
ð12Þ
0
where tf P 0 is an terminal time; c > 0 is a prescribed scalar representing performance index; W P 0 is a weighting matrix; d P 0 is a weighting scalar related to the initial conditions. Set
W¼
W
W
W
W
:
ð13Þ
Then, the inequality (12) can be rewritten as
E
Z
tf
nðtÞT WnðtÞdt
0
6 c2
Z
tf
rðtÞT rðtÞdt þ d:
ð14Þ
0
Furthermore, since W P 0, there must exist a matrix W 0 such that W ¼ W T0 W 0 . Therefore, the matrix W in (13) can be rewritten as
W¼
I I
W T0 W 0 ½ I
I ¼ W T0 W 0 :
ð15Þ
In order to obtain our main results, the following definitions and lemmas are employed throughout our paper. Definition 1 [27]. For system (1) with uðtÞ ¼ 0 is said to be robustly stochastically stable in mean square if for any there is a dðeÞ > 0 such that
EjxðtÞj2 < e;
t > 0;
for any initial conditions and all admissible uncertainties, when
sup EjxðtÞj2 < dðeÞ: l6s60
e > 0,
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If, in addition,
lim EjxðtÞj2 ¼ 0;
t!1
for any initial conditions and all admissible uncertainties, then system (1) with uðtÞ ¼ 0 is said to be robustly stochastically asymptotically stable in mean square. Definition 2. The robust H1 state tracking requirements of system (1), (2) associated with reference system 5,6 are said to be met, if there exists a state-feedback controller given in (7) such that the closed-loop augmented system (8) is robustly stochastically asymptotically stable in mean square with rðtÞ ¼ 0 and the state tracking error eðtÞ (11) and the bounded disturbance rðtÞ in (8) satisfy the relationship described in (12) or (14) for all admissible uncertainties. Lemma 1 [17]. For any constant matrix T 2 Rmm ; T ¼ T T > 0, scalar c > 0, vector function x : ½0; c ! Rm such that the integrals in the following are well defined, then
c
Z
c
xðbÞT T xðbÞdb P
Z
0
T Z
c
c
xðbÞdb T
0
xðbÞdb:
ð16Þ
0
Lemma 2 [25]. Let f 1 ; f 2 ; . . . ; f N : Rm ! R have positive values in an open subsets D of Rm . Then, the reciprocally convex combination of f i over D satisfies
X1
min P
fai jai >0;
a ¼1g i i i
ai
f i ðtÞ ¼
X
X f i ðtÞ þ max g ij ðtÞ g i;j ðtÞ
i
ð17Þ
i–j
subject to
(
"
f i ðtÞ g ij ðtÞ g ij ðtÞ : R ! R; g ij ðtÞ ¼ g ji ðtÞ; g ji ðtÞ f j ðtÞ
#
)
m
P0 :
ð18Þ
Lemma 3 [9]. Let A; D; E be real constant matrices with appropriate dimensions, matrix JðtÞ satisfies J T ðtÞJðtÞ 6 I. For any such that P1 eDDT > 0, then
DJðtÞE þ ET J T ðtÞDT 6 e1 DDT þ eET E
e > 0,
ð19Þ
1
ðA þ DJðtÞEÞT PðA þ DJðtÞEÞ 6 e1 ET E þ AT ðP1 eDDT Þ A: 3. Main results For simplicity of vector and matrix representation, we define
h
CðtÞ ¼ nðtÞT
nðt m1 d1 Þ
T
T
nðt m1 d1 Þ m
" T
T
T
vðtÞ ¼ CðtÞ nðt d1 Þ nðt dðtÞÞ nðt d2 Þ
ej ¼ 02n2ðj1Þn
kj ¼ 0nðj1Þn
I2n In
02n2ðmþ6jÞn ;
0nðmþ6jÞn ;
T
Z
iT 2mn1
t
;
T
f ðsÞ ds
1d tm 1
j ¼ 1; 2; . . . ; m þ 6:
Z
td1
tdðtÞ
T
f ðsÞ ds
Z
tdðtÞ
td2
#T T
f ðsÞ ds
; 2ðmþ6Þn1
ð20Þ
3.1. Tracking performance analysis Firstly, the robust H1 tracking performance analysis for the system (1) is considered in the following theorem, which is proposed in a set of matrix inequalities. Theorem 1. Given an integer m P 1, and scalars l; d2 P d1 > 0, c > 0, the robust H1 state tracking requirements are met, if there exist positive-definite matrices P > 0; Q 1 > 0; Q 2 > 0; Q 3 > 0; R1 > 0; R2 > 0; S1 > 0, S2 > 0, and matrices U; L1 ; L2 ; L3 such that the following matrix inequalities hold
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
W¼
R2
U
UT
R2
2
P 0;
ð21Þ
HðtÞ eT1 PBr ðtÞ
6 6 6 NðtÞ ¼ 6 6 6 4
L1
L2
c2 I
0
0
S1
0
S2
L3
3
2
3
b T AðtÞ
2
b T AðtÞ
3T
2
b CðtÞT
7 6 7 6 7 6 0 7 6 Br ðtÞT 7 6 Br ðtÞT 7 6 0 7 6 7 6 7 6 7 7 7 6 6 0 7 þ 6 0 7H1 6 0 7 þ 6 6 0 7 6 7 6 7 6 4 0 0 5 4 0 5 4 0 5 S2 0 0 0
3
2
b CðtÞT
3T
7 6 7 7 6 0 7 7 6 7 7H2 6 0 7 <0; 7 6 7 7 6 7 5 4 0 5
ð22Þ
0
where
b b T Pe1 þ PT Q 1 P1 PT Q 1 P2 þ eT Q 2 emþ1 ð1 lÞeT Q 2 emþ2 þ AðtÞ HðtÞ ¼ eT1 P AðtÞ mþ1 mþ2 1 2 þ eT1 Q 3 e1 eTmþ3 Q 3 emþ3 eTmþ4 R1 emþ4 PT3 WP3 þ L1 P4 þ PT4 LT1 þ L2 P5 þ PT5 LT2 þ L3 P6 þ PT6 LT3 þ eT1 We1 ; T T P1 ¼ eT1 eT2 eTm ; P2 ¼ eT2 eT3 eTmþ1 ; T T P3 ¼ emþ5 eTmþ6 ; P4 ¼ e1 e2 emþ4 ;
P5 ¼ emþ1 emþ2 emþ5 ; P6 ¼ emþ2 emþ3 emþ6 ;
ð23Þ
b AðtÞ ¼ ½AðtÞ þ Bu ðtÞK1 e1 þ ½Ad ðtÞ þ Bu ðtÞK2 emþ2 ; 2 d1 2 b R1 þ ðd2 d1 Þ R2 ; CðtÞ ¼ CðtÞe1 þ DðtÞemþ2 ; H1 ¼ m d1 H2 ¼ P þ S1 þ ðd2 d1 ÞS2 : m Proof. For convenience, we set
f ðtÞ ¼ ðAðtÞ þ Bu ðtÞK1 ÞnðtÞ þ ðAd ðtÞ þ Bu ðtÞK2 Þnðt dðtÞÞ þ Br ðtÞrðtÞ; gðtÞ ¼ CðtÞnðtÞ þ DðtÞnðt dðtÞÞ; then, system (1) becomes
dnðtÞ ¼ f ðtÞdt þ gðtÞdxðtÞ:
ð24Þ
On one hand, applying Schur complement lemma to (22), it is easily verified that
"
NðtÞ0 ¼
HðtÞ0 eT1 PBr ðtÞ
#
"
b T AðtÞ
#
"
b T AðtÞ
#T < 0;
ð25Þ
1 T 1 T T HðtÞ0 ¼ HðtÞ þ b CðtÞT H2 b CðtÞ þ L1 S1 1 L1 þ L2 S2 L2 þ L3 S2 L3 :
ð26Þ
c2 I
þ
Br ðtÞT
H1
Br ðtÞT
where
On the other hand, we choose Lyapunov–Krasovskii functional as
VðtÞ ¼
4 X V i ðtÞ;
ð27Þ
i¼1
where
V 1 ðtÞ ¼ nðtÞT PnðtÞ; Z t Z T V 2 ðtÞ ¼ C ðsÞ Q C ðsÞds þ 1 d t m1
td1
nðsÞT Q 2 nðsÞds þ
tdðtÞ
Z
t
nðsÞT Q 3 nðsÞds;
td2
Z Z t Z d1 Z t d1 0 T f ðsÞ R f ðsÞdsdh þ ðd d Þ f ðsÞT R2 f ðsÞdsdh; 1 2 1 m dm1 tþh tþh d2 Z 0 Z t Z d1 Z t gðsÞT S1 gðsÞdsdh þ gðsÞT S2 gðsÞdsdh: V 4 ðtÞ ¼ d
V 3 ðtÞ ¼
m1
tþh
d2
tþh
It is easy to get that there exist positive scalars k1 > 0 and k2 > 0 such that
k1 jnðtÞj2 6 VðnðtÞ; rt ; tÞ 6 k2 sup jnðhÞj2 : t2d6h6t
ð28Þ
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
Therefore D
Vðnð0Þ; r 0 ; 0Þ 6 d¼k2 sup jnðhÞj2 :
ð29Þ
2d6h60
Using Ito’s formula, we can obtain the derivation of VðtÞ for the system (24) as following
dVðtÞ ¼
4 X
LV i ðtÞ þ 2nðtÞT PgðtÞdxðtÞ;
ð30Þ
i¼1
where L is the weak infinitesimal generator of the random process and
b vðtÞ þ 2vðtÞT eT PBr ðtÞrðtÞ þ vðtÞT b LV 1 ðtÞ ¼ 2nðtÞT Pf ðtÞ þ gðtÞT PgðtÞ ¼ 2vðtÞT eT1 P AðtÞ CðtÞT P b CðtÞvðtÞ; 1 T d1 d1 T T þ nðt d1 Þ Q 2 nðt d1 Þ ð1 lÞnðt dðtÞÞ Q 2 nðt dðtÞÞ LV 2 ðtÞ 6 CðtÞT Q 1 CðtÞ C t Q 1C t m m T
þ nðtÞT Q 3 nðtÞ nðt d2 Þ Q 3 nðt d2 Þ; h i ¼ vðtÞT PT1 Q 1 P1 PT2 Q 1 P2 þ eTmþ1 Q 2 emþ1 ð1 lÞeTmþ2 Q 2 emþ2 þ eT1 Q 3 e1 eTmþ3 Q 3 emþ3 vðtÞ; " 2 # Z Z td1 d1 d1 t 2 R1 þ ðd2 d1 Þ R2 f ðtÞ f ðsÞT R1 f ðsÞds ðd2 d1 Þ f ðsÞT R2 f ðsÞds d m m t m1 td2 Z t Z td1 h iT h i b vðtÞ þ Br ðtÞrðtÞ H1 AðtÞ b vðtÞ þ Br ðtÞrðtÞ d1 ¼ AðtÞ f ðsÞT R1 f ðsÞds ðd2 d1 Þ f ðsÞT R2 f ðsÞds; d m t m1 td2
LV 3 ðtÞ ¼ f ðtÞT
Z t Z td1 d1 T gðsÞ S gðsÞds gðsÞT S2 gðsÞds; S1 þ ðd2 d1 ÞS2 gðtÞ 1 d m t m1 td2 Z t Z td1 d1 T CðtÞvðtÞ gðsÞ S gðsÞds gðsÞT S2 gðsÞds: CðtÞT S1 þ ðd2 d1 ÞS2 b ¼ vðtÞT b 1 d m t m1 td2
LV 4 ðtÞ ¼ gðtÞT
ð31Þ
By Lemma 1, we obtain
d1 m
Z
t
Z
T
d
t m1
f ðsÞ R1 f ðsÞds 6
!T
t d
f ðsÞds
Z
R1
t m1
!
t
f ðsÞds
d
t m1
¼ vðtÞT eTmþ4 R1 emþ4 vðtÞ:
ð32Þ
Meanwhile, using Lemma 2, there exists matrix U satisfying (21) such that
ðd2 d1 Þ
Z
"
td1
T
f ðsÞ R2 f ðsÞds ¼ ðd2 d1 Þ
Z
td2
ðd2 d1 Þ 6 dðtÞ d1 2 R td 6 6 4
1
tdðtÞ
R tdðtÞ td2
td1
T
f ðsÞ R2 f ðsÞds ðd2 d1 Þ
Z
#T
td1
f ðsÞds
R2
"Z
tdðtÞ
f ðsÞds f ðsÞds
3T 7 5
td1
tdðtÞ
"
R2 UT
T
f ðsÞ R2 f ðsÞds
td2
tdðtÞ
"Z
#
tdðtÞ
#
ðd2 d1 Þ f ðsÞds d2 dðtÞ
"Z
#T
tdðtÞ
f ðsÞds
td2
R2
"Z
tdðtÞ
td2
# f ðsÞds
ð33Þ
2 3 # R td1 f ðsÞds tdðtÞ 6 7 T T 4R 5 ¼ vðtÞ P3 WP3 vðtÞ; tdðtÞ R2 f ðsÞds td2 U
Furthermore, using the Newton-Leibniz formula, we obtain
"
# Z t Z t d1 2vðtÞ L1 nðtÞ n t f ðsÞds gðsÞdwðsÞ ¼ 0; d d m t m1 t m1 T
" 2vðtÞT L2 nðt d1 Þ nðt dðtÞÞ
Z
td1
f ðsÞds
Z
tdðtÞ
" T
2vðtÞ L3 nðt dðtÞÞ nðt d2 Þ
Z
gðsÞdwðsÞ ¼ 0;
tdðtÞ
tdðtÞ
td2
#
td1
f ðsÞds
Z
tdðtÞ
td2
# gðsÞdwðsÞ ¼ 0;
ð34Þ
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
that is
2vðtÞT L1 P4 vðtÞ 2vðtÞT L1 2vðtÞT L2 P5 vðtÞ 2vðtÞT L2
Z
t
gðsÞdwðsÞ ¼ 0;
d
t m1
Z
td1
gðsÞdwðsÞ ¼ 0;
ð35Þ
tdðtÞ
2vðtÞT L3 P6 vðtÞ 2vðtÞT L3
Z
tdðtÞ
gðsÞdwðsÞ ¼ 0:
td2
According to Lemma 1, it can be verified that T
2vðtÞ L1 T
2vðtÞ L2
Z
t d
gðsÞdwðsÞ 6 vðtÞ
T
t m1
Z
td1
T L1 S1 1 L1
gðsÞdwðsÞ 6 vðtÞ
T
tdðtÞ T
2vðtÞ L3
Z
tdðtÞ
gðsÞdwðsÞ 6 vðtÞ
td2
vðtÞ þ
T L2 S1 2 L2
T
"Z
#T
t
gðsÞdwðsÞ
d
t m1
vðtÞ þ
T L3 S1 2 L3
"Z
S1
"Z
gðsÞdwðsÞ
S2
"Z
tdðtÞ
vðtÞ þ
"Z
gðsÞdwðsÞ ;
d
t m1
#T
td1
#
t
#
td1
gðsÞdwðsÞ ;
tdðtÞ
#T
tdðtÞ
gðsÞdwðsÞ
S2
"Z
td2
tdðtÞ
# gðsÞdwðsÞ :
ð36Þ
td2
It is noted that
E
(Z
)
t d
t m1
(Z
gðsÞT S1 gðsÞds
tdðtÞ
¼E
8" < Z :
) T
d
gðsÞdwðsÞ
t m1
8" < Z
td1
S1
#9 = gðsÞdwðsÞ ; d ; t m1
"Z
#T
t
"Z
td1
#9 =
gðsÞdwðsÞ S2 gðsÞdwðsÞ : tdðtÞ ; tdðtÞ 8 (Z ) " #T "Z #9 = < Z tdðtÞ tdðtÞ tdðtÞ T gðsÞ S2 gðsÞds ¼ E gðsÞdwðsÞ S2 gðsÞdwðsÞ : E ; : td2 td2 td2 E
gðsÞ S2 gðsÞds
#T
t
¼E
ð37Þ
td2
Then, from (31)–(37), we can obtain
E fLVðnðtÞ; i; tÞg 6 vðtÞT HðtÞ0 eT1 We1 vðtÞ þ 2vðtÞT eT1 PBr ðtÞrðtÞ h iT h i b vðtÞ þ Br ðtÞrðtÞ H1 AðtÞ b vðtÞ þ Br ðtÞrðtÞ : þ AðtÞ
ð38Þ
To get the H1 tracking performance, we calculate that
n o n o vðtÞ T vðtÞ ; E LVðnðtÞ; i; tÞ þ nðtÞT WnðtÞ c2 rðtÞT rðtÞ ¼ E LVðnðtÞ; i; tÞ þ vðtÞT eT1 We1 vðtÞ c2 rðtÞT rðtÞ 6 NðtÞ0 rðtÞ rðtÞ ð39Þ 0
where NðtÞ is defined in (25). Then, it can be concluded from (25) and (39) that there exists a sufficiently small scalar c > 0 such that
n o E LVðnðtÞ; i; tÞ þ nðtÞT WnðtÞ c2 rðtÞT rðtÞ 6 cjnðtÞj2 :
ð40Þ
Therefore, applying the Dynkin’s formula to (40), we have
E
Z
tf
eðtÞT WeðtÞdt
6 c2
0
Z
tf
jrðtÞj2 dt þ Vðnð0Þ; r 0 ; 0Þ:
0
Noting the (29), therefore, the state tracking error eðtÞ in (11) and the bounded disturbance rðtÞ in (5) satisfy the relationship described in (14) for all admissible uncertainties. Furthermore, when rðtÞ ¼ 0, it follows from (40) that
LVðnðtÞ; i; tÞ 6 cjnðtÞj2 :
ð41Þ
Based on (41) and Definition 1, using the same proof in [27], we can easily get that augmented system (8) with rðtÞ ¼ 0 is robustly stochastically asymptotically stable in mean square with the state-feedback controller given in (7). This completes the proof. h
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
363
Remark 2. To achieve the improved results, some effective techniques are included in the proof Theorem 1, such as partition of the lower bound technique, free-weighting matrices method and reciprocally convex approach, which are helpful to reduce the conservativeness of the derived result. Remark 3. It should be pointed out that, the inequality (22) is not in the form of linear matrix inequality(LMI) due to the existence of some nonlinear terms. However, it is easily to be transformed into linear matrix inequality by using the Lemma 3 and Schur complement lemma. In order to compare our result with the recent ones in the literature, we consider the system (1) without input and disturbance terms, that is
dxðtÞ ¼ ½AðtÞxðtÞ þ Ad ðtÞxðt dðtÞÞdt þ ½CðtÞxðtÞ þ DðtÞxðt dðtÞÞdxðtÞ;
ð42Þ
Then, we can obtain a novel delay-dependent stability criterion for system (42). Corollary 1. Given an integer m P 1, and scalars l; d2 P d1 > 0, system (42) is robustly stochastically stable in the mean square, if there exist positive-definite matrices P > 0; Q 1 > 0; Q 2 > 0, Q 3 > 0; R1 > 0, R2 > 0; S1 > 0; S2 > 0, and matrices U; L1 ; L2 ; L3 , and positive scalar e1 > 0; e2 > 0, e3 > 0, such that (21) and the following LMIs holds
2
D L1 6 6 S1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
L2 0
T
L3 0
k1 PM 0
T
T
ðk1 AT þ kmþ2 ATd ÞH1 0
0 0
T
T
ðk1 C T þ kmþ2 DT ÞH2 0
S2
0
0
0
0
0
S2
0
0
0
0
e1 I
0
0
0
H1
H1 M
0
e2 I
0
H2
3 0 7 0 7 7 0 7 7 7 0 7 7 0 7 7 < 0; 7 0 7 7 0 7 7 7 H2 M 5
ð43Þ
e3 I
where H1 ; H2 are defined in (23) and T T b TQ1P b1P b TQ1P b 2 þ kT Q 2 kmþ1 ð1 lÞkT Q 2 kmþ2 D ¼ k1 PðAk1 þ Ad kmþ2 Þ þ ðAk1 þ Ad kmþ2 Þ Pk1 þ P mþ1 mþ2 1 2 T T T T b T WP b 3 þ L1 P b4þP b T LT þ L2 P b5þP b T LT þ k1 Q 3 k1 kmþ3 Q 3 kmþ3 kmþ4 R1 kmþ4 kmþ4 WR1 kmþ4 P 3 4 1 5 2
b6þP b T LT þ e1 ðNa k1 þ N ad kmþ2 ÞT ðNa k1 þ Nad kmþ2 Þ þ e2 ðN a k1 þ Nad kmþ2 ÞT ðNa k1 þ Nad kmþ2 Þ þ L3 P 6 3 T
þ e3 ðNc k1 þ N d kmþ2 Þ ðNc k1 þ Nd kmþ2 Þ; h
b 1 ¼ kT kT kT P 1 2 m h
T b 3 ¼ kT P kmþ6 mþ5
iT
iT
h b 2 ¼ kT ;P 2
T
k3
T
kmþ1
iT
;
b 4 ¼ k1 k2 kmþ4 ; ;P
b 6 ¼ kmþ2 kmþ3 kmþ6 : b 5 ¼ kmþ1 kmþ2 kmþ5 ; P P Proof. Choosing the same Lyapunov functional as (27) and using Lemma 3, the above criterion can be easily obtained. If there are no uncertainties, system (42) reduces to the following form
dxðtÞ ¼ ½AxðtÞ þ Ad xðt dðtÞÞdt þ ½CxðtÞ þ Dxðt dðtÞÞdxðtÞ:
ð44Þ h
ð45Þ
Then, the following corollary can be easily obtained. Corollary 2. Given a integer m P 1, and scalars l; d2 P d1 > 0, system (45) is stochastically stable in the mean square, if there exist positive-definite matrices P > 0; Q 1 > 0; Q 2 > 0, Q 3 > 0; R1 > 0; R2 > 0; S1 > 0; S2 > 0, and matrices U; L1 ; L2 ; L3 , such that (21) and the following LMIs holds
2
N
6 6 6 4
3
L1
L2
L3
S1
0
S2
0 7 7 bT b þb C T H2 b C < 0; 7 þ A H1 A 0 5
S2
ð46Þ
364
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
where H1 ; H2 are defined in (23) and
b TQ1P b1P b TQ1P b 2 þ kT Q 2 kmþ1 ð1 lÞkT Q 2 kmþ2 þ kT Q 3 k1 N ¼ kT1 PðAk1 þ Ad kmþ2 Þ þ ðAk1 þ Ad kmþ2 ÞT Pk1 þ P mþ1 mþ2 1 1 2 T T b T WP b 3 þ L1 P b4þP b T LT þ L2 P b5þP b T LT þ L3 P b6þP b T LT ; kmþ3 Q 3 kmþ3 kmþ4 R1 kmþ4 P 3 4 1 5 2 6 3
b ¼ ½ Ak1 þ Ad kmþ2 A b C ¼ ½ Ck1 þ Dkmþ2
0 0 0 ; ð47Þ
0 0 0 :
_ is unknown, we can get the corresponding Remark 4. If the time delay dðtÞ is not differential or the upper bound of dðtÞ criteria by setting Q 2 ¼ 0 in Theorem 1. For simplicity, we omit them here. 3.2. Tracking controller design In this section, the design of tracking controller (7) for system (1) is proposed in the following theorem. Theorem 2. Consider the uncertain stochastic system with time delay (1) and the reference system in (5). Given a integer m P 1, scalars l; d2 P d1 > 0; c > 0, and semi-positive-definite weighting matrix W P 0 in (12), there exists a memory state-feedback controller in (7) such that the H1 tracking requirements are achieved, if there exist positive-definite scalars e1 > 0; e2 > 0; e3 > 0, b 1 > 0; Q b 2 > 0; Q b 3 > 0; R b > 0; Q b 1 > 0; R b 2 > 0; b b K b 1; K b 2 with L1 ; b L2 ; b L 3 ; U; positive-definite matrices P S 1 > 0; b S 2 > 0, and b appropriate dimensions such that the following LMIs hold
" b ¼ W 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
b2 R bT U
b U b2 R
K11 eT1 Br c2 I
# P 0;
ð48Þ
b L1
b L2
b L3
K16
0
0
N br
T
K17
K18
K19
K1;10
N br
BTr
0
0
T
0 b S1
0
0
0
0
0
0
0
b S2
0
0
0
0
0
0 b S2
0 e1 I
0 0
0 0
0 0
0 0
e2 I
0 0 0 K88 0 0 e3 I 0 K10;10
K1;11
3
7 0 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 < 0; 7 0 7 7 7 0 7 7 0 7 7 7 0 5
ð49Þ
I
where
K11
K16 K18 K1;10 K88
T b T b b b 1 þ eT B u K b mþ2 þ eT Bu K b 1 e1 þ sym eT Ad Pe b 2 emþ2 þ PT Q ¼ sym eT1 A Pe 1 1 1 1 1 P1 P2 Q 1 P2 þ emþ1 Q 2 emþ1
T b T T b 2 emþ2 þ eT Q b b b b b b ð1 lÞeTmþ2 Q 1 3 e1 emþ3 Q 3 emþ3 emþ4 R 1 emþ4 P3 WP3 þ sym L 1 P4 þ L 2 P5 þ L 3 P6 þe1 eT1 MMT e1 ; T T b T þ eT K b T bT T bT T ¼ K17 ¼ eT1 PN a 1 1 N bu þ emþ2 PN ad þ emþ2 K 2 N bu ; T T b T þ eT K b T b T þ eT PN b T; bT T bT T ¼ eT1 PA K19 ¼ eT1 PN 1 1 B u þ emþ2 PAd þ emþ2 K 2 B u ; mþ2 c d b T þ eT PD b T ; K1;11 ¼ eT PW b T; ¼ eT1 PC mþ2 1 0 2 d1 b 1 þ ðd2 d1 Þ2 R b2 2P b þ e2 MMT ; R ¼ m
K10;10 ¼
d1 m
b b þ e3 MMT : S 1 þ ðd2 d1 Þb S2 P ð50Þ
When the LMIs in (48)–(50) are feasible, the controller gains of (7) can be obtained by
b 1 ; b iP Ki ¼ K
i ¼ 1; 2:
ð51Þ
365
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
b 1 Y 1 ; R1 ¼ P b 1 ; Y ¼ diag P b 1 R b1 P b 1 ; R2 ¼ P b 1 R b2 P b 1 ; S1 ¼ P b 1 b b 1 ; S2 ¼ P b 1 b b 1 ; b P b P b ; Q 1 ¼ Y 1 Q Proof. Set P ¼ P S1 P S2 P 1 b1 b 1 1 b1 b 1 1 b1 b 1 1 b b 1 b U ¼ P U P ; L1 ¼ Y L 1 P ; L2 ¼ Y L 2 P ; L3 ¼ Y L 3 P . Pre-multiplying and post-multiplying (48) by matrix b 1 ; P b 1 Þ, it can be easily verified that diagð P
" 06
b 1 P 0
#"
0 b 1 P
b U b R2
b2 R bT U
#"
b 1 P
0 b 1 P
0
#
" ¼
b 1 P 0
0 b 1 P
#"
b b 2P PR b b PU P
b P b PU b b PR2 P
#"
b 1 P 0
0 b 1 P
#
¼
R2
U
U
R2
b iP b 2 P; b This means that the condition in (48) implies the one in (21). Meanwhile, noting H1 6 PH i condition holds from (49)
2
K11
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
eT1 Br
b L1
b L2
b L3
K16
c2 I
0
0
0
N br
b S1
0
0
S2 b
K17
K18
K19
K1;10
K1;11
N br
BTr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
e3 I
0
T
T
b S2
e1 I
0
0
e2 I
0
K
0 88
K
0 0 10;10
0
:
ð52Þ
i ¼ 1; 2, the following
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 < 0; 7 7 7 7 7 7 7 7 7 7 7 7 7 5
ð53Þ
I
where T K088 ¼ H1 1 þ e2 MM ;
ð54Þ
T K010;10 ¼ H1 2 þ e3 MM ;
and H1 ; H2 are defined in (23). Then, using Schur complement lemma to (53), we obtain
2
3
3 2 T 3 2 K16 e1 M 7 7 6 6 T 7 6 0 7 6 N br 7 c2 I 0 0 0 7 7 7 7 6 6 7 7 7 T 6 1 6 T 7 b þ e e þ 0 0 0 0 N br 1 6 0 7 M e1 0 0 7 S1 1 6 0 7 K16 7 7 6 6 7 7 6 6 0 7 7 b 0 5 5 4 4 0 5 S2 0 0 S2 b 3 3 2 2 K17 K18 6 T 7 6 T 7 6 N br 7 6 Br 7 7 7
6 6 T 7 T 7 1 6 6 T 1 þ e1 N br 0 0 0 þ 6 0 7 H1 e2 MM K18 Br 0 0 0 2 6 0 7 K17 7 7 6 6 6 0 7 6 0 7 5 5 4 4
K011
6 6 6 6 6 6 6 6 4
b L1
eT1 Br
b L2
b L3
0 3
2
K19 7 6 6 0 7 7 6 7 T 6 þ e1 3 6 0 7 K19 7 6 6 0 7 5 4 0
0 0 0
ð55Þ
0 3
2
K1;10 7 6 7 60 7
6 h T 7 1 6 T 1 K1;10 7 H2 e3 MM 0 0 0 0 þ 60 7 6 7 60 5 4
i 0 0 0 0 < 0;
0
where
T b T b b b 1 þ eT B u K b mþ2 þ eT Bu K b 1 e1 þ sym eT Ad Pe b 2 emþ2 þ PT Q K011 ¼ sym eT1 A Pe 1 1 1 1 1 P1 P2 Q 1 P2 þ emþ1 Q 2 emþ1 T b T T b 2 emþ2 þ eT Q b b b ð1 lÞeTmþ2 Q 1 3 e1 emþ3 Q 3 emþ3 emþ4 R 1 emþ4 P3 WP3
b T W 0 P: b L 1 P4 þ b L 2 P5 þ b L 3 P6 þ eT1 PW þ sym b 0
ð56Þ
366
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
Then, applying Lemma 3 to (55) yields
3 2 3 2 3T 2 3 2 3T b b K18 ðtÞ K18 ðtÞ K1;10 ðtÞ K1;10 ðtÞ L2 L3 K11 ðtÞ eT1 Br ðtÞ bL 1 7 6 7 6 7 7 6 7 6 6 c2 I 0 0 0 7 6 BTr ðtÞ 7 6 BTr ðtÞ 7 6 6 0 7 6 0 7 7 6 7 6 7 7 6 7 6 6 7 6 7 7 6 7 6 6 6 b 0 7 S1 0 7 þ 6 0 7H1 6 0 7 þ 6 0 7H2 6 0 7 6 7 6 7 6 7 7 6 7 6 6 b 0 0 0 0 5 5 5 5 5 4 4 4 4 4 S2 0 b 0 0 0 0 S2 3 2 T 3 3 2 0 2 T b b b K16 e1 M K11 e1 Br L 1 L2 L3 T 7 7 7 6 6 6 0 0 7 6 0 7 6 c2 I 0 6N 7 7 6 7 6 T 6 br 7 T T 7 7 6 6 ¼6 b 0 7 S1 0 7 þ 6 0 7FðtÞ K16 N br 0 0 0 þ 6 0 7FðtÞ M e1 0 0 0 0 6 7 6 7 7 6 6 4 4 0 5 b S2 0 5 4 0 5 0 0 b S2 T T T T þ K18 Br 0 0 0 MFðtÞ K17 N br 0 0 0 H1 K18 Br 0 0 0 þ MFðtÞ KT17 N br 0 0 0 nh i h ioT n h i o þ KT1;10 0 0 0 0 þ MFðtÞ KT1;9 0 0 0 0 H2 KT10 0 0 0 0 þ MFðtÞ KT1;9 0 0 0 0 < 0; 2
ð57Þ where
b 1 þ eT Bu ðtÞ K b mþ2 þ eT Bu ðtÞ K b 1 e1 þ sym eT Ad ðtÞ Pe b 2 emþ2 K11 ðtÞ ¼ sym eT1 AðtÞ Pe 1 1 1 T T T b T b 1 P1 PT Q b b b b þ PT1 Q 2 1 P2 þ emþ1 Q 2 emþ1 ð1 lÞemþ2 Q 2 emþ2 þ e1 Q 3 e1 emþ3 Q 3 emþ3
b 3 þ sym b b 1 emþ4 PT WP b P; b L 1 P4 þ b L 2 P5 þ b L 3 P6 þ eT1 PW eTmþ4 R 3 T T b T ðtÞ þ eT K b T b T b T K18 ðtÞ ¼ eT1 PA 1 1 B u ðtÞ þ emþ2 PAd ðtÞ þ emþ2 K 2 B u ðtÞ; b T ðtÞ þ eT PD b T ðtÞ: K1;10 ðtÞ ¼ eT PC 1
mþ2
ð58Þ
Pre-multiplying and post-multiplying (58) by matrix
2
Y 1 6 6 0 6 6 0 6 6 4 0 0
0 I 0
0 0 b 1 P
0 0
0
0
0 b 1 P
0
0
0
3 0 7 0 7 7 0 7 7; 7 0 5 b 1 P
it is obvious that the condition (31) in Theorem 1 is hold. Then, by Theorem 1, the robust H1 tracking requirements are obtained. This completes the proof. h If there are no uncertainties in system (1), that is
dxðtÞ ¼ ½AxðtÞ þ Ad xðt dðtÞÞ þ Bu uðtÞ þ Br rðtÞdt þ ½CxðtÞ þ Dxðt dðtÞÞdxðtÞ;
ð59Þ
By the same proof, we have the following corollary. Corollary 3. Consider the stochastic system (59) and the reference system in (5). For given integer m P 1, scalars l, d2 P d1 > 0; c > 0, and semi-positive-definite weighting matrix W P 0 in (12), there exists a memory state-feedback controller in b 1 > 0; b > 0; Q (7) such that the H1 tracking requirements are achieved, if there exist positive-definite matrices P b b b b b b b b b b b b Q 2 > 0; Q 3 > 0; R 1 > 0; R 2 > 0; S 1 > 0; S 2 > 0, and L 1 ; L 2 ; L 3 ; U; K 1 ; K 2 such that (48) and the following LMIs hold
2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
b 11 eT Br K 1
b L1
b L2
b L3
b 16 K
b 1;7 K
c2 I
0
0
0
BTr
0
b S1
0
0
0
0
b S2
0
0
0
0
0
b 66 K
0
b S2
b 77 K
b 1;8 K
3
7 0 7 7 7 7 0 7 7 7 0 7 7 < 0; 7 0 7 7 7 0 7 7 7 0 7 5
I
ð60Þ
367
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
where
T b T b 11 ¼ sym eT A PL b b b 1 e1 þ eT Bu K b mþ2 þ eT Bu K b 1 e1 þ sym eT Ad Pe b 2 emþ2 þ PT Q K 1 1 1 1 1 1 P1 P2 Q 1 P2 þ emþ1 Q 2 emþ1
T b T T b 2 emþ2 þ eT Q b b b b b b ð1 lÞeTmþ2 Q 1 3 e1 emþ3 Q 3 emþ3 emþ4 R 1 emþ4 P3 WP3 þ sym L 1 P4 þ L 2 P5 þ L 3 P6 ; T T b 16 ¼ eT PA b T þ eT K b T bT T bT T K 1 1 1 B u þ emþ2 PAd þ emþ2 K 2 B u ; b 1;8 ¼ eT PW b 1;7 ¼ eT PC b T þ eT PD b T; K b T; K 0 1 mþ2 1 2 b 66 ¼ d1 R b 1 þ ðd2 d1 Þ2 R b 2 2 P; b K m b 77 ¼ d1 b b K S 1 þ ðd2 d1 Þb S 2 P: m
ð61Þ
When the LMIs in (48) and (60) are feasible, the controller gains of (7) can be obtained by
b 1 ; b iP Ki ¼ K
i ¼ 1; 2:
ð62Þ
4. Numerical Examples Example 1. Consider system (42) with following parameters
2:1829 0 1:0293 0:4123 0:0210 0:2123 ; Ad ¼ ; C¼ ; A1 ¼ 0 3:2092 2:9212 0:1234 0:0212 0:0532 0:1029 0 0:5021 0:2293 0:2345 0 ; M¼ ; Na ¼ ; D¼ 0 0:1212 0:2932 0:2283 0 0:1282 0:1123 0:1233 0:5124 0:4234 0:9213 0:7123 ; Nc ¼ ; Nd ¼ : Nad ¼ 0:1242 0:1134 0:2123 0:2234 0:2223 0:2123 To demonstrate the effectiveness of our approach, the allowable maximum values of d2 such that the system (42) is robustly stochastically stable in mean square are listed in Table 1. To demonstrate the effectiveness of our approach, results are compared with the recent ones in Table 1. Remark 5. From Table 1, it is clear the upper bounds obtained in Corollary 1 for different l are larger than the corresponding ones in [34]. That means that the result of Corollary 1 is less conservative than the criterion in [34]. Clearly, the conservatism of the results is considerably reduced when the partitioning number m increases. Example 2. Consider the stochastic system (1) with the following matrix parameters
A¼
;
1:1829
0:1000
0:1102 0:1934
0:2292 0:1924
Ad ¼
1:0293
0:4123
;
Bu ¼
0:2000
0:2943
; 0:1923 0:1000 0:9212 1:0234 0:0210 0:2123 0:0121 0:2293 Br ¼ ; C¼ ; D¼ ; 0:0930 0:2054 0:0212 0:0532 0:2932 0:2283 1:2922 0:8283 0:3983 0:2823 0:0011 0:0090 E¼ ; Ed ¼ ; M¼ ; 0:9135 1:4821 0:8291 0:1934 0:0012 0:0032 0:0012 0:0017 0:0041 0:0021 0:0031 0:0045 Na ¼ ; Nad ¼ ; Nbu ¼ ; 0:0002 0:0014 0:0018 0:0082 0:0012 0:0203 0:0053 0:0071 0:0029 0:0003 0:0067 0:0012 ; Nc ¼ ; Nd ¼ ; N br ¼ 0:0312 0:0013 0:0101 0:0132 0:0019 0:0103 0:1000 0 W0 ¼ ; d1 ¼ 0:1; d2 ¼ 0:2; l ¼ 0:3; c ¼ 2: 0 0:1000
Table 1 The maximum allowable delay bound of d2 for different l with d1 ¼ 0:3, Method
l ¼ 0:7
l ¼ 0:75
l ¼ 0:78
l ¼ 0:8
l ¼ 0:85
Theorem 2 in [34] (m ¼ 1) Theorem 2 in [34] (m ¼ 2) Corollary 1 in this paper (m ¼ 1) Corollary 1 in this paper (m ¼ 2)
1.4921 1.5983 1.6379 1.8632
0.8592 0.8709 0.8897 0.9739
0.7924 0.8023 0.8153 0.8314
0.7495 0.7725 0.7940 0.8123
0.7183 0.7694 0.7739 0.7986
368
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
Set m ¼ 1, then, solving the LMIs (48) and (60) in Corollary 3, obtains
2
14:8745
1:1026 0:4009 0:3532
6 6 1:1026 b¼6 P 6 4 0:4009
15:3164
0:0229
0:0229
10:6279
3
2
7 0:1115 7 7; 7 1:9967 5
0:0383 0:1480 23:6824
6 6 0:0151 b1 ¼ 6 R 6 4 0:1542 0:1300 2
19:1541
6 6 0:1074 b S1 ¼ 6 6 4 0:1640
0:0151 0:3036
24:9692
7 0:3083 7 7; 7 2:4753 5
0:3083
2:4753
23:5688
0:1074 0:1640 0:1186 19:0718
0:1179
0:1179
19:8811
0:0578
1:6463
1:1790
0:0466
0:0053
6 6 0:0054 1:1305 b ¼6 U 6 4 0:0473 0:0351 "
0:0303
0:0017
2
0:6923
0:5763 1:2565
47:3179 2:5829 1:9273
24:366
1:1032 1:8852 40:0704
0:0837
0:2377
1:2244
24:6804
18:4135
0:0242 0:1224 0:0998
2
6 6 0:0242 b S2 ¼ 6 6 4 0:1224 0:0998 " b1 ¼ K
2:4580
0:2205
18:3720
0:1874
0:1874
19:2921
0:1837
1:1410
b 1 b 2P K2 ¼ K
13:3970 24:6642 18:2564 19:7096
e1 ¼ 17:1180; e2 ¼ 17:1187; e3 ¼ 17:1080:
0:0200 2:9762 1:4814 1:4909 ; 0:9520 1:6704 1:4138 1:4458 4:2761 2:3857 0:3461 0:1917 ¼ : 0:4382 3:1192 0:2860 0:1763
2
0.5 x1 y1
1
x2 y2
0
0
−1 −2
0
50
100
2
−0.5
0
50
100
1 tracking error e1
tracking error e2
1
0.5
0
0
−1
−0.5
−2
0
50
18:7248
# ;
100
−1
0
50
Fig. 1. The responses of open-loop systems.
100
3
7 0:1837 7 7; 7 1:1410 5
45:7386 18:6443 20:1577
and b L1; b L2 ; b L 3 are omitted for simplicity. Therefore, the controller gains of (7) can be obtained as
b 1 ¼ b 1P K1 ¼ K
6:7328 3
0:0954 0:0837 25:3828
18:9973 3
7 0:0371 7 7; 7 0:0143 5
3
7 0:2377 7 7; 7 1:2244 5
0:0001
0:0732
0:1954
6 6 0:0704 24:3616 b2 ¼ 6 R 6 4 0:0954 0:2205
3
7 0:0578 7 7; 7 1:6463 5
60:9034 31:8547 1:6362 3:2547 3:0266
0:2072 3
0:3036
0:1258
0:1954 0:8845 0:7717 4:0868 3 2 9:2671 0:3759 0:1284 0:2072 7 6 6 0:3759 9:6585 1:0400 1:1032 7 b3 ¼ 6 7; Q 7 6 4 0:1284 1:0400 7:6997 1:8852 5
4:1325
23:7301
0:1186 2
b2 ¼ K
1:2925
0:1542 0:1300
0:8153
7 6 6 0:8153 8:2852 0:8326 0:8845 7 b1 ¼ 6 7; Q 7 6 4 0:1258 0:8326 4:5729 0:7717 5
0:3532 0:1115 1:9967 11:3186 3 2 4:8365 0:1520 0:0695 0:0383 7 6 6 0:1520 4:9255 0:1563 0:1480 7 b2 ¼ 6 7; Q 7 6 1:2925 5 4 0:0695 0:1563 5:3544 2
7:6531
# ;
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
2
0.4 x1 y2
1
0
0
−0.2
−1 −2
x2 y2
0.2
−0.4 0
50
100
2
−0.6
0
100
1 tracking error e1
tracking error e2
1
0.5
0
0
−1
−0.5
−2
50
0
50
100
−1
0
50
100
Fig. 2. The responses of closed-loop systems.
Furthermore, we assume
dðtÞ ¼ ð1 þ j sinðtÞjÞ=10; 0:5 cosðtÞ 0 JðtÞ ¼ ; 0 0:5 cosðtÞ 2 sinð0:5tÞ ; rðtÞ ¼ 2 cosð0:1tÞhðtÞ 2hðtÞ cosð0:3tÞ ; v ðtÞ ¼ 0:3 sinð0:2tÞ where hðtÞ is a step signal function. Meanwhile, let the initial conditions of system (1) and (2) be /ðtÞ ¼ ½ 0:2 0:2 ; wðtÞ ¼ ½ 0 0 , then, the responses of real system states, reference system states and states of tracking error for the open-loop system are shown in Fig. 1. The corresponding responses of closed-loop system are shown in Fig. 2. It is easy to see from the figures that the real system states tend to track the reference system states effectively with the designed controller.
Remark 6. From Figs. 1 and 2, it is easy to see that the states of the closed-loop real systems are better than the open-loop ones to follow the reference systems states, and the errors of closed-loop systems are much smaller than the ones of openloop systems, which show the effectiveness of obtained criterion.
5. Conclusion The problem of robust H1 tracking control has been investigated in this paper. New delay-dependent conditions for the addressed problem have been obtained by using delay-partitioning-dependent Lyapunov–Krasovskii functionals together with reciprocally convex approach. Numerical examples have also been provided to illustrate the effectiveness of the proposed methods. Acknowledgment The work of J. Xia was supported National Natural Science Foundation of China under Grant 61403178, 61403228, 61104117. Zijin Intelligent Program, Nanjing University of Science and Technology, Nanjing, China, under Grant ZJ0104. Also, the work of J.H. Park was supported by the 2014 Yeungnam University Research Grant. References [1] E. Boukas, S. Xu, J. Lam, Robust On Stability and Stabilizability of Singular Stochastic Systems with Delays, J. Optim. Theory Appl. 127 (2005) 249–262. [2] H. Gao, J. Lam, Z. Wang, Discrete bilinear stochastic systems with time-varying delay: stability analysis and control synthesis, Chaos, Solitons Fractals 34 (2007) 394–404. [3] Y. He, G.P. Liu, D. Rees, New delay-dependent stability criteria for neutral networks with time-varying delay, IEEE Trans. Neural Networks 8 (2007) 310–314.
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[4] J.H. Park, Delay-dependent criterion for asymptotic stability of a class of neutral equations, Appl. Math. Lett. 17 (2004) 1203–1206. [5] J.H. Park, O. Kwon, Novel stability criterion of time delay systems with nonlinear uncertainties, Appl. Math. Lett. 18 (2005) 683–688. [6] J.H. Park, Design of a dynamic output feedback controller for a class of neutral systems with discrete and distributed delays, IEE Proc. Control Theory Appl. 151 (2004) 610–614. [7] S. Xu, J. Lam, B. Zhang and Y, Zou. New insight into delay-dependent stability of time-delay systems. International Journal of Robust and Nonlinear Control, (in press). http://dx.doi.org/10.1002/rnc.3120. [8] Q. Song, J. Cao, Passivity of uncertain neural networks with both leakage delay and time-varying delay, Nonliear Dyn. 67 (2012) 1659–1707. [9] J. Xia, C. Sun, B. Zhang, New robust H1 control for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method, J. Franklin Inst. 349 (2012) 741–769. 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Kwon, P. Park, Delay-dependent robust H1 control for uncertain systems with a state-delay, Automatica 40 (2004) 65–72. [17] K. Gu, V. Kharitonov, J. Chen, Stability of time-delay systems, Birkhauser (2003). [18] Y. He, M. Wu, J. She, G. Liu, Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE Trans. Autom. Control 49 (2004) 828–832. [19] S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems, Int. J. Sys. Sci. 39 (2008) 1095–1113. [20] Z. Wu, H. Su, J. Chu, Delay-dependent H1 control for singular Markovian jump systems with time delay, Optimal Control Appl. Methods 30 (2009) 443–461. [21] Q. Han, D. Yue, Absolute stability of Lur’e systems with time-varying delay, IET Control Theory Appl. 1 (2007) 854–859. [22] Y. He, G. Liu, D. Rees, Augmented Lyapunov functional for the calculation of stability interval for time-varying delay, IET Control Theory. Appl. 1 (2007) 381–386. [23] H. Shao, New delay-dependent stability criteria for systems with interval delay, Automatica 45 (2009) 744–749. [24] R. Yang, Z. Zhang, P. Shi, Exponential stability on stochastic neural networks with discrete interval and distributed delays, IEEE Trans. Neural Netw. 2 (2010) 169–175. [25] P. Park, J. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (2011) 235–238. [26] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica 49 (2013) 2860–2866. [27] S. Xu, T. Chen, Robust H1 control for uncertain stochastic systems with state delay, IEEE Trans. Autom. Control 47 (2002) 2089–2094. [28] H. Gao, J. Lam, C. Wang, Robust energy-to peak filter design for stochastic time-delay systems, Sys. Control Lett. 55 (2006) 101–111. [29] Z. Wu, P. Shi, H. Su, J. Chu, Asynchronous l2 l1 filtering for discrete-time stochastic Markovian jump systems with randomly occurred sensor nonlinearities, Automatica 51 (2014) 180–186. [30] H. Li, H. Gao, P. Shi, X. Zhao, Fault-tolerant control of Markovian jump stochastic systems via augmented sliding mode observer approach, Automatica 50 (2014) 1825–1834. [31] J. Xia, S. Xu, B. Song, Delay-dependent L2 L1 filter design for stochastic time-delay systems, Sys. Control Lett. 56 (2007) 579–587. [32] H. Li, B. Chen, Q. Zhou, C. Lin, Delay-dependent robust stability for stochastic time-delay systems with polytopic uncertainties, Int. J. Robust Nonlinear Control 18 (2008) 1482–1492. [33] L. Huang, X. Mao, Robust delayed-state-feedback stabilization of uncertain stochastic systems, Automatica 45 (2009) 1332–1339. [34] C. Wang, Y. Shen, Delay partitioning approach to robust stability analysis for uncertain stochastic systems with interval time varying delay, IET Control Theory Appl. 6 (2012) 875–883. [35] Y. Ji, J. Qiu, Robust stability problems of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays, Appl. Math. Comput. 238 (2014) 70–81. [36] W. Ren, Y. Cao, Distributed Coordination of Multi-agent Networks, Springer-verlag, London, 2011. [37] G. Wen, G. Hu, W. Yu, J. Cao, G. Chen, Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs, Sys. Control Lett. 62 (2013) 1151–1158. [38] E. Boukas, On reference model tracking for Markov jump systems, Int. J. Sys. Sci. 40 (2009) 393–401. [39] G. Nakura, On noncausal H1 tracking control for linear discrete-time Markovian jump systems, In proc. IEEE International Conference on Control Application, Yokohama, Japan, September 2010, 1981–1986. [40] Z. Chen, Y. Li, Robust output tracking control for Markovian jump time-delay systems with uncertain mode transition rates, in Proc.: the 26th Chinese Control Conference, Zhangjiajie, China, July 2007, 545–548. 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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
H1 tracking of uncertain stochastic time-delay systems: Memory state-feedback controller design Jianwei Xia a,b, Ju H. Park b,⇑, Tae H. Lee b,⇑, Baoyong Zhang c a
School of Mathematics Science, Liaocheng University, Shandong 252000, China Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 712-749, Republic of Korea c School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China b
a r t i c l e
i n f o
Keywords: Stochastic systems Robust H1 tracking control Stochastic stability Time delay
a b s t r a c t The problem of state robust H1 tracking control for uncertain stochastic systems with interval time-varying delay is considered in this paper. The target is to design a memory state feedback controller such that the resulting closed-loop augmented system is robustly stochastically asymptotically stable and the states of original systems can follow the reference signal with given H1 performance. Firstly, by constructing delay-partitioningdependent Lyapunov–Krasovskii functional with reciprocally convex approach, a delaydependent condition guaranteeing the robust H1 tracking performance is proposed in a set of matrix inequalities. Secondly, new stability criteria for stochastic systems with time-varying delay are given with less conservativeness than some recent results. Thirdly, memory state feedback controller is designed via a set of linear matrix inequalities (LMIs) which can be solved by Matlab LMI toolbox. At last, numerical simulations are addressed to show the effectiveness of the proposed methods. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In the past several decades, the study for stochastic systems has drawn much attention due to their significant rolesin many branches of engineering and science applications, such as missile autopilot control, satellites attitude control and chemical process control, etc [1,2]. It is well known that time delay is usually inevitable encountered and often the main reason for instability and poor performance for almost all practical models [3–13]. Meanwhile, there has been a growing interest in the stability analysis for delayed system in recent years. For the reason that the delay-dependent results are usually less conservative than the delay-independent ones,much more consideration has been focused on the technique design to achieve delay-dependent stability conditions[14–26]. Naturally, a great deal of results related to stability analysis and synthesis for stochastic systems with time delays have been reported in the recent literature. For example, the delay-independent and delay-dependent conditions of exponential stability, robust stabilization, robust H1 control, H1 filtering and L2 L1 filtering were proposed in [27–35], respectively. Especially, the authors in [30] provided a novel fault-tolerant control method for Markovian jump stochastic systems and the results can be well applied to the real application systems. On the other hand, state/output tracking is an important issue of the system and control theory, which has been extensively applied in robot control, flight control, signal processing and other practical fields [36,37]. The target of tracking
⇑ Corresponding authors. E-mail addresses: [email protected] (J. Xia), [email protected] (J.H. Park), [email protected] (T.H. Lee), [email protected] (B. Zhang). http://dx.doi.org/10.1016/j.amc.2014.10.029 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
control problem is to design a controllers such that the state of the closed-loop system follows a prescribed reference signal. And some results of this topic have been made in this field in recent years. For example, the problem of state H1 tracking control for continuous and discrete linear systems were considered in [38,39]. When time delay was considered, delayindependent and delay-dependent sufficient conditions of state H1 tracking problem were obtained for time-delay Markovian jump systems and switched systems in [40–42], respectively. When the stochastic terms were considered in the systems, the output feedback tracking control and adaptive neural tracking control for stochastic systems were considered in [43,44]. However, to the best of authors’ knowledge, there is few effort to study the state H1 tracking control problem for uncertain stochastic systems with time-varying delays, which motivates our present study. In this paper, we address the problem of memory state robust H1 tracking control for uncertain stochastic systems interval time-varying delays. By a novel delay-partitioning-dependent Lyapunov–Krasovskii functional with reciprocally convex approach, a delay-dependent condition guaranteeing the robust H1 tracking performance is obtained. Meanwhile, improved stability criteria for stochastic systems with time-varying delay are proposed with less conservativeness. Then, existence conditions and design methods of the desired controller are constructed by solving a set of linear matrix inequalities. Some numerical examples will show the verification of our design methods. Notation:Throughout the paper, for symmetric matrices X and Y, the notation X P Y (respectively, X > Y) means that the matrix X Y is positive semi-definite (respectively, positive definite); I is the identity matrix with appropriate dimension; M T represents the transpose of the matrix M; Rn denotes the ndimensional Euclidean space; 0mn represents a zero matrix with m n dimensions; jj jj denotes the Euclidean norm for vector or the spectral norm of matrices; symðAÞ denotes A þ AT ; ðX; F; P) is a probability space, where X is the sample space, F is the
r-algebra of subsets of the sample space, and
P is the probability measure on F; and L2F 0 ð½h; 0; Rn Þ denotes the family of all F 0 measurable Cð½h; 0; Rn Þ-valued random variables n ¼ fnðhÞ : h 6 h 6 0g such that suph6h60 EjnðhÞj2 g < 1, where Efg stands for the expectation operator with respect to some probability measure P. The notations X > 0(P 0) is used to denote a symmetric positive-definite (positive-semidefinite) matrix. In symmetric block matrices or complex matrix expressions, we use an asterisk to represent a term that is induced by symmetry, and diagfg stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. System description and preliminaries Consider the following uncertain stochastic system with time delay
dxðtÞ ¼ ½AðtÞxðtÞ þ Ad ðtÞxðt dðtÞÞ þ Bu ðtÞuðtÞ þ Br ðtÞrðtÞdt þ ½CðtÞxðtÞ þ DðtÞxðt dðtÞÞdxðtÞ;
ð1Þ
xðtÞ ¼ /ðtÞ; t 2 ½2d2 ; 0;
ð2Þ
where x 2 Rn is state vector; uðtÞ 2 Rp is control input vector; rðtÞ 2 Rq is bounded disturbance; dðtÞ is time differential func_ 6 l, where d1 ; d2 and l are known constant scalars; tion representing time-varying delay and satisfies 0< d1 6 dðtÞ 6 d2 ; dðtÞ wðtÞ is a one-dimension Brownian motion satisfying EfdwðtÞg ¼ 0 and EfdwðtÞ2 g ¼ dt; /ðtÞ is a vector-valued initial continuous function defined on the interval ½2d2 ; 0; the uncertain matrices AðtÞ; Ad ðtÞ; Bu ðtÞ; Br ðtÞ; CðtÞ and DðtÞ are assumed to
½ AðtÞ Ad ðtÞ Bu ðtÞ Br ðtÞ CðtÞ DðtÞ ¼ ½ A Ad
Bu
Br
C
D þ MJðtÞ½ Na
Nad
Nbu
N br
Nc
Nd ;
ð3Þ
where A; Ad ; Bu ; Br ; C; D; M; N a N ad ; N bu ; N br ; N c , and N d are real constant matrices. JðtÞ is an unknown norm-bounded time-varying matrix satisfying
J T ðtÞJðtÞ 6 I:
ð4Þ
The reference signal is assumed to be generated by the following deterministic system
_ yðtÞ ¼ EyðtÞ þ Ed yðt dðtÞÞ þ v ðtÞ;
ð5Þ
yðtÞ ¼ wðtÞ; 8t 2 ½2d2 ; 0;
ð6Þ
where yðtÞ 2 Rn is reference signal state vector; v ðtÞ is a bounded reference input; and E and Ed are known real matrices with appropriate dimensions. Without loss of generality, it is assumed that this reference system is stable. Design memory state feedback controller in the following form
uðtÞ ¼ K 1 xðtÞ þ K 2 xðt dðtÞÞ þ F 1 yðtÞ þ F 2 yðt dðtÞÞ;
ð7Þ
where K 1 ; K 2 ; F 1 and F 2 are the controller gains to be determined. Remark 1. It is well known that, because more information is considered, memory controllers usually lead to the better performances than memoryless ones for the designed systems. In this paper, we discuss a synthesis problem of memory controller which guarantee the closed-loop augmented system is robustly stochastically asymptotically stable and the original states can follow the reference signal with given H1 performance. Applying the controller (7) into the system (1), we obtain an augmented closed-loop system as following
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
dnðtÞ ¼ ½ðAðtÞ þ Bu ðtÞK1 ÞnðtÞ þ ðAd ðtÞ þ Bu ðtÞK2 Þnðt dðtÞÞ þ Br ðtÞrðtÞdt þ ½CðtÞnðtÞ þ DðtÞnðt dðtÞÞdwðtÞ;
uðtÞ ¼
/ðtÞ
wðtÞ
; 8t 2 ½2d2 ; 0;
ð8Þ ð9Þ
where
" nðtÞ ¼
xðtÞ yðtÞ
#
"
;
rðtÞ ¼
rðtÞ
# ;
v ðtÞ "
A 0
#
K1 ¼ ½ K 1 "
M
F 1 ;
K2 ¼ ½ K 2
F 2 ;
#
þ JðtÞ½ Na 0 ; 0 E 0 " # " # Ad 0 M Ad ðtÞ ¼ Ad þ MJðtÞN ad ¼ þ JðtÞ½ Nad 0 ; 0 Ed 0 " # " # Bu M Bu ðtÞ ¼ Bu þ MJðtÞN bu ¼ þ JðtÞN bu ; 0 0 " # " # Br 0 M þ JðtÞ½ Nbr 0 ; Br ðtÞ ¼ Br þ MJðtÞN br ¼ 0 I 0 " # " # C 0 M CðtÞ ¼ C þ MJðtÞN c ¼ þ JðtÞ½ Nc 0 ; 0 0 0 " # " # D 0 M þ JðtÞ½ Nd 0 ; DðtÞ ¼ D þ MJðtÞN d ¼ 0 0 0 AðtÞ ¼ A þ MJðtÞN a ¼
ð10Þ
and uðtÞ denotes the initial condition of system (8). Define state tracking error as
eðtÞ ¼ xðtÞ yðtÞ;
ð11Þ
and denote the tracking performance constraint as
E
Z
tf
eðtÞT WeðtÞdt
6 c2
0
Z
tf
rðtÞT rðtÞdt þ d;
ð12Þ
0
where tf P 0 is an terminal time; c > 0 is a prescribed scalar representing performance index; W P 0 is a weighting matrix; d P 0 is a weighting scalar related to the initial conditions. Set
W¼
W
W
W
W
:
ð13Þ
Then, the inequality (12) can be rewritten as
E
Z
tf
nðtÞT WnðtÞdt
0
6 c2
Z
tf
rðtÞT rðtÞdt þ d:
ð14Þ
0
Furthermore, since W P 0, there must exist a matrix W 0 such that W ¼ W T0 W 0 . Therefore, the matrix W in (13) can be rewritten as
W¼
I I
W T0 W 0 ½ I
I ¼ W T0 W 0 :
ð15Þ
In order to obtain our main results, the following definitions and lemmas are employed throughout our paper. Definition 1 [27]. For system (1) with uðtÞ ¼ 0 is said to be robustly stochastically stable in mean square if for any there is a dðeÞ > 0 such that
EjxðtÞj2 < e;
t > 0;
for any initial conditions and all admissible uncertainties, when
sup EjxðtÞj2 < dðeÞ: l6s60
e > 0,
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If, in addition,
lim EjxðtÞj2 ¼ 0;
t!1
for any initial conditions and all admissible uncertainties, then system (1) with uðtÞ ¼ 0 is said to be robustly stochastically asymptotically stable in mean square. Definition 2. The robust H1 state tracking requirements of system (1), (2) associated with reference system 5,6 are said to be met, if there exists a state-feedback controller given in (7) such that the closed-loop augmented system (8) is robustly stochastically asymptotically stable in mean square with rðtÞ ¼ 0 and the state tracking error eðtÞ (11) and the bounded disturbance rðtÞ in (8) satisfy the relationship described in (12) or (14) for all admissible uncertainties. Lemma 1 [17]. For any constant matrix T 2 Rmm ; T ¼ T T > 0, scalar c > 0, vector function x : ½0; c ! Rm such that the integrals in the following are well defined, then
c
Z
c
xðbÞT T xðbÞdb P
Z
0
T Z
c
c
xðbÞdb T
0
xðbÞdb:
ð16Þ
0
Lemma 2 [25]. Let f 1 ; f 2 ; . . . ; f N : Rm ! R have positive values in an open subsets D of Rm . Then, the reciprocally convex combination of f i over D satisfies
X1
min P
fai jai >0;
a ¼1g i i i
ai
f i ðtÞ ¼
X
X f i ðtÞ þ max g ij ðtÞ g i;j ðtÞ
i
ð17Þ
i–j
subject to
(
"
f i ðtÞ g ij ðtÞ g ij ðtÞ : R ! R; g ij ðtÞ ¼ g ji ðtÞ; g ji ðtÞ f j ðtÞ
#
)
m
P0 :
ð18Þ
Lemma 3 [9]. Let A; D; E be real constant matrices with appropriate dimensions, matrix JðtÞ satisfies J T ðtÞJðtÞ 6 I. For any such that P1 eDDT > 0, then
DJðtÞE þ ET J T ðtÞDT 6 e1 DDT þ eET E
e > 0,
ð19Þ
1
ðA þ DJðtÞEÞT PðA þ DJðtÞEÞ 6 e1 ET E þ AT ðP1 eDDT Þ A: 3. Main results For simplicity of vector and matrix representation, we define
h
CðtÞ ¼ nðtÞT
nðt m1 d1 Þ
T
T
nðt m1 d1 Þ m
" T
T
T
vðtÞ ¼ CðtÞ nðt d1 Þ nðt dðtÞÞ nðt d2 Þ
ej ¼ 02n2ðj1Þn
kj ¼ 0nðj1Þn
I2n In
02n2ðmþ6jÞn ;
0nðmþ6jÞn ;
T
Z
iT 2mn1
t
;
T
f ðsÞ ds
1d tm 1
j ¼ 1; 2; . . . ; m þ 6:
Z
td1
tdðtÞ
T
f ðsÞ ds
Z
tdðtÞ
td2
#T T
f ðsÞ ds
; 2ðmþ6Þn1
ð20Þ
3.1. Tracking performance analysis Firstly, the robust H1 tracking performance analysis for the system (1) is considered in the following theorem, which is proposed in a set of matrix inequalities. Theorem 1. Given an integer m P 1, and scalars l; d2 P d1 > 0, c > 0, the robust H1 state tracking requirements are met, if there exist positive-definite matrices P > 0; Q 1 > 0; Q 2 > 0; Q 3 > 0; R1 > 0; R2 > 0; S1 > 0, S2 > 0, and matrices U; L1 ; L2 ; L3 such that the following matrix inequalities hold
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
W¼
R2
U
UT
R2
2
P 0;
ð21Þ
HðtÞ eT1 PBr ðtÞ
6 6 6 NðtÞ ¼ 6 6 6 4
L1
L2
c2 I
0
0
S1
0
S2
L3
3
2
3
b T AðtÞ
2
b T AðtÞ
3T
2
b CðtÞT
7 6 7 6 7 6 0 7 6 Br ðtÞT 7 6 Br ðtÞT 7 6 0 7 6 7 6 7 6 7 7 7 6 6 0 7 þ 6 0 7H1 6 0 7 þ 6 6 0 7 6 7 6 7 6 4 0 0 5 4 0 5 4 0 5 S2 0 0 0
3
2
b CðtÞT
3T
7 6 7 7 6 0 7 7 6 7 7H2 6 0 7 <0; 7 6 7 7 6 7 5 4 0 5
ð22Þ
0
where
b b T Pe1 þ PT Q 1 P1 PT Q 1 P2 þ eT Q 2 emþ1 ð1 lÞeT Q 2 emþ2 þ AðtÞ HðtÞ ¼ eT1 P AðtÞ mþ1 mþ2 1 2 þ eT1 Q 3 e1 eTmþ3 Q 3 emþ3 eTmþ4 R1 emþ4 PT3 WP3 þ L1 P4 þ PT4 LT1 þ L2 P5 þ PT5 LT2 þ L3 P6 þ PT6 LT3 þ eT1 We1 ; T T P1 ¼ eT1 eT2 eTm ; P2 ¼ eT2 eT3 eTmþ1 ; T T P3 ¼ emþ5 eTmþ6 ; P4 ¼ e1 e2 emþ4 ;
P5 ¼ emþ1 emþ2 emþ5 ; P6 ¼ emþ2 emþ3 emþ6 ;
ð23Þ
b AðtÞ ¼ ½AðtÞ þ Bu ðtÞK1 e1 þ ½Ad ðtÞ þ Bu ðtÞK2 emþ2 ; 2 d1 2 b R1 þ ðd2 d1 Þ R2 ; CðtÞ ¼ CðtÞe1 þ DðtÞemþ2 ; H1 ¼ m d1 H2 ¼ P þ S1 þ ðd2 d1 ÞS2 : m Proof. For convenience, we set
f ðtÞ ¼ ðAðtÞ þ Bu ðtÞK1 ÞnðtÞ þ ðAd ðtÞ þ Bu ðtÞK2 Þnðt dðtÞÞ þ Br ðtÞrðtÞ; gðtÞ ¼ CðtÞnðtÞ þ DðtÞnðt dðtÞÞ; then, system (1) becomes
dnðtÞ ¼ f ðtÞdt þ gðtÞdxðtÞ:
ð24Þ
On one hand, applying Schur complement lemma to (22), it is easily verified that
"
NðtÞ0 ¼
HðtÞ0 eT1 PBr ðtÞ
#
"
b T AðtÞ
#
"
b T AðtÞ
#T < 0;
ð25Þ
1 T 1 T T HðtÞ0 ¼ HðtÞ þ b CðtÞT H2 b CðtÞ þ L1 S1 1 L1 þ L2 S2 L2 þ L3 S2 L3 :
ð26Þ
c2 I
þ
Br ðtÞT
H1
Br ðtÞT
where
On the other hand, we choose Lyapunov–Krasovskii functional as
VðtÞ ¼
4 X V i ðtÞ;
ð27Þ
i¼1
where
V 1 ðtÞ ¼ nðtÞT PnðtÞ; Z t Z T V 2 ðtÞ ¼ C ðsÞ Q C ðsÞds þ 1 d t m1
td1
nðsÞT Q 2 nðsÞds þ
tdðtÞ
Z
t
nðsÞT Q 3 nðsÞds;
td2
Z Z t Z d1 Z t d1 0 T f ðsÞ R f ðsÞdsdh þ ðd d Þ f ðsÞT R2 f ðsÞdsdh; 1 2 1 m dm1 tþh tþh d2 Z 0 Z t Z d1 Z t gðsÞT S1 gðsÞdsdh þ gðsÞT S2 gðsÞdsdh: V 4 ðtÞ ¼ d
V 3 ðtÞ ¼
m1
tþh
d2
tþh
It is easy to get that there exist positive scalars k1 > 0 and k2 > 0 such that
k1 jnðtÞj2 6 VðnðtÞ; rt ; tÞ 6 k2 sup jnðhÞj2 : t2d6h6t
ð28Þ
361
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
Therefore D
Vðnð0Þ; r 0 ; 0Þ 6 d¼k2 sup jnðhÞj2 :
ð29Þ
2d6h60
Using Ito’s formula, we can obtain the derivation of VðtÞ for the system (24) as following
dVðtÞ ¼
4 X
LV i ðtÞ þ 2nðtÞT PgðtÞdxðtÞ;
ð30Þ
i¼1
where L is the weak infinitesimal generator of the random process and
b vðtÞ þ 2vðtÞT eT PBr ðtÞrðtÞ þ vðtÞT b LV 1 ðtÞ ¼ 2nðtÞT Pf ðtÞ þ gðtÞT PgðtÞ ¼ 2vðtÞT eT1 P AðtÞ CðtÞT P b CðtÞvðtÞ; 1 T d1 d1 T T þ nðt d1 Þ Q 2 nðt d1 Þ ð1 lÞnðt dðtÞÞ Q 2 nðt dðtÞÞ LV 2 ðtÞ 6 CðtÞT Q 1 CðtÞ C t Q 1C t m m T
þ nðtÞT Q 3 nðtÞ nðt d2 Þ Q 3 nðt d2 Þ; h i ¼ vðtÞT PT1 Q 1 P1 PT2 Q 1 P2 þ eTmþ1 Q 2 emþ1 ð1 lÞeTmþ2 Q 2 emþ2 þ eT1 Q 3 e1 eTmþ3 Q 3 emþ3 vðtÞ; " 2 # Z Z td1 d1 d1 t 2 R1 þ ðd2 d1 Þ R2 f ðtÞ f ðsÞT R1 f ðsÞds ðd2 d1 Þ f ðsÞT R2 f ðsÞds d m m t m1 td2 Z t Z td1 h iT h i b vðtÞ þ Br ðtÞrðtÞ H1 AðtÞ b vðtÞ þ Br ðtÞrðtÞ d1 ¼ AðtÞ f ðsÞT R1 f ðsÞds ðd2 d1 Þ f ðsÞT R2 f ðsÞds; d m t m1 td2
LV 3 ðtÞ ¼ f ðtÞT
Z t Z td1 d1 T gðsÞ S gðsÞds gðsÞT S2 gðsÞds; S1 þ ðd2 d1 ÞS2 gðtÞ 1 d m t m1 td2 Z t Z td1 d1 T CðtÞvðtÞ gðsÞ S gðsÞds gðsÞT S2 gðsÞds: CðtÞT S1 þ ðd2 d1 ÞS2 b ¼ vðtÞT b 1 d m t m1 td2
LV 4 ðtÞ ¼ gðtÞT
ð31Þ
By Lemma 1, we obtain
d1 m
Z
t
Z
T
d
t m1
f ðsÞ R1 f ðsÞds 6
!T
t d
f ðsÞds
Z
R1
t m1
!
t
f ðsÞds
d
t m1
¼ vðtÞT eTmþ4 R1 emþ4 vðtÞ:
ð32Þ
Meanwhile, using Lemma 2, there exists matrix U satisfying (21) such that
ðd2 d1 Þ
Z
"
td1
T
f ðsÞ R2 f ðsÞds ¼ ðd2 d1 Þ
Z
td2
ðd2 d1 Þ 6 dðtÞ d1 2 R td 6 6 4
1
tdðtÞ
R tdðtÞ td2
td1
T
f ðsÞ R2 f ðsÞds ðd2 d1 Þ
Z
#T
td1
f ðsÞds
R2
"Z
tdðtÞ
f ðsÞds f ðsÞds
3T 7 5
td1
tdðtÞ
"
R2 UT
T
f ðsÞ R2 f ðsÞds
td2
tdðtÞ
"Z
#
tdðtÞ
#
ðd2 d1 Þ f ðsÞds d2 dðtÞ
"Z
#T
tdðtÞ
f ðsÞds
td2
R2
"Z
tdðtÞ
td2
# f ðsÞds
ð33Þ
2 3 # R td1 f ðsÞds tdðtÞ 6 7 T T 4R 5 ¼ vðtÞ P3 WP3 vðtÞ; tdðtÞ R2 f ðsÞds td2 U
Furthermore, using the Newton-Leibniz formula, we obtain
"
# Z t Z t d1 2vðtÞ L1 nðtÞ n t f ðsÞds gðsÞdwðsÞ ¼ 0; d d m t m1 t m1 T
" 2vðtÞT L2 nðt d1 Þ nðt dðtÞÞ
Z
td1
f ðsÞds
Z
tdðtÞ
" T
2vðtÞ L3 nðt dðtÞÞ nðt d2 Þ
Z
gðsÞdwðsÞ ¼ 0;
tdðtÞ
tdðtÞ
td2
#
td1
f ðsÞds
Z
tdðtÞ
td2
# gðsÞdwðsÞ ¼ 0;
ð34Þ
362
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
that is
2vðtÞT L1 P4 vðtÞ 2vðtÞT L1 2vðtÞT L2 P5 vðtÞ 2vðtÞT L2
Z
t
gðsÞdwðsÞ ¼ 0;
d
t m1
Z
td1
gðsÞdwðsÞ ¼ 0;
ð35Þ
tdðtÞ
2vðtÞT L3 P6 vðtÞ 2vðtÞT L3
Z
tdðtÞ
gðsÞdwðsÞ ¼ 0:
td2
According to Lemma 1, it can be verified that T
2vðtÞ L1 T
2vðtÞ L2
Z
t d
gðsÞdwðsÞ 6 vðtÞ
T
t m1
Z
td1
T L1 S1 1 L1
gðsÞdwðsÞ 6 vðtÞ
T
tdðtÞ T
2vðtÞ L3
Z
tdðtÞ
gðsÞdwðsÞ 6 vðtÞ
td2
vðtÞ þ
T L2 S1 2 L2
T
"Z
#T
t
gðsÞdwðsÞ
d
t m1
vðtÞ þ
T L3 S1 2 L3
"Z
S1
"Z
gðsÞdwðsÞ
S2
"Z
tdðtÞ
vðtÞ þ
"Z
gðsÞdwðsÞ ;
d
t m1
#T
td1
#
t
#
td1
gðsÞdwðsÞ ;
tdðtÞ
#T
tdðtÞ
gðsÞdwðsÞ
S2
"Z
td2
tdðtÞ
# gðsÞdwðsÞ :
ð36Þ
td2
It is noted that
E
(Z
)
t d
t m1
(Z
gðsÞT S1 gðsÞds
tdðtÞ
¼E
8" < Z :
) T
d
gðsÞdwðsÞ
t m1
8" < Z
td1
S1
#9 = gðsÞdwðsÞ ; d ; t m1
"Z
#T
t
"Z
td1
#9 =
gðsÞdwðsÞ S2 gðsÞdwðsÞ : tdðtÞ ; tdðtÞ 8 (Z ) " #T "Z #9 = < Z tdðtÞ tdðtÞ tdðtÞ T gðsÞ S2 gðsÞds ¼ E gðsÞdwðsÞ S2 gðsÞdwðsÞ : E ; : td2 td2 td2 E
gðsÞ S2 gðsÞds
#T
t
¼E
ð37Þ
td2
Then, from (31)–(37), we can obtain
E fLVðnðtÞ; i; tÞg 6 vðtÞT HðtÞ0 eT1 We1 vðtÞ þ 2vðtÞT eT1 PBr ðtÞrðtÞ h iT h i b vðtÞ þ Br ðtÞrðtÞ H1 AðtÞ b vðtÞ þ Br ðtÞrðtÞ : þ AðtÞ
ð38Þ
To get the H1 tracking performance, we calculate that
n o n o vðtÞ T vðtÞ ; E LVðnðtÞ; i; tÞ þ nðtÞT WnðtÞ c2 rðtÞT rðtÞ ¼ E LVðnðtÞ; i; tÞ þ vðtÞT eT1 We1 vðtÞ c2 rðtÞT rðtÞ 6 NðtÞ0 rðtÞ rðtÞ ð39Þ 0
where NðtÞ is defined in (25). Then, it can be concluded from (25) and (39) that there exists a sufficiently small scalar c > 0 such that
n o E LVðnðtÞ; i; tÞ þ nðtÞT WnðtÞ c2 rðtÞT rðtÞ 6 cjnðtÞj2 :
ð40Þ
Therefore, applying the Dynkin’s formula to (40), we have
E
Z
tf
eðtÞT WeðtÞdt
6 c2
0
Z
tf
jrðtÞj2 dt þ Vðnð0Þ; r 0 ; 0Þ:
0
Noting the (29), therefore, the state tracking error eðtÞ in (11) and the bounded disturbance rðtÞ in (5) satisfy the relationship described in (14) for all admissible uncertainties. Furthermore, when rðtÞ ¼ 0, it follows from (40) that
LVðnðtÞ; i; tÞ 6 cjnðtÞj2 :
ð41Þ
Based on (41) and Definition 1, using the same proof in [27], we can easily get that augmented system (8) with rðtÞ ¼ 0 is robustly stochastically asymptotically stable in mean square with the state-feedback controller given in (7). This completes the proof. h
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
363
Remark 2. To achieve the improved results, some effective techniques are included in the proof Theorem 1, such as partition of the lower bound technique, free-weighting matrices method and reciprocally convex approach, which are helpful to reduce the conservativeness of the derived result. Remark 3. It should be pointed out that, the inequality (22) is not in the form of linear matrix inequality(LMI) due to the existence of some nonlinear terms. However, it is easily to be transformed into linear matrix inequality by using the Lemma 3 and Schur complement lemma. In order to compare our result with the recent ones in the literature, we consider the system (1) without input and disturbance terms, that is
dxðtÞ ¼ ½AðtÞxðtÞ þ Ad ðtÞxðt dðtÞÞdt þ ½CðtÞxðtÞ þ DðtÞxðt dðtÞÞdxðtÞ;
ð42Þ
Then, we can obtain a novel delay-dependent stability criterion for system (42). Corollary 1. Given an integer m P 1, and scalars l; d2 P d1 > 0, system (42) is robustly stochastically stable in the mean square, if there exist positive-definite matrices P > 0; Q 1 > 0; Q 2 > 0, Q 3 > 0; R1 > 0, R2 > 0; S1 > 0; S2 > 0, and matrices U; L1 ; L2 ; L3 , and positive scalar e1 > 0; e2 > 0, e3 > 0, such that (21) and the following LMIs holds
2
D L1 6 6 S1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
L2 0
T
L3 0
k1 PM 0
T
T
ðk1 AT þ kmþ2 ATd ÞH1 0
0 0
T
T
ðk1 C T þ kmþ2 DT ÞH2 0
S2
0
0
0
0
0
S2
0
0
0
0
e1 I
0
0
0
H1
H1 M
0
e2 I
0
H2
3 0 7 0 7 7 0 7 7 7 0 7 7 0 7 7 < 0; 7 0 7 7 0 7 7 7 H2 M 5
ð43Þ
e3 I
where H1 ; H2 are defined in (23) and T T b TQ1P b1P b TQ1P b 2 þ kT Q 2 kmþ1 ð1 lÞkT Q 2 kmþ2 D ¼ k1 PðAk1 þ Ad kmþ2 Þ þ ðAk1 þ Ad kmþ2 Þ Pk1 þ P mþ1 mþ2 1 2 T T T T b T WP b 3 þ L1 P b4þP b T LT þ L2 P b5þP b T LT þ k1 Q 3 k1 kmþ3 Q 3 kmþ3 kmþ4 R1 kmþ4 kmþ4 WR1 kmþ4 P 3 4 1 5 2
b6þP b T LT þ e1 ðNa k1 þ N ad kmþ2 ÞT ðNa k1 þ Nad kmþ2 Þ þ e2 ðN a k1 þ Nad kmþ2 ÞT ðNa k1 þ Nad kmþ2 Þ þ L3 P 6 3 T
þ e3 ðNc k1 þ N d kmþ2 Þ ðNc k1 þ Nd kmþ2 Þ; h
b 1 ¼ kT kT kT P 1 2 m h
T b 3 ¼ kT P kmþ6 mþ5
iT
iT
h b 2 ¼ kT ;P 2
T
k3
T
kmþ1
iT
;
b 4 ¼ k1 k2 kmþ4 ; ;P
b 6 ¼ kmþ2 kmþ3 kmþ6 : b 5 ¼ kmþ1 kmþ2 kmþ5 ; P P Proof. Choosing the same Lyapunov functional as (27) and using Lemma 3, the above criterion can be easily obtained. If there are no uncertainties, system (42) reduces to the following form
dxðtÞ ¼ ½AxðtÞ þ Ad xðt dðtÞÞdt þ ½CxðtÞ þ Dxðt dðtÞÞdxðtÞ:
ð44Þ h
ð45Þ
Then, the following corollary can be easily obtained. Corollary 2. Given a integer m P 1, and scalars l; d2 P d1 > 0, system (45) is stochastically stable in the mean square, if there exist positive-definite matrices P > 0; Q 1 > 0; Q 2 > 0, Q 3 > 0; R1 > 0; R2 > 0; S1 > 0; S2 > 0, and matrices U; L1 ; L2 ; L3 , such that (21) and the following LMIs holds
2
N
6 6 6 4
3
L1
L2
L3
S1
0
S2
0 7 7 bT b þb C T H2 b C < 0; 7 þ A H1 A 0 5
S2
ð46Þ
364
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
where H1 ; H2 are defined in (23) and
b TQ1P b1P b TQ1P b 2 þ kT Q 2 kmþ1 ð1 lÞkT Q 2 kmþ2 þ kT Q 3 k1 N ¼ kT1 PðAk1 þ Ad kmþ2 Þ þ ðAk1 þ Ad kmþ2 ÞT Pk1 þ P mþ1 mþ2 1 1 2 T T b T WP b 3 þ L1 P b4þP b T LT þ L2 P b5þP b T LT þ L3 P b6þP b T LT ; kmþ3 Q 3 kmþ3 kmþ4 R1 kmþ4 P 3 4 1 5 2 6 3
b ¼ ½ Ak1 þ Ad kmþ2 A b C ¼ ½ Ck1 þ Dkmþ2
0 0 0 ; ð47Þ
0 0 0 :
_ is unknown, we can get the corresponding Remark 4. If the time delay dðtÞ is not differential or the upper bound of dðtÞ criteria by setting Q 2 ¼ 0 in Theorem 1. For simplicity, we omit them here. 3.2. Tracking controller design In this section, the design of tracking controller (7) for system (1) is proposed in the following theorem. Theorem 2. Consider the uncertain stochastic system with time delay (1) and the reference system in (5). Given a integer m P 1, scalars l; d2 P d1 > 0; c > 0, and semi-positive-definite weighting matrix W P 0 in (12), there exists a memory state-feedback controller in (7) such that the H1 tracking requirements are achieved, if there exist positive-definite scalars e1 > 0; e2 > 0; e3 > 0, b 1 > 0; Q b 2 > 0; Q b 3 > 0; R b > 0; Q b 1 > 0; R b 2 > 0; b b K b 1; K b 2 with L1 ; b L2 ; b L 3 ; U; positive-definite matrices P S 1 > 0; b S 2 > 0, and b appropriate dimensions such that the following LMIs hold
" b ¼ W 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
b2 R bT U
b U b2 R
K11 eT1 Br c2 I
# P 0;
ð48Þ
b L1
b L2
b L3
K16
0
0
N br
T
K17
K18
K19
K1;10
N br
BTr
0
0
T
0 b S1
0
0
0
0
0
0
0
b S2
0
0
0
0
0
0 b S2
0 e1 I
0 0
0 0
0 0
0 0
e2 I
0 0 0 K88 0 0 e3 I 0 K10;10
K1;11
3
7 0 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 < 0; 7 0 7 7 7 0 7 7 0 7 7 7 0 5
ð49Þ
I
where
K11
K16 K18 K1;10 K88
T b T b b b 1 þ eT B u K b mþ2 þ eT Bu K b 1 e1 þ sym eT Ad Pe b 2 emþ2 þ PT Q ¼ sym eT1 A Pe 1 1 1 1 1 P1 P2 Q 1 P2 þ emþ1 Q 2 emþ1
T b T T b 2 emþ2 þ eT Q b b b b b b ð1 lÞeTmþ2 Q 1 3 e1 emþ3 Q 3 emþ3 emþ4 R 1 emþ4 P3 WP3 þ sym L 1 P4 þ L 2 P5 þ L 3 P6 þe1 eT1 MMT e1 ; T T b T þ eT K b T bT T bT T ¼ K17 ¼ eT1 PN a 1 1 N bu þ emþ2 PN ad þ emþ2 K 2 N bu ; T T b T þ eT K b T b T þ eT PN b T; bT T bT T ¼ eT1 PA K19 ¼ eT1 PN 1 1 B u þ emþ2 PAd þ emþ2 K 2 B u ; mþ2 c d b T þ eT PD b T ; K1;11 ¼ eT PW b T; ¼ eT1 PC mþ2 1 0 2 d1 b 1 þ ðd2 d1 Þ2 R b2 2P b þ e2 MMT ; R ¼ m
K10;10 ¼
d1 m
b b þ e3 MMT : S 1 þ ðd2 d1 Þb S2 P ð50Þ
When the LMIs in (48)–(50) are feasible, the controller gains of (7) can be obtained by
b 1 ; b iP Ki ¼ K
i ¼ 1; 2:
ð51Þ
365
J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
b 1 Y 1 ; R1 ¼ P b 1 ; Y ¼ diag P b 1 R b1 P b 1 ; R2 ¼ P b 1 R b2 P b 1 ; S1 ¼ P b 1 b b 1 ; S2 ¼ P b 1 b b 1 ; b P b P b ; Q 1 ¼ Y 1 Q Proof. Set P ¼ P S1 P S2 P 1 b1 b 1 1 b1 b 1 1 b1 b 1 1 b b 1 b U ¼ P U P ; L1 ¼ Y L 1 P ; L2 ¼ Y L 2 P ; L3 ¼ Y L 3 P . Pre-multiplying and post-multiplying (48) by matrix b 1 ; P b 1 Þ, it can be easily verified that diagð P
" 06
b 1 P 0
#"
0 b 1 P
b U b R2
b2 R bT U
#"
b 1 P
0 b 1 P
0
#
" ¼
b 1 P 0
0 b 1 P
#"
b b 2P PR b b PU P
b P b PU b b PR2 P
#"
b 1 P 0
0 b 1 P
#
¼
R2
U
U
R2
b iP b 2 P; b This means that the condition in (48) implies the one in (21). Meanwhile, noting H1 6 PH i condition holds from (49)
2
K11
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
eT1 Br
b L1
b L2
b L3
K16
c2 I
0
0
0
N br
b S1
0
0
S2 b
K17
K18
K19
K1;10
K1;11
N br
BTr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
e3 I
0
T
T
b S2
e1 I
0
0
e2 I
0
K
0 88
K
0 0 10;10
0
:
ð52Þ
i ¼ 1; 2, the following
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 < 0; 7 7 7 7 7 7 7 7 7 7 7 7 7 5
ð53Þ
I
where T K088 ¼ H1 1 þ e2 MM ;
ð54Þ
T K010;10 ¼ H1 2 þ e3 MM ;
and H1 ; H2 are defined in (23). Then, using Schur complement lemma to (53), we obtain
2
3
3 2 T 3 2 K16 e1 M 7 7 6 6 T 7 6 0 7 6 N br 7 c2 I 0 0 0 7 7 7 7 6 6 7 7 7 T 6 1 6 T 7 b þ e e þ 0 0 0 0 N br 1 6 0 7 M e1 0 0 7 S1 1 6 0 7 K16 7 7 6 6 7 7 6 6 0 7 7 b 0 5 5 4 4 0 5 S2 0 0 S2 b 3 3 2 2 K17 K18 6 T 7 6 T 7 6 N br 7 6 Br 7 7 7
6 6 T 7 T 7 1 6 6 T 1 þ e1 N br 0 0 0 þ 6 0 7 H1 e2 MM K18 Br 0 0 0 2 6 0 7 K17 7 7 6 6 6 0 7 6 0 7 5 5 4 4
K011
6 6 6 6 6 6 6 6 4
b L1
eT1 Br
b L2
b L3
0 3
2
K19 7 6 6 0 7 7 6 7 T 6 þ e1 3 6 0 7 K19 7 6 6 0 7 5 4 0
0 0 0
ð55Þ
0 3
2
K1;10 7 6 7 60 7
6 h T 7 1 6 T 1 K1;10 7 H2 e3 MM 0 0 0 0 þ 60 7 6 7 60 5 4
i 0 0 0 0 < 0;
0
where
T b T b b b 1 þ eT B u K b mþ2 þ eT Bu K b 1 e1 þ sym eT Ad Pe b 2 emþ2 þ PT Q K011 ¼ sym eT1 A Pe 1 1 1 1 1 P1 P2 Q 1 P2 þ emþ1 Q 2 emþ1 T b T T b 2 emþ2 þ eT Q b b b ð1 lÞeTmþ2 Q 1 3 e1 emþ3 Q 3 emþ3 emþ4 R 1 emþ4 P3 WP3
b T W 0 P: b L 1 P4 þ b L 2 P5 þ b L 3 P6 þ eT1 PW þ sym b 0
ð56Þ
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Then, applying Lemma 3 to (55) yields
3 2 3 2 3T 2 3 2 3T b b K18 ðtÞ K18 ðtÞ K1;10 ðtÞ K1;10 ðtÞ L2 L3 K11 ðtÞ eT1 Br ðtÞ bL 1 7 6 7 6 7 7 6 7 6 6 c2 I 0 0 0 7 6 BTr ðtÞ 7 6 BTr ðtÞ 7 6 6 0 7 6 0 7 7 6 7 6 7 7 6 7 6 6 7 6 7 7 6 7 6 6 6 b 0 7 S1 0 7 þ 6 0 7H1 6 0 7 þ 6 0 7H2 6 0 7 6 7 6 7 6 7 7 6 7 6 6 b 0 0 0 0 5 5 5 5 5 4 4 4 4 4 S2 0 b 0 0 0 0 S2 3 2 T 3 3 2 0 2 T b b b K16 e1 M K11 e1 Br L 1 L2 L3 T 7 7 7 6 6 6 0 0 7 6 0 7 6 c2 I 0 6N 7 7 6 7 6 T 6 br 7 T T 7 7 6 6 ¼6 b 0 7 S1 0 7 þ 6 0 7FðtÞ K16 N br 0 0 0 þ 6 0 7FðtÞ M e1 0 0 0 0 6 7 6 7 7 6 6 4 4 0 5 b S2 0 5 4 0 5 0 0 b S2 T T T T þ K18 Br 0 0 0 MFðtÞ K17 N br 0 0 0 H1 K18 Br 0 0 0 þ MFðtÞ KT17 N br 0 0 0 nh i h ioT n h i o þ KT1;10 0 0 0 0 þ MFðtÞ KT1;9 0 0 0 0 H2 KT10 0 0 0 0 þ MFðtÞ KT1;9 0 0 0 0 < 0; 2
ð57Þ where
b 1 þ eT Bu ðtÞ K b mþ2 þ eT Bu ðtÞ K b 1 e1 þ sym eT Ad ðtÞ Pe b 2 emþ2 K11 ðtÞ ¼ sym eT1 AðtÞ Pe 1 1 1 T T T b T b 1 P1 PT Q b b b b þ PT1 Q 2 1 P2 þ emþ1 Q 2 emþ1 ð1 lÞemþ2 Q 2 emþ2 þ e1 Q 3 e1 emþ3 Q 3 emþ3
b 3 þ sym b b 1 emþ4 PT WP b P; b L 1 P4 þ b L 2 P5 þ b L 3 P6 þ eT1 PW eTmþ4 R 3 T T b T ðtÞ þ eT K b T b T b T K18 ðtÞ ¼ eT1 PA 1 1 B u ðtÞ þ emþ2 PAd ðtÞ þ emþ2 K 2 B u ðtÞ; b T ðtÞ þ eT PD b T ðtÞ: K1;10 ðtÞ ¼ eT PC 1
mþ2
ð58Þ
Pre-multiplying and post-multiplying (58) by matrix
2
Y 1 6 6 0 6 6 0 6 6 4 0 0
0 I 0
0 0 b 1 P
0 0
0
0
0 b 1 P
0
0
0
3 0 7 0 7 7 0 7 7; 7 0 5 b 1 P
it is obvious that the condition (31) in Theorem 1 is hold. Then, by Theorem 1, the robust H1 tracking requirements are obtained. This completes the proof. h If there are no uncertainties in system (1), that is
dxðtÞ ¼ ½AxðtÞ þ Ad xðt dðtÞÞ þ Bu uðtÞ þ Br rðtÞdt þ ½CxðtÞ þ Dxðt dðtÞÞdxðtÞ;
ð59Þ
By the same proof, we have the following corollary. Corollary 3. Consider the stochastic system (59) and the reference system in (5). For given integer m P 1, scalars l, d2 P d1 > 0; c > 0, and semi-positive-definite weighting matrix W P 0 in (12), there exists a memory state-feedback controller in b 1 > 0; b > 0; Q (7) such that the H1 tracking requirements are achieved, if there exist positive-definite matrices P b b b b b b b b b b b b Q 2 > 0; Q 3 > 0; R 1 > 0; R 2 > 0; S 1 > 0; S 2 > 0, and L 1 ; L 2 ; L 3 ; U; K 1 ; K 2 such that (48) and the following LMIs hold
2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
b 11 eT Br K 1
b L1
b L2
b L3
b 16 K
b 1;7 K
c2 I
0
0
0
BTr
0
b S1
0
0
0
0
b S2
0
0
0
0
0
b 66 K
0
b S2
b 77 K
b 1;8 K
3
7 0 7 7 7 7 0 7 7 7 0 7 7 < 0; 7 0 7 7 7 0 7 7 7 0 7 5
I
ð60Þ
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
where
T b T b 11 ¼ sym eT A PL b b b 1 e1 þ eT Bu K b mþ2 þ eT Bu K b 1 e1 þ sym eT Ad Pe b 2 emþ2 þ PT Q K 1 1 1 1 1 1 P1 P2 Q 1 P2 þ emþ1 Q 2 emþ1
T b T T b 2 emþ2 þ eT Q b b b b b b ð1 lÞeTmþ2 Q 1 3 e1 emþ3 Q 3 emþ3 emþ4 R 1 emþ4 P3 WP3 þ sym L 1 P4 þ L 2 P5 þ L 3 P6 ; T T b 16 ¼ eT PA b T þ eT K b T bT T bT T K 1 1 1 B u þ emþ2 PAd þ emþ2 K 2 B u ; b 1;8 ¼ eT PW b 1;7 ¼ eT PC b T þ eT PD b T; K b T; K 0 1 mþ2 1 2 b 66 ¼ d1 R b 1 þ ðd2 d1 Þ2 R b 2 2 P; b K m b 77 ¼ d1 b b K S 1 þ ðd2 d1 Þb S 2 P: m
ð61Þ
When the LMIs in (48) and (60) are feasible, the controller gains of (7) can be obtained by
b 1 ; b iP Ki ¼ K
i ¼ 1; 2:
ð62Þ
4. Numerical Examples Example 1. Consider system (42) with following parameters
2:1829 0 1:0293 0:4123 0:0210 0:2123 ; Ad ¼ ; C¼ ; A1 ¼ 0 3:2092 2:9212 0:1234 0:0212 0:0532 0:1029 0 0:5021 0:2293 0:2345 0 ; M¼ ; Na ¼ ; D¼ 0 0:1212 0:2932 0:2283 0 0:1282 0:1123 0:1233 0:5124 0:4234 0:9213 0:7123 ; Nc ¼ ; Nd ¼ : Nad ¼ 0:1242 0:1134 0:2123 0:2234 0:2223 0:2123 To demonstrate the effectiveness of our approach, the allowable maximum values of d2 such that the system (42) is robustly stochastically stable in mean square are listed in Table 1. To demonstrate the effectiveness of our approach, results are compared with the recent ones in Table 1. Remark 5. From Table 1, it is clear the upper bounds obtained in Corollary 1 for different l are larger than the corresponding ones in [34]. That means that the result of Corollary 1 is less conservative than the criterion in [34]. Clearly, the conservatism of the results is considerably reduced when the partitioning number m increases. Example 2. Consider the stochastic system (1) with the following matrix parameters
A¼
;
1:1829
0:1000
0:1102 0:1934
0:2292 0:1924
Ad ¼
1:0293
0:4123
;
Bu ¼
0:2000
0:2943
; 0:1923 0:1000 0:9212 1:0234 0:0210 0:2123 0:0121 0:2293 Br ¼ ; C¼ ; D¼ ; 0:0930 0:2054 0:0212 0:0532 0:2932 0:2283 1:2922 0:8283 0:3983 0:2823 0:0011 0:0090 E¼ ; Ed ¼ ; M¼ ; 0:9135 1:4821 0:8291 0:1934 0:0012 0:0032 0:0012 0:0017 0:0041 0:0021 0:0031 0:0045 Na ¼ ; Nad ¼ ; Nbu ¼ ; 0:0002 0:0014 0:0018 0:0082 0:0012 0:0203 0:0053 0:0071 0:0029 0:0003 0:0067 0:0012 ; Nc ¼ ; Nd ¼ ; N br ¼ 0:0312 0:0013 0:0101 0:0132 0:0019 0:0103 0:1000 0 W0 ¼ ; d1 ¼ 0:1; d2 ¼ 0:2; l ¼ 0:3; c ¼ 2: 0 0:1000
Table 1 The maximum allowable delay bound of d2 for different l with d1 ¼ 0:3, Method
l ¼ 0:7
l ¼ 0:75
l ¼ 0:78
l ¼ 0:8
l ¼ 0:85
Theorem 2 in [34] (m ¼ 1) Theorem 2 in [34] (m ¼ 2) Corollary 1 in this paper (m ¼ 1) Corollary 1 in this paper (m ¼ 2)
1.4921 1.5983 1.6379 1.8632
0.8592 0.8709 0.8897 0.9739
0.7924 0.8023 0.8153 0.8314
0.7495 0.7725 0.7940 0.8123
0.7183 0.7694 0.7739 0.7986
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
Set m ¼ 1, then, solving the LMIs (48) and (60) in Corollary 3, obtains
2
14:8745
1:1026 0:4009 0:3532
6 6 1:1026 b¼6 P 6 4 0:4009
15:3164
0:0229
0:0229
10:6279
3
2
7 0:1115 7 7; 7 1:9967 5
0:0383 0:1480 23:6824
6 6 0:0151 b1 ¼ 6 R 6 4 0:1542 0:1300 2
19:1541
6 6 0:1074 b S1 ¼ 6 6 4 0:1640
0:0151 0:3036
24:9692
7 0:3083 7 7; 7 2:4753 5
0:3083
2:4753
23:5688
0:1074 0:1640 0:1186 19:0718
0:1179
0:1179
19:8811
0:0578
1:6463
1:1790
0:0466
0:0053
6 6 0:0054 1:1305 b ¼6 U 6 4 0:0473 0:0351 "
0:0303
0:0017
2
0:6923
0:5763 1:2565
47:3179 2:5829 1:9273
24:366
1:1032 1:8852 40:0704
0:0837
0:2377
1:2244
24:6804
18:4135
0:0242 0:1224 0:0998
2
6 6 0:0242 b S2 ¼ 6 6 4 0:1224 0:0998 " b1 ¼ K
2:4580
0:2205
18:3720
0:1874
0:1874
19:2921
0:1837
1:1410
b 1 b 2P K2 ¼ K
13:3970 24:6642 18:2564 19:7096
e1 ¼ 17:1180; e2 ¼ 17:1187; e3 ¼ 17:1080:
0:0200 2:9762 1:4814 1:4909 ; 0:9520 1:6704 1:4138 1:4458 4:2761 2:3857 0:3461 0:1917 ¼ : 0:4382 3:1192 0:2860 0:1763
2
0.5 x1 y1
1
x2 y2
0
0
−1 −2
0
50
100
2
−0.5
0
50
100
1 tracking error e1
tracking error e2
1
0.5
0
0
−1
−0.5
−2
0
50
18:7248
# ;
100
−1
0
50
Fig. 1. The responses of open-loop systems.
100
3
7 0:1837 7 7; 7 1:1410 5
45:7386 18:6443 20:1577
and b L1; b L2 ; b L 3 are omitted for simplicity. Therefore, the controller gains of (7) can be obtained as
b 1 ¼ b 1P K1 ¼ K
6:7328 3
0:0954 0:0837 25:3828
18:9973 3
7 0:0371 7 7; 7 0:0143 5
3
7 0:2377 7 7; 7 1:2244 5
0:0001
0:0732
0:1954
6 6 0:0704 24:3616 b2 ¼ 6 R 6 4 0:0954 0:2205
3
7 0:0578 7 7; 7 1:6463 5
60:9034 31:8547 1:6362 3:2547 3:0266
0:2072 3
0:3036
0:1258
0:1954 0:8845 0:7717 4:0868 3 2 9:2671 0:3759 0:1284 0:2072 7 6 6 0:3759 9:6585 1:0400 1:1032 7 b3 ¼ 6 7; Q 7 6 4 0:1284 1:0400 7:6997 1:8852 5
4:1325
23:7301
0:1186 2
b2 ¼ K
1:2925
0:1542 0:1300
0:8153
7 6 6 0:8153 8:2852 0:8326 0:8845 7 b1 ¼ 6 7; Q 7 6 4 0:1258 0:8326 4:5729 0:7717 5
0:3532 0:1115 1:9967 11:3186 3 2 4:8365 0:1520 0:0695 0:0383 7 6 6 0:1520 4:9255 0:1563 0:1480 7 b2 ¼ 6 7; Q 7 6 1:2925 5 4 0:0695 0:1563 5:3544 2
7:6531
# ;
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J. Xia et al. / Applied Mathematics and Computation 249 (2014) 356–370
2
0.4 x1 y2
1
0
0
−0.2
−1 −2
x2 y2
0.2
−0.4 0
50
100
2
−0.6
0
100
1 tracking error e1
tracking error e2
1
0.5
0
0
−1
−0.5
−2
50
0
50
100
−1
0
50
100
Fig. 2. The responses of closed-loop systems.
Furthermore, we assume
dðtÞ ¼ ð1 þ j sinðtÞjÞ=10; 0:5 cosðtÞ 0 JðtÞ ¼ ; 0 0:5 cosðtÞ 2 sinð0:5tÞ ; rðtÞ ¼ 2 cosð0:1tÞhðtÞ 2hðtÞ cosð0:3tÞ ; v ðtÞ ¼ 0:3 sinð0:2tÞ where hðtÞ is a step signal function. Meanwhile, let the initial conditions of system (1) and (2) be /ðtÞ ¼ ½ 0:2 0:2 ; wðtÞ ¼ ½ 0 0 , then, the responses of real system states, reference system states and states of tracking error for the open-loop system are shown in Fig. 1. The corresponding responses of closed-loop system are shown in Fig. 2. It is easy to see from the figures that the real system states tend to track the reference system states effectively with the designed controller.
Remark 6. From Figs. 1 and 2, it is easy to see that the states of the closed-loop real systems are better than the open-loop ones to follow the reference systems states, and the errors of closed-loop systems are much smaller than the ones of openloop systems, which show the effectiveness of obtained criterion.
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