Volume 2010, Article ID 494607, 14 pagesdoi:10.1155/2010/494607 Research Article Riccati Equations and Delay-Dependent BIBO Stabilization of Stochastic Systems with Mixed Delays and Nonl
Trang 1Volume 2010, Article ID 494607, 14 pages
doi:10.1155/2010/494607
Research Article
Riccati Equations and Delay-Dependent BIBO
Stabilization of Stochastic Systems with Mixed
Delays and Nonlinear Perturbations
Xia Zhou and Shouming Zhong
School of Mathematical Sciences, University of Electronic Science and Technology of China,
Chengdu, Sichuan 611731, China
Correspondence should be addressed to Xia Zhou,zhouxia44185@sohu.com
Received 21 August 2010; Accepted 9 December 2010
Academic Editor: T Bhaskar
Copyrightq 2010 X Zhou and S Zhong This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
The mean square BIBO stability is investigated for stochastic control systems with mixed delays and nonlinear perturbations The system with mixed delays is transformed, then a class of suitable Lyapunov functionals is selected, and some novel delay-dependent BIBO stabilization in mean square criteria for stochastic control systems with mixed delays and nonlinear perturbations are obtained by applying the technique of analyzing controller and the method of existing a positive definite solution to an auxiliary algebraic Riccati matrix equation A numerical example is given to illustrate the validity of the main results
1 Introduction
In recent years, Bounded-Input Bounded-OutputBIBO stabilization has been investigated
by many researchers in order to track out the reference input signal in real world, see
1 6 and some references therein On the other hand, because of the finite switching speed, memory effects, and so on, time delays are unavoidable in technology and nature, commonly exist in various mechanical, chemical engineering, physical, biological, and economic systems They can make the concerned control system be of poor performance and instable, which cause the hardware implementation of the control system to become difficult It is necessary to introduce the distributed delay in control systems, which can describe mathematical modeling of many biological phenomena, for instance, in prey-predator systems, see 7 9 And so, BIBO stabilization analysis for mixed delays and nonlinear systems is of great significance
In10,11, the sufficient condition for BIBO stabilization of the control system with
no delays was proposed by the Bihari-type inequality In12,13, employing the parameters
Trang 2technique and the Gronwall inequality investigated the BIBO stability of the system without distributed time delays In14–16, based on Riccati-equations, by constructing appropriate Lyapunov functions, some BIBO stabilization criteria for a class of delayed control systems with nonlinear perturbations were established In17, the BIBO stabilization problem of a class of piecewise switched linear system was further investigated However, up to now, these previous results have been assumed to be in deterministic systems, including continuous time deterministic systems and discrete time deterministic systems, but seldom in stochastic systemssee 18, Fu and Liao get several mean square BIBO stabilization criteria in terms of Razumikhin technique and comparison principle In practice, stochastic control systems are more applicable to problems that are environmentally noisly in nature or related to biological realities Thus, the BIBO stabilization analysis problems for stochastic case are necessary
Up to now, to the best of authors knowledge, the method of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation is only used to deal with the BIBO stabilization for deterministic differential equations 14–17, not for stochastic differential equations
Motivated by the above discussions, the main aim of this paper is to study the BIBO stabilization in mean square for the stochastic control system with mixed delays and nonlinear perturbations Based on the technique of analyzing controller and transforming
of the system, various suitable Lyapunov functionals are selected, different Riccati matrix equations are established, and some sufficient conditions guaranteeing BIBO stabilization
in mean square are obtained Finally, a numerical example is provided to demonstrate the effectiveness of the derived results
2 Problem Formulation and Preliminaries
Consider the stochastic control system described by the following equation:
dx t
Ax t B1x t − h1 B2
t
t −h2
x sds ft, xt Dut
dt
C1x t − τ1 C2
t
t −τ2
x sds
d wt, t ≥ t0 ≥ 0,
y t Hxt,
x θ ϕθ ∈ C b
F 0t0− τ, t0; R n , θ ∈ t0− τ, t0,
2.1
where xt, ut, yt are the state vector, control input, control output of the system, respectively τ1 > 0, h1 > 0 are discrete time delays, and h2 > 0, τ2 > 0 are distributed
time delays, τ max{τ1, τ2, h1, h2} A, B1, B2, C1, C2, D, H are constant matrices with
appropriate dimensional, and C2, D are nonsingular matrices wt w1t, w2t, , w n t
is an n-dimensional standard Brownian motion defined on a complete probability space
Ω, F, {F t}t≥0, P with a natural filtration {F t}t≥0 f t, xt is the nonlinear vector-valued
perturbation bounded in magnitude as
f t, xt ≤ αxt, 2.2
where α is known positive constant.
Trang 3To obtain the control law described by2.1 of tracking out the reference input of the system, we let the controller be in the form of
u t u1t u2t u3t, 2.3
with
u1t K1txt rt,
u2t K2txt − h1 rt − h1,
u3t K3txt − h2 rt − h2,
2.4
where K1t, K2t, K3t are the feedback gain matrices, rt, rt − h1, rt − h2 are the reference inputs
To derive our main results, we need to introduce the following definitions and lemmas
Definition 2.1see 18 A vector function rt r1t, r2t, , r n t T is said to be an element
of L n
∞if r∞ supt ∈t0, ∞rt < ∞, where · denotes the Euclid norm in R n or the norm of a matrix.
Definition 2.2see 18 The nonlinear stochastic control system 2.2 is said to be BIBO stabilized
in mean square if one can construct a controller2.5 such that the output yt satisfies
Eyt2
≤ N1 N2r2
where N1, N2are positive constants
Definition 2.3 L-operator Let Lyapunov functional V : C−τ, 0; R n ×R → R; its infinitesimal
operator, L, acting on functional V is defined by
LV xt, t lim
Δ → 0 sup 1
ΔEV xt Δ, t Δ − V xt, t. 2.6
Lemma 2.4 see 19 For any constant symmetric matrix M ∈ R n ×n , M M T > 0, scalar r > 0, vector function g : 0, r → R n , such that the integrations in the following are well defined:
r
r
0
g T sMgsds ≥
r
0
g sds
T M
r
0
g sds
Lemma 2.5 see 20 Let x, y ∈ R n and any n × n positive-definite matrix Q > 0 Then, one has
2x T y ≤ x T Q−1x y T Qy. 2.8
Trang 43 BIBO Stabilization for Nonlinear Stochastic Systems
Transform the original system2.1 to the following system:
d
x t B1
t
t −h1
x sds
A B1xt B2
t
t −h2
x sds ft, xt
dt
C1x t − τ1 C2
t
t −τ2
x sds
d wt Dutdt, t ≥ t0≥ 0,
y t Hxt,
x θ ϕθ ∈ C b
F 0t0− τ, t0; R n , θ ∈ t0− τ, t0.
3.1
Theorem 3.1 The nonlinear stochastic control system 2.1 or 3.1 with the control law 2.3 is
BIBO stabilized in mean square if h1B1 < 1 and there exist symmetric positive-definite matrices
R i > 0, i 1, 2, , 10, and Q1> 0 such that
λminQ1 − 2αP > 0 3.2
and P is the symmetric positive solution of the Riccati equation
P A B1 A B1T P PΣ1P Ξ1 Δ1 −Q1, 3.3
where
Σ1 B2R−11 B T
2 2DR10D T R−1
5 R−1
6 DR10D T P B1R−17 B T
1P DR10D T
B1R−18 B T
1 B1R−19 B T
1,
Ξ1 h2
2R1 R3 h2
1R2 R7 R8 R9 R5 R6 τ2
2R4,
Δ1 A B1T P B1R−12 B T
1P A B1 C T
1P C1 C T
1P C2R−14 C T
2P C1
τ2
2C T2P C2 h2
1B T1P B2R−13 B2T P B1,
K1 R10D T P, K2 K3 D−1.
3.4
Proof We define a Lyapunov functional V t, xt as
V t, xt V1t, xt V2t V3t V4t V5t V6t V7t, 3.5
Trang 5V1t, xt x t B1
t
t −h1
x sds
T
P x t B1
t
t −h1
x sds
,
V2t
t
t −h1
x T sR5 PB1R−18 B T1P
x sds,
V3t
t
t −h2
x T sR6 PB1R−19 B T
1P
x sds,
V4t
t
t −τ1
x T sC T
1P C1 C T
1P C2R−14 C T
2P C1
x sds,
V5t h2
t
t −h2
s − t h2x T sR1 R3xsds,
V6t τ2
t
t −τ2
s − t τ2x T sR4 C T
2P C2
x sds,
V7t h1
t
t −h1
s − t h1x T sR2 R7 R8 R9 B T
1P B2R−13 B T2P B1
x sds.
3.6
Taking the operatorL of V1t, xt along the trajectory of system 3.1, by Lemmas2.4and
2.5, we have
LV1t, xt 2
x t B1
t
t −h1
x sds T
× P
A B1xt B2
t
t −h2
x sds Dut ft, xt
1
2trace
C1x t − τ1 C2
t
t −τ2
x sds
2P
C1x t − τ1 C2
t
t −τ2
x sds
≤ x T t P A B1 A B1T P PB2R−11 B2T P
A B1T P B1R−12 B1T P A B1 x t 2x T P Df t, xt
h2
t
t −h2
x T sR1 R3xsds 2
t
t −h1
x T sdsB T
1P Du t
x T t − τ1C T1P C1 C T
1P C2R−14 C2T P C1
x t − τ1
h1
t
t −h1
x T sR2 B T
1P B2R−13 B T
2P B1
x sds
2
t
t −h1
x T sdsB T
1P f t, xt 2x T P Du t τ2
t
t −τ2
x T sR4 C T
2P C2
ds.
3.7
Trang 6By theLemma 2.5,2.3 and 3.5, we conclude
LV1t, xt ≤ x T t P A B1 A B1T P PB2R−11 B T2P PR−1
A B1T P B1R−12 B T
1P A B1 2PDR10D T P
PR−1
6 P PDR10D T P B1R−17 B T1P DR10D T P x t
x T t − τ1C1T P C1 C T
1P C2R−14 C T2P C1
x t − τ1
x T t − h1R5 PB1R−18 B T1P
x t − h1
x T t − h2R6 PB1R−19 B T
1P
x t − h2
h1
t
t −h1
x T sR2 B T
1P B2R−13 B2 T P B1 R7 R8 R9
x sds
τ2
t
t −τ2
x T sR4 C T
2P C2
ds h2
t
t −h2
x T sR1 R3xsds 2αP h1B T
1Pxt2 6PD h1B T
1Prt∞xt.
3.8
Taking the operatorL of V i t, i 2, 3, , 7 along the trajectory of system 3.1, we get
LV2t x T tR5 PB1R−18 B1T P
x t
− x T t − h1R5 PB1R−18 B T1P
x t − h1,
LV3t x T tR6 PB1R−19 B1T P
x t
− x T t − h2R6 PB1R−19 B T
1P
x t − h2,
LV4t x T tC1T P C1 C T
1P C2R−14 C T2P C1
x t
− x T t − τ1C1T P C1 C T
1P C2R−14 C T2P C1
x t − τ1,
LV5t h2
2x T tR1 R3xt − h2
t
t −h2
x T sR1 R3xsds,
LV6t τ2
2x T tR4 C T
2P C2
x t − τ2
t
t −τ2
x T sR4 C T
2P C2
x sds,
LV7t h2
1x T tR2 R7 R8 R9 B T
1P B2R−13 B2T P B1
x t
− h1
t
t −h1
x T sR2 R7 R8 R9 B T
1P B2R−13 B2T P B1
x sds.
3.9
Trang 7Combining3.8 and 3.9, we have
LV t, xt ≤ x T t P A B1 A B1T P PB2R−1B T
2P PR−1
A B1T P B1R−12 B T1P A B1 2PDR10D T P R5
PDR10D T P B1R−17 B1T P DR10D T P h2
2R1 R3 R6
PB1R−18 B1T P PB1R−19 B T1P C T
1P C2R−14 C T2P C1
h2 1
R2 B T
1P B2R−13 B2T P B1 R7 R8 R9
C T
1P C1
τ2 2
R4 C T
2P C2
PR−1
5 P x t 2αh1B T
1P xt2
2αPxt2 6PD h1B T
1Prt∞xt
≤ −λminQ1 − 2αP h1B T
1Pxt2
6PD h1B T
1Prt∞xt.
3.10
Let ρ1 λminQ1 − 2αP h1B T
1P , ρ2 6PD h1B T
1P rt∞; we have
LV t, xt ≤ −ρ1xt2 ρ2xt. 3.11 Set
β1 λmaxP h1λmaxPB1 h1λmax
B T1P
h2
1λmax
B T1P B1
,
β2 h1λmax
R5 PB1R−18 B1T P
, β3 h2λmax
R6 PB1R−19 B1T P
,
β4 τ1λmax
C T1P C1 C T
1P C2R−14 C T2P C1
, β6 τ3
2λmax
R4 C T
2P C2
,
β5 h3
2λmaxR1 R3, β7 h3
1λmax
R2 B T
1P B2R−13 B T2P B1 R7 R8 R9
,
3.12
under an assumption that V t, xt ≤ V t0, x t0 for all t ≥ t0, then
λminPE
x t B1
t
t −h1
x sds
2
≤ V t, xt ≤ V t0, x t0 ≤7
i1
β iEϕ θ2
, 3.13
so
E
x t B1
t
t −h x sds
2
≤
7
i1β iEϕ θ2
λminP . 3.14
Trang 8Thus, according to21, Theorem 1.3 page 331, we have
Ext2≤
1 h1B1
1− h1B1
27
i1β i Eϕθ2
λminP . 3.15
If not, there exist t > t0, such that V t, xt ≥ V s, xs for all s ∈ t0, t, and one has
D EV t, xt ≥ 0. 3.16
In view of Ito’s formula, we obtain
D EV t, xt ELV t, xt. 3.17
By3.16 and 3.17, it is easy to derive that
0≤ D EV t, xt ELV t, xt ≤ −ρ1Ext2 ρ2Ext, 3.18
soExt ≤ ρ2/ρ1 By3.18, we can conclude that
Ext2≤ ρ2
ρ1Ext ≤
ρ
2
ρ1
2
By3.15 and 3.19, we get
Ext2≤ ρ22
ρ21
1 h1B1
1− h1B1
27
i1β i Eϕθ2
λminP , 3.20
Thus
Eyt2≤ H2Ext2
≤ H
2ρ2 2
ρ12
1 h1B1
1− h1B1
2H27
i1β i Eϕθ2
λminP
≤ N1 N2r2
∞,
3.21
where
N1
1 h1B1
1− h1B1
27
i1β i H2
λminP Eϕθ2, N2 36H
2
PD h1B T
1P
λminQ1 − 2αP − h1B T
1P.
3.22
ByDefinition 2.2, the nonlinear stochastic control system3.1 with the control law 2.3 is said to be BIBO stabilized in mean square This completes the proof
Trang 9If we transform the original system2.1 to the following system
d
x t B2
t
t −h2
s − t h2xsds
A h2B2xt B1x t − h1 ft, xtdt Dutdt
C1x t − τ1 C2
t
t −τ2
x sds
d wt, t ≥ t0≥ 0
y t Hxt
x θ ϕθ ∈ C b
F 0t0− τ, t0; R n , θ ∈ t0− τ, t0,
3.23
we can get the following result
Theorem 3.2 The nonlinear stochastic control system 3.23 with the control law 2.3 is BIBO
stabilized in mean square if there exist symmetric positive-definite matrices S i > 0, i 1, 2, , 10,
and Q2> 0 such that
λminQ2 − 2α P > 0 3.24
and P is the symmetric positive solution of the Riccati equation
P A h2B2 A h2B2T P PΣ2P Δ2 Ξ2 −Q2, 3.25
where
Σ2 B1S−12 B T1 2DS10D T S−1
5 S−1
6 ,
Δ2 A h2B2T P B 2S−1
1 B T2PA h2B2 D T P S −1
7 P D τ2
2C T2P C 2
D T P S −1
8 P D C T
1P C 1 C T
1P C 2S−1
4 C T2P C 1 D T P S −1
9 P D
B T
1P B 2S−1
3 B T
2P B 1,
Ξ2 S2 S5 S6 1
3h
4
2S1 S3 S7 S8 S9 τ2
2S4,
K1 S10D T P , K2 K3 D−1.
3.26
Proof We define a Lyapunov functional V t, xt as
V t, xt V1t, xt V2t V3t V4t V5t V6t, 3.27
Trang 10V1t, xt x t B2
t
t −h2
s − t h2xsds
T
P x t B2
t
t −h2
s − t h2xsds
,
V2t
t
t −h1
x T sS2 S5 B T
1P B 2S−1
3 B T
2P B 1 D T P S −1
8 P D x sds,
V3t
t
t −h2
x T sS6 D T P S −1
9 P D x sds,
V4t
t
t −τ1
x T sC T1P C 1 C T
1P C 2S−1
4 C T2P C 1x sds,
V5t τ2
t
t −τ2
s − t τ2x T sS4 C T
2P C 2x sds,
V6t 1
3
t
t −h2
s − t h23x T sS1 S3 S7 S8 S9xsds.
3.28
Let ρ1 λminQ2−2α1 h2
2 P , ρ2 61 h2
2 P D rt∞ The rest of the proof is essentially
as that ofTheorem 3.1, and hence is omitted This completes the proof
Remark 3.3 If we transform the original system2.1 to the following system
d
x t B1
t
t −h1
x sds B2
t
t −h2
s − t h2xsds
Ax t ft, xt Dutdt
C1x t − τ1 C2
t
t −τ2
x sds
dwt, t ≥ t0≥ 0
y t Hxt
x θ ϕθ ∈ C b
F 0t0− τ, t0; R n , θ ∈ t0− τ, t0,
3.29
using the same process ofTheorem 3.1, we can get the corresponding BIBO stability in mean square results Here we omitted it
Theorem 3.4 The nonlinear stochastic control system 2.1 with the control law 2.3 is BIBO
stabilized in mean square if there exist symmetric positive-definite matricesΩi > 0, i 1, 2, , 6,
and Q3> 0 such that
λminQ3 − 2α P > 0 3.30
...λminP . 3 .14
Trang 8Thus, according to21, Theorem...
1P.
3.22
ByDefinition 2.2, the nonlinear stochastic control system3.1 with the control law 2.3 is said to be BIBO stabilized in mean square This completes the proof...
3.29
using the same process ofTheorem 3.1, we can get the corresponding BIBO stability in mean square results Here we omitted it
Theorem 3.4 The nonlinear stochastic control