Observer-based SMC for stochastic systems with disturbance driven by fractional Brownian motion

Abstract

This paper investigates the problem of disturbance-observer-based sliding mode control for stabilization of stochastic systems driven by fractional Brownian motion (fBm). By proposing a novel disturbance observer, an integral-type sliding surface is put forward with the estimated disturbance error confined within a certain value. Meanwhile, by virtue of fractional infinitesimal operator and linear matrix inequality, a sufficient criterion is derived to guarantee the asymptotic stability of obtained sliding mode dynamics. Further, an observer-based sliding mode controller is designed to ensure finite-time reachability of state trajectories onto the predefined sliding surface. Lastly, an illustrative example is utilized to verify the reliability and applicability of the proposed control strategy.

Full text

Discrete and Continuous Dynamical Systems - Series S
Vol. 15, No. 11, November 2022, pp. 3261–3274
doi:10.3934/dcdss.2022027
OBSERVER-BASED SMC FOR STOCHASTIC SYSTEMS WITH
DISTURBANCE DRIVEN BY FRACTIONAL
BROWNIAN MOTION
Xin Meng and Cunchen Gao
School of Mathematical Sciences
Ocean University of China
Qingdao, China
Baoping Jiang∗
School of Electronic and Information Engineering
Suzhou University of Science and Technology
Suzhou, China
Hamid Reza Karimi∗
Department of Mechanical Engineering
Politecnico di Milano
Milan, Italy
Abstract. This paper investigates the problem of disturbance-observer-based
sliding mode control for stabilization of stochastic systems driven by frac-
tional Brownian motion (fBm). By proposing a novel disturbance observer,
an integral-type sliding surface is put forward with the estimated disturbance
error confined within a certain value. Meanwhile, by virtue of fractional infin-
itesimal operator and linear matrix inequality, a sufficient criterion is derived
to guarantee the asymptotic stability of obtained sliding mode dynamics. Fur-
ther, an observer-based sliding mode controller is designed to ensure finite-time
reachability of state trajectories onto the predefined sliding surface. Lastly, an
illustrative example is utilized to verify the reliability and applicability of the
proposed control strategy.
1. Introduction. Recently, the stability analysis and stabilization of the stochastic
systems perturbed by Gaussian noise become a hot issue in engineering and science
fields. Particularly, in some specific areas, the effect of fractional Gaussian noise
imposed on complex systems is more complicated than classical noise, for example,
in the turbulence propagation model of pollutants, it is better to use fractional
Brownian noises to describe the character of turbulent [4]. It’s worth pointing out
that the fractional Gaussian noise is an incremental process of the fractional Brow-
nian motion (fBm) [1,20], which has been widely used in many scientific fields.
So far, the hotspot of stochastic systems has moved from the classical Gaussian
noise to the fBm. However, the Itˆo formula couldn’t be used in solving the solu-
tion for stochastic systems with fBm anymore, which is different from the classical
2020 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.
Key words and phrases. Stochastic systems, sliding mode control, fractional infinitesimal op-
erator, disturbance observer.
∗Corresponding author: Baoping Jiang and Hamid Reza Karimi.
3261

3262 XIN MENG, CUNCHEN GAO, BAOPING JIANG AND HAMID REZA KARIMI
stochastic systems. Therefore, to solve this problem, Khandani et al. designed a
novel fractional infinitesimal (FIN) operator which has been widely used in various
scientific areas, more details could be found in [8,10]. In [16], Lu and Zhang also
investigated the robust H∞control problem of a linear uncertain stochastic system
by FIN operator.
As we know, sliding mode control (SMC) [22,17,27] plays an important role
in theoretical research and engineering technology for complex systems, such as
in the field of automotive control systems [27], UAV swarm motion [13], secure
communications [3], amplifier circuit systems [21], fractional-order systems [18],
semi-Markovian jump systems [6], neural networks [25] and observer design [7,23,
15], etc. In [25], Yan et al. investigated stochastic exponential stability problems
for switched neural networks. In [14], by using a state-observer framework, a novel
passivity-based SMC scheme for an uncertain stochastic system was presented. In
[19], Parvizian and Khandani considered the problem of exponential mean-square
stabilization for stochastic systems driven by fBm through SMC approach.
The core of disturbance-observer control (DOC) strategy is that the disturbance
states could be estimated by the proposed disturbance observer. Generally speaking,
disturbances and uncertainties exist in all the above control systems and may bring
bad effects on the system performance. To deal with the uncertainty effect, some
researchers have paid attention to the observer-based robust control methods. It’s
worth noting that the DOC strategy has been widely used in solving the problem of
mismatched uncertainties in some control systems [12], such as stochastic systems
[24], fractional-order systems [2], underactuated robotic systems [5], and so on.
For instance, Wang et al. [2] presented a DOC scheme for linear systems with
mismatched uncertainties based on the fractional SMC approach. However, results
reporting on the stability analysis and controller design issue of stochastic systems
with mismatched disturbances driven by fBm via the disturbance-observer-based
SMC approach have not been considered yet. Hence, in comparison to the relevant
[8,10,7,14], the main contributions of the paper are listed below:
(1) A disturbance observer scheme is designed to estimate the mismatched dis-
turbance of the considered fractional stochastic system. From the point view of
system analysis, compared with existing works on DOC strategy in [23,12], it is
helpful to derive a disturbance error vector is norm-bounded;
(2) Depending on the proposed DOC strategy, a novel sliding mode surface func-
tion and a finite-time control law are given. Further, a criterion is established to
ensure the system is robustly asymptotic stable, and the reaching-time of state
trajectories onto the sliding surface is also given.
Notations. The symmetric positive definite matrix, transposed matrix and inverse
matrix denote as A>0,ATand A−1, respectively. λmax(∗) and λmin (∗) are the
maximum and minimum of matrix eigenvalues, respectively. k∗k represents the
Euclidean norm.
2. System description. Consider the stochastic system driven by fractional Gauss-
ian noise as follows:
˙
x(t) = A(t)x(t) + C(t)x(t)WH(t) (1)
where x(t)∈Rnis the state vector, and WH(t) denotes the fractional Gaussian
noise with H∈(1
2,1). The FIN operator LHwith H∈(1
2,1) is presented in [10] as
follows:

OBSERVER-BASED SMC FOR STOCHASTIC SYSTEMS WITH DISTURBANCE ... 3263
LH=∂
∂t +∂
∂xA(t)x(t) + xT(t)CT(t)∂2
∂x2Zt
0
φ(t, s)C(s)dsx(t)(2)
where φ(t, s) satisfies φ(t, s) = H(2H−1)|s−t|2H−2.
Definition 2.1. [7] The trivial solution of (1) is stochastically asymptotically stable
if there exists a function V(x, t)∈C2,1such that:
(V(0, t)=0, V (x, t)≥κ(kxk), Vx(x, t)C(t)x(t)∈ L(0, T )
LHV(x, t)<0(3)
for all (x, t)∈Rn×[0,+∞), and L(0, T ) denotes the family of stochastic processes
on [0, T ].
Remark 1. Compared with the classical systems, Definition 2.1 requires an ad-
ditional condition on Lyapunov function, that is Vx(x, t)C(t)x(t)∈ L(0, T ), which
makes the selection of an appropriate Lyapunov function more complicated.
Consider the following system driven by fractional Brownian motion (fBm):
˙
x(t) = A(t)x(t) + Ad(t)x(t−τ) + B(u(t) + δ(t)) + C0x(t)WH(t)(4)
where u(t)∈Rmis the control vector, δ(t)∈Rmis the disturbance vector, A(t) =
A+∆A(t), Ad(t) = Ad+ ∆Ad(t), rank(C0)< n, and τdenotes the constant time-
delay. The uncertainties [∆A(t) ∆Ad(t)] = MF(t)[N Nd], with FT(t)F(t)≤I.
As pointed in [24], the disturbance vector δ(t) meets
δ(t) = Cz(t),˙
z(t) = Gz(t) (5)
where z(t)∈Rp. Similar to classical stochastic systems, we have
dx(t) = A(t)x(t) + Ad(t)x(t−τ) + B(u(t) + δ(t))dt+C0x(t)dBH(t)(6)
where BH(t) denotes the fBm with H∈(1
2,1).
Remark 2. In [11], Khasminskii proved the stability condition can be reduced to
LV(t, x)≤0, hence, Lis the conventional infinitesimal operator when H= 0.5.
In other words, the classical stochastic system with Bm is a specific case of the
stochastic system with fBm.
Remark 3. In 2012, Zeng et al. first presented some conditions to transform
nonlinear stochastic systems with fBm to linear systems in [26]. However, it should
be pointed that Khandani et al. proved that Zeng’s results were incorrect, and some
comments have been presented in [26]. More details about the FIN operator can be
found in [8,9].
Lemma 2.2. [23]Assume that Ais Hurwitz and has ndistinct eigenvalues. Let
Xbe a nonsingular matrix such that XAX−1=diag(λ1,· · ·, λn), then keAtk ≤
δeλmax (A)t, where δ=kX−1kkXk.
Lemma 2.3. [7]Assume that there exists a positive-definite function V(t)such that
V(t0)≥0,˙
V(t)≤αV (t)−βV η(t), t ≥t0.(7)
with α > 0,β > 0and 0< η < 1. Then V(t)can converge to zero at least in a
finite-time:
tr=t0+1
α(1 −η)ln αV 1−η(t0) + β
β.(8)

3264 XIN MENG, CUNCHEN GAO, BAOPING JIANG AND HAMID REZA KARIMI
Assumption 2.4. The system state x(t)is available for measurement and norm-
bounded, i.e. kx(t)k ≤ ε, where εis a positive constant.
3. Main results. In this part, we will design a disturbance observer to estimate
the real value of the disturbance vector δ(t) of the system (6). Then, we consider
the problem of robust stability analysis and controller design for the fractional-order
stochastic system (6) by using the proposed DOC strategy.
3.1. Disturbance observer. We design the follow disturbance observer
(dp(t) = (GLˆ
z(t) + LAx(t) + LAdx(t−τ) + LBu(t))dt
ˆ
δ(t) = Cˆ
z(t),ˆ
z(t) = p(t)−Lx(t)(9)
where ˆ
δ(t) is the estimation of the disturbance δ(t) in (6), p(t) is the auxiliary
vector, Lis the gain matrix. In addition, Lis designed to ensure GL=G+LBC
is Hurwitz.
Let ez(t) = z(t)−ˆ
z(t) the disturbance estimation error, from (5),(6) and (9), it
can be easily obtained that the disturbance error system satisfies that
dez(t)=dz(t)−dp(t) + Ldx(t)
=Gz(t)dt−GLˆ
z(t) + LAx(t) + LAdx(t−τ) + LBu(t)dt
+LA(t)x(t) + Ad(t)x(t−τ) + B(u(t) + δ(t))dt+LC0x(t)dBH(t)
=GLez(t) + L∆A(t)x(t) + L∆Ad(t)x(t−τ)dt+LC0x(t)dBH(t)
(10)
Remark 4. It can be easily seen that the error dynamic (10) may be a fractional
Itˆo process which depends on the gain matrix L. Hence, in [8], Khandani et al.
presented a stronger restricting condition, i.e. by selecting an appropriate non-zero
gain matrix to ensure the stochastic term equals to zero. Particularly, we can dope
out from (10) that if rand(C0)< n, there must exist a non-zero gain matrix L
such that the condition LC0=0holds. It also implies that ez(t) maybe not an Itˆo
process. Therefore, ez(t) is semi-martingale and its time-derivative can be obtained.
Lemma 3.1. For the error system (10), choose Lto ensure LC0= 0 holds, hence
(10) can be rewritten as
dez(t)
dt=GLez(t) + π(t) (11)
where π(t) = L(∆A(t)x(t) + ∆Ad(t)x(t−τ)). Then, we can obtain that ez(t)is
norm-bounded, i.e. kez(t)k ≤ ζ, where ζis a positive scalar.
Proof. Taking integration of (11) from 0 to t, i.e.
ez(t) = eGLt·ez(0) + Zt
0
eGL·(t−τ)π(τ)dτ(12)
Then, based on the Lemma 2.2 and Assumption 2.4, we have
kez(t)k ≤ keGLtkkez(0)k+

Zt
0
eGL(t−τ)π(τ)dτ


≤eλmax(GL)tkez(0)k+Zt
0

eGL(t−τ)

kπ(τ)kdτ

OBSERVER-BASED SMC FOR STOCHASTIC SYSTEMS WITH DISTURBANCE ... 3265
≤eλmax(GL)tkez(0)k+Zt
0

eGL(t−τ)



L(∆A(t)x(t)+∆Ad(t)x(t−τ))

dτ
≤eλmax(GL)tkez(0)k+εkLMNk+kLMNdkZt
0

eGL(t−τ)

dτ
≤eλmax(GL)tkez(0)k+εkLMNk+kLMNdkZt
0
δeλmax (GL)(t−τ)dτ
≤eλmax(GL)tkez(0)k+εkLMNk+kLMNdk
δ
λmax(GL)(eλmax (GL)t−1)
(13)
where, δ=kG−1
LkkGLk. Because GLis a Hurwite matrix, i.e. Re(λmax (GL)) <0
and 0 < eλmax (GL)t≤1, we can obtain that
kez(t)k ≤ ζ(14)
where ζ=kez(0)k+ε(kLMNk+kLMNdk)
δ
λmax(GL).
3.2. Stability analysis. For stochastic system (6), the sliding mode surface func-
tion could be designed as follows,
s(t) = Px(t)−Zt
0
P(A+BK)x(s)ds(15)
where Pand K∈Rm×n. Particularly, Pis chosen to ensure PB ∈Rm×mis
nonsingular, and Kis selected to guarantee A+BK is Hurwitz.
Remark 5. In general, for the disturbance error system (10) and sliding surface
function (15), there exist two different methods to analyze the stabilization problem.
In [7,8], by selecting an appropriate gain matrix to ensure the sliding mode surface
was a classical Itˆo process, Khandani et al. investigated the stochastic stability
issue of uncertain systems with time-delay which driven by fBm. By using this
special technique, it is beneficial to obtain the reachability criterion which is similar
to the classical stochastic system. While, another complex method to consider the
stability problem was also proposed in [10].
Combine with (6), we know that s(t) can be rewritten as
s(t) = PZt
0hA(s)x(s) + Ad(s)x(s−τ) + B(u(s) + Cz(s))ids
+Zt
0
PC0x(s)dBH(s)−Zt
0
P(A+BK)x(s)ds
(16)
which implies that s(t) is a fractional Itˆo process. As previously mentioned, assume
the condition PC0= 0 holds. Then from ˙
s(t) = 0, we could calculate the equivalent
control law ueq(t), i.e.
ueq(t) = Kx(t)−(PB)−1P[∆Ax(t) + Ad(t)x(t−τ)] −cˆ
z(t)(17)
Substituting (17) into (6), we can obtain that
dx(t) = (Ak+I0∆A(t))x(t) + I0Ad(t)x(t−τ) + BCez(t)dt+C0x(t)dBH(t)
(18)
where Ak=A+BK,I0=I−B(PB)−1P. Denote ¯
X(t)=[xT(t),eT
z(t)]T, hence,
(10) and (18) are given as follows:
d¯
X(t) = ¯
A(t)¯
X(t) + ¯
Ad(t)¯
X(t−τ)dt+¯
C0¯
X(t)dBH(t)(19)

3266 XIN MENG, CUNCHEN GAO, BAOPING JIANG AND HAMID REZA KARIMI
where
¯
A(t) = Ak+I0∆A(t)BC
L∆A(t)GL,¯
Ad(t) = I0Ad(t)0
L∆Ad(t)0,¯
C0=C00
0 0 .
Theorem 3.2. (Stability criterion) Assuming that the mismatched disturbance
states (6) can be estimated by the observer (9). The sliding surface function is
given in (15). If there exist symmetric matrices R1>0,R2>0,Q1>0,Q2>0,
and scalar η > 0such that the following conditions hold:
LC0=0,PC0=0,(20)






Π11 R1BC R1I0Ad0 R1I0M
*Π22 0 0 R2LM
* * Π33 0 0
* * * −Q20
* * * * −ηI






<0(21)
where Π11 =Q1+ sym{R1Ak}+εNTN,Π22 =Q2+ sym{R2GL},Π33 =
−Q1+εNT
dNd. Then, close-loop system (19) is robustly asymptotically stable in
probability.
Proof. Utilize the following Lyapunov function
V(t, ¯
X(t)) = e−λRt
0Rs
0φ(τ,s)dτdsh¯
XT(t)R¯
X(t) + Zt
t−τ
¯
XT(s)Q¯
X(s)dsi(22)
where R= diag(R1,R2), Q= diag(Q1,Q2), λ > 0 is a constant. Following the
method in [20], V(t, ¯
X(t)) satisfies those conditions, i.e. V(t, 0) = 0 and
∂V
∂¯
X¯
C0¯
X(t) = e−λRt
0Rs
0φ(τ,s)dτds×2¯
XT(t)R¯
C0¯
X(t)∈ L(0, T )(23)
Then, by using the FIN operator (2), we have that
LHV=e−λRt
0Rs
0φ(τ,s)dτds(¯
XT(t)Q¯
X(t)−¯
XT(t−τ)Q¯
X(t−τ)
+ 2 ¯
XT(t)R¯
X(t)[ ¯
A(t)¯
X(t) + ¯
Ad(t)¯
X(t−τ)]
+Zt
0
φ(τ, t)dτn−λ¯
XT(t)R¯
X(t)−λZt
t−τ
¯
XT(s)Q¯
X(s)ds
+ 2 ¯
XT(t)¯
CT
0R¯
C0¯
X(t)o)
(24)
Select 0 < λ ≤λmax (2 ¯
CT
0R¯
C0)/λmin(R), then
λ¯
XT(t)R¯
X(t)+2¯
XT(t)¯
CT
0R¯
C0¯
X(t)≤0(25)
therefore,
LHV≤e−λRt
0Rs
0φ(τ,s)dτdsn¯
XT(t)Q¯
X(t)−¯
XT(t−τ)Q¯
X(t−τ)
+ 2 ¯
XT(t)R¯
X(t)¯
A(t)¯
X(t) + ¯
Ad(t)¯
X(t−τ)o
=e−λRt
0Rs
0φ(τ,s)dτdsh¯
XT(t)¯
XT(t−τ)iΓ"¯
XT(t)
¯
XT(t−τ)#
(26)

OBSERVER-BASED SMC FOR STOCHASTIC SYSTEMS WITH DISTURBANCE ... 3267
where Γ=Q+ sym{R¯
A(t)}R¯
Ad(t)
*−Q.
Substituting ¯
A(t) and ¯
Ad(t) into Γ, we have that
Γ=



Γ11 Γ12 Γ13 0
*Γ22 R2L∆Ad(t)0
* * −Q10
* * * −Q2




with Γ11 =Q1+ sym{R1Ak+R1I0∆A(t)},Γ12 =R1BC + ∆AT(t)LTR2,Γ13 =
R1I0Ad+R1I0∆Ad(t), Γ22 =Q2+ sym{R2GL}.
In addition, note that
Γ=Λ+ΦF(t)Ψ+ (ΦF(t)Ψ)T<0(27)
with
Λ=



Λ11 R1BC R1I0Ad0
*Λ22 0 0
* * Λ33 0
* * * −Q2




,Φ=



R1I0M
R2LM
0
0




,ΨT=



NT
0
NT
d
0




.
Hence, there exists a constant η > 0, (27) is equivalent to
Λ+η−1ΦΦT+ηΨTΨ<0(28)
Therefore, we will obtain that (28) and (21) are equivalent by using the Schur
complement. This completes the proof.
3.3. Controller design. In this part, the following theorem is provided to obtain
a sufficient condition on the finite-time stability of stochastic system (6) driven by
fBm.
Theorem 3.3. (Reachability criterion) For stochastic system (6), the sliding sur-
face function s(t)is given in (15). Then, the state trajectories of system (6) can be
driven onto the sliding surface s(t) = 0in finite-time and maintain sliding motion
by utilizing the following controller:
u(t) = Kx(t)−(PB)−1P[Ax(t) + Ad(t)x(t−τ)] −Cˆ
z(t)
−γ(PB)−1s(t)−ξ(PB)−1sign(s(t)),(29)
where Pand Lcan be calculated from Theorem 3.2. γ > 0and ξ > kPBCkζ > 0.
Proof. The Lyapunov function is chosen as
V(t) = 1
2sT(t)s(t) (30)
Then, using (15) and (29), its derivative is
˙
V(t) = sT(t)nPA(t)x(t) + Ad(t)x(t−τ) + B[Kx(t)
−(PB)−1P[∆Ax(t) + Ad(t)x(t−τ)]
−Cˆ
z(t)−γ(PB)−1s(t)−ξ(PB)−1sign(s(t))] + Cz(t))−P(A+BK)x(t)o
=sT(t)−γs(t)−ξsign(s(t)) + PBCez(t)
≤ −γks(t)k2−(ξ− kPBCk·kez(t)k)ks(t)k
(31)

3268 XIN MENG, CUNCHEN GAO, BAOPING JIANG AND HAMID REZA KARIMI
Based on the Lemma 3.1, there exists a ζ > 0 such that kez(t)k ≤ ζholds by the
observer (9). Hence, if γ > 0 and ξ > kPBCkζ, we can easily obtain that ˙
V(t)<0.
Therefore, for any s(t)6=0, we have
˙
V(t)≤ −γks(t)k2−(ξ− kPBCkζ)ks(t)k=−γV (t)−(ξ− kPBCkζ)pV(t)(32)
Hence, based on the Lemma 2.3, the reaching-time can be calculated
tfin =2
γln γpV(0) + (ξ− kPBCkζ)
ξ− kPBCkζ(33)
Remark 6. The proposed disturbance-observer-based control strategy ensures ro-
bust stability of the fractional-order stochastic system driven by fBm. As mentioned
in Remark 1, the proposed SMC method is also appropriate for the classical sto-
chastic systems by choosing the Hurst parameter H= 0.5, and similar stabilization
conditions can be derived by using the conventional infinitesimal operator proposed
in [4]. Furthermore, the condition (20) guarantees the sliding surface is determin-
istic for the stability analysis, the merit of which is that the fractional stochastic
perturbation bounded as a priori condition is not necessary now. Lastly, by using
Lemma 2.3, the reaching-time of the sliding surface (15) can be easily derived based
on eq.(33) from Theorem 3.3.
Remark 7. As mentioned above, it is assumed that rank(C0)< n holds in this
paper, hence there exist non-zero gain matrices Land Psuch that the condition
(20) holds, which also means that the sliding surface s(t) is semi-martingale, that
is to say, ˙
s(t) can be obtained. While, if rank(C0) = nholds, then s(t) becomes a
fractional Itˆo process. Therefore, compared with the general sliding mode surface
form, it will be more complicated to obtain the reachability criterion. For such case,
Khandani et al. [10] have proved that the state trajectories of fractional stochastic
systems will be forced to reach a region in finite time by using fractional Itˆo formula.
4. Numerical examples. In order to illustrate the effectiveness of the proposed
control strategy, let’s consider the water quality standards model borrowed from [4,
14]. The mathematical model is shown in (6), and some parameters are considered
as follows:
A=−1 1
−2−3,B=1
−2,Ad=0−0.1
0.5 1 ,C=−0.25
0.65 T
,
C0=0.1−0.5
0.05 −0.25 ,G=0 2.8
−1.8 0 ,M=0.1
0.1,
N=0
−0.2T
,Nd=0.2
−0.2T
, F (t) = sin(t), τ = 0.2.
where x(t)∈R2denotes the pollutant concentrations, and u(t)∈R1denotes the
control input.
Then, according to the conditions (20) and (21), select γ= 0.1, ξ= 1 and
P=BT,K=−1.2636
0.9031 T
,L=1.2446 −2.4892
−1.9230 3.8460 .

OBSERVER-BASED SMC FOR STOCHASTIC SYSTEMS WITH DISTURBANCE ... 3269
Obviously, it can be easily calculated that LC0=0,PC0=0,PB is nonsingu-
lar, AKand GLare Hurwitz. Hence, the sliding mode surface could be calculated
s(t) = [1 −2]x(t)−Zt
0
[−3.3180 11.5155]x(s)ds
Then, the feasible solutions could be calculated by solving the LMI (21), i.e.
R1=41.7153 12.7769
12.7769 23.4137 ,R2=93.2751 90.4950
90.4950 113.7350 ,
Q1=89.5075 −2.5178
−2.5178 88.2153 ,Q2=90.4810 −0.3894
−0.3894 91.3671 , ε = 90.8356.
1) Give a sample path, Fig.1 shows that the curve of the fBm with H= 0.6,
and the state trajectories x(t)∈R2of stochastic system (6) without controller are
presented in Fig.2.
2) Choose 10 independent fBm paths, the red line denotes the independent path
and the blue line denotes the average of the those trajectories. Hence, Fig.3 shows
the curves of real and estimated values of the disturbance vectors. Meanwhile, Fig.4
plots the state trajectories x(t)∈R3of stochastic system (6) under this proposed
SMC strategy (29). Lastly, the curves of sliding mode surface and control signal
are shown in Fig.5.
3) Under the sample path of fBm shown in Fig.1, the state trajectories x(t) are
depicted in Fig.6 and Fig.7 with time-delay τ= 0.2,0.4,···,1.0. It can be seen for
the results that these varying delays have little impact on the system performance
under the presented SMC strategy.
0123456
-6
-4
-2
0
2
4
6
8
Figure 1. A sample path of fBm with H= 0.6.

3270 XIN MENG, CUNCHEN GAO, BAOPING JIANG AND HAMID REZA KARIMI
0123456
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Figure 2. State trajectories of open-loop systems.
Figure 3. The curves of disturbance estimation.
5. Conclusions. The SMC problem of stochastic systems with disturbances driven
by fBm has been investigated in this paper. A DOC strategy has been designed

OBSERVER-BASED SMC FOR STOCHASTIC SYSTEMS WITH DISTURBANCE ... 3271
Figure 4. State trajectories of closed-loop systems.
0123456
0
0.5
1
1.5
2
0123456
-0.6
-0.4
-0.2
0
0.2
Figure 5. The curves of sliding mode surface function and control signal.
to estimate the real value of the disturbance vector at first. Then, by using the
FIN operator and the Lyapunov approach, some sufficient conditions of robustly

3272 XIN MENG, CUNCHEN GAO, BAOPING JIANG AND HAMID REZA KARIMI
0123456
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Figure 6. State trajectories x1(t) with τ= 0.2,0.4,···,1.0, respectively.
0123456
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Figure 7. State trajectories x2(t) with τ= 0.2,0.4,···,1.0, respectively.
asymptotically stable have been derived. Meanwhile, it has also shown that the
trajectories of stochastic systems could arrive at the sliding mode surface in a finite

OBSERVER-BASED SMC FOR STOCHASTIC SYSTEMS WITH DISTURBANCE ... 3273
time by using the proposed controller. At last, to demonstrate the validity and
feasibility of the obtained results, an illustrative example has been provided.
Acknowledgments. This work is supported by The National Natural Science
Foundation of China under grant 62003231; partially supported by The Natural Sci-
ence Foundation of Shandong/Jiangsu Province under grants no. ZR2019MF027,
BK20200989, in part by the China Scholarship Council under Grant 202006330047,
in part by the NCF for colleges and universities in Jiangsu Province under Grant
20KJB120005.
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Received August 2021; revised December 2021; early access February 2022.
E-mail address:xmeng1529@163.com
E-mail address:ccgao123@126.com
E-mail address:baopingj@163.com
E-mail address:hamidreza.karimi@polimi.it