# Sliding Mode Control of Uncertain Neutral Stochastic Systems with Multiple Delays

Authors: Dilan Chen; Weidong Zhang
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

This paper is concerned with the sliding mode control for uncertain
stochastic neutral systems with multiple delays. A switching surface is
adopted first. Then, by means of linear matrix inequalities (LMIs), a sufficient
condition is derived to ensure the global stochastic stability of the
stochastic system in the sliding mode for all admissible uncertainties. The synthesized
sliding mode controller guarantees the existence of the sliding mode.


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## Body

## 1. Introduction

Time delay
occurs due to the finite capabilities of information processing and data
transmission among various parts of the system. The phenomena of time delay are
often encountered in various relevant systems, such as HIV infection with drug
therapy, aircraft stabilization, chemical engineering systems, inferred
grinding model, manual control, neural network, nuclear reactor, population
dynamic model, rolling mill, ship stabilization, and systems with lossless
transmission lines. It is well known that time delay factors always lead to
poor performance. Hence, problems of stability analysis and stabilization of
dynamical systems with time delays in the state variables and/or control inputs
have received considerable interest for more than three decades [1–6].In practice, systems are almost always innately
“noisy”. Therefore, in order to model a system realistically, a degree of
randomness must be incorporated into the model. Thus, a class of stochastic
systems has received great attention in the past decade [7]. On the other hand,
it has been shown that a lot of practical systems can be modeled by using
functional differential equations of the neutral type [8, 9]. However, the
mathematical model always contains some uncertain elements. Therefore,
uncertain systems have been extensively studied in the past years [10–12].To cope with the problem of stability of uncertain
stochastic neutral delay systems, most of the research focused on the retarded
functional differential equations and it also seems that few results are
available on the variable structure control.Sliding mode control (SMC) is a particular type of
variable structure control. It provides an effective alternative to deal with
the nonlinear dynamic systems. The main feature of SMC is its easy realization,
control of independent motion, insensitivity to variation in plant parameters
or external perturbations, and wide variety of operational models [13–15].The purpose of this paper lies in the design of SMC
for a class of uncertain stochastic neutral delay systems. A switching surface,
which makes it easy to guarantee the stability of the uncertain stochastic
neutral delay systems in the sliding mode, is first proposed. By means of
linear matrix inequalities (LMIs), a sufficient condition is given such that
the stochastic dynamics in the specified switching surface is globally
stochastically stable. And then, based on this switching
surface, a synthesized SMC law is derived to guarantee the existence of the
composite sliding motion. Finally, a numerical example is illustrated to
demonstrate the validity of the proposed SMC.


## 2. Problem Formulation

Consider the
following neutral stochastic system with uncertainties and multiple
delays:(2.1)d[Ex(t)−Cx(t−τ)]=[(A+ΔA(t))x(t)+(Ad+ΔAd(t))x(t−h(t))+Bu(t)]dt+[ΔE(t)x(t)+ΔEd(t)x(t−h(t))]dw(t),x(t)=ϕ(t),t∈[−H,0],where x(t)∈ℝn is the state, u(t)∈ℝm is the control input, τ is the constant delay, h(t) is the time-varying differentiable bounded
delay satisfying 0≤h(t)≤hM, h(t)Δ≤hD<1, w(t) is an m-dimensional Brownian motion, H=max⁡{τ,hM}.
It is assumed that ϕ(t) is the initial condition which is continuous, t∈[−H,0].
In system (2.1), E∈ℝn×n, C∈ℝn×n, A∈ℝn×n, Ad∈ℝn×n, B∈ℝn×m  are known real constant matrices. ΔA(t), ΔAd(t), ΔE(t),
and ΔEd(t) represent the structured uncertainties in (2.1),
which are assumed to be of the forms(2.2)ΔA(t)=MF1(t)NA,ΔAd(t)=MF2(t)NAd,[ΔE(t),ΔEd(t)]=MF3(t)[NE,NEd],M, NA, NAd, NE,
and NEd are some given constant matrices, Fl(t)(l=1,2,3) are unknown real time-varying matrices which
have the following structure:Fl(t)=blockdiag{δl1(t)Irl1,…,δlk(t)Irlk,Fl1(t),…,Fls(t)}, δli∈ℝ, |δli|≤1, 1≤i≤k, and Flj⊤Flj≤I, 1≤j≤s.We define the setsΔl as(2.3)Δl={Fl⊤(t)Fl(t)≤I,FlNl=NlFl,∀Nl∈Σl},where Σl={Nl=blockdiag[Nl1,…,Nlk,nl1Ifl1,…,nlsIfls]}, Nli are invertible for 1≤i≤k, and nlj∈ℝ, nlj≠0 for 1≤j≤s.The following useful lemmas will be used to derive the
desired LMI-based stability criteria.Lemma 2.1 (See [1]).
The LMI[S11S12*S22]<0, with S11=S11⊤, S22=S22⊤,
is equivalent to(2.4)S22<0,S11−S12S22−1S12⊤<0.Lemma 2.2 (See [11]).
LetD, E, Fl be real matrices of appropriate dimensions and Fl∈Δl.
Then for any block-structured matrix Nl∈Σl,(2.5)DFlE+(DFlE)⊤≤D(NlNl⊤)D⊤+E⊤(NlNl⊤)−1E.Lemma 2.3 (See [11]).
LetA, D, E, Fl,
and P be real matrices of appropriate dimensions with P=P⊤>0 and Fl∈Δl. Then for any block-structured matrix Nl∈Σl satisfying P−1−D(NlNl⊤)D⊤>0, one
has(2.6)(A+DFlE)⊤P(A+DFlE)≤A⊤(P−1−DNlNl⊤D⊤)−1A+E⊤(NlNl⊤)−1E.Lemma 2.4 (See [11]).
For anyz,y∈ℝn and for any symmetric positive-definite matrix X∈ℝn×n, (2.7)−2z⊤y≤z⊤X−1z+y⊤Xy.Definition 2.5 (See [14]).
The nominal stochastic
time-delay system of form (2.1) with u(t)=0 is said to be mean-square asymptotically
stable if(2.8)lim⁡t→∞𝔼|x(t)|2=0.Definition 2.6 (See [14]).
The uncertain time delay
system of the form (2.1) is robustly mean square stabilized if the nominal system
is mean-square asymptotically stable for all admissible uncertainties.In order to simplify the treatment of the problem, the
operatorℑ:C([−τ,0],ℝn)→ℝn is defined to be(2.9)ℑ(xt)=Ex(t)−Cx(t−τ).The stability of the operator ℑ is defined as follows.Definition 2.7 (See [9]).
The operatorℑ is said to be stable if the zero solution of
the homogeneous difference equation(2.10)ℑ(xt)=0,t≥0,X0=φ∈{ψ∈C[−τ,0]:ℑψ=0}is uniformly asymptotically
stable.Ifrank(E)=m<n,
then it is easy to find that there exist nonsingular constant matrices K and S,
such that(2.11)KES−1=(E1000),KCS−1=(C100C2),where E1 is an m×m nonsingular matrix, C1 and C2 are m×m and (n−m)×(n−m) constant matrices, respectively.Lemma 2.8 (See [9]).
The operatorℑ is stable if ∥E1−1C1∥<1 and |C2|≠0, where E1, C1,
and C2 are defined as in (2.11) and ∥⋅∥ is any matrix norm.


## 3. Switching Surface and Controller Design

In this work,
we choose the switching function as follows:(3.1)S(t)=D[Ex(t)−Cx(t−τ)]+σ(t),where the auxiliary variable σ(t) satisfies the following:(3.2)dσ=−D[(A+BK)x(t)+Adx(t−h(t))]dt−D[ΔEx(t)+ΔEdx(t−h(t))]dw(t),where D∈ℝm×n and K∈ℝm×n are constant matrices. The matrix K is chosen such that the matrix A+BK is Hurwitz, and the matrix D is to be designed later so that DB is nonsingular. As long as the system operates
in the sliding mode, it satisfies the equations S(t)=0 and SΔ(t)=0 [13].Therefore, the equivalent controlueq(t) in the sliding manifold is given
by(3.3)ueq=−(DB)−1D[(ΔA(t)−BK)x(t)+ΔAd(t)x(t−h(t))].Substituting (3.3) into system
(2.1), the following equivalent sliding mode dynamics can be
obtained:(3.4)d[Ex(t)−Cx(t−τ)]=[(A+BK+ΔA(t)−B(DB)−1DΔA(t))x(t)+(Ad+ΔAd(t)−B(DB)−1DΔAd)x(t−h(t))]dt+[ΔEx(t)+ΔEdx(t−h(t))]dw(t).Now, we proceed to the first
task which is to analyze the robustly stochastic
stability of the sliding motion described by (3.4), and
derive a sufficient condition by means of the
linear matrix inequality method.


## 4. Robust Stabilization in the Mean Square Sense

Theorem 4.1.
Consider the equivalent sliding mode dynamics (3.4). If the operator ℑ is stable and there exist symmetric
positive-definite matrices X, Q1,
Q2, T1, T2, T3, T4, T5,     and T6 satisfying the following LMIs:
(4.1)Π=[Π11Π12Π13E⊤XBE⊤XME⊤XM000*Π22−C⊤XAd000C⊤XBC⊤XMC⊤XM**Π33000000***−B⊤XB200000****−T10000*****−T2000******−B⊤XB200*******−T30*******0−T4],where Π11=E⊤X(A+BK)+(A+BK)⊤XE+Q1+Q2+NA⊤(T1+T3+T5)NA+NE⊤T6NE, Π12=−(A+BK)⊤XC, Π13=E⊤XAd+NEd⊤T4NEd, Π22=−Q1+NA⊤T2NA, Π33=−(1−hD)Q1+NAd⊤(T1+T3+T5)NAd+NEd⊤T4NEd, (4.2)[−XXM*−T6]<0,(4.3)[−X2XM*−T5]<0,then the uncertain time delay
system of the form (2.1) with the switching surface (3.1) is robustly
stochastically stable and sliding mode matrix D=B⊤X.
In the above LMIs, T takes the form of NlNlT for Nl∈Σl.Proof.
Choose a Lyapunov functional candidateV(x(t),t) as(4.4)V(xt)=(ℑxt)⊤(t)P(ℑxt)+∫t−τtx⊤(s)Q1x(s)ds+∫t−h(t)tx⊤(s)Q2x(s)ds.Then, the averaged derivative is
given by the following expression:(4.5)LV(xt)=2[Ex(t)−Cx(t−τ)]⊤X[(A+BK+ΔA(t)−B(DB)−1DΔA(t))x(t)+(Ad+ΔAd(t)−B(DB)−1DΔAd(t))x(t−h(t))]+[ΔE(t)x(t)+ΔEd(t)x(t−h(t))]⊤P[ΔE(t)x(t)+ΔEd(t)x(t−h(t))]+x⊤(t)Q1x(t)−x⊤(t−τ)Q1x(t−τ)+x⊤(t)Q2x(t)−(1−hΔ(t))x⊤(t−h(t))Q2x(t−h(t)).Using Lemma 2.1, inequality (4.2)
is equivalent to(4.6)X−1−MT6−1M⊤>0.Hence, it follows from Lemma 2.3
that(4.7)[ΔE(t)x(t)+ΔEd(t)x(t−h(t))]⊤X[ΔE(t)x(t)+ΔEd(t)x(t−h(t))]={MF3(t)[NEx(t)+NEdx(t−h(t))]}⊤X{MF3(t)[NEx(t)+NEdx(t−h(t))]}≤[NEx(t)+NEdx(t−h(t))]⊤T6[NEx(t)+NEdx(t−h(t))]=x⊤(t)NE⊤T6NEx(t)+x⊤(t)NE⊤T6NEdx(t−h(t))+x⊤(t−h(t))NEd⊤T6NEx(t)+x⊤(t−h(t))NEd⊤T6NEdx(t−h(t)).Note that D=B⊤X,
and it follows form Lemma 2.4 that(4.8)−2x⊤(t)E⊤XB(DB)−1DΔA(t)x(t)≤x⊤[E⊤XB(B⊤XB)−1B⊤XE+ΔA⊤(t)XΔA(t)]x(t),−2x⊤(t)E⊤XB(DB)−1DΔAd(t)x(t−h(t))≤x⊤E⊤XB(B⊤XB)−1B⊤XEx(t)+x⊤(t−h(t))ΔAd⊤(t)XΔAd(t)x(t−h(t)),2x⊤(t−τ)C⊤XB(DB)−1DΔA(t)x(t)≤x⊤(t−τ)C⊤XB(B⊤XB)−1B⊤XCx(t−τ)+x⊤(t)ΔA⊤(t)XΔA(t)x(t),2x⊤(t−τ)C⊤XB(DB)−1DΔAd(t)x(t−h(t))≤x⊤(t−τ)C⊤XB(B⊤XB)−1B⊤XCx(t−τ)+x⊤(t−h(t))ΔAd⊤(t)XΔAd(t)x(t−h(t)).Substituting (4.7) and (4.8)
into (4.5), we obtain(4.9)LV(x(t))≤X⊤(t)ΞX(t),where Ξ=Ξ1+M1F1(t)N1+N1⊤F1(t)M1⊤+M1F2(t)N2+N2⊤F2(t)M1⊤+M2F1(t)N1+N1⊤F1(t)M2⊤+M2F2(t)N2+N2⊤F2(t)M2⊤, X=[x⊤(t),x⊤(t−τ),x⊤(t−h(t))]⊤,
(4.10)Ξ1=[Ξ11−(A+BK)⊤XCE⊤XAd+NEd⊤T6NEd*Ξ22−C⊤XAd**Ξ33],Ξ11=E⊤X(A+BK)+(A+BK)⊤XE+NE⊤T6NE+2ΔA⊤(t)XΔA(t)+Q1+Q2+2E⊤XB(B⊤XB)−1B⊤XE,Ξ22=−Q1+2C⊤XB(B⊤XB)−1B⊤XC,Ξ33=−(1−hD)Q2+2ΔAd⊤(t)XΔAd(t)+NEd⊤T6NEd,M1=[E⊤XM00],M2=[0−C⊤XM0],N1=[NA00],N2=[00NAd].Using Lemmas 2.1 and 2.2, we have(4.11)Ξ≤Ξ′=[Ξ11′Π12Π13E⊤XBE⊤XME⊤XM00000*Π22−C⊤XAd000C⊤XBC⊤XMC⊤XM00**Ξ33′00000000***−B⊤XB20000000****−T1000000*****−T200000******−B⊤XB20000*******−T3000********−T400*********−X20*********0−X2]+M3F¯(t)N3+N3⊤F¯⊤(t)M3T,Ξ11′=E⊤X(A+BK)+(A+BK)⊤XE+Q1+Q2+NA⊤(T1+T3)NA+NE⊤T6NE,Ξ33′=−(1−hD)Q1+NAd(T2+T4)NAd+NEd⊤T6NEd,M3=[NA000000000000NAd00000000]⊤,F¯(t)=[F1⊤(t)00F2⊤(t)],N3=[000000000M⊤X00000000000M⊤X].With Lemma 2.1, we can see that Ξ′<0 is equivalent to LMIs (4.1)–(4.3).
According to Ito^'s formula, system (2.1) is robustly
stochastically stable. This completes the proof.


## 5. Sliding Mode Control

We now design
an SMC law such that the reachability of the specified switching surface is
ensured.Theorem 5.1.
Consider the uncertain stochastic time delay system (2.1). Suppose that the
switching function is given as (3.1) with D=B⊤X,
where X is the solution of LMIs (4.1)–(4.3). Then the
reachability of the sliding surface s(t)=0 can be guaranteed by the following SMC
law:(5.1)u(t)=Kx(t)−ρ(t)sgn(s(t)),where the switching gain ρ(t) is given as(5.2)ρ(t)=λ+∥(DB)−1DM∥×(∥NAx(t)∥+∥NAdx(t−h(t))∥)with λ>0.Proof.
A Lyapunov functional candidateV(t) is defined as(5.3)V(t)=12S⊤(t)(DB)−1S(t).Hence we have(5.4)VΔ(t)=S⊤(t)(DB)−1SΔ(t)=S⊤(t)(DB)−1D[ΔA(t)x(t)+ΔAd(t)x(t−h(t))−Bρ(t)sgn(s(t))]≤∥S(t)∥∥(DB)−1DM∥×(∥NAx(t)∥+∥NAdx(t−h(t))∥)−ρ(t)∥S(t)∥≤−λ∥S(t)∥<0for∥S(t)∥≠0.This completes the proof.


## 6. An Illustrative Example

Consider
neutral stochastic systems (2.1) with(6.1)A=[2.31.223.4],B=[1001],C=[0.10.300.1],M=[0.200.3−0.01],E=I,Ad=[0.500.20.3],NA=[0.100.10.2],NAd=[−0.100.30.3],NE=[0.40−0.010.4],NEd=[0.2100.1−0.1],h(t)=0.1sin⁡t.We select matrix K=[−11.3000−1.2000−2.0000−12.4000]. Using Matlab LMI control toolbox to solve the
LMIs (4.1)–(4.3), we obtain the following:(6.2)X=[0.1752−0.0038−0.00380.1985],Q1=[1.1255−0.0186−0.01861.2138],Q2=[1.0613−0.0605−0.06051.2145],T1=[1.16700.00010.00011.1623],T2=[1.1661−0.0007−0.00071.1652],T3=[1.16700.00010.00011.1623],T4=[1.16470.00250.00251.1700],T5=[1.16700.00010.00011.1623],T6=[1.1580−0.0046−0.00461.1705].


## 7. Conclusions

In this paper,
we have investigated the sliding mode control problem for uncertain stochastic
neutral systems with multiple delays. The stability criteria are
expressed by means of LMIs, which can be readily tested by some
standard numerical packages. Therefore, the developed result is practical.



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*Source: 761342-2008-04-23.xml*