# Sliding Mode Control Based on Observer for a Class of State-Delayed

Authors: Zhaolan He; Xue Wang; Zongwei Gao; Jingjie Bai
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with a state observer-based sliding mode control design
methodology for a class of continuous-time state-delayed switched systems with
unmeasurable states and nonlinear uncertainties. The advantages of the proposed
scheme mainly lie in which it eliminates the need for state variables to be full
accessible and parameter uncertainties to be satisfied with the matching
condition. Firstly, a state observer is constructed, and a sliding surface is
designed. By matrix transformation techniques, combined with Lyapunov function
and sliding surface function, a sufficient condition is given to ensure
asymptotic stability of the overall closed-loop systems composed of the observer
dynamics and the estimation error dynamics. Then, reachability of sliding
surface is investigated. At last, an illustrative numerical example is presented
to prove feasibility of the proposed approaches.


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## Body

## 1. Introduction

A switched system consists of a finite number of subsystems described by a class of
differential or difference equations and a logical law that is used to orchestrate
switching between these subsystems [1]. As
increasing demand, this theory is widely developed, which means that many working
systems can be modeled as switched systems, such as automated highway systems [2].It is well known that different switching strategies produce different systems
stability and performance, and the systems states cannot be directly measured.
Accordingly, choosing a suitable switching law that stabilizes switched systems and
designing an observer become an important problem [3]. Various methods of observer design have been successfully proposed,
such as algebraic transfer function and singular-value decomposition. However,
designing state observer turns out to be much more difficult when the system is
nonlinear [4] and uncertain [5]. In [6], this problem of state observer is considered for discrete time delay
switched systems with Lipschitz’s nonlinearity and random switching law.Since 1916, sliding mode control (SMC) has been proven to be an effective robust
control strategy for hybrid or uncertain systems [7]. SMC belongs to variable structure control (VSC), which utilizes a
discontinuous control to force the state trajectories of the system to some specific
sliding surfaces on which the system acquires desired properties such as decay
speed, disturbance rejection capability, and robustness. Developments on SMC involve
uncertain systems [8, 9], time-delay systems [10, 11], fuzzy systems [12, 13],
and Markovian jump systems [14–18].Full state feedback is not always available and measurable in many practical systems.
In Wu’s work [19] a new robust
stability condition based on observer is proposed for a class of uncertain nonlinear
neutral delay systems by using the sliding mode control theory combined with
reaching law technique. In He’s paper [20], a sliding mode control strategy of uncertain switched linear
systems based on observer is presented. The average dwell time is introduced and the
matching condition of parameter uncertainties needs not to be satisfied. As the
paper [19, 20] improved, this paper presents a new observer design for a class of
uncertain nonlinear state-delayed switched systems by using the sliding mode control
theory.In the present paper, the motivation of our work is to introduce an SMC scheme based
on observer for a class of switched systems that are uncertain nonlinear
state-delayed systems with unmeasurable state, unknown nonlinear function, and
mismatching parameter uncertainties. This is a new problem in SMC and switched
systems research areas. In this work, the sliding mode observer for each subsystem
is designed to estimate full state. By matrix transformation techniques, a
sufficient condition is proposed to ensure Lyapunov asymptotic stability of the
overall closed-loop switched system. In addition, the derived SMC law is provided to
guarantee reachability of the designed sliding surface. Finally, an example is given
to prove the feasibility of the proposed approaches.


## 2. Problem Formulation and Preliminaries

### 2.1. Switched Uncertain Nonlinear State-Delayed Systems

In this work, consider the following uncertain nonlinear state-delayed switched
systems:(1)x˙(t)=[Aσ(t)+ΔAσ(t)(t)]x(t)+[Adσ(t)+ΔAdσ(t)(t)]x(t-d)+B[uσ(t)(t)+fσ(t)(x(t),t)],y(t)=Cx(t),x(t)=φ(t),t∈[-d,0],
where x(t)∈Rn
denotes the vector of continuous-time state variables; uσ(t)(t)∈Rm
denotes the vector of control inputs; y(t)∈Rp
denotes the system outputs; fσ(t)(x(t),t)∈Rm
denotes an unknown nonlinearity; Aσ(t),
Adσ(t),
B,
and C are
known real constant matrixes of corresponding dimension and the matrix
B
is of full column rank for arbitrary i∈I;
ΔAσ(t)(t) and
ΔAdσ(t)(t) are unknown
time-varying system parameter uncertainties; σ(t):[0,∞)→I  {1,…,N} is switching
signal which is assumed to be a piecewise constant function of time
t;
k
denotes the total number of switching modes. In this paper, the notations
tirin
and tirout
are used to denote the time at which, for the rth,
the ith
subsystem is switched in and out, respectively, that is,
σ(tirin+)=σ(tirout-)=i.
With these notations, i
is used to replace σ(t) for tirin≤t<tirout.
d
is a known constant delay; φ(t) denotes a differentiable
vector-valued initial function on [-d,0].The following assumptions are useful for the development of our work.Assumption 1.
Each of the subsystems is completely observable.Assumption 2.
The uncertain parameters are of the form(2)(ΔAi(t)ΔAdi(t))=EiFi(t)(HiHdi).ΔAi(t) and
ΔAdi(t) are composed by
Ei,
Hi,
Hdi,
and Fi(t), where
Ei,
Hi,
and Hdi
are constant matrices and Fi(t) is a
time-varying matrix function satisfying (3)FiT(t)Fi(t)≤I.Remark 1.
Obviously, by comparison with Assumption2, the matching condition that the following
function must be satisfied is a stronger condition: (4)(ΔAi(t)ΔAdi(t))=B(Mi(t)Mdi(t)).Assumption 3.
There exists a known scalar functionρ(t,y) such
that the nonlinearities fi(x(t),t), for
arbitrary i∈I, satisfy (5)||fi(x(t),t)||≤ρi(t,y).


### 2.2. Sliding Mode Control

According to the sliding mode control theory, a sliding mode function is chosen
as follows:(6)S(x,t)=BTXx(t),
where X∈Rn×n
is a positive definite matrix that will be designed.We know that an ideal sliding mode within the state-estimate space exists if
there exists a finite timets,
such that (7)S(t)=0,S˙(t)=0,t≥tsorSTS˙(t)<0.The second function of (7) will
be used to prove the accessibility condition. And in a conventional SMC design,
a class of equivalent control laws and some switching control terms can be
chosen to undertake that the accessibility condition STS˙<0
is satisfied.Then, from(8)S˙=∂S∂x∂x∂t=BTXx˙(t)=BTX{[Ai+ΔAi(t)]x(t)+[Adi+ΔAdi(t)]x(t-d).+B[ui(t)+fi(x(t),t)]},
we get (9)BTX{[Ai+ΔAi(t)]x(t)+[Adi+ΔAdi(t)]x(t-d).zZ+B[ui(t)+fi(x(t),t)]}=0.The solutionui(t) of (7), namely, sliding mode control
law, can be broken down into two parts: equivalent control law
ueqi(t) and
switching control term uci(t), where the
equivalent control ueqi(t) is designed
for a class of certain systems without nonlinear disturbance and the switching
control uci(t) is robust
control of nonlinear uncertain systems.So,ueqi(t) and
uci(t) are designed
as the following functions (10)
and (11), respectively:
(10)ueqi(t)=-(BTXB)-1BTX(Aix(t)+Adix(t-d)),(11)uci(t)=-[(ρi(t,y)+γ1i)+γ2i]sgn(S(t)),
where, γ1i,
γ2i
are real positive scalars to be specified and function (10) is the solution of the
following equation: (12)BTX[Aix(t)+Adix(t-d)+Bueqi(t)]=0.Remark 2.
It should be noticed that the sliding surface function defined in (6) does not switch with
switching single (so B
not Bi
and X
not Xi
are designed). It means that there is a unique nonswitched sliding surface
in order to avoid repetitive jumps of the state trajectories between sliding
surfaces leading to instability and chattering.


### 2.3. Some Lemmas

For further analysis, some lemmas are given that are useful for stability
analysis of the sliding mode dynamics and the development of other theorems.Lemma 3 (see [21]).
LetE,  F,
and H
be real matrices of appropriate dimensions, with FTF≤I;
then one has that for any scalar ε>0,
(13)EFH+HTFTET≤ε-1EET+εHTH.Lemma 4 (Schur complement).
LetS=[S11S12S21S22] be symmetrical
matrix; then the following three functions are equivalent:
(14)S<0,S11<0,S22-S12TS11-1S12<0,S22<0,S11-S12S22-1S12T<0.Lemma 5 (see [22]).
Given matrixesG∈Rn×n
and U∈Rn×m.
Assume that  U
has full rank m<n
and G=GT.
Then, δUUT-G>0
for some scalar δ
if and only if   U~TGU~<0
where U~
is any matrix whose columns from basis of the null space of
UT.When the SMC is designed, we always suppose that all of the system states are
available. But, this assumption is hardly satisfied on the practical viewpoint.
Hence, in the next chapter, the sliding mode control-based observer will be
designed.


## 2.1. Switched Uncertain Nonlinear State-Delayed Systems

In this work, consider the following uncertain nonlinear state-delayed switched
systems:(1)x˙(t)=[Aσ(t)+ΔAσ(t)(t)]x(t)+[Adσ(t)+ΔAdσ(t)(t)]x(t-d)+B[uσ(t)(t)+fσ(t)(x(t),t)],y(t)=Cx(t),x(t)=φ(t),t∈[-d,0],
where x(t)∈Rn
denotes the vector of continuous-time state variables; uσ(t)(t)∈Rm
denotes the vector of control inputs; y(t)∈Rp
denotes the system outputs; fσ(t)(x(t),t)∈Rm
denotes an unknown nonlinearity; Aσ(t),
Adσ(t),
B,
and C are
known real constant matrixes of corresponding dimension and the matrix
B
is of full column rank for arbitrary i∈I;
ΔAσ(t)(t) and
ΔAdσ(t)(t) are unknown
time-varying system parameter uncertainties; σ(t):[0,∞)→I  {1,…,N} is switching
signal which is assumed to be a piecewise constant function of time
t;
k
denotes the total number of switching modes. In this paper, the notations
tirin
and tirout
are used to denote the time at which, for the rth,
the ith
subsystem is switched in and out, respectively, that is,
σ(tirin+)=σ(tirout-)=i.
With these notations, i
is used to replace σ(t) for tirin≤t<tirout.
d
is a known constant delay; φ(t) denotes a differentiable
vector-valued initial function on [-d,0].The following assumptions are useful for the development of our work.Assumption 1.
Each of the subsystems is completely observable.Assumption 2.
The uncertain parameters are of the form(2)(ΔAi(t)ΔAdi(t))=EiFi(t)(HiHdi).ΔAi(t) and
ΔAdi(t) are composed by
Ei,
Hi,
Hdi,
and Fi(t), where
Ei,
Hi,
and Hdi
are constant matrices and Fi(t) is a
time-varying matrix function satisfying (3)FiT(t)Fi(t)≤I.Remark 1.
Obviously, by comparison with Assumption2, the matching condition that the following
function must be satisfied is a stronger condition: (4)(ΔAi(t)ΔAdi(t))=B(Mi(t)Mdi(t)).Assumption 3.
There exists a known scalar functionρ(t,y) such
that the nonlinearities fi(x(t),t), for
arbitrary i∈I, satisfy (5)||fi(x(t),t)||≤ρi(t,y).


## 2.2. Sliding Mode Control

According to the sliding mode control theory, a sliding mode function is chosen
as follows:(6)S(x,t)=BTXx(t),
where X∈Rn×n
is a positive definite matrix that will be designed.We know that an ideal sliding mode within the state-estimate space exists if
there exists a finite timets,
such that (7)S(t)=0,S˙(t)=0,t≥tsorSTS˙(t)<0.The second function of (7) will
be used to prove the accessibility condition. And in a conventional SMC design,
a class of equivalent control laws and some switching control terms can be
chosen to undertake that the accessibility condition STS˙<0
is satisfied.Then, from(8)S˙=∂S∂x∂x∂t=BTXx˙(t)=BTX{[Ai+ΔAi(t)]x(t)+[Adi+ΔAdi(t)]x(t-d).+B[ui(t)+fi(x(t),t)]},
we get (9)BTX{[Ai+ΔAi(t)]x(t)+[Adi+ΔAdi(t)]x(t-d).zZ+B[ui(t)+fi(x(t),t)]}=0.The solutionui(t) of (7), namely, sliding mode control
law, can be broken down into two parts: equivalent control law
ueqi(t) and
switching control term uci(t), where the
equivalent control ueqi(t) is designed
for a class of certain systems without nonlinear disturbance and the switching
control uci(t) is robust
control of nonlinear uncertain systems.So,ueqi(t) and
uci(t) are designed
as the following functions (10)
and (11), respectively:
(10)ueqi(t)=-(BTXB)-1BTX(Aix(t)+Adix(t-d)),(11)uci(t)=-[(ρi(t,y)+γ1i)+γ2i]sgn(S(t)),
where, γ1i,
γ2i
are real positive scalars to be specified and function (10) is the solution of the
following equation: (12)BTX[Aix(t)+Adix(t-d)+Bueqi(t)]=0.Remark 2.
It should be noticed that the sliding surface function defined in (6) does not switch with
switching single (so B
not Bi
and X
not Xi
are designed). It means that there is a unique nonswitched sliding surface
in order to avoid repetitive jumps of the state trajectories between sliding
surfaces leading to instability and chattering.


## 2.3. Some Lemmas

For further analysis, some lemmas are given that are useful for stability
analysis of the sliding mode dynamics and the development of other theorems.Lemma 3 (see [21]).
LetE,  F,
and H
be real matrices of appropriate dimensions, with FTF≤I;
then one has that for any scalar ε>0,
(13)EFH+HTFTET≤ε-1EET+εHTH.Lemma 4 (Schur complement).
LetS=[S11S12S21S22] be symmetrical
matrix; then the following three functions are equivalent:
(14)S<0,S11<0,S22-S12TS11-1S12<0,S22<0,S11-S12S22-1S12T<0.Lemma 5 (see [22]).
Given matrixesG∈Rn×n
and U∈Rn×m.
Assume that  U
has full rank m<n
and G=GT.
Then, δUUT-G>0
for some scalar δ
if and only if   U~TGU~<0
where U~
is any matrix whose columns from basis of the null space of
UT.When the SMC is designed, we always suppose that all of the system states are
available. But, this assumption is hardly satisfied on the practical viewpoint.
Hence, in the next chapter, the sliding mode control-based observer will be
designed.


## 3. Observer-Based Sliding Mode Control

In this section, a sliding mode observer is designed to provide the estimate of state
vector, and then a sliding mode controller is synthesized based on state estimates.
Furthermore, by applying the sliding mode control and the multiple Lyapunov function
technique, a sufficient condition is given to ensure the asymptotic stability of the
overall closed-loop state-delayed system. Finally, we guarantee that the sliding
modes within both the state estimate space and state estimation errors pace are
attained, respectively.


### 3.1. Sliding Mode Observer Design

The sliding mode observer to be designed, for state-delayed system (1)–(4), has the form of (6), which can be given by
(15)x^˙(t)=Aix^(t)+Adix^(t-d)+Bui(t)+Buei(t)+Li(y(t)-Cx^(t)),
where x^(t) denotes the estimate of
system state x(t) and
Li∈Rn×p
is the observer feedback matrix to be computed later. uei
is a robust control term used to eliminate impact of nonlinear
fi(x(t),t), given by
(16)uei(t)=(ρi(t,y)+γ1i)sgn(Se(t)).The corresponding state-estimation error dynamics is given by(17)e˙(t)=(Ai-LiC+ΔAi(t))e(t)+(Adi-ΔAdi(t))e(t-d)+ΔAi(t)x^(t)+ΔAdi(t)x^(t-d)-Bi(uei(t)-fi(x(t),t)).According to the sliding mode control theory and (6), Se(e(t),t)=BTXe(t).In (6), it is assumed that matrix
X
satisfies (18)BTX=NC.
We obtain (19)Se(e(t),t)=NCe(t)=NC(x(t)-x^(t))=N(y(t)-Cx^(t)).This estimation error dynamics corresponds to a nonlinear uncertain state-delayed
system, which is dependent on the observer feedback matrixLi
and state estimates x^(t),
x^(t-d). This means
that stability analysis of the error dynamics (17) is dependent on the observer dynamics. So, when
designing the sliding mode observer, the overall closed-loop system composed of
(15) and (17) must be considered to
guarantee system stability and accessibility of both the sliding surface
Se(e(t),t)=BTXe(t)=N(y(t)-Cx^(t))=0
in state-estimation error space and the sliding surface
S^(x^(t),t)=BTXx^(t)=0
in the state estimate space. Namely, as (7), the following function should be satisfied:
(20)S^(t)=0,S^˙(t)=0,t≥ts.Hence, as functions (10) and
(11), equivalent control
law and switching control term are designed as follows, respectively:
(21)ueqi(t)=-(BTXB)-1BTX(Aix^(t)+Adix^(t-d)),uci(t)=-[(ρi(t,y)+γ1i)+γ2i]sgn(S^(t)),
where γ1i
and γ2i
are positive design constants, which should be chosen suitably because
approaching rate and chattering of sliding mode face can be influenced by
it.Hence, we obtain the following sliding mode controller:(22)ui(t)=ueqi(t)+uci(t).


### 3.2. Stability of Closed-Loop System with Observer

Summing up, considering the design of sliding mode observer and synthesis of
sliding mode controller, a sufficient condition for asymptotic stability of the
overall closed-loop system can be given as follows.Theorem 6.
If there exist matricesYi,N,X>0,
Q1i>0,
and Q2i>0,
and scalars δ1i>0,
and δ2i>0
satisfying(23)(Π1iXAdi+δ1iHiTHdiCTYiT+δ1iHiTHiδ1iHiTHdiXEAdiTX+δ1iHdiTHiΠ2iδ1iHdiTHiδ1iHdiTHdi0YiC+δ1iHiTHiδ1iHiTHdiΠ3iXAdi+δ1iHiTHdi0δ1iHdiTHiδ1iHdiTHdiAdiTX+δ1iHdiTHiΠ4i0ETX000-δ1iI)≤0, (24)BTX=NC,
where (25)Π1i=XAi+AiTX-YiC-CTYiT+Q1i+δ1iHiTHi-δ2iBBT,Π2i=-Q1i+δ1iHdiTHdi-δ2iBBT,Π3i=AiTX+XAi+Q2i+δ1iHiTHi-δ2iBBT,Π4i=-Q2i+δ1iHdiTHdi-δ2iBBT,
then the sliding mode control law (21)-(22)
guarantees that the combined closed-loop switched system is asymptotically
stable for the switching signal satisfies (26)σ(t)=i=arg{min(Vi(t))},i∈I.
Furthermore, an observer feedback matrix is given byLi=X-1Yi.Proof.
Firstly, using the method of matrix transformation [23] function (15) is transformed, and the transformation matrix and the
associated vector z^(t) are defined as
(27)z^(t)=(z^1(t)z^2(t))=Wx^(t),
where z^1(t)∈Rn-m,
z^2(t)∈Rm,
B~
is an orthogonal complement of B,
which means B~TB=0.
So B
is designed to a nonsingular matrix. It is easily known that
W-1=(X-1B~B) and
z^2(t)=(BTXB)-1s^(t) due to
the following function: (28)((B~TX-1B~)-1B~T(BTXB)-1BTX)(X-1B~B)=((B~TX-1B~)-1B~TX-1B~(B~TX-1B~)-1B~TB(BTXB)-1BTXX-1B~(BTXB)-1BTXB)=(I00I)=I.
Because of the sliding surfaces^(x,t)=0,
we obtain z^2(t)=0.
So (n-m)
reduced-order sliding mode dynamic system as follows:
(29)z^˙1=(B~TX-1B~)-1B~T(Ai-LiC)X-1B~z^1(t)+(B~TX-1B~)-1B~TAdiX-1B~z^1(t-d)+(B~TX-1B~)-1B~TLiCX-1B~z1(t).
Similar to the previous method, function (17) is transformed as (30)e˙z1(t)=(B~TX-1B~)-1B~T(Ai-LiC+ΔAi(t))X-1B~ez1(t)+(B~TX-1B~)-1B~TΔAi(t)X-1B~v^1(t)+(B~TX-1B~)-1B~T(Adi+ΔAdi(t))X-1B~ez1(t-d)+(B~TX-1B~)-1B~TΔAdi(t)X-1B~v^1(t-d).
Then, the Lyapunov functional is chosen as(31)Vi(t)=ez1T(t)(B~TX-1B~)ez1(t)+∫t-dtez1T(τ)(B~TQ-1iB~)ez1(τ)dτ+z^1T(t)(B~TX-1B~)z^1(t)+∫t-dtz^1T(τ)(B~TQ-2iB~)z^1(τ)dτ,V˙i(t)=e˙z1T(t)(B~TX-1B~)ez1(t)+ez1T(t)(B~TX-1B~)e˙z1(t)+(ez1T(t)(B~TQ-1iB~)ez1(t))t-dt+z^˙1T(t)(B~TX-1B~)z^1(t)+z^1T(t)(B~TX-1B~)z^˙1(t)+(z^1T(t)(B~TQ-2iB~)z^1(t))t-dt.
Taking the time derivative along the state trajectories of (15) and (17), and considering (21)-(22), it follows that
(32)V˙i(t)=ZT(t)Gi(t)Z(t),
where (33)Gi(t)=(X-1B~0000X-1B~0000X-1B~0000X-1B~)T(G1i+G2i(t))×(X-1B~0000X-1B~0000X-1B~0000X-1B~),Z(t)=(ez1(t)ez1(t-d)z^1(t)z^1(t-d)),G1i=(Π-1iXAdiCTLiTX0AdiTX-Q1i00XLiC0Π-2iXAdi00AdiTX-Q2i),G2i(t)=(XE000)Fi(t)(HiHdiHiHdi)+((XE000)Fi(t)(HiHdiHiHdi))T,Π-1i=(Ai-LiC)TX+X(Ai-LiC)+Q1i,Π-2i=AiTX+XAi+Q2i,Q1i=XQ-1iX,Q2i=XQ-2iX.
From (32), it is easily seen
that V˙i(t)<0
if Gi(t)<0
for Z≠0.
So, we just prove the inequality Gi(t)<0
is satisfied.
By Lemma3, the previous matrix
inequality holds for Fi(t)
satisfying FiT(t)Fi(t)≤I
if there exists a constant δ1i>0
such that (34)G2i(t)≤G-2i=δ-1(XE000)(ETX000)+δ(HiTHdiTHiTHdiT)(HiHdiHiHdi).
So,Gi<0
only if (35)(X-1B~0000X-1B~0000X-1B~0000X-1B~)T(G1i+G-2i)×(X-1B~0000X-1B~0000X-1B~0000X-1B~)<0.
By Lemma 4, inequality (35) is
equivalent to the following inequality (36): (36)(X-1B~0000X-1B~0000X-1B~0000X-1B~)T×G3i(X-1B~0000X-1B~0000X-1B~0000X-1B~)<0,
where(37)G3i=(Π1iXAdi+δ1iHiTHdiCTYiT+δ1iHiTHiδ1iHiTHdiXEAdiTX+δ1iHdiTHiΠ2iδ1iHdiTHiδ1iHdiTHdi0YiC+δ1iHiTHiδ1iHiTHdiΠ3iXAdi+δ1iHiTHdi0δ1iHdiTHiδ1iHdiTHdiAdiTX+δ1iHdiTHiΠ4i0ETX000-δ1iI).
By Lemma5 and given
Li=X-1Yi,
LMI (23) is obtained. That
means V˙i(t)<0
only if there exist matrices Yi,N,X>0,
Q1i>0,
and Q2i>0
and scalars δ1i>0,
and δ2i>0
satisfying LMI (23).Remark 7.
As a result of the given Theorem6, the observer-based SMC stable problem becomes a linear matrix
inequality feasibility problem. Consider the linear equality condition
BTX=NC,
where N,X>0
satisfies LMI (23), which
can be equivalently converted to (38)tr((BTX-NC)T(BTX-NC))=0.
Introduce the condition(39)(BTX-NC)T(BTX-NC)≤βI,
and Lemma 4 gives
(40)(-βIXB-CTNTBTX-NC-βI)≤0.
Hence, it is now changed to a problem in which a global solution of the
following minimization is found: (41)minβsubjectto(23)and(40).
So, it is a minimization problem relating linear objective and LMI
constraints; it has an infimum and when infimum equals zero, the
observed-based SMC problem is solvable.


### 3.3. Accessibility Condition

Finally, we prove the accessibility of sliding surfacesS^(x^(t),t)=0
in the state-estimate space and Se(e(t),t)=0
in estimation error space. So, Theorem 8 is given.Theorem 8.
If there exist matricesYi,N,X>0,
Q1i>0,
Q2i>0
and scalars δ1i>0,
and δ2i>0
satisfying (23) with (24), and observer feedback
matrix Li=X-1Yi,
then the sliding mode control law (26) with (23)
and (24) guarantees that
the sliding motion is attained on the sliding surfaces
S^(x^(t),t)=0
and Se(e(t),t)=0,
respectively.Proof.
As (7),
S^(x^(t),t)=0
and Se(e(t),t)=0
can be replaced by S^TS^˙(t)<0
and SeTS˙e(t)<0.
We knowS^TS^˙(t)<0 is
the same as S^T(BTXB)-1S^˙(t)<0, so
S^TS^˙(t)<0 and
SeTS˙e(t)<0 can
be replaced by S^T(BTXB)-1S^˙(t)-1<0 and
SeT(BTXB)-1S˙e(t)<0 to
make proving simpler: (42)S^T(t)(BTXB)-1S^˙(t)=S^T(t)(BTXB)-1BTX[Aix^(t)+Adix^(t-d)+Bui(t)S^T(t)(BTXB)-1BTXZZ+Buei(t)+Li(y(t)-Cx^(t))]=S^T(t)(BTXB)-1BTX=.×(Aix^(t)+Adix^(t-d)+LiCe(t)+Bueqi)=.-S^T(t)(uci+uei)≤∥S^(t)∥[∥(BTXB)-1-I∥∥S^(t)∥ZZ×(∥BTXAi∥∥x^(t)∥+∥BTXAdi∥∥x^(t-d)∥)∥S^(t)∥zZ+∥(BTXB)-1∥∥BTXLiC∥∥e(t)∥]≤.-∥S^(t)∥[(ρi(t,y)+γ1i)+γ2i]≤.+∥S^(t)∥(ρi(t,y)+γ1i)≤∥S^(t)∥[(-γ2i+α1iα2i)∥x^(t)∥≤.+α1iα3i∥x^(t-d)∥+α4iα5i∥e(t)∥],SeT(t)(BTXB)-1S˙e(t)=SeT(t)(BTXB)-1BTX[(Ai-LiC+ΔAi(t))e(t)=SeT(t)(BTXB)-1BTZX+(Adi+ΔAdi(t))e(t-d)=SeT(t)(BTXB)-1BTZX-B(uei(t)-fi(x(t),t))=SeT(t)(BTXB)-1BTzX+ΔAix^(t)+ΔAdix^(t-d)]=-SeT(t)(uei(t)-fi(x(t),t))=+SeT(t)(BTXB)-1BTX[(Ai-LiC+ΔAi(t))e(t)=z+SeT(t)(BTXB)-1BTX+(Adi+ΔAdi(t))e(t-d)=+SeT(t)(BTXB)-1BTX+ΔAix^(t)+ΔAdix^(t-d)]≤∥Se(t)∥ρi(t,y)+∥Se(t)∥∥(BTXB)-1∥≤.×[∥BTX(Ai-LiC)∥∥e(t)∥+∥BTXE∥∥Hi∥∥e(t)∥≤.zZ+∥BTXAdi∥∥e(t-d)∥+∥BTXE∥∥Hdi∥∥e(t-d)∥≤zZ.+∥BTXE∥∥Hi∥∥x^(t)∥+∥BTXE∥∥Hdi∥∥x^(t-d)∥]≤∥Se(t)∥[-γ1i+α4i(β1i-β2i)∥e(t)∥≤∥Se(t)∥z+(α3i+β3i)∥e(t-d)∥+β2i∥x^(t)∥≤∥Se(t)∥.+β3i∥x^(t-d)∥]
with α1i=||(BTXB)-1-I||, α2i=||BTXAi||, α3i=||BTXAdi||, α4i=||(BTXB)-1||, α5i=||BTXLiC||, β1i=||BTX(Ai-LiC)||,  β2i=||BTXE||||Hi||,  β3i=||BTXE||||Hdi||.
In the state space composed of state estimate vector and estimation error
vector, we define the following domain:(43)Ωδi={Z∈R4n:α1iα2i∥x^(t)∥+α1iα3i∥x^(t-d)∥Z∈R4nz:+α4iα5i∥e(t)∥<γ2i-δ,Z∈R4nz:α4i(β1i-β2i)∥e(t)∥+β2i∥x^(t)∥Z∈R4nz:+(α3i+β3i)∥e(t-d)∥Z∈R4nz:+β3i∥x^(t-d)∥<γ1i-δ}
with constant δ
satisfying 0<δ<min{γ1i,γ2i}, so
from (42), in the domain
Ωδi,
we have S^TS^˙(t)<0
and SeTS˙e(t)<0.
So, the sliding surfacesS^(x^(t),t)=0
in the state-estimate space and Se(e(t),t)=0
in estimation error space can be accessed.Remark 9.
Only when the state trajectories enter theΩδi,
could S^(t)=0
and Se(t)=0
happen. The region in which sliding motion takes place is usually referred
to as sliding patch [24]. In the
present scheme, it can be seen that size of the sliding patch depends on
design constants γ1i,γ2i.


## 3.1. Sliding Mode Observer Design

The sliding mode observer to be designed, for state-delayed system (1)–(4), has the form of (6), which can be given by
(15)x^˙(t)=Aix^(t)+Adix^(t-d)+Bui(t)+Buei(t)+Li(y(t)-Cx^(t)),
where x^(t) denotes the estimate of
system state x(t) and
Li∈Rn×p
is the observer feedback matrix to be computed later. uei
is a robust control term used to eliminate impact of nonlinear
fi(x(t),t), given by
(16)uei(t)=(ρi(t,y)+γ1i)sgn(Se(t)).The corresponding state-estimation error dynamics is given by(17)e˙(t)=(Ai-LiC+ΔAi(t))e(t)+(Adi-ΔAdi(t))e(t-d)+ΔAi(t)x^(t)+ΔAdi(t)x^(t-d)-Bi(uei(t)-fi(x(t),t)).According to the sliding mode control theory and (6), Se(e(t),t)=BTXe(t).In (6), it is assumed that matrix
X
satisfies (18)BTX=NC.
We obtain (19)Se(e(t),t)=NCe(t)=NC(x(t)-x^(t))=N(y(t)-Cx^(t)).This estimation error dynamics corresponds to a nonlinear uncertain state-delayed
system, which is dependent on the observer feedback matrixLi
and state estimates x^(t),
x^(t-d). This means
that stability analysis of the error dynamics (17) is dependent on the observer dynamics. So, when
designing the sliding mode observer, the overall closed-loop system composed of
(15) and (17) must be considered to
guarantee system stability and accessibility of both the sliding surface
Se(e(t),t)=BTXe(t)=N(y(t)-Cx^(t))=0
in state-estimation error space and the sliding surface
S^(x^(t),t)=BTXx^(t)=0
in the state estimate space. Namely, as (7), the following function should be satisfied:
(20)S^(t)=0,S^˙(t)=0,t≥ts.Hence, as functions (10) and
(11), equivalent control
law and switching control term are designed as follows, respectively:
(21)ueqi(t)=-(BTXB)-1BTX(Aix^(t)+Adix^(t-d)),uci(t)=-[(ρi(t,y)+γ1i)+γ2i]sgn(S^(t)),
where γ1i
and γ2i
are positive design constants, which should be chosen suitably because
approaching rate and chattering of sliding mode face can be influenced by
it.Hence, we obtain the following sliding mode controller:(22)ui(t)=ueqi(t)+uci(t).


## 3.2. Stability of Closed-Loop System with Observer

Summing up, considering the design of sliding mode observer and synthesis of
sliding mode controller, a sufficient condition for asymptotic stability of the
overall closed-loop system can be given as follows.Theorem 6.
If there exist matricesYi,N,X>0,
Q1i>0,
and Q2i>0,
and scalars δ1i>0,
and δ2i>0
satisfying(23)(Π1iXAdi+δ1iHiTHdiCTYiT+δ1iHiTHiδ1iHiTHdiXEAdiTX+δ1iHdiTHiΠ2iδ1iHdiTHiδ1iHdiTHdi0YiC+δ1iHiTHiδ1iHiTHdiΠ3iXAdi+δ1iHiTHdi0δ1iHdiTHiδ1iHdiTHdiAdiTX+δ1iHdiTHiΠ4i0ETX000-δ1iI)≤0, (24)BTX=NC,
where (25)Π1i=XAi+AiTX-YiC-CTYiT+Q1i+δ1iHiTHi-δ2iBBT,Π2i=-Q1i+δ1iHdiTHdi-δ2iBBT,Π3i=AiTX+XAi+Q2i+δ1iHiTHi-δ2iBBT,Π4i=-Q2i+δ1iHdiTHdi-δ2iBBT,
then the sliding mode control law (21)-(22)
guarantees that the combined closed-loop switched system is asymptotically
stable for the switching signal satisfies (26)σ(t)=i=arg{min(Vi(t))},i∈I.
Furthermore, an observer feedback matrix is given byLi=X-1Yi.Proof.
Firstly, using the method of matrix transformation [23] function (15) is transformed, and the transformation matrix and the
associated vector z^(t) are defined as
(27)z^(t)=(z^1(t)z^2(t))=Wx^(t),
where z^1(t)∈Rn-m,
z^2(t)∈Rm,
B~
is an orthogonal complement of B,
which means B~TB=0.
So B
is designed to a nonsingular matrix. It is easily known that
W-1=(X-1B~B) and
z^2(t)=(BTXB)-1s^(t) due to
the following function: (28)((B~TX-1B~)-1B~T(BTXB)-1BTX)(X-1B~B)=((B~TX-1B~)-1B~TX-1B~(B~TX-1B~)-1B~TB(BTXB)-1BTXX-1B~(BTXB)-1BTXB)=(I00I)=I.
Because of the sliding surfaces^(x,t)=0,
we obtain z^2(t)=0.
So (n-m)
reduced-order sliding mode dynamic system as follows:
(29)z^˙1=(B~TX-1B~)-1B~T(Ai-LiC)X-1B~z^1(t)+(B~TX-1B~)-1B~TAdiX-1B~z^1(t-d)+(B~TX-1B~)-1B~TLiCX-1B~z1(t).
Similar to the previous method, function (17) is transformed as (30)e˙z1(t)=(B~TX-1B~)-1B~T(Ai-LiC+ΔAi(t))X-1B~ez1(t)+(B~TX-1B~)-1B~TΔAi(t)X-1B~v^1(t)+(B~TX-1B~)-1B~T(Adi+ΔAdi(t))X-1B~ez1(t-d)+(B~TX-1B~)-1B~TΔAdi(t)X-1B~v^1(t-d).
Then, the Lyapunov functional is chosen as(31)Vi(t)=ez1T(t)(B~TX-1B~)ez1(t)+∫t-dtez1T(τ)(B~TQ-1iB~)ez1(τ)dτ+z^1T(t)(B~TX-1B~)z^1(t)+∫t-dtz^1T(τ)(B~TQ-2iB~)z^1(τ)dτ,V˙i(t)=e˙z1T(t)(B~TX-1B~)ez1(t)+ez1T(t)(B~TX-1B~)e˙z1(t)+(ez1T(t)(B~TQ-1iB~)ez1(t))t-dt+z^˙1T(t)(B~TX-1B~)z^1(t)+z^1T(t)(B~TX-1B~)z^˙1(t)+(z^1T(t)(B~TQ-2iB~)z^1(t))t-dt.
Taking the time derivative along the state trajectories of (15) and (17), and considering (21)-(22), it follows that
(32)V˙i(t)=ZT(t)Gi(t)Z(t),
where (33)Gi(t)=(X-1B~0000X-1B~0000X-1B~0000X-1B~)T(G1i+G2i(t))×(X-1B~0000X-1B~0000X-1B~0000X-1B~),Z(t)=(ez1(t)ez1(t-d)z^1(t)z^1(t-d)),G1i=(Π-1iXAdiCTLiTX0AdiTX-Q1i00XLiC0Π-2iXAdi00AdiTX-Q2i),G2i(t)=(XE000)Fi(t)(HiHdiHiHdi)+((XE000)Fi(t)(HiHdiHiHdi))T,Π-1i=(Ai-LiC)TX+X(Ai-LiC)+Q1i,Π-2i=AiTX+XAi+Q2i,Q1i=XQ-1iX,Q2i=XQ-2iX.
From (32), it is easily seen
that V˙i(t)<0
if Gi(t)<0
for Z≠0.
So, we just prove the inequality Gi(t)<0
is satisfied.
By Lemma3, the previous matrix
inequality holds for Fi(t)
satisfying FiT(t)Fi(t)≤I
if there exists a constant δ1i>0
such that (34)G2i(t)≤G-2i=δ-1(XE000)(ETX000)+δ(HiTHdiTHiTHdiT)(HiHdiHiHdi).
So,Gi<0
only if (35)(X-1B~0000X-1B~0000X-1B~0000X-1B~)T(G1i+G-2i)×(X-1B~0000X-1B~0000X-1B~0000X-1B~)<0.
By Lemma 4, inequality (35) is
equivalent to the following inequality (36): (36)(X-1B~0000X-1B~0000X-1B~0000X-1B~)T×G3i(X-1B~0000X-1B~0000X-1B~0000X-1B~)<0,
where(37)G3i=(Π1iXAdi+δ1iHiTHdiCTYiT+δ1iHiTHiδ1iHiTHdiXEAdiTX+δ1iHdiTHiΠ2iδ1iHdiTHiδ1iHdiTHdi0YiC+δ1iHiTHiδ1iHiTHdiΠ3iXAdi+δ1iHiTHdi0δ1iHdiTHiδ1iHdiTHdiAdiTX+δ1iHdiTHiΠ4i0ETX000-δ1iI).
By Lemma5 and given
Li=X-1Yi,
LMI (23) is obtained. That
means V˙i(t)<0
only if there exist matrices Yi,N,X>0,
Q1i>0,
and Q2i>0
and scalars δ1i>0,
and δ2i>0
satisfying LMI (23).Remark 7.
As a result of the given Theorem6, the observer-based SMC stable problem becomes a linear matrix
inequality feasibility problem. Consider the linear equality condition
BTX=NC,
where N,X>0
satisfies LMI (23), which
can be equivalently converted to (38)tr((BTX-NC)T(BTX-NC))=0.
Introduce the condition(39)(BTX-NC)T(BTX-NC)≤βI,
and Lemma 4 gives
(40)(-βIXB-CTNTBTX-NC-βI)≤0.
Hence, it is now changed to a problem in which a global solution of the
following minimization is found: (41)minβsubjectto(23)and(40).
So, it is a minimization problem relating linear objective and LMI
constraints; it has an infimum and when infimum equals zero, the
observed-based SMC problem is solvable.


## 3.3. Accessibility Condition

Finally, we prove the accessibility of sliding surfacesS^(x^(t),t)=0
in the state-estimate space and Se(e(t),t)=0
in estimation error space. So, Theorem 8 is given.Theorem 8.
If there exist matricesYi,N,X>0,
Q1i>0,
Q2i>0
and scalars δ1i>0,
and δ2i>0
satisfying (23) with (24), and observer feedback
matrix Li=X-1Yi,
then the sliding mode control law (26) with (23)
and (24) guarantees that
the sliding motion is attained on the sliding surfaces
S^(x^(t),t)=0
and Se(e(t),t)=0,
respectively.Proof.
As (7),
S^(x^(t),t)=0
and Se(e(t),t)=0
can be replaced by S^TS^˙(t)<0
and SeTS˙e(t)<0.
We knowS^TS^˙(t)<0 is
the same as S^T(BTXB)-1S^˙(t)<0, so
S^TS^˙(t)<0 and
SeTS˙e(t)<0 can
be replaced by S^T(BTXB)-1S^˙(t)-1<0 and
SeT(BTXB)-1S˙e(t)<0 to
make proving simpler: (42)S^T(t)(BTXB)-1S^˙(t)=S^T(t)(BTXB)-1BTX[Aix^(t)+Adix^(t-d)+Bui(t)S^T(t)(BTXB)-1BTXZZ+Buei(t)+Li(y(t)-Cx^(t))]=S^T(t)(BTXB)-1BTX=.×(Aix^(t)+Adix^(t-d)+LiCe(t)+Bueqi)=.-S^T(t)(uci+uei)≤∥S^(t)∥[∥(BTXB)-1-I∥∥S^(t)∥ZZ×(∥BTXAi∥∥x^(t)∥+∥BTXAdi∥∥x^(t-d)∥)∥S^(t)∥zZ+∥(BTXB)-1∥∥BTXLiC∥∥e(t)∥]≤.-∥S^(t)∥[(ρi(t,y)+γ1i)+γ2i]≤.+∥S^(t)∥(ρi(t,y)+γ1i)≤∥S^(t)∥[(-γ2i+α1iα2i)∥x^(t)∥≤.+α1iα3i∥x^(t-d)∥+α4iα5i∥e(t)∥],SeT(t)(BTXB)-1S˙e(t)=SeT(t)(BTXB)-1BTX[(Ai-LiC+ΔAi(t))e(t)=SeT(t)(BTXB)-1BTZX+(Adi+ΔAdi(t))e(t-d)=SeT(t)(BTXB)-1BTZX-B(uei(t)-fi(x(t),t))=SeT(t)(BTXB)-1BTzX+ΔAix^(t)+ΔAdix^(t-d)]=-SeT(t)(uei(t)-fi(x(t),t))=+SeT(t)(BTXB)-1BTX[(Ai-LiC+ΔAi(t))e(t)=z+SeT(t)(BTXB)-1BTX+(Adi+ΔAdi(t))e(t-d)=+SeT(t)(BTXB)-1BTX+ΔAix^(t)+ΔAdix^(t-d)]≤∥Se(t)∥ρi(t,y)+∥Se(t)∥∥(BTXB)-1∥≤.×[∥BTX(Ai-LiC)∥∥e(t)∥+∥BTXE∥∥Hi∥∥e(t)∥≤.zZ+∥BTXAdi∥∥e(t-d)∥+∥BTXE∥∥Hdi∥∥e(t-d)∥≤zZ.+∥BTXE∥∥Hi∥∥x^(t)∥+∥BTXE∥∥Hdi∥∥x^(t-d)∥]≤∥Se(t)∥[-γ1i+α4i(β1i-β2i)∥e(t)∥≤∥Se(t)∥z+(α3i+β3i)∥e(t-d)∥+β2i∥x^(t)∥≤∥Se(t)∥.+β3i∥x^(t-d)∥]
with α1i=||(BTXB)-1-I||, α2i=||BTXAi||, α3i=||BTXAdi||, α4i=||(BTXB)-1||, α5i=||BTXLiC||, β1i=||BTX(Ai-LiC)||,  β2i=||BTXE||||Hi||,  β3i=||BTXE||||Hdi||.
In the state space composed of state estimate vector and estimation error
vector, we define the following domain:(43)Ωδi={Z∈R4n:α1iα2i∥x^(t)∥+α1iα3i∥x^(t-d)∥Z∈R4nz:+α4iα5i∥e(t)∥<γ2i-δ,Z∈R4nz:α4i(β1i-β2i)∥e(t)∥+β2i∥x^(t)∥Z∈R4nz:+(α3i+β3i)∥e(t-d)∥Z∈R4nz:+β3i∥x^(t-d)∥<γ1i-δ}
with constant δ
satisfying 0<δ<min{γ1i,γ2i}, so
from (42), in the domain
Ωδi,
we have S^TS^˙(t)<0
and SeTS˙e(t)<0.
So, the sliding surfacesS^(x^(t),t)=0
in the state-estimate space and Se(e(t),t)=0
in estimation error space can be accessed.Remark 9.
Only when the state trajectories enter theΩδi,
could S^(t)=0
and Se(t)=0
happen. The region in which sliding motion takes place is usually referred
to as sliding patch [24]. In the
present scheme, it can be seen that size of the sliding patch depends on
design constants γ1i,γ2i.


## 4. Simulation

In this section, the following switch system will be used to illustrate the proposed
sliding mode control scheme-based observer is feasible:(44)x˙(t)=[Aσ(t)+ΔAσ(t)(t)]x(t)+[Adσ(t)+ΔAdσ(t)(t)]x(t-d)+B[uσ(t)(t)+fσ(t)(x(t),t)],y(t)=Cx(t),
where (45)A1=(-240-7-1.5001-1),A2=(-1.20.902-2001.4-1.3),Ad1=(-0.30.10.10.10.1000.10.2),Ad2=(0.10.20.30.120.09000.10.2),B=(211211),C=(132220),E1=E2=(0.10.20.200.20.1000.3),H1=H2=(0.10.20.100.20.30.20.20.1),Hd1=Hd2=(0.200.100.10.10.10.10.2),f1(x(t),t)=(0.4sin(x1(t))0.2cos(x2(t))),f2(x(t),t)=(0.3sin(x2(t))0.3cos(x3(t))),F1(t)=(sin(t)000sin(t)000cos(t)),F2(t)=(cos(t)000sin(t)0001),d(t)=2sin(t)+1,d=3,x(t)=(1-10)T,t∈[-2,0],x(0)=(1-10.5)T.Then, the linear inequalities (23)
and (24) with (41) are solved, and we obtain:
(46)X=(2.8965-0.8585-5.1634-0.85851.79451.2445-5.16341.244510.6332),N=(0.7754-0.50223.9794-3.9817),Y1=(-4.30620.7478-3.32640.1157-1.9617-1.3973),Y2=(-0.60940.41160.5758-0.6553-0.0320-0.1995),L1=(-16.38870.3210-4.40450.2188-7.6272-0.0012),L2=(-1.53030.62090.1156-0.2876-0.75960.3164),S^(t)=BTXx^(t)=(0.00910.00380.0017-0.00900.00600.0145)x^(t),ui(t)=ueqi(t)+uci(t),uei(t)=(ρi(t,y)+γ1i)sgn(Se(t)),uci(t)=-[(ρi(t,y)+γ1i)+γ2i]sgn(S^(t)),ueqi(t)=-(BTXB)-1BTX(Aix^(t)+Adix^(t-d)),ueq1(t)=(2.000-3.0000-1.00001.00002.16671.0000)x^(t)+(-0.30000.00000.10000.1667-0.06670.1333)x^(t-d),ueq2(t)=(1.20000.5000-1.3000-1.4667-0.13331.3000)x^(t)+(-0.1000-0.1000-0.10000.02670.0033-0.0000)x^(t-d).Letγ11=γ12=60,
γ21=γ22=40,
x^(t)=(000)T,
t∈[-2,0],
x^(0)=(000)T.
To restrain control signals from quiver, sgn(S^(t)) is
replaced by S^(t)/(||S^(t)||+0.01), and
Se(t) is replaced by
Se(t)/(||Se(t)||+0.01).
ρi(t,y) is very tiny,
compared with γ1i,
so it is ignored.Figures1–4 testify the simulation results. Figures 1 and 2 show the time
evolution of states and confirm asymptotic stability of the respective switched
system. According to Figure 1, we know that
the opened-loop switched system is instability and oscillation. Nevertheless, Figure
2 shows that the closed-loop switched
system is stability. So it can be seen that state response of the closed-loop
switched system is clearly superior to the opened-loop switched system.Figure 1
State response of the opened-loop switched system.Figure 2
State response of the closed-loop switched system.Figure 3
State response of the observer system.Figure 4
Sliding mode variables.And we check that, from Figure3, the present
observer based on sliding mode control scheme effectively eliminated effects of
parameter uncertainties and nonlinearities and guaranteed asymptotic stability of
the closed-loop system. Figure 4 shows that
the sliding surface is accessibility but continue chattering. The chattering of the
sliding mode variable S^(t) is treated as reduced-order
compensation of dynamic system.


## 5. Conclusion

In this paper, an observer-based sliding mode control strategy is presented for a
class of uncertain nonlinear state-delayed switched systems. Through this work,
state-delayed switched system with immeasurable states and nonlinear uncertainties
get better performance. Not only are the parameter uncertainties not needed to
satisfy matching condition, but also the availabilities of all system states are no
longer required. Meanwhile, a unique nonswitched sliding surface is designed in
order to avoid repetitive jumps of the state trajectories between sliding surfaces
leading to instability and chattering.The weakness is that only the switching signal satisfiesσ(t)=i=arg{min(Vi(t))}, can the switched
system be asymptotically stable. It is expected that the results developed in the
paper can be extended to common cases that the underlying systems are involved in
Markovian switching signal [25–27] or arbitrary switching signal.



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*Source: 614878-2013-04-15.xml*