# Singular Value Decomposition-Based Method for Sliding Mode Control and Optimization of Nonlinear Neutral Systems

Authors: Heli Hu; Dan Zhao; Qingling Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The sliding mode control and optimization are investigated for a class of nonlinear
neutral systems with the unmatched nonlinear term. In the framework of Lyapunov stability theory,
the existence conditions for the designed sliding surface and the stability boundα∗ are derived via
twice transformations. The further results are to develop an efficient sliding mode control law with
tuned parameters to attract the state trajectories onto the sliding surface in finite time and remain
there for all the subsequent time. Finally, some comparisons are made to show the advantages of our
proposed method.


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## Body

## 1. Introduction

Time delays often arise in the processing state, input, or related variables of dynamic systems. Inparticular, when the state derivative also contains time-delay, the considered systems are called as neutral systems [1]. The outstanding characteristic of neutral systems is the fact that such systems contain the same highest order derivatives for the state vector x(t), at both time t and past time(s) ts≤t. Many engineering systems can be represented as neutral equation [2–8], such as population ecology [9], distributed networks containing lossless transmission lines [10], heat exchangers, and robots in contact with rigid environments [11]. The delay-dependent stability criteria for neutral stochastic systems with time delay in state are studied in [4, 5]. The difference is that the former is about exponential stability; the other is on robust stochastic stability, stabilization, and H∞ control. Furthermore, a robust H∞ reduced-order filter and a memory state feedback control are developed to establish the improved stability criteria for neutral systems in [6, 7], respectively. In [12], a periodic output feedback is studied in the context of infinite-dimensional linear systems modeled by neutral functional differential equations, and the main work only focuses on stabilization of neutral systems with delayed control. The stability and H∞ performance analysis, the reliable stabilization, and the finite-time H∞ control for uncertain switched neutral systems are investigated in [13–15], respectively.On the other hand, as an important robust control approach, sliding mode control strategy has many advantages such as fast response, insensitiveness to parametric uncertainties and external disturbances, and unnecessariness for online identification. Hence, the sliding mode control for dynamic systems has received attention extensively [16–24]. Xia et al. propose a novel approach combining SMC and ESO and utilize the backstepping technique to control the attitude of a nonlinear missile system in [16]. In [17], a stable integral type fractional-order sliding surface is introduced to stabilize and synchronize a class of fractional-order chaotic systems. In [18, 19], Xia et al. investigate the robust sliding control for uncertain continuous systems and discrete-time systems with constant delays, respectively. In [20, 21], the PSMC approach and the PDSMO approach are utilized, respectively, to deal with the fault-tolerant control question for uncertain systems. It is worth pointing out that the nonlinear terms are subject to the matched condition in [20]. The same requirement also appears in [25, 26] for neutral systems. In addition, the matched condition is required for the parametric uncertainty in the input in [27].To the best of the authors’ knowledge, the sliding control and optimization for uncertain neutral systems have not yet been investigated, which motivates the present study. One contribution of this paper is the optimization for the upper bound of the unmatched nonlinear term. The other contribution lies in the introduction of the scalarsKX and KQ2 to prevent the unacceptably high gains. In this paper, the state transformations based on singular value decomposition and the descriptor system model are utilized to obtain the existence conditions for the designed sliding surface and the upper bound α*. Inparticular, the achieved bound α* secures the quadratic stability of the sliding motion with all α satisfying α-<α<α*. In addition, an efficient control law is designed to attract the state trajectories onto the sliding surface in finite time and remain there for all the subsequent time. In order to reduce the chattering on the sliding surface, the tuned parameters are introduced into the developed control law. One example under three cases is given to illustrate our proposed method and make some comparisons. Compared with [27], our results have the advantages of the more rapid convergent rate and the less chattering phenomena.The sliding mode control problem formulation is described in Section2. In Section 3, the sliding surface design and the reaching motion control design are developed. An example under three cases and some compared results are provided in Section 4. Finally, conclusion is given in Section 5.


## 2. Problem Formulation

Consider the following class of nonlinear neutral systems:(1)x˙(t)-Aηx˙(t-η(t))=[A+ΔA(t)]x(t)+[Aσ+ΔAσ(t)]x(t-σ(t))+Bu(t)+h(x),x(t)=ϕ(t),t∈[-l0,0],
where x(t)∈ℜn is the state of the system and u(t)∈ℜm is the input vector. A∈ℜn×n, Aσ∈ℜn×n, Aη∈ℜn×n, and B∈ℜn×m are known constant matrices, and h:ℜn→ℜn represents a piecewise-continuous nonlinear function satisfying h(0)=0. It is assumed that the nonlinear term h(x) can be bounded by a quadratic inequality
(2)hT(x)h(x)≤α2xTHTHx,
where H is a constant matrix and α>0 is a scalar parameter. ϕ(t) is the initial condition. σ(t) and η(t) are the time-varying delays. Assume that there exist constants f0, g0, l0, f, and g satisfying
(3)0≤σ(t)≤f0,0≤η(t)≤g0,σ˙(t)≤f<1,η˙(t)≤g<1,max{f0,g0}=l0.
Time-varying parametric uncertainties ΔA(t) and ΔAσ(t) are assumed to be of the following form:
(4)ΔA(t)=EF(t)D,ΔAσ(t)=EσFσ(t)Dσ,
where E, D, Eσ, and Dσ are constant matrices of appropriate dimensions and F(t) and Fσ(t) are the unknown matrix function satisfying FT(t)F(t)≤I and FσT(t)Fσ(t)≤I for all t≥0.Without loss of the generality, assume thatrank(B)=m. One can easily get the singular value decomposition of B(5)B=[U1U2][Σ0(n-m)×m]VT,
where Σ∈ℜm×m is a diagonal positive definite matrix; U1∈ℜn×m, U2∈ℜn×(n-m), and V∈ℜm×m are unitary matrices.The following state transformation is similar to [18, 20, 28]. Choose z=Γx, where Γ=[U1TU2T], another form of the system (1) can be obtained
(6)z˙(t)-A-ηz˙(t-η(t))=[A-+ΔA-(t)]z(t)+[A-σ+ΔA-σ(t)]z(t-σ(t))+[B10]u(t)+h-(Γ-1z),z(t)=ϕ-(t),t∈[-l0,0],
where A-η=ΓAηΓ-1, A-=ΓAΓ-1, ΔA-=ΓΔAΓ-1, A-σ=ΓAσΓ-1, ΔA-σ=ΓΔAσΓ-1, B1=ΣVT, h-(Γ-1z)=Γh(x)=[U1TU2T]h(x)=[h-1(Γ-1z)h-2(Γ-1z)]=[U1Th(Γ-1z)U2Th(Γ-1z)], and ϕ-(t)=Γϕ(t).Furthermore, the system (6) can be rewritten as
(7)z˙1(t)-A-η11z˙1(t-η(t))-A-η12z˙2(t-η(t))=[A-11+ΔA-11(t)]z1(t)+[A-12+ΔA-12(t)]z2(t)+[A-σ11+ΔA-σ11(t)]z1(t-σ(t))+[A-σ12+ΔA-σ12(t)]z2(t-σ(t))+B1u(t)+h-1(Γ-1z),z˙2(t)-A-η21z˙1(t-η(t))-A-η22z˙2(t-η(t))=[A-21+ΔA-21(t)]z1(t)+[A-22+ΔA-22(t)]z2(t)+[A-σ21+ΔA-σ21(t)]z1(t-σ(t))+[A-σ22+ΔA-σ22(t)]z2(t-σ(t))+h-2(Γ-1z),z1(t)=ϕ-1(t),t∈[-l0,0],z2(t)=ϕ-2(t),t∈[-l0,0],
where z1(t)∈ℜm, z2(t)∈ℜn-m, A-11=U1TAU1, A-12=U1TAU2, A-21=U2TAU1,A-22=U2TAU2, h-1(Γ-1z)=U1Th(Γ-1z) and h-2(Γ-1z)=U2Th(Γ-1z) and the others can be obtained easily according to the same way.On account of the second equation of (7) that represents the sliding motion dynamics of (6), one can construct the following sliding surface:
(8)S(t)=[IC]z(t)=z1(t)+Cz2(t)=0,
where C∈ℜm×(n-m) is the gain to be designed.Based on (8) and the second equation of (7), one can get the following sliding motion:
(9)z˙2(t)-[A-η22-A-η21C]z˙2(t-η(t))=[A-22-A-21C+ΔA-22(t)+ΔA-21(t)C]z2(t)+[A-σ22-A-σ21C+ΔA-σ22(t)+ΔA-σ21(t)C]×z2(t-σ(t))+h-2(Γ-1z),z2(t)=ϕ-2(t),t∈[-l0,0].In view of (2), (8), and h-2(Γ-1z) in (9), one can obtain that
(10)h-2T(Γ-1z)h-2(Γ-1z)=hT(Γ-1z)U2U2Th(Γ-1z)≤α2zT(t)Γ-THTHΓ-1z(t)≤α2z2T(t)[-CTI]H-TH-[-CI]z2(t),
where H-=HΓ-1.Our purpose is to design a sliding surfaceS(t) and a reaching motion control law u(t) such that the following holds:(i)
sliding motion (9) is quadratically stable with upper bound α*; that is, for any given α-, find out the upper bound α* so that for all α satisfying α-<α<α*, sliding motion (9) is quadratically stable;(ii)
the system (7) is asymptotically stable under the reaching control law u(t).Lemma 1 (see [4]).
For any constant matrixM∈ℜq×l, inequality
(11)2uTMv≤ruTMGMTu+1rvTG-1v,u∈ℜq,v∈ℜl
holds for any pair of symmetric positive definite matrix G∈ℜl×l and positive number r>0.Lemma 2 (see [29]).
LetZ, X, S, and Y be matrices of appropriate dimensions. Assume that Z is symmetric and STS≤I, then Z+XSY+YTSTXT<0 if and only if there exists a scalar ε>0 satisfying
(12)Z+εXXT+ε-1YTY=Z+ε-1(εX)(εX)T+ε-1YTY<0.


## 3. Main Result

### 3.1. Sliding Surface Design for Nonlinear Neutral Systems

Theorem 3.
The sliding motion (9) is quadratically stable with upper bound α* if the following optimization problem
(13)minγ+KX+KQ2
subject to
(14)Ω=[Ω11Ω12Ω13Ω14Ω15Ω16Ω170Ω19*Ω22Ω23Ω24Ω250Ω2700**Ω330000Ω380***Ω4400000****Ω550000*****Ω66000******Ω7700*******Ω880********Ω99]<0(15)[-Q2A-η22Q2-A-η21X*-Q2]<0(16)γ-1α-2<0(17)[-KXIXT*-I]<0(18)[Q2I*KQ2I]>0
has a solution set (positive scalars γ,ε1,ε2,KQ2, and KX, positive definite matrices Q1,Q2,Q3, and Q4, and matrix X). Moreover, the gain C of the sliding surface (8) and the upper bound α* are obtained as follows:
(19)C=XQ2-1,α*=1γ,
where α- is a given scalar parameter in (2) and Ω11=A-22Q2-A-21X+Q2A-22T-XTA-21T+Q3,Ω12=Q1-Q2+Q2A-22T-XTA-21T,Ω13=A-σ22Q2-A-σ21X,Ω14=A-η22Q2-A-σ21X,Ω15=ε1U2TE,Ω16=Q2U2TDT-XTU1TDT,Ω17=ε2U2TEσ,Ω19=[-XTQ2]H-T,Ω22=-Q2-Q2+Q4+I,Ω23=Ω13,Ω24=Ω14,Ω25=Ω15,Ω27=Ω17,Ω33=-(1-f)Q3,Ω38=Q2U2TDσT-XTU1TDσT,Ω44=-(1-g)Q4,Ω55=-ε1I,Ω66=-ε1I,Ω77=-ε2I,Ω88=-ε2I, and Ω99=-γI.Proof.
First, transform the sliding motion (9) to the equivalent descriptor system form as follows:
(20)[I000][z˙2(t)y˙(t)]=[y(t)ξ(t)],
where
(21)ξ(t)=-y(t)+[A-η22-A-η21C]y(t-η(t))+[A-22-A-21C+ΔA-22(t)+ΔA-21(t)C]z2(t)+[A-σ22-A-σ21C+ΔA-σ22(t)+ΔA-σ21(t)C]×z2(t-σ(t))+h-2(Γ-1z).
ChooseP2=Q2-1, P1=Q2-1Q1Q2-1, P3=Q2-1Q3Q2-1, and P4=Q2-1Q4Q2-1, and construct the following Lyapunov functional
(22)V(z2(t),y(t),t)=[z2T(t)yT(t)][I000][P10P2P2][z2(t)y(t)]+∫t-σ(t)t‍z2T(s)P3z2(s)ds+∫t-η(t)t‍y(s)P4y(s)ds.
Obviously,V(z2(t),y(t),t)>0 for all [z2T(t)yT(t)]≠0. Applying (3), (4), Lemma 1, and the decomposition techniques of matrix to the derivative of V(z2(t),y(t),t) along the trajectories of the sliding motion (20), yield
(23)V˙(z2(t),y(t),t)≤ζTΥζ,
where
(24)ζT=[z2T(t)yT(t)z2T(t-σ(t))yT(t-η(t))],Υ=[Υ11Υ12Υ13Υ14*Υ22Υ23Υ24**Υ33Υ34***Υ44]+[P2TU2TEP2TU2TE00]×F(t)[DU2-DU1C000]+[P2TU2TEσP2TU2TEσ00]Fσ(t)[00DσU2-DσU1C0]+[U2TDT-CTU1TDT000]FT(t)[ETU2P2ETU2P200]+[00U2TDσT-CTU1TDσT0]FσT(t)[EσTU2P2EσTU2P200],Υ11=P2(A-22-A-21C)+(A-22-A-21C)TP2+P3+(1/γ)[-CTI]H-TH-[-CI],Υ12=P1-P2+(A-22-A-21C)TP2,Υ13=P2(A-σ22-A-σ21C),Υ14=P2(A-η22-A-η21C),Υ22=-P2-P2+P4+P2P2,Υ23=Υ13,Υ24=Υ14,Υ33=-(1-f)P3,Υ44=-(1-g)P4,γ=1/α2.
Pre- and postmultiplyingΥ in (23) by YT and Y, where YT=diag{P2-1,P2-1,P2-1,P2-1}, we obtain the following matrix:
(25)Υ-=[Υ-11Υ-12Υ-13Υ-14*Υ-22Υ-23Υ-24**Υ33Υ34***Υ-44]+[U2TEU2TE00]F(t)[DU2Q2-DU1X000]+[U2TEσU2TEσ00]Fσ(t)[00DσU2Q2-DσU1X0]+[Q2U2TDT-XTU1TDT000]FT(t)[ETU2ETU200]+[00Q2U2TDσT-XTU1TDσT0]FσT(t)[EσTU2EσTU200],
where Υ-11=A-22Q2-A-21X+Q2A-22-XTA-21T+Q3+(1/γ)[-XTQ2]H-TH-[-XQ2],Υ-12=Q1-Q2+Q2A-22T-XTA-21T,Υ-13=A-σ22Q2-A-σ21X,Υ-14=A-η22Q2-A-η21X,Υ22=-Q2-Q2+Q4+I,Υ-23=Υ-13,Υ-24=Υ-14,Υ-33=-(1-f)Q3,Υ-44=-(1-g)Q4, and γ=1/α2.
By Schur complement lemma and Lemma2, the condition Ω<0 in (14) is equivalent to Υ-<0 in (25). Obviously, Υ-<0 means Υ<0. From the condition Υ<0 in (23), there exists a constant ρ>0, such that
(26)V˙(z2(t),y(t),t)≤-ρ∥z2(t)∥2.
According to Schur complement lemma, LMI (15) is equivalent to the following inequality:
(27)-Q2+(A-η22Q2-A-η21X)Q2-1(A-η22Q2-A-η21X)T<0
that is,
(28)-Q2+(A-η22-A-η21C)Q2(A-η22T-CTA-η21T)<0.
Thus, one can obtain that(29)ρ(A-η22-A-η21C)=∥A-η22-A-η21C∥<1.
This implies that the sliding motion (9) is Lipschitzian in the term A-η22-A-η21C with Lipschitz constant less than 1 [29].
By conditions (22) and (26) and ∥A-η22-A-η21C∥<1, one can conclude that the sliding motion (9) with (3) and (4) is quadratically stable.
Furthermore, for some desired valueα- satisfying (2), the design of gain C can be formulated as an LMI problem in γ, ε1, ε2, KQ2, KX, Q1, Q2, Q3, Q4, and X in (13). Also, one can obtain the upper bound α* which guarantees the quadratic stability for the sliding motion (9) with all α satisfying α-<α≤α* in (2) if the LMI optimization is feasible. This completes the proof.Remark 4.
It should be pointed out that the norm of the gain matrix is implicitly bounded by (17) and (18), which imply that ∥C∥≤KXKQ2. This is necessary in order to prevent the unacceptably high gains.Remark 5.
It is obvious that (14), (15), (16), (17), and (18) are a group of LMIs with respect to solution variables; various efficient convex algorithms can be used to ascertain the LMI solutions. In this paper, we utilize Matlab’s LMI Toolbox [30] to solve the convex optimization problem to obtain directly the gain C as C=XQ2-1.Remark 6.
If the convex optimization problem is solvable, the boundα* guarantees the quadratic stability for the sliding motion (9) with all α satisfying α-<α<α* in (2).


### 3.2. Reaching Motion Control Design for Nonlinear Neutral Systems

Theorem 7.
Suppose that the optimization problem (13) has solutions: positive scalars γ, ε1, ε2, KQ2, and KX, positive definite matrices Q1,Q2,Q3, and Q4, and matrix X, and the sliding surface is given by (8). Then, the trajectory of the closed-loop system (7) can be driven onto the sliding surface in finite time with the control
(30)u(t)=-B1-1[KS+θsign(S)+C-A-z(t)+C-A-σz(t-σ(t))+C-A-ηz˙(t-η(t))+diag(sign(S))×(N1+N2+N3)C-A-σz(t-σ(t))(t-σ(t))],
where
(31)diag(sign(S))=diag{sign(s1),sign(s2),…,sign(sm)},C-=[IC]=[c-1c-2⋮c-m],N1=|C-ΓEDΓ-1z(t)|=[N11N21⋮Nm1]=[|c-1ΓEDΓ-1z(t)||c-2ΓEDΓ-1z(t)|⋮|c-mΓEDΓ-1z(t)|],N2=|C-ΓEσDσΓ-1z(t-σ(t))|=[N12N22⋮Nm2]=[|c-1ΓEσDσΓ-1z(t-σ(t))||c-2ΓEσDσΓ-1z(t-σ(t))|⋮|c-mΓEσDσΓ-1z(t-σ(t))|],N3=1γ|C-ΓHΓ-1z(t)|=[N13N23⋮Nm3]=[1γ|c-1ΓHΓ-1z(t)|1γ|c-2ΓHΓ-1z(t)|⋮1γ|c-mΓHΓ-1z(t)|],K=diag{k1,k2,…,km},θ=diag{θ1,θ2,…,θm}, Ki, and θi are positive constants.Proof.
From the sliding surface (8), one can compute the derivative of S as follows:
(32)S˙=[IC]z˙(t)=C-A-ηz˙(t-η(t))+C-[A-+ΔA-(t)]z(t)+C-[A-σ+ΔA-σ(t)]z(t-σ(t))+B1u(t)+C-h-(Γ-1z).
Replacing u(t) in (32) with u(t) in (30) yields that
(33)S˙=-KS-θsign(S)-[diag(sign(S))N1-C-ΓEF(t)DΓ-1z(t)]-[diag(sign(S))N2-C-ΓEσFσ(t)DσΓ-1z(t-σ(t))]-[diag(sign(S))N3-C-h-(Γ-1z)].
That is, for each one si in S, we have
(34)s˙i=-kisi-θisign(si)-[sign(si)Ni1-c-iΓEF(t)DΓ-1z(t)]-[sign(si)Ni2-c-iΓEσFσ(t)DσΓ-1z(t-σ(t))]-[sign(si)Ni3-c-ih-(Γ-1z)].
In view of(35)c-iΓEF(t)DΓ-1z(t)≤|c-iΓEDΓ-1z(t)|=Ni1,c-iΓEσFσ(t)DσΓ-1z(t)≤|c-iΓEσDσΓ-1z(t)|=Ni2,c-ih-(Γ-1z)≤|c-iΓHΓ-1z(t)|=Ni3,
thus one can conclude that
(36)s˙i<0,si>0,s˙i>0,si<0
which means that the trajectory of the system (7) with the control law (30) can be driven onto the sliding surface in finite time and remain there in the subsequent time. This completes the proof.


## 3.1. Sliding Surface Design for Nonlinear Neutral Systems

Theorem 3.
The sliding motion (9) is quadratically stable with upper bound α* if the following optimization problem
(13)minγ+KX+KQ2
subject to
(14)Ω=[Ω11Ω12Ω13Ω14Ω15Ω16Ω170Ω19*Ω22Ω23Ω24Ω250Ω2700**Ω330000Ω380***Ω4400000****Ω550000*****Ω66000******Ω7700*******Ω880********Ω99]<0(15)[-Q2A-η22Q2-A-η21X*-Q2]<0(16)γ-1α-2<0(17)[-KXIXT*-I]<0(18)[Q2I*KQ2I]>0
has a solution set (positive scalars γ,ε1,ε2,KQ2, and KX, positive definite matrices Q1,Q2,Q3, and Q4, and matrix X). Moreover, the gain C of the sliding surface (8) and the upper bound α* are obtained as follows:
(19)C=XQ2-1,α*=1γ,
where α- is a given scalar parameter in (2) and Ω11=A-22Q2-A-21X+Q2A-22T-XTA-21T+Q3,Ω12=Q1-Q2+Q2A-22T-XTA-21T,Ω13=A-σ22Q2-A-σ21X,Ω14=A-η22Q2-A-σ21X,Ω15=ε1U2TE,Ω16=Q2U2TDT-XTU1TDT,Ω17=ε2U2TEσ,Ω19=[-XTQ2]H-T,Ω22=-Q2-Q2+Q4+I,Ω23=Ω13,Ω24=Ω14,Ω25=Ω15,Ω27=Ω17,Ω33=-(1-f)Q3,Ω38=Q2U2TDσT-XTU1TDσT,Ω44=-(1-g)Q4,Ω55=-ε1I,Ω66=-ε1I,Ω77=-ε2I,Ω88=-ε2I, and Ω99=-γI.Proof.
First, transform the sliding motion (9) to the equivalent descriptor system form as follows:
(20)[I000][z˙2(t)y˙(t)]=[y(t)ξ(t)],
where
(21)ξ(t)=-y(t)+[A-η22-A-η21C]y(t-η(t))+[A-22-A-21C+ΔA-22(t)+ΔA-21(t)C]z2(t)+[A-σ22-A-σ21C+ΔA-σ22(t)+ΔA-σ21(t)C]×z2(t-σ(t))+h-2(Γ-1z).
ChooseP2=Q2-1, P1=Q2-1Q1Q2-1, P3=Q2-1Q3Q2-1, and P4=Q2-1Q4Q2-1, and construct the following Lyapunov functional
(22)V(z2(t),y(t),t)=[z2T(t)yT(t)][I000][P10P2P2][z2(t)y(t)]+∫t-σ(t)t‍z2T(s)P3z2(s)ds+∫t-η(t)t‍y(s)P4y(s)ds.
Obviously,V(z2(t),y(t),t)>0 for all [z2T(t)yT(t)]≠0. Applying (3), (4), Lemma 1, and the decomposition techniques of matrix to the derivative of V(z2(t),y(t),t) along the trajectories of the sliding motion (20), yield
(23)V˙(z2(t),y(t),t)≤ζTΥζ,
where
(24)ζT=[z2T(t)yT(t)z2T(t-σ(t))yT(t-η(t))],Υ=[Υ11Υ12Υ13Υ14*Υ22Υ23Υ24**Υ33Υ34***Υ44]+[P2TU2TEP2TU2TE00]×F(t)[DU2-DU1C000]+[P2TU2TEσP2TU2TEσ00]Fσ(t)[00DσU2-DσU1C0]+[U2TDT-CTU1TDT000]FT(t)[ETU2P2ETU2P200]+[00U2TDσT-CTU1TDσT0]FσT(t)[EσTU2P2EσTU2P200],Υ11=P2(A-22-A-21C)+(A-22-A-21C)TP2+P3+(1/γ)[-CTI]H-TH-[-CI],Υ12=P1-P2+(A-22-A-21C)TP2,Υ13=P2(A-σ22-A-σ21C),Υ14=P2(A-η22-A-η21C),Υ22=-P2-P2+P4+P2P2,Υ23=Υ13,Υ24=Υ14,Υ33=-(1-f)P3,Υ44=-(1-g)P4,γ=1/α2.
Pre- and postmultiplyingΥ in (23) by YT and Y, where YT=diag{P2-1,P2-1,P2-1,P2-1}, we obtain the following matrix:
(25)Υ-=[Υ-11Υ-12Υ-13Υ-14*Υ-22Υ-23Υ-24**Υ33Υ34***Υ-44]+[U2TEU2TE00]F(t)[DU2Q2-DU1X000]+[U2TEσU2TEσ00]Fσ(t)[00DσU2Q2-DσU1X0]+[Q2U2TDT-XTU1TDT000]FT(t)[ETU2ETU200]+[00Q2U2TDσT-XTU1TDσT0]FσT(t)[EσTU2EσTU200],
where Υ-11=A-22Q2-A-21X+Q2A-22-XTA-21T+Q3+(1/γ)[-XTQ2]H-TH-[-XQ2],Υ-12=Q1-Q2+Q2A-22T-XTA-21T,Υ-13=A-σ22Q2-A-σ21X,Υ-14=A-η22Q2-A-η21X,Υ22=-Q2-Q2+Q4+I,Υ-23=Υ-13,Υ-24=Υ-14,Υ-33=-(1-f)Q3,Υ-44=-(1-g)Q4, and γ=1/α2.
By Schur complement lemma and Lemma2, the condition Ω<0 in (14) is equivalent to Υ-<0 in (25). Obviously, Υ-<0 means Υ<0. From the condition Υ<0 in (23), there exists a constant ρ>0, such that
(26)V˙(z2(t),y(t),t)≤-ρ∥z2(t)∥2.
According to Schur complement lemma, LMI (15) is equivalent to the following inequality:
(27)-Q2+(A-η22Q2-A-η21X)Q2-1(A-η22Q2-A-η21X)T<0
that is,
(28)-Q2+(A-η22-A-η21C)Q2(A-η22T-CTA-η21T)<0.
Thus, one can obtain that(29)ρ(A-η22-A-η21C)=∥A-η22-A-η21C∥<1.
This implies that the sliding motion (9) is Lipschitzian in the term A-η22-A-η21C with Lipschitz constant less than 1 [29].
By conditions (22) and (26) and ∥A-η22-A-η21C∥<1, one can conclude that the sliding motion (9) with (3) and (4) is quadratically stable.
Furthermore, for some desired valueα- satisfying (2), the design of gain C can be formulated as an LMI problem in γ, ε1, ε2, KQ2, KX, Q1, Q2, Q3, Q4, and X in (13). Also, one can obtain the upper bound α* which guarantees the quadratic stability for the sliding motion (9) with all α satisfying α-<α≤α* in (2) if the LMI optimization is feasible. This completes the proof.Remark 4.
It should be pointed out that the norm of the gain matrix is implicitly bounded by (17) and (18), which imply that ∥C∥≤KXKQ2. This is necessary in order to prevent the unacceptably high gains.Remark 5.
It is obvious that (14), (15), (16), (17), and (18) are a group of LMIs with respect to solution variables; various efficient convex algorithms can be used to ascertain the LMI solutions. In this paper, we utilize Matlab’s LMI Toolbox [30] to solve the convex optimization problem to obtain directly the gain C as C=XQ2-1.Remark 6.
If the convex optimization problem is solvable, the boundα* guarantees the quadratic stability for the sliding motion (9) with all α satisfying α-<α<α* in (2).


## 3.2. Reaching Motion Control Design for Nonlinear Neutral Systems

Theorem 7.
Suppose that the optimization problem (13) has solutions: positive scalars γ, ε1, ε2, KQ2, and KX, positive definite matrices Q1,Q2,Q3, and Q4, and matrix X, and the sliding surface is given by (8). Then, the trajectory of the closed-loop system (7) can be driven onto the sliding surface in finite time with the control
(30)u(t)=-B1-1[KS+θsign(S)+C-A-z(t)+C-A-σz(t-σ(t))+C-A-ηz˙(t-η(t))+diag(sign(S))×(N1+N2+N3)C-A-σz(t-σ(t))(t-σ(t))],
where
(31)diag(sign(S))=diag{sign(s1),sign(s2),…,sign(sm)},C-=[IC]=[c-1c-2⋮c-m],N1=|C-ΓEDΓ-1z(t)|=[N11N21⋮Nm1]=[|c-1ΓEDΓ-1z(t)||c-2ΓEDΓ-1z(t)|⋮|c-mΓEDΓ-1z(t)|],N2=|C-ΓEσDσΓ-1z(t-σ(t))|=[N12N22⋮Nm2]=[|c-1ΓEσDσΓ-1z(t-σ(t))||c-2ΓEσDσΓ-1z(t-σ(t))|⋮|c-mΓEσDσΓ-1z(t-σ(t))|],N3=1γ|C-ΓHΓ-1z(t)|=[N13N23⋮Nm3]=[1γ|c-1ΓHΓ-1z(t)|1γ|c-2ΓHΓ-1z(t)|⋮1γ|c-mΓHΓ-1z(t)|],K=diag{k1,k2,…,km},θ=diag{θ1,θ2,…,θm}, Ki, and θi are positive constants.Proof.
From the sliding surface (8), one can compute the derivative of S as follows:
(32)S˙=[IC]z˙(t)=C-A-ηz˙(t-η(t))+C-[A-+ΔA-(t)]z(t)+C-[A-σ+ΔA-σ(t)]z(t-σ(t))+B1u(t)+C-h-(Γ-1z).
Replacing u(t) in (32) with u(t) in (30) yields that
(33)S˙=-KS-θsign(S)-[diag(sign(S))N1-C-ΓEF(t)DΓ-1z(t)]-[diag(sign(S))N2-C-ΓEσFσ(t)DσΓ-1z(t-σ(t))]-[diag(sign(S))N3-C-h-(Γ-1z)].
That is, for each one si in S, we have
(34)s˙i=-kisi-θisign(si)-[sign(si)Ni1-c-iΓEF(t)DΓ-1z(t)]-[sign(si)Ni2-c-iΓEσFσ(t)DσΓ-1z(t-σ(t))]-[sign(si)Ni3-c-ih-(Γ-1z)].
In view of(35)c-iΓEF(t)DΓ-1z(t)≤|c-iΓEDΓ-1z(t)|=Ni1,c-iΓEσFσ(t)DσΓ-1z(t)≤|c-iΓEσDσΓ-1z(t)|=Ni2,c-ih-(Γ-1z)≤|c-iΓHΓ-1z(t)|=Ni3,
thus one can conclude that
(36)s˙i<0,si>0,s˙i>0,si<0
which means that the trajectory of the system (7) with the control law (30) can be driven onto the sliding surface in finite time and remain there in the subsequent time. This completes the proof.


## 4. Numerical Example

In this section, three examples are presented to illustrate the design approach of sliding surface and reaching motion control and show their advantages.Consider parts of parameters in [27]
(37)A=[10.30-30.100.10-2],Aσ=[0.20.10.10.1000.1-0.10],B=[5-7-9835],Aη=[-0.100.100.50000.02],E=[0.10-0.2],Eσ=[-0.200.1],D=[-0.020.020.10],Dσ=[0.10.010.02].


### 4.1. Case 1

For the system (1) in our paper, when
(38)H=[0.3-0.3-0.10.10.5-0.10.20.1-0.5],σ(t)=0.1(2+sin(t)),η(t)=0.2(1+cos(t))
and the desired parameter α-=0.1, by solving the optimization problem (13) subject to (14), (15), (16), (17), and (18), we can get following results:
(39)γ=0.2779,ε1=9.9589,ε2=1.8862,KQ2=0.3980,KX=0.0574,Q1=3.5006,Q2=2.5130,Q3=0.6958,Q4=3.8877,X=[0.1515-0.1847].
Thus, the gain C and the upper bound α* are computed as
(40)C=XQ2-1=[0.0603-0.0735],α*=1γ=1.8969.Under the following initial condition(41)ϕ(0)=[-0.50.1-0.2],
when the tuned parameters K and θ in (30) are chosen as
(42)K=[3.8974005.7103],θ=[0.001000.002],
the simulation results are shown in Figures 1–6 based on the above parameters. From Figures 1 and 2, one can see that the nonlinear neutral systems (1) and (6) without control are divergent. From Figures 3 and 4, one can see that the nonlinear neutral system (6) with control (30) is indeed well stabilized. The control signal u(t) of the system (6) and the sliding surface (8) are rather smooth in Figures 5 and 6.Figure 1
State response of the open-loop system (1) in Case 1.Figure 2
State response of the open-loop system (6) in Case 1.Figure 3
State response of the closed-loop system (1) in Case 1.Figure 4
State response of the closed-loop system (6) in Case 1.Figure 5
Control signal in (30) in Case 1.Figure 6
Trajectories of sliding variable in sliding surface (8) in Case 1.


### 4.2. Case 2

In order to compare with [27], we need guarantee no difference for the considered system firstly. Because in [27] the system (1) contains the term B+ΔB(t), where ΔB(t)=Bδ(t), and does not have the term h(t), and also the delays are constant, for the system (1) in our paper, matrix L is needed between B and u(t), where L=I+δ(t). When L is contained in our system (1), the theoretical results should make some modification; that is, the reaching motion control u(t) should have the following form:
(43)u(t)=-(B1L)*[KS+θsign(S)+C-A-z(t)+C-A-σz×(t-σ(t))+C-A-ηz˙(t-η(t))+diag(sign(S))(N1+N2+N3)],
where (•)* denotes the pseudoinverse of the argument •.In addition, to remove the nonlinear termh(t), we chose h(t)=[000]T, which means that the parameter α in (2) does not need to be optimized.As for the constant delays, their derivatives,f and g, are equal to zero.Utilizing the parameters in [27] to solve LMIs (14) and (15), one can get following results:
(44)ε1=53.0488,ε2=51.8644,Q1=77.2200,Q2=46.7611,Q3=46.0459,Q4=48.8738,X=[13.6121-12.1425].
Thus, the gain C is computed as
(45)C=XQ2-1=[0.2911-0.2597].Under the initialϕ(0)=[10-1] in [27], when the tuned parameters K and θ in (30) are chosen as
(46)K=[6002],θ=[0.0001000.0001],
the simulation results are presented in Figures 7, 8, 9, and 10. Seen from Figures 7–10 in our paper, the convergent time is about 10 seconds. However, the convergent time is about 16 seconds from Figures  2–4 in [27]. Compared with Figures  2–4 in [27], one can conclude that the convergent rate of our results is faster.Figure 7
State response of the closed-loop system (1) in Case 2.Figure 8
State response of the closed-loop system (6) in Case 2.Figure 9
Control signal in (30) in Case 2.Figure 10
Trajectories of sliding variable in sliding surface (8) in Case 2.


### 4.3. Case 3

Under the initialϕ(0)=[0-11] in [27], when the tuned parameters K and θ in (30) are chosen as
(47)K=[1.5005],θ=[0.0001000.0001]
and other parameters are similar to [27], the simulation results are presented in Figures 11, 12, 13, and 14.Figure 11
State response of the closed-loop system (1) in Case 3.Figure 12
State response of the closed-loop system (6) in Case 3.Figure 13
Control signal in (30) in Case 3.Figure 14
Trajectories of sliding variable in sliding surface (8) in Case 3.Seen from Figures11–14 in our paper, the convergent time is about 20 seconds. However, the convergent time is about 30 seconds from Figures  5–7 in [27]. Also, the amplitudes of the corresponding control signal u(t) are 0.45 and 3.5, respectively. Therefore, one can conclude that the convergent rate of our results is faster and the amplitude of control signal u(t) of our results is smaller.


## 4.1. Case 1

For the system (1) in our paper, when
(38)H=[0.3-0.3-0.10.10.5-0.10.20.1-0.5],σ(t)=0.1(2+sin(t)),η(t)=0.2(1+cos(t))
and the desired parameter α-=0.1, by solving the optimization problem (13) subject to (14), (15), (16), (17), and (18), we can get following results:
(39)γ=0.2779,ε1=9.9589,ε2=1.8862,KQ2=0.3980,KX=0.0574,Q1=3.5006,Q2=2.5130,Q3=0.6958,Q4=3.8877,X=[0.1515-0.1847].
Thus, the gain C and the upper bound α* are computed as
(40)C=XQ2-1=[0.0603-0.0735],α*=1γ=1.8969.Under the following initial condition(41)ϕ(0)=[-0.50.1-0.2],
when the tuned parameters K and θ in (30) are chosen as
(42)K=[3.8974005.7103],θ=[0.001000.002],
the simulation results are shown in Figures 1–6 based on the above parameters. From Figures 1 and 2, one can see that the nonlinear neutral systems (1) and (6) without control are divergent. From Figures 3 and 4, one can see that the nonlinear neutral system (6) with control (30) is indeed well stabilized. The control signal u(t) of the system (6) and the sliding surface (8) are rather smooth in Figures 5 and 6.Figure 1
State response of the open-loop system (1) in Case 1.Figure 2
State response of the open-loop system (6) in Case 1.Figure 3
State response of the closed-loop system (1) in Case 1.Figure 4
State response of the closed-loop system (6) in Case 1.Figure 5
Control signal in (30) in Case 1.Figure 6
Trajectories of sliding variable in sliding surface (8) in Case 1.


## 4.2. Case 2

In order to compare with [27], we need guarantee no difference for the considered system firstly. Because in [27] the system (1) contains the term B+ΔB(t), where ΔB(t)=Bδ(t), and does not have the term h(t), and also the delays are constant, for the system (1) in our paper, matrix L is needed between B and u(t), where L=I+δ(t). When L is contained in our system (1), the theoretical results should make some modification; that is, the reaching motion control u(t) should have the following form:
(43)u(t)=-(B1L)*[KS+θsign(S)+C-A-z(t)+C-A-σz×(t-σ(t))+C-A-ηz˙(t-η(t))+diag(sign(S))(N1+N2+N3)],
where (•)* denotes the pseudoinverse of the argument •.In addition, to remove the nonlinear termh(t), we chose h(t)=[000]T, which means that the parameter α in (2) does not need to be optimized.As for the constant delays, their derivatives,f and g, are equal to zero.Utilizing the parameters in [27] to solve LMIs (14) and (15), one can get following results:
(44)ε1=53.0488,ε2=51.8644,Q1=77.2200,Q2=46.7611,Q3=46.0459,Q4=48.8738,X=[13.6121-12.1425].
Thus, the gain C is computed as
(45)C=XQ2-1=[0.2911-0.2597].Under the initialϕ(0)=[10-1] in [27], when the tuned parameters K and θ in (30) are chosen as
(46)K=[6002],θ=[0.0001000.0001],
the simulation results are presented in Figures 7, 8, 9, and 10. Seen from Figures 7–10 in our paper, the convergent time is about 10 seconds. However, the convergent time is about 16 seconds from Figures  2–4 in [27]. Compared with Figures  2–4 in [27], one can conclude that the convergent rate of our results is faster.Figure 7
State response of the closed-loop system (1) in Case 2.Figure 8
State response of the closed-loop system (6) in Case 2.Figure 9
Control signal in (30) in Case 2.Figure 10
Trajectories of sliding variable in sliding surface (8) in Case 2.


## 4.3. Case 3

Under the initialϕ(0)=[0-11] in [27], when the tuned parameters K and θ in (30) are chosen as
(47)K=[1.5005],θ=[0.0001000.0001]
and other parameters are similar to [27], the simulation results are presented in Figures 11, 12, 13, and 14.Figure 11
State response of the closed-loop system (1) in Case 3.Figure 12
State response of the closed-loop system (6) in Case 3.Figure 13
Control signal in (30) in Case 3.Figure 14
Trajectories of sliding variable in sliding surface (8) in Case 3.Seen from Figures11–14 in our paper, the convergent time is about 20 seconds. However, the convergent time is about 30 seconds from Figures  5–7 in [27]. Also, the amplitudes of the corresponding control signal u(t) are 0.45 and 3.5, respectively. Therefore, one can conclude that the convergent rate of our results is faster and the amplitude of control signal u(t) of our results is smaller.


## 5. Conclusion

The sliding mode control and optimization problem of nonlinear neutral systems with time-varying delays are complex and challenging. In the framework of the Lyapunov stability theory, based on the methods of singular value decomposition and descriptor system model transformation, the sliding surface and the reaching control law are designed, which can be obtained by solving the optimization problem. Also, the upper boundα* that guarantees the quadratic stability of sliding motion (9) for all α satisfying α-<α<α* is derived. The numerical example has shown the validity of the present design and the advantages of the schemes over the existing results in the literature.In future research, it is expected to investigate the optimization of reaching motion control for a class of nonlinear neutral systems with time-varying delays.



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*Source: 357531-2013-04-09.xml*