# Asymptotic Stability Analysis and Optimality Algorithm for Uncertain Neutral Systems with Saturation

Authors: Xinghua Liu
Journal: ISRN Applied Mathematics
 (2014)

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## Abstract

The certain and uncertain neutral systems with time-delay and saturating actuator are considered in this paper. In order to analyse and optimize the system, auxiliary functions are presented based on additive decomposition approach and the relationship among them is discussed. As the novel stability criterion, two sufficient conditions are obtained for asymptotic stability of the neutral systems. Furthermore, the paper gives the stability analysis algorithm and optimality algorithm to optimize the result. Finally, from the two-stage dissolution tank of solid caustic soda in a chemical plant, three numerical examples are implemented to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

Delay is often inevitable in various practical systems; examples include population ecology [1], steam or water pipes, heat exchanges [2], and many others [3–5]. In the control engineering language, these delays can be categorized as state delay, input or output delay (retarded systems), delay in the state derivative (neutral systems), and so forth. Guaranteeing the stability of systems with delay is one core design objective both in theory and in practice. Particularly, in terms of neutral systems, the focus has mainly been on systems with identical delays in neutral and discrete terms [6–10]. Results also exist that depend only on the size of the discrete delays but not on the size of the neutral delays [11–13].Besides delays, the saturated controller is apt to cause instability as well. In the presence of actuator saturation, the problem of estimating asymptotic stability regions for linear systems subject to it has been studied by many researches in the past years in [14]. Generally speaking, the existing methods for estimating the stability regions for linear systems with saturating actuators are based on the concept of Lyapunov level set. LMI optimization-based approaches were proposed to estimate the stability regions by using quadratic Lyapunov functions and the Lur’s-type Lyapunov functions in [15–19].For the studies in response to both issues of delay and saturation, the sufficient conditions for systems with delay and saturated actuator are obtained in [18, 20–22]; Lyapunov-Krasovskii functional is employed to investigate the delay-dependent robust stabilization for uncertain neutral systems with saturated actuators in [20]; a controller is constructed in terms of linear matrix inequalities using descriptor model transformation in [18], just to name a few. However, this paper wants to provide a new method to find the system stability region and give the optimality algorithm to obtain the largest region in this method. Besides, it gives the application in the chemical process of the plant.In this paper, a novel Lyapunov functional is proposed based on the delay-dividing approach, which leads to less conservative stability conditions for linear systems with time-delay and saturated actuator. This is done by introducing auxiliary functions based on the additive decomposition approach [23]. We also propose an algorithm to obtain the optimal auxiliary function. Finally, we design a two-stage dissolution tank of the chemical process by modeling it as a neutral delay system with actuator saturation and demonstrate the effectiveness of the proposed method.Notations.
⋆ denotes the symmetric part to be a symmetric block matrix, ℝn denotes the n dimensional Euclidean space, and ℝm×n is the set of all real m×n matrices. I is the identity matrix with proper dimensions. C0 is the set of all continuous functions from [-τ,0] to ℝn where τ is a constant representing the neutral time-delay. AT is the transpose of matrix A. |A|=[|aij|], with A=[aij]. For real symmetric matrices X and Y, the notation X≥Y (respectively, X>Y) means that the matrix X-Y is positive semidefinite (respectively, positive definite). λmax⁡(min⁡)(A) is the eigenvalue of matrix A with maximum (minimum) real part. ∥v∥ is the Euclidean norm of vector v, ∥v∥=(vTv)1/2, while ∥A∥ is spectral norm of matrix A, ∥A∥=[λmax⁡(ATA)]1/2. ρ represents the domain of attraction. diag⁡{⋯} denotes a block-diagonal matrix decided by the corresponding elements in the brace and finally B(ρ)={y∈Rn:∥y∥≤ρ}.


## 2. Problem Statement and Preliminaries

The following neutral system with time-delay and actuator saturation is considered:(1)x˙(t)-Dx˙(t-τ)=Ax(t)+Bx(t-h)+CSat(u(t)),
where x(t)∈ℝn is the system state and u(t) is the control input. h>0 is the constant discrete time delay and τ>0 is the constant neutral time-delay. A, B, C, and D are known real constant parameter matrices of appropriate dimensions with ∥D∥<1. Sat(·) is used to denote the standard saturation function defined for u=[u1,u2,…,um]:
(2)Sat(u)=[Sat(u1),Sat(u2),…,Sat(um)]T,Sat(ui)={ui+ifui>ui+uiotherwise-ui+ifui<-ui+.The following linear state feedback is to be designed:(3)u(t)=-Kx(t),
where the linear state feedback gain K=[K1T,K2T,…,KmT]T; Ki∈ℝn is an n-dimensional row vector.Here we have slightly abused the notation by usingSat to denote both a scalar valued and a vector valued function. We have also assumed a unity saturation level for the saturation function without loss of generality.DefineDz(u(t))=u(t)-Sat(u(t)) where Dz(u(t))=[dz(u1),dz(u2),…,dz(um)]T and
(4)dz(ui)={ui-ui+ifui>ui+0otherwiseui+ui+ifui<-ui+.The saturated system can now be written as follows:(5)x˙(t)-Dx˙(t-τ)=Acx(t)+Bx(t-h)+CDz(Kx(t)),
where Ac=A-CK.The neutral system (5) then leads to the following by model transformation:
(6)ddt[ℒ(x(t))]=A^cx(t)+CDz(Kx),
where A^c=Ac+B=A+B-CK and ℒ(x(t))=x(t)-Dx(t-τ)+B∫t-ht‍x(s)ds.The following definitions and lemmas are required before proceeding with the main contributions presented in the next section.Definition 1 (see [24]).
The operatorℒ:C0→ℝn is said to be stable if the zero solution of the homogeneous difference equation
(7)ℒ(x(t))=0,t≥0,x0=ψ∈{φ∈C0:ℒφ=0}
is uniformly asymptotically stable. The stability of operator ℒ is necessary for the stability of neutral system (1) with (3), which is always satisfied when ∥h|B|+D∥<1.Lemma 2 (see [24]).
For any matrixB,D∈ℝm×n, if  ∥h|B|+D∥<1, then the operator ℒ:C0→ℝn with
(8)ℒ(x(t))=x(t)+B∫t-ht‍x(s)ds-Dx(t-τ)
is stable.Lemma 3 (see [25]).
LetU∈ℝn×n and  V∈ℝn×n, and let x∈ℝn. Then we have
(9)xTUVx≤∥UV∥xTx≤∥U∥∥V∥xTx,2xTUVx≤xT(1γUUT+γVTV)x,∀γ>0.Definition 4 (see [23]).
Auxiliary functionsW1(ρ1) and W2(ρ2) are described in the following
(10)W1(ρ1)={∑i=1m‍(1-ui+∥Ki∥ρ1)2KiTKiif∥Ki∥ρ1>ui+,0otherwise,W2(ρ2)={∑i=1m‍(ρ24ui+)2(KiTKi)2if∥Ki∥ρ2>ui+,0otherwise.


## 3. Main Results

In this section, we firstly construct the auxiliary functionsW3(ρ3),W4(ρ4),…,Wn(ρn) using the geometric method and with them present a new delay-dependent stabilization criterion. Then the relationship between those auxiliary functions is explored which helps to obtain the optimal Wi(ρi).Consider the polynomial functiony=axn which is tangent with saturated function y=x-ui+ in positive axis. a can be firstly determined by the geometric relation and auxiliary polynomial functions can be obtained.Let(11)y=axn,y=x-ui+.The slope of polynomial functiony=axn equals to 1 at the tangent point. Thus we have
(12)y˙=anxn-1=1.It can be determined by solving the resulting equations simultaneously and we obtain(13)a=(n-1ui+)n-11nn.Noting in the polynomial function, we replacex with ui here so that they have the same independent variable with dz(ui). Consider
(14)y=(n-1ui+)n-11nnuin.Thus we compare the nonlinear functiondz(ui) with the polynomial function above. For simplicity, we consider the cases n=2 and n=3. As shown in Figure 1, the graphics of quadratic and cubic polynomial functions are above the graphic of |dz(ui)|.Figure 1
Two polynomial functions arey=(1/4ui+)ui2 and y=(4/27)(1/ui+)2ui3 (when ui+=2) and the piecewise linear function is dz(ui).Furthermore, according to the nature of the polynomial function whenn=4,5,…, the graphic of function y=((n-1)/ui+)n-1(1/nn)uin is also above the graphic of |dz(ui)|. So we have the following inequality:
(15)|dz(ui)|≤(n-1ui+)n-11nnuin,
where n=2,3,4,….Lemma 5.
Whenn=2,3,4,…, the following inequality holds:
(16)DzT(Kx(t))Dz(Kx(t))≤xTWn(ρn)x,
where
(17)Wn(ρn)={∑i=1m‍((n-1)ρnnn/(n-1)ui+)2n-2(KiTKi)nif∥Ki∥ρn>ui+0otherwise.Proof.
From (15) we obtain that
(18)DzT(Kx)Dz(Kx)=∑i=1m‍dz2(ui)≤∑i=1m‍(n-1ui+)2n-21n2nui2n≤∑i=1m‍(n-1ui+)2n-2ghhhh×1n2nxTKiTKix⋯xTKiTKix≤∑i=1m‍(n-1ui+)2n-2ghhhh×1n2n∥xxT∥n-1xTKiTKi⋯KiTKix≤∑i=1m‍(n-1ui+)2n-21n2nρn2n-2xT(KiTKi)nx≤xTWn(ρn)x.
This completes the proof.


### 3.1. Asymptotic Stability for Certain Neutral System

Theorem 6.
The neutral system with time-delay and actuator saturation as described in (1) and (3) is asymptotic stability if ∥h|B|+D∥<1 and there exist scalars γi>0, i=1,2,3, P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0(i=1,2,3,4,5,6), and T~i>0(i=3,4) such that the following symmetric linear matrix inequality holds:
(19)Φ=[Φ11Φ120Φ14Φ15Φ160⋆Φ22Φ230000⋆⋆Φ330000⋆⋆⋆Φ44Φ4500⋆⋆⋆⋆Φ5500⋆⋆⋆⋆⋆Φ660⋆⋆⋆⋆⋆⋆Φ77]<0,
where
(20)Wn(ρ)={∑i=1m‍((n-1)ρnnn/(n-1)ui+)2n-2(KiTKi)nif∥Ki∥ρn>ui+,0otherwise,Φ11=A^cTP+PA^c+1γ1P^+(γ1+γ2+γ3)∥CTC∥Wi(ρi)+Q11+R11+T1+T2+h22(T~3+T5)+τ22(T~4+T6),Φ22=Q22-Q11,Φ33=-Q22-T1,Φ44=R22-R11,Φ55=1γ2DTP^D-R22-T2,Φ66=1γ3BTP^B-T3,Φ77=-T4,Φ12=Q12,Φ14=R12,Φ15=-A^cTPD,Φ16=A^cTPB,Φ23=-Q12,Φ45=-R12.Proof.
Define a legitimate Lyapunov functional candidate as follows:(21)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))+V6(x(t)),
where
(22)V1(x(t))=ℒT(x(t))Pℒ(x(t)),V2(x(t))=∫t-(h/2)t‍[x(s)x(s-h2)]T[Q11Q12⋆Q22]ghhhhh×[x(s)x(s-h2)]ds+∫t-(τ/2)t‍[x(s)x(s-τ2)]Tghhhhhh×[R11R12⋆R22][x(s)x(s-τ2)]ds,V3(x(t))=∫t-ht‍xT(s)T1x(s)ds+∫t-τt‍xT(s)T2x(s)ds,V4(x(t))=∫t-ht‍[∫st‍xT(s1)ds1]T3[∫st‍x(s2)ds2]ds,V5(x(t))=∫t-τt‍[∫st‍xT(s1)ds1]T4[∫st‍xT(s2)ds2]ds,(23)V6(x(t))=12∫t-ht‍(s-t+h)2xT(s)T5x(s)ds+12∫t-τt‍(s-t+τ)2xT(s)T6x(s)ds,
where P>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, and Ti>0, i=1,2,3,4,5,6.
Then(24)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))+V˙6(x(t)).
By (19)-(22), we obtain that
(25)V˙1(x(t))=2{xT(t)-xT(t-τ)DT+[∫t-ht‍xT(s)ds]BT}×P[A^cx(t)+CDz(Kx)]=2xTPA^cx+2xTPCDz(Kx)-2xTA^cTPDx(t-τ)-2xT(t-τ)DTPCDz(Kx)+2xT(t)A^cTPB[∫t-ht‍x(s)ds]+2[∫t-ht‍xT(s)ds]BTPCDz(Kx)≤xT[A^cTP+PA^c+1γ1P^ghh+(γ1+γ2+γ3)∥CTC∥Wi(ρi)1γ1]x+1γ2xT(t-τ)DTP^Dx(t-τ)+1γ3[∫t-ht‍xT(s)ds]BTP^B[∫t-ht‍x(s)ds]-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-ht‍x(s)ds],
where P^>0, since P^=P2. Consider
(26)V˙2(x(t))=[x(t)x(t-h2)]T[Q11Q12⋆Q22][x(t)x(t-h2)]-[x(t-h2)x(t-h)]T[Q11Q12⋆Q22][x(t-h2)x(t-h)]+[x(t)x(t-τ2)]T[R11R12⋆R22][x(t)x(t-τ2)]-[x(t-τ2)x(t-τ)]T[R11R12⋆R22][x(t-τ2)x(t-τ)],V˙3(x(t))=xT(t)T1x(t)-xT(t-h)T1x(t-h)+xT(t)T2x(t)-xT(t-τ)T2x(t-τ),V˙4(x(t))=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]+∫t-htxT(t)T3[∫stx(s2)ds2]ds+∫t-ht[∫stx(s2)ds2]T3x(t)ds=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]+∫t-ht(s-t+h)xT(t)T3x(s)ds+∫t-ht‍(s-t+h)xT(s)T3x(t)ds=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]-∫t-ht‍(s-t+h)(T3x(t)-T5x(s))Tghhhh×T5-1(T3x(t)-T5x(s))ds+h22xT(t)T3T5-1T3x(t)+∫t-ht‍(s-t+h)xT(s)T5x(s)ds≤-[∫t-ht‍xT(s1)ds1]T3[∫t-htx(s2)ds2]+h22xT(t)T~3x(t)+∫t-ht‍(s-t+h)xT(s)T5x(s)ds,
where T~3>0, since T~3=T3T5-1T3.
Similarly we have(27)V˙5(x(t))≤-[∫t-τtxT(s1)ds1]T4[∫t-τtx(s2)ds2]+τ22xT(t)T~4x(t)+∫t-τt(s-t+τ)xT(s)T6x(s)ds,
where T~4>0, since T~4=T4T6-1T4. Consider
(28)V˙6(x(t))=12h2xT(t)T5x(t)-∫t-ht(s-t+h)xT(s)T5x(s)ds+12τ2xT(t)T6x(t)-∫t-τt‍(s-t+τ)xT(s)T6x(s)ds.
Substituting these into (24), the time-derivative of V has new upper bound as follows:
(29)V˙(x(t))≤ξT(t)Φξ(t),
where(30)ξT(t)=[xT(t)xT(t-h2)xT(t-h)xT(t-τ2)xT(t-τ)∫t-htxT(s)ds∫t-τtxT(s)ds];Φ is defined as stated in (19).
If linear matrix inequality (19) is feasible, then we can get V˙(x(t))<0, for all x∈B(ρ). Therefore, if constant scalar h, constant parameter matrices B,D such that ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0(i=1,2,3,4,5,6), and T~i>0(i=3,4) satisfying (19) for real scalars γi>0(i=1,2,3), from Hale and Verduyn Lunel [24], we can draw the neutral system which can be described by (1) and (3) is asymptotic stability. This completes the proof.Remark 7.
W
i
(
ρ
i
)
(
i
=
1,2
,
…
,
n
) are created by the parameter ρi which is a measure tool for domain of attraction. With these functions, we obtain the novel stability criterion. However, in Theorem 6 we need to look for the largest value of ρi with the optimal Wi(ρi). These can be seen in Section 3.3 below.Remark 8.
Theorem6 gives a delay-dependent stability criterion for neutral system with (1) and (3) using a delay-dividing approach. The delay differential conditions in other works, such as in [26], are usually more strict. These facts mean that our result is less conservative than some previous approaches.The delay-dependent stability criterion for system (1) with τ≡h is presented in the following corollary.Corollary 9.
The neutral systems (1) and (3) with τ≡h are asymptotic stability if ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, Ti>0(i=1,2,3), and T~2>0 such that the following symmetric linear matrix inequality holds for real constant scalars γi>0, i=1,2,3:
(31)Φ¯=[Φ¯11Φ¯12Φ¯13Φ¯14⋆Φ¯22Φ¯230⋆⋆Φ¯330⋆⋆⋆Φ¯44]<0,
where Wi(ρi), i=1,2,3,…,n, are defined as before. Consider
(32)Φ¯11=A^cTP+PA^c+1γ1P^+(γ1+γ2+γ3)∥CTC∥Wi(ρi)+Q11+T1+h22(T~2+T3),Φ¯22=Q22-Q11,Φ¯33=1γ2DTP^D-Q22-T1,Φ¯44=1γ3BTP^B-T2,Φ¯12=Q12,Φ¯13=-A^cTPD,Φ¯14=A^cTPB,Φ¯23=-Q12.Proof.
Define a legitimate Lyapunov functional candidate as(33)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t)),
where
(34)V1(x(t))=ℒT(x(t))Pℒ(x(t)),V2(x(t))=∫t-(h/2)t[x(s)x(s-h2)]T[Q11Q12⋆Q22][x(s)x(s-h2)]ds,V3(x(t))=∫t-htxT(s)T1x(s)ds,V4(x(t))=∫t-ht‍[∫stxT(s1)ds1]T2[∫stx(s2)ds2]ds,V5(x(t))=12∫t-ht(s-t+h)2xT(s)T3x(s)ds,
where P>0, Q=[Qij]2×2, Qij>0, and Ti>0, i=1,2,3.
According to (34) we obtain
(35)V˙1(x(t))≤xT[A^cTP+PA^c+1γ1P^ghh+(γ1+γ2+γ3)∥CTC∥Wi(ρi)1γ1]x+1γ2xT(t-h)DTP^Dx(t-h)+1γ3[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]-2xT(t)A^cTPDx(t-h)+2xT(t)A^cTPB[∫t-ht‍x(s)ds],
where P^>0, since P^=P2. Consider
(36)V˙2(x(t))=[x(t)x(t-h2)]T[Q11Q12⋆Q22][x(t)x(t-h2)]-[x(t-h2)x(t-h)]T[Q11Q12⋆Q22][x(t-h2)x(t-h)],V˙3(x(t))=xT(t)T1x(t)-xT(t-h)T1x(t-h),V˙4(x(t))≤-[∫t-htxT(s1)ds1]T2[∫t-htx(s2)ds2]+h22xT(t)T~2x(t)+∫t-ht‍(s-t+h)xT(s)T3x(s)ds,
where T~2>0, since T~2=T2T3-1T2. Consider
(37)V˙5(x(t))=12h2xT(t)T3x(t)-∫t-ht(s-t+h)xT(s)T3x(s)ds.
Then, the time-derivative ofV has new upper bound as follows:
(38)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))≤ξ¯T(t)Φ¯ξ¯(t),
where Φ¯ is defined as stated in (31) and
(39)ξ¯T(t)=[xT(t)xT(t-h2)xT(t-h)∫t-htxT(s)ds].
The corollary can then be proved following [24].


### 3.2.  Asymptotic Stability for Uncertain Neutral System

Consider the following uncertain neutral system with time-delay and actuator saturation:(40)x˙(t)-Dx˙(t-τ)=(A+ΔA(t))x(t)+Bx(t-h)+(C+ΔC(t))Sat(u(t)),
where ΔA(t) and ΔC(t) stand for the uncertainties. For simplicity, the constant parameter matrices A, B, C, and D are square matrices. The spectral norm bound of the unknown uncertainties is
(41)∥ΔA(t)∥≤α,∥ΔC(t)∥≤β,∀t≥0.Using the nonlinear functionDz(·), rewrite the uncertain neutral system as follows:
(42)x˙(t)-Dx˙(t-τ)=(Ac+ΔA(t)-ΔC(t)K)x(t)+Bx(t-h)+(C+ΔC(t))Dz(Kx(t)),
where Ac is defined as the same with certain neutral system with time-delay and actuator saturation. In particular, when ∥ΔA(t)∥=0 and ∥ΔC(t)∥=0, the uncertain neutral system becomes the certain case.Similarly, we employ the operatorℒ:C0→ℝn with
(43)ℒ(x(t))=x(t)+B∫t-htx(s)ds-Dx(t-τ).The following transformed system is then obtained:(44)ddtℒ(x(t))=A¯cx(t)+(C+ΔC(t))Dz(Kx(t)),
where A¯c=A^c+ΔA(t)-ΔC(t)K=A+B-CK+ΔA(t)-ΔC(t)K.Theorem 10.
The uncertain neutral system in (40) with feedback control (3) is asymptotic stability if ∥h|B|+D∥<1 and there exist scalars γi>0, i=1,2,3,4,5,6, P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0(i=1,2,3,4,5,6), and T~i>0(i=3,4) such that the following symmetric linear matrix inequality holds:
(45)Ξ=[Ξ11Ξ120Ξ14Ξ15Ξ160⋆Ξ22Ξ230000⋆⋆Ξ330000⋆⋆⋆Ξ44Ξ4500⋆⋆⋆⋆Ξ5500⋆⋆⋆⋆⋆Ξ660⋆⋆⋆⋆⋆⋆Ξ77]<0,
where
(46)Ξ11=A^cTP+PA^c+1γ1P^+1γ2P^+(γ1+γ3+γ5)(α2+2αβ∥K∥+β2∥K∥2)I+(γ2+γ4+γ6)(β2+2β∥C∥+∥C∥2)Wi(ρi)+Q11+R11+T1+T2+h22(T~3+T5)+τ22(T~4+T6),Ξ22=Q22-Q11,Ξ33=-Q22-T1,Ξ44=R22-R11,Ξ55=(1γ3+1γ4)DTP^D-R22-T2,Ξ66=(1γ5+1γ6)BTP^B-T3,Ξ77=-T4,Ξ12=Q12,Ξ14=R12,Ξ15=-A^cTPD,Ξ16=A^cTPB,Ξ23=-Q12,Ξ45=-R12.Proof.
Define the legitimate Lyapunov functional candidate as(47)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))+V6(x(t)),
where V1(x(t)), V2(x(t)), V3(x(t)), V4(x(t)), V5(x(t)), and V6(x(t)) are the same as in Theorem 6.
The time-derivative ofV(x(t)) along the trajectories of closed system (44) is given by the following:
(48)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))+V˙6(x(t)).
Then
(49)V˙1(x(t))=2{[∫t-htxT(s)ds]xT(t)-xT(t-τ)DTh+[∫t-htxT(s)ds]BT}×P[A¯cx(t)+(C+ΔC)Dz(Kx)]=2xTPA¯cx+2xTP(C+ΔC)Dz(Kx)-2xTA¯cTPDx(t-τ)-2xT(t-τ)DT×P(C+ΔC)Dz(Kx)+2xT(t)A¯cTPB[∫t-htx(s)ds]+2[∫t-htxT(s)ds]BTP(C+ΔC)Dz(Kx)=2xTPA^cx+2xTP(ΔA-ΔCK)x+2xTP(C+ΔC)Dz(Kx)-2xT(t-τ)DTPA^cx(t)-2xT(t-τ)DTP(ΔA-ΔCK)x-2xT(t-τ)DTP(C+ΔC)Dz(Kx)+2[∫t-htxT(s)ds]BTPA^cx+2[∫t-htxT(s)ds]BTP(ΔA-ΔCK)x+2[∫t-htxT(s)ds]BTP(C+ΔC)Dz(Kx)≤xT[A^cTP+PA^c]x-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-htx(s)ds]+1γ1xTP^x+γ1xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ2xTP^x+γ2DzT(Kx)(C+ΔC)T×(C+ΔC)Dz(Kx)+1γ3xT(t-τ)DTP^Dx(t-τ)+γ3xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ4xT(t-τ)DTP^Dx(t-τ)+γ4DzT(Kx)(C+ΔC)T(C+ΔC)Dz(Kx)+1γ5[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]+γ5xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ6[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]+γ6DzT(Kx)(C+ΔC)T(C+ΔC)Dz(Kx)≤xT{(A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I](A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I]+[1γ2P^+γ2(β2+2β∥C∥+∥C∥2)Wi(ρ)]+γ3(α2+2αβ∥K∥+β2∥K∥2)I+γ4(β2+2β∥C∥+∥C∥2)Wi(ρi)+γ5(α2+2αβ∥K∥+β2∥K∥2)I+γ6(β2+2β∥C∥+∥C∥2)Wi(ρi)(A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I]}x+xT(t-τ)(1γ3DTP^D+1γ4DTP^D)x(t-τ)+[∫t-htxT(s)ds][1γ5BTP^B+1γ6BTP^B]×[∫t-htx(s)ds]-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-htx(s)ds],
where P^>0, since P^=P2.
V
˙
2
(
x
(
t
)
), V˙3(x(t)), V˙4(x(t)), V˙5(x(t)), and V˙6(x(t)) are obtained similarly as in Theorem 6. Substituting these into (48), the time-derivative of V has new upper bound as follows:
(50)V˙(x(t))≤ξT(t)Ξξ(t),
where(51)ξT(t)=[xT(t)xT(t-h2)xT(t-h)xT(t-τ2)xT(t-τ)∫t-htxT(s)ds∫t-τtxT(s)ds];Ξ is defined as stated in (45).
If linear matrix inequality (45) is feasible, then V˙(x(t))<0, for all x∈B(ρ). The theorem can then be proved following [24].Theorem10 provides new asymptotic stability conditions for the uncertain neutral systems in (40) and (3). The following corollary is presented as a special case of the theorem.Corollary 11.
The uncertain neutral system in (40) and (3) with τ≡h is asymptotic stability if ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, Ti>0(i=1,3), and T~2>0 such that the following symmetric linear matrix inequality holds for real constant scalars γi>0, i=1,2,3,4,5,6:
(52)Ξ¯=[Ξ¯11Ξ¯12Ξ¯13Ξ¯14⋆Ξ¯22Ξ¯230⋆⋆Ξ¯330⋆⋆⋆Ξ¯44]<0,
where Wi(ρ), i=1,2,3,…,n are defined as before and
(53)Ξ¯11=A^cTP+PA^c+1γ1P^+1γ2P^+(γ1+γ3+γ5)(α2+2αβ∥K∥+β2∥K∥2)I+(γ2+γ4+γ6)(β2+2β∥C∥+∥C∥2)Wi(ρi)+Q11+T1+h22(T~2+T3),Ξ¯22=Q22-Q11,Ξ¯33=(1γ3+1γ4)DTP^D-Q22-T1,Ξ¯44=(1γ5+1γ6)BTP^B-T2,Ξ¯12=Q12,Ξ¯13=-A^cTPD,Ξ¯14=A^cTPB,Ξ¯23=-Q12.Proof.
Choose a legitimate Lyapunov functional candidate as(54)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))
which are the same as (34).
V
˙
1
(
x
(
t
)
) can be evaluated similarly as in Theorem 10 and Corollary 9. The proof can be readily obtained.


### 3.3.  The Algorithm withW2(ρ2) and the Algorithm to Solve the Optimal Wi(ρi)

From Definition4, it is seen that W1(ρ1) are different from W2(ρ2), W3(ρ3), and W4(ρ4),…,Wn(ρn). We compare between W2(ρ2), W3(ρ3), and W4(ρ4),…,Wn(ρn) and intend to reduce the conservativeness of the result. To that end, we should obtain ρ2 in the first place. In what follows we present the stability analysis algorithm with W2(ρ2) to solve ρ2.Step 1.
Giveα,β,K.Step 2.
Set positive valuesγi(i=1,2,3,4,5,6).Step 3.
Initializeρ2.Step 4.
WithW2(ρ2), solve linear matrix inequality (45) by Matlab LMI Toolbox.Step 5.
If the solution satisfies the stability condition, go to Step6; otherwise, reduce ρ2 and go to Step 3.Step 6.
Increaseρ2 and go to Step 3.Step 7.
End.Remark 12.
The above algorithm is stated with respect to uncertain systems (40) and (3). Other cases can be dealt with similarly.After definingρ2, we replace W2(ρ2) by Wi(ρi) to reduce the conservativeness. We analyse these functions W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn) and find the optimal Wi(ρi) to obtain the maximum ρi among ρ2,ρ3,…,ρn.Theorem 13.
GivenW2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn), if there exists j≤n, such that ρj-1≥ρk and ∥Wj(ρj-1)∥≤∥Wk(ρk)∥, for all k=2,3,…,j-1, then we have
(55)∥Ki∥ρj-1ui+≤fk(j),∀k=2,3,…,j-1,
where ρj is the domain attraction obtained by Wj(ρj) and
(56)fk(j)=(k-1)(k-1)/(j-k)kk/(j-k)×jj/(j-k)(j-1)(j-1)/(j-k).Proof.
Recall the definition ofWj(ρj), we know it can be expressed in the following equality:
(57)Wj(ρj)=∑i=1m‍((j-1)ρjjj/(j-1)ui+)2j-2(KiTKi)j.
If we have ρj-1≥ρk, for all k=2,3,…,j-1, then from the above equality we obtain
(58)∥Wk(ρk)∥≤∥Wk(ρj-1)∥,∀k=2,3,…,j-1.
With regard to Wj(ρj-1), we have
(59)Wj(ρj-1)=∑i=1m‍((j-1)ρj-1jj/(j-1)ui+)2j-2(KiTKi)k(KiTKi)j-k=∑i=1m‍[(k-1)ρj-1kk/(k-1)ui+]2k-2(KiTKi)k(KiTKi)j-kgh×(ρj-1ui+)2(j-k)k2k(k-1)2k-2(j-1)2j-2j2j=∥Ki∥2(j-k)(ρj-1ui+)2(j-k)×k2k(k-1)2k-2(j-1)2j-2j2jWk(ρj-1).
If we have ∥Wj(ρj-1)∥≤∥Wk(ρk)∥, then we can obtain
(60)∥Wj(ρj-1)∥=∥Ki∥2(j-k)(ρj-1ui+)2(j-k)k2k(k-1)2k-2×(j-1)2j-2j2j∥Wk(ρj-1)∥≤∥Wk(ρk)∥.
By (58), we have
(61)(∥Ki∥ρj-1ui+)2(j-k)k2k(k-1)2k-2(j-1)2j-2j2j≤1,
where j=3,4,5,…, and k=2,3,…,j-1.
Then we easily obtain(62)∥Ki∥ρj-1ui+≤fk(j)=(k-1)(k-1)/(j-k)kk/(j-k)×jj/(j-k)(j-1)(j-1)/(j-k)
which completes the proof.Remark 14.
Theorem13 provides us a criterion to find the optimal Wj(ρj). That is, if condition (55) cannot be satisfied, the procedure to find the optimal one should be ended. For example, when ρ4 is obtained, we compute ∥Ki∥ρ4/ui+ and the following inequalities need to be satisfied:
(63)∥Ki∥ρ4ui+≤f2(5),∥Ki∥ρ4ui+≤f3(5),∥Ki∥ρ4ui+≤f4(5).
Otherwise, the algorithm terminates with ρ4.The following optimality algorithm is used to obtain the optimalWn(ρn).Step 1.
Set positive valueδ (small value, for example, 10-5).Step 2.
Solveρ2 based on the above stability analysis algorithm with W2(ρ2).Step 3.
Compute∥Ki∥ρ2/ui+, if (∥Ki∥ρ2/ui+)-fk(j)≤f2(3), then go to Step 4; otherwise, go to Step 7.Step 4.
Setρj+1=ρj, repeat the stability analysis algorithm with Wj+1(ρj) then obtain ρj+1 iteratively.Step 5.
Compute∥Ki∥ρj+1/ui+, if
(64)∥Ki∥ρj+1ui+≤fk(j+2),|∥Ki∥ρj+1ui+-fk(j+2)|>δ,∀k=2,3,…,j+1
then go to Step 6; otherwise go to Step 7.Step 6.
Setj=j+1 and go to Step 4.Step 7.
End.


## 3.1. Asymptotic Stability for Certain Neutral System

Theorem 6.
The neutral system with time-delay and actuator saturation as described in (1) and (3) is asymptotic stability if ∥h|B|+D∥<1 and there exist scalars γi>0, i=1,2,3, P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0(i=1,2,3,4,5,6), and T~i>0(i=3,4) such that the following symmetric linear matrix inequality holds:
(19)Φ=[Φ11Φ120Φ14Φ15Φ160⋆Φ22Φ230000⋆⋆Φ330000⋆⋆⋆Φ44Φ4500⋆⋆⋆⋆Φ5500⋆⋆⋆⋆⋆Φ660⋆⋆⋆⋆⋆⋆Φ77]<0,
where
(20)Wn(ρ)={∑i=1m‍((n-1)ρnnn/(n-1)ui+)2n-2(KiTKi)nif∥Ki∥ρn>ui+,0otherwise,Φ11=A^cTP+PA^c+1γ1P^+(γ1+γ2+γ3)∥CTC∥Wi(ρi)+Q11+R11+T1+T2+h22(T~3+T5)+τ22(T~4+T6),Φ22=Q22-Q11,Φ33=-Q22-T1,Φ44=R22-R11,Φ55=1γ2DTP^D-R22-T2,Φ66=1γ3BTP^B-T3,Φ77=-T4,Φ12=Q12,Φ14=R12,Φ15=-A^cTPD,Φ16=A^cTPB,Φ23=-Q12,Φ45=-R12.Proof.
Define a legitimate Lyapunov functional candidate as follows:(21)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))+V6(x(t)),
where
(22)V1(x(t))=ℒT(x(t))Pℒ(x(t)),V2(x(t))=∫t-(h/2)t‍[x(s)x(s-h2)]T[Q11Q12⋆Q22]ghhhhh×[x(s)x(s-h2)]ds+∫t-(τ/2)t‍[x(s)x(s-τ2)]Tghhhhhh×[R11R12⋆R22][x(s)x(s-τ2)]ds,V3(x(t))=∫t-ht‍xT(s)T1x(s)ds+∫t-τt‍xT(s)T2x(s)ds,V4(x(t))=∫t-ht‍[∫st‍xT(s1)ds1]T3[∫st‍x(s2)ds2]ds,V5(x(t))=∫t-τt‍[∫st‍xT(s1)ds1]T4[∫st‍xT(s2)ds2]ds,(23)V6(x(t))=12∫t-ht‍(s-t+h)2xT(s)T5x(s)ds+12∫t-τt‍(s-t+τ)2xT(s)T6x(s)ds,
where P>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, and Ti>0, i=1,2,3,4,5,6.
Then(24)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))+V˙6(x(t)).
By (19)-(22), we obtain that
(25)V˙1(x(t))=2{xT(t)-xT(t-τ)DT+[∫t-ht‍xT(s)ds]BT}×P[A^cx(t)+CDz(Kx)]=2xTPA^cx+2xTPCDz(Kx)-2xTA^cTPDx(t-τ)-2xT(t-τ)DTPCDz(Kx)+2xT(t)A^cTPB[∫t-ht‍x(s)ds]+2[∫t-ht‍xT(s)ds]BTPCDz(Kx)≤xT[A^cTP+PA^c+1γ1P^ghh+(γ1+γ2+γ3)∥CTC∥Wi(ρi)1γ1]x+1γ2xT(t-τ)DTP^Dx(t-τ)+1γ3[∫t-ht‍xT(s)ds]BTP^B[∫t-ht‍x(s)ds]-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-ht‍x(s)ds],
where P^>0, since P^=P2. Consider
(26)V˙2(x(t))=[x(t)x(t-h2)]T[Q11Q12⋆Q22][x(t)x(t-h2)]-[x(t-h2)x(t-h)]T[Q11Q12⋆Q22][x(t-h2)x(t-h)]+[x(t)x(t-τ2)]T[R11R12⋆R22][x(t)x(t-τ2)]-[x(t-τ2)x(t-τ)]T[R11R12⋆R22][x(t-τ2)x(t-τ)],V˙3(x(t))=xT(t)T1x(t)-xT(t-h)T1x(t-h)+xT(t)T2x(t)-xT(t-τ)T2x(t-τ),V˙4(x(t))=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]+∫t-htxT(t)T3[∫stx(s2)ds2]ds+∫t-ht[∫stx(s2)ds2]T3x(t)ds=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]+∫t-ht(s-t+h)xT(t)T3x(s)ds+∫t-ht‍(s-t+h)xT(s)T3x(t)ds=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]-∫t-ht‍(s-t+h)(T3x(t)-T5x(s))Tghhhh×T5-1(T3x(t)-T5x(s))ds+h22xT(t)T3T5-1T3x(t)+∫t-ht‍(s-t+h)xT(s)T5x(s)ds≤-[∫t-ht‍xT(s1)ds1]T3[∫t-htx(s2)ds2]+h22xT(t)T~3x(t)+∫t-ht‍(s-t+h)xT(s)T5x(s)ds,
where T~3>0, since T~3=T3T5-1T3.
Similarly we have(27)V˙5(x(t))≤-[∫t-τtxT(s1)ds1]T4[∫t-τtx(s2)ds2]+τ22xT(t)T~4x(t)+∫t-τt(s-t+τ)xT(s)T6x(s)ds,
where T~4>0, since T~4=T4T6-1T4. Consider
(28)V˙6(x(t))=12h2xT(t)T5x(t)-∫t-ht(s-t+h)xT(s)T5x(s)ds+12τ2xT(t)T6x(t)-∫t-τt‍(s-t+τ)xT(s)T6x(s)ds.
Substituting these into (24), the time-derivative of V has new upper bound as follows:
(29)V˙(x(t))≤ξT(t)Φξ(t),
where(30)ξT(t)=[xT(t)xT(t-h2)xT(t-h)xT(t-τ2)xT(t-τ)∫t-htxT(s)ds∫t-τtxT(s)ds];Φ is defined as stated in (19).
If linear matrix inequality (19) is feasible, then we can get V˙(x(t))<0, for all x∈B(ρ). Therefore, if constant scalar h, constant parameter matrices B,D such that ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0(i=1,2,3,4,5,6), and T~i>0(i=3,4) satisfying (19) for real scalars γi>0(i=1,2,3), from Hale and Verduyn Lunel [24], we can draw the neutral system which can be described by (1) and (3) is asymptotic stability. This completes the proof.Remark 7.
W
i
(
ρ
i
)
(
i
=
1,2
,
…
,
n
) are created by the parameter ρi which is a measure tool for domain of attraction. With these functions, we obtain the novel stability criterion. However, in Theorem 6 we need to look for the largest value of ρi with the optimal Wi(ρi). These can be seen in Section 3.3 below.Remark 8.
Theorem6 gives a delay-dependent stability criterion for neutral system with (1) and (3) using a delay-dividing approach. The delay differential conditions in other works, such as in [26], are usually more strict. These facts mean that our result is less conservative than some previous approaches.The delay-dependent stability criterion for system (1) with τ≡h is presented in the following corollary.Corollary 9.
The neutral systems (1) and (3) with τ≡h are asymptotic stability if ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, Ti>0(i=1,2,3), and T~2>0 such that the following symmetric linear matrix inequality holds for real constant scalars γi>0, i=1,2,3:
(31)Φ¯=[Φ¯11Φ¯12Φ¯13Φ¯14⋆Φ¯22Φ¯230⋆⋆Φ¯330⋆⋆⋆Φ¯44]<0,
where Wi(ρi), i=1,2,3,…,n, are defined as before. Consider
(32)Φ¯11=A^cTP+PA^c+1γ1P^+(γ1+γ2+γ3)∥CTC∥Wi(ρi)+Q11+T1+h22(T~2+T3),Φ¯22=Q22-Q11,Φ¯33=1γ2DTP^D-Q22-T1,Φ¯44=1γ3BTP^B-T2,Φ¯12=Q12,Φ¯13=-A^cTPD,Φ¯14=A^cTPB,Φ¯23=-Q12.Proof.
Define a legitimate Lyapunov functional candidate as(33)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t)),
where
(34)V1(x(t))=ℒT(x(t))Pℒ(x(t)),V2(x(t))=∫t-(h/2)t[x(s)x(s-h2)]T[Q11Q12⋆Q22][x(s)x(s-h2)]ds,V3(x(t))=∫t-htxT(s)T1x(s)ds,V4(x(t))=∫t-ht‍[∫stxT(s1)ds1]T2[∫stx(s2)ds2]ds,V5(x(t))=12∫t-ht(s-t+h)2xT(s)T3x(s)ds,
where P>0, Q=[Qij]2×2, Qij>0, and Ti>0, i=1,2,3.
According to (34) we obtain
(35)V˙1(x(t))≤xT[A^cTP+PA^c+1γ1P^ghh+(γ1+γ2+γ3)∥CTC∥Wi(ρi)1γ1]x+1γ2xT(t-h)DTP^Dx(t-h)+1γ3[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]-2xT(t)A^cTPDx(t-h)+2xT(t)A^cTPB[∫t-ht‍x(s)ds],
where P^>0, since P^=P2. Consider
(36)V˙2(x(t))=[x(t)x(t-h2)]T[Q11Q12⋆Q22][x(t)x(t-h2)]-[x(t-h2)x(t-h)]T[Q11Q12⋆Q22][x(t-h2)x(t-h)],V˙3(x(t))=xT(t)T1x(t)-xT(t-h)T1x(t-h),V˙4(x(t))≤-[∫t-htxT(s1)ds1]T2[∫t-htx(s2)ds2]+h22xT(t)T~2x(t)+∫t-ht‍(s-t+h)xT(s)T3x(s)ds,
where T~2>0, since T~2=T2T3-1T2. Consider
(37)V˙5(x(t))=12h2xT(t)T3x(t)-∫t-ht(s-t+h)xT(s)T3x(s)ds.
Then, the time-derivative ofV has new upper bound as follows:
(38)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))≤ξ¯T(t)Φ¯ξ¯(t),
where Φ¯ is defined as stated in (31) and
(39)ξ¯T(t)=[xT(t)xT(t-h2)xT(t-h)∫t-htxT(s)ds].
The corollary can then be proved following [24].


## 3.2.  Asymptotic Stability for Uncertain Neutral System

Consider the following uncertain neutral system with time-delay and actuator saturation:(40)x˙(t)-Dx˙(t-τ)=(A+ΔA(t))x(t)+Bx(t-h)+(C+ΔC(t))Sat(u(t)),
where ΔA(t) and ΔC(t) stand for the uncertainties. For simplicity, the constant parameter matrices A, B, C, and D are square matrices. The spectral norm bound of the unknown uncertainties is
(41)∥ΔA(t)∥≤α,∥ΔC(t)∥≤β,∀t≥0.Using the nonlinear functionDz(·), rewrite the uncertain neutral system as follows:
(42)x˙(t)-Dx˙(t-τ)=(Ac+ΔA(t)-ΔC(t)K)x(t)+Bx(t-h)+(C+ΔC(t))Dz(Kx(t)),
where Ac is defined as the same with certain neutral system with time-delay and actuator saturation. In particular, when ∥ΔA(t)∥=0 and ∥ΔC(t)∥=0, the uncertain neutral system becomes the certain case.Similarly, we employ the operatorℒ:C0→ℝn with
(43)ℒ(x(t))=x(t)+B∫t-htx(s)ds-Dx(t-τ).The following transformed system is then obtained:(44)ddtℒ(x(t))=A¯cx(t)+(C+ΔC(t))Dz(Kx(t)),
where A¯c=A^c+ΔA(t)-ΔC(t)K=A+B-CK+ΔA(t)-ΔC(t)K.Theorem 10.
The uncertain neutral system in (40) with feedback control (3) is asymptotic stability if ∥h|B|+D∥<1 and there exist scalars γi>0, i=1,2,3,4,5,6, P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0(i=1,2,3,4,5,6), and T~i>0(i=3,4) such that the following symmetric linear matrix inequality holds:
(45)Ξ=[Ξ11Ξ120Ξ14Ξ15Ξ160⋆Ξ22Ξ230000⋆⋆Ξ330000⋆⋆⋆Ξ44Ξ4500⋆⋆⋆⋆Ξ5500⋆⋆⋆⋆⋆Ξ660⋆⋆⋆⋆⋆⋆Ξ77]<0,
where
(46)Ξ11=A^cTP+PA^c+1γ1P^+1γ2P^+(γ1+γ3+γ5)(α2+2αβ∥K∥+β2∥K∥2)I+(γ2+γ4+γ6)(β2+2β∥C∥+∥C∥2)Wi(ρi)+Q11+R11+T1+T2+h22(T~3+T5)+τ22(T~4+T6),Ξ22=Q22-Q11,Ξ33=-Q22-T1,Ξ44=R22-R11,Ξ55=(1γ3+1γ4)DTP^D-R22-T2,Ξ66=(1γ5+1γ6)BTP^B-T3,Ξ77=-T4,Ξ12=Q12,Ξ14=R12,Ξ15=-A^cTPD,Ξ16=A^cTPB,Ξ23=-Q12,Ξ45=-R12.Proof.
Define the legitimate Lyapunov functional candidate as(47)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))+V6(x(t)),
where V1(x(t)), V2(x(t)), V3(x(t)), V4(x(t)), V5(x(t)), and V6(x(t)) are the same as in Theorem 6.
The time-derivative ofV(x(t)) along the trajectories of closed system (44) is given by the following:
(48)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))+V˙6(x(t)).
Then
(49)V˙1(x(t))=2{[∫t-htxT(s)ds]xT(t)-xT(t-τ)DTh+[∫t-htxT(s)ds]BT}×P[A¯cx(t)+(C+ΔC)Dz(Kx)]=2xTPA¯cx+2xTP(C+ΔC)Dz(Kx)-2xTA¯cTPDx(t-τ)-2xT(t-τ)DT×P(C+ΔC)Dz(Kx)+2xT(t)A¯cTPB[∫t-htx(s)ds]+2[∫t-htxT(s)ds]BTP(C+ΔC)Dz(Kx)=2xTPA^cx+2xTP(ΔA-ΔCK)x+2xTP(C+ΔC)Dz(Kx)-2xT(t-τ)DTPA^cx(t)-2xT(t-τ)DTP(ΔA-ΔCK)x-2xT(t-τ)DTP(C+ΔC)Dz(Kx)+2[∫t-htxT(s)ds]BTPA^cx+2[∫t-htxT(s)ds]BTP(ΔA-ΔCK)x+2[∫t-htxT(s)ds]BTP(C+ΔC)Dz(Kx)≤xT[A^cTP+PA^c]x-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-htx(s)ds]+1γ1xTP^x+γ1xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ2xTP^x+γ2DzT(Kx)(C+ΔC)T×(C+ΔC)Dz(Kx)+1γ3xT(t-τ)DTP^Dx(t-τ)+γ3xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ4xT(t-τ)DTP^Dx(t-τ)+γ4DzT(Kx)(C+ΔC)T(C+ΔC)Dz(Kx)+1γ5[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]+γ5xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ6[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]+γ6DzT(Kx)(C+ΔC)T(C+ΔC)Dz(Kx)≤xT{(A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I](A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I]+[1γ2P^+γ2(β2+2β∥C∥+∥C∥2)Wi(ρ)]+γ3(α2+2αβ∥K∥+β2∥K∥2)I+γ4(β2+2β∥C∥+∥C∥2)Wi(ρi)+γ5(α2+2αβ∥K∥+β2∥K∥2)I+γ6(β2+2β∥C∥+∥C∥2)Wi(ρi)(A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I]}x+xT(t-τ)(1γ3DTP^D+1γ4DTP^D)x(t-τ)+[∫t-htxT(s)ds][1γ5BTP^B+1γ6BTP^B]×[∫t-htx(s)ds]-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-htx(s)ds],
where P^>0, since P^=P2.
V
˙
2
(
x
(
t
)
), V˙3(x(t)), V˙4(x(t)), V˙5(x(t)), and V˙6(x(t)) are obtained similarly as in Theorem 6. Substituting these into (48), the time-derivative of V has new upper bound as follows:
(50)V˙(x(t))≤ξT(t)Ξξ(t),
where(51)ξT(t)=[xT(t)xT(t-h2)xT(t-h)xT(t-τ2)xT(t-τ)∫t-htxT(s)ds∫t-τtxT(s)ds];Ξ is defined as stated in (45).
If linear matrix inequality (45) is feasible, then V˙(x(t))<0, for all x∈B(ρ). The theorem can then be proved following [24].Theorem10 provides new asymptotic stability conditions for the uncertain neutral systems in (40) and (3). The following corollary is presented as a special case of the theorem.Corollary 11.
The uncertain neutral system in (40) and (3) with τ≡h is asymptotic stability if ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, Ti>0(i=1,3), and T~2>0 such that the following symmetric linear matrix inequality holds for real constant scalars γi>0, i=1,2,3,4,5,6:
(52)Ξ¯=[Ξ¯11Ξ¯12Ξ¯13Ξ¯14⋆Ξ¯22Ξ¯230⋆⋆Ξ¯330⋆⋆⋆Ξ¯44]<0,
where Wi(ρ), i=1,2,3,…,n are defined as before and
(53)Ξ¯11=A^cTP+PA^c+1γ1P^+1γ2P^+(γ1+γ3+γ5)(α2+2αβ∥K∥+β2∥K∥2)I+(γ2+γ4+γ6)(β2+2β∥C∥+∥C∥2)Wi(ρi)+Q11+T1+h22(T~2+T3),Ξ¯22=Q22-Q11,Ξ¯33=(1γ3+1γ4)DTP^D-Q22-T1,Ξ¯44=(1γ5+1γ6)BTP^B-T2,Ξ¯12=Q12,Ξ¯13=-A^cTPD,Ξ¯14=A^cTPB,Ξ¯23=-Q12.Proof.
Choose a legitimate Lyapunov functional candidate as(54)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))
which are the same as (34).
V
˙
1
(
x
(
t
)
) can be evaluated similarly as in Theorem 10 and Corollary 9. The proof can be readily obtained.


## 3.3.  The Algorithm withW2(ρ2) and the Algorithm to Solve the Optimal Wi(ρi)

From Definition4, it is seen that W1(ρ1) are different from W2(ρ2), W3(ρ3), and W4(ρ4),…,Wn(ρn). We compare between W2(ρ2), W3(ρ3), and W4(ρ4),…,Wn(ρn) and intend to reduce the conservativeness of the result. To that end, we should obtain ρ2 in the first place. In what follows we present the stability analysis algorithm with W2(ρ2) to solve ρ2.Step 1.
Giveα,β,K.Step 2.
Set positive valuesγi(i=1,2,3,4,5,6).Step 3.
Initializeρ2.Step 4.
WithW2(ρ2), solve linear matrix inequality (45) by Matlab LMI Toolbox.Step 5.
If the solution satisfies the stability condition, go to Step6; otherwise, reduce ρ2 and go to Step 3.Step 6.
Increaseρ2 and go to Step 3.Step 7.
End.Remark 12.
The above algorithm is stated with respect to uncertain systems (40) and (3). Other cases can be dealt with similarly.After definingρ2, we replace W2(ρ2) by Wi(ρi) to reduce the conservativeness. We analyse these functions W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn) and find the optimal Wi(ρi) to obtain the maximum ρi among ρ2,ρ3,…,ρn.Theorem 13.
GivenW2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn), if there exists j≤n, such that ρj-1≥ρk and ∥Wj(ρj-1)∥≤∥Wk(ρk)∥, for all k=2,3,…,j-1, then we have
(55)∥Ki∥ρj-1ui+≤fk(j),∀k=2,3,…,j-1,
where ρj is the domain attraction obtained by Wj(ρj) and
(56)fk(j)=(k-1)(k-1)/(j-k)kk/(j-k)×jj/(j-k)(j-1)(j-1)/(j-k).Proof.
Recall the definition ofWj(ρj), we know it can be expressed in the following equality:
(57)Wj(ρj)=∑i=1m‍((j-1)ρjjj/(j-1)ui+)2j-2(KiTKi)j.
If we have ρj-1≥ρk, for all k=2,3,…,j-1, then from the above equality we obtain
(58)∥Wk(ρk)∥≤∥Wk(ρj-1)∥,∀k=2,3,…,j-1.
With regard to Wj(ρj-1), we have
(59)Wj(ρj-1)=∑i=1m‍((j-1)ρj-1jj/(j-1)ui+)2j-2(KiTKi)k(KiTKi)j-k=∑i=1m‍[(k-1)ρj-1kk/(k-1)ui+]2k-2(KiTKi)k(KiTKi)j-kgh×(ρj-1ui+)2(j-k)k2k(k-1)2k-2(j-1)2j-2j2j=∥Ki∥2(j-k)(ρj-1ui+)2(j-k)×k2k(k-1)2k-2(j-1)2j-2j2jWk(ρj-1).
If we have ∥Wj(ρj-1)∥≤∥Wk(ρk)∥, then we can obtain
(60)∥Wj(ρj-1)∥=∥Ki∥2(j-k)(ρj-1ui+)2(j-k)k2k(k-1)2k-2×(j-1)2j-2j2j∥Wk(ρj-1)∥≤∥Wk(ρk)∥.
By (58), we have
(61)(∥Ki∥ρj-1ui+)2(j-k)k2k(k-1)2k-2(j-1)2j-2j2j≤1,
where j=3,4,5,…, and k=2,3,…,j-1.
Then we easily obtain(62)∥Ki∥ρj-1ui+≤fk(j)=(k-1)(k-1)/(j-k)kk/(j-k)×jj/(j-k)(j-1)(j-1)/(j-k)
which completes the proof.Remark 14.
Theorem13 provides us a criterion to find the optimal Wj(ρj). That is, if condition (55) cannot be satisfied, the procedure to find the optimal one should be ended. For example, when ρ4 is obtained, we compute ∥Ki∥ρ4/ui+ and the following inequalities need to be satisfied:
(63)∥Ki∥ρ4ui+≤f2(5),∥Ki∥ρ4ui+≤f3(5),∥Ki∥ρ4ui+≤f4(5).
Otherwise, the algorithm terminates with ρ4.The following optimality algorithm is used to obtain the optimalWn(ρn).Step 1.
Set positive valueδ (small value, for example, 10-5).Step 2.
Solveρ2 based on the above stability analysis algorithm with W2(ρ2).Step 3.
Compute∥Ki∥ρ2/ui+, if (∥Ki∥ρ2/ui+)-fk(j)≤f2(3), then go to Step 4; otherwise, go to Step 7.Step 4.
Setρj+1=ρj, repeat the stability analysis algorithm with Wj+1(ρj) then obtain ρj+1 iteratively.Step 5.
Compute∥Ki∥ρj+1/ui+, if
(64)∥Ki∥ρj+1ui+≤fk(j+2),|∥Ki∥ρj+1ui+-fk(j+2)|>δ,∀k=2,3,…,j+1
then go to Step 6; otherwise go to Step 7.Step 6.
Setj=j+1 and go to Step 4.Step 7.
End.


## 4. Application and Numerical Examples

The two-stage system has been applied to many factories worldwide. For the interest of efficiency, a lot of equipment such as converter, cylinder, and ejector is designed in the two-stage form. The natural, social, and economic systems also exhibit the two-stage mode. A typical example is the two-stage ditch design which is a conservation tool supported by the conservancy in Indiana. The advantages of a two-stage ditch against the typical agricultural ditch include both improved drainage function and ecological function. The two-stage design improves ditch stability by reducing water flow and the need for maintenance, saving both labor and money.Consider Figure2 which shows two-stage dissolution tank in the chemical process. The solute in the hopper is transported by the conveyor belt and falls into DT1 within a certain time. The solute is part of the solution in DT1; then within a certain time the solution in DT1 and the undissolved solute flow into DT2 to continue to dissolve and dilute.Figure 2
Two-stage dissolution tank.Under normal circumstances, the dilute fluid flow of DT1 and the velocity of the conveyor belt are constant. The dilute fluid flow of DT2 is controllable. In practical system design, the pipeline of the DT2 dilute fluid is thin and the regulating action of DT2 is tiny, which can thus be represented by the saturating formSat(u(t)). Let λ1(t) and λ2(t) be the solution concentration of DT1 and DT2 and λ10 and λ20 the solution concentration of DT1 and DT2 when the system reaches an equilibrium, respectively. x1(t)=λ1(t)-λ10 and x2(t)=λ2(t)-λ20 are the considered system states and the dilute fluid flow into DT2 u(t) is the control input. The latter is of the saturating form Sat(u(t)) because the dilute fluid flow into DT2 u(t) is a fine adjustment role. The chemical plant puts the solute into DT1 with constant speed; that is, the velocity of the conveyor belt is constant in a period of time interval. In addition, the velocity of the dilute fluid flow into DT1 is also constant in a period of time interval.


### 4.1. No Uncertainty Case

Example 1.
The model for the above described system is(65)x˙(t)-[0.2000.2]x˙(t-τ)=[11.50.3-2]x(t)+[0-100]x(t-h)[01]Sat(u(t)),
where the saturation limit is u+=15. The discrete time-delay h=0.2 and neutral time-delay τ=0.4.
It is seen that(66)∥h|B|+D∥=∥0.2|[0-100]|+[0.2000.2]∥=0.1047<1
and the operator ℒ:C0→ℝn with
(67)ℒ(x(t))=x(t)+B∫t-htx(s)ds-Dx(t-τ)
is stable.
The following linear state feedback control is used and the same as [18]:
(68)u=-Kx=[-0.13250.0153]x
To begin with the stability analysis algorithm, we haveρ2=161.73, and compute ∥K∥ρ2/u+=1.4383. It is seen that
(69)1<∥K∥ρ2u+≤f2(3)=2716≐1.6875.
According to the optimality algorithm, let initial dataρ3=ρ2. Repeating the stability algorithm with W3(ρ3), we obtain ρ3=163.46. Similarly, we compute ∥K∥ρ3/u+ and (∥K∥ρ3/u+)=1.4537.
It is seen that(70)∥K∥ρ3u+<f2(4)≐1.5396,but∥K∥ρ3u+>f3(4)≐1.4047
which terminates the algorithm. In this example, we obtain that the optimal auxiliary function is W3(ρ3). By Theorem 13, we must verify ∥W3(ρ3)∥≤∥W2(ρ2)∥, which is true as ∥W3(ρ3)∥=0.0017 and ∥W2(ρ2)∥=0.0023. So we infer that W3(ρ3) is the optimal auxiliary function among W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn) and ρ3 is the largest one among ρ2,ρ3,ρ4,…,ρn.
The comparison of these results can be seen in Figure3. Notice that ρ is not the domain of attraction.
We obtain the largest parameterρi with optimal Wi(ρi) to enlarge the stability region. Thus the chemical plant can make the conveyor belt and dilute fluid flow of DT1 speed up accordingly. In this way, this chemical plant can improve the production efficiency and reduce the cost of production.Figure 3
Attraction domain expanding schematic diagram: the largest parameterρ3 with optimal W3(ρ3) is in red pentagram, with W2(ρ2) in blue, and another one with W4(ρ4) in red.However, in some circumstances, we cannot enlarge the domain of attraction by replacingW2(ρ2) with Wi(ρi), i=3,4,…,n. For example, let us see the following example.Example 2.
Consider the following linear neutral system with time-delay and actuator saturation:(71)x˙(t)-[0.2000.2]x˙(t-τ)=[0.1-0.10.1-3.0]x(t)+[0.5-100.5]x(t-h)+[0.7000.1]Sat(u(t)),
where the saturation limit values are u1+=10 and u2+=4. The discrete time-delay h=0.2 and neutral time-delay τ=0.1.
Compute∥h|B|+D∥ and we also know that the operator ℒ:C0→ℝn with
(72)ℒ(x(t))=x(t)+B∫t-htx(s)ds-Dx(t-τ)
is stable. Then we obtain the linear state feedback control without considering the saturation using optimal control law in [27]. Consider
(73)u=-Kx=-[0.84560.06270.01421.3754]x.
To begin with the stability analysis algorithm, we haveρ2=17.84, and compute
(74)∥K1∥ρ2u1+=1.5127,∥K2∥ρ2u2+=6.1780.
It is seen that
(75)1<∥K1∥ρ2u1+≤f2(3)=2716≐1.6875,but∥K2∥ρ2u2+>f2(3)
which terminates the algorithm. So we infer that W2(ρ2) is the optimal auxiliary function among W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn). ρ2 has been the largest one among ρ2,ρ3,ρ4,…,ρn and we cannot maximize the value of ρi.
The comparison of these results can be seen in Figure4.Figure 4
Attraction domain expanding schematic diagram: the largest parameterρ2 with optimal W2(ρ2) is in red and another one with W3(ρ3) in blue.


### 4.2. Uncertainty Case

In this subsection, we consider the existence of uncertainty due to outside interference. The numerical example demonstrates the effectiveness of the proposed algorithm.Example 3.
Consider system (65) with norm bounded uncertainties:
(76)∥ΔA(t)∥≤0.1,∥ΔB(t)∥≤0.1.
It is known that this uncertainty decreases the initial state bound that guarantees the asymptotic stability of the system (65). To compare with the above case, the feedback control (68) is applied and we use (45) in Theorem 10.
Using the stability analysis algorithm and the optimal algorithm, we see thatW3(ρ3) is the optimal auxiliary function among W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn). ρ3=43.57 is the largest one among ρ2,ρ3,ρ4,…,ρn. This is shown in Figure 5.Figure 5
Attraction domain expanding schematic diagram: the largest parameterρ3 with optimal W3(ρ3) is in red pentagram, with W2(ρ2) in blue, and another one with W4(ρ4) in red.


## 4.1. No Uncertainty Case

Example 1.
The model for the above described system is(65)x˙(t)-[0.2000.2]x˙(t-τ)=[11.50.3-2]x(t)+[0-100]x(t-h)[01]Sat(u(t)),
where the saturation limit is u+=15. The discrete time-delay h=0.2 and neutral time-delay τ=0.4.
It is seen that(66)∥h|B|+D∥=∥0.2|[0-100]|+[0.2000.2]∥=0.1047<1
and the operator ℒ:C0→ℝn with
(67)ℒ(x(t))=x(t)+B∫t-htx(s)ds-Dx(t-τ)
is stable.
The following linear state feedback control is used and the same as [18]:
(68)u=-Kx=[-0.13250.0153]x
To begin with the stability analysis algorithm, we haveρ2=161.73, and compute ∥K∥ρ2/u+=1.4383. It is seen that
(69)1<∥K∥ρ2u+≤f2(3)=2716≐1.6875.
According to the optimality algorithm, let initial dataρ3=ρ2. Repeating the stability algorithm with W3(ρ3), we obtain ρ3=163.46. Similarly, we compute ∥K∥ρ3/u+ and (∥K∥ρ3/u+)=1.4537.
It is seen that(70)∥K∥ρ3u+<f2(4)≐1.5396,but∥K∥ρ3u+>f3(4)≐1.4047
which terminates the algorithm. In this example, we obtain that the optimal auxiliary function is W3(ρ3). By Theorem 13, we must verify ∥W3(ρ3)∥≤∥W2(ρ2)∥, which is true as ∥W3(ρ3)∥=0.0017 and ∥W2(ρ2)∥=0.0023. So we infer that W3(ρ3) is the optimal auxiliary function among W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn) and ρ3 is the largest one among ρ2,ρ3,ρ4,…,ρn.
The comparison of these results can be seen in Figure3. Notice that ρ is not the domain of attraction.
We obtain the largest parameterρi with optimal Wi(ρi) to enlarge the stability region. Thus the chemical plant can make the conveyor belt and dilute fluid flow of DT1 speed up accordingly. In this way, this chemical plant can improve the production efficiency and reduce the cost of production.Figure 3
Attraction domain expanding schematic diagram: the largest parameterρ3 with optimal W3(ρ3) is in red pentagram, with W2(ρ2) in blue, and another one with W4(ρ4) in red.However, in some circumstances, we cannot enlarge the domain of attraction by replacingW2(ρ2) with Wi(ρi), i=3,4,…,n. For example, let us see the following example.Example 2.
Consider the following linear neutral system with time-delay and actuator saturation:(71)x˙(t)-[0.2000.2]x˙(t-τ)=[0.1-0.10.1-3.0]x(t)+[0.5-100.5]x(t-h)+[0.7000.1]Sat(u(t)),
where the saturation limit values are u1+=10 and u2+=4. The discrete time-delay h=0.2 and neutral time-delay τ=0.1.
Compute∥h|B|+D∥ and we also know that the operator ℒ:C0→ℝn with
(72)ℒ(x(t))=x(t)+B∫t-htx(s)ds-Dx(t-τ)
is stable. Then we obtain the linear state feedback control without considering the saturation using optimal control law in [27]. Consider
(73)u=-Kx=-[0.84560.06270.01421.3754]x.
To begin with the stability analysis algorithm, we haveρ2=17.84, and compute
(74)∥K1∥ρ2u1+=1.5127,∥K2∥ρ2u2+=6.1780.
It is seen that
(75)1<∥K1∥ρ2u1+≤f2(3)=2716≐1.6875,but∥K2∥ρ2u2+>f2(3)
which terminates the algorithm. So we infer that W2(ρ2) is the optimal auxiliary function among W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn). ρ2 has been the largest one among ρ2,ρ3,ρ4,…,ρn and we cannot maximize the value of ρi.
The comparison of these results can be seen in Figure4.Figure 4
Attraction domain expanding schematic diagram: the largest parameterρ2 with optimal W2(ρ2) is in red and another one with W3(ρ3) in blue.


## 4.2. Uncertainty Case

In this subsection, we consider the existence of uncertainty due to outside interference. The numerical example demonstrates the effectiveness of the proposed algorithm.Example 3.
Consider system (65) with norm bounded uncertainties:
(76)∥ΔA(t)∥≤0.1,∥ΔB(t)∥≤0.1.
It is known that this uncertainty decreases the initial state bound that guarantees the asymptotic stability of the system (65). To compare with the above case, the feedback control (68) is applied and we use (45) in Theorem 10.
Using the stability analysis algorithm and the optimal algorithm, we see thatW3(ρ3) is the optimal auxiliary function among W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn). ρ3=43.57 is the largest one among ρ2,ρ3,ρ4,…,ρn. This is shown in Figure 5.Figure 5
Attraction domain expanding schematic diagram: the largest parameterρ3 with optimal W3(ρ3) is in red pentagram, with W2(ρ2) in blue, and another one with W4(ρ4) in red.


## 5. Conclusions

By model transformation and Lyapunov method, delay-dependent criteria for a class of uncertain neutral delay systems with saturation are derived in terms of the spectral radius and LMIs. Both conditions for asymptotic stability and algorithms are presented. These theoretical findings are successfully verified via a two-stage dissolution tank in the chemical plant. Potentially, the result is of great significance to the chemical production process.



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*Source: 805798-2014-03-27.xml*