# New Delay-Dependent Stability Criteria for Uncertain Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

Authors: Hamid Reza Karimi; Mauricio Zapateiro; Ningsu Luo
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

The problem of stability analysis for a class of neutral systems with mixed time-varying neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent. The conditions are presented in terms of linear matrix inequalities (LMIs) and can be efficiently solved using convex programming techniques. Two numerical examples are given to illustrate the efficiency of the proposed method.


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## Body

## 1. Introduction

Delay (or memory) systems represent a class of
infinite-dimensional systems largely used to describe propagation and transport
phenomena or population dynamics [1–3]. Delay differential systems are
assuming an increasingly important role in many disciplines like economic, mathematics, science, and engineering. For instance, in economic systems, delays appear in a natural way since decisions and effects are separated by
some time interval. The presence of a delay in a system may be the result of
some essential simplification of the corresponding process model. The problem
of delay effects on the stability of systems including delays in the state,
and/or input is a problem of recurring interest since the delay presence may
induce complex behaviors (oscillation, instability, bad performances) for the
schemes [1, 2]. Some improved methods pertaining to the problems of
determining robust stability criteria and robust control design of uncertain
time-delay systems have been reported; see, for example, [4, 5] and the
references cited therein. When dealing with time-varying delays and the
reduction of the level of design conservatism, one has to select appropriate
Lyapunov-Krasovskii functional (LKF) with moderate number of terms [6].Neutral delay systems constitute a more general class than
those of the retarded type. Stability of these systems proves to be a more
complex issue because the system involves the derivative of the delayed state. Especially in the past few decades, increased attention has been devoted to the
problem of robust delay-independent stability or delay-dependent stability and
stabilization via different approaches (e.g., model transformation
techniques [2, 7–9], the improved bounding techniques [10, 11], and the properly chosen
Lyapunov-Krasovskii functionals [12, 13])
for a number of different neutral systems with delayed state and/or input, parameter uncertainties, and nonlinear perturbations (see, e.g., [14–25]
and the references therein).Among the existing results on neutral delay
systems, the linear matrix inequality (LMI) approach is an efficient method to
solve many problems such as stability analysis, stabilization [9, 15, 26, 27], H∞ control problems [28–30], filter designs [31, 32], and guaranteed-cost (observer-based) control [33–39]. Besides, for neutral
systems with mixed neutral and discrete delays, most of the aforementioned
methods can only provide neutral-delay-independent and discrete-delay-dependent
results. Furthermore, the subject of the robust stability and feedback
stabilization of continuous- and discrete-time systems (within the framework LMI) under additive perturbations
which are nonlinear functions in time and state of the systems are investigated
in  [40, 41], respectively.In the recent
literature on neutral systems, He et al. in [42] proposed a new approach to
analyze the stability of the systems with mixed delays by incorporating some
free-weighting matrices, and the less conservative criteria, which were both
discrete-delay-dependent and neutral-delay-dependent, were obtained without
considering the model transformations. However, some of the free matrices did
not serve to reduce the conservatism of the results that were obtained. Moreover, in [9, 20], the authors studied the problem of the robust stability of
neutral systems with nonlinear parameter perturbations and mixed time-varying
neutral and discrete delays and presented neutral-delay-independent stability
criteria, that cannot be directly applied to the systems with different
time-varying neutral, discrete, and distributed delays. Furthermore, from the
published results, it appears that general results pertaining to neutral
systems with mixed time-varying neutral, discrete, and distributed delays and nonlinear parameter
perturbations are few and restricted; see [9, 10, 18, 20, 42] where most of
the efforts were virtually neutral-delay-range-independent or were not centered
on distributed delays.In
this paper, we develop new stability criteria for the stability analysis of the
neutral systems with nonlinear parameter perturbations based on a descriptor model
transformation. The dynamical system under consideration consists of
time-varying neutral, discrete, and distributed delays without any restriction
on upper bounds of derivatives of time-varying delays. By introducing a novel Lyapunov-Krasovskii functional and combining the
descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable
change of variables, new sufficient
conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent, and
distributed-delay-dependent. The conditions are presented in terms of LMIs and can be easily solved by existing convex
optimization techniques. Two numerical examples are given to demonstrate the less conservatism of
the proposed results over some existence results in the literature.Notations. The superscript 'T' stands for matrix transposition; ℜn denotes the n-dimensional Euclidean space; ℜn×m is the set of all real m by n matrices. ∥⋅∥ refers to the Euclidean vector norm or the
induced matrix 2-norm. col{⋯} and diag{⋯} represent, respectively, a column vector and a
block diagonal matrix, and the operator sym(A) represents A+AT. λmin(A) and λmax(A) denote, respectively, the smallest and largest
eigenvalue of the square matrix A. The notation P>0 means that P is real symmetric and positive definite; the
symbol ∗ denotes the elements below the main diagonal
of a symmetric block matrix.


## 2. Problem
Description

Consider a class of linear neutral systems with different time-varying
neutral, discrete, and distributed delays and nonlinear parameter perturbations represented by(2.1)x˙(t)−Cx˙(t−τ(t))=Ax(t)+Bx(t−h(t))+G1f1(t,x(t))+G2f2(t,x(t−h(t)))+G3∫t−r(t)tf3(θ,x(θ))dθ+G4f4(t,x˙(t−τ(t))),x(t)=ϕ(t),t∈[−κ,0], where κ:=max{h2,τ1,r1}, and x(t)∈ℜn is the state vector. The time-varying vector
valued initial function ϕ(t) is a continuously differentiable functional, and the time-varying delays h(t), τ(t), and r(t) are functions satisfying, respectively,
(2.2a)0<h1≤h(t)≤h2,|h˙(t)|≤h3<∞,(2.2b)0<τ(t)≤τ1,|τ˙(t)|≤τ2<∞,(2.2c)0<r(t)≤r1,|r˙(t)|≤r2<∞.
The
time-varying vector-valued functions fi:ℜ+×ℜn→ℜni (i=1,…,4) are continuous and satisfy fi(t,0)=0, and the Lipschitz conditions, that is, ∥fi(t,x0)−fi(t,y0)∥≤∥Ui(x0−y0)∥ for all t and for all x0,y0∈ℜn such that Ui are some known matrices.Remark 2.1.
In this case,h(t) is called an interval-like or
range-like time-varying delay [14]. It is also noted that this kind of
time-delay describes the real situation in many practical engineering systems. For example, in the field of networked control systems, the network
transmission induced delays (either from the sensor to the controller or from
the controller to the plant) can be assumed to satisfy (2.2a) without loss of generality [43, 44].Throughout
the paper, the following assumptions are needed to enable the
application of Lyapunov’s method for the stability of neutral systems [1]:(A1)
let the difference
operatorD:C([−κ,0],ℜn)→ℜn given by Dxt=x(t)−Cx(t−τ(t)) be delay-independently stable with respect to
all delays. A sufficient condition for (A1) is that(A2)
all the eigenvalues of the matrixC are inside the unit circle.Before ending this section, we recall the following
lemmas, which will be used in the proof of our main results.Lemma 2.2 (see [9]).
For any arbitrary column
vectorsa(s),b(s)∈ℜp, any matrix W∈ℜp×p, and positive-definite matrix H∈ℜp×p, the following inequality holds: (2.3)−2∫t−r(t)tb(s)Ta(s)ds≤∫t−r(t)t[a(s)b(s)]T[HHW∗(HW+I)TH−1(HW+I)][a(s)b(s)]ds.Lemma 2.3 (see [45]).
Given matricesY=YT,D, E, and F of appropriate
dimensions with FTF≤I, then the following matrix inequality holds:(2.4)Y+sym(DFE)<0, for all F if and only
if there exists a scalar ε>0 such that (2.5)Y+εDDT+ε−1ETE<0.


## 3. Main Results

In this section, new delay-range-dependent sufficient conditions for the
asymptotic stability of the neutral system (2.1) are presented. By utilizing
the Leibniz-Newton
formula, the following two zero equations hold:
(3.1a)L1x(t)−L1x(t−h(t))−L1∫t−h(t)tx˙(s)ds=0,(3.1b)L2x(t)−L2x(t−τ(t))−L2∫t−τ(t)tx˙(s)ds=0,
then, we can represent the system (2.1) as
(3.2)x˙(t)−Cx˙(t−τ(t))=A˜x(t)+B˜x(t−h(t))+G1f1(t,x(t))+G2f2(t,x(t−h(t)))−L2x(t−τ(t))−L1∫t−h(t)tx˙(s)ds−L2∫t−τ(t)tx˙(s)ds+G3∫t−r(t)tf3(θ,x(θ))dθ+G4f4(t,x˙(t−τ(t))), with A˜:=A+L1+L2 and B˜:=B−L1 where the
matrices L1,L2∈ℜn×n will be chosen in the following theorem.Theorem 3.1.
Under (A1), for given scalarsγ,h1,h2,τ1,r1>0,h3,τ2,r2, the neutral system (2.1) is asymptotically stable, if there exist some scalars δ,α1,α2, matrices P2,{Ni}i=120,Y1,Y2, and positive-definite matrices P1,{Qi}i=14,{Ri}i=14,H˜1,H˜2, such that the following LMI is feasible: (3.3)Π=[Π11Π12Π13Π14Π15Π16Π17Π18Π19Π1,10Π1,11∗Π22Π230−BTN17T0Π27−BTN19TΠ29Π2,100∗∗Π33000Π370Π3900∗∗∗Π4400−N15T0−N16TΠ4,100∗∗∗∗Π55Π56−CTN18T−N17G3−CTN20T00∗∗∗∗∗Π66Π67Π68Π6900∗∗∗∗∗∗Π77−N18G3−N18G4Π7,100∗∗∗∗∗∗∗Π88−G3TN20T00∗∗∗∗∗∗∗∗−sym(N20G4)Π9,100∗∗∗∗∗∗∗∗∗Π10,100∗∗∗∗∗∗∗∗∗∗Π11,11]<0 with (3.4)Π11=sym([P2TA+(1+α1)Y1+(1+α2)Y2P1−P2Tδ(P2TA+(1+α1)Y1+(1+α2)Y2)−δP2T])+Ω,Π12=[P2TB−(1+α1)Y1+N2T−N1−N5+N9δP2TB−δ(1+α1)Y1],Π13=[N5N900],Π14=[−(1+α2)Y2+N14T−N13−δ(1+α2)Y2],Π15=[P2TC−ATN17TδP2TC+N17T],Π16=[P2TG1P2TG2δP2TG1δP2TG2],Π17=[N3T+N15T−ATN18TN18T],Π18=[P2TG3−ATN19TδP2TG3+N19T],Π19=[P2TG4+N4T+N16T−ATN20TδP2TG4+N20T],Π1,10=[h12N1h12N5h12N9τ1N130000],Π1,11=[h2(α1+1)H˜1τ1(α1+1)H˜2h2[0I][Y1δY1]Tτ1[0I][Y2δY2]T],Π22=U2TU2−(1−h3)R3−sym(N2+N6−N10),Π23=[N6N10],Π27=−N3T−N7T+N11T−BTN18T,Π29=−N4T−N8T+N12T−BTN20T,Π2,10=[h12N2h12N6h12N100],Π33=diag{−R1,−R2},Π37=[N7T−N11T],Π39=[N8T−N12T],Π44=−(1−τ2)Q1−sym(N14),Π4,10=[000τ1N14],Π55=−(1−τ2)Q2−sym(N17C)+U4TU4,Π56=[−N17G1−N17G2],Π66=diag{−I,−I},Π67=[−G1TN18T−G2TN18T],Π68=[N19G1N19G2]T,Π69=[−G1TN20T−G2TN20T],Π77=−I+r12Q4,Π88=−(1−r2)Q4−sym(N19G3),Π4,10=[000τ1N14],Π7,10=[h12N3h12N7h12N11τ1N15],Π9,10=[h12N4h12N8h12N12τ1N16],Π10,10=diag{−h12R4,−h12R5,−h12R4,−τ1Q3},Π1,11=diag{−h2H˜1,−τ1H˜2,−h2H˜1,−τ1H˜2}, where Ω=diag{Q1+∑i=13Ri+U1TU1+U3TU3+sym(N1+N13),Q2+τ1Q3+h12R4+h2R5}.Proof.
Firstly, we represent (3.2) in an
equivalent descriptor model form as (3.5)x˙(t)=η(t),0=−η(t)+A˜x(t)+Cη(t−τ(t))+B˜x(t−h(t))−L2x(t−τ(t))+G1f1(t,x(t))+G2f2(t,x(t−h(t)))−L1∫t−h(t)tη(s)ds−L2∫t−τ(t)tη(s)ds+G3∫t−r(t)tf3(θ,x(θ))dθ+G4f4(t,η(t−τ(t))). Define the Lyapunov-Krasovskii functional (3.6)V(t)=∑i=16Vi(t), where (3.7)V1(t)=x(t)TP1x(t):=ξ(t)TTPξ(t),V2(t)=∫t−τ(t)t[x(s)TQ1x(s)+η(s)TQ2η(s)]ds+∫t−h(t)tx(s)TR3x(s)ds+∑i=12∫t−hitx(s)TRix(s)ds,V3(t)=∫−h2−h1∫t+θtη(s)TR4η(s)dsdθ+∫−h20∫t+θtη(s)TR5η(s)dsdθ,V4(t)=∫−τ10∫t+θtη(s)TQ3η(s)dsdθ,V5(t)=∫−h20∫t+θtη(s)T[0L1]TH1[0L1]η(s)dsdθ+∫−τ10∫t+θtη(s)T[0L2]TH2[0L2]η(s)dsdθ,V6(t)=∫t−r(t)t[∫stf3(θ,x(θ))Tdθ]Q4[∫stf3(θ,x(θ))Tdθ]ds+∫0r1∫t−st(θ−t+s)f3(θ,x(θ))TQ4f3(θ,x(θ))dθds, with ξ(t):=col{x(t),η(t)}, T=diag{I,0}, and P=[P10P2P3], where P1=P1T>0.
On the other
hand, noting thatV(ϕ(t),t)≥λmin(P1)∥ϕ(0)∥2. According to [34], using the Cauchy-Schwarz
inequality and after
some manipulations, we obtain (3.8)V(ϕ(t),t)≤V(ϕ(0),0)≤ρ[∥ϕ(0)∥2+∫−κ0∥ϕ˙(θ)∥2dθ], where ρ:=max(ρ1,ρ2) with (3.9)ρ1:=λmax(P1)+2τ1λmax(Q1)+2h1λmax(R1)+2h2λmax(R2)+2h2λmax(R3)+3r12λmax(U3TQ4U3),ρ2:=2τ12λmax(Q1)+λmax(Q2)+2h22λmax(R3)+2h12λmax(R1)+2h22λmax(R2)+h2λmax(R5)+(h1+h2)λmax(R4)+τ1λmax(Q3)+h2λmax([0L1]TH1[0L1])+τ1λmax([0L2]TH2[0L2])+113r13λmax(U3TQ4U3).  Differentiating V1(t) along the system trajectory becomes (3.10)V˙1(t)=2x(t)TP1ξ˙(t)=2ξ(t)TPT[x˙(t)0]=2ξ(t)TPT{A¯ξ(t)+[0C]η(t−τ(t))+[0B˜]x(t−h(t))−[0L2]x(t−τ(t))+[0G1]f1(t,x(t))+[0G2]f2(t,x(t−h(t)))+G3∫t−r(t)tf3(θ,x(θ))dθ+G4f4(t,η(t−τ(t)))}+β1(t)+β2(t), where (3.11)A¯:=[0IA˜−I],β1(t)=−2∫t−h(t)tξ(t)TPT[0L1]η(s)ds,β2(t)=−2∫t−τ(t)tξ(t)TPT[0L2]η(s)ds.
Using Lemma2.2 for a(s)=col{0,Li}ξ(s) and b=Pcol{ξ(t),η(t)}, we obtain (3.12)β1(t)≤h2ξ(t)TPT(W1TH1+I)TH1−1(W1TH1+I)Pξ(t)+2ξ(t)TPTW1TH1[0L1](x(t)−x(t−h(t)))+∫t−h2tη(s)T[0L1]TH1[0L1]η(s)ds,β2(t)≤τ1ξ(t)TPT(W2TH2+I)TH2−1(W2TH2+I)Pξ(t)+2ξ(t)TPTW2TH2[0L2](x(t)−x(t−τ(t)))+∫t−τ1tη(s)T[0L2]TH2[0L2]η(s)ds.   Differentiating the second Lyapunov term in (3.6) gives (3.13)V˙2(t)=x(t)T(Q1+∑i=13Ri)x(t)+η(t)TQ2η(t)−(1−h˙(t))xT(t−h(t))R3x(t−h(t))−(1−τ˙(t))xT(t−τ(t))Q1x(t−τ(t))−(1−τ˙(t))ηT(t−τ(t))Q2η(t−τ(t))−∑i=12x(t−hi)TRix(t−hi)≤x(t)T(Q1+∑i=13Ri)x(t)+η(t)TQ2η(t)−(1−h3)xT(t−h(t))R3x(t−h(t))−(1−τ2)xT(t−τ(t))Q1x(t−τ(t))−(1−τ2)ηT(t−τ(t))Q2η(t−τ(t))−∑i=12x(t−hi)TRix(t−hi), and the time
derivative of the third term of V(t) in (3.6) is (3.14)V˙3(t)=η(t)T(h12R4+h2R5)η(t)−∫t−h2t−h1η(s)TR4η(s)ds−∫t−h2tη(s)TR5η(s)ds≤η(t)T(h12R4+h2R5)η(t)−∫t−h2t−h(t)η(s)TR4η(s)ds−∫t−h(t)t−h1η(s)T(R4+R5)η(s)ds, and, similarly, (3.15)V˙4(t)=τ1η(t)TQ3η(t)−∫t−τ1tη(s)TQ3η(s)ds≤τ1η(t)TQ3η(t)−∫t−τ(t)tη(s)TQ3η(s)ds, and also the
time derivative of the fifth and sixth terms of V(t) in (3.6) are, respectively, (3.16)V˙5(t)=η(t)T(h2[0L1]TH1[0L1]+τ1[0L2]TH2[0L2])η(t)−∫t−h2tη(s)T[0L1]TH1[0L1]η(s)ds−∫t−τ1tη(s)T[0L2]TH2[0L2]η(s)ds,(3.17)V˙6(t)=−(1−r˙(t))[∫t−r(t)tf3(θ,x(θ))Tdθ]Q4[∫t−r(t)tf3(θ,x(θ))dθ]+2∫t−r(t)tf3(t,x(t))TQ4[∫stf3(θ,x(θ))dθ]ds+∫0r1sf3(t,x(t))TQ4f3(t,x(t))ds−∫0r1∫t−stf3(θ,x(θ))TQ4f3(θ,x(θ))dθds≤∫t−r(t)t(θ−t+r(t))[f3(t,x(t))TQ4f3(t,x(t))+f3(θ,x(θ))TQ4f3(θ,x(θ))]dθ+∫0r1sf3(t,x(t))TQ4f3(t,x(t))ds−(1−r2)[∫t−r(t)tf3(θ,x(θ))Tdθ]Q4[∫t−r(t)tf3(θ,x(θ))dθ]−∫t−r1t(θ−t+r1)f3(θ,x(θ))TQ4f3(θ,x(θ))dθ=r12f3(t,x(t))TQ4f3(t,x(t))−(1−r2)[∫t−r(t)tf3(θ,x(θ))Tdθ]Q4[∫t−r(t)tf3(θ,x(θ))dθ]. For nonlinear functions fi(⋅), we have (3.18)0≤−f1(t,x(t))Tf1(t,x(t))+x(t)TU1TU1x(t),0≤−f2(t,x(t−h(t)))Tf2(t,x(t−h(t)))+x(t−h(t))TU2TU2x(t−h(t)),0≤−f3(t,x(t))Tf3(t,x(t))+x(t)TU3TU3x(t),0=−f4(t,η(t−τ(t)))Tf4(t,η(t−τ(t)))+η(t−τ(t))TU4TU4η(t−τ(t)).     Moreover, from the Leibniz-Newton formula and the system (2.1), the following equations
hold for any matrices {Ni}i=110 with appropriate dimensions: (3.19)2ϑT(t)χ1(x(t)−x(t−h(t))−∫t−h(t)tη(s)ds)=0,2ϑT(t)χ2(x(t−h1)−x(t−h(t))−∫t−h(t)t−h1η(s)ds)=0,2ϑT(t)χ3(x(t−h(t))−x(t−h2)−∫t−h2t−h(t)η(s)ds)=0,2ϑT(t)χ4(x(t)−x(t−τ(t))−∫t−τ(t)tη(s)ds)=0,2ϑT(t)χ5(η(t)−Cη(t−τ(t))−Ax(t)−Bx(t−h(t))−G1f1(t,x(t))−G2f2(t,x(t−h(t)))−G3∫t−r(t)tf3(θ,x(θ))dθ−G4f4(t,η(t−τ(t))))=0,     where (3.20)χ1:=[N1T,0,N2T,0,…,0︸6elements,N3T,0,N4T]T,χ2:=[N5T,0,N6T,0,…,0︸6elements,N7T,0,N8T]Tχ3:=[N9T,0,N10T,0,…,0︸6elements,N11T,0,N12T]T,χ4:=[N13T,0,…,0︸4elements,N14T,0,…,0︸3elements,N15T,0,N16T]T,χ5:=[0,…,0︸6elements,N17T,0,0,N18T,N19T,N20T]T,ϑ(t):=col{x(t),η(t),x(t−h(t)),x(t−h1),x(t−h2),x(t−τ(t)),η(t−τ(t)),f1(t,x(t)),f2(t,x(t−h(t))),f3(t,x(t)),∫t−r(t)tf3(θ,x(θ))dθ,f4(t,η(t−τ(t)))}.       Using the
obtained derivative terms (3.10)–(3.17) and adding the right- and the left-hand
sides of (3.18) and (3.19) into V˙(t), the following result is obtained: (3.21)V˙(t)=∑i=16V˙i(t)≤ϑ(t)TΣϑ(t)−∫t−h(t)t−h1(ϑT(t)χ1+ηT(s)R4)R4−1(ϑT(t)χ1+ηT(s)R4)Tds−∫t−h(t)t−h1(ϑT(t)χ2+ηT(s)R5)R5−1(ϑT(t)χ2+ηT(s)R5)Tds−∫t−h(t)t−h1(ϑT(t)χ3+ηT(s)R4)R4−1(ϑT(t)χ3+ηT(s)R5)Tds−∫t−τ(t)t(ϑT(t)χ4+ηT(s)Q3)Q3−1(ϑT(t)χ4+ηT(s)Q3)Tds, where Σ:=Π^+h12χ1R4−1χ1T+h12χ2R5−1χ2T+h12χ3R4−1χ3T+τ1χ4Q3−1χ4T, and the matrix Π^ is given by (3.22)Π^=[Π^11Π^12Π13Π^14Π^15Π^16Π17Π^18Π^19∗Π22Π230−BTN17T0Π27−BTN19TΠ29∗∗Π33000Π370Π39∗∗∗Π4400−N15T0−N16T∗∗∗∗Π55Π56−CTN18T−N17G3−CTN20T∗∗∗∗∗Π66Π67Π68Π69∗∗∗∗∗∗Π77−N18G3−N18G4∗∗∗∗∗∗∗Π88−G3TN20T∗∗∗∗∗∗∗∗−sym(N20G4)] with (3.23)Π^11=sym(PT(A¯+(W1TH1[0L1]+W2TH2[0L2])[I0]))+PT[h2(W1TH1+I)TH1−1(W1TH1+I)+τ1(W2TH2+I)TH2−1(W2TH2+I)]P+[0I](h2[0L1]TH1[0L1]+τ1[0L2]TH2[0L2])[0I]T+Ω,Π^12=PT[0B˜]−PTW1TH1[0L1]+[N2T−N1−N5+N90],Π^14=−PT[0L2]−PTW2TH2[0L2]+[N14T−N130],Π^15=PT[0C]+[−ATN17TN17T],Π^16=PT[00G1G2],Π^18=PT[0G3]+[−ATN19TN19T],Π^19=PT[0G4]+[N4T+N16T−ATN20TN20T]. Now, if Σ<0 holds, then V˙(t)<0 which means that the neutral system (2.1) is asymptotically stable. By applying the Schur
complement, the matrix inequity Σ<0 results in (3.24)[Π˜11Π^12Π13Π^14Π^15Π^16Π17Π^18Π^19Π1,10Π^1,11∗Π22Π230−BTN17T0Π27−BTN19TΠ29Π2,100∗∗Π33000Π370Π3900∗∗∗Π4400−N15T0−N16TΠ4,100∗∗∗∗Π55Π56−CTN18T−N17G3−CTN20T00∗∗∗∗∗Π66Π67Π68Π^6900∗∗∗∗∗∗Π77−N18G3−N18G4Π7,100∗∗∗∗∗∗∗Π88−G3TN20T00∗∗∗∗∗∗∗∗−sym(N20G4)Π9,100∗∗∗∗∗∗∗∗∗Π10,100∗∗∗∗∗∗∗∗∗∗Π^11,11]<0 with (3.25)Π˜11=sym(PT(A¯+(W1TH1[0L1]+W2TH2[0L2])[I0]))+Ω,Π^1,11=[h2PT(W1TH1+I)TH¯1τ1PT(W2TH2+I)TH¯2h2[0I][0L1]Tτ1[0I][0L2]T],Π^11,11=diag{−h2H¯1,−τ1H¯2,−h2H¯1,−τ1H¯2}, where H¯i=Hi−1(i=1,2).
Following [34, 35], we choose P3=δP2, δ∈R, where δ is a tuning scalar parameter (which may be
restrictive). Let (3.26)ζ=diag{I,…,I︸16elements,PT,…,PT︸4elements}.Premultiplying ζ and postmultiplying ζT to the matrix inequality (3.24) and considering Yi:=P2TLi, H˜i:=PTH¯iP, and HiWi=αiI (i=1,2) to eliminate the nonlinearities in the matrix inequality, we obtain the LMI
(3.3). Moreover, from (2.1) and the fact that x(t) is square integrable on [0,∞), it follows that Dηt∈L2n[0,∞). Under (A1), the later implies that η(t−τ(t))∈L2n[0,∞). Therefore, by [1, Theorem 1.6], we conclude that the neutral
system (2.1) is asymptotically stable.Remark 3.2.
The
results given in Theorem3.1 are derived for system (2.1) with time-varying delays h(t), τ(t), and r(t) satisfying (2.2a), (2.2b), and (2.2c), where the derivatives
of the time-varying delays are available. However, in many situations, the
information on the derivative of time-varying delays is unknown a prior. In such circumstances, the corresponding delay-rate-independent stability
analysis results for time-delays only satisfying (3.27)0<h1≤h(t)≤h2<∞,0<τ(t)≤τ1<∞,0<r(t)≤r1<∞, can be easily obtained by
setting Q1=Q2=Q4=R3=0 in Theorem 3.1.Remark 3.3.
The reduced conservatism of Theorem3.1
benefits from the construction of the new Lyapunov-Krasovskii
functional in (3.6), utilizing Leibniz-Newton formula, using a free-weighting matrix technique, and no
bounding technique is needed to estimate the inner product of the involved
crossing terms (see, e.g., [12, 20]). It can be easily seen that
results of this paper are quite different from most existing results in the recent
literature in the following
perspectives.
(a) Theoretically stability analysis of neutral
systems with different time-varying neutral, discrete, and distributed delays is
much more complicated, especially, for the case where the delays are
time-varying and different.
(b) In this paper, the derived sufficient
conditions are convex,  neutral-delay-dependent, discrete-delay-range-dependent, and
distributed-delay-dependent, which
make the treatment in the present paper more general with less conservative in
compare to most existing results in the literature which are independent of the
neutral or distributed delays; see for instance [21, 22, 38].


## 4. Uncertainty
Characterization

In this section, we will discuss the uncertainty
characterization for the linear neutral system (2.1) with different time-varying
neutral, discrete, and distributed delays and nonlinear parameter perturbations.


### 4.1. Polytopic Uncertainty

The first
class of uncertainty frequently encountered in practice is the polytopic
uncertainty [2]. In this case, the matrices of the system (2.1) are not exactly
known, except that they are within a compact set Ω denoting (4.1)Ω=[CABG1G2G3]. We assume that (4.2)Ω=∑j=1NsjΩj for some
scalars sj satisfying (4.3)0≤sj≤1,∑j=1Nsj=1, where the N vertices of the polytope are described by (4.4)Ωj=[C(j)A(j)B(j)G1(j)G2(j)G3(j)]. In order to take into account the polytopic
uncertainty in the system (2.1), we derive the following result from applying the
same transformation that was used in deriving Theorem 3.1.Theorem 4.1.
Under
(A1), for given scalarsγ,h1,h2,τ1,r1>0,h3,τ2,r2, if the uncertainty set Ω is polytopic with vertices Ωj, j=1,2,…,N, then the system described by (2.1)), (2.2a), (2.2b), (2.2c), and (4.2)–(4.4) is asymptotically stable if
there exist some scalars δ,α1,α2, matrices P2,{Ni}i=120,Y1,Y2, and positive-definite matrices P1,{Qi}i=14,{Ri}i=14,H˜1,H˜2 such that LMI (3.3) is satisfied for all (4.5)[CABG1G2G3]=[C(j)A(j)B(j)G1(j)G2(j)G3(j)],j=1,2,…,N.Proof.
It follows
directly from the proof of Theorem3.1 and using properties of (4.2)–(4.4).


### 4.2. Norm-Bounded Uncertainty

There are also other uncertainties that cannot
be reasonably modeled by a polytopic uncertainty set with a number of
vertices. In such a case, it is assumed that the deviation of the system
parameters of an uncertain system from their nominal values is norm bounded [2]. In our case, consider the time-varying structured uncertain neutral system (4.6)x˙(t)−(C+ΔC(t))x˙(t−τ(t))=(A+ΔA(t))x(t)+(B+ΔB(t))x(t−h(t))+(G1+ΔG1(t))f1(t,x(t))+(G2+ΔG2(t))f2(t,x(t−h(t)))+(G3+ΔG3(t))∫t−r(t)tf3(θ,x(θ))dθ+(G4+ΔG4(t))f4(t,x˙(t−τ(t))),x(t)=ϕ(t),t∈[−κ,0], where the time-varying
structured uncertainties are said to be admissible if the following form holds: (4.7)[ΔC(t)ΔA(t)ΔB(t)ΔG1(t)ΔG2(t)ΔG3(t)ΔG4(t)]=M1Δ(t)⌊LcLaLbLg1Lg2Lg3Lg4⌋, where Lc,La,Lb,Lg1,Lg2,Lg3,Lg4 are constant matrices with
appropriate dimensions, Δ(t) is an unknown, real, and
possibly time-varying matrix with Lebesgue measurable elements, and its
Euclidean norm satisfies (4.8)∥Δ(t)∥≤1,∀t. In this
section, we modify (A1)-(A2) in order to enable the application of
Lyapunov’s method for the stability of the time-varying structured
uncertain neutral system (4.6) as follows:(A′1)
let the difference
operatorD:C([−κ,0],ℜn)→ℜn given by Dxt=x(t)−(C+ΔC(t))x(t−τ(t)) be delay-independently stable with respect to
all delays. A sufficient condition for (A′1) is that(A′2)
all the eigenvalues of the matrixC+ΔC(t) are inside the unit circle, that is, ∥C+ΔC(t)∥<1.Theorem 4.2.
Under(A′1), for given scalars γ,h1,h2,τ1,r1>0,h3,τ2,r2, the neutral system (4.6) with admissible uncertainties (4.7)) and (4.8) is
robustly asymptotically stable if there
exist some scalars {λi}i=14>0, δ,α1,α2, matrices P2,{Ni}i=120,Y1,Y2, and positive-definite matrices P1,{Qi}i=14,{Ri}i=14,H˜1,H˜2, such that the following LMI is feasible: (4.9)[ΠΨ1Ψ2∗−Ψ30∗∗−Ψ3]<0, where Ψ1=[Γd1Γd2Γd3Γd4], Ψ2=[λ1Γe1Tλ2Γe2Tλ3Γe3Tλ4Γe4T], and Ψ3=diag{λ1I,λ2I,λ3I,λ4I} with (4.2)Γd1=[M1TP2δM1TP20⋯0︸18elements]T,Γe1=[La0Lb0⋯0︸3elementsLcLg1Lg20Lg3Lg40⋯0︸8elements],Γd2=[0⋯0︸6elementsM1TN17T0⋯0︸13elements]T,Γe2=[−La0−Lb0⋯0︸3elements−Lc0⋯0︸13elements],Γd3=[0⋯0︸7elements−M1TN17T0⋯0︸12elements]T,Γe3=[0⋯0︸7elementsLg1Lg20Lg3Lg40⋯0︸8elements],Γd4=[0⋯0︸9elementsM1TN19T0⋯0︸10elements]T,Γe4=[00−Lb0⋯0︸4elements−Lg1−Lg20−Lg3−Lg40⋯0︸8elements].Proof.
If the matricesC,A,B,G1,G2,G3,G4 in (3.3) are replaced with C+M1Δ(t)Lc, A+M1Δ(t)La, B+M1Δ(t)Lb, G1+M1Δ(t)Lg1, G2+M1Δ(t)Lg2, G3+M1Δ(t)Lg3, and G4+M1Δ(t)Lg4, respectively, then (3.3) with the admissible uncertainties (4.7) is equivalent to
the following condition: (4.11)Π+∑i=14sym(ΓdiTΔ(t)Γei)<0. By Lemma 2.3, a necessary and sufficient condition for (4.11) is that there
exist some {λi}i=14>0 such that (4.12)Π+∑i=14[λi−1ΓdiTΓdi+λiΓeiTΓei]<0. Applying
Schur complements, we find that (4.12) is equivalent to (4.9).Remark 4.3.
It
is noted that
our approach is
different from that in the reference [20] in several perspectives.
(a) The
system structure in [20] considers a system with different time-varying neutral
and discrete delays and in compare to our case do not center on time-varying distributed delays, that is, the
results in [20] cannot
be directly applied to the systems with different time-varying neutral, discrete, and distributed delays.
(b) Their system only considers the case that
the range of the time-varying delay h(t) is from 0 to an upper bound in compare to our
case that the time-varying discrete delay (h(t)) lies in a range, in which the lower bound is not 0.
(c) The derived neutral-delay-range-independent
conditions and using the inequality bounding technique [11, Lemma 1] employed
for some cross terms encountered in their analysis conditions may produce
conservative results in comparison with the present paper.


## 4.1. Polytopic Uncertainty

The first
class of uncertainty frequently encountered in practice is the polytopic
uncertainty [2]. In this case, the matrices of the system (2.1) are not exactly
known, except that they are within a compact set Ω denoting (4.1)Ω=[CABG1G2G3]. We assume that (4.2)Ω=∑j=1NsjΩj for some
scalars sj satisfying (4.3)0≤sj≤1,∑j=1Nsj=1, where the N vertices of the polytope are described by (4.4)Ωj=[C(j)A(j)B(j)G1(j)G2(j)G3(j)]. In order to take into account the polytopic
uncertainty in the system (2.1), we derive the following result from applying the
same transformation that was used in deriving Theorem 3.1.Theorem 4.1.
Under
(A1), for given scalarsγ,h1,h2,τ1,r1>0,h3,τ2,r2, if the uncertainty set Ω is polytopic with vertices Ωj, j=1,2,…,N, then the system described by (2.1)), (2.2a), (2.2b), (2.2c), and (4.2)–(4.4) is asymptotically stable if
there exist some scalars δ,α1,α2, matrices P2,{Ni}i=120,Y1,Y2, and positive-definite matrices P1,{Qi}i=14,{Ri}i=14,H˜1,H˜2 such that LMI (3.3) is satisfied for all (4.5)[CABG1G2G3]=[C(j)A(j)B(j)G1(j)G2(j)G3(j)],j=1,2,…,N.Proof.
It follows
directly from the proof of Theorem3.1 and using properties of (4.2)–(4.4).


## 4.2. Norm-Bounded Uncertainty

There are also other uncertainties that cannot
be reasonably modeled by a polytopic uncertainty set with a number of
vertices. In such a case, it is assumed that the deviation of the system
parameters of an uncertain system from their nominal values is norm bounded [2]. In our case, consider the time-varying structured uncertain neutral system (4.6)x˙(t)−(C+ΔC(t))x˙(t−τ(t))=(A+ΔA(t))x(t)+(B+ΔB(t))x(t−h(t))+(G1+ΔG1(t))f1(t,x(t))+(G2+ΔG2(t))f2(t,x(t−h(t)))+(G3+ΔG3(t))∫t−r(t)tf3(θ,x(θ))dθ+(G4+ΔG4(t))f4(t,x˙(t−τ(t))),x(t)=ϕ(t),t∈[−κ,0], where the time-varying
structured uncertainties are said to be admissible if the following form holds: (4.7)[ΔC(t)ΔA(t)ΔB(t)ΔG1(t)ΔG2(t)ΔG3(t)ΔG4(t)]=M1Δ(t)⌊LcLaLbLg1Lg2Lg3Lg4⌋, where Lc,La,Lb,Lg1,Lg2,Lg3,Lg4 are constant matrices with
appropriate dimensions, Δ(t) is an unknown, real, and
possibly time-varying matrix with Lebesgue measurable elements, and its
Euclidean norm satisfies (4.8)∥Δ(t)∥≤1,∀t. In this
section, we modify (A1)-(A2) in order to enable the application of
Lyapunov’s method for the stability of the time-varying structured
uncertain neutral system (4.6) as follows:(A′1)
let the difference
operatorD:C([−κ,0],ℜn)→ℜn given by Dxt=x(t)−(C+ΔC(t))x(t−τ(t)) be delay-independently stable with respect to
all delays. A sufficient condition for (A′1) is that(A′2)
all the eigenvalues of the matrixC+ΔC(t) are inside the unit circle, that is, ∥C+ΔC(t)∥<1.Theorem 4.2.
Under(A′1), for given scalars γ,h1,h2,τ1,r1>0,h3,τ2,r2, the neutral system (4.6) with admissible uncertainties (4.7)) and (4.8) is
robustly asymptotically stable if there
exist some scalars {λi}i=14>0, δ,α1,α2, matrices P2,{Ni}i=120,Y1,Y2, and positive-definite matrices P1,{Qi}i=14,{Ri}i=14,H˜1,H˜2, such that the following LMI is feasible: (4.9)[ΠΨ1Ψ2∗−Ψ30∗∗−Ψ3]<0, where Ψ1=[Γd1Γd2Γd3Γd4], Ψ2=[λ1Γe1Tλ2Γe2Tλ3Γe3Tλ4Γe4T], and Ψ3=diag{λ1I,λ2I,λ3I,λ4I} with (4.2)Γd1=[M1TP2δM1TP20⋯0︸18elements]T,Γe1=[La0Lb0⋯0︸3elementsLcLg1Lg20Lg3Lg40⋯0︸8elements],Γd2=[0⋯0︸6elementsM1TN17T0⋯0︸13elements]T,Γe2=[−La0−Lb0⋯0︸3elements−Lc0⋯0︸13elements],Γd3=[0⋯0︸7elements−M1TN17T0⋯0︸12elements]T,Γe3=[0⋯0︸7elementsLg1Lg20Lg3Lg40⋯0︸8elements],Γd4=[0⋯0︸9elementsM1TN19T0⋯0︸10elements]T,Γe4=[00−Lb0⋯0︸4elements−Lg1−Lg20−Lg3−Lg40⋯0︸8elements].Proof.
If the matricesC,A,B,G1,G2,G3,G4 in (3.3) are replaced with C+M1Δ(t)Lc, A+M1Δ(t)La, B+M1Δ(t)Lb, G1+M1Δ(t)Lg1, G2+M1Δ(t)Lg2, G3+M1Δ(t)Lg3, and G4+M1Δ(t)Lg4, respectively, then (3.3) with the admissible uncertainties (4.7) is equivalent to
the following condition: (4.11)Π+∑i=14sym(ΓdiTΔ(t)Γei)<0. By Lemma 2.3, a necessary and sufficient condition for (4.11) is that there
exist some {λi}i=14>0 such that (4.12)Π+∑i=14[λi−1ΓdiTΓdi+λiΓeiTΓei]<0. Applying
Schur complements, we find that (4.12) is equivalent to (4.9).Remark 4.3.
It
is noted that
our approach is
different from that in the reference [20] in several perspectives.
(a) The
system structure in [20] considers a system with different time-varying neutral
and discrete delays and in compare to our case do not center on time-varying distributed delays, that is, the
results in [20] cannot
be directly applied to the systems with different time-varying neutral, discrete, and distributed delays.
(b) Their system only considers the case that
the range of the time-varying delay h(t) is from 0 to an upper bound in compare to our
case that the time-varying discrete delay (h(t)) lies in a range, in which the lower bound is not 0.
(c) The derived neutral-delay-range-independent
conditions and using the inequality bounding technique [11, Lemma 1] employed
for some cross terms encountered in their analysis conditions may produce
conservative results in comparison with the present paper.


## 5. Numerical
Results

In this section, two examples are provided to
illustrate the effectiveness of the results obtained in the previous sections.Example 5.1.
Consider the neutral system (2.1) with
the following matrices adopted from [9]: (5.1)A=[−1.20.1−0.1−1],B=[−0.60.7−1−0.8],G1=G2=I,G3=0, and the nonlinear parameter perturbations ∥f1(t,x(t))∥≤η1∥x(t)∥ and ∥f2(t,x(t−h(t)))∥≤η2∥x(t−h(t))∥, where η1,η2≥0.Case 1.
Assume thatC=0, h1=0, and G4=0. Applying the criteria in [10, 18, 21, 22] and Theorem 3.1 in this paper, the maximum value of h2 for stability of system under consideration is
listed in Table 1. Furthermore, in the case of h3=0.5 and η1=η2=0.1 by the criteria in [20], the nominal system is
asymptotically stable for any h(t) satisfying h2=1.0097. It is easy to see that the stability criterion in this paper gives a much less
conservative result than that in [10, 20–22], and also the maximum
value of h2 decreases as ηi increases. Moreover, unlike the results obtained in [10, 18, 21, 22] our
approach can also
consider the case h3≥1 and handle fast time-varying delays
completely.Table 1
Comparative results forh2.
η1=0,η2=0.1η1=0.1,η2=0.1h3=0h3=0.5h3=0h3=0.5Results of [10]0.68110.54670.61290.4950Results of [18]1.32790.67431.25030.5716Results of [21]2.7421.1421.8751.009Results of [22]3.7441.4712.4431.299Results of this paper3.82051.63502.71051.3580Case 2.
Assume thatC=diag{0.1,0.1}, G4=I, h1=0,h3=τ2=0.5,  and ∥f4(t,x˙(t−τ(t)))∥≤η3∥x˙(t−τ(t))∥. We
also calculate the maximum delay bounds h2 that guarantee the asymptotic stability
of the system under consideration by Theorem 3.1 in this paper and developed
methods in [9, 20] for different values of the parameter η3, as listed in Table 2. It can be seen from the table that h2 decreases as η3 increases, and the
stability condition in Theorem 3.1 of this paper is less conservative than that
in [9, 20].Table 2
Comparative results forh2.
η3=0.0η3=0.1η3=0.2η3=0.3Results of [9]0.74370.51310.31120.1398Results of [20]0.77490.56580.38590.2357Results of this paper0.84290.69030.45040.3015Example 5.2 (see [46]).
Consider the system (4.6) with the following matrices: (5.2)A=[−200−3],G1=[0.5−0.1−0.2−0.3],G2=[0.2−0.40.10.2],G3=[0.40.30.10.2],La=[000.20.2],Lg1=[000.10.1],Lg2=[000.30.3],Lg3=[000.20.2],B=C=G4=Lc=Lb=Lg4=0,M1=I,U1=U2=U3=I. Recently, Kwon and Park in [46]
derived the stability bound of h(t)=r(t) with h3=r2=1 as 0<h(t)≤1.97 with the parameters above for the system (4.6). However, by applying Theorem 4.2 to the system under consideration, one can see
that our criterion is feasible for 0<h(t)≤2.0105. And for the condition h3=1, for fixed h2 or r2, the upper bounds of delays h2 and r2 are shown in Tables 3 and 4, respectively. From Tables 3 and 4, it can be seen that Theorem 4.2 provides a condition
for guaranteeing stability with respect to given delay bounds h(t) and r(t).Table 3
Upper bounds of delaysh2 with respect to r2.
r20.10.30.50.70.9h21.60251.58751.52151.40501.1510Table 4
Upper bounds of delaysr2 with respect to  h2.
h20.10.30.50.70.9r21.25601.23581.21251.19501.1045


## 6. Conclusion

The problem of stability analysis has been presented in this paper for a class of neutral systems with different
time-varying neutral, discrete, and distributed delays and nonlinear parameter perturbations using an appropriate Lyapunov-Krasovskii
functional. By combining the descriptor model transformation, the
Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new feasibility conditions, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent, have been
developed to ensure that the considered
system is asymptotically stable. The conditions were presented in terms of linear matrix inequalities
(LMIs) and solved by existing convex
optimization techniques. Two numerical examples were given to demonstrate the
less conservatism of the proposed results over some existence results in the
literature.



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*Source: 759248-2009-03-29.xml*