The delay-range-dependent stochastic stability for uncertain neutral Markovian jump systems with interval time-varying delays is studied in this paper. The uncertainties under consideration are assumed to be time varying but norm bounded. To begin with the nominal systems, a novel augmented Lyapunov functional which contains some triple-integral terms is introduced. Then, by employing some integral inequalities and the nature of convex combination, some less conservative stochastic stability conditions are presented in terms of linear matrix inequalities without introducing any free-weighting matrices. Finally, numerical examples are provided to demonstrate the effectiveness and to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature.
1. Introduction
Time delays are frequently encountered in various engineering systems, such as chemical or process control systems, networked control systems, and manufacturing systems. To sum up, delays can appear in the state, input, or output variables (retarded systems), as well as in the state derivative (neutral systems). In fact, neutral delay systems constitute a more general class than those of the retarded type because such systems can be found in places such as population ecology [1], distributed neural networks [2], heat exchangers, and robots in contact with rigid environments [3]. Since it is shown that the existence of delays in a dynamic system may result in instability, oscillations, or poor performances [3–5], the stability of time-delay systems has been an important problem of recurring interest for many years. Existing results on this topic can be roughly classified into two categories, namely, delay-independent criteria [6] and delay-dependent criteria, and it is generally recognized that the latter cases are less conservative. Actually, the stability of neutral time-delay systems proves to be a more complex issue as well as singular systems [7–9] because the systems involve the derivative of the delayed state. So considerable attention has been devoted to the problem of robust delay-independent stability or delay-dependent stability and stabilization via different approaches for linear neutral systems with delayed state input and parameter uncertainties. Results are mainly presented based on Lyapunov-Krasovskii (L-K) method; see, for example, [10–16] and the references therein. However, there is room for further investigation because the conservativeness of the neutral systems can be further reduced by a better technique.
On the other hand, with the development of science and technology, many practical dynamics, for example, solar thermal central receivers, robotic manipulator systems, aircraft control systems, economic systems, and so on, experience abrupt changes in their structures, whose parameters are caused by phenomena such as component failures or repairs, changes in subsystem interconnections, and sudden environmental changes. This class of systems is more appropriate to be described as Markovian jump systems (MJSs), which can be regarded as a special class of hybrid systems with finite operation modes. The system parameters jump among finite modes, and the mode switching is governed by a Markov process to represent the abrupt variation in their structures and parameters. With so many applications in engineering systems, a great deal of attention has been paid to the stability analysis and controller synthesis for Markovian jump systems (MJSs) in recent years. Many researchers have made a lot of progress on Markovian jump delay systems and Markovian jump control theory; see, for example, [17–23] and references therein for more details. However, a few of these papers have considered the effect of delay on the stability or stabilization for the corresponding neutral systems. Besides, to the best of the authors’ knowledge, it seems that the problem of stochastic stability for neutral Markovian jumping systems with interval time-varying delays has not been fully investigated and it is very challenging. Motivated by the previous description, this paper investigates the stochastic stability of neutral Markovian jumping systems with interval time-varying delays to seek less conservative stochastic stability conditions than some previous ones.
In order to simplify the treatment of the problem, in this paper, we first investigate the nominal systems and construct a new augmented Lyapunov functional containing some triple-integral terms to reduce conservativeness. By some integral inequalities and the nature of convex combination, the delay-range-dependent stochastic stability conditions are derived for the nominal neutral systems with Markovian jump parameters and interval time-varying delays. Then, the results are extended to the corresponding uncertain case on the basis of obtained conditions. In addition, these conditions are expressed in linear matrix inequalities (LMIs), which can be easily checked by utilizing the LMI Toolbox in MATLAB. Numerical examples are given to show the effectiveness and reduced conservativeness over some previous references.
The main contributions of this paper can be summarized as follows: (1) the proposed Lyapunov functional contains some triple-integral terms which is very effective in the reduction of conservativeness, and has not been used in any of the existing literatures in the same context before; (2) the delay-range-dependent stability conditions are obtained in terms of LMIs without introducing any free-weighting matrices besides the Lyapunov matrices, which will reduce the number of variables and decrease the complexity of computation; (3) the proposed results are expressed in a new representation and proved to be less conservative than some existing ones.
The remainder of this paper is organized as follows: Section 2 contains the problem statement and preliminaries; Section 3 presents the main results; Section 4 provides a numerical example to verify the effectiveness of the results; Section 5 draws a brief conclusion.
1.1. Notations
In this paper, ℝn denotes the n dimensional Euclidean space and ℝm×n is for the set of all m×n matrices. The notation X<Y (X>Y), where X and Y are both symmetric matrices, means that X-Y is negative (positive) definite. I denotes the identity matrix with proper dimensions. λmax(min)(A) is the eigenvalue of matrix A with maximum (minimum) real part. For a symmetric block matrix, we use the sign * to denote the terms introduced by symmetry. ℰ stands for the mathematical expectation, and ∥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. C([-ρ,0],ℝn) is the space of continuous function from [-ρ,0] to ℝn. In addition, if not explicitly stated, matrices are assumed to have compatible dimensions.
2. Problem Statement and Preliminaries
Given a probability space {Ω,ℱ,P} where Ω is the sample space, ℱ is the algebra of events and P is the probability measure defined on ℱ. {rt,t≥0} is a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in a finite set S={1,2,3,…,N}, with the mode transition probability matrix
(1)P(rt+Δt=j∣rt=i)={πijΔt+o(Δt)i≠j,1+πiiΔt+o(Δt)i=j,
where Δt>0, limΔt→0(o(Δt)/Δt)=0, and πij≥0(i,j∈S,i≠j) denote the transition rate from mode i to j. For any state or mode i∈S, we have
(2)πii=-∑j=1,j≠iNπij.
In this paper, we consider the following uncertain neutral systems with Markovian jump parameters and time-varying delay over the space {Ω,ℱ,P} as follows:
(3)x˙(t)-C(rt)x˙(t-τ)=[A(rt)+ΔA(rt)]x(t)+[B(rt)+ΔB(rt)]x(t-d(t)),(4)x(s)=φ(s),rs=r0,s∈[-ρ,0],
where x(t)∈ℝn is the system state and τ>0 is a constant neutral delay. It is assumed that the time-varying delay d(t) satisfies
(5)0<d1≤d(t)≤d2,d˙(t)≤μ,
where d1<d2 and μ≥0 are constant real values. The initial condition φ(s) is a continuously differentiable vector-valued function. The continuous norm of φ(s) is defined as
(6)∥φ∥c=maxs∈[-ρ,0]|φ(s)|,ρ=max{τ,d2};A(rt)∈ℝn×n, B(rt)∈ℝn×n, and C(rt)∈ℝn×n are known mode-dependent constant matrices, while ΔA(rt)∈ℝn×n and ΔB(rt)∈ℝn×n are uncertainties. For notational simplicity, when rt=i∈S, A(rt), ΔA(rt), B(rt), ΔB(rt), and C(rt) are, respectively, denoted as Ai, ΔAi, Bi, ΔBi, and Ci. Throughout this paper, the parametric matrix ∥Ci∥<1 and the admissible parametric uncertainties are assumed to satisfy the following condition:
(7)[ΔAi(t)ΔBi(t)]=HiFi(t)[EAiEBi],
where Hi, EAi, and EBi are known mode-dependent constant matrices with appropriate dimensions and Fi(t) is an unknown and time-varying matrix satisfying
(8)FiT(t)Fi(t)≤I,∀t.
Particularly, when we consider Fi(t)=0, we get the nominal systems which can be described as
(9)x˙(t)-Cix˙(t-τ)=Aix(t)+Bix(t-d(t)).
Before proceeding further, the following assumptions, definitions, and lemmas need to be introduced.
Assumption 1.
The system matrix Ai(foralli∈S) is Hurwitz matrix with all the eigenvalues having negative real parts for each mode. The matrix Hi(foralli∈S) is chosen as a full row rank matrix.
Assumption 2.
The Markov process is irreducible, and the system mode rt is available at time t.
With regard to neutral systems, the operator 𝔇:C([-ρ,0],ℝn)→ℝn is defined to be
(10)𝔇(xt)=x(t)-Cx(t-τ).
Then, the stability of operator 𝔇 is defined as follows.
Definition 3 (see [<xref ref-type="bibr" rid="B13">4</xref>]).
The operator 𝔇 is said to be stable if the homogeneous difference equation
(11)𝔇(xt)=0,t≥0,x0=ψ∈{ϕ∈C([-ρ,0],ℝn):𝔇ϕ=0}
is uniformly asymptotically stable. In order to guarantee the stability of the operator 𝔇, one has assumed that ∥Ci∥<1 as previosuly mentioned, which was introduced in [24].
Definition 4 (see [<xref ref-type="bibr" rid="B40">25</xref>]).
The systems which are described in (3) are said to be stochastically stable if there exists a positive constant Υ such that
(12)ℰ{∫0∞∥x(rt,t)∥2dt∣φ(s),s∈[-ρ,0],r0}<Υ.
Definition 5 (see [<xref ref-type="bibr" rid="B37">26</xref>]).
In the Euclidean space {ℝn×S×R+}, where x(t)∈ℝn, rt∈S, and t∈R+, one introduces the stochastic Lyapunov-Krasovskii function of system (3) as V(x(t),rt=i,t>0)=V(xt,i,t), the infinitesimal generator satisfying
(13)𝔏V(x(t),i,t)=limΔt→01Δt[ℰ{V(x(t+Δt),rt+Δt,t+Δt)∣x(t)=x,rt=i}ppppppppp-V(x(t),i,t)ℰ{V(x(t+Δt),rt+Δt,t+Δt)∣x(t)=x,rt=i}]=∂∂tV(x(t),i,t)+∂∂xV(x(t),i,t)x˙(t)+∑j=1NπijV(x(t),j,t).
Lemma 6 (see [<xref ref-type="bibr" rid="B11">27</xref>, <xref ref-type="bibr" rid="B15">28</xref>]).
For any constant matrix H=HT>0 and scalars τ2>τ1>0 such that the following integrations are well defined, then
(14)(a)-(τ2-τ1)∫t-τ2t-τ1xT(s)Hx(s)ds≤-[∫t-τ2t-τ1xT(s)ds]H[∫t-τ2t-τ1x(s)ds],(b)-12(τ22-τ12)∫-τ2-τ1∫t+θtxT(s)Hx(s)dsdθ≤-[∫-τ2-τ1∫t+θtxT(s)dsdθ]H[∫-τ2-τ1∫t+θtx(s)dsdθ].
Lemma 7 (see [<xref ref-type="bibr" rid="B14">19</xref>]).
Suppose that 0≤τm≤τ(t)≤τM, Ξ1, Ξ2, and Ω are constant matrices of appropriate dimensions, then
(15)(τ(t)-τm)Ξ1+(τM-τ(t))Ξ2+Ω<0
if and only if (τM-τm)Ξ1+Ω<0 and (τM-τm)Ξ2+Ω<0 hold.
Lemma 8 (see [<xref ref-type="bibr" rid="B17">29</xref>]).
For given matrices Q=QT, M, and N with appropriate dimensions,
(16)Q+MF(t)N+NTFT(t)MT<0
for all F(t) satisfying FT(t)F(t)≤I if and only if there exists a scalar δ>0 such that
(17)Q+δ-1MMT+δNNT<0.
Lemma 9 (see [<xref ref-type="bibr" rid="B9">30</xref>]).
Given constant matrices Ω1, Ω2, and Ω3, where Ω1=Ω1T and Ω2=Ω2T>0, then Ω1+Ω3TΩ2-1Ω3<0 if and only if
(18)[Ω1Ω3T*-Ω2]<0or[-Ω2Ω3T*Ω1]<0.
3. Main Results
In this section, we first consider the nominal systems described by (9) and extend to the uncertain case. The following theorems present sufficient conditions to guarantee the stochastic stability for the neutral systems with Markovian jump parameters and time-varying delays.
3.1. Stochastic Stability for the Nominal SystemsTheorem 10.
For the given finite set S of modes with transition rates matrix, scalars d1, d2, τ, and μ, the neutral systems with Markovian jump parameters and time-varying delays as described by (9) are stochastically stable if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0(i∈S), Qj>0(j=1,2,3,4,5), and Rk>0(k=1,2,…,11) such that the following linear matrix inequalities hold:
(19)Πi1+ΓTMΓ<0,Πi2+ΓTMΓ<0,
where
(20)Πi1=Πi0-2(e2-e4)R7(e2T-e4T)-(e3-e2)R7(e3T-e2T)-e8R9e8T-2e9R9e9T,Πi2=Πi0-(e2-e4)R7(e2T-e4T)-2(e3-e2)R7(e3T-e2T)-2e8R9e8T-e9R9e9T,
where
(21)Πi0=e1Ye1T-e2[(1-μ)R3]e2T+e1PiBie2T+e2BiTPie1T+e1PiCie11T+e11CiTPie1T+e3(R2+R3-R1)e3T-e4R2e4T+e5(R5-R4)e5T-e6R5e6T-e7R8e7T-e10Q1e10T-e11Q2e11T-e12Q3e12T-(e1-e10)Q4(e1T-e10T)-(e1-e3)R6(e1T-e3T)-(τe1-e12)Q5(τe1T-e12T)-(d1e1-e7)R10(d1e1T-e7T)-(d12e1-e8-e9)R11(d12e1T-e8T-e9T),
where ei{i=1,2,…,12} are block entry matrices; for instance,
(22)e2T=[0I0000000000],Y=AiTPi+PiAi+∑j=1Nπijpj+Q1+τ2Q3+R1+d12R8+d122R9,M=Q2+τ2Q4+τ44Q5+R4+d12R6+d122R7+d144R10+dm2R11,Γ=[AiBi00000000Ci0],d12=d2-d1,dm=12(d22-d12).
Proof.
Construct the novel Lyapunov functional as follows:
(23)V(x(t),i,t)=Vr(xt,i)+Vτ(xt,i)+Vd1(xt,i)+Vd2(xt,i)+Vd3(xt,i),
where
(24)Vr(xt,i)=xT(t)Pix(t),Vτ(xt,i)=∫t-τtxT(s)Q1x(s)ds+∫t-τtx˙T(s)Q2x˙(s)ds+∫-τ0∫t+θtxT(s)[τQ3]x(s)dsdθ+∫-τ0∫t+θtx˙T(s)[τQ4]x˙(s)dsdθ+∫-τ0∫θ0∫t+λtx˙T(s)[τ22Q5]x˙(s)dsdλdθ,Vd1(xt,i)=∫t-d1txT(s)R1x(s)ds+∫t-d2t-d1xT(s)R2x(s)ds+∫t-d(t)t-d1xT(s)R3x(s)ds+∫t-d1tx˙T(s)R4x˙(s)ds+∫t-d2t-d1x˙T(s)R5x˙(s)ds,Vd2(xt,i)=∫-d10∫t+θtx˙T(s)[d1R6]x˙(s)dsdθ+∫-d2-d1∫t+θtx˙T(s)[d12R7]x˙(s)dsdθ+∫-d10∫t+θtxT(s)[d1R8]x(s)dsdθ+∫-d2-d1∫t+θtxT(s)[d12R9]x(s)dsdθ,Vd3(xt,i)=∫-d10∫θ0∫t+λtx˙T(s)[d122R10]x˙(s)dsdλdθ+∫-d2-d1∫θ0∫t+λtx˙T(s)[dmR11]x˙(s)dsdλdθ.
From Definition 5, taking 𝔏 as its infinitesimal generator along the trajectory of system (9), then from (23) and (24) we get the following equalities and inequalities:
(25)𝔏V(x(t),i,t)=𝔏Vr(xt,i)+𝔏Vτ(xt,i)+𝔏Vd1(xt,i)+𝔏Vd2(xt,i)+𝔏Vd3(xt,i),(26)𝔏Vr(xt,i)=2[xT(t)AiT+xT(t-d(t))BiT+x˙T(t-τ)CiT]Pix(t)+∑j=1NπijxT(t)Pjx(t),(27)𝔏Vτ(xt,i)=xT(t)[Q1+τ2Q3]x(t)+x˙T(t)[Q2+τ2Q4+τ44Q5]x˙(t)-xT(t-τ)Q1x(t-τ)-x˙T(t-τ)Q2x˙(t-τ)-∫t-τtxT(s)[τQ3]x(s)ds-∫t-τtx˙T(s)[τQ4]x˙(s)ds-∫-τ0∫t+θtx˙(s)[τ22Q5]x˙(s)dsdθ,𝔏Vd1(xt,i)=xT(t)R1x(t)+x˙T(t)R4x˙(t)+xT(t-d1)[R2+R3-R1]x(t-d1)+x˙T(t-d1)[R5-R4]x˙(t-d1)-x˙T(t-d2)R5x˙(t-d2)-xT(t-d2)R2x(t-d2)-(1-d˙(t))xT(t-d(t))R3x(t-d(t)),𝔏Vd2(xt,i)=xT(t)[d12R8+d122R9]x(t)+x˙T(t)[d12R6+d122R7]x˙(t)-∫t-d1tx˙T(s)[d1R6]x˙(s)ds-∫t-d2t-d1x˙T(s)[d12R7]x˙(s)ds-∫t-d1txT(s)[d1R8]x(s)ds-∫t-d2t-d1xT(s)[d12R9]x(s)ds,𝔏Vd3(xt,i)=x˙T(t)[d144R10+dm2R11]x˙(t)-∫-d10∫t+θtx˙T(s)[d122R10]x˙(s)dsdθ-∫-d2-d1∫t+θtx˙T(s)[dmR11]x˙(s)dsdθ.
Let us define
(28)ξ(t)=col{∫t-d2t-d(t)x(t)x(t-d(t))x(t-d1)x(t-d2)x˙(t-d1)ppppppx˙(t-d2)∫t-d1tx(s)ds∫t-d(t)t-d1x(s)dspppppp∫t-d2t-d(t)x(s)dsx(t-τ)x˙(t-τ)∫t-τtx(s)ds}.
Applying (a) of Lemma 6, we obtain
(29)-∫t-τtxT(s)[τQ3]x(s)ds≤-[∫t-τtxT(s)ds]Q3[∫t-τtx(s)ds]=-ξT(t)e12Q3e12Tξ(t).
Following the same procedure, we also obtain the inequalities as follows:
(30)-∫t-τtx˙T(s)[τQ4]x˙(s)ds≤-ξT(t)(e1-e10)Q4(e1T-e10T)ξ(t),-∫t-d1tx˙T(s)[d1R6]x˙(s)ds≤-ξT(t)(e1-e3)R6(e1T-e3T)ξ(t),-∫t-d1txT(s)[d1R8]x(s)ds≤-[∫t-d1txT(s)ds]R8[∫t-d1tx(s)ds].
Applying (b) of Lemma 6, we have
(31)-∫-τ0∫t+θtx˙T(s)[τ22Q5]x˙(s)dsdθ≤-[∫-τ0∫t+θtx˙T(s)dsdθ]Q5[∫-τ0∫t+θtx˙(s)dsdθ]=-[τxT(t)-∫t-τtxT(s)ds]Q5[τx(t)-∫t-τtx(s)ds]=-ξT(t)(τe1-e12)Q5(τe1T-e12T)ξ(t).
Then the following inequalities are obtained by the same technique:
(32)-∫-d10∫t+θtx˙T(s)[d122R10]x˙(s)dsdθ≤-ξT(t)(d1e1-e7)R10(d1e1T-e7T)ξ(t),-∫-d2-d1∫t+θtx˙T(s)[dmR11]x˙(s)dsdθ≤-ξT(t)(d12e1-e8-e9)R11(d12e1T-e8T-e9T)ξ(t).
Let λ(t)=(d(t)-d1)/d12, then we have
(33)-∫t-d2t-d1x˙T(s)[d12R7]x˙(s)ds=-d12∫t-d2t-d(t)x˙T(s)R7x˙(s)ds-d12∫t-d(t)t-d1x˙T(s)R7x˙(s)ds=-(d2-d(t))∫t-d2t-d(t)x˙T(s)R7x˙(s)ds-(d(t)-d1)∫t-d2t-d(t)x˙T(s)R7x˙(s)ds-(d(t)-d1)∫t-d(t)t-d1x˙T(s)R7x˙(s)ds-(d2-d(t))∫t-d(t)t-d1x˙T(s)R7x˙(s)ds=-ξT(t)(e2-e4)R7(e2T-e4T)ξ(t)-d(t)-d1d2-d(t)ξT(t)(e2-e4)R7(e2T-e4T)ξ(t)-ξT(t)(e3-e2)R7(e3T-e2T)ξ(t)-d2-d(t)d(t)-d1ξT(t)(e3-e2)R7(e3T-e2T)ξ(t)≤-ξT(t)(e2-e4)R7(e2T-e4T)ξ(t)-λ(t)ξT(t)(e2-e4)R7(e2T-e4T)ξ(t)-ξT(t)(e3-e2)R7(e3T-e2T)ξ(t)-(1-λ(t))ξT(t)(e3-e2)R7(e3T-e2T)ξ(t)=-(1+λ(t))ξT(t)(e2-e4)R7(e2T-e4T)ξ(t)-(2-λ(t))ξT(t)(e3-e2)R7(e3T-e2T)ξ(t).
Consistent with the technique of (33), we obtain
(34)-∫t-d2t-d1xT(s)[d12R9]x(s)ds≤-(2-λ(t))ξT(t)(e8R9e8T)ξ(t)pppppp-(1+λ(t))ξT(t)(e9R9e9T)ξ(t),
considering
(35)x˙T(t)Mx˙(t)=(Aix(t)+Bix(t-d(t))+Cix˙(t-τ))T×M(Aix(t)+Bix(t-d(t))+Cix˙(t-τ))=ξT(t)ΓTMΓξ(t),
where M, Γ have been defined as before.
We take the previous equalities and inequalities (26)–(35) into (25); thus, we finally get
(36)𝔏V(x(t),i,t)≤ξT(t)[λ(t)Πi1+(1-λ(t))Πi2+ΓTMΓ]×ξ(t).
Since 0≤λ(t)≤1, by utilizing Lemma 7, we know that λ(t)Πi1+(1-λ(t))Πi2+ΓTMΓ<0 is equivalent to (19). So we choose
(37)β=maxi∈S,λ(t)∈[0,1]λmax[λ(t)Πi1+(1-λ(t))Πi2+ΓTMΓ].
Then β<0 and
(38)𝔏V(x(t),i,t)≤β∥ξ(t)∥2≤β∥x(t)∥2.
According to (38), from Dynkin's formula [31], we obtain
(39)ℰ{V(x(t),i,t)}-V(x0,r0)≤βℰ{∫0t∥x(s)∥2ds}.
Let t→∞, then we have
(40)limt→∞ℰ{∫0t∥x(s)∥2ds}≤(-β)-1V(x0,r0).
From Definition 4, we know that the systems described by (9) are stochastically stable. This completes the proof.
Remark 11.
Theorem 10 provides a delay-range-dependent stochastic stability criterion for nominal neutral systems with interval time-varying delays and Markovian jump parameters as described by (9). By utilizing a new Lyapunov functional, the less conservative criterion is obtained in terms of LMIs and it can be verified in Section 4.
Remark 12.
In the same context of the stochastic stability for neutral systems with Markovian jumping parameters and time-varying delays, the type of augmented Lyapunov functional has not been used in any of the existing literatures. Compared with the existing Lyapunov functional, the proposed one (23) contains some triple-integral terms, which is very effective in the reduction of conservativeness in [28]. Besides, the information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms, such as ∫t-d2t-d1xT(s)R2x(s)ds and ∫t-d(t)t-d1xT(s)R3x(s)ds.
In many circumstances, the information on the delay derivative may not be available. That is, μ is usually unknown in the real systems. So we give the following result as a corollary which can be obtained from Theorem 10 by setting R3=0.
Corollary 13.
For the given finite set S of modes with transition rates matrix, scalars d1, d2, and τ, the neutral systems with Markovian jump parameters and time-varying delays as described by (9) are stochastically stable if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0(i∈S), Qj>0(j=1,2,3,4,5), and Rk>0(k=1,2,4,5,…,11) such that the following linear matrix inequalities hold:
(41)Π~i1+ΓTMΓ<0,Π~i2+ΓTMΓ<0,
where
(42)Π~i1=Π~i0-2(e2-e4)R7(e2T-e4T)-(e3-e2)R7(e3T-e2T)-e8R9e8T-2e9R9e9T,Π~i2=Π~i0-(e2-e4)R7(e2T-e4T)-2(e3-e2)R7(e3T-e2T)-2e8R9e8T-e9R9e9T,
where
(43)Π~i0=e1Ye1T+e1PiBie2T+e2BiTPie1T+e1PiCie11T+e11CiTPie1T+e3(R2-R1)e3T-e4R2e4T+e5(R5-R4)e5T-e6R5e6T-e7R8e7T-e10Q1e10T-e11Q2e11T-e12Q3e12T-(e1-e10)Q4(e1T-e10T)-(e1-e3)R6(e1T-e3T)-(τe1-e12)Q5(τe1T-e12T)-(d1e1-e7)R10(d1e1T-e7T)-(d12e1-e8-e9)R11(d12e1T-e8T-e9T),
and other notations are the same as Theorem 10.
3.2. Stochastic Stability for the Uncertain Neutral Markovian Jump Systems
In this subsection, we consider the uncertain case which can be described by (3). Based on Theorem 10, we obtain the following theorem to guarantee the stochastic stability for the uncertain neutral systems with interval time-varying delays and Markovian jump parameters.
Theorem 14.
For the given finite set S of modes with transition rates matrix, scalars d1, d2, τ, and μ, the uncertain neutral systems with Markovian jump parameters and time-varying delays as described by (3) are stochastically stable if the operator 𝔇 is stable, and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0(i∈S), Qj>0(j=1,2,3,4,5), and Rk>0(k=1,2,…,11) such that the following matrix inequalities hold:
(44)[Πi1+1δ1e1PiHiHiTPie1T+δ1εεT(ΓT+1δ1e1PiHiHiT)M*1δ1MHiHiTM-M]<0,(45)[Πi2+1δ2e1PiHiHiTPie1T+δ2εεT(ΓT+1δ2e1PiHiHiT)M*1δ2MHiHiTM-M]<0,
where εT=EAie1T+EBie2T, Πi1, Πi2, Γ, and M have been defined in Theorem 10.
Proof.
On the basis of Theorem 10, we directly replace Ai and Bi with Ai+ΔAi(t), Bi+ΔBi(t) and obtain
(46)Πi1(t)+ΓT(t)MΓ(t)<0,(47)Πi2(t)+ΓT(t)MΓ(t)<0,
where
(48)Πi1(t)=Πi1+e1[ΔAiT(t)Pi+PiΔAi(t)]e1T+e1PiΔBi(t)e2T+e2ΔBiT(t)Pie1T,Πi2(t)=Πi2+e1[ΔAiT(t)Pi+PiΔAi(t)]e1T+e1PiΔBi(t)e2T+e2ΔBiT(t)Pie1T.
Considering (46) and combining the uncertainties condition (7) by Lemma 9, we have
(49)[Πi1ΓTMMΓ-M]+[e1Pi0M0]HiFi(t)εT+εFiT(t)HiT[Pie1TM00]<0.
With (8), by Lemma 8 from (49), we obtain
(50)[Πi1ΓTMMΓ-M]+1δ1[e1Pi0M0]HiHiT[Pie1TM00]+δ1[εεT000]<0.
Obviously, (50) is equivalent to (44). Similarly, considering (46) and following the same procedure, we can get (45). Finally, following from the latter proof of Theorem 10, we know that the uncertain neutral systems with Markovian jump parameters and time-varying delay as described by (3) are stochastically stable. This completes the proof.
Remark 15.
It should be noted that (44) and (45) can be viewed as linear matrix inequalities by introducing new variables. That is, define matrices
(51)Pi(1)=1δ1PiHiHiTPi,Pi(2)=1δ2PiHiHiTPi,PiM(1)=1δ1PiHiHiTM,PiM(2)=1δ2PiHiHiTM,M(1)=1δ1MHiHiTM,M(2)=1δ2MHiHiTM,
where Hi, i∈S, are known constant matrices and have been defined in (7). Then (44) and (45) can be easily solved by the LMI Toolbox in MATLAB.
Remark 16.
It should be mentioned that Theorem 14 is an extension of Theorem 10 to uncertain neutral Markovian jump systems with interval time-varying delays. It provides a stochastic delay-range-dependent stability criterion for (3) and it will be verified to be less conservative than some existing ones in Section 4.
Consistent with the nominal systems, in the uncertain case, we have the following result as a corollary if the information on the delay derivative μ may not be available. The corollary is also obtained by setting R3=0 in Theorem 14.
Corollary 17.
For the given finite set S of modes with transition rates matrix, scalars d1, d2, and τ, the uncertain neutral systems with Markovian jump parameters and time-varying delays as described by (3) are stochastically stable if the operator 𝔇 is stable, and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0(i∈S), Qj>0(j=1,2,3,4,5), and Rk>0(k=1,2,4,5,…,11) such that the following symmetric matrix inequalities hold:
(52)[Π~i1+1δ1e1PiHiHiTPie1T+δ1εεT(ΓT+1δ1e1PiHiHiT)M*1δ1MHiHiTM-M]<0,[Π~i2+1δ2e1PiHiHiTPie1T+δ2εεT(ΓT+1δ2e1PiHiHiT)M*1δ2MHiHiTM-M]<0,
where Π~i1 and Π~i2 have been defined in Corollary 13, the remaining notations are the same as Theorem 14.
4. Numerical Examples
In this section, numerical examples are given to show that the proposed theoretical results in this paper are effective and less conservative than some previous ones in the literature.
Example 1.
Consider the nominal system in the form of (9) described as follows:
(53)x˙(t)-Cix˙(t-0.1)=Aix(t)+Bix(t-d(t)),
where i∈S={1,2} and the mode switching is governed by the rate matrix [πij]2×2=[-554-4], which is described by Figure 1(54)A1=[25-2-3],B1=[-0.30.5-0.2-0.3],C1=[-0.200.1-0.2],A2=[-5-1.62-4],B2=[-0.30.5-0.2-0.3],C2=[-0.10.20-0.2].
Given the time-varying delay d(t)=0.5(2+sin3t), from the graph of d(t) with t∈[0,2π] in Figure 2, we easily obtain d1=0.5 and d2=1.5. In addition, we have d˙(t)=1.5sin2tcost and its maximum μ=3/3.
Operation modes of Example 1.
Interval time-varying delay d(t) of Example 1.
By Theorem 10, with the help of LMI toolbox in MATLAB, we solve (19) and get a group of matrices for the solution to guarantee the stochastic stability for the system (53) as follows: for simplicity, we only list the matrices for Pi, i∈S={1,2}, Qj, j=1,2,…,5. (55)P1=[1.4857-0.3614*0.6329],P2=[2.0645-0.3761*0.5086],Q1=[0.7342-0.1546*0.2978],Q2=[0.4083-0.1873*0.3652],Q3=[0.4165-0.2137*0.3056],Q4=[0.4576-0.2539*0.3684],Q5=[0.2673-0.0845*0.1766].
Therefore, it can be seen that the system (9) is determined to be stochastically stable by Theorem 10.
Example 2.
As said in the literature [32], with the abrupt variation in its structures and parameters, we can present the partial element equivalent circuit (PEEC) model as a stochastic jump one. Then, a general form of PEEC model is given by (3), where we assume that the neutral delay of the PEEC model is constant. Consider the stochastic neutral partial element equivalent circuit (PEEC) model described by the following equation:
(56)x˙(t)-Cix˙(t-0.3)=(Ai+ΔA(t))x(t)+(Bi+ΔB(t))x(t-d(t)),
where i∈S={1,2} and the mode switching is governed by the rate matrix [πij]2×2=[-443-3], which is described by Figure 3(57)A1=[-500-6],B1=[-1.60-1.8-1.5],C1=0.5I,H1=[0.20.2],A2=[-400-5],B2=[-20-0.9-1.2],C2=0.3I,H2=[0-0.3],EA1=[0.20],EA2=[00.2],EB1=[-0.30.3],EB2=[0.20.2].
Given ∥Fi(t)∥<1 and the time-varying delay d(t)=0.5(2+cos3t), from the graph of d(t) with t∈[0,2π] in Figure 4, we easily obtain d1=0.5 and d2=1.5. In addition, we have d˙(t)=-1.5cos2tsint and its maximum μ=3/3.
Operation modes of Example 2.
Interval time-varying delay d(t) of Example 2.
By Theorem 14, with the help of LMI toolbox in MATLAB, we solve (44) and (45) and get a group of matrices for the solution to guarantee the stochastic stability for the system (56) as follows: for simplicity, we only list the matrices for Pi, i∈S={1,2}, Qj, j=1,2,…,5. (58)P1=[2.4327-0.4713*0.7846],P2=[2.7685-0.4617*0.7432],Q1=[0.5122-0.1558*0.3976],Q2=[0.6083-0.1898*0.3976],Q3=[0.3164-0.2058*0.4751],Q4=[0.2563-0.1584*0.3476],Q5=[0.1574-0.0713*0.1798].
Therefore, according to Theorem 14, the uncertain neutral PEEC system presented by (3) is stochastically stable.
Example 3.
In the study of practical electrical circuit systems, a small test circuit which consists of a partial element equivalent circuit (PEEC) was considered in [33], which can be described as the following form:
(59)x˙(t)-Cx˙(t-τ)=Ax(t)+Bx(t-d).
Compared with (9), (59) can be regarded as i∈S={1} and d(t)=d. So we have d1=d2=d, μ=0 and utilize Theorem 10 to compute the maximum discrete delay for system stability.
Remark 18.
It should be pointed out that we require d1≠d2 and d1≠0 in order to conveniently organize this paper. But from the results of the theorems and corollaries in this paper, we know that they are applicable to many special cases, such as d(t)≡d, d1=d2, or d1=0, τ=0. Actually, we just need to delete the corresponding integral terms in the Lyapunov functional and obtain homologous results.
Consider (59) with the following parameters:
(60)A=[-0.90.20.1-0.9],B=[-1.1-0.2-0.1-1.1],C=[-0.200.2-0.1].
For given τ, by Theorem 10, the maximum d, which satisfies the LMIs in (19), can be calculated by solving a quasiconvex optimization problem. This neutral system was considered in references [34–36]. The results on the maximum upper bound of d are compared in Table 1.
From Table 1, we know that the maximum upper bound of delay d=2.3026 in this paper by setting τ=0.1, while the maximum upper bound of delay d=1.7100 for [36], d=2.1229 for [34], and d=2.2951 for [35]. The results are also given by setting τ=0.5 and τ=1, and it is found that the maximum upper bound in this paper is larger than those in [34–36]. So it can be demonstrated that the stability condition Theorem 10 in this paper yields less conservative results than the previous ones.
Maximum upper bound of d with different neutral delay τ.
Methods
τ=0.1
τ=0.5
τ=1
He et al. [36]
1.7100
1.6718
1.6543
Han [34]
2.1229
2.1229
2.1229
Li et al. [35]
2.2951
2.3471
2.3752
Theorem 10
2.3026
2.3547
2.3835
Example 4.
Further consideration on Example 3, if we take the parameter uncertainties commonly existing in the modeling of a real system into account, a general form of PEEC model is given by
(61)x˙(t)-Cx˙(t-τ)=(A+ΔA(t))x(t)+(B+ΔB(t))x(t-d),
where A, B, and C are given in Example 3 and the uncertain matrices ΔA(t) and ΔB(t) satisfy
(62)∥ΔA(t)∥≤κ,∥ΔB(t)∥≤κ,κ≥0.
Moreover, in the form of (7) and (8), we assume that
(63)H=κI,EA=EB=I,0≤κ≤1.
Consider (61), for given τ and κ, by Theorem 14, the maximum upper bound of d, which satisfies the LMIs in (44) and (45), can be calculated by solving a quasiconvex optimization problem. When τ=1.0, Table 2 gives the comparisons of the maximum allowed delay of d for various parameters κ in different methods.
From Table 2, provided that τ=1.0, we know that the maximum upper bound of delay d=1.5316 in this paper by setting κ=0.10, while the maximum upper bound of delay d=1.3864 for [37], d=1.4385 for [36], and d=1.5047 for [38]. The results are also given by setting κ=0.15, κ=0.20, and κ=0.25, and it is found that the maximum upper bound in this paper is larger than those in [36–38]. So it can be seen that the delay-range-dependent stability condition Theorem 14 in this paper is less conservative than some earlier reported ones in the literature.
Maximum upper bound of d with τ=1.0 and different parameter κ.
Methods
κ=0.10
κ=0.15
κ=0.20
κ=0.25
Han [37]
1.3864
1.2705
1.1607
1.0456
He et al. [36]
1.4385
1.3309
1.2396
1.1547
Xu et al. [38]
1.5047
1.4052
1.2998
1.2136
Theorem 14
1.5316
1.4089
1.3028
1.2217
Example 5.
In this example, to compare the stochastic stability result in Theorem 10 with those in [23, 39, 40], we consider the nominal system (9) with Crt=0 and d1=0. In fact, we consider here that there are no longer neutral delay systems, for given (9) with the following parameters:
(64)A1=[-3.490.81-0.65-3.27],A2=[-2.490.291.34-0.02],B1=[-0.86-1.29-0.68-2.07],B2=[-2.830.50-0.84-1.01],C1=C2=0,Pij=[πij]2×2,i,j∈S={1,2}.
As described previously, for given π22=-0.8, different values of π11 and different values of μ, by Theorem 10 and the maximum d2, which satisfies the LMIs in (19), can be calculated by solving a quasiconvex optimization problem. Tables 3 and 4 give the contrastive results.
From Table 3, provided that μ=0, we know that the maximum upper bound of delay d2=0.6853 in this paper by setting π11=-0.10, while the maximum upper bound of delay d2=0.5012 for [40], d2=0.5012 for [39], and d2=0.6797 for [23]. The results are also given by setting π11=-0.50, π11=-0.80, and π11=-1.00, and it is found that the maximum upper bound of delay d2 in this paper is larger than those in [23, 39, 40]. So it also can be shown that the stochastic stability result in Theorem 10 is less conservative than those results in [23, 39, 40].
From Table 4, provided that μ=1.5, we know that the maximum upper bound of delay d2=0.3953 in this paper by setting π11=-0.10, while the methods in [39, 40] cannot be applicable to the case μ≥1, and the maximum upper bound of delay d2=0.3860 for [23]. So it can be shown that Theorem 10 in this paper is less conservative and can be applied to the time-varying delay without the requirement on μ<1.
Maximum upper bound of d2 with μ=0 and different parameter π11.
Methods
π11=-0.10
π11=-0.50
π11=-0.80
π11=-1.00
Cao et al. [40]
0.5012
0.4941
0.4915
0.4903
Chen et al. [39]
0.5012
0.4941
0.4915
0.4903
Xu et al. [23]
0.6797
0.5794
0.5562
0.5465
Theorem 10
0.6853
0.5874
0.5625
0.5574
Maximum upper bound of d2 with μ=1.5 and different parameter π11.
Methods
π11=-0.10
π11=-0.50
π11=-0.80
π11=-1.00
Cao et al. [40]
—
—
—
—
Chen et al. [39]
—
—
—
—
Xu et al. [23]
0.3860
0.3656
0.3487
0.3378
Theorem 10
0.3953
0.3746
0.3502
0.3449
5. Conclusions
In this paper, some new delay-range-dependent conditions have been provided to guarantee the stochastic stability of the neutral systems with Markovian jumping parameters and interval time-varying delays. A novel augmented Lyapunov-Krasovskii functional which contains some triple-integral terms is constructed. By some integral inequalities and the nature of convex combination, some less conservative delay-range-dependent stochastic stability criteria are obtained. Numerical examples are given to demonstrate the effectiveness and less conservativeness of our result.
Acknowledgments
This work was supported in part by the National Key Scientific Research Project (61233003), the National Natural Science Foundation of China (60935001, 61174061, 61074033, and 60934006), and the Fundamental Research Funds for the Central Universities.
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