This paper studies the problem of finite-time H∞ control for time-delayed Itô stochastic systems with Markovian switching. By using the appropriate Lyapunov-Krasovskii functional and free-weighting matrix techniques, some sufficient conditions of finite-time stability for time-delayed stochastic systems with Markovian switching are proposed. Based on constructing new Lyapunov-Krasovskii functional, the mode-dependent state feedback controller for the finite-time H∞ control is obtained. Simulation results illustrate the effectiveness of the proposed method.
1. Introduction
Finite-time stability is different from the usual Lyapunov stability. Lyapunov stability is always used to deal with the asymptotic pattern of system trajectories by applying the steady-state behavior of control dynamics over an infinite-time interval [1]. Often Lyapunov asymptotic stability is not enough for practical applications, because there are some cases where large values of the state are not acceptable, for instance, in the presence of saturations [2]. Lyapunov asymptotic stability depicts steady-state performance of a dynamic system, and it could not reflect transient state performance [3]. A finite-time stable system may not be Lyapunov stable, and a Lyapunov stable system may not be finite-time stable. To study the transient performances of a system, the concept of finite-time stability was introduced by Dorato in [4]. Finite-time stability (or short-time stability) is also called finite-time boundness. A system is said to be finite-time stable if, once a time interval is fixed, its state does not exceed some bounds during this time interval. Because the working time of many systems such as communication network system, missile system, and robot control system is short, people are more interested in finite-time stability of these systems.
Early results on finite-time stability are mostly confined to the stability analysis and lack of design and comprehensiveness of control systems (see [5–9]). During the nineteen seventies, scholars began to discuss the control design method of finite-time stabilization (see [10–13]). In recent years, the development of the theory of linear matrix inequalities promotes the research on finite-time stability and makes this research field a new breakthrough [14–20].
In particular, for systems with time delay or Markov switching or random disturbance, there are some significant research results on finite-time stability and stabilization. For example, finite-time stability and stabilization problem for Itô stochastic systems was studied in [21–27], finite-time stability and stabilization problem for Markovian jump systems was studied in [28–31], and finite-time stability and stabilization problem for time-delay systems was studied in [2, 32].
With the development of finite-time [33] stability, the problem of finite-time H∞ control has received a lot of attention [1, 3, 34–39]. For example, using the average dwell time method and the multiple Lyapunov-like function technique, some sufficient conditions are proposed to guarantee the finite-time properties for the switched Itô stochastic systems in the form of matrix inequalities and a state feedback controller for the finite-time H∞ control problem is also obtained in [36]. Delay-dependent observer-based H∞ finite-time control for switched systems with time-varying delay was investigated in [34]. The robust finite-time H∞ control problem for a class of uncertain switched neutral systems with unknown time-varying disturbance was developed in [3]. The problem of robust finite-time H∞ control of singular Itô stochastic systems via static output feedback was addressed in [38]. However, the systems discussed in [3, 34, 36] are general switched systems rather than Markovian jump systems. Markovian jump systems [40–45] (also called systems with Markovian switching) are frequently used to model the dynamics behavior of the process in which variable parameters or structures subject to random abrupt changes occur, for example, sudden environment changes, system noises, subsystem switching, and failures that occurred in interconnections or components and executor faults [46]. On the other hand, most work on the problem of finite-time control focused on the determination of linear or nonlinear system. As is known, stochastic modeling plays an important role in many branches of science and engineering (see [47, 48]). At present, the research of finite-time control for Itô stochastic system is still at the beginning stage. To the best of the authors' knowledge, the problem of finite-time H∞ control for time-delayed Itô stochastic systems with Markovian switching has not been investigated, which motivated our study.
In this paper, we will focus on the finite-time H∞ state feedback control problem for time-delayed Itô stochastic systems with Markovian switching. The aim is to find a state feedback controller
(1)ui(t)=Kix(t),t∈[0,T],i=1,2,…,N
for system (2) such that the corresponding closed-loop system is finite-time stochastically bounded with a weighted H∞ performance γ. The rest of the paper is organized as follows. In Section 2, problem description and some definitions are given. In Section 3, finite-time stochastic stability and bounded conditions for time-delayed Itô stochastic systems with Markovian switching are presented. The corresponding results of finite-time stochastic H∞ control problem for time-delayed Itô stochastic systems with Markovian switching are proposed in Section 4. An illustrative example is given in Section 5, and conclusions are given in Section 6.
Notation.
Throughout this paper, if not explicit, matrices are assumed to have compatible dimensions. The notation M>(≥,<,≤)0 means that the symmetric matrix M is positive-definite (positive-semidefinite, negative, and negative-semidefinite). λmin(·) and λmax(·) denote the minimum and the maximum eigenvalue of the corresponding matrix. ∥·∥ represents the Euclidean norm for vector or the spectral norm of matrices. I refers to an identity matrix of appropriate dimensions. E{·} stands for the mathematical expectation. The symbol “*” within a matrix denotes a term that is induced by symmetry.
2. Problem Description
In this paper, we consider the following time-delayed stochastic systems with Markovian switching:
(2)dx(t)=[A(rt)x(t)+A1(rt)x(t-τ(t))+B1(rt)u(t)+E1(rt)v(t)]dt+[H(rt)x(t)+H1(rt)x(t-τ(t))+B2(rt)u(t)+E2(rt)v(t)]dw(t),z(t)=C(rt)x(t)+C1(rt)x(t-τ(t))+D1(rt)u(t),x(t)=φ(t),t∈[-τ,0],
where x(t)∈ℝn is the system state, u(t)∈ℝl is the control input, z(t)∈ℝp is the control output, and v(t)∈ℝq is exogenous disturbance that satisfies ∫0TvT(t)v(t)dt≤d(d≥0). A(rt), A1(rt), B1(rt), E1(rt), H(rt), H1(rt), B2(rt), E2(rt), C(rt), C1(rt), and D1(rt) are known mode-dependent constant matrices with appropriate dimensions. w(t) is a zero-mean real scalar Wiener process on a complete probability space (Ω,ℱ,P) with a natural filtration {ℱt}t≥0, where Ω is the sample space, ℱ is the σ-algebras of sets of the sample space, and P is the probability measure on ℱ. φ(t) is an initial condition. It is known that system (2) has a unique solution, denoted by x(t)=x(t,φ). τ(t) is the time-varying delay and satisfies 0≤τ(t)<τ, τ˙(t)≤h, where τ, h are constants.
The jump parameter rt(t≥0) is a continuous-time discrete-state Markov stochastic process taking values on a finite set Λ={1,2,…,N} with transition rate matrix Π={Πij} given by
(3)Pr=Pr{rt+Δt=j∣rt=i}={ΠijΔt+o(Δt),i≠j1+ΠijΔt+o(Δt),i=j,
where limΔt→0+(o(Δt)/Δt)=0, Πij≥0, for i≠j, and ∑j=1,j≠iNΠij=-Πii, for i,j∈Λ.
Definition 1.
For given time-constant T>0, system (2) with u(t)=0 and v(t)=0 is said to be stochastically finite-time stable with respect to (c1,c2,T,Ri), where c1<c2, Ri>0, if
(4)supt∈[-τ,0]φT(t)Riφ(t)≤c1⟹E{xT(t)Rix(t)}<c2,∀t∈[0,T],i∈Λ.
Definition 2.
For given time-constant T>0, system (2) with u(t)=0 is said to be finite-time stochastically bounded with respect to (c1,c2,T,Ri,d), where c1<c2, Ri>0, if
(5)supt∈[-τ,0]φT(t)Riφ(t)≤c1⟹E{xT(t)Rix(t)}<c2,∀t∈[0,T],i∈Λ,∀v(t):∫0TvT(t)v(t)dt≤d.
Definition 3.
For given time-constant T>0, γ>0, system (2) with u(t)=0 is said to be H∞ finite-time stochastically bounded with respect to (c1,c2,T,Ri,d), where c1<c2, Ri>0, if
system (2) is finite-time stochastically bounded with respect to (c1,c2,T,Ri,d);
under zero-initial condition, the output z(t) satisfies
(6)E{∫0TzT(t)z(t)dt}<γ2∫0TvT(t)v(t)dt.
Definition 4.
For given time-constant T>0, γ>0, systems (2) are said to be finite-time stabilizable with H∞ disturbance attenuation level γ, if there exists a controller ui(t)=Kix(t) such that
(i) the corresponding closed-loop system is finite-time stochastically bounded with respect to (c1,c2,T,Ri,d);
(ii) under zero-initial condition, (6) holds for any v(t) satisfying ∫0TvT(t)v(t)dt≤d.
Lemma 5.
Given constant matrices Ω1, Ω2, and Ω3 with appropriate dimensions, where Ω1=Ω1T, 0<Ω2=Ω2T, then Ω1+Ω3TΩ2-1Ω3<0 if and only if
(7)[Ω1Ω3TΩ3-Ω2]<0.
3. Finite-Time Stochastic Stability and Bounded Analysis
In this section, we consider the systems (2) with u(t)=0:
(8)dx(t)=[A(rt)x(t)+A1(rt)x(t-τ(t))+E1(rt)v(t)]dt+[H(rt)x(t)+H1(rt)x(t-τ(t))+E2(rt)v(t)]dw(t),z(t)=C(rt)x(t)+C1(rt)x(t-τ(t)),x(t)=φ(t),t∈[-τ,0].
Let V(x(t),rt,t) be the stochastic Lyapunov Krasovskii functional; define its weak infinitesimal operator as
(9)ℒV(x(t),rt,t)=limΔt→01Δt[E{V(x(t+Δt),rt+Δt,t+Δt)∣x(t),rt}-V(x(t),rt,t)].
Theorem 6.
System (2) with u(t)=0 is finite-time stochastically bounded with respect to (c1,c2,T,Ri,d), where c1<c2, Ri>0, if there exist positive-definite symmetric matrices Pi, Ni, Q, and W and positive scalars α, λ1, λ2, and λ3, such that the following conditions hold:
We denote that rt=i. For convenience, we also denote A(rt), A1(rt), B1(rt), E1(rt), H(rt), H1(rt), B2(rt), E2(rt), C(rt), C1(rt), and D1(rt) as Ai, A1i, B1i, E1i, Hi, H1i, B2i, E2i, Ci, C1i, and D1i. Take the Lyapunov-Krasovskii functional for systems (8) as
(14)V(x(t),i,t)=xT(t)Pix(t)+∫t-τ(t)teα(t-s)xT(s)Qx(s)ds≜V1i(t)+V2i(t),
where Pi>0 is the given mode-dependent symmetric positive-definite matrix for each mode i∈Λ and Q is the symmetric positive-definite matrix.
Along the trajectory of system (8), we have
(15)ℒV1i(t)=xT(t)(-αPi+∑j=1NΠijPjAiTPi+PiAi+HiTPiHi-αPi+∑j=1NΠijPj)x(t)+2xT(t)(PiA1i+HiTPiH1i)x(t-τ(t))+2xT(t)PiE1iv(t)+2xT(t)HiTPiE2iv(t)+xT(t-τ(t))H1iTPiH1ix(t-τ(t))+2xT(t-τ(t))H1iTPiE2iv(t)-vT(t)Wv(t)+αxT(t)Pix(t)+vT(t)Wv(t),
where W>0.
Consider the following:
(16)ℒV2i(t)=xT(t)Qx(t)-(1-τ˙(t))eατ(t)×xT(t-τ(t))Qx(t-τ(t))+α∫t-τ(t)teα(t-s)xT(s)Qx(s)ds≤xT(t)Qx(t)-Φ(h)xT×(t-τ(t))Qx(t-τ(t))+α∫t-τ(t)teα(t-s)xT(s)Qx(s)ds,
where
(17)Φ(h)={1-h,h≤1(1-h)eατ,h>1.
Set y(t)=Aix(t)+A1ix(t-τ(t))+E1iv(t), Ni>0; we have
(18)2xT(t-τ(t))Ni[y(t)-Aix(t)-A1ix(t-τ(t))-E1iv(t)]=0,(19)2yT(t)Ni[Aix(t)+A1ix(t-τ(t))+E1iv(t)-y(t)]=0.
From (15) to (19), we obtain
(20)ℒV(x(t),i,t)<ξT(t)Ωξ(t)+αV(x(t),i,t)+vT(t)Wv(t),
where(21)ξT(t)=[xT(t),xT(t-τ(t)),vT(t),yT(t)],Ω=[AiTPi+PiAi+HiTPiHi-αPi+Q+∑j=1NΠijPjPiA1i+HiTPiH1i-AiTNiPiE1i+HiTPiE2iAiTNiT*H1iTPiH1i-Φ(h)Q-NiA1iH1iTPiE2i-NiE1iNi+A1iTNi**-WE1iTNiT***-Ni].
Using weak infinitesimal operator and (8), we can get
(22)d[e-αtV(x(t),i,t)]=-αe-αtV(x(t),i,t)dt+e-αtdV(x(t),i,t)=e-αt(ℒV(x(t),i,t)-αV(x(t),i,t))dt+2e-αtxT(t)Pi[Hix(t)+H1ix(t-τ(t))+E2iv(t)]dw(t).
By integrating both sides of (22) from 0 to t, taking expectations, and by (10)–(12), it follows that
(23)E{V(x(t),i,t)}<eαtE{V(x(0),r0,0)}+∫0teα(t-s)vT(s)Wv(s)ds≤eαtxT(0)Pix(0)+eαt∫-τ0eαsxT(s)Qx(s)ds+eαT∫0te-αsvT(s)Wv(s)ds≤eαTλ2xT(0)Rix(0)+eαTλ3∫-τ0eαsxT(s)Rix(s)ds+eαT∫0tvT(s)Wv(s)ds≤eαTλ2c1+eαTλ3τeατc1+λmax(W)eαTd.
On the other hand, by (11), it is easy to see that
(24)E{V(x(t),i,t)}>E{xT(t)Pix(t)}≥λ1E{xT(t)Rix(t)}.
Now, (24) together with (13) and (23) implies that
(25)E{xT(t)Rix(t)}<c2.
The proof is completed.
Remark 7.
It should be pointed out that the upper bound h of the derivative of time-varying delay τ(t) in this paper allows h≤1 or h>1. When h≤1, we have (τ˙(t)-1)eατ(t)≤h-1. When h>1, we have (τ˙(t)-1)eατ(t)<(h-1)eατ whether 1<τ˙(t)<h or τ˙(t)<1<h. So the function Φ(h) in (16) is introduced. It should be noted that the upper bound h in [49] only allows h<1. Moreover, as explained above, the inequality amplification result on (14) in [49] is not true. So our results can be applied to more general systems.
Remark 8.
From (13), we can obtain the upper bound τmax of the delay τ(t); that is,
(26)τmax=λ1c2/eαT-λ2c1-dλmax(W)λ3c1.
Remark 9.
Assuming that W≤λ4I, for certain τ and α, by Lemma 5, we can obtain the following linear matrix inequalities (LMIs) that are equivalent to condition (13):
(27)[-λ1c2e-αTλ2c1λ3τc1eατλ4d*-λ200**-λ30***-λ4]<0.
Corollary 10.
System (8) with v(t)=0 is stochastically finite-time stable with respect to (c1,c2,T,Ri), where c1<c2, Ri>0, if there exist positive-definite symmetric matrices Pi, Q, and Ni and positive scalars α, λ1, λ2, and λ3, such that the following conditions hold:(28)[AiTPi+PiAi+HiTPiHi-αPi+∑j=1NΠijPjPiA1i+HiTPiH1i-AiTNiAiTNiT*H1iTPiH1i-Φ(h)Q-NiA1iNi+A1iTNi**-Ni]<0,λ1Ri≤Pi≤λ2Ri,0<Q≤λ3Ri,eαTλ2c1+eαTλ3τeατc1<λ1c2.
4. Finite-Time Stochastic H∞ Control
In this section, we consider the problem of finite-time stochastic H∞ control for time-delayed Itô stochastic systems with Markovian switching. We consider the mode-dependent controller u(t)=Kix(t), t∈[0,T], where Ki is the state feedback gain that has to be determined. Applying the state feedback controller into system (2) and denoting rt=i, we can obtain the corresponding closed-loop system as follows:
(29)dx(t)=[A~ix(t)+A1ix(t-τ(t))+E1iv(t)]dt+[H~ix(t)+H1ix(t-τ(t))+E2iv(t)]dw(t),z(t)=C~ix(t)+C1ix(t-τ(t)),x(t)=φ(t),t∈[-τ,0],
where A~i=Ai+B1iKi, H~i=Hi+B2iKi, and C~i=Ci+D1iKi.
Theorem 11.
System (29) is finite-time stabilizable with H∞ disturbance attenuation level γ¯, if there exist positive-definite symmetric matrices Pi, Q, and N~i and positive scalars α, λ1, λ2, and λ3, such that conditions (11)-(12) and the following conditions hold:(30)[A~iTPi+PiA~i-αPi+Q+∑j=1NΠijPjPiA1i-A~iTN~iTPiE1iA~iTN~iTC~iTH~iT*-Φ(h)Q-N~iA1i-N~iE1iN~i+A1iTN~iC1iTH1iT**-γ2IE1iTN~iT0E2iT***-N~i00****-I0*****-PiT]<0,(31)eαTλ2c1+eαTλ3τeατc1+γ2deαT<λ1c2.
Proof.
Choose the Lyapunov-Krasovskii functional for systems (29) as
(32)V(x(t),i,t)=xT(t)Pix(t)+∫t-τ(t)teα(t-s)xT(s)Qx(s)ds+∫0teα(t-s)vT(s)E2iTPiE2iv(s)ds≜V1i(t)+V2i(t)+V3i(t),
where Pi>0 is the given mode-dependent symmetric positive-definite matrix for each mode i∈Λ and Q is the symmetric positive-definite matrix.
Along the trajectory of system (29), we have
(33)ℒV1i(t)=xT(t)(-αPi+∑j=1NΠijPjA~iTPi+PiA~i+H~iTPiH~i-αPi+∑j=1NΠijPj)x(t)+2xT(t)(PiA1i+H~iTPiH1i)x(t-τ(t))+2xT(t)PiE1iv(t)+2xT(t)H~iTPiE2iv(t)+xT(t-τ(t))H1iTPiH1ix(t-τ(t))+2xT(t-τ(t))H1iTPiE2iv(t)-γ2vT(t)v(t)+αxT(t)Pix(t)+γ2vT(t)v(t),ℒV2i(t)≤xT(t)Qx(t)-Φ(h)xT(t-τ(t))Qx(t-τ(t))+αV2i(t),ℒV3i(t)=vT(t)E2iTPiE2iv(t)+α∫0teα(t-s)vT(s)E2iTPiE2iv(s)ds.
Set y~(t)=A~ix(t)+A1ix(t-τ(t))+E1iv(t), N~i>0; we have
(34)2xT(t-τ(t))N~i[y~(t)-A~ix(t)-A1ix(t-τ(t))-E1iv(t)]=0,2y~T(t)N~i[A~ix(t)+A1ix(t-τ(t))+E1iv(t)-y~(t)]=0.
From (33) to (34), we obtain
(35)ℒV(x(t),i,t)<ξ~T(t)Ω~ξ~(t)+αV(x(t),i,t)+γ2vT(t)v(t)-zT(t)z(t),
where(36)ξ~T(t)=[xT(t),xT(t-τ(t)),vT(t),y~T(t)],Ω~=[A~iTPi+PiA~i+H~iTPiH~i-αPi+∑j=1NΠijPj+C~iTC~iPiA1i+H~iTPiH1i-A~iTN~i+C~iTC~1iPiE1i+H~iTPiE2iA~iTN~iT*H1iTPiH1i-Φ(h)Q-N~iA1i+C1iTC1iH1iTPiE2i-N~iE1iN~i+A1iTN~i**-γ2E2iTPiE2iE1iTN~iT***-N~i].
Using Lemma 5, we have that (30) is equivalent to Ω~<0. Then (35) becomes
(37)ℒV(x(t),i,t)<αV(x(t),i,t)+γ2vT(t)v(t)-zT(t)z(t).
Under zero initial condition, we have
(38)0<e-αTE{V(x(t),i,t)}<E{∫0Te-αs(γ2vT(s)v(s)-zT(s)z(s))ds}.
Thus
(39)E{∫0Te-αszT(s)z(s)ds}<γ2E{∫0Te-αsvT(s)v(s)ds},E{∫0TzT(s)z(s)ds}<γ2eαTE{∫0Te-αsvT(s)v(s)ds}<γ2eαTE{∫0TvT(s)v(s)ds}.
Let γ¯=eαTγ; then γ¯ is H∞ performance index. When z(t)=0, similar to the proof of Theorem 6, it can be obtained that
(40)E{xT(t)Rix(t)}≤eαTλ2c1+eαTλ3τeατc1+γ2eαTdλ1.
From (31), we can get
(41)E{xT(t)Rix(t)}<c2.
The proof is completed.
Theorem 12.
System (29) is finite-time stabilizable with H∞ disturbance attenuation level γ¯, if there exist positive-definite symmetric matrices Xi, Q~i, Q^i, and N^i, appropriate dimensions matrices Yi, and positive scalars α, λ1, λ2, and λ3, such that conditions (11)-(12), (31) and the following conditions hold:(42)[AiXi+XiAiT+B1iYi+YiTB1iT-αXi+Q^i+∑j=1NΠijXjA1iN^i-XiAiT-YiTB1iTE1iXiAiT+YiTB1iTXiCiT+YiTD1iTXiHiT+YiTB2iT*-Φ(h)Q~i-A1iN^i-E1iN^i+N^iA1iTN^iC1iTN^iH1iT**-γ2N^iN^iE1iT0E2iT***-N^i00****-I0*****-Xi]<0.
Moreover, a state feedback controller gain is given by Ki=YiXi-1.
Proof.
Replacing A~i, H~i, and C~i in (30) with Ai+B1iKi, Hi+B2iKi, and Ci+D1iKi, then premultiplying and postmultiplying it by diag{Pi-1,N~i-1,I,N~i-1,I,I}, and denoting Pi-1=Xi, Yi=KXi, XiTQXi=Q^i, N~i-1=N^i, and N~i-1QN~i-1=Q~i, we can obtain (42).
The proof is completed.
Remark 13.
Replacing λ4 in (27) with γ2, then it is equivalent to (31). For certain λ1 and λ2, all the conditions of Theorem 12 can be expressed as linear matrix inequalities. In this way, finite-time H∞ state feedback stabilization conditions for time-delayed Itô stochastic systems with Markovian switching are based entirely on linear matrix inequalities. In the practical application of dynamical systems, we can obtain the controller effectively with the help of LMI toolbox in MATLAB.
Remark 14.
In order to obtain the finite-time H∞ stabilization conditions based on LMIs for time-delayed Itô stochastic systems with Markovian switching, new Lyapunov-Krasovskii functional (32) is introduced.
Remark 15.
In the sense of Lyapunov stability, the problem of H∞ control for systems with Markovian switching and time delay has attracted a lot of research (e.g., see [40, 41]). Different from these studies, this paper focuses on this problem under the sense of finite-time stability. The latter is suitable for transient performance of actual systems such as communication network system, missile system, and robot control system.
5. Illustrative Example
In this section, we will discuss one example to illustrate our results.
Example 16.
Consider time-delayed Itô stochastic systems with Markovian switching (29) with the following parameters:
(43)A1=[-0.122-1],A2=[-210-2],A11=[-0.10-0.1-0.1],A12=[0.20.10.10.1]C1=C11=[-0.200-0.2],C2=C12=[0.2000.1],H11=[-0.10-0.1-0.1],H12=[0.10.10.10],H1=[0.10.20.20.3],H2=[0.10.20.10.3],E21=[0.10.3-0.20.4],E22=[0.10.3-0.10.1],B11=[1612213],B12=[128103],B21=[0.10.3-20.4],B22=[0.10.300.1],E11=[0.010.020.10.2],E12=[0.30.20.10],D11=[0.10.10.20.3],D12=[0.20.210.1],Π=[-668-8].
Denote transition probabilities by p1 and p2. By using [p1p2]Π=0 and p1+p2=1, we can obtain p1=4/7 and p2=3/7. Figure 1 shows the Markovian switching signal within 100 times according to the above transition probabilities.
Choose α=0.1, τ=0.1, h=1, c1=1, c2=4, T=10, R1=R2=I, γ=3, and d=0.1. Then, solving conditions (41), (11), (12), and (31) in Theorem 12 for λ1=2 and λ2=2.01 yields
(44)K1=[0.1209-0.1828-0.17430.0778],K2=[-0.11060.38900.3831-0.7077],λ3=0.0012.
The state trajectories of the closed-loop system are shown in Figure 2. It is easy to see that the system is finite-time stochastically bounded.
Markovian switching signal.
State trajectory of the closed-loop system.
6. Conclusions
In this paper, finite-time stochastic stability and finite-time stochastic H∞ control problem for time-delayed Itô stochastic systems with Markovian switching are investigated with Lyapunov-Krasovskii functional approach and free-weighting matrix techniques. Some criteria are established. One example is given for illustration.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities under Grants 2012ZM0059 and 2012ZM0079, the Natural Science Foundation of Guangdong Province under Grant 10251064101000008, and the National Natural Science Foundation of China under Grant 61273126. The authors would like to thank the editors and anonymous reviewers for their constructive comments and suggestions for improving the quality of the work.
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