Research article

Two accelerated gradient-based iteration methods for solving the Sylvester matrix equation AX + XB = C

  • Received: 25 September 2024 Revised: 17 November 2024 Accepted: 29 November 2024 Published: 12 December 2024
  • MSC : 15A24, 65F30

  • In this paper, combining the precondition technique and momentum item with the gradient-based iteration algorithm, two accelerated iteration algorithms are presented for solving the Sylvester matrix equation AX+XB=C. Sufficient conditions to guarantee the convergence properties of the proposed algorithms are analyzed in detail. Varying the parameters of these algorithms in each iteration, the corresponding adaptive iteration algorithms are also provided, and the adaptive parameters can be explicitly obtained by the minimum residual technique. Several numerical examples are implemented to illustrate the effectiveness of the proposed algorithms.

    Citation: Huiling Wang, Nian-Ci Wu, Yufeng Nie. Two accelerated gradient-based iteration methods for solving the Sylvester matrix equation AX + XB = C[J]. AIMS Mathematics, 2024, 9(12): 34734-34752. doi: 10.3934/math.20241654

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  • In this paper, combining the precondition technique and momentum item with the gradient-based iteration algorithm, two accelerated iteration algorithms are presented for solving the Sylvester matrix equation AX+XB=C. Sufficient conditions to guarantee the convergence properties of the proposed algorithms are analyzed in detail. Varying the parameters of these algorithms in each iteration, the corresponding adaptive iteration algorithms are also provided, and the adaptive parameters can be explicitly obtained by the minimum residual technique. Several numerical examples are implemented to illustrate the effectiveness of the proposed algorithms.



    Most of the physical models are nonlinear in nature, and they frequently involve many complex input-output relationships. Over the years, different nonlinear control techniques were developed to obtain the actual behavior of the nonlinear models [1,2]. The Takagi–Sugeno (T–S) fuzzy model approach is recognized as an effective tool for approximation of complex nonlinear systems among several control methods [3]. The development and application of the T–S fuzzy model have greatly increased for the study of nonlinear systems. The universal approximation principle, which states that a T–S fuzzy model may estimate any smooth nonlinear system with any degree of certainty, made it possible to use a T–S fuzzy model to investigate nonlinear systems. The T–S can represent a nonlinear system into local linear models through nonlinear membership functions so that established stability and control theories can be applied directly [4,5,6]. The major purpose of the T–S fuzzy control technique is that the stability and control architectures can be transformed into the linear matrix inequality (LMI) framework. [7].

    Time delay is inherently present in most engineering systems, and it is one of the primary reasons of instability and performance degradation [8,9]. Numerous results addressing the synthesis and analysis of T–S fuzzy systems with variable time-delay conditions have been obtained as a result of the industrial need [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. The Lyapunov second approach can be used to analyze the stability of time-delay systems in two ways, (i) delay-dependent stability [8,9,10,11,12,13,14], and (ii) delay-independent stability [15]. A Lyapunov–Krasovskii functional (LKF) with an integral term is developed to obtain an efficient delay-dependent stability condition. Constructing a suitable LKF and estimating the integral term in the derivative of the LKF is the most common strategy for minimizing conservativeness. So far, various methods have been presented for processing the integral term and reducing the conservatism, such as model transformation method [10], free-weighting matrix approach was proposed in [11], Jensen's inequality technique in [5,8,17], reciprocal convex lemma in [12], Wirtinger-based inequality [13], auxiliary-function-based inequality [14]. In order to further improve the stability conditions of time-delay systems, quadratic function negative-determination lemma was introduced in [16,18].

    Apart from the various bounding inequalities, constructing a appropriate LKF is another key point. In the recent years, delay-partitioning LKFs and augmented LKFs are broadly studied and successful results have been achieved [21,22]. However, introducing too many free matrices makes computing LMIs immensely challenging, increasing processing complexity. Recently, a line integral fuzzy Lyapunov function was used to analyze the stability of T–S fuzzy systems in [23]. In the meantime, the concept was utilized for LMI based control design in [24]. Although the line integral Lyapunov function can be utilized to avoid time derivatives, it can also result in bilinear matrix inequalities in the controller design, which can be difficult to extrapolate for higher-order systems [25,26,27,28,29,30]. In the recent years, augmented LKF technique and Bessel-Legendre ploynomial based integral inequality have been used to obtain the less conservative results for T–S fuzzy TDS in [31,32,33,34,35,36,37,38,39,40]. As a result, a less conservative criterion can be obtained by selecting a suitable LKF and developing a new integral inequality. A unique LKF construction method, called delay-product-type (DPT) functional approach, was recently introduced and analyzed in [31,34,36,39], taking into account both conservatism and computational burden. Specifically, this LKF construction method multiplies time-varying delay terms with integral and non-integral terms, resulting in a LKF with additional time-delay information. Also, because the restriction on certain places is eased when using the DPT functional method, the LKF have a more comprehensive form. Thus, by combining the new Bessel-Legendre polynomial-based relaxed integral inequality with the augmented and DPT LKF, it is possible to obtain a less conservative admissibility condition for T–S fuzzy systems with time-varying delays. This is the motivation of this paper.

    The aim of this paper is to study the stability of T–S fuzzy systems with time-varying delays. The main contributions of this study are listed as follows:

    (i) A less conservative stability condition is established for T–S fuzzy TDS by constructing a suitable DPT augmented LKF. The proposed LKF has the motivation that, situations of various delay derivative nature can be handled with ease and less complexity.

    (ii) Bessel-Legendre polynomial based relaxed integral inequality is used to estimate the integral terms coming out from the derivative of LKF.

    (iii) The advantages of the proposed stability criteria are demonstrated using three numerical examples and a comparison of maximum delay upper bound results with various recent stability criteria.

    Notation: Throughout this paper, AT and A1 stands for the transpose and the inverse of the matrix respectively, Rn denotes the n-dimensional Euclidean space; Rn×m is the set of all n×m real matrices; P>0 means that the matrix is positive definite; Sym(A) is defined as A+AT; for any square matrix A and B, define diag{A,B}=[A00B]. The notation I stands for the identity matrix and [ABC] stands for [ABBTC].

    Consider the following nonlinear system with time delay as

    ˙η(t)=f(t,η(t),η(tγ(t))),t0,η(t)=ϕ(t),γ2t0, (2.1)

    where η(t)Rn is the state vector, f is a non-linear function, η(t)=ϕ(t) denote the initial condition on [γ2,0] and γ(t) is the time-varying delay differential function.

    The T–S fuzzy model of the system given in (2.1) can be described by following IF-THEN form as

    Rule i: IF θ1(t) is Gi1 and.... and θp(t)is Gip THEN

    ˙η(t)=Aiη(t)+Aγiη(tγ(t)),t0,η(t)=ϕ(t),γ2t0, (2.2)

    where θ1(t),θ2(t),.....,θp(t) are the premises variables, Gij are the fuzzy membership functions with i=1,2,3,..,r,j=1,2,3,..,p, the scalars r and p indicates the number of fuzzy IF-THEN rules and number of premise variable, respectively. Ai,Aγi are known system matrices of appropriate dimensions. The delay differential function γ(t) satisfy the following:

    0<γ1γ(t)γ2,ν1˙γ(t)ν2, (2.3)

    where γ1,γ2,ν1 and ν2 are given positive scalars represent the lower and upper bound of γ(t) and ˙γ(t), respectively.

    If θj(t)=θ0j are given, where θ0j are singletons, then for each ith fuzzy rule, the aggregation of the fuzzy rule using fuzzy 'min' operator can be expressed as

    λi(θ(t))=(Gi1(θ1(t))...Gip(θp(t))),i=1,2,..,r, (2.4)

    where Gi1(θ1(t)),....,Gip(θp(t)) is the grade of the membership of θ1(t),....,θp(t) in Gij.

    By fuzzy blending technique, the final output of (2.2) is calculated as

    ˙η(t)=ri=1λi(θ(t)){Aiη(t)+Aγiη(tγ(t))}ri=1λi(θ(t))=ri=1wi(θ(t)){Aiη(t)+Aγiη(tγ(t))}, (2.5)

    where wi(θ(t))=λi(θ(t))ri=1λi(θ(t)),t and i=1,2,....,r, is called the fuzzy weighting function and λi(θ(t))=pj=1Gij(θj(t)). Since λi(θ(t))>0, it holds that wi(θ(t))0 and ri=1wi(θ(t))=1 for all i=1,2,...,r. Further, wi(θ(t)) will be denoted as wi for simplicity.

    The objective of this study is to derive a delay-range-dependent stability condition for T–S fuzzy time delay system (2.5). Following lemmas are used to obtain our main result.

    Lemma 1. ([40]) For a positive definite matrix R>0, and any continuously differentiable function η(.):[δ1,δ2]Rn, the following inequality holds

    δ12δ2δ1˙ηT(ρ)R˙η(ρ)dρΨT1RΨ1+3ΨT2RΨ2+5ΨT3RΨ3+7ΨT4RΨ4, (2.6)

    where

    Ψ1=[η(δ2)η(δ1)],δ12=(δ2δ1),Ψ2=[η(δ2)+η(δ1)2δ12δ2δ1η(ρ)dρ],Ψ3=[η(δ2)η(δ1)+6δ12δ2δ1η(ρ)dρ12δ212δ2δ1δ2θη(ρ)dρdθ],Ψ4=[η(δ2)+η(δ1)12δ12δ2δ1η(ρ)dρ+60δ212δ2δ1δ2θη(ρ)dρdθ120δ312δ2δ1δ2θδ2λη(ρ)dρdλdθ].

    Lemma 2. ([40]) Let η(t) be a continuously differentiable function, RRn×n be real symmetric positive definite matrix and ψ1(t),ψ2(t)R4n×n are real vectors, ϵ1,ϵ2[0,1],δ1δ(t)δ2 are positive real scalars and τ satisfies τ(0,1). If there exists any real symmetric matrices M1,M2R4n×4n and any appropriately dimensioned matrices X1,X2R4n×4n,

    [M1X1ϵ2M1]0,[ϵ1M2X2M2]0, (2.7)

    such that the inequality

    δ12tδ1tδ2˙ηT(ρ)R˙η(ρ)dρχT(t)[ψT1(+(1τ)M1+ϵ1(1τ)2τM2)ψ1Sym{ψT1(τX1+(1τ)X2)ψ2}ψT2(+τM2+ϵ2τ21τM1)ψ2]χ(t), (2.8)

    holds, where

    χ(t)=[ηT(tδ1),ηT(tδ(t)),ηT(tδ2),1δ(t)δ1tδ1tδ(t)ηT(ρ)dρ,1δ1δ(t)tδ(t)tδ2ηT(ρ)dρ,2(δ(t)δ1)2tδ1tδ(t)tδ1θηT(ρ)dρdθ,2(δ2δ(t))2tδ(t)tδ2tδ(t)θηT(ρ)dρdθ,6(δ(t)δ1)3tδ1tδ(t)tδ1θtδ1ληT(ρ)dρdλdθ,6(δ2δ(t))3tδ(t)tδ2tδ(t)θtδ(t)ληT(ρ)dρdλdθ],ψ1=[σT1σT2σT3σT4]T,ψ2=[σT5σT6σT7σT8]T,=diag{R,3R,5R,7R},σ1=(¯e1¯e2),σ2=(¯e1+¯e22¯e4),σ3=(¯e1¯e2+6¯e46¯e6),σ5=(¯e2¯e3),σ6=(¯e2+¯e32¯e5),σ7=(¯e2¯e3+6¯e56¯e7),σ4=(¯e1+¯e212¯e4+30¯e620¯e8),σ8=(¯e2+¯e312¯e5+30¯e720¯e9),¯eq=[0n×(q1)nIn×n0n×(9q)n],q=1,2,...,9.

    Remark 1. It is worth noting that Lemma 2 is coupled to the two predefined individual factors ϵ1 and ϵ2. ϵ1 and ϵ2 can be determined independently because they are distinct of each other and unrestricted.

    Lemma 3. ([19]) Let f(s)=a2s2+a1s+a0, where s[h1,h2] and a0,a1,a2R. Supposethat the following conditions are satisfied

    (i)f(h1)<0,(ii)f(h2)<0,(iii)a2(h2h1)2+f(h1)<0.

    Then, f(s)<0.

    In this section, an improved DRD stability condition is established for the T–S fuzzy TDS (2.5). The following notations are defined for simplicity:

    ξ(t)=[ηT(t),ηT(tγ1),ηT(tγ(t)),ηT(tγ2),˙ηT(t),˙ηT(tγ1),˙ηT(tγ(t)),˙ηT(tγ2),2γ1ttγ1ηT(ρ)dρ,1γ(t)γltγltγ(t)xT(ρ)dρ,1γ2γ(t)tγ(t)tγ2ηT(ρ)dρ,2γ21ttγ1tθηT(ρ)dρdθ,2(γ(t)γ1)2tγ1tγ(t)tγ1θηT(ρ)dρdθ,2(γ2γ(t))2tγ(t)tγ2tγ(t)θηT(ρ)dρdθ6γ31ttγ1tθtληT(ρ)dρdλdθ,6(γ(t)γ1)3tγ1tγ(t)tγ1θtγ1ληT(ρ)dρdλdθ6(γ2γ(t))3tγ(t)tγ2tγ(t)θtγ(t)ληT(ρ)dρdλdθ]T,γ12=(γ2γ1),α=γ(t)γ1γ12,¯ξ(t)=[ηT(t),ηT(tγ(t)),ηT(tγ2),˙ηT(t),˙ηT(tγ(t)),˙ηT(tγ2),1γ(t)ttγ(t)xT(ρ)dρ,1γ2γ(t)tγ(t)tγ2ηT(ρ)dρ,2(γ(t))2ttγ(t)tθηT(ρ)dρdθ,2(γ2γ(t))2tγ(t)tγ2tγ(t)θηT(ρ)dρdθ6(γ(t))3ttγ(t)tθtληT(ρ)dρdλdθ,6(γ2γ(t))3tγ(t)tγ2tγ(t)θtγ(t)ληT(ρ)dρdλdθ]T,¯α=γ(t)γ2,ˆep=[0n×(p1)nIn×n0n×(6p)n],p=1,2,...,6,eq=[0n×(q1)nIn×n0n×(17q)n],q=1,2,...,17,˜er=[0n×(r1)nIn×n0n×(3r)n],r=1,2,3,¯es=[0n×(s1)nIn×n0n×(12s)n],s=1,2,...,12.

    Theorem 1. For given scalars γ1,γ2,ν1 and ν2, the T–S fuzzy TDS (2.5) with (2.3), is asymptotically stable if there exist symmetric positive definite matrices P1R6n×6n,P2R4n×4n,QlR2n×2n,(l=1,2,3,4), RkRn×n, symmetric matrices MkR4n×4n, and any matrices YkR4n×4n(k=1,2) and Np(p=1,2,3) with suitable dimension such that the following LMIs are satisfied for all ˙γ(t)[ν1,ν2],i=1,2,...,r:

    P1+γ1ΥTP2Υ>0,P1+γ2ΥTP2Υ>0, (3.1)
    [ˆR2M1Y1ˆR2M1]0,[ˆR2Y2ˆR2M2]0, (3.2)
    [ˆR2M1Y1ˆR2]0,[ˆR2M2Y2ˆR2M2]0, (3.3)

    and

    f(0,˙γ(t))<0,f(1,˙γ(t))<0, (3.4)
    Ξ2+f(0,˙γ(t))<0, (3.5)

    where

    Proof. Choose the delay-dependent LKF as follows:

    V(ηt)=4q=1Vq(ηt), (3.6)

    where

    V1(ηt)=ϖT1(t)P1ϖ1(t),V2(ηt)=γ(t)ϖT2(t)P2ϖ2(t),V3(ηt)=ttγ1ϖT3(s)Q1ϖ3(s)ds+ttγ2ϖT3(s)Q2ϖ3(s)ds+tγ1tγ(t)ϖT3(s)Q3ϖ3(s)ds+tγ(t)tγ2ϖT3(s)Q4ϖ3(s)ds,V4(ηt)=γ10γ1tt+λ˙ηT(s)R1˙η(s)dsdλ+γ12γ1γ2tt+λ˙ηT(s)R2˙η(s)dsdλ,

    with

    ϖ1(t)=[ηT(t),ttγ1ηT(s)ds,tγ1tγ(t)ηT(s)ds,tγ(t)tγ2ηT(s)ds,2γ10γ1tt+ληT(s)dsdλ,6γ210γ10λtt+θηT(s)dsdθdλ]T,ϖ3(t)=[ηT(t),˙ηT(t)]T,ϖ2(t)=[ηT(t),ttγ1ηT(s)ds,tγ1tγ(t)ηT(s)ds,tγ(t)tγ2ηT(s)ds]T.

    In order to meet the stability condition of the fuzzy system (2.5) using the Lyapunov method, first we show the positive definiteness of V(ηt). From the LKF V1(ηt) and V2(ηt), we deduce

    V1(ηt)+V2(ηt)=ϖT1(t)[P1+γ(t)ΥTP2Υ]ϖ1(t)=ϖT1(t)[P1+(γ1+αγ12)ΥTP2Υ]ϖ1(t), (3.7)

    where Υ is defined after (3.5).

    Therefore, if the LMIs (3.1) is holds; then V1(ηt)+V2(ηt)>ϵηt2 should be fulfilled with ϵ>0. Hence, the positivity of V(ηt) ensure when the LMIs in (3.1) and Ql,Rk>0(l=1,2,3,4,k=1,2) hold.

    Finding the time derivative of (3.3) along with the trajectory of (2.5), one can obtain

    ˙V(ηt)=4q=1˙Vq(ηt), (3.8)

    where

    ˙V1(ηt)=2ϖT1(t)P1[˙η(t)η(t)η(tγ1)η(tγ1)(1˙γ(t))η(tγ(t))(1˙γ(t))η(tγ(t))η(tγ2)2η(t)2γ1ttγ1η(s)ds3η(t)6γ21ttγ1tλη(s)dsdλ]=2ξT(t)[(ΔT1a+αΔT1b+(1α)ΔT1c)P1Δ2]ξ(t)=ξT(t)[Sym{(ΔT1a+ΔT1c)P1Δ2}+αSym{(ΔT1bΔT1c)P1Δ2}]ξ(t). (3.9)

    Similarly, we obtain

    ˙V2(ηt)=˙γ(t)ϖT2(t)P2ϖ2(t)+2γ(t)ϖT2(t)P2˙ϖ2(t)=ξT(t)[˙γ(t)(ΔT3a+αΔT3b+(1α)ΔT3c)P2(Δ3a+αΔ3b+(1α)Δ3c)+(γ1+αγ12)Sym{(ΔT3a+αΔT3b+(1α)ΔT3c)P2Δ4}]ξ(t)=ξT(t)[˙γ(t)(ΔT3a+ΔT3c)P2(Δ3a+Δ3c)+γ1Sym{(ΔT3a+ΔT3c)P2Δ4}+α{γ1Sym{(ΔT3bΔT3c)P2Δ4}+˙γ(t)Sym{(ΔT3a+ΔT3c)P2(Δ3bΔ3c)}+γ12Sym{(ΔT3a+ΔT3c)P2Δ4}}+α2{(ΔT3bΔT3c)P2(Δ3bΔ3c)+γ12Sym{(ΔT3bΔT3c)P2Δ4}}]ξ(t), (3.10)
    (3.11)
    ˙V4(ηt)=˙ηT(t)(γ21R1+γ212R2)˙η(t)γ1ttγ1˙ηT(s)R1˙η(s)dsγ12tγ1tγ2˙ηT(s)R2˙η(s)ds, (3.12)

    where Δl(l=2,4,5...,8),Δ1a,Δ1b,Δ1c,Δ3a,Δ3b, and Δ3c can be found after (3.5).

    The first integral term in the right hand side (RHS) of (3.12) contains only constant limits of integration, Lemma 1 yields

    γ1ttγ1˙ηT(s)R1˙η(s)dsξT(t)[ΔT9R1Δ9+3ΔT10R1Δ10+5ΔT11R1Δ11+7ΔT12R1Δ12]ξ(t). (3.13)

    Next, treating second integral term in the RHS of (3.13) containing uncertain limit of integration. According to Lemma 2, we choose ϵ1=α and ϵ2=(1α)(α[0,1]).

    If the LMIs [ˆR2M1Y1ˆR2(1α)M2] and [ˆR2αM1Y2ˆR2M2] is satisfied for all α[0,1], then approximate the second integral terms in the RHS of (3.13) Lemma 2 is applied, yields

    (3.14)

    where Δl(l=9,10,...,20),Ψ1,Ψ2 and ˆR2 are defined after (3.5).

    For any free weighting matrices Np(p=1.2.3) with suitable dimension, the following zero equation holds

    0=2ri=1wi[ηT(t)N1+ηT(tγ(t))N2+˙ηT(t)N3]×[Aiη(t)+Aγiη(tγ(t))˙η(t)]=ri=1wiξT(t)[Sym{Δ21Θi}]ξ(t), (3.15)

    where

    Δ21=[eT1N1+eT3N2+eT5N3],Θi=[Aie1+Aγie3e5].

    Then, by substituting (3.9)–(3.15) in (3.8), we obtain

    (3.16)

    where f(α,˙γ(t)) is defined after (3.5).

    Note that, the RHS of (3.16) depends on the two parameters α[0,1] and ˙γ(t)[ν1,ν2]. Since ri=1wi=1 and the RHS of (3.16) is quadratic with respect to α, so by Lemma 3 we can easily obtain the LMIs in (3.4) and (3.5). Thus, if the LMIs (3.4) and (3.5) along with constraint (3.1)–(3.3) are holds, then it implies that ˙V(ηt)<ϵηt2,forϵ>0, which in turn guaranteed the asymptotic stability of the fuzzy system (2.5) as per Lyapunov–Krasovskii Theorem. This completes the proof of Theorem 1.

    Remark 2. If γ1=0, then γ12=γ2, then Theorem 1 is no more applicable to find the maximum delay upper bound γ2 for stability of the T–S fuzzy TDS (2.5). The following Corollary is formulated to deal with this circumstance.

    Corollary 1. Given scalars γ2,ν1 and ν2, the T–S fuzzy TDS (2.5) with 0γ(t)γ2,ν1˙γ(t)ν2 is asymptotically stable if there exist symmetric positive definite matrices ¯P1,¯P2R3n×3n,¯QlR2n×2n,(l=1,2,3), RRn×n, symmetric matrices ¯MkR4n×4n, and any matrices ¯YkR4n×4n(k=1,2) and ¯Np(p=1,2,3) with suitable dimension such that the following LMIs are satisfied for all ˙γ(t)[ν1,ν2],i=1,2,...,r:

    ¯P1+γ2¯ΥT¯P2¯Υ>0, (3.17)
    [ˆR¯M1¯Y1ˆR¯M1]0,[ˆR¯Y2ˆR¯M2]0, (3.18)
    [ˆR¯M1¯Y1ˆR]0,[ˆR¯M2¯Y2ˆR¯M2]0, (3.19)

    and

    ¯f(0,˙γ(t))<0,¯f(1,˙γ(t))<0, (3.20)
    ¯Ξ2+¯f(0,˙γ(t))<0, (3.21)

    where

    ¯f(0,˙γ(t))=¯f(¯α,˙γ(t))|¯α=0,¯f(1,˙γ(t))=¯f(¯α,˙γ(t))|¯α=1,¯f(¯α,˙γ(t))=¯α2¯Ξ2+¯α¯Ξ1+(¯Ξ0+Sym{¯Δ14¯Θi)},¯Ξ0=Sym{(¯ΔT1a+¯ΔT1c)¯P1¯Δ2}+˙γ(t)(¯ΔT1a+¯ΔT1c)¯P2(¯Δ1a+¯Δ1c)+¯eT4(γ22R)¯e4+¯ΔT3(¯Q1+¯Q2)¯Δ3+(1˙γ(t))¯ΔT4(¯Q2+Q3)¯Δ4+¯ΔT5(¯Q1¯Q3)¯Δ5¯ΨT1(ˆR+¯M1+¯M2)¯Ψ1¯ΨT2ˆR¯Ψ2Sym{¯ΨT1¯Y2¯Ψ2},¯Ξ1=Sym{(¯ΔT1b¯ΔT1c)¯P1¯Δ2}+˙γ(t)Sym{(¯ΔT1a+¯ΔT1c)¯P2(¯Δ1b¯Δ1c)}¯ΨT2¯M2¯Ψ2+γ2Sym{(¯ΔT1a+¯ΔT1c)¯P2¯Δ2}¯ΨT1(¯M12¯M2)¯Ψ1Sym{¯ΨT1(¯Y1¯Y2)¯Ψ2},¯Ξ2=˙γ(t)(¯ΔT1b¯ΔT1c)¯P2(¯Δ1b¯Δ1c)+γ2Sym{(¯ΔT1b¯ΔT1c)¯P2¯Δ2}¯ΨT1¯M2¯Ψ1¯ΨT2¯M1¯Ψ2,¯Υ=[˜eT1,˜eT2,˜eT3]T,ˆR=diag{R,3R,5R,7R},¯Δ1a=[¯eT1,0,0]T,¯Δ1b=[0,γ2¯eT7,0]T,¯Δ1c=[0,0,γ2¯eT8]T,¯Δ2=[¯eT4,(¯e1(1˙γ(t))¯e2)T,((1˙γ(t))¯e2¯e3)T]T,¯Δ3=[¯eT1,¯eT4]T,¯Δ4=[¯eT2,¯eT5]T,¯Δ5=[¯eT3,¯eT6]T,¯Ψ1=[¯ΔT6,¯ΔT7,¯ΔT8,¯ΔT9]T,¯Ψ2=[¯ΔT10,¯ΔT11,¯ΔT12,¯ΔT13]T,¯Δ6=(¯e1¯e2),¯Δ7=(¯e1+¯e22¯e7),¯Δ8=(¯e1¯e2+6¯e76¯e9),¯Δ9=(¯e1+¯e212¯e7+30¯e920¯e11),¯Δ10=(¯e2¯e3),¯Δ11=(¯e2+¯e32¯e8),¯Δ13=(¯e2+¯e312¯e8+30¯e1020¯e12),¯Δ12=(¯e2¯e3+6¯e86¯e10),¯Δ14=[¯eT1N1+¯eT2N2+¯eT4N3],¯Θi=[Ai¯e1+Aγi¯e2¯e4].

    Proof. Let us consider the LKF as follows:

    ¯V(ηt)=4q=1¯Vq(ηt), (3.22)

    where

    ¯V1(ηt)=¯ϖT1(t)¯P1¯ϖ1(t),¯V2(ηt)=γ(t)¯ϖT1(t)¯P2¯ϖ1(t),¯V3(ηt)=ttγ2¯ϖT2(s)¯Q1¯ϖ2(s)ds+ttγ(t)¯ϖT2(s)¯Q2¯ϖ2(s)ds+tγ(t)tγ2¯ϖT2(s)¯Q3¯ϖ2(s)ds,¯V4(ηt)=γ20γ2tt+λ˙ηT(s)R˙η(s)dsdλ,

    with

    ¯ϖ1(t)=[ηT(t),ttγ(t)ηT(s)ds,tγ(t)tγ2ηT(s)ds]T,¯ϖ2(t)=[ηT(t),˙ηT(t)]T.

    Now, take the similar steps as in Theorem 1 yields the LMIs in (3.17)–(3.21). The analysis is skipped here because it is straightforward.

    Three numerical examples are given in this section to demonstrate the reduction of conservativeness of our proposed method numerically.

    Example 1. Consider the following T–S fuzzy TDS [31]:

    ˙η(t)=2i=1wi{Aiη(t)+Aγiη(tγ(t))}, (4.1)

    where

    A1=[2000.9],A2=[10.501],Aγ1=[1011],Aγ2=[100.11],

    and the membership functions are defined as w1=11+e2η1(t),w2=1w1.

    For given ν1=0,ν2=0.1, delay upper bound γ2 is calculated by using Theorem 1 with different values γ1. The obtained delay bound results are presented in Table 1 along with the results of some recent stability conditions. Further, numerical simulation for the system (4.1) is carried out and it is consider that γ(t)=0.5768sin(ωt)+1.3768 is a very slow varying sine signal (ω=3.141rad/secorf=0.5Hz). Figure 1 validates the fact that, the system states are asymptotically stable for the initial condition η(0)=[2,2]T.

    Table 1.  Admissible upper bound γ2 for various γ1 and ν=0.1 for Example 1.
    γ1/Method [5] [6] [20] [30] [31] Theorem 1
    0.8 1.6539 1.7633 1.7828 1.857 NA 1.9536
    1.0 1.8069 1.7718 1.8106 1.868 1.9405 1.9956

     | Show Table
    DownLoad: CSV
    Figure 1.  State responses of the system given in Example 1 with γ1=0.8,γ2=1.9536.

    Example 2. Consider the T–S fuzzy TDS (2.5) with a two plant rule and the parameters are as follows:

    A1=[3.20.602.1],A2=[1013],Aγ1=[10.902],Aγ2=[0.9011.6].

    The membership functions are chosen as w1=11+e2η1(t),w2=1w1. Let γ1=0 and ν=ν2=ν1. For given various ν, the maximum allowable upper bounds of γ2 for Corollary 1 are obtained according to Remark 2. The results of Corollary 1 are given in Table 2 along with several recent results from the literatures in [32,33,34,35,36]. By choosing γ1=0,γ2=1.6519 and η(0)=[6,4]T, Figure 2 shows the state trajectory η(t). The state responses clearly indicate that T–S fuzzy system considered in Example 2 is asymptotically stable.

    Table 2.  Maximum delay bound γ2 for various ν and γ1=0 for Example 2.
    Method ν=0.03 ν=0.1 ν=0.5
    [32] 0.8771 0.7687 0.7584
    [33] 0.9281 0.8092 0.7671
    [34] (R1=R2=0) 1.8328 1.3857 1.2186
    [34] 1.9137 1.4354 1.3123
    [35] 2.4291 1.7493 1.6355
    [36] (Theorem 1 with (I)) 2.9931 1.8916 1.4594
    [36] (Theorem 2 with (I)) 2.6160 1.6084 1.3409
    Corollary 1 3.0130 1.6519 0.9925

     | Show Table
    DownLoad: CSV
    Figure 2.  State responses of the system given in Example 2.

    Remark 3. Based on the results presented in Table 1 and Table 2, it can be seen that the proposed Theorem 1 and Corollary 1 of this paper are less conservative than the existing results [5,6,20,30,31,32,33,34,35,36], which shows the effectiveness and superiority of the method.

    Example 3. Consider the T–S fuzzy TDS given in [27] as

    ˙η(t)=2i=1wi{Aiη(t)+Aγiη(tγ(t))}, (4.2)

    where

    A1=[2.10.10.20.9],A2=[1.90.00.21.1],Aγ1=[1.10.10.80.9],Aγ2=[0.90.01.11.2],

    and the membership functions are w1=11+e2η1(t),w2=1w1.

    By solving the LMIs in Corollary 1, delay upper bounds (γ2) obtained for given different values of ν with γ1=0 and the results are given in Table 3. From Table 3, it is found that the results presented in this paper is less conservative than previous researches [27,34,37,38,39]. Figure 3 show the state responses of the fuzzy system given in Example 3 with γ1=0,γ2=2.5198 and η(0)=[2,1]T.

    Table 3.  Delay bound γ2 with various ν=ν2=ν1 and γ1=0 for Example 3.
    Method ν=0.1 ν=0.5
    [34] 3.42 2.02
    [27] 3.5518 2.3204
    [37] 4.2044 2.0685
    [38] 4.324 2.226
    [39] 5.2300 3.3454
    Corollary 1 5.4985 2.5198

     | Show Table
    DownLoad: CSV
    Figure 3.  State responses of the system given in Example 3.

    Remark 4. The Lyapunov conditions for finite-time stability of impulsive systems is proposed in [41], where the settling-time is well estimated via impulsive signals. In [42], the authors addressed a class of nonlinear systems with delayed impulses, where the double effects (i.e., negative and positive effects) of time delays in impulses are fully and systematically considered. In the framework of Lyapunov conditions, in [43], the authors proposed a novel Zeno-free event-triggered impulsive control strategy for uniform stability and asymptotic stability, where a class of forced impulse sequences was introduced freely. Also, singular systems have found widespread use in circuits, power systems, economic models, interconnected systems, and neural network models in recent years [44,45,46,47]. Further research topics would be considered to extend the main results of this paper to design an event-triggered control scheme and filter design for the T–S fuzzy TDS or singular network systems with induced network delays.

    An improved DRD stability criteria in a LMI framework has been proposed in this paper for T–S fuzzy TDS. A new stability condition that successfully reduces conservativeness is obtained by constructing a suitable DPT LKF and estimating the derivative of LKF using the Bessel–Legendre polynomial-based relaxed integral inequality. Moreover, three numerical examples are given that compare maximum acceptable delay bounds to highlight the advantages and usefulness of the proposed criteria.

    This work was supported by Grant-in-Aid for Research Activity Start-up No. 20K23328, funded by Japan Society for the Promotion of Science (JSPS) and in part by the Hiroshima University KIBANKEIHI grant No. OBNK14, Japan.

    The authors declare that there is no conflict of interest regarding the publication of this paper.



    [1] A. L. Andrew, Eigenvectors of certain matrices, Linear Algebra Appl., 7 (1973), 151–162. http://dx.doi.org/10.1016/0024-3795(73)90049-9 doi: 10.1016/0024-3795(73)90049-9
    [2] Z. Z. Bai, G. H. Golub, J. Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1–32. http://dx.doi.org/10.1007/s00211-004-0521-1 doi: 10.1007/s00211-004-0521-1
    [3] Z. Z. Bai, On hermitian and skew-hermitian splitting iteration methods for continuous sylvester equations, J. Comput. Math., 29 (2011), 185–198. http://dx.doi.org/10.4208/jcm.1009-m3152 doi: 10.4208/jcm.1009-m3152
    [4] Z. Z. Bai, M. Benzi, F. Chen, Z. Q. Wang, Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal., 33 (2013), 343–369. http://dx.doi.org/10.1093/imanum/drs001 doi: 10.1093/imanum/drs001
    [5] A. Bhaya, E. Kaszkurewicz, Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method, Neural Networks, 17 (2004), 65–71. http://dx.doi.org/10.1016/S0893-6080(03)00170-9 doi: 10.1016/S0893-6080(03)00170-9
    [6] Z. B. Chen, X. S. Chen, Conjugate gradient-based iterative algorithm for solving generalized periodic coupled Sylvester matrix equation, J. Franklin I., 359 (2022), 9925–9951. http://dx.doi.org/10.1016/j.jfranklin.2022.09.049 doi: 10.1016/j.jfranklin.2022.09.049
    [7] M. Dehghan, A. Shirilord, The double-step scale splitting method for solving complex Sylvester matrix equation, Comp. Appl. Math., 38 (2019), 146. http://dx.doi.org/10.1007/s40314-019-0921-6 doi: 10.1007/s40314-019-0921-6
    [8] F. Ding, T. W. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE T. Automat. Contr., 50 (2005), 1216–1221. http://dx.doi.org/10.1109/TAC.2005.852558 doi: 10.1109/TAC.2005.852558
    [9] F. Ding, P. X. Liu, J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput., 197 (2008), 41–50. http://dx.doi.org/10.1016/j.amc.2007.07.040 doi: 10.1016/j.amc.2007.07.040
    [10] F. Ding, X. H. Wang, Q. J. Chen, Y. S. Xiao, Recursive least squares parameter estimation for a class of output nonlinear systems based on the model decompositions, Circuits Syst. Signal Process., 35 (2016), 3323–3338. http://dx.doi.org/10.1007/s00034-015-0190-6 doi: 10.1007/s00034-015-0190-6
    [11] X. Dong, X. H. Shao, H. L. Shen, A new SOR-like method for solving absolute value equations, Appl. Numer. Math., 156 (2020), 410–421. http://dx.doi.org/10.1016/j.apnum.2020.05.013 doi: 10.1016/j.apnum.2020.05.013
    [12] K. Du, C. C. Ruan, X. H. Sun, On the convergence of a randomized block coordinate descent algorithm for a matrix least squares problem, Appl. Math. Lett., 124 (2022), 107689. http://dx.doi.org/10.1016/j.aml.2021.107689 doi: 10.1016/j.aml.2021.107689
    [13] W. Fan, C. Gu, Z. Tian, Jacobi-gradient iterative algorithms for Sylvester matrix equations, 14th Conference of the International Linear Algebra Society, Shanghai, China, 2007, 16–20.
    [14] C. Q. Gu, H. Y. Xue, A shift-splitting hierarchical identification method for solving Lyapunov matrix equations, Linear Algebra Appl., 430 (2009), 1517–1530. http://dx.doi.org/10.1016/j.laa.2008.01.010 doi: 10.1016/j.laa.2008.01.010
    [15] B. H. Huang, W. Li, A modified SOR-like method for absolute value equations associated with second order cones, J. Comput. Appl. Math., 400 (2022), 113745. http://dx.doi.org/10.1016/j.cam.2021.113745 doi: 10.1016/j.cam.2021.113745
    [16] X. Li, H. F. Huo, A. L. Yang, Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations, Comput. Math. Appl., 75 (2018), 1095–1106. http://dx.doi.org/10.1016/j.camwa.2017.10.028 doi: 10.1016/j.camwa.2017.10.028
    [17] M. S. Mehany, Q. W. Wang, Three symmetrical systems of coupled Sylvester-like quaternion matrix equations, Symmetry, 14 (2022), 550. http://dx.doi.org/10.3390/sym14030550 doi: 10.3390/sym14030550
    [18] Q. Niu, X. Wang, L. Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control, 13 (2011), 461–464. http://dx.doi.org/10.1002/asjc.328 doi: 10.1002/asjc.328
    [19] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Comp. Math. Math. Phys., 4 (1964), 1–17. http://dx.doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
    [20] S. G. Shafiei, M. Hajarian, An iterative method based on ADMM for solving generalized Sylvester matrix equations, J. Franklin I., 359 (2022), 8155–8170. http://dx.doi.org/10.1016/j.jfranklin.2022.07.049 doi: 10.1016/j.jfranklin.2022.07.049
    [21] Z. L. Tian, M. Y. Tian, C. Q. Gu, X. N. Hao, An accelerated Jacobi-gradient based iterative algorithm for solving Sylvester matrix equations, Filomat, 31 (2017), 2381–2390. http://dx.doi.org/10.2298/FIL1708381T doi: 10.2298/FIL1708381T
    [22] Z. L. Tian, Y. D. Wang, Y. H. Dong, S. Y. Wang, New results of the IO iteration algorithm for solving Sylvester matrix equation, J. Franklin I., 359 (2022), 8201–8217. http://dx.doi.org/10.1016/j.jfranklin.2022.08.018 doi: 10.1016/j.jfranklin.2022.08.018
    [23] Q. W. Wang, R. Y. Lv, Y. Zhang, The least-squares solution with the least norm to a system of tensor equations over the quaternion algebra, Linear Multilinear A., 70 (2022), 1942–1962. http://dx.doi.org/10.1080/03081087.2020.1779172 doi: 10.1080/03081087.2020.1779172
    [24] Q. W. Wang, X. Wang, A system of coupled two-sided Sylvester-type tensor equations over the quaternion algebra, Taiwanese J. Math., 24 (2020), 1399–1416. http://dx.doi.org/10.11650/tjm/200504 doi: 10.11650/tjm/200504
    [25] Y. J. Xie, C. F. Ma, The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation, Appl. Math. Comput., 273 (2016), 1257–1269. http://dx.doi.org/10.1016/j.amc.2015.07.022 doi: 10.1016/j.amc.2015.07.022
    [26] A. L. Yang, Y. Cao, Y. J. Wu, Minimum residual Hermitian and skew-Hermitian splitting iteration method for non Hermitian positive definite linear systems, BIT Numer. Math., 59 (2019), 299–319. http://dx.doi.org/10.1007/s10543-018-0729-6 doi: 10.1007/s10543-018-0729-6
    [27] A. L. Yang, On the convergence of the minimum residual HSS iteration method, Appl. Math. Lett., 94 (2019), 210–216. http://dx.doi.org/10.1016/j.aml.2019.02.031 doi: 10.1016/j.aml.2019.02.031
    [28] J. F. Yin, Q. Y. Dou, Generalized preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, J. Comput. Math., 30 (2012), 404–417. http://dx.doi.org/10.4208/jcm.1201-m3209 doi: 10.4208/jcm.1201-m3209
    [29] X. F. Zhang, Q. W. Wang, Developing iterative algorithms to solve Sylvester tensor equations, Appl. Math. Comput., 409 (2021), 126403. http://dx.doi.org/10.1016/j.amc.2021.126403 doi: 10.1016/j.amc.2021.126403
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