Research article

Reachable set estimation for 2-D switched nonlinear positive systems with impulsive effects and bounded disturbances described by the Roesser model

  • Received: 05 September 2023 Revised: 07 January 2024 Accepted: 04 February 2024 Published: 14 May 2024
  • The reachable set estimation for two-dimensional (2-D) switched nonlinear positive systems (SNPSs) with bounded disturbances given by the Roesser model is investigated in this paper, in which both the time-varying delays and lagged impulsive effects are taken into account. By applying the average dwell time (ADT) technique, we provide a sufficient condition for the presence of a ball such that any solution of the system converges exponentially within it. An accurate estimate of the convergence rate is provided. We also extend the result to 2-D SNPSs with multi-directional delays, general 2-D switched linear systems, and 2-D SPNSs with heterogeneous delays. Finally, an example is worked out to demonstrate the effectiveness of the main result.

    Citation: Hongyu Ma, Dadong Tian, Mei Li, Chao Zhang. Reachable set estimation for 2-D switched nonlinear positive systems with impulsive effects and bounded disturbances described by the Roesser model[J]. Mathematical Modelling and Control, 2024, 4(2): 152-162. doi: 10.3934/mmc.2024014

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  • The reachable set estimation for two-dimensional (2-D) switched nonlinear positive systems (SNPSs) with bounded disturbances given by the Roesser model is investigated in this paper, in which both the time-varying delays and lagged impulsive effects are taken into account. By applying the average dwell time (ADT) technique, we provide a sufficient condition for the presence of a ball such that any solution of the system converges exponentially within it. An accurate estimate of the convergence rate is provided. We also extend the result to 2-D SNPSs with multi-directional delays, general 2-D switched linear systems, and 2-D SPNSs with heterogeneous delays. Finally, an example is worked out to demonstrate the effectiveness of the main result.



    Two-dimensional (2-D) systems are systems which can be used to model real-world engineering systems, and examples such as multi-dimensional digital filtering and circuit analysis [1,2] can be described by 2-D systems. As special types of 2-D systems, the Roesser model and and the Fornasini-Marchesini (FM) model have been given special attention because of their structures and applications [3,4,5]. In recent years, stability theory and control synthesis of 2-D systems have been extensively studied [6,7,8,9].

    Switched positive systems are comprised of a series of positive subsystems, and for any switching signal the states remain nonnegative if the initial conditions are nonnegative. Switched positive systems possess some properties of both switched systems and positive systems, so it is of great interest to study switched positive systems applying the methods which are used to discuss the positive systems, such as, co-positive Lyapunov functions approach (see the researches [10,11,12,13]). [14] considered the problem of the existence of common linear co-positive Lyapunov functions for one-dimensional switched positive systems, and the authors introduced multiple linear co-positive Lyapunov functions in [15]. Meanwhile, some practical systems in engineering are described by 2-D switched positive systems, for example, the thermal process with multiple models. The theory of the 2-D switched positive systems have been widely studied in recent years. In [16], the problems of exponential stability for 2-D switched positive systems were considered. The authors investigated the robust observer design for 2-D switched positive systems in [17]. In [18], a necessary and sufficient condition for the asymptotic stability of switched 2-D fractional order positive systems described by the Roesser model is established. Sufficient conditions for the stabilization by state feedback controllers for positive 2-D fractional order sub-systems were reported by [19]. [20] studied the stability problem of uncertain 2-D switched positive systems. Robust stability conditions of 2-D positive systems employing saturation conditions have been reported in [21].

    However, the majority of existing research is focused on 2-D switched linear positive systems (SLPSs), and the theory for 2-D switched nonlinear positive systems (SNPSs) is considerably less developed. The methods for studying SLPSs, such as linear copositive functions, are no longer applicable to SNPSs. The pioneering work on the stability analysis of a class of 2-D SNPSs was reported by [22]. Additionally, impulsive phenomena and external disturbances often occur in many real systems of which their states are subject to abrupt changes at certain moments. The research on impulsive systems has emerged in a variety of practical problems, such as in biology and communication networks [23,24]. Moreover, time-delay phenomena widely exists in practical engineering and it is one of the important reasons for system performance deterioration and instability [25,26]. Since the activated subsystem is changed at switched and impulsive instants, it is more complicated to makes the system analysis due to the existence of delays for the 2-D SNPSs with impulsive effects. To the best of our knowledge, few studies have attempted to conduct the estimation of reachable sets for the 2-D SNPSs subject to unknown disturbances and delayed impulse effects.

    In this paper, we consider the reachable set estimation for 2-D SNPSs given by the Roesser model with unknown exogenous disturbances. Both the systems delay and delayed impulse effects are considered. The contributions of this article are as follows:

    First, by applying the multiple max-separable Lyapunov functions approach, we present an explicit sufficient condition for the presence of a ball such that any solution of the system converges exponentially within it for bounded directional delays and delayed impulse effects.

    Second, if impulsive matrices and external disturbances are set to zero, then the considered system of this study reduces to existing one in [22]. Therefore, the existing results can be seen as a special case of this article. An accurate estimate of the convergence rate is also provided.

    Finally, we also extend the result to 2-D SNPSs with multiple directional delays, general 2-D switched linear systems, and 2-D SNPSs with heterogeneous time-varying delays.

    The rest of this article is organized as follows. Some necessary notations, definitions, and problem formulation are presented in Section 2. Our main results and the proofs are provided in Section 3. Section 4 gives an example to justify the efficiency of the obtained results, and the conclusions are stated in Section 5.

    R and N represent the sets of real and natural numbers, respectively, N0=N{0}. Rn is the set of n-dimensional real vectors, and

    Rn+:={xRn,xj0,1jn}.

    For x,yRn, denoted by xy (xy, xy), if xjyj (xj>yj, xj<yj) for 1jn. Given a positive vector ξ0,

    ||x||ξ=max1jn|xj|ξj.

    Denote the weighted l norm of xRn. Set

    ||x||=max1jn|xj|.

    Rn×n represents n×n-dimensional real matrices. En and On denote the identity matrix and zero matrix, respectively.

    In this paper, we consider 2-D SPNSs with lagged impulsive effects:

    {[xh(k+1,l)xv(k,l+1)]=fσ(k,l)[xh(k,l)xv(k,l)]+gσ(k,l)[xh(kτh(k),l)xv(k,lτv(l))] +ω(k,l),k+lεr,[xh(k,l)xv(k,l)]=Fσ(k,l)[xh(kdh(k),l)xv(k,ldv(l))],k+l=εr, (2.1)

    where xh(k,l)Rn1 and xv(k,l)Rn2 stand for horizontal and vertical state vectors, respectively. x(k,l)Rn represents the whole state with n=n1+n2. σ(k,l): N0×N0M={1,2,3,,m} is the switching rule. For any PM, the vector fields fp,gp: RnRn are continuous on Rn. The diagonal matrix

    FP=diag{FP11,FP11,,FPnn}

    is called the impulsive matrix, and we assume FPii>0 for all 1in. The exogenous disturbances are denoted by ω(k,l): N0×N0Rn.

    It is assumed in this study that the switching rule σ(k,l) relies on ε, that is, if

    k+l=˜k+˜l=ε,

    then σ(k,l)=σ(˜k,˜l). The switching sequence is stated as follows:

    (ε0,σ(ε0)), (ε1,σ(ε1)),,(εr,σ(εr)),,

    where εr=kr+lr. The σ(εr)-th subsystem is activated when k+l[εr,εr+1). We suppose system delays τh(k),τv(l) and impulsive delays dh(k),dv(l) are all bounded. Therefore, there exist nonnegative real numbers ^τh,^τv,^dh,^dv such that

    0τh(k)^τh,  0τv(l)^τv,0dh(k)^dh,  0dv(l)^dv,kdh(k)^τh, ldv(l)^τv.

    Denote

    τmax=max(^τh,^τv),dmax=max(^dh,^dv).

    The initial conditions are presented as follows:

    {xh(k,l)=h(k,l),^τhk0,0l¯h,xh(k,l)=0,^τhk0,l>¯h,xv(k,l)=v(k,l),^τvl0,0k¯v,xv(k,l)=0,^τvl0,k>¯v. (2.2)

    where ¯h and ¯v are positive real numbers, and h(k,l), v(k,l) are given positive vectors. Let

    ˆh(r)=maxpMsupτhk0h(k,r)ξpn1

    and

    ˆv(s)=maxpMsupτvl0v(s,l)ξpn2,

    where

    ξpn1=[En1  On1×n2] ξp,ξpn2=[On2×n1  En2] ξp.

    Definition 2.1. The impulsive switched system (2.1) is said to be positive if xh(k,l)0 and xv(k,l)0 hold for any nonnegative boundary condition h(k,l)Rn1,v(k,l)Rn2 and any nonnegative disturbance ω(k,l).

    Definition 2.2. A vector field f: RnRn is called homogeneous of degree one if for any xRn and λ>0,

    f(λx)=λf(x).

    g is defined to be order-preserving on Rn+ if g(x)g(y) for any x,yRn+ satisfying xy.

    Definition 2.3. For any nonnegative integers i,j and i0,j0 with

    i+j=εε0=i0+j0

    and any switching signal σ, let Nσ(ε0,ε) denote the number of switching times during the period [ε0,ε). If there exist two constants N0>0 and τε>0 such that

    Nσ(ε0,ε)N0+εε0τε,

    then τε is referred to as the average dwell time (ADT) of the switching signal σ and N0 is the chatter bound. In this paper, we choose N0=0.

    Definition 2.4 Consider a certain type of ADT switching signals. System (2.1) is said to converge exponentially within a ball if there exist constants a0, b>0, 0<c<1, and 0<γ<1 such that

    x(k,l)ξa+b(lr=0ˆh(r)γr+1+ks=0ˆv(s)γs+1)ck+l,

    where ξ0 is given vector.

    Remark 2.1. It follows from the boundary condition (2.2) that

    lr=0ˆh(r)γr+1+ks=0ˆv(s)γs+1

    is bounded by

    ˉhr=0ˆh(r)γr+1+ˉvs=0ˆv(s)γs+1.

    First, two necessary assumptions are proposed on the system (2.1).

    Assumption 3.1. fp and gp are order-preserving on Rn+ and homogeneous of degree one for any pM.

    Assumption 3.2. ω(k,l)0 are external disturbances and satisfy

    ω(k,l)γk+lˉω,

    where γ and ˉω are positive constants.

    Remark 3.1 It follows from Assumptions 3.1 and 3.2 that system (2.1) is positive for any nonnegative initial condition under arbitrary switching.

    Theorem 3.1. Let Assumptions 3.1 and 3.2 hold. If for any pM, there exists a vector ξP0 such that

    fp(ξp)+gp(ξp)ξp,

    then any solution of system (2.1) converges exponentially within a ball under suitable ADT switching. The ADT switching signals satisfy

    τε>lnαβlnγ,

    where

    β=max1inˉξiξi_

    with

    ˉξi=maxpMξpi,ξi_=minpMξpi

    and

    F=maxpM,1inFpii,γ=maxpM,1inγpi

    with γpi satisfying

    fpi(ξp)+γτmaxpigpi(ξp)γpiξpi=0, (3.1)

    and

    α={γdmaxF,if γdmaxF1,1,if γdmaxF<1.

    Proof. Let x(k,l)ξσ(k,l) be the multiple max-separable Lyapunov function. First, the variable transformation is introduced. Set

    [xh(k,l)xv(k,l)]=[γk+l00γk+l][yh(k,l)yv(k,l)], (3.2)

    then system (2.1) is reduced to

    {[yh(k+1,l)yv(k,l+1)]=γ1fσ(k,l)[yh(k,l)yv(k,l)]+gσ(k,l)([γτh(k)100γτv(l)1][yh(kτh(k),l)yv(k,lτv(l))])        +γkl1ω(k,l),   k+lεr,[yh(k,l)yv(k,l)]=Fσ(k,l)([γdh(k)00γdv(l)][yh(kdh(k),l)yv(k,ldv(l))]),   k+l=εr.

    A set of functions with respect to γ are defined by

    upi(γ)=fpi(ξp)+γτmaxgpi(ξp)γξpi, (3.3)

    where pM,i=1,2,3,,n, then upi decreases precisely monotonically for γ and upi tends to infinity as γ approaches zero. Following from

    fp(ξp)+gp(ξp)ξp,

    we can get upi(1)<0. This implies (3.3) has a solution γpi(0,1). Let

    γ=maxpMmax1inγpi,

    then 0<γ<1 and upi(γ)0. Therefore,

    fp(ξp)+γτmaxgp(ξp)γξp,pM. (3.4)

    When k+l[ε0,ε1), we have σ(k,l)=σ(ε0).

    In the following, we demonstrate for any k+l[ε0,ε1)

    y(k,l)ξσ(0,0)Φ0+[(k+l)(k0+l0)](γ1ˉωξmin), (3.5)

    where

    ξmin=minpM,1inξpi

    and

    Φ0=lr=0ˆh(r)γr+1+ks=0ˆv(s)γs+1.

    From (3.2), we have x(0,0)=y(0,0), which implies

    y(0,0)ξσ(0,0)max{ˆh(0),ˆv(0)}.

    Furthermore, we can get

    y(0,0)ξσ(0,0)ˆh(0)γ+ˆv(0)γ+[(k0+l0)(k0+l0)](γ1ˉωξmin).

    Therefore, (3.5) is true when k+l=0. Assume (3.5) holds for all (k,l) satisfying k+lu, where u[ε0,ε11), uN. In the following, we demonstrate that (3.5) is also true for u+1. From the definition of l, we have

    y(k,l)ξσ(0,0)Φ0+[(k+l)(k0+l0)](γ1ˉωξmin), (3.6)

    where k+lu. Since fσ(0,0) and gσ(0,0) satisfy the Assumption 3.1, from (3.4) and (3.6), we can get

    [yh(k+1,l)yv(k,l+1)]γ1fσ(0,0)([Φ0+[(k+l)(k0+l0)]γ1ˉωξmin]ξσ(0,0))+γkl1γk+lˉωξminξσ(0,0)+gσ(0,0)([Φ0+[(k+l)(k0+l0)]γ1ˉωξmin][γτh(k)1,00,γτv(l)1][ξhσ(0,0)ξvσ(0,0)])γ1(Φ0+[(k+l)(k0+l0)]γ1ˉωξmin)fσ(0,0)(ξσ(0,0))+γ1ˉωξminξσ(0,0)+[Φ0+[(k+l)(k0+l0)]γ1ˉωξmin][γτmax1,00,γτmax1]gσ(0,0)(ξσ(0,0))γ1[Φ0+[(k+l)(k0+l0)]γ1ˉωξmin][fσ(0,0)(ξσ(0,0))+γτmaxgσ(0,0)(ξσ(0,0))]+γ1ˉωξmin ξσ(0,0)(Φ0+[(k+l)(k0+l0)]γ1ˉωξmin+γ1ˉωξmin)ξσ(0,0)=(Φ0+[(k+l+1)(k0+l0)]γ1ˉωξmin)ξσ(0,0), (3.7)

    where k0+l0=0. Note that (3.7) is true whether or not k+lτh(k) and k+lτv(l) are non-negative. It follows from system (2.1) that

    yh(k,l+1)=[En10n1×n2][yh(k,l+1)yv(k1,l+2)]

    and

    yv(k+1,l)=[0n2×n1En2][yh(k+2,l1)yv(k+1,l)].

    Then, based on the preceding analysis, it is not difficult to prove

    yh(k,l+1)(Φ0+[(k+l+1)(k0+l0)]γ1ˉωξmin)[En10n1×n2]ξσ(0,0),yv(k+1,l)(Φ0+[(k+l+1)(k0+l0)]γ1ˉωξmin)[0n2×n1En2]ξσ(0,0). (3.8)

    As γ1ˉωξmin is non-negative and Φ is nondecreasing in k,l, Φ0+[(k+l)(k0+l0)]γ1ˉωξmin is nondecreasing in k,l. Combining the Eqs (3.7) and (3.8) yields

    y(k,l)(Φ0+[(k+l+1)(k0+l0)]γ1ˉωξmin)ξσ(0,0),

    where k+l=u+1. This implies that

    y(k,l)ξσ(0,0)Φ0+[(k+l)(k0+l0)]γ1ˉωξmin, (3.9)

    where k+l=u+1. Then, when k+l=ε1, we have

    [yh(k,l)yv(k,l)]=Fσ(k,l)([γdh(k),00,γdv(l)][yh(kdh(k),l)yv(k,ldv(l))])γdmaxFσ(k,l)[yh(kdh(k),l)yv(k,ldv(l))]γdmaxF(Φ0+[(k+l)(k0+l0)]γ1ˉωξmin)ξσ(0,0).

    Note that

    α={γdmaxF,if γdmaxF1,1,if γdmaxF<1,y(k,l)(Φ0+[(k+l)(k0+l0)]γ1ˉωξmin)ξσ(0,0),

    where k+l[ε0,ε1), which leads to

    [yh(k,l)yv(k,l)]α[Φ0+[(k+l)(k0+l0)]γ1ˉωξmin]ξσ(0,0),

    where k+l[ε0,ε1). Therefore, we can get

    y(k,l)ξσ(0,0)α[Φ0+[(k+l)(k0+l0)]γ1ˉωξmin], (3.10)

    where k+l[ε0,ε1). Denote σ(k1,l1)=σ(ε1) as the switching instant, that is, k+l=ε1. From the definition of l, we can get

    y(k1,l1)ξσ(k1,l1)=max1jnyj(k1,l1)ξσ(k1,l1)j=max1jnξσ(0,0)jξσ(k1,l1)jyj(k1,l1)ξσ(0,0)jmax1jnˉξjξ_jyj(k1,l1)ξσ(0,0)jβy(k1,l1)ξσ(0,0).

    As a result of (3.10), it is clear that

    y(k1,l1)ξσ(k1,l1)βα[Φ0+[(k1+l1)(k0+l0)]γ1ˉωξmin].

    Let

    Φ1=βα[Φ0+[(k1+l1)(k0+l0)]γ1ˉωξmin].

    Thus,

    y(k1,l1)(Φ1+[(k1+l1)(k1+l1)]γ1ˉωξmin)ξσ(k1,l1). (3.11)

    Similar to the preceding analysis, the following inequality holds

    y(k,l)ξσ(k1,l1)α[Φ1+[(k+l)(k1+l1)]γ1ˉωξmin],

    where k+l[ε1,ε2). Furthermore, we have

    y(k,l)ξσ(km1,lm1)α[Φm1+[(k+l)(km1+lm1)]γ1ˉωξmin], (3.12)

    where k+l[εm1,εm). Let

    Φm=βα[Φm1+[(km+lm)(km1+lm1)]γ1ˉωξmin].

    Then, we have

    y(k,l)ξσ(km,lm)α[Φm+[(k+l)(km+lm)]γ1ˉωξmin],

    where k+l[εm,εm+1). According to the definition of Φi, combining (3.11) and (3.12) leads to

    y(k,l)ξσ(km,lm)α(βα[Φm1+[(km+lm)(km1+lm1)]γ1ˉωξmin])+α([(k+l)(km+lm)]γ1ˉωξmin)=βα2[Φm1+(εmεm1)γ1ˉωξmin]+[(k+l)εm]γ1αˉωξmin=βα2[βα(Φm2+(εm1εm2)γ1ˉωξmin)+(εmεm1)γ1ˉωεmin]+(k+lεm)γ1αˉωξmin=β2α3Φm2+β2α3(εm1εm2)γ1ˉωεmin+βα2(εmεm1)γ1ˉωεmin+(k+lεm)γ1αˉωξmin =βmαm+1Φ0+βmαm+1(ε1ε0)γ1ˉωξmin++(k+lεm)γ1αˉωξminβmαm+1Φ0+α(k+l)(βmαm+βm1αm1++1)γ1ˉωξmin=βmαmαΦ0+(k+l)α1βm+1αm+11αβγ1ˉωξmin,

    where k+l[εm,εm+1).

    Obviously,

    mk+lτε,

    where

    τε>lnαβlnγ.

    Hence, we get

    y(k,l)ξσ(km,lm)(βα)k+lτεαΦ0+α(k+l)(βα)k+lτεβα1βαγ1ˉωξmin.

    We can deduce from (3.2) that

    x(k,l)ξσ(km,lm)γk+l(αβ)k+lτεαΦ0+βα2(k+l)γk+l(αβ)k+lτεβα1γ1ˉωξmin=((αβ)1τεγ)k+lαΦ0+βα2(k+l)(γ(αβ)1τε)k+lβα1γ1ˉωξmin=(e1nαβτε+lnγ)k+lαΦ0+βα2βα1γ1ˉωξmin(k+l)(elnαβτε+lnγ)k+l.

    Denote

    b=αandc=elnαβτε+lnγ.

    Furthermore, if we let

    f(x)=xcx(0<c<1),

    then

    fmax=f(1lnc)=1c1lnclnc.

    Hence,

    (k+l) ck+lfmax,k,lN0.

    Let

    a=r1ˉωβα2(βα1)ξmin1c1lnclnc

    and

    ˉξ=[ˉξ1,ˉξ2,,ˉξn].

    Then we have

    x(k,l)ˉξx(k,l)ξσ(km,lm)a+b Φ0ck+l.

    That is, system (2.1) converges exponentially within a ball.

    Remark 3.2. Comparing with the main result given in [22], the external disturbances and impulsive effects are considered. If we let \bmω(k,l)0 and impulsive matrix FP0 in Theorem 1, then any solution of system (2.1) under the switching signal with ADT

    τε>lnαβlnγ

    satisfying

    x(k,l)ˉξx(k,l)ξσ(km,lm)b Φ0ck+l.

    That is, Theorem 3.1 in this paper reduces to [22,Theorem 2].

    Remark 3.3. It follows from the proof of Theorem 3.1 that the convergence rate is related to the parameter γ. On the other hand, γpi is the unique solution of the Eq (3.1). Obviously, γpi is monotonically increasing in ^τh and ^τv, and γpi approaches to one as max(^τh,^τv) tends to infinity. This implies that system delays have an impact on the convergence rate.

    In the following, we extend the impulse matrix to the nonlinear case.

    Corollary 3.1. If the impulse matrix

    FP=diag{FP11(x),FP22(x),,FPnn(x)}

    is bounded for any FPii(x), i=1,2,,n, then system (2.1) converges exponentially within a ball under a class of ADT switching signals.

    Proof. Let

    F=suppM,1insupx|FPii(x)|.

    Then, Corollary 3.1 can be derived from Theorem 3.1.

    Consider 2-D SNPSs with multiple time-varying delays

    {[xh(k+1,l)xv(k,l+1)]=fσ(k,l)[xh(k,l)xv(k,l)]+Ns=1gsσ(k,l)[xh(kτhs(k),l)xv(k,lτvs(l))]                  +ω(k,l),    k+lεr,[xh(k,l)xv(k,l)]=Qz=1F zσ(k,l)[xh(kdhz(k),l)xv(k,ldvz(l))],   k+l=εr, (3.13)

    where the delay functions τhs(k), τvs(l), dhz(k), and dvz(l) satisfy 0τhs(k)ˉτhs, 0τvs(l)ˉτvs, 0dhz(k)ˉdhz, 0dvz(l)ˉdvz, s{1,2,,N}, z{1,2,,Q}.

    Now, we give the reachable set estimation for the system (3.13).

    Theorem 3.2. Let Assumptions 3.1 and 3.2 hold and the impulse matrix FzP be bounded for any F zPii(x), i=1,2,n. For any pM, if there exists a vector ξP0 satisfying

    fp(ξp)+Ns=1gsp(ξp)ξp,

    then each solution of system (3.13) converges exponentially within a ball with ADT switching satisfying

    τε>lnαβlnγ,

    where

    α={Qz=1γdzmaxFz,ifQz=1γdzmaxFz1,1,ifQz=1γdzmaxFz<1,τsmax=max(ˉτhs,ˉτvs),dzmax=max(ˉdhz,ˉdvz),Fz=suppM,1insupx|F zpii(x)|

    and

    γ=maxpM,0inγpi

    with γpi satisfying

    fpi(ξp)+Ns=1γτsmaxpigspi(ξp)γpiξpi=0.

    Proof. The same variable transformation as stated in Theorem 3.1 is also used. Then, according to similar analysis to (3.9), one can verify that

    y(k,l)ξσ(0,0)Φ0+[(k+l)(k0+l0)]γ1ˉωξmin,k+l=u+1.

    As k+l=ε1, we have

    [yh(k,l)yv(k,l)]=Qz=1F zσ(k,l)([γdhz(k),00,γdvz(l)][yh(kdhz(k),l)yv(k,ldvz(l))])Qz=1γdzmaxF zσ(k,l)[yh(kdhz(k),l)yv(k,ldvz(l))]Qz=1γdzmaxFz(Φ0+[(k+l)(k0+l0)]γ1ˉωξmin)ξσ(0,0).

    Then, it follows from the definition of α that

    y(k,l)ξσ(0,0)α[Φ0+[(k+l)(k0+l0)]γ1ˉωξmin],

    where k+l[ε0,ε1). The rest of the proof can be analyzed applying the same arguments as in the proof of Theorem 3.1. It will be omitted here.

    Theorem 3.2 can be generalized to general 2-D switched linear systems.

    {[xh(k+1,l)xv(k,l+1)]=Aσ(k,l)[xh(k,l)xv(k,l)]+ω(k,l)+Ns=1Bsσ(k,l)[xh(kτhs(k),l)xv(k,lvsτ(l))],k+lεr,[xh(k,l)xv(k,l)]=Qz=1F zσ(k,l)[xh(kdhz(k),l)xv(k,ldvz(l))],   k+l=εr. (3.14)

    Denote

    |Ap|=[|apij|]n×n,|Bsp|=[|b(s)pij|]n×n.

    Theorem 3.3. If for any pM, there exists a vector ξP0 such that

    (|Ap|+Ns=1|Bsp|)ξpξp,

    then any solution of the system (3.14) converges exponentially within a ball under certain ADT switching. The ADT switching signals satisfy

    τε>lnαβlnγ,

    where

    γ=maxpMmax1inγpi

    with γpi

    nj=1|apij|ξpj+Ns=1(γτsmaxpinj=1|b(s)pij|ξpj)γpiξpi=0.

    Proof. It is simple to check that

    [|xh(k+1,l)||xν(k,l+1)|]|Aσ(k,l)|[|xh(k,l)||xv(k,l)|]+Ns=1|Bsσ(k,l)|[|xh(kτhs(k),l)||xv(k,lτvs(l))|]+ω(k,l).

    Then, the method to prove Theorem 3.3 is similar to that of Theorem 3.1, and it is omitted.

    Consider 2-D SPNSs with heterogeneous time-varying delays.

    {xhi(k+1,l)=fσ(k,l)i[xh(k,l)xv(k,l)]+ωi(k,l)+gσ(k,l)i[(xh1(kτih1(k),l)xhn1(kτihn1(k),l))(xv1(k,lτiv1(l))xvn2(k,lτivn2(l)))],k+lεr,xhi(k,l)=Fσ(k,l)i[(xh1(kdih1(k),l)xhn1(kdihn1(k),l))(xv1(k,ldiv1(l))xvn2(k,ldivn2(l)))],k+l=εr. (3.15)

    xhj(k,l) and xvj(k,l) represent the j-th element of the vector functions xh(k,l) and xv(k,l), respectively. The delay functions are non-negative and have an upper bound. Denote

    τmax=max(τih1(k),,τihn1(k),τiv1(l),,τivn2(l), i=1,2,,n),dmax=max(dih1(k),,dihn1(k),div1(l),,divn2(l), i=1,2,,n).

    Supposing Assumptions 3.1 and 3.2 hold, we can get the following result.

    Theorem 3.4. If for any pM, there exists a vector ξP0 such that

    fp(ξp)+gp(ξp)ξp,

    then system (3.15) converges exponentially within a ball under appropriate ADT switching. Furthermore, the ADT switching signals satisfy

    τε>lnαβlnγ,

    where α,β,γ are defined in Theorem 3.1.

    Proof. Since the heterogeneous time-varying delays are bounded, Theorem 3.4 can be proved by using the same method used in the proof of Theorem 3.1.

    Consider the system (3.15) consisting of two subsystems with

    f1([xh(k,l)xv(k,l)])=[0.140.160.250.1][xh(k,l)xv(k,l)]+(xh(k,l))2+(xv(k,l))2[0.010.05],g1([xh(k,l)xv(k,l)])=[0.625xh(k,l)xv(k,l)(2.3xh(k,l))2+(xv(k,l))20.5xh(k,l)xv(k,l)(xh(k,l))2+(xv(k,l))2],f2([xh(k,l)xv(k,l)])=[0.30.230.20.4][xh(k,l)xv(k,l)]+(xh(k,l))2+(2xv(k,l))2[0.020.04],g2([xh(k,l)xv(k,l)])=[0.22xh(k,l)xv(k,l)(2.3xh(k,l))2+(xv(k,l))20.1xh(k,l)xv(k,l)(xh(k,l))2+(xv(k,l))2],F1=[0.5001.02],F2=[1.01000.8],ω(k,l)=0.25[|sin(k)||cos(l)|].

    Obviously, the vector fields f1, f2, g1, and g2 are homogeneous of degree one and order preserving. F1, F2, and ω(k,l) are bounded. It is determined that there exist vectors

    ξ1=[1.09,1.09]Tandξ2=[0.8,1.15]T

    such that

    (fi+gi)ξiξi.

    Let

    τh(k)=1+3sin(π2k),τv(l)=1+3cos(π2l)

    and

    dh(k)=1+|sin(π2k)|,dv(l)=1+|cos(π2l)|.

    It follows from Eq (3.1) that

    γ11=0.8821, γ12=0.9181, γ21=0.8982, γ22=0.7595.

    We pick γ=0.9181. Then, according to Theorem 1, the SPNS converges exponentially within a ball under ADT switching τε6.48. Figure 1 shows the ADT switching signal. Figures 2 and 3 provide the estimates for xh(k,l) and xv(k,l) under the switching signal τε=7, respectively.

    Figure 1.  The ADT switching signal.
    Figure 2.  The estimate for xh(k,l).
    Figure 3.  The estimate for xv(k,l).

    The reachable set estimation for 2-D SNPSs in the Roesser model with unknown exogenous disturbances are studied. System delays and delayed impulse effects are all considered in the involved systems. For bounded directional delays and delayed impulse effects, an explicit sufficient is presented for the presence of a ball such that any solution of the system converge exponentially within it. The existing result can be seen as a special case of this article. Finally, we also extend the result to 2-D SNPSs with multiple directional delays, general 2-D switched linear systems, and 2-D SNPSs with heterogeneous time-varying delays.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    All authors declare that there are no conflicts of interest in this paper.



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