
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+:={x∈Rn,xj≥0,1≤j≤n}. |
For x,y∈Rn, denoted by x≥y (x≫y, x≪y), if xj≥yj (xj>yj, xj<yj) for 1≤j≤n. Given a positive vector ξ≫0,
||x||ξ∞=max1≤j≤n|xj|ξj. |
Denote the weighted l∞ norm of x∈Rn. Set
||x||∞=max1≤j≤n|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(k−τh(k),l)xv(k,l−τv(l))] +ω(k,l),k+l≠εr,[xh(k,l)xv(k,l)]=Fσ(k,l)[xh(k−dh(k),l)xv(k,l−dv(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×N0→M={1,2,3,…,m} is the switching rule. For any P∈M, the vector fields fp,gp: Rn→Rn are continuous on Rn. The diagonal matrix
FP=diag{FP11,FP11,…,FPnn} |
is called the impulsive matrix, and we assume FPii>0 for all 1≤i≤n. The exogenous disturbances are denoted by ω(k,l): N0×N0→Rn.
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
0≤τh(k)≤^τh, 0≤τv(l)≤^τv,0≤dh(k)≤^dh, 0≤dv(l)≤^dv,k−dh(k)≥−^τh, l−dv(l)≥−^τv. |
Denote
τmax=max(^τh,^τv),dmax=max(^dh,^dv). |
The initial conditions are presented as follows:
{xh(k,l)=h(k,l),−^τh≤k≤0,0≤l≤¯h,xh(k,l)=0,−^τh≤k≤0,l>¯h,xv(k,l)=v(k,l),−^τv≤l≤0,0≤k≤¯v,xv(k,l)=0,−^τv≤l≤0,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)=maxp∈Msup−τh≤k≤0‖h(k,r)‖ξpn1∞ |
and
ˆv(s)=maxp∈Msup−τv≤l≤0‖v(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: Rn→Rn is called homogeneous of degree one if for any x∈Rn and λ>0,
f(λx)=λf(x). |
g is defined to be order-preserving on Rn+ if g(x)≥g(y) for any x,y∈Rn+ satisfying x≥y.
Definition 2.3. For any nonnegative integers i,j and i0,j0 with
i+j=ε≥ε0=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+ε−ε0τε, |
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 a≥0, b>0, 0<c<1, and 0<γ<1 such that
‖x(k,l)‖ξ∞≤a+b(l∑r=0ˆh(r)γr+1+k∑s=0ˆv(s)γs+1)ck+l, |
where ξ≫0 is given vector.
Remark 2.1. It follows from the boundary condition (2.2) that
l∑r=0ˆh(r)γr+1+k∑s=0ˆv(s)γs+1 |
is bounded by
ˉh∑r=0ˆh(r)γr+1+ˉv∑s=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 p∈M.
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 p∈M, there exists a vector ξP≫0 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=max1≤i≤nˉξiξi_ |
with
ˉξi=maxp∈Mξpi,ξi_=minp∈Mξpi |
and
F=maxp∈M,1≤i≤nFpii,γ=maxp∈M,1≤i≤nγpi |
with γpi satisfying
fpi(ξp)+γ−τmaxpigpi(ξp)−γpiξpi=0, | (3.1) |
and
α={γ−dmaxF,if γ−dmaxF≥1,1,if γ−dmaxF<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)]=γ−1fσ(k,l)[yh(k,l)yv(k,l)]+gσ(k,l)([γ−τh(k)−100γ−τv(l)−1][yh(k−τh(k),l)yv(k,l−τv(l))]) +γ−k−l−1ω(k,l), k+l≠εr,[yh(k,l)yv(k,l)]=Fσ(k,l)([γ−dh(k)00γ−dv(l)][yh(k−dh(k),l)yv(k,l−dv(l))]), k+l=εr. |
A set of functions with respect to γ are defined by
upi(γ)=fpi(ξp)+γ−τmaxgpi(ξp)−γξpi, | (3.3) |
where ∀p∈M,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=maxp∈Mmax1≤i≤nγpi, |
then 0<γ<1 and upi(γ)≤0. Therefore,
fp(ξp)+γ−τmaxgp(ξp)≤γξp,∀p∈M. | (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)](γ−1ˉωξmin), | (3.5) |
where
ξmin=minp∈M,1≤i≤nξpi |
and
Φ0=l∑r=0ˆh(r)γr+1+k∑s=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)](γ−1ˉωξmin). |
Therefore, (3.5) is true when k+l=0. Assume (3.5) holds for all (k,l) satisfying k+l≤u, where u∈[ε0,ε1−1), u∈N. 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)](γ−1ˉωξmin), | (3.6) |
where k+l≤u. 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)]≤γ−1fσ(0,0)([Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin]ξσ(0,0))+γ−k−l−1γk+lˉωξminξσ(0,0)+gσ(0,0)([Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin]∗[γ−τh(k)−1,00,γ−τv(l)−1][ξhσ(0,0)ξvσ(0,0)])≤γ−1(Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin)fσ(0,0)(ξσ(0,0))+γ−1ˉωξminξσ(0,0)+[Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin]∗[γ−τmax−1,00,γ−τmax−1]gσ(0,0)(ξσ(0,0))≤γ−1[Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin]∗[fσ(0,0)(ξσ(0,0))+γ−τmaxgσ(0,0)(ξσ(0,0))]+γ−1ˉωξmin ξσ(0,0)≤(Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin+γ−1ˉωξmin)ξσ(0,0)=(Φ0+[(k+l+1)−(k0+l0)]γ−1ˉωξmin)ξσ(0,0), | (3.7) |
where k0+l0=0. Note that (3.7) is true whether or not k+l−τh(k) and k+l−τv(l) are non-negative. It follows from system (2.1) that
yh(k,l+1)=[En10n1×n2][yh(k,l+1)yv(k−1,l+2)] |
and
yv(k+1,l)=[0n2×n1En2][yh(k+2,l−1)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)]γ−1ˉωξmin)[En10n1×n2]ξσ(0,0),yv(k+1,l)≤(Φ0+[(k+l+1)−(k0+l0)]γ−1ˉωξmin)[0n2×n1En2]ξσ(0,0). | (3.8) |
As γ−1ˉωξmin is non-negative and Φ is nondecreasing in k,l, Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin is nondecreasing in k,l. Combining the Eqs (3.7) and (3.8) yields
y(k,l)≤(Φ0+[(k+l+1)−(k0+l0)]γ−1ˉωξmin)ξσ(0,0), |
where k+l=u+1. This implies that
‖y(k,l)‖ξσ(0,0)∞≤Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin, | (3.9) |
where k+l=u+1. Then, when k+l=ε1, we have
[yh(k,l)yv(k,l)]=Fσ(k,l)([γ−dh(k),00,γ−dv(l)][yh(k−dh(k),l)yv(k,l−dv(l))])≤γ−dmaxFσ(k,l)[yh(k−dh(k),l)yv(k,l−dv(l))]≤γ−dmaxF(Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin)ξσ(0,0). |
Note that
α={γ−dmaxF,if γ−dmaxF≥1,1,if γ−dmaxF<1,y(k,l)≤(Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin)ξσ(0,0), |
where k+l∈[ε0,ε1), which leads to
[yh(k,l)yv(k,l)]≤α[Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin]ξσ(0,0), |
where k+l∈[ε0,ε1). Therefore, we can get
‖y(k,l)‖ξσ(0,0)∞≤α[Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin], | (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)∞=max1≤j≤nyj(k1,l1)ξσ(k1,l1)j=max1≤j≤nξσ(0,0)jξσ(k1,l1)jyj(k1,l1)ξσ(0,0)j≤max1≤j≤nˉξjξ_jyj(k1,l1)ξσ(0,0)j≤β‖y(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)]γ−1ˉωξmin]. |
Let
Φ1=βα[Φ0+[(k1+l1)−(k0+l0)]γ−1ˉωξmin]. |
Thus,
y(k1,l1)≤(Φ1+[(k1+l1)−(k1+l1)]γ−1ˉωξmin)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)]γ−1ˉωξmin], |
where k+l∈[ε1,ε2). Furthermore, we have
‖y(k,l)‖ξσ(km−1,lm−1)∞≤α[Φm−1+[(k+l)−(km−1+lm−1)]γ−1ˉωξmin], | (3.12) |
where k+l∈[εm−1,εm). Let
Φm=βα[Φm−1+[(km+lm)−(km−1+lm−1)]γ−1ˉωξmin]. |
Then, we have
‖y(k,l)‖ξσ(km,lm)∞≤α[Φm+[(k+l)−(km+lm)]γ−1ˉωξmin], |
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[Φm−1+[(km+lm)−(km−1+lm−1)]γ−1ˉωξmin])+α([(k+l)−(km+lm)]γ−1ˉωξmin)=βα2[Φm−1+(εm−εm−1)γ−1ˉωξmin]+[(k+l)−εm]γ−1αˉωξmin=βα2[βα(Φm−2+(εm−1−εm−2)γ−1ˉωξmin)+(εm−εm−1)γ−1ˉωεmin]+(k+l−εm)γ−1αˉωξmin=β2α3Φm−2+β2α3(εm−1−εm−2)γ−1ˉωεmin+βα2(εm−εm−1)γ−1ˉωεmin+(k+l−εm)γ−1αˉωξmin =βmαm+1Φ0+βmαm+1(ε1−ε0)γ−1ˉωξmin+⋯+(k+l−εm)γ−1αˉωξmin≤βmαm+1Φ0+α(k+l)(βmαm+βm−1αm−1+⋯+1)γ−1ˉωξmin=βmαmαΦ0+(k+l)α1−βm+1αm+11−αβγ−1ˉωξmin, |
where k+l∈[εm,εm+1).
Obviously,
m≤k+lτε, |
where
τε>−lnαβlnγ. |
Hence, we get
‖y(k,l)‖ξσ(km,lm)∞≤(βα)k+lτεαΦ0+α(k+l)(βα)k+lτεβα−1βαγ−1ˉωξmin. |
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+lτεβα−1γ−1ˉωξmin=((αβ)1τεγ)k+lαΦ0+βα2(k+l)(γ(αβ)1τε)k+lβα−1γ−1ˉωξmin=(e1nαβτε+lnγ)k+lαΦ0+βα2βα−1γ−1ˉωξmin(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+l≤fmax,k,l∈N0. |
Let
a=−r−1ˉωβα2(βα−1)ξmin1c1lnclnc |
and
ˉξ=[ˉξ1,ˉξ2,…,ˉξn]. |
Then we have
‖x(k,l)‖ˉξ∞≤‖x(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 FP≡0 in Theorem 1, then any solution of system (2.1) under the switching signal with ADT
τε>−lnαβlnγ |
satisfying
‖x(k,l)‖ˉξ∞≤‖x(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=supp∈M,1≤i≤nsupx|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(k−τhs(k),l)xv(k,l−τvs(l))] +ω(k,l), k+l≠εr,[xh(k,l)xv(k,l)]=∑Qz=1F zσ(k,l)[xh(k−dhz(k),l)xv(k,l−dvz(l))], k+l=εr, | (3.13) |
where the delay functions τhs(k), τvs(l), dhz(k), and dvz(l) satisfy 0≤τhs(k)≤ˉτhs, 0≤τvs(l)≤ˉτvs, 0≤dhz(k)≤ˉdhz, 0≤dvz(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 p∈M, if there exists a vector ξP≫0 satisfying
fp(ξp)+N∑s=1gsp(ξp)≪ξp, |
then each solution of system (3.13) converges exponentially within a ball with ADT switching satisfying
τε>−lnαβlnγ, |
where
α={∑Qz=1γ−dzmaxFz,if∑Qz=1γ−dzmaxFz≥1,1,if∑Qz=1γ−dzmaxFz<1,τsmax=max(ˉτhs,ˉτvs),dzmax=max(ˉdhz,ˉdvz),Fz=supp∈M,1≤i≤nsupx|F zpii(x)| |
and
γ=maxp∈M,0≤i≤nγpi |
with γpi satisfying
fpi(ξp)+N∑s=1γ−τsmaxpigspi(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)]γ−1ˉωξmin,k+l=u+1. |
As k+l=ε1, we have
[yh(k,l)yv(k,l)]=Q∑z=1F zσ(k,l)([γ−dhz(k),00,γ−dvz(l)][yh(k−dhz(k),l)yv(k,l−dvz(l))])≤Q∑z=1γ−dzmaxF zσ(k,l)[yh(k−dhz(k),l)yv(k,l−dvz(l))]≤Q∑z=1γ−dzmaxFz(Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin)ξσ(0,0). |
Then, it follows from the definition of α that
‖y(k,l)‖ξσ(0,0)∞≤α[Φ0+[(k+l)−(k0+l0)]γ−1ˉωξmin], |
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)+N∑s=1Bsσ(k,l)[xh(k−τhs(k),l)xv(k,lvs−τ(l))],k+l≠εr,[xh(k,l)xv(k,l)]=Q∑z=1F zσ(k,l)[xh(k−dhz(k),l)xv(k,l−dvz(l))], k+l=εr. | (3.14) |
Denote
|Ap|=[|apij|]n×n,|Bsp|=[|b(s)pij|]n×n. |
Theorem 3.3. If for any p∈M, there exists a vector ξP≫0 such that
(|Ap|+N∑s=1|Bsp|)ξp≪ξp, |
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=maxp∈Mmax1≤i≤nγpi |
with γpi
n∑j=1|apij|ξpj+N∑s=1(γ−τsmaxpin∑j=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)|]+N∑s=1|Bsσ(k,l)|[|xh(k−τhs(k),l)||xv(k,l−τvs(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(k−τih1(k),l)⋯xhn1(k−τihn1(k),l))⊤(xv1(k,l−τiv1(l))⋯xvn2(k,l−τivn2(l)))⊤],k+l≠εr,xhi(k,l)=Fσ(k,l)i[(xh1(k−dih1(k),l)⋯xhn1(k−dihn1(k),l))⊤(xv1(k,l−div1(l))⋯xvn2(k,l−divn2(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 p∈M, there exists a vector ξP≫0 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)ξi≪ξi. |
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 τε≥6.48. 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.
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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