
This paper investigates iterative learning control for Caputo fractional-order systems with one-sided Lipschitz nonlinearity. Both open- and closed-loop P-type learning algorithms are proposed to achieve perfect tracking for the desired trajectory, and their convergence conditions are established. It is shown that the algorithms can make the output tracking error converge to zero along the iteration axis. A simulation example illustrates the application of the theoretical findings, and shows the effectiveness of the proposed approach.
Citation: Hanjiang Wu, Jie Huang, Kehan Wu, António M. Lopes, Liping Chen. Precise tracking control via iterative learning for one-sided Lipschitz Caputo fractional-order systems[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 3095-3109. doi: 10.3934/mbe.2024137
[1] | Tao Xie, Xing Xiong . Finite-time synchronization of fractional-order heterogeneous dynamical networks with impulsive interference via aperiodical intermittent control. AIMS Mathematics, 2025, 10(3): 6291-6317. doi: 10.3934/math.2025287 |
[2] | Chengqiang Wang, Xiangqing Zhao, Yang Wang . Finite-time stochastic synchronization of fuzzy bi-directional associative memory neural networks with Markovian switching and mixed time delays via intermittent quantized control. AIMS Mathematics, 2023, 8(2): 4098-4125. doi: 10.3934/math.2023204 |
[3] | Qinghua Zhou, Li Wan, Hongbo Fu, Qunjiao Zhang . Exponential stability of stochastic Hopfield neural network with mixed multiple delays. AIMS Mathematics, 2021, 6(4): 4142-4155. doi: 10.3934/math.2021245 |
[4] | Zhifeng Lu, Fei Wang, Yujuan Tian, Yaping Li . Lag synchronization of complex-valued interval neural networks via distributed delayed impulsive control. AIMS Mathematics, 2023, 8(3): 5502-5521. doi: 10.3934/math.2023277 |
[5] | Dong Pan, Huizhen Qu . Finite-time boundary synchronization of space-time discretized stochastic fuzzy genetic regulatory networks with time delays. AIMS Mathematics, 2025, 10(2): 2163-2190. doi: 10.3934/math.2025101 |
[6] | Rakkiet Srisuntorn, Wajaree Weera, Thongchai Botmart . Modified function projective synchronization of master-slave neural networks with mixed interval time-varying delays via intermittent feedback control. AIMS Mathematics, 2022, 7(10): 18632-18661. doi: 10.3934/math.20221025 |
[7] | Zhengqi Zhang, Huaiqin Wu . Cluster synchronization in finite/fixed time for semi-Markovian switching T-S fuzzy complex dynamical networks with discontinuous dynamic nodes. AIMS Mathematics, 2022, 7(7): 11942-11971. doi: 10.3934/math.2022666 |
[8] | Tian Xu, Jin-E Zhang . Intermittent control for stabilization of uncertain nonlinear systems via event-triggered mechanism. AIMS Mathematics, 2024, 9(10): 28487-28507. doi: 10.3934/math.20241382 |
[9] | Xingxing Song, Pengfei Zhi, Wanlu Zhu, Hui Wang, Haiyang Qiu . Exponential synchronization control of delayed memristive neural network based on canonical Bessel-Legendre inequality. AIMS Mathematics, 2022, 7(3): 4711-4734. doi: 10.3934/math.2022262 |
[10] | Saravanan Shanmugam, Mohamed Rhaima, Hamza Ghoudi . Exponential synchronization analysis for complex dynamical networks with hybrid delays and uncertainties under given control parameters. AIMS Mathematics, 2023, 8(12): 28976-29007. doi: 10.3934/math.20231484 |
This paper investigates iterative learning control for Caputo fractional-order systems with one-sided Lipschitz nonlinearity. Both open- and closed-loop P-type learning algorithms are proposed to achieve perfect tracking for the desired trajectory, and their convergence conditions are established. It is shown that the algorithms can make the output tracking error converge to zero along the iteration axis. A simulation example illustrates the application of the theoretical findings, and shows the effectiveness of the proposed approach.
In recent years, an enormous advance of complex networks (CNS) has been witnessed on account of their extensive applications in many areas such as unmanned aerial vehicles, communication systems, power systems, transportation networks, ecological networks, and so forth [1,2,3,4]. A large number of interesting achievements about dynamical properties of CNS have sprung up [5,6,7,8,9]. It can be observed that CNS in the majority of the above findings are supposed to be single-link. In contrast to single-link complex, multilayer complex networks (MCNS) can more accurately simulate the real systems, which are more general and practical due to possessing sophisticated structures. For instance, a transportation network is composed of highways, ships, railways, and airplanes, and it can be seen as one type of multilayer complex network. Apparently, the transmission speed and cost are different among these ways, which signifies that each interaction mode has different weights. On the other hand, great importance can also be attached to social networks. There are various kinds of contact ways for people to communicate with others including phone, mail, and internet, which leads to multilayer networks. Particularly, large group decision-making plays a significant role in social networks. In [10], large group decision-making with a rough integrated asymmetric cloud model under a multi-granularity linguistic environment was considered, and large group emergency decision-making with bi-directional trust in social networks was studied based on a probabilistic hesitant fuzzy integrated cloud approach in [11], where strong uncertainty and randomness were tackled effectively. Since the MCNS may better depict the connection and interaction among different layer networks, the investigation of MCNS has important theoretical and realistic significance.
Synchronization is a typical dynamic characteristic of MCNS, which means all the states of different nodes of networks can evolve in one common mode with various different initial data, and plenty of relevant results about MCNS have been reported. Particularly, in [12], global exponential synchronization of multi-link memristive neural networks with time-varying delays was analyzed by designing state feedback controllers. In [13], the practical fixed-time synchronization issue of MCNS was addressed based on an intermittent event-triggered control strategy by utilizing the proposed new practical fixed-time stability lemma. In [14], bipartite synchronization of multilayer signed networks was investigated under aperiodic intermittent-based adaptive dynamic event-triggered control, where the control gains and the triggering parameters were adjusted with the system states. Furthermore, the discussions of various synchronizations were extended to fractional MCNS with the help of impulsive control [15,16]. On the other hand, in the real world, due to the existence of random disturbances and abrupt variations in systems' structures and parameters, stochastic Markovian switching systems were introduced to simulate such systems. Recently, stability and synchronization of stochastic MCNS with Markovian switching have aroused scholars' interests [17,18,19,20,21,22,23]. In [17], pth moment exponential stability for a class of stochastic complex multi-link networks with Markovian switching and multi-delayed impulsive effects was examined by virtue of the Razumikhin approach. In [18], combining with Kirchhoff's matrix tree theorem, finite-time synchronization of stochastic MCNS with Markovian switching was studied via a novel quantized aperiodically intermittent control. Moreover, in [19], the pth moment exponential synchronization issue of stochastic delayed MCNS with semi-Markov jump under aperiodically intermittent control was solved based on the Lyapunov method and graph theory, and several sufficient criteria were derived. Nevertheless, the preceding existing results mainly focus on pth moment synchronization of stochastic MCNS with Markovian switching rather than almost sure exponential synchronization, which sparks the appearance of this work.
It is worth noting that stochastic noise can be used as the control input [24,25]. For instance, in social networks, by virtue of public opinion control, which can be viewed as noise control, a certain behavior mode of the entire society can keep the consensus in a special direction [26]. Particularly, white noise was incorporated initially to stabilize the neural networks by Liao and Mao [27]. Soon afterward, stochastic stabilization of nonlinear systems was considered by utilizing an intermittent Brownian noise or aperiodical intermittent Brownian noise in [28,29]. Meanwhile, when the discrete-time feedback control was introduced to the drift term of systems, mean-square exponential stabilization of stochastic nonlinear systems was discussed in [30]. While the discrete-time feedback control was introduced to the diffusion term of systems, the almost sure exponential stabilization issue was solved in [31]. Subsequently, the studies about discrete feedback control and noise stabilization were further developed in [32,33,34]. Based on the previous findings, a new periodically intermittent discrete observation control scheme was proposed, by which the issue of mean-square exponential synchronization of stochastic neural networks was coped with in [35]. Furthermore, under the similar control strategy in [35], almost sure exponential stabilization of deterministic neural networks and hybrid neural networks was analyzed in [36,37]. More recently, aperiodically intermittent discrete observation noise control (AIDONC) as one kind of discontinuous control scheme was designed by adjusting the distributions of work/rest intervals. Since feedback control based on intermittent discrete observations on current states are imposed, the control time and control frequency can be reduced. Consequently, energy consumption and control cost can be greatly saved. Besides, efficiency and feasibility of implementation of the control strategy will be improved significantly. By using aperiodically intermittent discrete observation noise control (AIDONC), synchronization of complex networks was investigated in [38,39]. For MCNS, it can be observed that in [21,22], the intermittent control was imposed on the drift term and several criteria on pth moment exponential synchronization were presented. Additionally, in [23], some sufficient conditions on almost sure exponential synchronization of MCNS were given, but the control strategy AIDONC was not adopted.
Inspired by the above discussions, this paper will investigate the almost sure exponential synchronization of MCNS with Markovian switching via AIDONC. The main contributions are summarized below:
1) In this paper, Markovian switching and multilayer interaction factors are taken into account simultaneously, which make our system more general compared with the networks in literature [35,36,37,38,39]. Meanwhile, the network architecture may be undirected or directed, and consequently, the adjacency matrix could be symmetrical and asymmetrical.
2) Different from the work about pth moment exponential synchronization of network systems in [21,22], by utilizing M-matrix theory and various stochastic analysis techniques including the Itô formula, the Gronwall inequality, and the Borel-Cantelli lemma, almost sure exponential synchronization of multilayer coupled networks with Markovian switching is analyzed and the upper bound of the duration time is also estimated.
3) Compared with the work in [23], one kind of control strategy based on AIDONC is adopted, where the average control rate is integrated to depict the distributions of work/rest intervals of the control strategy from an overall perspective.
The remainder of this article is organized as follows. In Section 2, some necessary preliminaries and model descriptions are presented. The main findings about almost sure exponential synchronization for multilayer coupled networks with Markovian switching via aperiodically intermittent noise are derived in Section 3. In Section 4, a multilayer Chua's circuits network with Markovian switching is considered and some numerical simulations are carried out to validate the effectiveness of our theoretical results, and conclusions are drawn in the last section.
For convenience and simplicity, let us introduce some standard notations utilized in the following context. Let Rn be the n-dimensional Euclidean space. If x∈Rn, then |x| represents the Euclidean norm of x. Let IN be an N×N identity matrix. Besides, the superscript T is defined as the transpose of a vector or a matrix. For any real symmetric matrix, λmax(⋅) and λmin(⋅) denote the maximum and minimum eigenvalues of the given matrix, respectively. Let (Ω,F,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual condition, and B(t) represents one-dimensional Brownian motion defined on the above probability space.
In this section, we consider the multilayer coupled networks with Markovian switching and M nodes as follows:
dδk(t)=[(δk(t),t,r(t))+H∑h=1M∑m=1ρh(r(t))bhkm(r(t))Υδm(t)]dt+uk(t,r(t)),k=1,2,⋯⋯,M, | (2.1) |
where δk(t)=((δk1(t),δk2(t),…,δkn(t))T∈Rn denotes the vector of the kth node at time t. ui(t,r(t)) is the controller to be devised. G(δk(t),t,r(t))∈Rn:Rn×R+×S→Rn represents the nonlinear activation function with G(δk(0),r(0),0)=0. ρh(r(t))>0 is the coupling strength of the hth layer. Υ stands for an inner coupling matrix. The adjacency matrix (bhkm)M×M is the outer coupling configuration matrix, where bhkm>0 holds if node m directs links to k, and otherwise, bhkm=0(k≠m). Meanwhile, the diagonal elements bhkk=−M∑m=1,m≠kbhkm, which leads to M∑m=1bhkm=0. Let r(t),t≥0 signify a right-continuous Markov chain on the probability space (Ω,F,P) with initial date r(0)=r0∈S={1,2,⋯⋯,N}, and generator Ψ=(ψij)N×N given by
P(r(t+Δ)=j|r(t)=i)={ψijΔ+o(Δ),ifi≠j,1+ψijΔ+o(Δ),ifi=j, | (2.2) |
where Δ>0, ψii=−∑j≠iψij, and ψij≥0 denotes the transition rate from i to j.
Subsequently, the isolated node of the network is expressed as
ds(t)=G(s(t),t,r(t))dt, | (2.3) |
where s(t)=(s1(t),s2(t),…,sn(t))T∈Rn. Apparently, the isolated node is independent of other nodes. s(t) can be considered as the equilibrium state or stable objective trajectory, and in multi-agent dynamics, it stands for the leader. Let ηk(t)=δk(t)−s(t) be the synchronization error. Accordingly, the error system can be depicted as
dηk(t)=[ˆG(ηk(t),t,r(t))+H∑h=1M∑m=1ρh(r(t))bhkm(r(t))Υηi(t)]dt+uk(t,r(t)),k=1,2,⋯⋯,M, | (2.4) |
where ˆG(ηk(t),t,r(t))=G(ηk(t),t,r(t))−G(s(t),t,r(t)). In order to realize synchronization, the aperiodically intermittent feedback control based on discrete-time state observations is designed below:
uk(t)={βk(ηk(μ(t)),t,r(t))dw(t),t∈[ti,si),0,t∈[si,ti+1), | (2.5) |
where μ(t)=[t/τ]τ,τ>0, τ denotes the duration between two consecutive observations, and βk(ηk(μ(t),t,r(t))=Ck(r(t))ηk(μ(t)) represents the diffusion coefficient column vector. Time interval [ti,si) is the working time of the kth period while [si,ti+1) stands for the rest time. Furthermore, under aperiodically intermittent noise control, error system (2.4) can be rewritten by
{dη(t)=[˜G(η(t),t,r(t))+H∑h=1ρh(r(t))(Bh(r(t))⊗Υ)η(t)]dt+˜β(η(μ(t)),t,r(t))dw(t),t∈[ti,si),i=0,1,⋯⋯dη(t)=˜G(η(t),t,r(t))+H∑h=1ρh(r(t))(Bh(r(t))⊗Υ)η(t),t∈[si,ti+1),i=0,1,⋯⋯ | (2.6) |
where η(t)=(ηT1(t),ηT2(t),⋯⋯,ηTM(t))T, ˆG(η(t))=(˜GT(η1(t)),˜GT(η2(t)),⋯⋯,˜GT(ηM(t)))T∈RnM, and ˜β(η(μ(t)),t,r(t))=(ηT1(μ(t))CT1(r(t)),ηT2(μ(t))CT2(r(t)),⋯⋯,ηTM(μ(t))CTM(r(t)))T. Since multilayer networks, Markovian switching, and aperiodically intermittent controller based on discrete-time observation noise exist simultaneously, this makes them more complex to analyze the synchronization feature. In order to overcome the difficulties, the auxiliary systems with continuous-time observations are presented below,
{dz(t)=[˜G(z(t),t,r(t))+H∑h=1ρh(r(t))(Bh(r(t))⊗Υ)z(t)]dt+˜β(z(t),t,r(t))dw(t),t∈[ti,si),i=0,1,⋯⋯dz(t)=˜G(z(t),t,r(t))+H∑h=1ρh(r(t))(Bh(r(t))⊗Υ)z(t),t∈[si,ti+1),i=0,1,⋯⋯ | (2.7) |
where z(t)=δ(t)−s(t).
In what follows, several necessary definitions and assumptions are provided, which play a fundamental role in acquiring the theoretical results.
Definition 1. The multilayer coupled networks (2.1) can be almost surely exponentially synchronized if for any initial data η(t0)=η0, there exists a positive constant satisfying the following inequality
limt→∞sup1tlog|η(t)|<−κ,a.s. | (2.8) |
Assumption 1. Suppose that there exist two positive constants α∈(0,1) and N0 such that
N(t,s)≥α(t−s)−N0,∀t>s≥t0, | (2.9) |
where N(t,s) denotes the total control time length on [s,t), and N0 is the elasticity number.
Assumption 2. Suppose that there are positive constants li and σki such that following inequalities hold
|G(u1,t,i)−G(u2,t,i)|≤li|u1−u2|, | (2.10) |
for ∀u1,u2∈Rn, and i∈S.
In this section, some novel lemmas are first established by utilizing stochastic analysis techniques. Based on the derived lemmas, almost sure exponential synchronization of MCNS with Markovian switching are further examined via AIDONC, and some special cases are also discussed.
Lemma 3.1. Let Q=diag{χ1,χ2,⋯,χN}−Ψ denote an M-matrix, where q∈(0,1), χi=0.5q[(2−q)di−σi]−qli−q(H∑h=1ρh(i)λmax([Bh(i)⊗Υ]s)), and Ψ=(ψij)M×M. Under Assumptions 1 and 2, if ϵ=˜λα−˜ζ(1−α)>0, then the solution z(t) of Eq (2.7) satisfies that
E|z(t)|q≤K0E|z0|qe−ϵt, | (3.1) |
where σi=max{λmax(CT1(i)C1(i)),λmax(CT2(i)C2(i)),⋯⋯,λmax(CTM(i)CM(i))}, di=min{λ2min(C1(i)),λ2min(C2(i)),⋯⋯,λ2min(CM(i))}, Qθ=Q(θ1,θ2,⋯⋯,θN)T=(λ1,λ2,⋯⋯,λN)T>0, ˜λ=min1≤i≤N{λiθi}>0, ζi=0.5q[(2−q)di−σi]θi−λi, ˜ζ=max1≤i≤N{ζiθi}>0, θm=mini∈S{θi}, θM=maxi∈S{θi}, and K0=θMeN0(˜λ+˜ζ)θm. In other words, the trivial solution of Eq (2.7) is qth moment exponentially stable.
Proof. Noting that Q is an M-matrix, there exists a vector θ=(θ1,θ2,⋯⋯,θN)T>0 such that Qθ=(λ1,λ2,⋯⋯,λN)T>0. Choose the Lyapunov function V(z(t),t,i)=θi|z(t)|q,i∈S. Obviously, θm|z(t)|q≤V(z(t),t,i)≤θM|z(t)|q. By utilizing the Itô formula, when t∈[ti,si),i=0,1,2,⋯⋯, and we have that
LV(z(t),t,i)=qθi|z(t)|q−2zT(t)[˜G(z(t),t,i)+H∑h=1ρh(i)(Bh(i)⊗Υ)z(t)]+12qθi|z(t)|q−2|˜β(z(t),t,i)|2−12q(2−q)θi|z(t)|q−4|zT(t)˜β(z(t),t,i)|2+N∑j=1ψijθj|z(t)|q. | (3.2) |
According to Assumption 2, the following inequalities are calculated:
qθi|z(t)|q−2zT(t)ˆG(z(t),t,i)≤qθi|z(t)|q−1|ˆG(z(t),i,t)|≤qθili|z(t)|q, | (3.3) |
qθi|z(t)|q−2zT(t)(H∑h=1ρh(i)(Bh(i)⊗Υ)z(t))≤qθi(H∑h=1ρh(i)λmax([Bh(i)⊗Υ]s))|z(t)|q, | (3.4) |
|z(t)|q−2|˜β(z(t),t,i)|2≤max(λmax(CT1(i)C1(i)),λmax(CT2(i)C2(i)),⋯,λmax(CTM(i)CM(i)))|z(t)|q=σi|z(t)|q, | (3.5) |
|z(t)|q−4|zT(t)˜β(z(t),t,i)|2≥min{λ2min(C1(i)),λ2min(C2(i)),⋯⋯,λ2min(CM(i))}|z(t)|q=di|z(t)|q. | (3.6) |
Substituting inequalities (3.3)–(3.6) into equality (3.2) yields that
LV(z(t),t,i)≤[qθili+qθi(H∑h=1ρh(i)λmax([Bh(i)⊗Υ]s))+12qθiσi−12q(2−q)θidi+N∑j=1ψijθj]|z(t)|q. | (3.7) |
Since Qθ=(λ1,λ2,⋯,λN)T>0, when t∈[ti,si),i=0,1,2,⋯⋯, we have that
LV(z(t),t,i)≤−λi|z(t)|q≤−λiθiV(z(t),i,t)≤−min1≤i≤N{λiθi}V(z(t),i,t)=−˜λV(z(t),i,t). | (3.8) |
Moreover, when t∈[si,ti+1),i=0,1,2,⋯⋯, it can be calculated that
LV(z(t),t,i)≤[0.5q((2−q)d(i)−σ(i))θi−λi]|z(t)|q≤ζiθiV(z(t),t,i)≤max1≤i≤N{ζiθi}V(z(t),i,t)=˜ζV(z(t),i,t). | (3.9) |
Particularly, according to inequality (3.8), if t∈[t0,s0),t0=0, we can infer that
EV(z(t),t,r(t))≤θME|z0|qe−˜λt. | (3.10) |
Moreover, combining inequalities (3.9) and (3.10), it can be deduced that
EV(z(t),t,r(t))≤θME|z(s0)|qe˜ζ(t−s0)≤θME|z0|qe−˜λ(s0−t0)+˜ζ(t−s0). | (3.11) |
Repeating the iteration leads to the following result,
EV(z(t),t,r(t))≤θME|z0|qe−˜λi−1∑j=0(sj−tj)+˜ζi∑j=1(tj−sj−1)−˜λ(t−ti),t∈[ti,si), | (3.12) |
and
EV(z(t),t,r(t))≤θME|z0|qe−˜λi∑j=0(sj−tj)+˜ζi∑j=1(tj−sj−1)+˜ζ(t−si),t∈[si,ti+1), | (3.13) |
which means that
EV(z(t),t,r(t))≤θME|z0|qe−˜λNc(0,t)+˜ζNr(0,t). | (3.14) |
In light of Assumption 1, one has that Nc(t,0)≥αt−N0 and Nr(t,0)≥(1−α)t+N0. Consequently,
EV(z(t),t,r(t))≤θME|z0|qe−˜λ[αt−N0]+˜ζ[(1−α)t+N0]≤eN0(˜λ+˜ζ)θME|z0|qe−[˜λα−˜ζ(1−α)]t. | (3.15) |
Therefore, one has that
E|z(t)|q≤K0E|z0|qe−ϵt, | (3.16) |
where K0=θMeN0(˜λ+˜ζ)θm, and ϵ=˜λα−˜ζ(1−α)>0.
Lemma 3.2. Let q∈(0,1) and Assumption 2 hold. The following estimations are presented:
E|η(t)|2≤|η0|2e(2ˆl+ˆσ)t, | (3.17) |
E|η(t)−η(μ(t))|2≤K1(τ)e(2ˆl+ˆσ)t, | (3.18) |
E|η(t)−z(t)|q≤|η0|q[4(ˆl+ˆσ)K1(τ)2ˆl+ˆσ]q2|η0|qe(ˆl+ˆσ)qt, | (3.19) |
where ˆl=max1≤i≤N{li+H∑h=1ρh(i)λmax([Bh(i)⊗Υ]s)}, ˆσ=max0≤i≤N{σi},
σi=max{λmax(CT1(i)C1(i)),λmax(CT2(i)C2(i)),⋯,λmax(CTM(i)CM(i))},
K1(τ)={4τ[(max1≤i≤N{li})2+Hmax1≤i≤N{H∑h=1ρ2h(i)λmax((Bh(i)⊗Υ)T(Bh(i)⊗Υ))}]+2ˆσ}τ.
Proof. Error system (2.6) in matrix form can be formulated as
dη(t)=[˜G(η(t),t,r(t))+H∑h=1ρh(r(t))(Bh(r(t))⊗Υ)η(t)]dt+I(t)ˆβ(η(μ(t)),t,r(t))dB(t), | (3.20) |
where I(t)={1,t∈[ti,si),0,t∈[si,ti+1),i=0,1,2,⋯⋯. By employing the Itô formula, one has that
E|η(t)|2=|η0|2+E∫t0{2ηT(s)[˜G(η(s),s,r(s))+H∑h=1ρh(r(s))(Bh(r(s))⊗Υ)η(s)]+|ˆβ(η(μ(s)),s,r(s))|2I(s)}ds≤|η0|2+∫t02max1≤i≤N{li+H∑h=1ρh(i)λmax([Bh(i)⊗Υ]s)}E|η(s)|2ds+max1≤i≤N{σi}∫t0E|η(μ(s))|2ds. | (3.21) |
Moreover, it can be inferred that
sup0≤s≤tE|η(s)|2≤|η0|2+(2ˆl+ˆσ)∫t0(sup0≤s≤tE|η(s)|2)ds. | (3.22) |
Applying the Gronwall inequality yields that
sup0≤s≤tE|η(s)|2≤|η0|2e(2ˆl+ˆσ)t. | (3.23) |
Accordingly, it can be concluded that
sup0≤s≤tE|η(s)|2≤|η0|2e(2ˆl+ˆσ)t, | (3.24) |
which indicates that
E|η(t)|2≤|η0|2e(2ˆl+ˆσ)t. | (3.25) |
Obviously, it follows from Eq (3.20) that
η(t)−η(μ(t))=∫tμ(t)[˜G(η(s),s,r(s))+H∑h=1ρh(r(s))(Bh(r(s))⊗Υ)]η(s)ds | (3.26) |
+∫tμ(t)I(s)ˆβ(η(μ(s)),s,r(s))dw(s). | (3.27) |
Together with the Hölder inequality, it can be calculated as
E|η(t)−η(μ(t))|2≤2E|∫tμ(t)[˜G(η(s),s,r(s))+H∑h=1ρh(r(s))(Bh(r(s))⊗Υ)]η(s)ds|2+2E∫tμ(t)|ˆβ(η(μ(s)),s,r(s))|2ds≤4τ[(max1≤i≤N{li})2+Hmax1≤i≤N{H∑h=1ρ2h(i)λmax((Bh(i)⊗Υ)T(Bh(i)⊗Υ))}]∫tμ(t)E|η(s)|2ds+2ˆσ∫tμ(t)E|η(μ(s))|2ds≤K1(τ)τ∫tμ(t)sup0≤s≤tE|η(s)|2du≤K1(τ)|η0|2e(2ˆl+ˆσ)t. | (3.28) |
On the other hand, by utilizing the Itô formula again, one gets that
E|η(t)−z(t)|2=2E∫t0{[η(s)−z(s)]T[ˆG(η(s),s,r(s))−ˆG(z(s),s,r(s))+H∑h=1ρh(r(s))(Bh(r(s))⊗Υ)(η(s)−z(s))]+|ˆβ(η(μ(s)),s,r(s))−ˆβ(η(s),s,r(s))|2I(s)}ds≤2[max1≤i≤N{li}+max1≤i≤N{H∑h=1ρh(i)λmax([Bh(i)⊗Υ]s)}]∫t0E|η(s)−z(s)|2ds+max1≤i≤N{λmax(CT1(i)C1(i)),λmax(CT2(i)C2(i)),⋯⋯,λmax(CTM(i)CM(i))}×∫t0E|η(μ(s))−z(s)|2ds≤2ˆl∫t0E|η(s)−z(s)|2ds+ˆσ∫t0E|η(μ(s)−z(s)|2ds≤(2ˆl+2ˆσ)∫t0E|η(s)−z(s)|2ds+2ˆσ∫t0E|η(μ(s)−η(s)|2ds. | (3.29) |
Recalling assertion (3.18), it can be deduced that
∫t0E|η(μ(s)−η(s)|2ds≤K1(τ)|η0|2∫t0e(2ˆl+ˆσ)sds≤K1(τ)|η0|22ˆl+ˆσ[e(2ˆl+ˆσ)t−1]. | (3.30) |
Substituting inequality (3.30) into inequality (3.29) leads to
E|η(t)−z(t)|2≤(2ˆl+2ˆσ)∫t0E|η(s)−z(s)|2ds+2ˆσK1(τ)|η0|22ˆl+ˆσ[e(2ˆl+ˆσ)t−1]. | (3.31) |
Subsequently, by using the Gronwall inequality [38], it can be readily obtained that
E|η(t)−z(t)|2≤2ˆσK1(τ)|η0|22ˆl+ˆσ[e(2ˆl+ˆσ)t−1]+4ˆσ(ˆl+ˆσ)K1(τ)|η0|22ˆl+ˆσ∫t0e2(ˆl+ˆσ)(t−s)[e(2ˆl+ˆσ)s−1]ds≤[4(ˆl+ˆσ)K1(τ)2ˆl+ˆσ]|η0|2e2(ˆl+ˆσ)t. | (3.32) |
Hence, in light of the Hölder inequality, one can further acquire the following assertion:
E|η(t)−z(t)|q≤(E|η(t)−z(t)|2)q2≤[4(ˆl+ˆσ)K1(τ)2ˆl+ˆσ]q2|η0|qeq(ˆl+ˆσ)t. | (3.33) |
Theorem 3.1. Suppose that all of the conditions in Lemma 3.1 are satisfied. Let ξ∈(0,1). τ∗>0 is the unique positive root of the following equation:
ξ+K2(τ)e(ˆl+ˆσ)q(τ+1ϵlog(K0ξ))=1, | (3.34) |
where θm=min0≤i≤N{θi}, θM=max0≤i≤N{θi}, K2(τ)=[4(ˆl+ˆσ)K1(τ)2ˆl+ˆσ]q2, and ϵ,θi,ˆρ,ˆσ,K0, and K1 are the same as in Lemmas 3.1 and 3.2. If 0<τ<τ∗, then the multilayer complex network system (2.1) can achieve almost sure exponential synchronization with isolated node (2.3) via aperiodically intermittent noise control, i.e.,
limt→∞sup1tlog|η(t)|≤−2ˆδ5q<0,a.s. | (3.35) |
Proof. For given positive constant ξ∈(0,1), we can choose sufficiently large positive integer m0 such that
log(K0ξ)ϵτ≤m0<1+log(K0ξ)ϵτ. | (3.36) |
Combining the above inequality with Lemma 3.1, one gets that
K0e−m0ϵτ≤ξ,E|z(m0τ)|q≤K0e−m0ϵτ|η0|q≤ξ|η0|q. | (3.37) |
Noting that
ξ+K2(τ)e(ˆl+ˆσ)qm0τ<ξ+K2(τ)e(ˆl+ˆσ)q(τ+1ϵlog(K0ξ))=1, | (3.38) |
there exists a constant ˆδ>0 such that
ξ+K2(τ)e(ˆl+ˆσ)qm0τ<e−ˆδm0τ. | (3.39) |
For simplicity, let zm0=z(m0τ,η0,r0) and ηm0=η(m0τ,η0,r0). Combing Lemma 3.2 and the elementary inequality (u+v)q≤uq+vq, u≥0,v≥0,q∈(0,1), one can compute that
E|ηm0|q≤E|zm0|q+E|ηm0−zm0|q≤K0e−m0ϵτ|η0|q+|η0|qK2(τ)e(ˆl+ˆσ)qm0τ≤|η0|q[ξ+K2(τ)e(ˆl+ˆσ)qm0τ]≤|η0|qe−ˆδm0τ. | (3.40) |
When t≥mτ, it can be deduced that
E(|η2m0|q|Fm0τ)≤|ηm0|qe−ˆδm0τ≤|η0|qe−2ˆδm0τ. | (3.41) |
Similarly, one can derive that
E|ηim0|q≤E|η(i−1)m0|qe−ˆδm0τ≤⋯⋯≤E|ηm0|qe−ˆδ(i−1)m0τ≤|η0|qe−ˆδim0τi=1,2,3,⋯⋯. | (3.42) |
Let τ∈(0,τ∗). It follows from Eq (3.14) that
η(t)=η0+∫t0[˜G(η(s),s,r(s))+H∑h=1ρh(r(s))(Bh(r(s))⊗Υ)η(s)]ds+I(s)ˆβ(η(μ(s)),s,r(s))dB(s). |
By exploiting the Hölder inequality and the Burkholder-Davis-Gundy inequality, it can be estimated that
E(sup0≤t≤m0τ|η(t)|2)≤3|η0|2+3E{sup0≤t≤m0τ|∫t0˜G(η(s),s,r(s))+H∑h=1ρh(r(s))(Bh(r(s))⊗Υ))η(s)ds|2}+3E{sup0≤t≤m0τ|∫t0I(s)ˆβ(η(μ(s)),s,r(s))dB(s)|2}≤3|η0|2+6mτ[(max1≤i≤N{li})2+Hmax1≤i≤N{H∑i=1ρ2h(i)λmax(Bh(i)⊗Υ)T(Bh(i)⊗Υ))}].E(sup0≤t≤m0τ∫t0|η(s)|2ds)+12ˆσE(sup0≤t≤m0τ∫t0|η(μ(s))|2ds)≤3|η0|2+6m0τ[(max1≤i≤N{li})2+Hmax1≤i≤N{H∑i=1ρ2h(i)λmax((Bh(i)⊗Υ)T(Bh(i)⊗Υ))}+12ˆσ]∫m0τ0E(sup0≤t≤s|η(t)|2)ds. | (3.43) |
Moreover, by the Gronwall inequality, it can be attained that
E(sup0≤t≤m0τ|η(t)|2)≤3|η0|2eK3(τ)m0τ, | (3.44) |
where K3(τ)=6m0τ[(max1≤i≤N{li})2+Hmax1≤i≤N{∑Hi=1ρ2h(i)λmax((Bh(i)⊗Υ)T(Bh(i)⊗Υ))}+12ˆσ]. By virtue of the Hölder inequality, one gets that
E(sup0τ≤t≤m0τ|η(t)|q)≤3q2|η0|qeqK3(τ)mτ2=K4(τ)|η0|q, | (3.45) |
where K4(τ)=3q2eqK3(τ)mτ2. Accordingly, based on the time stationarity of stochastic differential equations, it can be derived that
E(supim0τ≤t≤(i+1)m0τ|η(t)|q|Fim0τ)≤K4(τ)|ηim0τ|q,i=0,1,2,⋯⋯. | (3.46) |
Combining (3.42) and (3.46) gives
E(supim0τ≤t≤(i+1)m0τ|η(t)|q)≤K4(τ)E|ηim0τ|q≤K4(τ)|η0|qe−ˆδim0τ. | (3.47) |
According to the Markov inequality, one gets that
P{supim0τ≤t≤(i+1)m0τ|η(t)|q≥e−0.4m0ˆδiτ}≤K4(τ)|η0|qe−0.6m0ˆδiτ,i=1,2,⋯⋯. | (3.48) |
As a result, by the Borel-Cantelli lemma, for almost all ω∈Ω, there exists a stochastic integer i0=i0(ω) satisfying
supi∗m0τ≤t≤(i∗+1)m0τ|η(t)|q<e−0.4m0ˆδi∗τ, | (3.49) |
for i∗>i0(ω). Hence, it can be concluded that
limt→∞sup1tlog|η(t)|≤−2ˆδ5q<0,a.s. | (3.50) |
which signifies that the multilayer complex network system (2.1) realizes almost sure exponential synchronization with isolated node (2.3) via aperiodically intermittent noise control.
Remark 1. Recently, stability and synchronization of multilayer complex networks have been discussed by developing various control approaches such as state feedback control, event-triggered control, impulsive control, and intermittent control in [12,14,16,21]. Compared with the previous findings, this paper has dealt with the almost sure exponential synchronization issue of MCNS by adopting aperiodically intermittent discrete observation noise control, where the stochastic noise can be viewed as control input in the diffusion term rather than the drift term [22]. In the considered control scheme, the average control rate has been integrated to depict the distributions of work/rest intervals of the control strategy from an overall perspective. In general, as the average control rate become larger, the control performance will be better.
Remark 2. Different from the work about pth moment exponential synchronization of network systems in [21,22], by utilizing M-matrix theory and various stochastic analysis techniques including the Itô formula, the Gronwall inequality, and the Borel-Cantelli lemma, almost sure exponential synchronization of multilayer coupled networks with Markovian switching has been analyzed via AIDONC. Although some sufficient conditions on almost sure exponential synchronization of MCNS have been given in [23], the control strategy AIDONC has not been adopted. Additionally, in [35,36], almost sure exponential synchronization and stabilization of neural networks have been investigated under AIDONC and the theoretical work has been extended to hybrid neural networks and complex networks [37,38,39]. In this paper, Markovian switching and multilayer interaction factors have been taken into account simultaneously, which make our network system more general.
Remark 3. In this paper, we utilize the stochastic operator LV and matrix multiplication to investigate the almost sure exponential synchronization of stochastic multilayer complex networks. Actually, fast algorithms about matrix multiplication are an interesting and hot topic which can be applied in contemporary Intel Xeon microprocessors. Particularly, in [40], a broad series of algorithms taking advantage of the efficiency of fast matrix multiplication algorithms in other mathematical and computer science operations have been reported. Extending the applications of these algorithms to complex networks remains a current challenge.
In particular, when τ=0 and H=1, the multilayer networks are reduced to the following single-layered networks.
dδk(t)=[G(δk(t),t,r(t))+ρ(r(t))M∑m=1bkm(r(t))Υδm(t)]dt+uk(t,r(t)),k=1,2,⋯⋯,M. | (3.51) |
Corollary 1. Let Q=diag{χ1,χ2,⋯⋯,χN}−Ψ denote an M-matrix, where q∈(0,1), χi=0.5q((2−q)d(i)−σ(i))−qli−q(ρ(i)λmax([B(i)⊗Υ]s)), and Ψ=(ψij)N×N. Under Assumptions 1 and 2, if ϵ=˜λα−˜ζ(1−α)>0, complex network (3.51) can achieve almost sure exponential synchronization with the isolated node (2.3), where σi=max{λmax(CT1(i)C1(i)),λmax(CT2(i)C2(i)),⋯⋯,λmax(CTM(i)CM(i))}, di=min{λ2min(C1(i)),λ2min(C2(i)),⋯⋯,λ2min(CM(i))}, Qθ=Q(θ1,θ2,⋯⋯,θN)T=(λ1,λ2,⋯⋯,λN)T>0, ˜λ=min1≤i≤N{λiθi}>0, ζi=0.5q[(2−q)di−σi]θi−λi, ˜ζ=max1≤i≤N{ζiθi}>0, θm=mini∈S{θi}, θM=maxi∈S{θi}, and K0=θMeN0(˜λ+˜ζ)θm.
In particular, when θ=(θ1,θ2,⋯⋯,θn)T=(1,1,⋯⋯,1)TandH=1, immediately, we acquire the following assertion.
Corollary 2. Let all the conditions in Theorem 1 hold except (λ1,λ2,⋯⋯,λn)T=(1,1,⋯⋯,1)T, χi=0.5q[(2−q)di−σi]−qli−q(ρ(i)λmax([B(i)⊗Υ]s)), and ˆl=max1≤i≤N{li+ρ(i)λmax([B(i)⊗Υ]s)}. If 0<τ<τ∗, then the multilayer coupled network system (3.51) can achieve almost sure exponential synchronization with isolated node (2.3) via aperiodically intermittent noise control, i.e.,
limt→∞sup1tlog|η(t)|≤−2ˆδ5q<0,a.s. | (3.52) |
Remark 4. Fractional differences can be seen as generalization or as an extension of classical calculus while fractional order derivatives are non-locally distributed, which can describe the memory and hereditary effects of complex processes and materials accurately. In recent years, many mathematical models in engineering and biological sciences have been proposed by using discrete or continuous fractional equations [41,42,43,44,45]. Particularly, in [41,42], some dynamic behaviors such as the existence and stability of solutions of worms in a wireless sensor model in the sense of the fractal fractional and fractional nabla difference COVID-19 model have been analyzed through function analysis and the Ulam-Hyers stability technique. Furthermore, some properties such as existence, uniqueness, controllability, stability of solutions of some equations including Hilfer fractional evolution equations, ABC-fuzzy-Volterra integro-differential equations, and coupled pantograph discrete fractional order difference equations have been discussed in [44,45]. Actually, fractional complex networks have been a current attractive topic. In the future, our work can be further extended to stochastic fractional complex networks.
In order to exhibit the feasibility of our theoretical findings, a famous Chua's circuits network is considered here, which has been extensively applied to various areas as an essential nonlinear electronic oscillator model. Initially, the single Chua's circuit is described as follows:
{˙V1(t)=−1R1C∗1V1(t)+1R1C∗1V2(t)−1C∗1Γ(V1(t));˙V2(t)=1R1C∗2V1(t)−1R1C∗2V2(t)+1C∗2I3(t);˙I3(t)=−1L0(V2(t)+R2I3(t)); | (4.1) |
where V1(t) and V2(t) denote the voltages across the capacitors C∗1 and C∗2. Meanwhile, I3 signifies the current through the inductance L0, and R1 and R2 stands for the linear resistors. Γ(V1(t)) denotes the current through the nonlinear resistor NR1, which can be recasted as Γ(V1(t))=ϖ1V1+0.5(ϖ2−ϖ1)(|V1+1|−|V1−1|),ϖ1=−0.05,ϖ2=−0.07. Soon afterward, we consider a multilayer coupled network of Chua's circuits with Markovian switching as follows:
(dδk1(t)dδk2(t)dδk3(t))=(˜F11(r(t))˜F12(r(t))0˜F21(r(t))˜F22(r(t))˜F23(r(t))0˜F32(r(t))˜F33(r(t)))(δk1(t)δk2(t)δk3(t))dt+(˜Γk(r(t))00)dt+H∑h=1M∑m=1ρh(r(t))bhkm(r(t))Υδm(t)dt+uk(t,r(t)), | (4.2) |
where δk(t)=(δk1(t),δk2(t),δk3(t))T=(Vk1,Vk2,Ik3)T, (k=1,2,3,4,5,6), ˜F11(r(t))=−1R1(r(t))C∗1(r(t)),˜F12(r(t))=−˜F11(r(t)), ˜F21(r(t))=1R1(r(t))C∗2(r(t)),˜F22(r(t))=−1R1(r(t))C∗2(r(t)),˜F23(r(t))=1C∗2(r(t)), ˜F32(r(t))=−1L0(r(t)), ˜F33(r(t))=−R2(r(t))L0(r(t)), and ˜Γk(r(t))=−Γ(δk1(t))C∗1(r(t)). Accordingly, the isolated node of the network is expressed as
ds(t)=G(s(t),t,r(t))dt. | (4.3) |
Meanwhile, controller uk(t,r(t)) is designed as
uk(t)={Ck(r(t))ηk(μ(t))dw(t),t∈[ti,si),0,t∈[si,ti+1), | (4.4) |
where ηk(t)=δk(t)−s(t) and μ(t)=[t/τ]τ,τ>0, and τ denotes the duration between two consecutive observations. It is supposed that Markov jump r(t)∈S={1,2} with generator
Ψ=(−441−1). |
Parameters M and H can be selected as M=6 and H=3. When r(t)=1, let C∗1(1)=2, C∗2(1)=4, R∗1(1)=2, R∗2(1)=0.5, L∗0(1)=4, ρ1(1)=0.6, ρ2(1)=0.5, and ρ3(1)=0.4. When r(t)=2, let C∗1(2)=4, C∗2(2)=2, R∗1(2)=2.5, R∗2(2)=1, L∗0(2)=5, ρ1(2)=0.4, ρ2(2)=0.6, and ρ3(2)=0.5. The values of six adjacency matrices B1(1),B2(1),B3(1), B1(2), B2(2), and B3(2) are given as
B1(1)=[−0.30.10.1000.10−0.20.100.1000−0.10.1000.10.10−0.30.100.1000−0.20.1000.10.10−0.2],B2(1)=[−0.20.100.1000−0.30.100.10.10.10−0.20.10000.10−0.200.10.1000−0.100.100.10.10−0.3],B3(1)=[−0.40.10.100.10.10.1−0.300.10.1000−0.20.100.1000−0.10.100.100.10−0.2000.100.10−0.2],B1(2)=[−0.30.100.100.10−0.20.100.100.10−0.30.100.100.10−0.20.10000.10−0.100.100.10.10−0.3],B2(2)=[−0.30.100.100.10−0.10.100000−0.20.10.1000.10.1−0.30.100.1000−0.20.100.100.10−0.2],B3(2)=[−0.200.100.100.1−0.3000.10.10.10−0.20.10000.10−0.20.100.100.10−0.30.100.100.10−0.2]. |
Furthermore, choose q=0.5,Υ=0.5I,α=0.8,Ck(r(t))=2I,ξ=0.9, and N0=0.01. By calculation, we can obtain l1=0.3215,l2=0.3561, ˆl=0.3633,ˆσ=4, χ1=0.3349,χ2=0.3184, λ1=λ2=1,θ1=3.1009,θ2=3.1105,ˆλ=0.3215,ˆζ=0.1785, and ϵ=˜λα−˜ζ(1−α)=0.2215>0. According to ξ+K2(τ)e(ˆl+ˆσ)q(τ+1ϵlog(K0ξ))=1, the upper bound of duration time τ is estimated as τ<τ∗=3.6213×10−5. Therefore, all of the conditions of Theorem 1 are satisfied, and the almost sure exponential synchronization between network Eq (4.2) and isolated node Eq (4.3) is realized. Figure 1 shows a right continuous Markov chain with initial data r(0)=2. Meanwhile, Figures 2–4 illustrate the synchronization sample trajectories ηi1,ηi2,ηi3,(i=1,2,3,4,5,6) between network Eq (4.2) and isolated node Eq (4.3), respectively. It can be observed from Figures 2–4 that the error system is stable under aperiodically intermittent discrete observation noise control, which means that almost sure exponential synchronization is achieved. Therefore, the numerical simulations validate the effectiveness of the theoretical findings. By increasing the average control rate and the control intensity, the synchronization performance including the exponential convergent rate will be further improved. In comparison, this method is one discontinuous control, which can save cost and promote efficiency.
In this paper, almost sure exponential synchronization of multilayer complex networks with Markovian switching via AIDONC is investigated. Different from the work about pth moment exponential synchronization of network systems in [21,22], by utilizing M-matrix theory and various stochastic analysis techniques including the Itô formula, the Gronwall inequality, and the Borel-Cantelli lemma, some criteria on almost sure exponential synchronization of multilayer coupled networks with Markovian switching are constructed and the upper bound of the duration time is also estimated. It is noted that the control strategy is based on aperiodically intermittent discrete observation noise, where the average control rate is integrated to depict the distributions of work/rest intervals of the control strategy from an overall perspective. Finally, some numerical simulations are exhibited to illustrate the effectiveness and feasibility of our analytical findings. Although the aperiodically intermittent discrete observation noise control strategy can reduce energy consumption and save control cost, the derived algebraic criteria on almost sure exponential synchronization generally are comparatively difficult to calculate. Meanwhile, the duration time is comparatively small, and its length needs to be extended by designing several optimization algorithms in order to demonstrate the feasibility in practise. In the future, based on the work in [46,47], the issue of stabilization and synchronization of coupled networks with semi-Markovian switching based on AIDONC will be further explored.
Li Liu: conceptualization, investigation, writing-original draft; Yinfang Song: funding acquisition, methodology, supervision; Hong Yu: software; Gang Zhang: writing-review and editing. All authors have read and approved the final version of the manuscript for publication.
This work was supported by the National Natural Science Foundation of China (62076039, 62273059).
The authors declare no conflict of interest.
[1] |
A. Maachou, R. Malti, P. Melchior, J. L. Battagliaand, A. Oustaloup, Nonlinear thermal system identification using fractional Volterra series, Control Eng. Pract., 29 (2014), 50–60. https://doi.org/10.1016/j.conengprac.2014.02.023 doi: 10.1016/j.conengprac.2014.02.023
![]() |
[2] |
P. E. Jacob, S. M. M. Alavi, A. Mahdi, S. J. Payne, D. A. Howey, Bayesian inference in non-Markovian state-space models with applications to battery fractional-order systems, IEEE Trans. Control Syst. Technol., 26 (2018), 497–506. https://doi.org/10.1109/TCST.2017.2672402 doi: 10.1109/TCST.2017.2672402
![]() |
[3] |
G. Tsirimokou, C. Psychalinos, A. S. Elwakil, K. N. Salama, Electronically tunable fully integrated fractional-order resonator, IEEE Trans. Circuits Syst. II Express Briefs, 65 (2018), 166–170. https://doi.org/10.1109/TCSII.2017.2684710 doi: 10.1109/TCSII.2017.2684710
![]() |
[4] |
E. S. A. Shahri, A. Alfi, J. A. T. Machado, Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation, Appl. Math. Modell., 81 (2020), 663–672. https://doi.org/10.1016/j.apm.2020.01.013 doi: 10.1016/j.apm.2020.01.013
![]() |
[5] |
C. Hou, X. Liu, H. Wang, Adaptive fault tolerant control for a class of uncertain fractional-order systems based on disturbance observer, Int. J. Robust Nonlinear Control, 30 (2020), 3436–3450. https://doi.org/10.1002/rnc.4950 doi: 10.1002/rnc.4950
![]() |
[6] |
S. Kamal, R. K. Sharma, T. N. Dinh, H. MS, B. Bandyopadhyay, Sliding mode control of uncertain fractional-order systems: A reaching phase free approach, Asian J. Control, 23 (2021), 199–208. https://doi.org/10.1002/asjc.2223 doi: 10.1002/asjc.2223
![]() |
[7] | J. X. Xu, Y. Tan, Linear and Nonlinear Iterative Learning Control, Springer-Verlag, Berlin, 2003. https://doi.org/10.1007/3-540-44845-4 |
[8] | Y. Q. Chen, K. L. Moore, On Dα-type iterative learning control, in Proceedings of the 40th IEEE Conference on Decision and Control, IEEE, (2001), 4451–4456. https://doi.org/10.1109/CDC.2001.980903 |
[9] | Y. Li, H. S. Ahn, Y. Q. Chen, Iterative learning control of a class of fractional order nonlinear systems, in Proceedings of the 2010 IEEE International Symposium on Intelligent Control, IEEE, (2010), 779–782. https://doi.org/10.1109/ISIC.2010.5612935 |
[10] |
Y. H. Lan, Iterative learning control with initial state learning for fractional order nonlinear systems, Comput. Math. Appl., 64 (2012), 3210–3216. https://doi.org/10.1016/j.camwa.2012.03.086 doi: 10.1016/j.camwa.2012.03.086
![]() |
[11] |
Y. Li, W. Jiang, Fractional order nonlinear systems with delay initerative learning control, Appl. Math. Comput., 257 (2015), 546–552. https://doi.org/10.1016/j.amc.2015.01.014 doi: 10.1016/j.amc.2015.01.014
![]() |
[12] |
L. Li, Lebesgue-p norm convergence of fractional-order PID-type iterative learning control for linear systems, Asian J. Control, 20 (2018), 483–494. https://doi.org/10.1002/asjc.1561 doi: 10.1002/asjc.1561
![]() |
[13] |
S. Liu, J. R. Wang, Fractional order iterative learning control with randomly varying trial lengths, J. Franklin Inst., 354 (2017), 967–992. https://doi.org/10.1016/j.jfranklin.2016.11.004 doi: 10.1016/j.jfranklin.2016.11.004
![]() |
[14] |
D. Luo, J. R. Wang, D. Shen, PDα-type distributed learning control for nonlinear fractional-order multiagent systems, Math. Methods Appl. Sci., 42 (2019), 4543–4553. https://doi.org/10.1002/mma.5677 doi: 10.1002/mma.5677
![]() |
[15] |
D. Luo, J. R. Wang, D. Shen, Iterative learning control for locally Lipschitz nonlinear fractional-order multi-agent systems, J. Franklin Inst., 357 (2020), 6671–6693. https://doi.org/10.1016/j.jfranklin.2020.04.032 doi: 10.1016/j.jfranklin.2020.04.032
![]() |
[16] |
J. Zhang, D. Meng, Convergence analysis of saturated iterative learning control systems with locally Lipschitz nonlinearities, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 4025–4035. https://doi.org/10.1109/TNNLS.2019.2951752 doi: 10.1109/TNNLS.2019.2951752
![]() |
[17] |
D. Meng, K. L. Moore, Contraction mapping-based robust convergence of iterative learning control with uncertain, locally-Lipschitz nonlinearity, IEEE Trans. Syst. Man Cybern.: Syst., 50 (2020), 442–454. https://doi.org/10.1109/TSMC.2017.2780131 doi: 10.1109/TSMC.2017.2780131
![]() |
[18] |
S. He, H. Fang, M. Zhang, F. Liu, Z. Ding, Adaptive optimal control for a class of nonlinear systems: the online policy iteration approach, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 549–558. https://doi.org/10.1109/TNNLS.2019.2905715 doi: 10.1109/TNNLS.2019.2905715
![]() |
[19] |
H. Fang, M. Zhang, S. He, X. Luan, F. Liu, Z. Ding, Solving the zero-sum control problem for tidal turbine system: an online reinforcement learning approach, IEEE Trans. Cybern., 53 (2023), 7635–7647. https://doi.org/10.1109/TCYB.2022.3186886 doi: 10.1109/TCYB.2022.3186886
![]() |
[20] |
S. Raghavan, J. K. Hedrick, Observer design for a class of nonlinear systems, Int. J. Control, 59 (1994), 515–528. https://doi.org/10.1080/00207179408923090 doi: 10.1080/00207179408923090
![]() |
[21] |
W. Yu, P. DeLellis, G. Chen, M. di Bernardo, J. Kurths, Distributed adaptive control of synchronization in complex networks, IEEE Trans. Autom. Control, 57 (2012), 2153–2158. https://doi.org/10.1109/TAC.2012.2183190 doi: 10.1109/TAC.2012.2183190
![]() |
[22] |
M. Hussain, M. Rehan, C. K. Ahn, M. Tufail, Robust antiwindup for one-sided Lipschitz systems subject to input saturation and applications, IEEE Trans. Ind. Electron., 65 (2018), 9706–9716. https://doi.org/10.1109/TIE.2018.2815950 doi: 10.1109/TIE.2018.2815950
![]() |
[23] |
G. Hu, Observers for one-sided Lipschitz non-linear systems, IMA J. Math. Control Inf., 23 (2006), 395–401. https://doi.org/10.1093/imamci/dni068 doi: 10.1093/imamci/dni068
![]() |
[24] |
M. Xu, G. Hu, Y. Zhao, Reduced-order observer for one-sided Lipschitz nonlinear systems, IMA J. Math. Control Inf., 26 (2009), 299–317. https://doi.org/10.1093/imamci/dnp017 doi: 10.1093/imamci/dnp017
![]() |
[25] |
Y. Zhao, J. Tao, N. Z. Shi, A note on observer design for one-sided Lipschitz nonlinear systems, Syst. Control Lett., 59 (2010), 66–71. https://doi.org/10.1016/j.sysconle.2009.11.009 doi: 10.1016/j.sysconle.2009.11.009
![]() |
[26] | M. Abbaszadeh, H. Marquez, Nonlinear observer design for one-sided Lipschitz systems, in Proceedings of the 2010 American Control Conference, IEEE, (2010), 5284–5289. https://doi.org/10.1109/ACC.2010.5530715 |
[27] |
W. Zhang, H. Su, H. Wang, Z. Han, Full-order and reduced-order observers for one-sided lipschitz nonlinear systems using Riccati equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4968–4977. https://doi.org/10.1016/j.cnsns.2012.05.027 doi: 10.1016/j.cnsns.2012.05.027
![]() |
[28] |
W. Zhang, H. Su, F. Zhu, S. P. Bhattacharyya, Improved exponential observer design for one-sided Lipschitz nonlinear systems, Int. J. Robust Nonlinear Control, 26 (2016), 3958–3973. https://doi.org/10.1002/rnc.3543 doi: 10.1002/rnc.3543
![]() |
[29] |
X. Cai, H. Gao, L. Liu, W. Zhang, Control design for one-sided Lipschitz nonlinear differential inclusions, ISA Trans., 53 (2014), 298–304. https://doi.org/10.1016/j.isatra.2013.12.005 doi: 10.1016/j.isatra.2013.12.005
![]() |
[30] |
W. Saad, A. Sellami, G. Garcia, H∞ control for uncertain one-sided Lipschitz nonlinear systems in finite frequency domain, Int. J. Robust Nonlinear Control, 30 (2020), 5712–5727. https://doi.org/10.1002/rnc.5101 doi: 10.1002/rnc.5101
![]() |
[31] |
J. Ren, J. Sun, J. Fu, Finite-time event-triggered sliding mode control for one-sided Lipschitz nonlinear systems with uncertainties, Nonlinear Dyn., 103 (2021), 865–882. https://doi.org/10.1007/s11071-020-06096-2 doi: 10.1007/s11071-020-06096-2
![]() |
[32] |
R. Agha, M. Rehan, C. K. Ahn, G. Mustafa, S. Ahmad, Adaptive distributed consensus control of one-sided Lipschitz nonlinear multiagents, IEEE Trans. Syst. Man Cybern.: Syst., 49 (2019), 568–578. https://doi.org/10.1109/TSMC.2017.2764521 doi: 10.1109/TSMC.2017.2764521
![]() |
[33] |
M. A. Razaq, M. Rehan, M. Tufail, C. K. Ahn, Multiple Lyapunov functions approach for consensus of one-sided Lipschitz multi-agents over switching topologies and input saturation, IEEE Trans. Circuits Syst. II Express Briefs, 67 (2020), 3267–3271. https://doi.org/10.1109/TCSII.2020.2986009 doi: 10.1109/TCSII.2020.2986009
![]() |
[34] |
P. Gu, S. Tian, Analysis of iterative learning control for one-sided Lipschitz nonlinear singular systems, J. Franklin Inst., 356 (2019), 196–208. https://doi.org/10.1016/j.jfranklin.2018.10.014 doi: 10.1016/j.jfranklin.2018.10.014
![]() |
[35] |
P. Gu, S. Tian, D-type iterative learning control for one-sided Lipschitz nonlinear systems, Int. J. Robust Nonlinear Control, 29 (2019), 2546–2560. https://doi.org/10.1002/rnc.4511 doi: 10.1002/rnc.4511
![]() |
[36] |
P. Gu, S. Tian, P-type iterative learning control with initial state learning for one-sided Lipschitz nonlinear systems, Int. J. Control Autom. Syst., 17 (2019), 2203–2210. https://doi.org/10.1007/s12555-018-0891-2 doi: 10.1007/s12555-018-0891-2
![]() |
[37] | I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. |
[38] |
M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 650–659. https://doi.org/10.1016/j.cnsns.2014.10.008 doi: 10.1016/j.cnsns.2014.10.008
![]() |
[39] |
Y. H. Lan, Y. Zhou, Non-fragile observer-based robust control for a class of fractional-order nonlinear systems, Syst. Control Lett., 62 (2013), 1143–1150. https://doi.org/10.1016/j.sysconle.2013.09.007 doi: 10.1016/j.sysconle.2013.09.007
![]() |
1. | Aslı Akyol İnada, Samaneh Arman, Babak Safaei, A novel review on the efficiency of nanomaterials for solar energy storage systems, 2022, 55, 2352152X, 105661, 10.1016/j.est.2022.105661 | |
2. | Muneesh Sethi, R.K. Tripathi, Birajashis Pattnaik, Sushil Kumar, Rohit Khargotra, Satish Chand, Abhishek Thakur, Recent developments in design of evacuated tube solar collectors integrated with thermal energy storage: A review, 2022, 52, 22147853, 1689, 10.1016/j.matpr.2021.11.324 | |
3. | Berrin YILMAZ, Evren OSMA, FARKLI DUYULUR ISI DEPOLAMA MADDELERİNİN DEŞARJ PERİYODUNA ETKİLERİNİN DENEYSEL ARAŞTIRILMASI, 2023, 24, 2147-0308, 39, 10.59314/tujes.1378892 | |
4. | Sally F. Naji, Hosham S. Anead, Khalid F. Sultan, 2024, 3092, 0094-243X, 050008, 10.1063/5.0199869 | |
5. | Lukman Ahmed Omeiza, Muhammad Abid, Yathavan Subramanian, Anitha Dhanasekaran, Saifullah Abu Bakar, Abul Kalam Azad, Challenges, limitations, and applications of nanofluids in solar thermal collectors—a comprehensive review, 2023, 1614-7499, 10.1007/s11356-023-30656-9 |