Symbol | Connotation |
R | the set of real numbers |
Rr | the real r-dimensional space |
diag{⋅} | diagonal matrix |
A ⊗ B | Kronecker product of matrices A and B |
|⋅| | absolute value |
xi,q, gi,q | xi,q(t), gi,q(⋅) |
In real-life experiments, collecting complete data is time-, finance-, and resources-consuming as stated by statisticians and analysts. Their goal was to compromise between the total time of testing, the number of units under scrutiny, and the expenditures paid through a censoring scheme. Comparing failure-censored schemes (Type-Ⅱ and Progressive Type-Ⅱ) to Time-censored schemes (Type-Ⅰ), it's worth noting that the former is time-consuming and is no more suitable to be applied in real-life situations. This is the reason why the Type-Ⅰ adaptive progressive hybrid censoring scheme has exceeded other failure-censored types; Time-censored types enable analysts to accomplish their trials and experiments in a shorter time and with higher efficiency. In this paper, the parameters of the inverse Weibull distribution are estimated under the Type-Ⅰ adaptive progressive hybrid censoring scheme (Type-Ⅰ APHCS) based on competing risks data. The model parameters are estimated using maximum likelihood estimation and Bayesian estimation methods. Further, we examine the asymptotic confidence intervals and bootstrap confidence intervals for the unknown model parameters. Monte Carlo simulations are carried out to compare the performance of the suggested estimation methods under Type-Ⅰ APHCS. Moreover, Markov Chain Monte Carlo by applying Metropolis-Hasting algorithm under the square error of loss function is used to compute Bayes estimates and related to the highest posterior density. Finally, two data sets are studied to illustrate the introduced methods of inference. Based on our results, we can conclude that the Bayesian estimation outperforms the maximum likelihood estimation for estimating the inverse Weibull parameters under Type-Ⅰ APHCS.
Citation: Wael S. Abu El Azm, Ramy Aldallal, Hassan M. Aljohani, Said G. Nassr. Estimations of competing lifetime data from inverse Weibull distribution under adaptive progressively hybrid censored[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 6252-6275. doi: 10.3934/mbe.2022292
[1] | Serge Nicaise, Cristina Pignotti . Asymptotic analysis of a simple model of fluid-structure interaction. Networks and Heterogeneous Media, 2008, 3(4): 787-813. doi: 10.3934/nhm.2008.3.787 |
[2] | Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić . Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks and Heterogeneous Media, 2007, 2(3): 397-423. doi: 10.3934/nhm.2007.2.397 |
[3] | M. Berezhnyi, L. Berlyand, Evgen Khruslov . The homogenized model of small oscillations of complex fluids. Networks and Heterogeneous Media, 2008, 3(4): 831-862. doi: 10.3934/nhm.2008.3.831 |
[4] | Grigory Panasenko, Ruxandra Stavre . Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks and Heterogeneous Media, 2008, 3(3): 651-673. doi: 10.3934/nhm.2008.3.651 |
[5] | Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi . Small solids in an inviscid fluid. Networks and Heterogeneous Media, 2010, 5(3): 385-404. doi: 10.3934/nhm.2010.5.385 |
[6] | Marina Dolfin, Mirosław Lachowicz . Modeling opinion dynamics: How the network enhances consensus. Networks and Heterogeneous Media, 2015, 10(4): 877-896. doi: 10.3934/nhm.2015.10.877 |
[7] | Boris Muha . A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks and Heterogeneous Media, 2014, 9(1): 191-196. doi: 10.3934/nhm.2014.9.191 |
[8] | Alessia Marigo . Equilibria for data networks. Networks and Heterogeneous Media, 2007, 2(3): 497-528. doi: 10.3934/nhm.2007.2.497 |
[9] | Michiel Bertsch, Carlo Nitsch . Groundwater flow in a fissurised porous stratum. Networks and Heterogeneous Media, 2010, 5(4): 765-782. doi: 10.3934/nhm.2010.5.765 |
[10] | Steinar Evje, Kenneth H. Karlsen . Hyperbolic-elliptic models for well-reservoir flow. Networks and Heterogeneous Media, 2006, 1(4): 639-673. doi: 10.3934/nhm.2006.1.639 |
In real-life experiments, collecting complete data is time-, finance-, and resources-consuming as stated by statisticians and analysts. Their goal was to compromise between the total time of testing, the number of units under scrutiny, and the expenditures paid through a censoring scheme. Comparing failure-censored schemes (Type-Ⅱ and Progressive Type-Ⅱ) to Time-censored schemes (Type-Ⅰ), it's worth noting that the former is time-consuming and is no more suitable to be applied in real-life situations. This is the reason why the Type-Ⅰ adaptive progressive hybrid censoring scheme has exceeded other failure-censored types; Time-censored types enable analysts to accomplish their trials and experiments in a shorter time and with higher efficiency. In this paper, the parameters of the inverse Weibull distribution are estimated under the Type-Ⅰ adaptive progressive hybrid censoring scheme (Type-Ⅰ APHCS) based on competing risks data. The model parameters are estimated using maximum likelihood estimation and Bayesian estimation methods. Further, we examine the asymptotic confidence intervals and bootstrap confidence intervals for the unknown model parameters. Monte Carlo simulations are carried out to compare the performance of the suggested estimation methods under Type-Ⅰ APHCS. Moreover, Markov Chain Monte Carlo by applying Metropolis-Hasting algorithm under the square error of loss function is used to compute Bayes estimates and related to the highest posterior density. Finally, two data sets are studied to illustrate the introduced methods of inference. Based on our results, we can conclude that the Bayesian estimation outperforms the maximum likelihood estimation for estimating the inverse Weibull parameters under Type-Ⅰ APHCS.
In recent years, the distributed consensus problem in the MASs has been widely studied by many scholars. Remarkably, leader-following consensus control, which is intend to design a control protocol according to local communication so that all followers can track the trajectory of the leader, has been applied to many engineering fields, such as robotic manipulators [1] and grid-connected microgrids [2]. Considering the existence of uncertainties, a large number of adaptive consensus tracking control strategies based on backstepping method have been proposed in [3,4,5,6]. Note that the results of the above studies have a common problem, that is, the "explosion of complexity" problem caused by repeated differentiations of the virtual control signals. To tackle this problem, command filter-based backstepping method was proposed in [7]. In line with this method, many excellent research results have been presented in [8,9,10]. For example, in [10], a novel adaptive command filter backstepping control scheme was advanced, and it was proved that asymptotic tracking could be achieved by a Lyapunov stability analysis. In addition, with regard to stability issues, various types of Lyapunov functions have been offered in [11,12,13,14] to analyze the stability and synchronization control. However, the above results mainly discuss stability in infinite time. In many industrial engineering fields, to improve convergence rate and convergence time, systems are required to achieve stable performance and tracking performance in finite time. Based on this, numerous finite time consensus tracking schemes were developed in [15,16,17,18,19,20,21,22]. Nevertheless, actuators of all agents mentioned above are required to operate healthily.
As is known to all, actuator failures often occur during the operation of many practical systems, which can lead to system performance degradation, or may bring adverse impact on the industrial production. In response to this, researchers have focused on fault-tolerant control to compensate the influence caused by actuator faults, see [23,24,25,26,27,28,29]. However, the residual fault control rates in the aforementioned literatures were always constants. There is a more practical class of faults composed of time-varying actuation efficiency and time-varying uncontrolled additional faults. In order to deal with such actuator faults, many research results have been presented in [30,31,32,33,34,35,36,37]. For example, in [34], by introducing some integrable auxiliary signals and a new contradiction argument, an adaptive consensus tracking control scheme was given for MASs with time-varying actuator faults. Unfortunately, the control signals of above-mentioned results are needed to have continuous transmission to actuators, which inevitably results in a waste of resources.
Recently, so as to utilize communication resources more effectively and reduce computational expenses, event-triggered control (ETC) has been investigated and has received attention increasingly. Different from the schemes based on event-triggered mechanism (ETM) in [38,39,40,41,42,43,44,45,46,47,49,50] where the threshold parameters were all fixed constants, DETC schemes with the help of adaptive backstepping method were developed in [48,51,52,53,54], and the threshold parameters were required to be adjusted dynamically. Furthermore, in [55], based on DETM, two intermittent control protocols were introduced depending on whether combined measurement or single measurement was used, which could diminish the update frequency of control protocol in the nonlinear MASs. However, to our knowledge, there are few research results on dynamic event-triggered adaptive finite-time consensus tracking for high-order nonlinear MASs with time-varying actuator faults, which motivates our study.
Inspired by the above mentioned results, this paper will study the problem of adaptive finite-time leader-following consensus via DETC for MASs with time-varying actuator faults. The main contributions are as follows:
1) Unlike the previous results in [27,28,29], the time-varying actuator faults are studied in this paper. An adaptive leader-following tracking control strategy is presented, which makes consensus errors converge to a small neighborhood of the origin in finite time.
2) To reduce the computational burden, command filter and compensating mechanism are introduced. Besides, different from the schemes based on ETM proposed in [38,39,40,41,42,43,44,45], DETC is proposed. The dynamic event-triggered controller with larger triggering time interval is designed for each follower, which greatly saves communication resources while successfully compensating for actuator efficiency.
Notation
Symbol | Connotation |
R | the set of real numbers |
Rr | the real r-dimensional space |
diag{⋅} | diagonal matrix |
A ⊗ B | Kronecker product of matrices A and B |
|⋅| | absolute value |
xi,q, gi,q | xi,q(t), gi,q(⋅) |
Consider a class of nonstrict-feedback nonlinear MASs, whose dynamics can be described by
{˙xi,q=xi,q+1+gi,q(xi), q=1,2,…,n−1˙xi,n=ui+gi,n(xi),yi=xi,1, i=1,2,…,N | (2.1) |
where xi=[xi,1,xi,2,…,xi,n]T∈Rn represents the ith follower's state, ui∈R and yi∈R denote the system input and output, respectively. gi,q(xi) (q=1,2,…,n) are uncertain smooth nonlinear functions satisfying gi,q(0)=0. The actuators of N followers may suffer from failures, which can be modeled as
ui(t)=ϱi(t)ˉui(t)+˘ui(t), i=1,2,…,N, | (2.2) |
where ϱi(t)∈R is an unknown time-varying actuation effectiveness, ˉui(t) represents a control signal which needs to be designed later and ˘ui(t) denotes an unknown additive fault.
Remark 1. In lots of literature on the MASs consensus control [27,28,29], there always exists a normal actuator, that is, ϱi(t)=1, ˘ui(t)=0. However, in practical applications, the actuator may be subject to fault. As shown in (2.2), the time-varying actuation effectiveness and additive fault are presented. In other words, the actuator may resume healthy operation or change from one type of fault to another.
The directed graph Y=(A,B) is composed of node set A={1,2,…,N} and edge set B⊆A×A, which is used to denote the interaction between N followers in this study. Define the connectivity matrix as L=[lip], namely, lip=1 in case of having a directed edge from p to i ((p,i)∈B), lip=0 otherwise. Besides, it is required that lii=0. The neighbors set of node i is expressed as Mi={p|(p,i)∈B}. K=diag{k1,k2,…,kN} is an in-degree matrix with ki=∑p∈Milip. Denote the Laplacian matrix as C=K−L. Define the pinning matrix W=diag{w1,w2,…,wN}, and wi=1 indicates there is a directed edge from the leader indexed by 0 to the ith follower, wi=0 otherwise. And the augmented graph ˉY can be obtained, which is composed of Y and edges between some followers and the leader. Additionally, we can get the matrix D=C+W.
Control objective: This study is to design an event-triggered controller for every follower with actuator faults, so that yi can track the output of leader yd where possible in finite time, and can successfully avoid Zeno phenomenon. To achieve the control objective, some preparatory knowledge will be given in the following.
Assumption 1. [15] The fixed directed graph ˉY includes a spanning tree with the leader as a root.
Assumption 2. The output yd and ˙yd are continuous, known and bounded functions.
Assumption 3. [31] ϱi(t) and ˘ui(t) are bounded for i=1,2,…,N, namely, there exist constants ϱimin>0 and ˘uimax>0, such that ϱimin≤ϱi(t)≤1 and |˘ui(t)|≤˘uimax.
Remark 2. Compared with [5], the condition required for the leader's trajectory signal yd is relaxed in Assumption 2. In other words, it is not required that yd and its derivatives up to the n-th order are bounded and continuous.
Remark 3. Assumption 3 indicates that the actuator has a limited actuation effectiveness. Furthermore, the existence of ϱimin implies the exclusion of cases where ϱi(t)=0 and ϱi(t) tends to 0. In other words, the ith follower can be affected by ˉui(t). Besides, it is required that ϱi(t) and ˘ui(t) are bounded, which can contribute to compensating the actuator fault in the following control design.
Lemma 1. [40] Define that ˉg(Z) is any continuous function over the compact set Ξ and ε>0, then there is always a radial basis function neural network (RBFNN) Φ∗TS(Z) such that
supZ∈Ξ|ˉg(Z)−ΦTS(Z)|≤ε, Z∈Ξ⊂Rr, | (2.3) |
where Φ∗=[Φ∗1,Φ∗2,…,Φ∗l]T∈Rl with node number l>1 represents the weight vector, and S(Z)=[s1(Z),s2(Z),…,sl(Z)]T. sj(Z)=exp[−(Z−ϑj)T(Z−ϑj)ℏ2](j=1,2,…,l) are Gaussian functions with the center of receptive field ϑj=[ϑj1,…,ϑjr]T and ℏ being the width of sj(Z). Besides, the ideal constant weight is denoted as Φ=argmin{supZ∈Ξ|ˉg(Z)−Φ∗TS(Z)|}.
Lemma 2. [40] Suppose ˘ϵk=[ϵ1,ϵ2,…,ϵk]T, where k is a positive integer. Let S(˘ϵk)=[s1(˘ϵk),s2(˘ϵk),…,sk(˘ϵk)]T be the RBF vector. Then, for positive integers m≤n, one has
ST(˘ϵn)S(˘ϵn)≤ST(˘ϵm)S(˘ϵm). | (2.4) |
Lemma 3. [44] The inequality 0≤|μ1|−μ1tanh(μ1σ1)≤0.2785σ1 holds for any σ1>0,μ1∈R.
Lemma 4. [31] Let δ1,δ2,…,δn∈R, 0<a<1, then the following inequality holds
(n∑i=1|δi|)a≤n∑i=1|δi|a≤n1−a(n∑i=1|δi|)a. | (2.5) |
Lemma 5. [16,29] For the system ˙x=f(x,u), if there exist constants λ, ξ>0, π∈(0,∞), a∈(0,1), h∈(0,1) and a continuous function V(x), such that
β1(‖x‖)≤V(x)≤β2(‖x‖),˙V(x)≤−λV(x)−ξVa(x)+π, |
where β1(.) and β2(.) are K∞-functions, then it is said that the system is practical finite-time stability. Besides, the settling time T satisfies
T≤1(1−a)λlnλV1−a(x0)+hξλ(π(1−h)ξ)1−aa+hξ. | (2.6) |
In this section, an event-triggered adaptive finite-time control scheme is proposed by backstepping method for system (2.1) subjected to actuator faults. To avoid "explosion of complexity" problem in the process of backstepping design, the first order command filter and compensating mechanism are introduced. The design framework is as follows.
First, define the tracking error and transformation of coordinates as
ςi,1=∑p∈Milip(yi−yp)+ωi(yi−yd), | (3.1) |
ςi,q=xi,q−ˇαi,q, | (3.2) |
where q=2,…,n. ˇαi,q is the output signal of command filter, given by
ψi,q˙ˇαi,q+ˇαi,q=αi,q−1, ˇαi,q(0)=αi,q−1(0), |
where αi,q−1 is as the input signal and ψi,q is a positive parameter. To deal with the impact of the unachieved portion (ˇαi,q−αi,q−1) caused by the command filter, the compensating signals ηi,q (q=1,2,…,n) are introduced as
˙ηi,1=−λi,1ηi,1+(ki+ωi)ηi,2+(ki+ωi)(ˇαi,2−αi,1)−di,1sgn(ηi,1), | (3.3) |
˙ηi,q=−λi,qηi,q+ηi,q+1+(ˇαi,q+1−αi,q)−di,qsgn(ηi,q)−ℓ4ηi,q, q=2,3,…,n−1, | (3.4) |
˙ηi,n=−λi,nηi,n−di,nsgn(ηi,n)−14ηi,n, | (3.5) |
where λi,q, di,q are positive parameters, and ηi,q(0)=0. Then, define the compensated tracking errors as χi,q=ςi,q−ηi,q.
In order to develop the following backstepping design process smoothly, we define the constant θi=max{‖Φi,q‖2,q=1,2,…,n}, i=1,2,…,N. Obviously, θi is an unknown constant because ‖Φi,q‖ are unknown. Let ˆθi be an estimation of θi, and the corresponding estimation error is defined as ˜θi=θi−ˆθi.
Next, the design procedure will be described detailedly based on backstepping method.
Step 1: From (2.1) and (3.1), we can get that the derivative of χi,1 is
˙χi,1=(ki+ωi)(ςi,2+ˇαi,2+gi,1)−∑p∈Milip(xp,2+gp,1)−ωi˙yd−˙ηi,1. | (3.6) |
Construct the candidate Lyapunov function as
Vi,1=χ4i,14+12ρi˜θ2i, |
where ρi is a positive parameter. Then from (3.6), we have
˙Vi,1=χ3i,1((ki+ωi)(ςi,2+ˇαi,2+gi,1)−∑p∈Milip(xp,2+gp,1)−ωi˙yd−˙ηi,1)−1ρi˜θi˙ˆθi. |
According to (3.3), one has
˙Vi,1=χ3i,1((ki+ωi)(χi,2+αi,1)+ˉgi,1(Zi,1)−ωi˙yd+λi,1ηi,1+di,1sgn(ηi,1))−1ρi˜θi˙ˆθi, | (3.7) |
where ˉgi,1(Zi,1)=(ki+ωi)gi,1−∑p∈Milip(xp,2+gp,1) with Zi,1=[xTi,xTp]T. In view of Lemma 1, by using RBFNNs to approximate function ˉgi,1(Zi,1), namely,
ˉgi,1(Zi,1)=ΦTi,1Si,1(Zi,1)+εi,1(Zi,1),|εi,1(Zi,1)|≤ˉεi,1, | (3.8) |
where εi,1(Zi,1) denotes the approximation error and ˉεi,1>0. Through employing Young's inequality and Lemma 2, one can obtain that
χ3i,1ˉgi,1(Zi,1)≤θi2b2i,1χ6i,1STi,1Si,1+b2i,12+34χ4i,1+ˉε4i,14, | (3.9) |
χ3i,1di,1sgn(ηi,1)≤χ6i,12+d2i,12, | (3.10) |
where Si,1=Si,1(ˉZi,1) with ˉZi,1=[xi,1,xp,1]T and bi,1>0 is a parameter to be designed. Then, substituting (3.9) and (3.10) into (3.7), we have
˙Vi,1≤χ3i,1((ki+ωi)(χi,2+αi,1)−ωi˙yd+λi,1ηi,1)−1ρi˜θi˙ˆθi+34χ4i,1+χ6i,12+θi2b2i,1χ6i,1STi,1Si,1+b2i,12+ˉε4i,14+d2i,12. | (3.11) |
The virtual controller is devised as
αi,1=1ki+ωi(−λi,1ςi,1−χ3i,12−χ3i,1ˆθiSTi,1Si,12b2i,1−ξi,1+ωi˙yd−34(1+ki+ωi)χi,1), | (3.12) |
where ξi,1>0 is a design parameter. It follows from (3.12) and Young's inequality that
˙Vi,1≤−λi,1χ4i,1−ξi,1χ3i,1+(ki+ωi)χ3i,1χi,2+˜θiρi(ρiχ6i,1STi,1Si,12b2i,1−˙ˆθi)−34(ki+ωi)χ4i,1+b2i,12+ˉε4i,14+d2i,12≤−λi,1χ4i,1−ξi,1χ3i,1+14(ki+ωi)χ4i,2+˜θiρi(ρiχ6i,1STi,1Si,12b2i,1−˙ˆθi)+b2i,12+ˉε4i,14+d2i,12. |
Step q (q=2,3,…,n−1): According to (3.2), one has
˙χi,q=ςi,q+1+ˇαi,q+1+gi,q−˙ˇαi,q−˙ηi,q. | (3.13) |
Choose the following candidate Lyapunov function as
Vi,q=Vi,q−1+14χ4i,q, |
then we can derive that
˙Vi,q=˙Vi,q−1+χ3i,q(χi,q+1+αi,q+ˉgi,q(Zi,q)−˙ˇαi,q+λi,qηi,q+di,qsgn(ηi,q)+ℓ4ηi,q), | (3.14) |
where ˉgi,q(Zi,q)=gi,q. Similarly, it is obtained from Lemma 1 that
ˉgi,q(Zi,q)=ΦTi,qSi,q(Zi,q)+εi,q(Zi,q),|εi,q(Zi,q)|≤ˉεi,q, | (3.15) |
where εi,q(Zi,q) is the approximation error and ˉεi,q>0. It follows from Young's inequality and Lemma 2 that
χ3i,qˉgi,q(Zi,q)≤θi2b2i,qχ6i,qSTi,qSi,q+b2i,q2+34χ4i,q+ˉε4i,q4, | (3.16) |
χ3i,qdi,qsgn(ηi,q)≤χ6i,q2+d2i,q2, | (3.17) |
where Si,p=Si,p(ˉZi,p) with ˉZi,p=[xi,1,xi,2,…,xi,p]T and bi,p>0 is a parameter. By plugging (3.16) and (3.17) into (3.14), one can obtain that
˙Vi,q≤˙Vi,q−1+χ3i,q(χi,q+1+αi,q−˙ˇαi,q+λi,qηi,q+ℓ4ηi,q)+θi2b2i,qχ6i,qSTi,qSi,q+34χ4i,q+χ6i,q2+b2i,q2+ˉε4i,q4+d2i,q2. |
Then the virtual controller αi,q can be constructed as
αi,q=−λi,qςi,q−χ3i,q2−χ3i,qˆθiSTi,qSi,q2b2i,q−ξi,q−32χi,q−ℓςi,q4+˙ˇαi,q, | (3.18) |
where ξi,q is a positive parameter to be designed. Using Young's inequality, we have
˙Vi,q≤−q∑v=1λi,vχ4i,v−q∑v=1ξi,vχ3i,v+14χ4i,q+1+˜θiρi(q∑v=1ρiχ6i,vSTi,vSi,v2b2i,v−˙ˆθi)+q∑v=1(b2i,v2+ˉε4i,v4+d2i,v2). |
Step n: In the last step, we will design the event-triggered controller based on actuator failures. The DETM is considered as follows:
ˉui(t)=ϖi(tiι),∀t∈[tiι,tiι+1), | (3.19) |
tiι+1=inf{t∈R| |zi(t)|≥γi(t)|ˉui(t)|+Δi}, | (3.20) |
˙γi(t)=−βiγ2i(t), | (3.21) |
where ϖi(t) is the control signal to be designed next, tiι denotes the update time, zi(t)=ϖi(t)−ˉui(t) is the measurement error, Δi>0 and βi>0 are parameters. Moreover, it can be seen that ∀γi(0)∈(0,1), we have γi(t)∈(0,1). According to (3.20), we can obtain that ∀t∈[tiι,tiι+1), |zi(t)|≤γi(t)|ˉui(t)|+Δi, then it is concluded that ϖi(t)=(1+ζi,1(t)γi(t))ˉui(t)+ζi,2(t)Δi with |ζi,1(t)|≤1 and |ζi,2(t)|≤1. Hence, ˉui(t) can be expressed as
ˉui(t)=ϖi(t)1+ζi,1(t)γi(t)−ζi,2(t)Δi1+ζi,1(t)γi(t). | (3.22) |
According to (3.2), one has
˙χi,n=ϱi(t)ˉui+˘ui+gi,n−˙ˇαi,n−˙ηi,n. | (3.23) |
Define the Lyapunov function candidate as
Vi,n=Vi,n−1+14χ4i,n. |
From (3.5), one has
˙Vi,n=˙Vi,n−1+χ3i,n(ϱi(t)ˉui+˘ui+ˉgi,n(Zi,n)−˙ˇαi,n+λi,nηi,n+di,nsgn(ηi,n)+14ηi,n), | (3.24) |
where ˉgi,n(Zi,n)=gi,n. And we can get from the similar process (3.15) that
ˉgi,n(Zi,n)=ΦTi,nSi,n(Zi,n)+εi,n(Zi,n),|εi,n(Zi,n)|≤ˉεi,n, | (3.25) |
where εi,n(Zi,n) represents the approximation error and ˉεi,n>0. From Young's inequality and Assumption 3, it can be derived that
χ3i,n˘ui≤34χ4i,n+14˘u4imax, | (3.26) |
χ3i,nˉgi,n(Zi,n)≤θi2b2i,nχ6i,nSTi,nSi,n+b2i,n2+34χ4i,n+ˉε4i,n4, | (3.27) |
χ3i,ndi,nsgn(ηi,n)≤χ6i,n2+d2i,n2, | (3.28) |
where Si,n=Si,n(ˉZi,n) with ˉZi,n=Zi,n and the design parameter bi,n>0. By substituting (3.22) and (3.26)–(3.28) into (3.24) yields
˙Vi,n≤˙Vi,n−1+χ3i,n(ϱi(t)(ϖi(t)1+ζi,1(t)γi(t)−ζi,2(t)Δi1+ζi,1(t)γi(t))−˙ˇαi,n+λi,nηi,n+14ηi,n)+32χ4i,n+θi2b2i,nχ6i,nSTi,nSi,n+χ6i,n2+14˘u4imax+b2i,n2+ˉε4i,n4+d2i,n2≤−n∑v=1λi,vχ4i,v−n∑v=1ξi,vχ3i,v+˜θiρi(n∑v=1ρiχ6i,vSTi,vSi,v2b2i,v−˙ˆθi)+n∑v=1(b2i,v2+ˉε4i,v4+d2i,v2+14˘u4imax)+χ3i,nϱi(t)(ϖi(t)1+ζi,1(t)γi(t)−ζi,2(t)Δi1+ζi,1(t)γi(t))+ϱimin|χ3i,nαi,n|, | (3.29) |
where αi,n=1ϱimin(−λi,nςi,n−χ3i,n2−χ3i,nˆθiSTi,nSi,n2b2i,n−ξi,n−32χi,n−ςi,n4+˙ˇαi,n) and ξi,n>0 is a parameter. The event-triggered controller ϖi(t) and the adaptive law ˆθi are devised in the following:
ϖi(t)=−(1+γi(t))(αi,ntanhχ3i,nαi,nσi+ˉoitanhχ3i,nˉoiσi), | (3.30) |
˙ˆθi=n∑v=1ρiχ6i,vSTi,vSi,v2b2i,v−Λiˆθi, | (3.31) |
where ˉoi>0 and ϱiminˉoi>Δi1−γi. Owing to 0<ϱimin≤ϱi(t)≤1, |ζi,1(t)|≤1, |ζi,2(t)|≤1 and xtanhx≥0, we can get that
ϱi(t)χ3i,nϖi(t)1+ζi,1(t)γi(t)≤−ϱiminχ3i,nαi,ntanhχ3i,nαi,nσi−ϱiminχ3i,nˉoitanhχ3i,nˉoiσi, | (3.32) |
ϱi(t)χ3i,nζi,2(t)Δi1+ζi,1(t)γi(t)≤|χ3i,nΔi1−γi|. | (3.33) |
According to (3.29)–(3.33) and Lemma 3, one has
˙Vi,n≤−n∑v=1λi,vχ4i,v−n∑v=1ξi,vχ3i,v+Λiρi˜θiˆθi+n∑v=1(b2i,v2+ˉε4i,v4+d2i,v2+14˘u4imax)+0.557ϱiminσi. | (3.34) |
Remark 4. As shown in (3.19)–(3.21), one of the characteristics of DETC is that the threshold parameter γi(t) can be dynamically adjusted. If supposing γi(0)=0, Δi≠0 and βi=0, (3.21) changes into tiι+1=inf{t∈R| |zi(t)|≥Δi}, which is the classical sample-data control. Moreover, if setting γi(0)≠0, Δi≠0 and βi=0, the proposed DETC becomes the static event-triggered control. Therefore, by comparison, DETC is more flexible. Besides, DETC has a larger average triggering time interval and fewer communication times, which can contribute to reducing the update frequency of the controller, saving communication resources and improving the utilization rate of resources.
Step n+1: For the compensating system, the following Lyapunov function is constructed as
Vi,n+1=14n∑q=1η4i,q. |
From (3.3)–(3.5), we have
˙Vi,n+1=−n∑q=1λi,qη4i,q−n∑q=1di,qsgn(ηi,q)η3i,q+(ki+ωi)η3i,1ηi,2+n−1∑q=2η3i,qηi,q+1−n−1∑q=2ℓ4η4i,q−14η4i,n+(ki+ωi)(ˇαi,2−αi,1)η3i,1+n−1∑q=2(ˇαi,q+1−αi,q)η3i,q. | (3.35) |
It follows from Young's inequality that
(ki+ωi)η3i,1ηi,2≤34(ki+ωi)η4i,1+14(ki+ωi)η4i,2,η3i,qηi,q+1≤34η4i,q+14η4i,q+1, |
and from [26], for q=1,2,…,n−1, it is gained that ‖ˇαi,q+1−αi,q‖≤τi,q in the time Ti1 where τi,q are positive constants. Hence, (3.35) can be written as
˙Vi,n+1≤−(λi,1−34(ki+ωi))η4i,1−n∑q=2(λi,q−34)η4i,q−n∑q=1di,q|ηi,q|3+(ki+ωi)τi,1|ηi,1|3+n−1∑q=2τi,q|ηi,q|3≤−λiVi,n+1−diV34i,n+1, | (3.36) |
where λi=4min{λi,1−34(ki+ωi),λi,q−34}, di=2√2(min{di,q}−max{(ki+ωi)τi,1,τi,q}) and di>0 can be satisfied by choosing suitable parameters. A candidate Lyapunov function is selected as Vi=Vi,n+Vi,n+1, then from (3.34), we have
˙Vi≤−n∑v=1ˉλi(14χ4i,v)−n∑v=1ˉξi(14χ4i,q)34−λiVi,n+1−diV34i,n+1+Λiρi˜θiˆθi+Πi, | (3.37) |
where ˉλi=4min{λi,v}, ˉξi=2√2min{ξi,v} and Πi=n∑v=1(b2i,v2+ˉε4i,v4+d2i,v2+14˘u4imax)+0.557ϱiminσi. Choose the whole Lyapunov function candidate as V=N∑i=1Vi, then the derivative of V can be gained that
˙V≤−N∑i=1n∑v=1ˉλi(14χ4i,v)−N∑i=1n∑v=1ˉξi(14χ4i,q)34−N∑i=1λiVi,n+1−N∑i=1diV34i,n+1+N∑i=1Λiρi˜θiˆθi+N∑i=1Πi. | (3.38) |
Theorem 1. Consider the MAS (2.1) with actuator faults (2.2) satisfying Assumption 1–3, if virtual controllers (3.12) and (3.18), actual controller (3.19), adaptive laws (3.31) and the compensating signals (3.3)–(3.5) are designed under event-triggered mechanism (3.20) and (3.21), then it can be guaranteed that 1) all signals in the closed-loop system are bounded; 2) the tracking errors converge into a small neighborhood of the origin in finite time; 3) Zeno behavior is effectively elimitated.
Proof: We can know from Young's inequality that
Λiρi˜θiˆθi≤−Λi2ρi˜θ2i+Λi2ρiθ2i≤−Λi8ρi˜θ2i−Λi(˜θ2i2ρi)34+πi, | (3.39) |
where πi=Λi4+Λi2ρiθ2i. With the help of Lemma 4 and by substituting (3.39) into (3.38), one has
˙V≤−N∑i=1n∑v=1ˉλi(14χ4i,v)−N∑i=1Λi8ρi˜θ2i−N∑i=1n∑v=1ˉξi(14χ4i,q)34−N∑i=1Λi(˜θ2i2ρi)34−λiVi,n+1−N∑i=1diV34i,n+1+Π≤−λV−ξV34+Π, | (3.40) |
where \Pi = \sum\limits^{N}_{i = 1}(\Pi_i+\pi_i) , \lambda = \min\{\bar{\lambda}_{i}, \frac{1}{4}\Lambda_i, \lambda_i\} and \xi = \min\{\bar{\xi}_{i}, \Lambda_i, d_i\} , i = 1, 2, \dots, n . It follows from (3.40) that \dot{V}\leq-\lambda V+\Pi , namely, V\leq (V(t_0)-\frac{\Pi}{\lambda})e^{-\lambda(t-t_0)}+\frac{\Pi}{\lambda}\leq V(t_0)+\frac{\Pi}{\lambda} . Therefore, all signals in the closed-loop system remain bounded. According to Lemma 5, we can conclude that V\leq (\frac{\Pi}{(1-h)\xi})^\frac{4}{3} in a finite time T for h\in (0, 1) . Thus, it is derived that |\varsigma_{i, q}|\leq 2\sqrt{2}\left(\frac{\Pi}{(1-h)\xi}\right)^\frac{1}{3} . Due to \varsigma_1 = \mathcal{D}\big(y-(1_N\otimes y_d)\big) with \varsigma_1 = [\varsigma_{1, 1}, \varsigma_{2, 1}, \dots, \varsigma_{N, 1}]^T and y = [y_{1, 1}, y_{2, 1}, \dots, y_{N, 1}]^T , thus |y_i-y_d|\leq \frac{2\sqrt{2}\left(\frac{\Pi}{(1-h)\xi}\right)^\frac{1}{3}}{\mu} , where \mu denotes the least singular value of \mathcal{D} . Moreover, T satisfies
\begin{align} T\leq\max\{T_{i, 1}\}+\frac{4}{\lambda}\ln\frac{\lambda V^{\frac{1}{4}}(t_0)+h\xi}{\lambda(\frac{\Pi}{(1-h)\xi})^\frac{1}{3}+h\xi}. \end{align} | (3.41) |
Next, we will demonstrate that Zeno phenomenon can be excluded under the proposed scheme, namely, there exists t^i_\star > 0 , such that t^i_{l+1}-t^i_l\geq t^i_\star with l\in Z^+ . Owing to z_i(t) = \varpi_i(t)-\bar{u}_i(t) , we have \forall t\in[t^i_l, t^i_{l+1}) ,
\begin{align} \frac{d}{dt}|z_i| = \frac{d}{dt}\sqrt{z_i \times z_i} = sgn(z_i)\dot{z}_i\leq|\dot{\varpi}_i|. \end{align} | (3.42) |
Furthermore, it follows from (3.30) that \varpi_i is differentiable and \dot{\varpi}_i is a continuous function of bounded signals. Hence, it can be obtained that |\dot{\varpi}_i|\leq\varpi^*_i , where \varpi^*_i > 0 is a constant. Because of z_i(t^i_l) = 0 and \lim_{t\rightarrow t^i_{l+1}}z_i(t) = \gamma_i|\bar{u}_i(t^i_l)|+\Delta_i , therefore t^i_\star\geq \frac{\gamma_i|\bar{u}_i(t^i_l)|+\Delta_i}{\varpi^*_i} . In conclusion, the Zeno behavior is prevented.
In this section, two simulation examples are given to verify the effectiveness of the proposed control scheme.
Example 1: Consider the following multi-agent system
\begin{eqnarray} \left\{ \begin{aligned} \dot{x}_{i, 1}& = x_{i, 2}+g_{i, 1}(x_{i, 1}, x_{i, 2}), \\ \dot{x}_{i, 2}& = u_{i}+g_{i, 2}(x_{i, 1}, x_{i, 2}), \ \ i = 1, 2, 3, 4, \end{aligned} \right. \end{eqnarray} | (4.1) |
where g_{1, 1} = 0.05\sin(x_{1, 1}-x_{1, 2}) , g_{1, 2} = 0.01\sin(x_{1, 1})\cos(x_{1, 2}) , g_{2, 1} = 0.01x_{2, 1}\cos(x_{2, 2}) , g_{2, 2} = 0.03\sin\big(0.5(x_{2, 1}-x_{2, 2})\big) , g_{3, 1} = 0.02\exp(-x_{3, 1})\cos(x_{3, 2}) , g_{3, 2} = 0.02\sin\big(0.3(x_{3, 1}-x_{3, 2})\big) , g_{4, 1} = 0.05\exp(-x_{4, 1})\cos(x_{4, 2}) , g_{4, 2} = 0.01\sin(x_{4, 1}-x_{4, 2}) . The actuator fault models for four followers are defined as
\begin{align} u_1 = \big(0.8+0.1\cos(-0.5t)\big)\bar{u}_1+0.3\sin t, \ \ u_2 = \big(0.7+0.1\sin(-0.6t)\big)\bar{u}_2+0.3\sin t, \\ u_3 = \big(0.6+0.2\sin(-0.5t)\big)\bar{u}_3+0.3\sin t, \ \ u_4 = \big(0.5+0.1\cos(-0.7t)\big)\bar{u}_4+0.3\sin t. \end{align} |
The communication topology of four followers and one leader is shown in Figure 1. And the Laplacian matrix and the pinning matrix can be described as
\begin{equation} \nonumber \mathcal{C} = \left(\begin{array}{cccc} 1 \quad & 0\quad & -1\quad & 0 \\ 0 \quad & 1\quad & -1\quad & 0 \\ 0 \quad & 0\quad & 0\quad & 0 \\ 0 \quad & -1\quad & 0\quad & 1\end{array} \right), \qquad \mathcal{W} = \left( \begin{array}{cccc} 0 \quad & 0\quad & 0\quad & 0 \\ 0 \quad & 0\quad & 0\quad & 0 \\ 0 \quad & 0\quad & 1\quad & 0 \\ 0 \quad & 0\quad & 0\quad & 0\end{array} \right). \end{equation} |
Moreover, the output of leader is y_d = \sin (0.5t) . Our control objective is to make the output of each follower y_i track y_d in finite time. NNs are used to approximate the unknown nonlinear functions \bar{g}_{i, q}(Z_{i, q}) (i = 1, 2, 3, 4, q = 1, 2) . \Phi^{\ast^T}_{i, q} S_{i, q}(Z_{i, q}) contains 11 nodes, and the centers \vartheta_j are evenly distributed in [-2.5, 2.5] with width of 2 . Besides, the initial conditions are selected as x(0) = [0.1, 0.1, 0.1, 0.2, 0.2, 0.1, 0.4, 0.3]^T , \gamma(0) = [0.5, 0.5, 0.3, 0.5]^T , \theta(0) = [0.1, 0.2, 0.3, 0.4]^T , \eta_{i, q}(0) = 0 and \check{\alpha}_{i, 2}(0) = 0 , i = 1, 2, 3, 4 , q = 1, 2 . To achieve the control objective, the design parameters are chosen as \lambda_{i, q} = 15 , b_{i, q} = 10 , \xi_{i, q} = 0.1 , d_{1, q} = d_{2, 1} = 0.25 , d_{2, 2} = d_{3, q} = d_{4, q} = 0.5 , \Lambda_i = 0.6 , \beta_i = 0.5 , \rho_1 = \rho_2 = \rho_4 = 15 , \rho_3 = 10 , \psi_i = 0.05 , \bar{o}_1 = \bar{o}_2 = \bar{o}_3 = 50 , \bar{o}_4 = 60 , \Delta_i = 3 and \sigma_i = 0.2 , i = 1, 2, 3, 4 , q = 1, 2 .
The simulation results under the proposed control strategy are depicted in Figures 2–7. As shown in Figure 2, each follower's output y_i can well track the leader's output y_d .
Figures 3 and 4 show the input signals \bar{u}_i of four followers. The trajectories of dynamic trigger time intervals t^i_{\iota+1}-t^i_\iota are exhibited in Figures 5 and 6, and the event-triggered numbers of four followers are shown in Table 2. It is clearly seen that the amount of computation and communication resources are considerably reduced. At last, Figure 7 displays the boundedness of the adaptive laws \hat{\theta}_i (i = 1, 2, 3, 4) .
Agent(node) | 1 | 2 | 3 | 4 |
The number of triggers | 81 | 100 | 71 | 229 |
Example 2: To prove that the proposed scheme is applicable in practice, the multiple single-link robot manipulator systems (SRMSs) are considered. According to [29], suppose that there are four followers and the SRMS is described as
\begin{align} M\ddot{q}_i+B\dot{q}_i+N\sin(\dot{q}_i) = I_i, \ \ i = 1, 2, 3, 4, \end{align} | (4.2) |
where M = \frac{J}{K_\tau}+\frac{mL^2_0}{3K_\tau}+\frac{M_0L^2_0}{K_\tau}+\frac{2M_0R^2_0}{5K_\tau} , N = \frac{mL_0G}{2K_\tau}+\frac{M_0L_0G}{K_\tau} and B = \frac{B_0}{K_\tau} , q_i and I_i are the angular position and motor armature current, respectively. In [29], the designed parameters M = 1, B = 1, N = 1 have been given. Define x_{i, 1} = q_i, x_{i, 2} = \dot{q}_i, u_i = I_i , (4.2) can be rewritten as
\begin{align} \dot{x}_{i, 1}& = x_{i, 2}+g_{i, 1}, \\ \dot{x}_{i, 2}& = u_i+g_{i, 2}, \end{align} | (4.3) |
where g_{i, 1} = 0, \ g_{i, 2} = 10\sin(x_{i, 1})-x_{i, 2} . The actuator fault models for four followers are defined as
\begin{align} u_1 = \big(0.6+0.1\cos(-0.5t)\big)\bar{u}_1+0.3\sin t, \ \ u_2 = \big(0.5+0.25\sin(-0.6t)\big)\bar{u}_2+0.3\sin t, \\ u_3 = \big(0.7+0.2\sin(-0.5t)\big)\bar{u}_3+0.3\sin t, \ \ u_4 = \big(0.3+0.25\cos(-0.7t)\big)\bar{u}_4+0.3\sin t. \end{align} |
The communication relationship between the four followers and the leader is depicted in Figure 8. Therefore, based on Figure 8, the Laplacian matrix \mathcal{C} and the pinning matrix \mathcal{W} are expressed as
\begin{equation} \nonumber \mathcal{C} = \left(\begin{array}{cccc} 1 \quad & -1\quad & 0\quad & 0 \\ 0 \quad & 1\quad & -1\quad & 0 \\ 0 \quad & -1\quad &1\quad & 0 \\ 0 \quad & 0\quad & -1\quad & 1\end{array} \right), \qquad \mathcal{W} = \left( \begin{array}{cccc} 0 \quad & 0\quad & 0\quad & 0 \\ 0 \quad & 0\quad & 0\quad & 0 \\ 0 \quad & 0\quad & 1\quad & 0 \\ 0 \quad & 0\quad & 0\quad & 0\end{array} \right). \end{equation} |
And the leader's output is y_d = 0.5\sin t . The initial conditions are the same as in Example 1, and the parameters are designed as \lambda_{1, q} = \lambda_{2, q} = \lambda_{4, 1} = 65 , \lambda_{3, q} = \lambda_{4, 2} = 60 , b_{i, q} = 10 , \xi_{i, q} = 0.01 , d_{i, q} = 0.01 , \Lambda_i = 2 , \beta_i = 2 , \rho_i = 20 , \psi_1 = \psi_2 = \psi_3 = 0.05 , \psi_4 = 0.12 , \bar{o}_1 = \bar{o}_2 = \bar{o}_4 = 50 , \bar{o}_3 = 45 , \Delta_1 = 4 , \Delta_2 = \Delta_3 = 1 , \Delta_4 = 5 and \sigma_i = 0.1 , i = 1, 2, 3, 4 , q = 1, 2 .
Figures 9–14 display the simulation results of our devised scheme. The output curves are shown in Figure 9, which indicates that the desired consensus can be achieved in finite time. Figure 10 illustrates the trajectories of the adaptive laws \hat{\theta}_i (i = 1, 2, 3, 4) . The trajectories of \bar{u}_i(i = 1, 2, 3, 4) are drawn in Figures 11 and 12, and the trigger time intervals are reflected in Figures 13 and 14. Furthermore, the event-triggered numbers of four followers are presented in Table 3. In conformity with the simulation results, the control scheme can be well applied in the SRMSs.
Agent(node) | 1 | 2 | 3 | 4 |
The number of triggers | 362 | 685 | 390 | 289 |
In this paper, with a view towards leader-following consensus tracking, an adaptive finite-time DETC scheme has been proposed for MASs with unknown time-varying actuator faults. Based on adaptive backstepping method, neural network approximation technique and command filter technique, the actuator efficiency has been compensated successfully. Moreover, the DETM has been given for each follower to reduce the update frequency of controller and mitigate the communication burden. In the light of the presented control scheme, leader-following consensus has been achieved in finite time and all signals of the closed-loop system are bounded. In the future, we will tend to study the consensus tracking problem of multiple leaders with time-varying actuator failures in switching topology.
We sincerely thank the editors and reviewers for their careful reading and valuable suggestions, which makes this paper a great improvement. This work was supported by the National Natural Science Foundation of China under Grants 61973148, and supported by Discipline with Strong Characteristics of Liaocheng University: Intelligent Science and Technology under Grant 319462208.
The authors declare there is no conflict of interest.
[1] |
B. Epstein, Truncated life tests in the exponential case, Ann. Math. Stat., 25 (1954), 555-564. https://doi.org/10.1214/aoms/1177728723 doi: 10.1214/aoms/1177728723
![]() |
[2] |
A. Childs, B. Chandrasekar, N. Balakrishnan, D. Kundu, Exact likelihood inference based on type Ⅰ and type Ⅱ hybrid censored samples from the exponential distribution, Ann. Inst. Stat. Math., 55 (2003), 319-330. https://doi.org/10.1007/BF02530502 doi: 10.1007/BF02530502
![]() |
[3] |
D. Kundu, A. Joarder, Analysis of type Ⅱ progressively hybrid censored data, Comput. Stat. Data Anal., 50 (2006a), 2509-2528. https://doi.org/10.1016/j.csda.2005.05.002 doi: 10.1016/j.csda.2005.05.002
![]() |
[4] |
D. Kundu, A. Joarder, Analysis of type Ⅱ progressively hybrid censored competing risks data, J. Mod. Appl. Stat. Methods, 5 (2006b), 186-204. https://doi.org/10.22237/jmasm/1146456780 doi: 10.22237/jmasm/1146456780
![]() |
[5] | A. Childs, B. Chandrasekar, N. Balakrishnan, Exact likelihood inference for exponential parameter under progressive hybrid censoring schemes, in Statistical Models and Methods for Biomedical and Technical Systems (eds. Vonta, F., Nikulin, M., Limnios, N., Huber-Carol), Boston Birkhauser, (2008), 323-334. |
[6] |
H. K. T. Ng, D. Kundu, P. S. Chan, Statistical analysis of exponential lifetimes under an adaptive Type-Ⅱ progressively censoring scheme, Nav. Res. Logist., 56 (2009), 687-698. https://doi.org/10.1002/nav.20371 doi: 10.1002/nav.20371
![]() |
[7] |
N. Balakrishnan, D. Kundu, Hybrid censoring: inference results and applications, Comput. Stat. Data Anal., 57 (2013), 166-209. https://doi.org/10.1016/j.csda.2012.03.025 doi: 10.1016/j.csda.2012.03.025
![]() |
[8] |
C. T. Lin, Y. L. Huang, On progressive hybrid censored exponential distribution, J. Stat. Comput. Simul., 82 (2012), 689-709. https://doi.org/10.1080/00949655.2010.550581 doi: 10.1080/00949655.2010.550581
![]() |
[9] |
C. T. Lin, C. C. Chou, Y. L. Huang, Inference for the Weibull distribution with progressive hybrid censoring, Comput. Stat. Data Anal., 56 (2012), 451-467. https://doi.org/10.1016/j.csda.2011.09.002 doi: 10.1016/j.csda.2011.09.002
![]() |
[10] |
D. V. Lindley, Approximate Bayesian method. Trab. Estandistica, 31 (1980), 223-237. https://doi.org/10.1007/BF02888353 doi: 10.1007/BF02888353
![]() |
[11] |
L. Tierney, J. B. Kadane, Accurate approximations for posterior moments and marginal densities, J. Am. Stat. Assoc., 81 (1986), 82-86. https://doi.org/10.1080/01621459.1986.10478240 doi: 10.1080/01621459.1986.10478240
![]() |
[12] |
M. Nassar, S. G. Nassr, S. Dey, Analysis of Burr type XⅡ distribution under step stress partially accelerated life tests with type Ⅰ and adaptive type Ⅱ progressively hybrid censoring schemes, Ann. Data Sci., 4 (2017), 227-248. https://doi.org/10.1007/s40745-017-0101-8 doi: 10.1007/s40745-017-0101-8
![]() |
[13] |
H. Okasha, A. Mustafa, E-Bayesian estimation for the Weibull distribution under adaptive type-Ⅰ progressive hybrid censored competing risks data, Entropy, 22 (2020), 1-20. https://doi.org/10.3390/e22080903 doi: 10.3390/e22080903
![]() |
[14] |
A. Helu, H. Samawi, Statistical analysis based on adaptive progressive hybrid censored data from Lomax distribution, Stat. Optim. Inf. Comput., 9 (2021), 789-808. https://doi.org/10.19139/soic-2310-5070-1330 doi: 10.19139/soic-2310-5070-1330
![]() |
[15] |
H. Okasha, Y. Lio, M. Albassam, On reliability estimation of Lomax distribution under adaptive type-Ⅰ progressive hybrid censoring scheme, Mathematics, 9 (2021), 1-40. https://doi.org/10.3390/math9222903 doi: 10.3390/math9222903
![]() |
[16] |
S. K. Ashour, M. M. A. Nassar, Inference for Weibull distribution under adaptive type-Ⅰ progressive hybrid censored competing risks data, Commun. Stat. Theory Methods, 46 (2016), 4756-4773. https://doi.org/10.1080/03610926.2015.1083111 doi: 10.1080/03610926.2015.1083111
![]() |
[17] |
M. Nassar, S. A. Dobbah, Analysis of reliability characteristics of bathtub-shaped distribution under adaptive type-Ⅰ progressive hybrid censoring, IEEE Access, 8 (2020), 181796-181806. https://doi.org/10.1109/ACCESS.2020.3029023 doi: 10.1109/ACCESS.2020.3029023
![]() |
[18] |
D. R. Cox, The analysis of exponentially distributed lifetimes with two types of failure, J. R. Stat. Soc. Ser. B, 21 (1959), 411-421. https://doi.org/10.1111/j.2517-6161.1959.tb00349.x doi: 10.1111/j.2517-6161.1959.tb00349.x
![]() |
[19] | M. J. Crowder, Classical Competing Risks, Chapman & Hall, 2001. |
[20] |
S. K. Ashour, M. M. A. Nassar, Analysis of exponential distribution under adaptive type-Ⅰ progressive hybrid censored competing risks data, Pak. J. Stat. Oper. Res., 10 (2014), 229- 245. https://doi.org/10.1234/pjsor.v10i2.705 doi: 10.1234/pjsor.v10i2.705
![]() |
[21] |
A. S. Hassan, S. G. Nassr, S. Pramanik, S. S. Maiti, Estimation in constant stress partially accelerated life tests for Weibull distribution based on censored competing risks data, Ann. Data Sci., 7 (2020), 45-62. https://doi.org/10.1007/s40745-019-00226-3 doi: 10.1007/s40745-019-00226-3
![]() |
[22] | S. G. Nassr, E. M. Almetwally, W. S. Abu El Azm, Statistical inference for the extended Weibull distribution based on adaptive type Ⅱ progressive hybrid censored competing risks data, Thail. Stat., 19 (2021), 547-564. |
[23] | A. Z, Keller, A. R. R. Kamath, Alternative reliability models for mechanical systems, in Proceeding of the third international conference on reliability and maintainability, (1982), 411-415. |
[24] |
P. Erto, M. Rapone, Non-informative and practical Bayesian confidence bounds for reliable life in the Weibull model, Reliab. Eng., 7 (1984), 181-191. https://doi.org/10.1016/0143-8174(84)90016-7 doi: 10.1016/0143-8174(84)90016-7
![]() |
[25] |
R. Calabria, G. Pulcini, Bayesian 2-sample prediction for the inverse Weibull distribution, Commun. Stat. Theory Methods, 23 (1994), 1811-1824. https://doi.org/10.1080/03610929408831356 doi: 10.1080/03610929408831356
![]() |
[26] |
M. Maswadah, Conditional confidence interval estimation for the inverse Weibull distribution based on censored generalized order statistics, J. Stat. Comput. Simul., 73 (2003), 887-898. https://doi.org/10.1080/0094965031000099140 doi: 10.1080/0094965031000099140
![]() |
[27] |
B. O. Oluyede, T. Yang, Generalizations of the inverse Weibull and related distributions with applications, Electron. J. Appl. Stat. Anal., 7 (2014), 94-116. https://doi.org/10.1285/i20705948v7n1p94 doi: 10.1285/i20705948v7n1p94
![]() |
[28] |
D. Kundu, H. Howlader, Bayesian inference and prediction of the inverse Weibull distribution for type-Ⅱ censored data, Comput. Stat. Data Anal., 54 (2010), 1547-1558. https://doi.org/10.1016/j.csda.2010.01.003 doi: 10.1016/j.csda.2010.01.003
![]() |
[29] |
R. Musleh, A. Helu, Estimation of the inverse Weibull distribution based on progressively censored data: Comparative study, Reliab. Eng. Syst. Saf., 131 (2014), 216-227. https://doi.org/10.1016/j.ress.2014.07.006 doi: 10.1016/j.ress.2014.07.006
![]() |
[30] |
K. S. Sultan, N. H. Alsadat, D. Kundu, Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-Ⅱ censoring, J. Stat. Comput. Simul., 84 (10) (2014), 2248-2265. https://doi.org/10.1080/00949655.2013.788652 doi: 10.1080/00949655.2013.788652
![]() |
[31] |
X. Peng, Z. Yan, Bayesian estimation and prediction for the inverse Weibull distribution under general progressive censoring, Commun. Stat. Theory Methods, 45 (2016), 624-635. https://doi.org/10.1080/03610926.2013.834452 doi: 10.1080/03610926.2013.834452
![]() |
[32] |
A. S. Hassan, S. G. Nassr, The inverse Weibull generator of distribution: properties and applications, J. Data Sci., 16 (2018), 723-742. https://doi.org/10.6339/JDS.201810_16(4).00004 doi: 10.6339/JDS.201810_16(4).00004
![]() |
[33] |
A. C. Cohen, Maximum likelihood estimation in the Weibull distribution based on complete and censored samples, Technometrics, 5 (1965), 327-329. https://doi.org/10.1080/00401706.1965.10490300 doi: 10.1080/00401706.1965.10490300
![]() |
[34] |
S. Dey, S. Singh, Y. M. Tripathi, A. Asgharzadeh, Estimation and prediction for a progressively censored generalized inverted exponential distribution, Stat. Methodol., 132 (2016), 185-202. https://doi.org/10.1016/j.stamet.2016.05.007 doi: 10.1016/j.stamet.2016.05.007
![]() |
[35] |
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, Equation of state calculations by fast computing machines, J. Chem Phys., 21 (1953), 1087-1091. https://doi.org/10.1063/1.1699114 doi: 10.1063/1.1699114
![]() |
[36] | C. P. Robert, G. Casella, Monte carlo statistical methods, Springier, 2004. |
[37] | D. V. Ravenzwaaij, P. Cassey, S. D. Brown, A simple introduction to Markov Chain Monte-Carlo sampling, Psychon. Bull. Rev., 25 (2018), 143-154. https://doi.org/0.3758/s13423-016-1015-8 |
[38] |
M. H. Chen, Q. M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Stat., 8 (1999), 69-92. https://doi.org/10.1080/10618600.1999.10474802 doi: 10.1080/10618600.1999.10474802
![]() |
[39] |
S. Dey, B. Pradhan, Generalized inverted exponential distribution under hybrid censoring, Stat. Methodol., 18 (2014), 101-114. https://doi.org/10.1016/j.stamet.2013.07.007 doi: 10.1016/j.stamet.2013.07.007
![]() |
[40] |
D. G. Hoel, A representation of mortality data by competing risks, Biometrics, 28 (1972), 475-488. https://doi.org/10.2307/2556161 doi: 10.2307/2556161
![]() |
Symbol | Connotation |
R | the set of real numbers |
R^r | the real r -dimensional space |
diag{ \cdot } | diagonal matrix |
A \otimes B | Kronecker product of matrices A and B |
|\cdot| | absolute value |
x_{i, q} , g_{i, q} | x_{i, q}(t) , g_{i, q}(\cdot) |
Agent(node) | 1 | 2 | 3 | 4 |
The number of triggers | 81 | 100 | 71 | 229 |
Agent(node) | 1 | 2 | 3 | 4 |
The number of triggers | 362 | 685 | 390 | 289 |
Symbol | Connotation |
R | the set of real numbers |
R^r | the real r -dimensional space |
diag{ \cdot } | diagonal matrix |
A \otimes B | Kronecker product of matrices A and B |
|\cdot| | absolute value |
x_{i, q} , g_{i, q} | x_{i, q}(t) , g_{i, q}(\cdot) |
Agent(node) | 1 | 2 | 3 | 4 |
The number of triggers | 81 | 100 | 71 | 229 |
Agent(node) | 1 | 2 | 3 | 4 |
The number of triggers | 362 | 685 | 390 | 289 |