Event-triggered output feedback control for Takagi-Sugeno fuzzy systems Yanpeng Guan1 , Qing-Long Han1,∗ and Chen Peng1,2 1. Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton, QLD 4702, Australia 2. School of Mechatronic Engineering and Automation, Shanghai University, 200072, PR China ∗ Corresponding author: Tel. +61 7 4930 9270; E-mail:
[email protected] Abstract—This paper considers the 𝐻∞ control problem for discrete-time Takagi-Sugeno (T-S) model fuzzy systems with event-triggered output feedback. The measurement output is transmitted to a fuzzy controller when the output error exceeds a pre-given threshold. The parallel distribution compensation (PDC) can not be used for controller design since the controller may not receive enough information about premise variables of the plant due to the event-triggered transmission scheme. A fuzzy dynamical output feedback controller is proposed to regularly generate the control input, which makes the controlled system stable with a certain 𝐻∞ disturbance attenuation level. A numerical example is given to show the effectiveness of the proposed approach.
I.
I NTRODUCTION
Event-triggered control has attracted increasing interest in the past few years since it can efficiently reduce the feedback data flow while maintaining a certain level of control performance compared to its time-triggered counterpart [1]– [6]. This advantage will be very appealing when the eventtriggered control strategy is applied in network connected systems, where the network resources, for example, battery power and network bandwidth, are usually limited. An eventtriggered scheduler is given in [6] to trigger control tasks when the norm of the state error achieves a state-dependent critical value. An impulsive system method is proposed in [7] to model and analyze the decentralized event-triggered control systems. A piecewise linear system approach is employed in [8] to study the model-based event-triggered control system. The event-triggered control mechanism is also applied in distributed and decentralized networked control systems [9]–[12]. It is shown in [13]–[15] that a time delay method can be used to model the continuous networked control systems dynamics with a discrete event-triggered transmission scheme. Takagi-Sugeno(T-S) model fuzzy systems have been extensively studied in the past decades since T-S fuzzy models can be used to effectively represent a large class of nonlinear systems. Many controller design problems for T-S model fuzzy systems have been considered in the literature [16]– [19]. The most frequently used control design method is the so-called parallel distribution compensation (PDC), by which the controller shares the same fuzzy premise variables and membership functions with the T-S fuzzy plant. It is
978-1-4799-0224-8/13/$31.00 ©2013 IEEE
shown that the PDC works well for the traditional point-topoint T-S model fuzzy systems, where signals are assumed to be transmitted instantly and accurately between system components. However, the general PDC may not apply when the fuzzy plant and the controller are connected by a communication network [15], [20], [21]. In this case, the information of the premise variables could not be guaranteed to arrive at the controller timely because of the network complexity, which leads to that the PDC can not be applied for fuzzy controller design. Several efforts have been made to deal with the case. By giving upper error bounds of the asynchronous membership functions, a fuzzy controller design method is proposed for networked-based T-S fuzzy systems in [21]. In [15], the PDC is applied under a strict assumption that the fuzzy controller could obtain the system state based on the system mechanism and control input. It is noted that the controller still can not get the exact system state information if the plant is exposed to an unexpected disturbance. In this paper, we study the event-triggered output feedback control problem for discrete-time T-S model fuzzy systems. On the one hand, most of the results about the eventtriggered control in the literature are obtained for continuous systems. To the best of authors’ knowledge, there is no existing work that considers the event-triggered discrete-time T-S model fuzzy systems. On the other hand, applying eventtriggered transmission scheme implies that the controller can not regularly receive output data from the fuzzy plant, which makes the traditional PDC control method may not work. These thoughts motivate the current study. In this paper, an event-triggered transmission scheme is proposed for the discrete-time T-S model fuzzy systems. The current measurement output data is transmitted to a fuzzy controller only when the output error exceeds a predetermined threshold. Since the controller can not get enough output data to share the same membership functions with the fuzzy plant, we attempt to propose a dynamical controller to generate control input regularly. Based on appropriately chosen premise variables and fuzzy sets for the controller, the fuzzy controller can operate separately to generate control input for the T-S model fuzzy plant. Moreover, a certain 𝐻∞ control performance level will be guaranteed. A method to obtain a group of feasible controller parameter matrices will be given.
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The organization of this paper is as follows. Section II presents the event-triggered output feedback T-S model fuzzy system and formulates the 𝐻∞ output feedback control problem. The stability and 𝐻∞ performance analysis for the event-triggered closed-loop system is given in Section III. A fuzzy output feedback controller design method is developed in Section IV. A simulation example is given to demonstrate the effectiveness of the proposed approach in Section V, and this paper is concluded in Section VI. II.
P ROBLEM F ORMULATION
In this section, we propose a framework for outputbased event-triggered discrete-time T-S model fuzzy control systems. Consider the following discrete-time T-S model with 𝑠 plant rules: Plant Rule 𝑖: IF 𝜉1 (𝑥(𝑘)) is 𝐹1𝑖 and 𝜉2 (𝑥(𝑘)) is 𝐹2𝑖 and . . . and 𝜉𝑟 (𝑥(𝑘)) is 𝐹𝑟𝑖 , THEN { 𝑥(𝑘 + 1) = 𝐴𝑖 𝑥(𝑘) + 𝐵1𝑖 𝜔(𝑘) + 𝐵2𝑖 𝑢(𝑘) 𝑦(𝑘) = 𝐶𝑖 𝑥(𝑘) + 𝐷𝑖 𝜔(𝑘) (1) 𝑧(𝑘) = 𝐿1𝑖 𝑥(𝑘) + 𝐿2𝑖 𝜔(𝑘) + 𝐿3𝑖 𝑢(𝑘) where 𝐹𝑗𝑖 is a fuzzy set and 𝑠 is the number of IF-THEN rules; 𝑥(𝑘) ∈ ℝ𝑛 , 𝜔(𝑘) ∈ ℝ𝑝 , 𝑧(𝑘) ∈ ℝ𝑞 , and 𝑦(𝑘) ∈ ℝ𝑚 are the state, disturbance, regulated output, and the measurement output respectively; 𝐴𝑖 , 𝐵1𝑖 , 𝐵2𝑖 , 𝐶𝑖 , 𝐷𝑖 𝐿1𝑖 𝐿2𝑖 and 𝐿3𝑖 are system matrices with appropriate dimensions; 𝜉1 (𝑥(𝑘)), . . . , 𝜉𝑟 (𝑥(𝑘)) are the premise variables. Then the T-S fuzzy system can be compactly represented by { 𝑥(𝑘 + 1) = 𝐴(ℎ)𝑥(𝑘) + 𝐵1 (ℎ)𝜔(𝑘) + 𝐵2 (ℎ)𝑢(𝑘) 𝑦(𝑘) = 𝐶(ℎ)𝑥(𝑘) + 𝐷(ℎ)𝜔(𝑘) (2) 𝑧(𝑘) = 𝐿1 (ℎ)𝑥(𝑘) + 𝐿2 (ℎ)𝜔(𝑘) + 𝐿3 (ℎ)𝑢(𝑘) where [ 𝐴(ℎ) 𝐶(ℎ) 𝐿1 (ℎ)
] [ 𝑠 𝐴𝑖 𝐵1𝑖 𝐵1 (ℎ) 𝐵2 (ℎ) ∑ 𝐷(ℎ) 0 ℎ𝑖 𝐶 𝑖 𝐷 𝑖 = 𝐿1𝑖 𝐿2𝑖 𝐿2 (ℎ) 𝐿3 (ℎ) 𝑖=1 ∏𝑟 𝜇 [𝜉 (𝑥(𝑘))] 𝑗=1 𝑖𝑗 𝑗 ℎ 𝑖 = ∑𝑠 ∏ 𝑟 , 𝑖 = 1, . . . , 𝑠 𝑙=1 𝑗=1 𝜇𝑙𝑗 [𝜉𝑗 (𝑥(𝑘))] ℎ := (ℎ1 , ℎ2 , . . . , ℎ𝑠 ) ∈ Ξ
𝐵2𝑖 0 𝐿3𝑖
]
transmitted output data. More specifically, the event-triggered transmission scheme can be mathematically expressed as 𝑡𝑘+1 = 𝑡𝑘 + min {𝑙 ∣ 𝑒𝑇 (𝑟𝑘,𝑙 )𝑒(𝑟𝑘,𝑙 ) ≥ 𝛿𝑦 𝑇 (𝑡𝑘 )𝑦(𝑡𝑘 )} 𝑙∈ℤ+
where ℤ+ is the set of positive integers; 𝑒(𝑟𝑘,𝑙 ) = 𝑦(𝑡𝑘 + 𝑙) − 𝑦(𝑡𝑘 ) 0 < 𝛿 < 1 is an adjustable factor of the threshold; 𝑡𝑘 is the 𝑘th transmission time instant. It is clear that {𝑦(𝑡𝑘 )}∞ 𝑘=1 is a subsequence of {𝑦(𝑘)}∞ 𝑘=1 . A typical control design method for the T-S model fuzzy systems usually takes a well-known PDC fuzzy controller, which shares the same premise variables and membership functions with the T-S fuzzy plant all the time. However, the PDC can not be applied to the proposed output-based eventtriggered T-S fuzzy model in this paper. Since the premise variables in (1) is not available to the controller due to the fact that only some of the measurement outputs are transmitted to the controller. In this paper, we employ the following fuzzy controller with 𝑠 controller rules: Controller Rule 𝑗: IF 𝜁1 (ˆ 𝑥(𝑘)) is 𝐺𝑗1 and . . . and 𝜁𝑟 (ˆ 𝑥(𝑘)) 𝑗 is 𝐺𝑟 , THEN { 𝑥 ˆ(𝑘 + 1) = 𝐴𝑐𝑗 𝑥 ˆ(𝑘) + 𝐵𝑐𝑗 𝑦(𝑡𝑘 ) (5) 𝑢(𝑘) = 𝐶𝑐𝑗 𝑥 ˆ(𝑘), 𝑘 ∈ [𝑡𝑘 , 𝑡𝑘+1 ) where 𝐴𝑐𝑗 , 𝐵𝑐𝑗 and 𝐶𝑐𝑗 are to be determined. Then the fuzzy controller can be compactly written as { ˆ 𝑥(𝑘) + 𝐵𝑐 (ℎ)𝑦(𝑡 ˆ 𝑥 ˆ(𝑘 + 1) = 𝐴𝑐 (ℎ)ˆ 𝑘) (6) ˆ 𝑢(𝑘) = 𝐶𝑐 (ℎ)ˆ 𝑥(𝑘), 𝑘 ∈ [𝑡𝑘 , 𝑡𝑘+1 ) with [ ] ∑ 𝑠 ˆ 𝐵𝑐 (ℎ) ˆ 𝐴𝑐 (ℎ) ˆ 𝑗 𝐴𝑐𝑗 ℎ = ˆ 𝐶𝑐𝑗 𝐶𝑐 (ℎ) 0 𝑗=1 ) ( ˆ2, . . . , ℎ ˆ 𝑠 ∈ Ξ. ˆ := ℎ ˆ 1, ℎ ℎ
[
in which 𝜇𝑖𝑗 [𝜉𝑗 (𝑥(𝑘))] is the grade of membership of 𝜉𝑗 (𝑥(𝑘)) in 𝐹𝑗𝑖 and Ξ is a set of basis functions satisfying ℎ𝑖 ≥ 0, 𝑖 = 1, . . . , 𝑠,
𝑠 ∑
ℎ𝑖 = 1.
(4)
] (7) (8)
For any 𝑘 ∈ [𝑡𝑘 , 𝑡𝑘+1 ), let
(3)
𝑖=1
𝐵𝑐𝑗 0
𝜖(𝑘) = 𝑦(𝑘) − 𝑦(𝑡𝑘 ).
(9)
𝑦(𝑡𝑘 ) = 𝑦(𝑘) − 𝜖(𝑘).
(10)
Then we have
The measurement output 𝑦(𝑘) is proposed to be transmitted through a network channel to a fuzzy controller. In order to alleviate the traffic burden and to save the limited network resources, we employ an event-triggered transmitter in this paper to determine whether or not the current measurement 𝑦(𝑘) should be transmitted. The basic idea is that 𝑦(𝑘) is released by the transmitter only if the difference between 𝑦(𝑘) and 𝑦(𝑡𝑘 ) is bigger enough, where 𝑦(𝑡𝑘 ) is the latest
By considering (2)-(10), we can get the following control closed-loop system:
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¯ 𝑥𝑘 + ℬ1 (ℎ)𝜔 ¯ 𝑘 + ℬ2 (ℎ)𝜖(𝑘) ¯ 𝑥 ¯(𝑘 + 1) = 𝒜(ℎ)¯ ¯ 𝑥𝑘 + ℒ2 (ℎ)𝜔 ¯ 𝑘 + ℒ3 (ℎ)𝜖(𝑘) ¯ 𝑧(𝑘) = ℒ1 (ℎ)¯
(11) (12)
¯ := (ℎ, ℎ) ˆ where 𝑥 ¯(𝑘) = col{𝑥(𝑘), 𝑥 ˆ(𝑘)}, ℎ [ ] ˆ ˆ 𝐴(ℎ) + 𝐵2 (ℎ)𝐷𝑐 (ℎ)𝐶(ℎ) 𝐵2 (ℎ)𝐶𝑐 (ℎ) ¯ 𝒜(ℎ) = ˆ ˆ 𝐵𝑐 (ℎ)𝐶(ℎ) 𝐴𝑐 (ℎ) [ ] ˆ ¯ = 𝐵1 (ℎ) + 𝐵2 (ℎ)𝐷𝑐 (ℎ)𝐷(ℎ) ℬ1 (ℎ) ˆ 𝐵𝑐 (ℎ)𝐷(ℎ) [ ] ˆ −𝐵2 (ℎ)𝐷𝑐 (ℎ) ¯ ℬ2 (ℎ) = ˆ −𝐵𝑐 (ℎ) ] [ ¯ = 𝐿1 (ℎ) + 𝐿3 (ℎ)𝐷𝑐 (ℎ)𝐶(ℎ) ˆ ˆ ℒ1 (ℎ) 𝐿3 (ℎ)𝐶𝑐 (ℎ)
¯𝑇 (𝑘)𝑃 −1 𝑥 ¯(𝑘) as the Lyapunov function. Then Take 𝑉𝑘 = 𝑥 the increment of 𝑉𝑘 along the solution of system (15) is ] {[ 𝑇 𝒜 (ℎ) 𝑇 𝑃 −1 [ 𝒜(ℎ) ℬ2 (ℎ) ] Δ𝑉𝑘 ∣(15) = 𝜁(𝑘) ℬ2𝑇 (ℎ) [ −1 ]} 𝑃 0 − 𝜁(𝑘) (16) 0 0
¯ = 𝐿2 (ℎ) + 𝐿3 (ℎ)𝐷𝑐 (ℎ)𝐷(ℎ) ˆ ℒ2 (ℎ) ¯ ˆ ℒ3 (ℎ) = −𝐿3 (ℎ)𝐷𝑐 (ℎ)
The purpose in what follows is to design a fuzzy controller in the form of (5) such that (i) (ii)
the system (11) with 𝜔(𝑘) = 0 is asymptotically ˆ ∈ Ξ; stable for any fuzzy basis function ℎ, ℎ the 𝐿2 -gain from the disturbance signal to the regulated output of the closed-loop system is less than a given scalar 𝛾 > 0, that is, under zero initial condition, ∥𝑧𝑘 ∥2 ≤ 𝛾 ∥𝜔𝑘 ∥2 for any nonzero 𝜔𝑘 ∈ 𝑙2 [0, ∞).
In the sequel, we will refer systems satisfying (I) and (II) to as stable and with 𝐻∞ performance 𝛾. III.
S TABILITY AND 𝐻∞ P ERFORMANCE A NALYSIS
In this section, we will give a sufficient condition to ensure that the closed-loop system (11)-(12) is stable with 𝐻∞ performance 𝛾. Theorem 1: For a given scalar 𝛾 > 0, with communication scheme (4), the closed-loop system (11)-(12) is stable with 𝐻∞ performance 𝛾, if there exists a real matrix 𝑃 > 0 ˆ∈Ξ such that for any ℎ, ℎ [ ] Π11 Π12 0, if there exist 𝑠 𝑠 𝑠 matrices {𝐸𝑖 }𝑖=1 , {𝐹𝑖 }𝑖=1 , {𝑄𝑖 }𝑖=1 , 𝑋 > 0, 𝑌 > 0 such that for all 𝑖, 𝑗 ∈ {1, . . . , 𝑠} [ ] Υ11 Υ12 Υ(𝑖𝑗) = 0 by 𝑋 −1 − 𝐼 , we obtain −𝑋 −1 + 𝑌 > 0 which implies that 𝐼−𝑋𝑌 is nonsingular. Hence there always exist two nonsingular matrices 𝑀 and 𝑁 such that 𝑀 𝑁 𝑇 = 𝐼 − 𝑋𝑌 .
respectively. Let
[ Ξ=
𝑋 𝑀𝑇
𝐼 0
𝑁 𝑉
where
(23)
] (24)
ˆ 𝑗 Υ(𝑖𝑗) < 0 ℎ𝑖 ℎ
(25)
𝑖=1 𝑗=1
Performing congruence transformation to (25) by 𝑑𝑖𝑎𝑔{Ξ−1 , 𝐼, 𝐼, Ξ−1 , 𝐼, 𝐼}, one can get (13) with consideration of (20)-(22). The result then follows from Theorem 1. □ It is noted that the matrix inequalities in Theorem 2 are not strict LMIs. Motivated by the idea in [20], we propose the following algorithm to solve the matrix inequalities in Theorem 2.
is 𝑀1 , THEN 𝑥𝑘+1 𝑦𝑘 𝑧𝑘 is 𝑀2 , THEN 𝑥𝑘+1 𝑦𝑘 𝑧𝑘
= = =
𝐴1 𝑥𝑘 + 𝐵11 𝜔𝑘 + 𝐵21 𝑢𝑘 𝐶1 𝑥 𝑘 + 𝐷1 𝜔𝑘 𝐿11 𝑥𝑘 + 𝐿21 𝜔𝑘 + 𝐿31 𝑢𝑘
= = =
𝐴2 𝑥𝑘 + 𝐵12 𝜔𝑘 + 𝐵22 𝑢𝑘 𝐶2 𝑥 𝑘 + 𝐷2 𝜔𝑘 𝐿12 𝑥𝑘 + 𝐿22 𝜔𝑘 + 𝐿32 𝑢𝑘
] 𝜎𝑇𝑠 0 1 − 𝜎𝑇𝑠 𝜂𝑇𝑠 1 − 𝑇𝑠 −𝑀1 𝑇𝑠 0 𝑚1 𝑇𝑠 1 − 𝑏𝑇𝑠 ] [ 𝜎𝑇𝑠 0 1 − 𝜎𝑇𝑠 𝜂𝑇𝑠 1 − 𝑇𝑠 −𝑀2 𝑇𝑠 0 𝑚2 𝑇𝑠 1 − 𝑏𝑇𝑠 [ ] [ ] 0.001 0.001 0.002 , 𝐵12 = 0.0015 0.008 0.007 [ ] 0.1 0 𝐵22 = 0 [ −0.1 −0.1 0.3 ] [ −0.05 −0.05 0.2 ] 0.001, 𝐿22 = 0.0015, 𝐿31 = 0.5, 𝐿32 = 0.4 [ ] [ ] 1 −1 3 1 −1 3.5 , 𝐶2 = −1 5 1 −1 4.5 1 [ ] [ ] 0.05 0.04 , 𝐷2 = 0.01 0.01 [
𝐴1
=
𝐴2
=
𝐵11
=
𝐵21
=
𝐿11 𝐿12 𝐿21
= = =
𝐶1
=
𝐷1
=
]
It follows from (19) that 𝑠 𝑠 ∑ ∑
D ESIGN E XAMPLE
Plant Rules:
The controller parameter matrices can be obtained by
Denote 𝑃 −1 and 𝑃 in Theorem 1 as [ ] [ 𝑋 𝑀 𝑌 −1 𝑃 = , 𝑃 = 𝑀𝑇 𝑈 𝑁𝑇
(26)
Solve (26) to get a group of feasible solution 𝑠 {𝐹𝑖 }𝑖=1 and 𝑌 . 𝑠 Use the obtained {𝐹𝑖 }𝑖=1 and 𝑌 to solve (19); if not feasible, go to Step 2.
Step 3.
hold, then there exists a fuzzy controller in the form of (5) such that the closed-loop system (11)-(12) with communication scheme (4) is stable with 𝐻∞ performance 𝛾. Furthermore, two nonsingular constant matrices 𝑀 and 𝑁 can always be obtained such that
𝐴𝑐𝑗 = 𝑀 −1 𝑄𝑗 𝑁 −𝑇 𝐵𝑐𝑗 = 𝑀 −1 𝐸𝑗 𝐶𝑐𝑗 = 𝐹𝑗 𝑁 −𝑇 , 𝑗 = 1, 2, . . . , 𝑠.
Step 1.
with [𝑚1 , 𝑚2 ] = [−20, 30] and (𝜎, 𝜂, 𝑏) = (10, 28, 8/3). The membership functions, ℎ1 and ℎ2 , are described respectively
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by ℎ1
(
(1) 𝑥𝑘
)
and
2
(1)
=
30−𝑥 exp(−( 50 𝑘 )2 ) (1) (1) 30−𝑥 20+𝑥 exp(−( 50 𝑘 )2 ) + exp(−( 50 𝑘 )2 )
x1
1.5
x
1
( ) ( ) (1) (1) ℎ2 𝑥 𝑘 = 1 − 𝑀1 𝑥 𝑘 .
2
x3
0.5 0
We choose the parameter of event-triggered transmission scheme (4) as 𝛿 = 0.1, 𝐻∞ performance level 𝛾 = 9.8. By using Theorem 2 and Algorithm 1, we obtain a fuzzy controller (5) with the following parameter matrices [ ] −0.0905 0.3063 0.0353 2.2253 0.4623 −0.0534 𝐴𝑐1 = 0.0438 −0.0410 0.9391 [ ] −0.2499 0.3169 0.0040 2.747 0.4299 −0.0213 𝐴𝑐2 = −0.0038 −0.018782 0.93875 [ ] −1.0539 0.4842 1.0622 −4.1929 𝐵𝑐1 = −3.0750 −0.8861 [ ] −1.005 0.42153 1.0352 −4.1309 𝐵𝑐2 = −2.9949 −0.7871 𝐶𝑐1 (ℎ) = [ −1.5252 0.0246 −0.0009 ] 𝐶𝑐2 (ℎ) = [ −1.44 0.0237 − 0.0045 ] .
−0.5 −1 −1.5 0
ˆ1 ℎ
(1) 𝑥 ˆ𝑘
)
=
exp(−( (1)
30−ˆ 𝑥 exp(−( 50 𝑘
)2 )
(1)
40−ˆ 𝑥𝑘 50
+
300
400
1.5 xc1 xc2
1
xc3
0.5 0
)2 ) (1)
10+ˆ 𝑥 exp(−( 50 𝑘
)2 )
−0.5
.
The initial condition and the disturbance signal are assumed to be 𝑥(0) = [1 − 1 − 1]𝑇 , 𝜔(𝑘) = 1/(1 + 𝑘), respectively. Fig. 1 and Fig. 2 illustrate the state response of the plant and controller, respectively. Fig. 3 shows the release time instants distribution of the event-triggered transmitter. It can be seen that the controlled fuzzy system is stable with the prescribed 𝐻∞ performance level although only a small proportion of the measurement output is triggered to be transmitted to the controller. Within the simulation time, the measurement are generated 400 times while 34 of them are transmitted to the fuzzy controller, which implies that the required transmission resources may be saved. VI.
200 Time
Fig. 1: State response of the controlled system.
ˆ 1 is supposed to be In this example, ℎ (
100
C ONCLUSION
This paper studies the event-triggered output feedback control problem for a class of discrete-time T-S model fuzzy systems. An event-triggered transmission scheme is proposed to reduce the feedback data transmission. The current measurement output is available to the controller only when a certain threshold exceeds, which leads to that controller may not receive enough information about the premise variables of the fuzzy plant. By appropriately choosing controller premise
−1 0
100
200 Time
300
400
Fig. 2: State response of the fuzzy controller.
variables and fuzzy sets, we developed a comparatively separate fuzzy controller, which is used to generate control input regularly. A design method is given to obtain the controller parameter matrices. A numerical example shows that with the proposed approach, the transmission frequency of system feedback data could be reduced while a certain level of 𝐻∞ disturbance attenuation performance is maintained. ACKNOWLEDGMENT This work was supported in part by the Australian Research Council Discovery Projects under Grant DP1096780; the Research Advancement Award Scheme (January 2010 December 2012) at Central Queensland University, Australia; and the Natural Science Foundation of China under Grants 61074024 and 61273114.
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30
Release interval
25 20 15 10 5 0 0
100
200 Release time
300
400
Fig. 3: Release time distribution.
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