- Email: [email protected]
Observer-based output-feedback control of large-scale networked fuzzy systems with two-channel event-triggering
Accepted Manuscript
Observer-Based Output-Feedback Control of Large-Scale Networked Fuzzy Systems with Two-Channel Event-Triggering Zhixiong Zhong, Yanzheng Zhu PII: DOI: Reference:
S0016-0032(17)30271-5 10.1016/j.jfranklin.2017.05.036 FI 3010
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
8 February 2017 26 April 2017 23 May 2017
Please cite this article as: Zhixiong Zhong, Yanzheng Zhu, Observer-Based Output-Feedback Control of Large-Scale Networked Fuzzy Systems with Two-Channel Event-Triggering, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.036
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Observer-Based Output-Feedback Control of Large-Scale Networked Fuzzy Systems with Two-Channel Event-Triggering∗
Abstract
CR IP T
Zhixiong Zhong† and Yanzheng Zhu‡
AN US
This paper concerns data transmissions for large-scale T-S fuzzy systems with event-triggering control, where each subsystem communicates its information via a two-channel network. We propose an event-triggering scheme in which two event-triggering mechanisms are used to verify the data transmissions. At first, a novel model transformation is presented, where the event-triggered control system is reconstructed as a constant-delay system with extra inputs and outputs. By using a relaxed Lyapunov-Krasovskii functional (LKF) without the requirement of positive definiteness for all Lyapunov matrices, and the scaled small gain (SSG) theorem, the co-design problem of desired observer and controller gains, event-triggering parameters, and the sampling period is resolved in the form of linear matrix inequalities (LMIs). It will be shown that the solution guarantees the stability of closed-loop fuzzy control system and the reductions of data communications in both the sensor-to-controller and controller-to-actuator channels. The proposed method is validated by using a numerical example.
Introduction
ED
1
M
Keywords: Large-scale fuzzy systems, two-channel event-triggering, observer-based outputfeedback, co-design.
AC
CE
PT
In real-world applications, it is not very easy to deal with the control problem of nonlinear systems because of strong nonlinearities contained in the plants. In recent decades, Takagi-Sugeno (T-S) model-based method has been validated that it can overcome the difficulties induced by the nonlinearities [1]. With the efforts of academic and engineering researchers, a great number of topics have been focused upon in latest years for T-S models, such as stability and performance analysis [2, 3, 4, 5], controller design [7, 8, 9], and filter design [10, 11]. For a given nonlinear system, the T-S model can be used to approximate it at any preciseness, and the mature linear control methods can be used to resolve the control problem of the T-S model. In addition, the control of large-scale systems refers to many practical applications including transportation networks, industrial production, power grids, communication systems, and so on. However, its control also brings great challenges because of the excessively processing data, different locations, as well as interconnections among subsystems ∗ The work described in this paper was supported in part by the Natural Science Foundation of Fujian Province, China under Grants 2017J01781; the National Natural Science Foundation of China (61603221); the Natural Science Foundation of Shandong Province (ZR2016FB11); the China Postdoctoral Science Foundation (2017M610437); the Special Foundation for Postdoctoral Science Foundation of Shandong Province (201601014); the Applied Research Project for Postdoctoral Researchers of Qingdao (2016109); Research Fund for the Taishan Scholar Project of Shandong Province of China; the Open Fund Project of Fujian Provincial Key Laboratory of Information Processing and Intelligent Control (Minjiang University) (No. MJUKF201731). † Zhixiong Zhong is with the High-voltage Key Laboratory of Fujian Province, Xiamen University of Technology, Xiamen 361024, China; and Fujian Provincial Key Laboratory of Information Processing and Intelligent Control (Minjiang University), Fuzhou 350121, China. Email: [email protected] ‡ Yanzheng Zhu (corresponding author) is with the College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, China. Email: [email protected]
1
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
[12]. More recently, the decentralization has been proved that it is an effective technique for the control of large-scale systems, and some results on decentralization control of large-scale nonlinear systems have been published using the T-S fuzzy model-based method, see, e.g., [13, 14, 15, 16]. On the other hand, networked control systems (NCSs) have received considerable attractions for their various superiorities, such as light weights, simple installation, simple maintenance, and low cost [17, 18, 19, 20, 21, 22]. However, its intrinsic imperfections, including packet dropouts, quantization errors, and time delays, could weaken its control performance and even may result in instability [23]. Currently, there have been lots of results focusing on these imperfections, see [23, 24, 25], and the references therein. It is noted that a tradeoff among these network-induced constraints always be required when designing an NCS. Specifically, sending larger packets, giving time-stamping of data, reducing quantization errors generally improve the control performance while inducing longer transmission delays [26]. Thus, it is important to develop new methods in the implementation of NCSs to make use of the limited network bandwidth more effectively. In the implementation of feedback control, the traditional policy transmits feedback signals in a periodic manner, which is called time-triggered control. This policy could result in collision or congestion or longer time delays in the network with limited bandwidths. In recent years, an increasing interest has been focused on the event-triggering control that aims at reduction in data transmissions. Its basic policy is to transmit control signals in the light of some prescribed events [27, 28, 29, 30, 31, 32]. In other words, not all feedback signals are necessarily transmitted at each sampling period. The event-based policy has appeared under a variety of aliases, such as event-triggered feedback [27, 28, 29], self-triggering feedback [31], interrupt-based feedback [30], and state-triggered feedback [32]. Moreover, the event-triggering scheme has been applied to the area of T-S fuzzy networked systems [33, 34, 35, 36, 37, 38]. To mention a few, the work in [34] proposed a novel event-triggering scheme for fuzzy Markov jump systems based on general switching policies, where each transiting rate allows to be unknown, or partly known. In [35, 36, 37], some results were obtained on the co-design problem with event-triggered state-feedback control for T-S fuzzy sampled-data systems. In [38], the controller design of fuzzy dynamic-output feedback was studied for discrete-time T-S fuzzy systems under an event-triggering scheme. We are aware of few attempts making on the decentralized event-triggering observer-based control for large-scale networked T-S fuzzy systems, which motivates us for the present work. This paper attempts to deal with the decentralized control problem for large-scale networked fuzzy systems with two-channel event-triggering, where the controller form is of observer-based output-feedback type, and each subsystem exchanges its information via networks. The objective is to design a decentralized event-triggering observer-based output-feedback controller, which guarantees the stability of closed-loop control system and the reduction of communication data in both the sensor-to-controller (S-C) and controller-to-actuator (C-A) channels. To this end, an event-triggering scheme is proposed in which both the measured output and control input are sampled periodically, and two event-triggering mechanisms (ETMs) are established respectively when the sampled measurement output and control input should be transmitted for each subsystem. Firstly, the closed-loop sampled-data control system is reconstructed as a continuous-time system with time-varying delays by using the input delays, then both the delay variations and event-triggered counterparts are modeled as the disturbances. By hauling out these disturbances, the uncertain continuous-time system is reformulated as a constant-delay system with extra inputs and outputs. Thus, based on the novel model, a relaxed LKF without the requirement of positive definition for all Lyapunov matrices is introduced, and combining with the virtue of SSG theorem, the co-design problem of desired observer and controller gains, event-triggering parameters, and the sampling period is reformulated in the form of LMIs. Finally, one numerical example is illustrated to verify the effectiveness of developed methods. The main contribution of this paper is threefold: i ) For purpose of reducing the data transmissions in both the S-C and C-A channels, and lowering the requirement of full state to be available for feedback control, the controller design to decentralized event-triggered observer-based outputfeedback is proposed for the large-scale T-S fuzzy system, and the premise variables between the 2
ACCEPTED MANUSCRIPT
2
CR IP T
plant and the proposed controller are not necessarily considered to be synchronous. ii ) Based on the input-delay and perturbation approaches, a novel model transformation is proposed in the paper. Compared with the method proposed in [14], the results with less conservativeness are derived via a relaxed LKF and the SSG theorem on the design of decentralized observer-based output-feedback controller in the event-triggered scheme. This means that the longer sampling intervals or larger specified threshold can be obtained. iii ) The co-design problem of obtaining observer and controller gains, event-triggering parameters, and the sampling period is resolved in the form of LMIs. Notations. Rn and Rn×m are the Euclidean space with n dimension, and the set with n × m matrices, respectively. P > 0 (≥ 0) denotes P is positive definite (positive semidefinite). Sym{A} is A + AT . In and 0m×n represent the n × n identity matrix and m × n zero matrix, respectively. The subscripts n and m × n are omitted when the size is irrelevant or can be known in the context. A−1 and AT are the inverse and transpose of the matrix A ∈
Problem Formulation
i
i i
i
AN US
This paper aims at a large-scale system which contains N nonlinear subsystems, where the i-th nonlinear subsystem can be represented by T-S fuzzy model in the following: l and f (x (t)) is F l and · · · and f (x (t)) is F l , THEN Plant Rule Ril : IF fi1 (xi (t)) is Fi1 i2 i ig i i2 ig N P x˙ (t) = Al x (t) + B u i F¯ijl xj (t) i i ˆi (tk ) + i i j=1 (1) j6=i y (t) = C l x (t), l ∈ L := {1, 2, . . . , r } , t ∈ [ti , ti ) i
k
k+1
PT
ED
M
where i ∈ N := {1, 2, . . . , N }; Ril is the l-th fuzzy inference rule; ri is the number of inference l (φ = 1, 2, . . . , g) are fuzzy sets; x (t) ∈ Rnxi is the system state; u ˆi (tik ) ∈ Rnui is the rules; Fiφ i control input applied to the system at the instant tik ; yi (t) ∈ Rnyi is the measured output; fi (xi (t)) := [fi1 (xi (t)), fi2 (xi (t)), . . . , fig (xi (t))] are some measurable state variables; (Ali , Bi , Cil ) denotes the lth local model, and the matrix Bi is of full column rank; F¯ijl is the interconnection of the i-th and j-th subsystems for the l-th local model. Q l and the normalized membership function µl [f (x (t))], Define the inferring fuzzy set Fil := gφ=1 Fiφ i i i one has Qg ri l X φ=1 µiφ [fiφ (xi (t))] l µi [fi (xi (t))] := Pri Qg ≥ 0, µli [fi (xi (t))] = 1, (2) ς µ [f (x (t))] i iφ ς=1 φ=1 iφ
CE
l=1
l . This paper will denote µl := where is the grade of membership of fiφ (xi (t)) in Fiφ i µli [fi (xi (t))] for simplify. Based on fuzzy blending, the i-th global T-S fuzzy dynamic model is N P x˙ (t) = A (µ )x (t) + B u i F¯ij (µi )xj (t) i i i i i ˆi (tk ) + j=1 (3) j6=i y (t) = C (µ )x (t), t ∈ [ti , ti ), i ∈ N
AC
µliφ [fiφ (xi (t))]
i
where
Ai (µi ) :=
i
ri X l=1
i
i
k
µli Ali , Ci (µi ) :=
ri X l=1
k+1
µli Cil , F¯ij (µi ) :=
ri X
µli F¯ijl .
(4)
l=1
Remark 1. Note that, a special large-scale fuzzy system with linear interconnection F¯ij has been investigated extensively in [14, 15, 16]. This paper considers the large-scale fuzzy system with nonlinear interconnection F¯ijl as shown in (1), which is more challenging than those with linear interconnection matrix F¯ij . 3
ACCEPTED MANUSCRIPT
Before moving on, we require the assumptions as below: Assumption 1. The sampler is clock-driven in each subsystem. Let hi denote the upper bound of sampling intervals for the i-th subsystem, it has tik+1 − tik ≤ hi , k ∈ N,
(5)
CR IP T
where hi > 0. Assumption 2. For each subsystem, it is assumed that both the S-C and C-A channels are closed via network-based communications. Assumption 3. The zero-order-hold (ZOH) is event-triggered, and it makes use of the latest sampling data and holds them until the next transmitting datum come. In the NCSs, the traditionally time-triggering implementation is undesirable because of the existence of the limited bandwidths. Here, for the purpose of reducing the data transmissions, inspired by the work in [27], the decentralized event-triggering fuzzy observer is proposed as follows: s and f (x (t)) is F s and · · · and f (x (t)) is F s , Plant Rule Ris : IF fi1 (xci (t)) is Fi1 i2 ci ig ci i2 ig THEN x˙ ci (t) = Asi xci (t) + Bi u ˆi tik + Lsi yˆi (tik ) − yci (t) (6) s i yci (t) = Ci xci (t), t ∈ [tk , tik+1 ), i ∈ N
AN US
where xci (t) ∈ Rnxi is the estimated state, yˆi (tik ) is the measured output applied to the observer at the instant tik , and Lsi ∈ Rnxi ×nyi , s ∈ Li , i ∈ N are observer gains to be designed with compatible dimensions. Similarly, the global decentralized event-triggering fuzzy observer can be given by µi ) yˆi (tik ) − yci (t) x˙ ci (t) = Ai (ˆ µi )xci (t) + Bi u ˆi tik + Li (ˆ (7) yci (t) = Ci (ˆ µi )xci (t), t ∈ [tik , tik+1 ), i ∈ N where
(8)
ED
M
ri ri P P s [f (x (t))] As , L (ˆ µ ˆsi [fi (xci (t))] Lsi , µ ˆ µ ) := A (ˆ µ ) := i i i ci i i i i s=1 s=1 ri ri P P µ ˆsi [fi (xci (t))] = 1, µ ˆsi [fi (xci (t))] Cis , Ci (ˆ µi ) := s=1 s=1 Qg µ ˆsiφ [fiφ (xci (t))] µ Qg ≥ 0. ˆsi [fi (xci (t))] := Pri φ=1 ˆςiφ [fiφ (xci (t))] ς=1 φ=1 µ
CE
PT
For simplicity, we define µ ˆsi := µ ˆsi [fci (xci (t))]. Based on the observer in (7), the estimated state xci (t) can be used by the following controller: (9) ui tik = Ki xci (tik ), t ∈ [tik , tik+1 ), i ∈ N ,
AC
where Ki ∈ Rnui ×nxi , i ∈ N are controller gains to be determined with compatible dimensions. For implementing the proposed controller given in (7) and (9), a structure diagram of the presented control system is shown in Figure 1, where the S-C and C-A channels are closed through network-based communications. The sensor system consists of a sampler SPs , a buffer BFs , and an ETMs . The BFs is to store the latest measured output transmitted successfully to the observer yˆi tik , and the ETMs determines whether or not to transmit the sampling output yi (tik ) to the observer. Hence, the measured output yi (t) is firstly sampled by the SPs at each sample period. Then, it is transmitted to the controller and is executed only when one preselected event is violated, which leads to a data reduction in the S-C channel. The controller system consists of a zero-order-hold ZOHc , an observer, a controller, a sampler SPc , a buffer BFc , and an ETMc . The BFc is to store the latest control input transmitted successfully to the plant u ˆi tik , and the ETMc determines whether or not i to transmit the sampling control input ui (tk ) to the actuator system. Thus, ui (tik ) is transmitted and implemented to the plant at each sample instant only when another prescribed event is violated, which leads to a data reduction in the C-A channel.
4
ACCEPTED MANUSCRIPT
Actuator System Actuator
Sensor System
yi (t)
Plant
yˆi (tik )
SPs yi (tik )
ZOHa
ETMs
BFs
Networks BFs u ˆi (tik )
Controller
ETMc ui (tik )
SPs
Observer
ui (t) xci (t)
Controller System
ZOHc
CR IP T
Networks
Figure 1: Observer-based control with two-channel event-triggering
AN US
To formalize the described solution, these two ETMs can operate as
ETMs : yi tik is sent ⇔ yi tik − yˆi tik−1 > σyi yi tik ,
ˆi tik−1 > σui ui tik , ETMc : ui tik is sent ⇔ ui tik − u
(10) (11)
M
where σyi ≥ 0 and σui ≥ 0 are two suitably chosen design parameters. Based on the operation given by (10) and (11), an event-triggering strategy is formulated as below:
> σyi yi ti , yi tik , when yi tik − yˆi tik−1 k
yˆi tik = (12) yˆi tik−1 , when yi tik − yˆi tik−1 ≤ σyi yi tik ,
> σui ui ti , ui tik , when ui tik − u ˆi tik−1 i k
u ˆi tk = (13) u ˆi tik−1 , when ui tik − u ˆi tik−1 ≤ σui ui tik .
AC
CE
PT
ED
Remark 2. As µli = µ ˆli , the decentralized event-triggered fuzzy observer in (7) reduces to a parallel distributed compensation (PDC) one. However, the premise variables between the T-S fuzzy systems in (1) and the event-triggered fuzzy observer in (7) are not always same in the network-based applications due to some limitations such as signal transformation and quantization, packet dropouts, time delays, etc. In other words, it is more realistic to consider the fuzzy observer with the estimated states. Moreover, in this case, the premise variables are no longer required to send via networks, which leads to the reduction in data transmissions. Pri Remark 3. When considering the fuzzy controller ui tik = ˆsi Kis xci (tik ), it is difficult to s=1 µ maintain a synchronous premise variable among the fuzzy system, the fuzzy observer, and the fuzzy controller. For simplification of controller design, the linear controller is proposed in (12) instead of a fuzzy one.
ˆli are unavailable, the asynchronous condition µli 6= µ ˆli Remark 4. As the knowledge on µli and µ generally reduces to a linear controller instead of a fuzzy one, which declines the control performance [39]. However, when the knowledge of µli and µ ˆli is available, a corresponding fuzzy controller can be obtained, and therefore can reduce the design conservatism.
3
Main Results
A new model transformation will be proposed in this section. Then, based on the novel model, a relaxed LKF without the requirement of all positive-definite Lyapunov matrices is introduced. Combined with the SSG technique, the co-design problem of calculating the observer and controller gains, the event-triggering parameters, and the sampling period is resolved in the form of LMIs. 5
ACCEPTED MANUSCRIPT
3.1
Model Reformulation
Based on the input delay method in [40], both the measured output and the control input are reconstructed as the following delayed forms, ei (t) = xi (t) − xci (t) , yˆi (tik ) = yˆi (t − ηi (t)), u ˆi (tik ) = u ˆi (t − ηi (t)),
(14)
0 ≤ ηi (t) < hi , t ∈ [tik , tik+1 ), k ∈ N.
(15)
where ηi (t) = t − tik . It follows from the Assumptions 1-3 that In addition, we model the event-triggered counterparts as [27]
CR IP T
wyi (t) = yˆi (t − ηi (t)) − yi (t − ηi (t)),
wui (t) = u ˆi (t − ηi (t)) − ui (t − ηi (t)).
(16) (17)
Combining with (3), (7), (9), (14), (16) and (17), the augmented closed-loop control system is given by x ¯˙ i (t) = Ai (µi , µ ˆi )¯ xi (t) + Aid (¯ µi , µ ˆi )¯ xi (t − ηi (t)) N X j=1 j6=i
R3 F¯ij (µi )xj (t),
AN US
+ R1 Bi wui (t) − R2 Li (ˆ µi )wyi (t) + where
ED
M
Ai (µi ) 0 , Ai (µi , µ ˆi ) = Ai (µi , µ ˆi ) + Li (ˆ µi )Ci (ˆ µi ) Ai (ˆ µi ) − Li (ˆ µi )Ci(ˆ µi ) xi (t) Bi Ki −Bi Ki , µi , µ ¯i ) = ,x ¯i (t) = Aid (ˆ ei (t) −L (ˆ µ )C (¯ µ ) 0 i i i i I 0 I , R2 = , R3 = , Ai (µi , µ ˆi ) = Ai (µi ) − Ai (ˆ µi ), R1 = 0 I I Q g ri P µ ¯f [fiφ (xi (tik ))] Qg iφ ς ≥ 0, µ ¯fi fi (xi tik ) = 1. ¯fi fi (xi tik ) := Pri φ=1 µ i t ) f (x µ ¯ ( ) ] [ i iφ ς=1 φ=1 iφ k f =1
(18)
(19)
AC
CE
PT
Besides, the delay variations can be modeled as [41] 2 1 wdi (t) = x ¯i (t − ηi (t)) − [¯ xi (t) + x ¯i (t − hi )] , (20) hi 2 " # (1) n o wdi (t) (1) (2) where wdi (t) = , wdi (t) , wdi (t) ∈
ACCEPTED MANUSCRIPT
where Ri1 denotes the forward subsystem with the operator {wui (t) , wyi (t) , wdi (t)} → {ξui (t) , ξyi (t) , ξdi (t)}, and Ri2 denotes the feedback interconnection with the operator {ξui (t) , ξyi (t) , ξdi (t)} → {wui (t) , wyi (t) , wdi (t)}, and {∆yi , ∆ui , ∆di } denotes the uncertain operator, respectively. Based on the interconnected system given by (21) and (22), the following lemma is provided at first. Lemma 1. Given the interconnection with two subsystems Ri1 in (21) and Ri2 in (22), the operators T T (t) ξ T (t) T (t) ξui ∆i : ξi (t) 7−→ wi (t) satisfy the property k∆i k∞ ≤ 1, where ξi (t) = ξyi , di T T T T wi (t) = wyi (t) wui (t) wdi (t) , ∆i = diag{∆yi , ∆ui , ∆di }.
CR IP T
Proof. It follows from the event-triggered strategy given in (12) and (13), one has kwyi (t)k = kˆ yi (t − ηi (t)) − yi (t − ηi (t))k
≤ σyi kyi (t − ηi (t))k
1 hi (1) 1
σyi Ci (¯ µi ) xi (t) + xi (t − hi ) + wdi (t)
2 2 2 = kξyi (t)k ,
kwui (t)k = kˆ ui (t − ηi (t)) − ui (t − ηi (t))k
AN US
≤ σui kui (t − ηi (t))k
= σui Ki I −I x ¯i (t − ηi (t))
1 1 hi
I −I x ¯ (t) + x ¯ (t − h ) + w (t) = σui K i i i di
i
2 2 2 = kξui (t)k .
(23)
(24)
0
1 = 2 hi
Z t Z 0
Z t
0
−hi
Z
PT
1 h2i
ED
M
Next, considering the zero initial conditions, and by using Jensen’s inequality (Lemma A1 in the Appendix), it follows from (20) that Z t T (α)wdi (α) dα wdi
≤
CE
1 = hi
AC
1 = hi
Z
0 0
−hi Z 0 −hi Z 0
hi
Z
0
Z
β
ρi (β) ξdi (α + β) dβ
0
0
−hi
ρi (β) ξdi (α + β) dβ dα
T ρ2i (β) ξdi (α + β) ξdi (α + β) dβ dα
−hi t T ξdi (α t+β
T Z
+ β) ξdi (α + β) dα dβ
T ξdi (α) ξdi (α) dα
dβ
Z t 1 T ≤ ξ (α) ξdi (α) dα dβ hi −hi 0 di Z t T = ξdi (α) ξdi (α) dα.
(25)
0
It is not difficult to see from (23)-(25) that k∆i k∞ ≤ 1, thus completing this proof.
3.2
Stability Analysis
This section first introduces the following LKF: V (t) =
N X i=1
7
Vi (t),
(26)
ACCEPTED MANUSCRIPT
with Vi (t) = x ¯Ti (t) Pi x ¯i (t) +
Z
t
t−hi
Z
x ¯Ti (α) Qi x ¯i (α) dα + hi
0
−hi
Z
t
t+β
¯˙ i (α) dαdβ, x ¯˙ Ti (α) Zi x
(27)
where the matrices {Pi , Qi , Zi } ∈ R2nxi ×2nxi , i ∈ N are symmetric, and Pi > 0, Zi > 0. Inspired by [42], the positive definiteness of matrix Qi in (27) is not required. The following lemma is given to ensure the positive property of V (t).
CR IP T
Lemma 2. Consider the LKF in (26), the inequality V (t) ≥ k¯ x (t)k2 is satisfied, where > 0, T T ¯1 (t) x ¯T2 (t) · · · x ¯TN (t) , if there exist positive-definite symmetric matrices {Pi , Zi } ∈ x ¯ (t) = x 2n ×2n xi , and symmetric matrices Q ∈ R2nxi ×2nxi , such that for all i ∈ N the following inequalities R xi i hold: 1 −Zi hi Pi + Zi > 0. (28) ? Qi + Zi Proof. Firstly, by using Lemma A1 in the Appendix (Jensen’s inequality), one gets 0
−hi
Z
t
t+β
≥ hi = hi
x ¯˙ Ti (α) Zi x ¯˙ i (α) dαdβ Z
0
−hi 0
Z
−hi hi
Z
−1 β
Z
t
t+β
Z T ˙x ¯i (α) dα Zi
˙x ¯i (α) dα dβ
AN US
hi
Z
t
t+β
−1 [¯ xi (t) − x ¯i (t + β)]T Zi [¯ xi (t) − x ¯i (t + β)] dβ β
ED
M
1 = hi [¯ xi (t) − x ¯i (t − β)]T Zi [¯ xi (t) − x ¯i (t − β)] dβ β 0 Z hi [¯ xi (t) − x ¯i (t − β)]T Zi [¯ xi (t) − x ¯i (t − β)] dβ ≥ 0 Z t [¯ xi (t) − x ¯i (α)]T Zi [¯ xi (t) − x ¯i (α)] dα. =
(29)
t−hi
It follows from (27) and (29) that t
PT
Z
Vi (t) ≥
t−hi
x ¯i (t) x ¯i (α)
T
1 hi Pi
+ Zi −Zi ? Qi + Zi
x ¯i (t) x ¯i (α)
dα.
(30)
CE
Therefore, it is easy to see from (28) that V (t) ≥ k¯ x (t)k2 . Thus, completing this proof. Based on the interconnection model in (21) and (22), and the LKF in (26), a sufficient condition for stability of the control system in (18) is validated by the following theorem.
AC
Theorem 1. Given the large-scale fuzzy system in (3) and the decentralized event-triggering observerbased controller given by (7) and (9), the stability of the control system in (18) can be validated, if there exist positive-definite symmetric matrices {Pi , Zi , Hi3 } ∈ R2nxi ×2nxi , Hi1 ∈ Rnyi ×nyi , Hi2 ∈ Rnui ×nui , symmetric matrices Qi ∈ R2nxi ×2nxi , matrix multipliers Gi ∈ R(8nxi +nyi +nui )×2nxi , and positive scalars {σui , σyi , εij (µi ), ε0 } , εij (µi ) ≤ ε0 , such that for all i ∈ N the following matrix inequalities hold:
+ Zi −Zi > 0, ? Qi + Zi 2 ΩT (¯ 2 ΩT H Ω + Sym {G A (µ , µ Θi + σyi µi ) + σui ¯i )} Gi R3 F¯ij (µi ) i i i ˆi , µ 1 µi )Hi1 Ω1 (¯ 2 i2 2 < 0, ? −Eij (µi )
8
1 hi Pi
(31) (32)
ACCEPTED MANUSCRIPT
hi µi , µ ¯i ) 2 Adi (ˆ
CR IP T
where 2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Qi − Zi 0 0 0 , Θ = i ? ? ? −Hi1 0 0 ? ? ? ? −Hi2 0 ? ? ? ? −Hi3 ? Ω1 (¯ µi )= 0 12 Ci (¯ µi ) 0 12 Ci (¯ µi ) 0 0 0 h2i Ci (¯ µi ) 0 , hi 1 1 1 1 Ω2 = 0 2 Ki − 2 Ki 2 Ki − 2 Ki 0 0 2 Ki − h2i Ki , Ai (µi , µ ˆi , µ ¯i ) = −I Ai (µi , µ µi ,µ ¯i ) 12 Adi (ˆ µi , µ ¯i ) −R2 Li (ˆ µi ) R1 Bi ˆi ) + 21 Adi (ˆ ¯ ¯ ¯ ¯ F (µ ) · · · F (µ ) · · · F (µ ) F (µ ) = , Θ = Q − Z + ε (N − 1) R1 R1T , i1 i i i i 0 iN ij,j6=i i i(1) ij i | {z } T N −1 T T I 0 0 I R = , R = , R3 = I I , 1 2 . E (µ ) = diag {ε (µ )I · · · ε (µ )I · · · ε (µ )I} ij i i1 i i iN ij,j6=i i {z } | N −1
Proof. By taking the time derivative of V (t), one has
AN US
V˙ i (t) ≤ 2¯ xTi (t) Pi x ¯˙ i (t) + x ¯Ti (t) Qi x ¯i (t) − x ¯Ti (t − hi ) Qi x ¯i (t − hi ) Z t ¯˙ Ti (t) Zi x ¯˙ i (t) − hi + h2i x x ¯˙ Ti (α) Zi x ¯˙ i (α) dα.
(33)
(34)
t−hi
Based on Lemma A1 given in the Appendix (Jensen’s inequality), it holds that Z t T Z t Z t x ¯˙ i (α) dα −hi x ¯˙ i (α) Zi x ¯˙ i (α) dα ≤ − x ¯˙ i (α) dα Zi t−hi
t−hi
M
t−hi
T
= − [¯ xi (t) − x ¯i (t − hi )] Zi [¯ xi (t) − x ¯i (t − hi )] .
(35)
0=
N X
2χTi
i=1
PT
(t) Gi Ai (µi , µ ˆi , µ ¯i )χi (t) +
where x ¯, y¯ ∈
Rn
N X i=1
2χTi
(t) Gi R3
T (t) w T (t) w T (t) x ¯˙ Ti (t) x ¯Ti (t − hi ) wyi ¯Ti (t) x ui di
CE
where χi (t) = in (33). Note that
ED
Define the matrix multipliers Gi ∈ <(8nxi +nyi +nui )×2nxi , i ∈ N , and it follows from (20) that
and the scalar κ > 0.
AC
inequality (37), one has
i=1
2χTi
(t) Gi R3
N X
F¯ij (µi )xj (t),
(36)
j=1 j6=i
T
, and Ai (µi , µ ˆi , µ ¯i ) is defined
2¯ xT y¯ ≤ κ−1 x ¯T x ¯ + κ¯ y T y¯,
Define the scalar parameter 0 < εij (µi ) ≤ ε0 , where εij (µi ) :=
N X
N X
(37) ri P
l=1
µli εlij , i ∈ N , and by using the
F¯ij (µi )xj (t)
j=1 j6=i
≤
N X N X
T T ¯T χTi (t) Gi R3 F¯ij (µi )ε−1 ij (µi )Fij (µi )R3 Gi χi (t) +
≤
N X N X
T T ¯T χTi (t) Gi R3 F¯ij (µi )ε−1 ij (µi )Fij (µi )R3 Gi χi (t) + ε0 (N − 1)
i=1 j=1 j6=i
i=1 j=1 j6=i
9
N X N X
εij (µi )xTj (t)xj (t)
i=1 j=1 j6=i
N X i=1
xTi (t)xi (t).
(38)
,
ACCEPTED MANUSCRIPT
Define systematic positive matrices {Hi1 , Hi2 , Hi3 } , i ∈ N , and given the following index: J(t) =
=
N X
Ji (t) i=1 N Z ∞ X i=1
0
T T T ξyi (t) Hi1 ξyi (t) − wyi (t) Hi1 wyi (t) + ξui (t) Hi2 ξui (t)
T T T −wui (t) Hi2 wui (t) + ξdi (t) Hi3 ξdi (t) − wdi (t) Hi3 wdi (t) dt.
(39)
CR IP T
When considering zero initial conditions, it is easy to see that V (0) = 0 and V (∞) ≥ 0. Then, it yields from (34)-(39) that J(t) ≤ J(t) + V (∞) − V (0) N Z ∞n X T T T = V˙ i (t) + ξyi (t) Hi1 ξyi (t) − wyi (t) Hi1 wyi (t) + ξui (t) Hi2 ξui (t) ≤
0
N Z X i=1
T T T −wui (t) Hi2 wui (t) + ξdi (t) Hi3 ξdi (t) − wdi (t) Hi3 wdi (t) dt
∞
χTi (t) Σi (µi , µ ˆi , µ ¯i )χi (t) dt,
0
AN US
i=1
with
(40)
2 T 2 T Σi (µi , µ ˆi , µ ¯i ) = Θi + σyi Ω1 (¯ µi )Hi1 Ω1 (¯ µi ) + σui Ω2 Hi2 Ω2
M
+ Sym {Gi Ai (µi , µ ˆi , µ ¯i )} +
N X j=1 j6=i
T T ¯ ¯T ε−1 ij (µi )Gi R3 Fij (µi )Fij (µi )R3 Gi ,
(41)
3.3
CE
PT
ED
where all notations have been defined in (33). By using Schur complement to (32), it is straightforward to get that Σi (µi , µ ˆi , µ ¯i ) < 0, which implies V˙ (t) < 0 and J(t) < 0. By using the SSG theorem (Lemma A2 in the Appendix), the stability of system (18) can be ensured, thus completing this proof. It is noticed that the conditions given in (31) are nonlinear matrix inequalities, which brings the difficulty of calculating the numerical solution. In the followings, by virtue of some matrix inequality convexification technique, the co-design results simultaneously including the controller gains, eventtriggered parameters, and the sampling period will be provided in terms of LMIs.
Co-Design
AC
In this paper, it supposes that the matrix Bi is of full column rank, thus one gets [43] Inui Ti Bi = , 0(nxi −nui )×nui where Ti ∈ Rnxi ×nxi , i ∈ N , are nonsingular matrices. Define Gi11 Gi12 Gi1 = Ti , 0 Gi13
where Gi11 ∈ Rnui ×nui , Gi12 ∈ Rnui ×n(xi−ui) , Gi13 ∈ Rn(xi−ui) ×n(xi−ui) .
10
(42)
(43)
ACCEPTED MANUSCRIPT
Based on the expressions in (42) and (43), the coupling term Gi1 Bi Ki can be transformed into the following LMIs condition, Gi11 Gi12 Gi1 Bi Ki = Ti Bi Ki 0 Gi13 Gi11 Ki = 0(nxi −nui )×nxi ¯i K = 0(nxi −nui )×nxi = Ki .
CR IP T
(44)
It should be pointed out that when the knowledge between µli and µ ˆli are unavailable, the condition µli 6= µ ˆli degrades into a linear controller instead of a fuzzy one, which produces the design conservatism. Currently, assuming that the knowledge between µli and µ ˆli are available, that is ρli ≤
µ ˆli ≤ ρ¯li , l µi
(45)
AN US
where ρli and ρ¯li are positive constant scalars. Combining the Theorem 1 with the assumption in (45), the co-design containing the fuzzy controller gains, the sampled period, and the event-triggered parameters, is given as follows.
ED
M
Theorem 2. Given the large-scale fuzzy system in (3). A decentralized event-triggering observerbased controller given by (7) and (9) exists, such that the stability of the control system in (18) is validated, if there exist positive-definite symmetric matrices {Pi , Zi , Mi } ∈ R2nxi ×2nxi , symmetric T ,L ¯ s ∈ Rnxi ×nyi , K ¯ i ∈ Rnui ×nxi , matrices Qi ∈ R2nxi ×2nxi , matrices {Gi1 , Gi2 } ∈ Rnxi ×nxi , Xils = Xisl i and positive scalars {ε0 , i1 , i2 , hi , σi , εijl } , εisl ≤ ε0 , such that for all (l, s, f ) ∈ Li , i ∈ N the following inequalities hold: 1 −Zi hi Pi + Zi > 0, (46) ? Qi + Zi ll ρ¯li Σllf i + Xi < 0,
AC
CE
PT
ll ρli Σllf i + Xi ls sl ρ¯si Σlsf ¯li Σslf i +ρ i + Xi + Xi l slf ls sl ρsi Σlsf i + ρi Σi + Xi + Xi ls sl ρsi Σlsf ¯li Σslf i +ρ i + Xi + Xi l slf ls sl ρ¯si Σlsf i + ρi Σi + Xi + Xi 11 Xi · · · Xi1ri
where
Σlsf = i
Hi2 − Sym {Gi11 } ? ?
.. .
Xiri 1
..
. ···
.. .
Xiri ri
(47)
< 0,
(48)
< 0,
(49)
< 0,
(50)
< 0,
(51)
< 0,
(52)
> 0,
(53)
¯2 0 σui Ω h iT h i 2 Ωf Θi + σyi Hi1 Ωf1 + Sym Πls Gi R3 F¯ijl , i 1 ?
and
11
−Eijl
(54)
ACCEPTED MANUSCRIPT
2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Q − Z 0 0 0 i i , Θi = ? ? ? −H 0 0 i1 ? ? ? ? −Hi2 0 ? ? ? ? ?i −Hi3 h f f f f h 1 1 Ω1 = 0 2 Ci 0 2 Ci 0 0 0 2i Ci 0 , Ω 1 ¯ ¯ 2= 0 1 K ¯ −1K ¯ ¯ i 0 0 hi K ¯ i − hi K ¯i , − 21 K 2 i 2 i 2 Ki 2 2 Θi(1) = Qi − Zi + ε0 (N − 1) R1 R1T , T T T R1 = I 0 , R2 = 0 I , R3 = I I , T T T Gi = hGi Gi 02nxi ×(4nxii+nyi +nui ) , l l l l ¯ ¯ ¯ ¯ Fij = Fi1 · · · Fij,j6=i · · · FiN , {z } | N −1 n o l l l = diag εl I , E · · · ε I · · · ε I n n n i1 iN ij,j6 = i xi xi xi ij {z } | 0 = −Gi1 0 0
1 2 Ki 1 ¯s f − 2 Li Ci 1 2 Ki 1 ¯s f − 2 Li Ci
0
CR IP T
Gi1 Ali + 21 K i l s s ¯ Gi2 Ai − Gi2 Ai + Li Cis − 12 Cif
0 −Gi2
Gi1 Ali + 21 K i ¯s Cs − 1 Cf Gi2 Ali − Gi2 Asi + L i i 2 i
0
−Gi2 0
0
− 21 Ki 0 1 − 2 Ki 0 0
M
−Gi1
0 ¯s −L i
0 ¯s −L i 0
ED
Πlsf i
AN US
N −1
and
(55)
Gi1 Bi 0 Gi1 Bi 0 0
hi 2 Ki hi ¯ s f − 2 Li Ci hi 2 Ki hi ¯ s f − 2 Li Ci
− 12 Ki ¯ sC s Gi2 As − L
− h2i Ki 0 hi − 2 Ki 0 0
0
i
i
i
− 12 Ki ¯ sC s Gi2 As − L i
i
i
0
.
(56)
PT
Moreover, the decentralized event-triggering observer and controller gains are given by −1 ¯ ¯ Ki = G−1 i11 Ki , Lis = Gi2 Lis .
(57)
AC
CE
Proof. By applying Schur complement to (32), we have −1 −Hi2 σui Ω2 0 2 ΩT (¯ ? Θi + σyi µi ) + Sym {Gi Ai (µi , µ ˆi , µ ¯i )} Gi R3 F¯ij (µi ) < 0. 1 µi ) Hi1 Ω1 (¯ ? ? −Eij (µi )
Define Wi = diag Gi11 I6nxi +nyi +nui I(N −1)nxi , and perform a congruence transformation to (57) by Wi , one has −1 T ¯2 −Gi11 Hi2 Gi11 σui Ω 0 2 ΩT (¯ ? Θi + σyi µi ) + Sym {Gi Ai (µi , µ ¯i , µ ˆi )} Gi R3 F¯ij (µi ) < 0, (58) 1 µi ) Hi1 Ω1 (¯ ? ? −Eij (µi ) 1 ¯ ¯ 2= 0 1 K ¯ −1K ¯ ¯ i 0 0 hi K ¯ i − hi K ¯i . where Ω − 21 K 2 i 2 i 2 Ki 2 2 For simplification of controller design, the matrix multipliers Gi can be directly designated as Gi =
GTi
GTi
02nxi ×(4nxi +nyi +nui ) 12
T
,i ∈ N
(59)
ACCEPTED MANUSCRIPT
where Gi = diag{Gi1 , Gi2 }, and {Gi1 , Gi2 } ∈ Rnxi ×nxi are nonsingular matrices. Note that −1 −GTi11 Hi2 Gi11 ≤ Hi2 − GTi11 − Gi11 .
(60)
By substituting (59) into (58) and extracting the fuzzy basis functions, together with the inequality in (60), one gets ri X ri X ri X µli µ ˆsi µ ¯fi Σlsf < 0, (61) i i=1 s=1 f =1
CR IP T
which implies (58), where Σlsf is defined in (54). i In terms of the assumption in (45), and by using Lemma A2, the inequalities (46)-(53) can be derived, thus completing this proof. Besides, as a special case, i.e., the fuzzy system in (3) and the event-triggering observer-based controller in (7) and (9) share the same premise variables. It maybe exists when the measurable premise variables transmitting to the controller are not closed through the communication networks. In this case, the corresponding LMI-based co-design results are given as below.
M
i
AN US
Theorem 3. Given the large-scale fuzzy system in (3). Then, a decentralized event-triggering observer-based controller given by (7) with the premise variables fi (xi (t)) and (9) exists, such that the stability of the control system in (18) is validated, if there exist positive-definite symmetric matrices {Pi , Zi , Mi } ∈ Rn2nxi ×2nxi , symmetricomatrices Qi ∈ R2nxi ×2nxi , matrices {Gi1 , Gi2 } ∈ Rnxi ×nxi , and positive scalars ε0 , i1 , i2 , hi , σi , εlij , εlij ≤ ε0 , such that for all (l, s, f ) ∈ Li , i ∈ N , the following inequalities hold: 1 −Zi hi Pi + Zi > 0, (62) ? Qi + Zi ¯ llf < 0, 1 ≤ l ≤ ri Σ (63)
¯ lsf = Σ i
Hi2 − Sym {Gi11 } ? ?
h
2 Ωf Θi + σyi 1
iT
(64)
¯ σui Ω 0 n o h 2i ¯ lsf Gi R3 F¯ijl , Hi1 Ωf1 + Sym Π i ?
(65)
−Eijl
2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Q − Z 0 0 0 i i , Θi = ? ? ? −H 0 0 i1 ? ? ? ? −Hi2 0 ? ? ? ? ?i −Hi3 h f f f f h 1 1 Ω1 = 0 2 Ci 0 2 Ci 0 0 0 2i Ci 0 , Ω 1 ¯ ¯ 2= 0 1 K ¯ −1K ¯ ¯ i 0 0 hi K ¯ i − hi K ¯i , − 21 K 2 i 2 i 2 Ki 2 2 Θi(1) = Qi − Zi + ε0 (N − 1) R1 R1T , T T T R1 = I 0 , R2 = 0 I , R3 = I I , T T T Gi = hGi Gi 02nxi ×(4nxii+nyi +nui ) , l l l l ¯ ¯ ¯ ¯ Fij = Fi1 · · · Fij,j6=i · · · FiN , | {z } N −1 n o l l l l E = diag , ε I · · · ε I · · · ε I n n n i1 iN ij,j6 = i xi xi xi ij | {z }
AC
CE
and
PT
where
ED
¯ lsf + Σ ¯ slf < 0, 1 ≤ l < s ≤ ri Σ i i
N −1
13
(66)
ACCEPTED MANUSCRIPT
Πlsf i
−Gi1
0 = −Gi1 0 0
Gi1 Ali + 12 Ki ¯s Cs − 1 Cf L i i 2 i
0 −Gi2
Gi1 Ali + 21 Ki ¯s Cs − 1 Cf L i i 2 i
0
−Gi2
1 2 Ki ¯ sC f − 21 L i i 1 K 2 i ¯ sC f − 21 L i i
0
0
0
− 21 Ki
0 ¯s −L i 0 ¯s −L i 0
0 − 21 Ki 0 0
Gi1 Bi 0 Gi1 Bi 0 0
− 12 Ki ¯ sC s Gi2 As − L i
i i 1 − 2 Ki ¯ sC s Gi2 Asi − L i i
0
hi 2 Ki hi ¯ s f − 2 Li Ci hi 2 Ki hi ¯ s f − 2 Li Ci
0
− h2i Ki 0 − h2i Ki 0 0
.
CR IP T
and
(67)
Furthermore, the decentralized event-triggered observer and controller gains are given by −1 ¯ ¯ Ki = G−1 i11 Ki , Lis = Gi2 Lis .
(68)
AN US
When considering the unknown information with µli 6= µ ˆli , the corresponding LMI-based result can be summarized in the following theorem.
? ?
¯ σ Ω 0 h iT uih 2 i n o 2 Ωf ¯ lf Hi1 Ωf1 + Sym Π Gi R3 F¯ijl , Θi + σyi 1 i
ED
¯ lf = Σ i
Hi2 − Sym {Gi11 }
i
?
(71)
−Eijl
2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Q − Z 0 0 0 i i , Θi = ? ? ? −H 0 0 i1 ? ? ? ? −Hi2 0 ? ? ? ? ? −H i i3 h f f f f h 1 1 i Ω = , 0 C 0 C 0 0 0 C 0 1 2 i 2 i 2 i h h Ω 1 1 1 1 i ¯ ¯ 2= 0 ¯ − K ¯ ¯ − K ¯ 0 0 ¯i , − 2i K 2 Ki 2 i 2 Ki 2 i 2 Ki Θi(1) = Qi − Zi + ε0 (N − 1) R1 R1T , T T T R1 = I 0 , R2 = 0 I , R3 = I I , T T T Gi = hGi Gi 02nxi ×(4nxii+nyi +nui ) , l l l l ¯ ¯ ¯ ¯ Fij = Fi1 · · · Fij,j6=i · · · FiN , | {z } N −1 n o l l l = diag εl I E , · · · ε I · · · ε I n n n i1 iN ij,j6 = i xi xi xi ij | {z }
AC
CE
and
PT
where
M
Theorem 4. Consider the large-scale fuzzy system in (3). Then, a decentralized event-triggered observer-based linear controller given by (7) and (9) exists, such that the stability of the control system in (18) is validated, if there exist positive-definite symmetric matrices {Pi , Zi , Mi } ∈ R2nxi ×2nxi , symmetric matrices Qi ∈ R2nxi ×2nxi , matrices {Gi1 , Gi2 } ∈ Rnxi ×nxi , and positive scalars {ε0 , i1 , i2 , hi , σi , εlij }, εlij ≤ ε0 , such that for all (l, f ) ∈ Li , i ∈ N , the following inequalities hold: 1 −Zi hi Pi + Zi > 0, (69) ? Qi + Zi ¯ lf < 0, Σ (70)
N −1
14
(72)
ACCEPTED MANUSCRIPT
and −Gi1
Gi1 Ali + 21 Ki ¯i Cl − 1 Cf L
0
0 = −Gi1 0 0
−Gi2
i
2 i l Gi1 Ai + 12 Ki ¯i Cl − 1 Cf L i 2 i
0
−Gi2 0
0
− 21 Ki
1 2 Ki 1¯ − 2 Li Cif 1 2 Ki 1¯ − 2 Li Cif
0 − 21 Ki 0 0
0
0 ¯i −L 0 ¯i −L 0
Gi1 Bi 0 Gi1 Bi 0 0
− 12 Ki ¯ iC l Gi2 Ai − L i
− 12 Ki
¯ iC l Gi2 Ai − L i 0
hi 2 Ki hi ¯ − 2 Li Cif hi 2 Ki hi ¯ − 2 Li Cif
0
− h2i Ki 0 − h2i Ki 0 0
.
CR IP T
Πlf i
(73)
Furthermore, the decentralized event-triggering linear observer and controller gains are given by
4
Numerical Example
(74)
AN US
−1 ¯ ¯ Ki = G−1 i11 Ki , Li = Gi2 Li .
For validating the proposed method, we consider the large-scale fuzzy system with two interconnected subsystems as below: l , THEN Plant Rule Ril : IF xi1 (t) is Fi1 N P x˙ i (t) = Al xi (t) + Bi u ˆi (ti ) + F¯ l xj (t) i
k
j=1 j6=i
ij
M
y (t) = C x (t), l = {1, 2} , i = {1, 2} i i i
0.6 1 1.1 , , A21 = 0 −1.6 −1 0.8 0 1.2 0 2 1 ¯ ¯ , , F12 = F12 = 0 0 0 0 0.5 B1 = , C1 = 1 0 , 0
ED
where
0.4 0
CE
PT
A11 =
AC
for the first subsystem, and
−1.1 0.6 −1 2 = , A2 = , −0.5 0.9 −0.7 0.01 0 0.02 0 1 2 ¯ ¯ F21 = , F21 = , 0 0 0 0 0.4 B2 = , C2 = 1 0 , 0 A12
0.4 1
for the second subsystem. Here, we assume that σy1 = 0.17, σu1 = 0.11, σy2 = 0.17, σu2 = 0.19, and hi = 0.01. The normalized membership functions are given in Figure 2. For the design results given in Theorem 2, it is clear to see that the transformation matrices 2 0 2.5 0 T1 = , T2 = . 0 1 0 1 15
ACCEPTED MANUSCRIPT
1 0.8 0.6 0.4 0.2 0 −ri1
0
|xi1 (t)|
Figure 2: Membership functions
CR IP T
rule i1 rule i2
ri1
AN US
Firstly, it can be checked that the method proposed in [14] fails to find a feasible solution. By Theorem 3, however, a feasible solution exists, and the corresponding controller gains are 9.1848 8.2148 L11 = , L12 = , K1 = −9.0822 −0.4312 , 0.0004 −0.0001 8.3478 8.8277 , K2 = −12.2723 −0.4519 . , L22 = L21 = 0.0005 0.0001
PT
ED
M
Based on the above solutions, and given the initial conditions x1 (0) = [1.1, 0]T and x2 (0) = [1.3, 0]T , Figures 3 and 4 show the responses of system states and their estimations tend to zero. Figure 5 implies that the number of data transmissions decreases from 100 to 31 in the S-C channel of the subsystem 1, and from 100 to 41 in the C-A channel of the subsystem 1. Figure 6 shows that the number of data transmissions decreases from 100 to 30 in the S-C channel of the subsystem 2, and from 100 to 32 in the C-A channel of the subsystem 2. Thus validating the effectiveness and availability of the developed method.
AC
CE
1.5
x11 x12 xc11 xc12
1
0.5 0
−0.5 −1 0
1
2 3 4 5 Time in seconds Figure 3: Responses of system states and estimations for subsystem 1
16
ACCEPTED MANUSCRIPT
1.5 x21 x22 xc21 xc22
1 0.5
CR IP T
0 −0.5 −1 0
2 3 4 5 Time in seconds Figure 4: Responses of system states and estimations for subsystem 2
AN US
1
ED
M
σy1 = 0.17 σu1 = 0.11
20
40 60 80 100 Time in samples Figure 5: Data transmissions in subsystem 1
CE
PT
0
AC
σy2 = 0.17 σu2 = 0.19
0
20
40 60 80 100 Time in samples Figure 6: Data transmissions in subsystem 2
17
ACCEPTED MANUSCRIPT
5
Conclusions
Appendix
CR IP T
This paper studied the decentralized output-feedback control problem for large-scale networked fuzzy systems in the event-triggering framework. A control scheme with the two-channel event-triggering was proposed. By using a novel model transformation, and introducing an LKF without the requirement of positive definition for all Lyapunov matrices, and combining with the virtue of SSG theorem, the co-design with less conservatism was derived in the form of LMIs to simultaneously obtain the controller gains, event-triggered parameters, and the sampled period. The simulation results show that the proposed controller guarantees the stability of the resulting control system while reducing data communications in both the S-C channel and the C-A channel.
Lemma A1. [44] For any positive-definite symmetric matrix M ∈ Rn×n , and scalars d2 > d1 ≥ 0, the following inequality holds d2
x (t) dt
d1
T
M
Z
d2
d1
Z x (t) dt ≤ (d2 − d1 )
d2
d1
xT (t) M x (t) dt.
AN US
Z
Lemma A2. [39] Assume that the membership functions satisfy ρl ≤ µµˆll ≤ ρ¯l , l ∈ L := {1, 2, . . . , r} . r r P P T , such that for all (l, s) ∈ L , the µl µ ˆs Σls < 0 holds if there exist matrices Xls = Xsl Then, l=1 s=1
following inequalities hold
ρ¯l Σll + Xll < 0,
ρl Σll + Xll < 0,
M
ρ¯s Σls + ρ¯l Σsl + Xls + Xsl < 0,
ρs Σls + ρl Σsl + Xls + Xsl < 0,
ED
ρs Σls + ρ¯l Σsl + Xls + Xsl < 0,
PT
ρ¯s Σls + ρl Σsl + Xls + Xsl < 0, X11 · · · X1r .. .. > 0. .. . . . Xr1 · · ·
Xrr
CE
Acknowledgment
AC
The authors are grateful to the Editor-in-Chief, Senior Editor Prof. Michael V. Basin, Associate Editor, and anonymous reviewers for their insightful comments, which helped to improve the quality of this work.
References
[1] G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 676-697, 2006. [2] C. Han, L. Wu, P. Shi, and Q. Zeng, “On dissipativity of Takagi-Sugeno fuzzy descriptor systems with time-delay,” Journal of the Franklin Institute, vol. 349, no. 10, pp. 3170-3184, 2012. [3] L. Zhang, T. Yang, P. Shi, and M. Liu, “Stability and stabilization of a class of discrete-time fuzzy systems with semi-Markov stochastic uncertainties,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2016, DOI: 10.1109/TSMC.2016.2527627. 18
ACCEPTED MANUSCRIPT
[4] Y. Liu and S. Lee, “Stability and stabilization of Takagi-Sugeno fuzzy systems via sampleddata and state quantized controller,” IEEE Transactions on Fuzzy Systems, vol. 24, no. 3, pp. 635-644, 2016. [5] H. Shen, J. Park, and Z. Wu, “Finite-time reliable L2 -L∞ /H∞ control for Takagi-Sugeno fuzzy systems with actuator faults,” IET Control Theory & Applications, vol. 8, no. 9, pp. 688-696, 2014.
CR IP T
[6] H. Shen, J. Park, and Z. Wu, “Extended dissipative state estimation for Markov jump neural networks with unreliable links,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 2, pp. 346-358, 2017. [7] J. Qiu, G. Feng, and H. Gao, “Nonsynchronized-state estimation of multichannel networked nonlinear systems with multiple packet dropouts via T-S fuzzy-affine dynamic models,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 1, pp. 75-90, 2011. [8] Y. Liu, J. Park, B. Guo, and Y. Shu, “Further results on stabilization of Chaotic systems based on fuzzy memory sampled-data control,” IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2017.2686364.
AN US
[9] L. Zhang, Z. Ning, and P. Shi, “Input-output approach to control for fuzzy Markov jump systems with time-varying delays and uncertain packet dropout rate,” IEEE Transactions on Cybernetics, vol. 45, no. 11, pp. 2449-2460, 2015. [10] Y. Liu, B. Guo, and J. Park, “Non-fragile H∞ filtering for delayed Takagi-Sugeno fuzzy systems with randomly occurring gain variations,”Fuzzy Sets and Systems, vol. 316, pp. 99-116, 2017.
M
[11] H. Zhang, Y. Shi, and A. Mehr, “On filtering for discrete-time Takagi-Sugeno fuzzy systems,”IEEE Transactions on Fuzzy Systems, vol. 20, no. 2, pp. 396-401, 2012.
ED
[12] L. Bakule, “Decentralized control: Status and outlook,” Annual Reviews in Control, vol. 38, no. 1, pp. 71-80, 2014. [13] C. Hua and S. Ding “Decentralized networked control system design using T-S fuzzy approach,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 1, pp. 9-21, 2012.
CE
PT
[14] H. Zhang, G. Yu, C. Zhou, and C. Dang, “Delay-dependent decentralized H∞ filtering for fuzzy interconnected systems with time-varying delay based on Takagi-Sugeno fuzzy model,” IET Control and Applications, vol. 7, no. 5, pp. 720-729, 2013.
AC
[15] Z. Zhong, S. Fu, T. Hayat, F. Alsaadi, and G. Sun, “Decentralized piecewise H∞ fuzzy filtering design for discrete-time large-scale nonlinear systems with time-varying delay,” Journal of the Franklin Institute, vol. 352, no. 9, pp. 3782-3807, 2015. [16] H. Zhang, C. Dang, and J. Zhang, “Decentralized fuzzy H∞ filtering for nonlinear interconnected systems with multiple time delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 40, no. 4, pp. 1197-1203, 2010. [17] X. He, Z. Wang, X. Wang, and D. Zhou, “Networked strong tracking filtering with multiple packet dropouts: algorithms and applications” IEEE Transactions on Industrial Electronics, vol. 61, no. 3, pp. 1454-1463, 2014. [18] Y. Xu, R. Lu, H. Peng, K. Xie, and A. Xue, “Asynchronous dissipative state estimation for stochastic complex networks with quantized jumping coupling and uncertain measurements” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 2, pp. 268-277, 2017.
19
ACCEPTED MANUSCRIPT
[19] X. He, Z. Wang, L. Qin, and D. Zhou, “Active fault tolerant control for an Internet-based networked three-tank system” IEEE Transactions on Control Systems Technology, vol. 24, no. 6, pp. 2150-2157, 2016. [20] Y. Shi and B. Yu, “Output feedback stabilization of networked control systems with random delays modeled by Markov chains” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1668-1674, 2009.
CR IP T
[21] Y. Xu, R. Lu, H. Peng, and S. Xie, “Filtering for fuzzy systems with multiplicative sensor noises and multi-density quantizer” IEEE transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2017.2702119. [22] X. He, Z. Wang, Y. Liu, D. Zhou, “Least-squares fault detection and diagnosis for networked sensing systems using direct state estimation approach” IEEE Transactions on Industrial Informatics, vol. 9, no. 3, pp. 1670-1679, 2013. [23] L. Zhang, Y. Zhu, Z. Ning, and X. Yin, “Resilient estimation for networked systems with variable communication capability,” IEEE Transactions on Automatic Control, vol. 61, no. 12, pp. 41504156, 2016.
AN US
[24] Q. Lu, L. Zhang, M. Basin, and H. Tian, “Analysis and synthesis for networked control systems with uncertain rate of packet losses,” Journal of the Franklin Institute, vol. 349, no. 7, pp. 2500-2514, 2012. [25] R. Yang, G. Liu, P. Shi, C. Thomas, and M. Basin, “Predictive output feedback control for networked control systems,” IEEE Transactions on Industrial Electronics, vol. 61, no. 1, pp. 513-520, 2014.
M
[26] W. Heemels, A. Teel, N. Wouw, and D. Neˇsi´c, “Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1781-1796, 2010.
ED
[27] W. Heemels and M. Donkers, “Model-based periodic event-triggered control for linear systems,” Automatica, vol. 49, no. 3, pp. 698-711, 2013.
PT
[28] M. Donkers and W. Heemels, “Output-based event-triggered control with guaranteed L∞ -gain and improved and decentralized event-triggering,” IEEE Transactions on Automatic Control, vol. 57, no. 6, pp. 1362-1376, 2012.
CE
[29] D. Yue, E. Tian, and Q. Han, “A delay system method for designing event-triggered controllers of networked control systems,” IEEE Transactions on Automatic Control, vol. 58, no. 2, pp. 475-481, 2013.
AC
[30] D. Hristu-Varsakelis and P. Kumar, “Interrupt-based feedback control over a shared communication medium,” In Proceedings of the 2002 IEEE Conference on Decision Control, pp. 3223-3228, 2002. [31] X. Wang and M. Lemmon, “Event-triggering in distributed networked control systems,”IEEE Transactions on Automatic Control, vol. 56, no. 3, pp. 586-601, 2011. [32] P. Tabuada and X. Wang, “Preliminary results on state-triggered scheduling of stabilizing control tasks,” In Proceedings of the 2006 IEEE Conference on Decision Control, pp. 282–287, 2006. [33] J. Cheng, J. Park, H. Wang, “Event-triggered H∞ control for T-S fuzzy nonlinear systems and its application to truck-trailer system” ISA Transactions, vol. 65, pp. 62-71, 2016.
20
ACCEPTED MANUSCRIPT
[34] J. Cheng, J. Park, L. Zhang, and Y. Zhu, “An asynchronous operation approach to eventtriggered control for fuzzy Markovian Jump systems with general switching policies,” IEEE Transactions on Fuzzy systems, DOI 10.1109/TFUZZ.2016.2633325. [35] C. Peng and Q. Han, “To transmit or not to transmit: a discrete event-triggered communication scheme for networked Takagi–Sugeno fuzzy systems,” IEEE Transactions on Fuzzy systems, vol. 21, no. 1, pp. 164-170, 2013.
CR IP T
[36] X. Jia, X. Chi, Q. Han, and N. Zheng, “Event-triggered fuzzy H∞ control for a class of nonlinear networked control systems using the deviation bounds of asynchronous normalized membership functions,”Information Sciences, vol. 259, no. 20, pp. 100-117, 2014. [37] D. He, X. Jia, X. Chi, and W. Ma, “Fuzzy H∞ tracking control for a class of nonlinear networked control systems with a discrete event-triggered communication scheme,” In Proceedings of the 2013 IEEE Conference on Control and Decision, pp. 2562-2567, 2013. [38] Y. Guan, Q. Han, and C. Peng, “Event-triggered output feedback control for Takagi-Sugeno fuzzy systems,” In Proceedings of the 2013 IEEE Conference on Industrial Electronics Society, pp. 5644-5649, 2013.
AN US
[39] C. Ario and A. Sala, “Extensions to “stability analysis of fuzzy control systems subject to uncertain grades of membership”,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, no. 2, pp. 558-563, 2008. [40] E. Fridman, “A refined input delay approach to sampled-data control,” Automatica, vol. 46, no. 2, pp. 421-427, 2010.
M
[41] K. Gu, Y. Zhang, and S. Xu, “Small gain problem in coupled differential-difference equations, time-varying delays, and direct Lyapunov method,” International Journal of Robust and Nonlinear Control, vol. 21, no. 4, pp. 429-451, 2011.
ED
[42] S. Xu, J. Lam, B. Zhang, and Y. Zou, “New insight into delay-dependent stability of time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 25, no. 7, pp. 961-970, 2015.
PT
[43] J. Qiu, G. Feng, and H. Gao, “Static output feedback H∞ control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 2, pp. 245-261, 2013.
AC
CE
[44] K. Gu, “An integral inequality in the stability problem of time-delay systems,” In Proceedings of the 2000 IEEE Conference on Decision and Control, pp. 2805-2810, 2000.
21
Observer-Based Output-Feedback Control of Large-Scale Networked Fuzzy Systems with Two-Channel Event-Triggering Zhixiong Zhong, Yanzheng Zhu PII: DOI: Reference:
S0016-0032(17)30271-5 10.1016/j.jfranklin.2017.05.036 FI 3010
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
8 February 2017 26 April 2017 23 May 2017
Please cite this article as: Zhixiong Zhong, Yanzheng Zhu, Observer-Based Output-Feedback Control of Large-Scale Networked Fuzzy Systems with Two-Channel Event-Triggering, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.036
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Observer-Based Output-Feedback Control of Large-Scale Networked Fuzzy Systems with Two-Channel Event-Triggering∗
Abstract
CR IP T
Zhixiong Zhong† and Yanzheng Zhu‡
AN US
This paper concerns data transmissions for large-scale T-S fuzzy systems with event-triggering control, where each subsystem communicates its information via a two-channel network. We propose an event-triggering scheme in which two event-triggering mechanisms are used to verify the data transmissions. At first, a novel model transformation is presented, where the event-triggered control system is reconstructed as a constant-delay system with extra inputs and outputs. By using a relaxed Lyapunov-Krasovskii functional (LKF) without the requirement of positive definiteness for all Lyapunov matrices, and the scaled small gain (SSG) theorem, the co-design problem of desired observer and controller gains, event-triggering parameters, and the sampling period is resolved in the form of linear matrix inequalities (LMIs). It will be shown that the solution guarantees the stability of closed-loop fuzzy control system and the reductions of data communications in both the sensor-to-controller and controller-to-actuator channels. The proposed method is validated by using a numerical example.
Introduction
ED
1
M
Keywords: Large-scale fuzzy systems, two-channel event-triggering, observer-based outputfeedback, co-design.
AC
CE
PT
In real-world applications, it is not very easy to deal with the control problem of nonlinear systems because of strong nonlinearities contained in the plants. In recent decades, Takagi-Sugeno (T-S) model-based method has been validated that it can overcome the difficulties induced by the nonlinearities [1]. With the efforts of academic and engineering researchers, a great number of topics have been focused upon in latest years for T-S models, such as stability and performance analysis [2, 3, 4, 5], controller design [7, 8, 9], and filter design [10, 11]. For a given nonlinear system, the T-S model can be used to approximate it at any preciseness, and the mature linear control methods can be used to resolve the control problem of the T-S model. In addition, the control of large-scale systems refers to many practical applications including transportation networks, industrial production, power grids, communication systems, and so on. However, its control also brings great challenges because of the excessively processing data, different locations, as well as interconnections among subsystems ∗ The work described in this paper was supported in part by the Natural Science Foundation of Fujian Province, China under Grants 2017J01781; the National Natural Science Foundation of China (61603221); the Natural Science Foundation of Shandong Province (ZR2016FB11); the China Postdoctoral Science Foundation (2017M610437); the Special Foundation for Postdoctoral Science Foundation of Shandong Province (201601014); the Applied Research Project for Postdoctoral Researchers of Qingdao (2016109); Research Fund for the Taishan Scholar Project of Shandong Province of China; the Open Fund Project of Fujian Provincial Key Laboratory of Information Processing and Intelligent Control (Minjiang University) (No. MJUKF201731). † Zhixiong Zhong is with the High-voltage Key Laboratory of Fujian Province, Xiamen University of Technology, Xiamen 361024, China; and Fujian Provincial Key Laboratory of Information Processing and Intelligent Control (Minjiang University), Fuzhou 350121, China. Email: [email protected] ‡ Yanzheng Zhu (corresponding author) is with the College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, China. Email: [email protected]
1
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
[12]. More recently, the decentralization has been proved that it is an effective technique for the control of large-scale systems, and some results on decentralization control of large-scale nonlinear systems have been published using the T-S fuzzy model-based method, see, e.g., [13, 14, 15, 16]. On the other hand, networked control systems (NCSs) have received considerable attractions for their various superiorities, such as light weights, simple installation, simple maintenance, and low cost [17, 18, 19, 20, 21, 22]. However, its intrinsic imperfections, including packet dropouts, quantization errors, and time delays, could weaken its control performance and even may result in instability [23]. Currently, there have been lots of results focusing on these imperfections, see [23, 24, 25], and the references therein. It is noted that a tradeoff among these network-induced constraints always be required when designing an NCS. Specifically, sending larger packets, giving time-stamping of data, reducing quantization errors generally improve the control performance while inducing longer transmission delays [26]. Thus, it is important to develop new methods in the implementation of NCSs to make use of the limited network bandwidth more effectively. In the implementation of feedback control, the traditional policy transmits feedback signals in a periodic manner, which is called time-triggered control. This policy could result in collision or congestion or longer time delays in the network with limited bandwidths. In recent years, an increasing interest has been focused on the event-triggering control that aims at reduction in data transmissions. Its basic policy is to transmit control signals in the light of some prescribed events [27, 28, 29, 30, 31, 32]. In other words, not all feedback signals are necessarily transmitted at each sampling period. The event-based policy has appeared under a variety of aliases, such as event-triggered feedback [27, 28, 29], self-triggering feedback [31], interrupt-based feedback [30], and state-triggered feedback [32]. Moreover, the event-triggering scheme has been applied to the area of T-S fuzzy networked systems [33, 34, 35, 36, 37, 38]. To mention a few, the work in [34] proposed a novel event-triggering scheme for fuzzy Markov jump systems based on general switching policies, where each transiting rate allows to be unknown, or partly known. In [35, 36, 37], some results were obtained on the co-design problem with event-triggered state-feedback control for T-S fuzzy sampled-data systems. In [38], the controller design of fuzzy dynamic-output feedback was studied for discrete-time T-S fuzzy systems under an event-triggering scheme. We are aware of few attempts making on the decentralized event-triggering observer-based control for large-scale networked T-S fuzzy systems, which motivates us for the present work. This paper attempts to deal with the decentralized control problem for large-scale networked fuzzy systems with two-channel event-triggering, where the controller form is of observer-based output-feedback type, and each subsystem exchanges its information via networks. The objective is to design a decentralized event-triggering observer-based output-feedback controller, which guarantees the stability of closed-loop control system and the reduction of communication data in both the sensor-to-controller (S-C) and controller-to-actuator (C-A) channels. To this end, an event-triggering scheme is proposed in which both the measured output and control input are sampled periodically, and two event-triggering mechanisms (ETMs) are established respectively when the sampled measurement output and control input should be transmitted for each subsystem. Firstly, the closed-loop sampled-data control system is reconstructed as a continuous-time system with time-varying delays by using the input delays, then both the delay variations and event-triggered counterparts are modeled as the disturbances. By hauling out these disturbances, the uncertain continuous-time system is reformulated as a constant-delay system with extra inputs and outputs. Thus, based on the novel model, a relaxed LKF without the requirement of positive definition for all Lyapunov matrices is introduced, and combining with the virtue of SSG theorem, the co-design problem of desired observer and controller gains, event-triggering parameters, and the sampling period is reformulated in the form of LMIs. Finally, one numerical example is illustrated to verify the effectiveness of developed methods. The main contribution of this paper is threefold: i ) For purpose of reducing the data transmissions in both the S-C and C-A channels, and lowering the requirement of full state to be available for feedback control, the controller design to decentralized event-triggered observer-based outputfeedback is proposed for the large-scale T-S fuzzy system, and the premise variables between the 2
ACCEPTED MANUSCRIPT
2
CR IP T
plant and the proposed controller are not necessarily considered to be synchronous. ii ) Based on the input-delay and perturbation approaches, a novel model transformation is proposed in the paper. Compared with the method proposed in [14], the results with less conservativeness are derived via a relaxed LKF and the SSG theorem on the design of decentralized observer-based output-feedback controller in the event-triggered scheme. This means that the longer sampling intervals or larger specified threshold can be obtained. iii ) The co-design problem of obtaining observer and controller gains, event-triggering parameters, and the sampling period is resolved in the form of LMIs. Notations. Rn and Rn×m are the Euclidean space with n dimension, and the set with n × m matrices, respectively. P > 0 (≥ 0) denotes P is positive definite (positive semidefinite). Sym{A} is A + AT . In and 0m×n represent the n × n identity matrix and m × n zero matrix, respectively. The subscripts n and m × n are omitted when the size is irrelevant or can be known in the context. A−1 and AT are the inverse and transpose of the matrix A ∈
Problem Formulation
i
i i
i
AN US
This paper aims at a large-scale system which contains N nonlinear subsystems, where the i-th nonlinear subsystem can be represented by T-S fuzzy model in the following: l and f (x (t)) is F l and · · · and f (x (t)) is F l , THEN Plant Rule Ril : IF fi1 (xi (t)) is Fi1 i2 i ig i i2 ig N P x˙ (t) = Al x (t) + B u i F¯ijl xj (t) i i ˆi (tk ) + i i j=1 (1) j6=i y (t) = C l x (t), l ∈ L := {1, 2, . . . , r } , t ∈ [ti , ti ) i
k
k+1
PT
ED
M
where i ∈ N := {1, 2, . . . , N }; Ril is the l-th fuzzy inference rule; ri is the number of inference l (φ = 1, 2, . . . , g) are fuzzy sets; x (t) ∈ Rnxi is the system state; u ˆi (tik ) ∈ Rnui is the rules; Fiφ i control input applied to the system at the instant tik ; yi (t) ∈ Rnyi is the measured output; fi (xi (t)) := [fi1 (xi (t)), fi2 (xi (t)), . . . , fig (xi (t))] are some measurable state variables; (Ali , Bi , Cil ) denotes the lth local model, and the matrix Bi is of full column rank; F¯ijl is the interconnection of the i-th and j-th subsystems for the l-th local model. Q l and the normalized membership function µl [f (x (t))], Define the inferring fuzzy set Fil := gφ=1 Fiφ i i i one has Qg ri l X φ=1 µiφ [fiφ (xi (t))] l µi [fi (xi (t))] := Pri Qg ≥ 0, µli [fi (xi (t))] = 1, (2) ς µ [f (x (t))] i iφ ς=1 φ=1 iφ
CE
l=1
l . This paper will denote µl := where is the grade of membership of fiφ (xi (t)) in Fiφ i µli [fi (xi (t))] for simplify. Based on fuzzy blending, the i-th global T-S fuzzy dynamic model is N P x˙ (t) = A (µ )x (t) + B u i F¯ij (µi )xj (t) i i i i i ˆi (tk ) + j=1 (3) j6=i y (t) = C (µ )x (t), t ∈ [ti , ti ), i ∈ N
AC
µliφ [fiφ (xi (t))]
i
where
Ai (µi ) :=
i
ri X l=1
i
i
k
µli Ali , Ci (µi ) :=
ri X l=1
k+1
µli Cil , F¯ij (µi ) :=
ri X
µli F¯ijl .
(4)
l=1
Remark 1. Note that, a special large-scale fuzzy system with linear interconnection F¯ij has been investigated extensively in [14, 15, 16]. This paper considers the large-scale fuzzy system with nonlinear interconnection F¯ijl as shown in (1), which is more challenging than those with linear interconnection matrix F¯ij . 3
ACCEPTED MANUSCRIPT
Before moving on, we require the assumptions as below: Assumption 1. The sampler is clock-driven in each subsystem. Let hi denote the upper bound of sampling intervals for the i-th subsystem, it has tik+1 − tik ≤ hi , k ∈ N,
(5)
CR IP T
where hi > 0. Assumption 2. For each subsystem, it is assumed that both the S-C and C-A channels are closed via network-based communications. Assumption 3. The zero-order-hold (ZOH) is event-triggered, and it makes use of the latest sampling data and holds them until the next transmitting datum come. In the NCSs, the traditionally time-triggering implementation is undesirable because of the existence of the limited bandwidths. Here, for the purpose of reducing the data transmissions, inspired by the work in [27], the decentralized event-triggering fuzzy observer is proposed as follows: s and f (x (t)) is F s and · · · and f (x (t)) is F s , Plant Rule Ris : IF fi1 (xci (t)) is Fi1 i2 ci ig ci i2 ig THEN x˙ ci (t) = Asi xci (t) + Bi u ˆi tik + Lsi yˆi (tik ) − yci (t) (6) s i yci (t) = Ci xci (t), t ∈ [tk , tik+1 ), i ∈ N
AN US
where xci (t) ∈ Rnxi is the estimated state, yˆi (tik ) is the measured output applied to the observer at the instant tik , and Lsi ∈ Rnxi ×nyi , s ∈ Li , i ∈ N are observer gains to be designed with compatible dimensions. Similarly, the global decentralized event-triggering fuzzy observer can be given by µi ) yˆi (tik ) − yci (t) x˙ ci (t) = Ai (ˆ µi )xci (t) + Bi u ˆi tik + Li (ˆ (7) yci (t) = Ci (ˆ µi )xci (t), t ∈ [tik , tik+1 ), i ∈ N where
(8)
ED
M
ri ri P P s [f (x (t))] As , L (ˆ µ ˆsi [fi (xci (t))] Lsi , µ ˆ µ ) := A (ˆ µ ) := i i i ci i i i i s=1 s=1 ri ri P P µ ˆsi [fi (xci (t))] = 1, µ ˆsi [fi (xci (t))] Cis , Ci (ˆ µi ) := s=1 s=1 Qg µ ˆsiφ [fiφ (xci (t))] µ Qg ≥ 0. ˆsi [fi (xci (t))] := Pri φ=1 ˆςiφ [fiφ (xci (t))] ς=1 φ=1 µ
CE
PT
For simplicity, we define µ ˆsi := µ ˆsi [fci (xci (t))]. Based on the observer in (7), the estimated state xci (t) can be used by the following controller: (9) ui tik = Ki xci (tik ), t ∈ [tik , tik+1 ), i ∈ N ,
AC
where Ki ∈ Rnui ×nxi , i ∈ N are controller gains to be determined with compatible dimensions. For implementing the proposed controller given in (7) and (9), a structure diagram of the presented control system is shown in Figure 1, where the S-C and C-A channels are closed through network-based communications. The sensor system consists of a sampler SPs , a buffer BFs , and an ETMs . The BFs is to store the latest measured output transmitted successfully to the observer yˆi tik , and the ETMs determines whether or not to transmit the sampling output yi (tik ) to the observer. Hence, the measured output yi (t) is firstly sampled by the SPs at each sample period. Then, it is transmitted to the controller and is executed only when one preselected event is violated, which leads to a data reduction in the S-C channel. The controller system consists of a zero-order-hold ZOHc , an observer, a controller, a sampler SPc , a buffer BFc , and an ETMc . The BFc is to store the latest control input transmitted successfully to the plant u ˆi tik , and the ETMc determines whether or not i to transmit the sampling control input ui (tk ) to the actuator system. Thus, ui (tik ) is transmitted and implemented to the plant at each sample instant only when another prescribed event is violated, which leads to a data reduction in the C-A channel.
4
ACCEPTED MANUSCRIPT
Actuator System Actuator
Sensor System
yi (t)
Plant
yˆi (tik )
SPs yi (tik )
ZOHa
ETMs
BFs
Networks BFs u ˆi (tik )
Controller
ETMc ui (tik )
SPs
Observer
ui (t) xci (t)
Controller System
ZOHc
CR IP T
Networks
Figure 1: Observer-based control with two-channel event-triggering
AN US
To formalize the described solution, these two ETMs can operate as
ETMs : yi tik is sent ⇔ yi tik − yˆi tik−1 > σyi yi tik ,
ˆi tik−1 > σui ui tik , ETMc : ui tik is sent ⇔ ui tik − u
(10) (11)
M
where σyi ≥ 0 and σui ≥ 0 are two suitably chosen design parameters. Based on the operation given by (10) and (11), an event-triggering strategy is formulated as below:
> σyi yi ti , yi tik , when yi tik − yˆi tik−1 k
yˆi tik = (12) yˆi tik−1 , when yi tik − yˆi tik−1 ≤ σyi yi tik ,
> σui ui ti , ui tik , when ui tik − u ˆi tik−1 i k
u ˆi tk = (13) u ˆi tik−1 , when ui tik − u ˆi tik−1 ≤ σui ui tik .
AC
CE
PT
ED
Remark 2. As µli = µ ˆli , the decentralized event-triggered fuzzy observer in (7) reduces to a parallel distributed compensation (PDC) one. However, the premise variables between the T-S fuzzy systems in (1) and the event-triggered fuzzy observer in (7) are not always same in the network-based applications due to some limitations such as signal transformation and quantization, packet dropouts, time delays, etc. In other words, it is more realistic to consider the fuzzy observer with the estimated states. Moreover, in this case, the premise variables are no longer required to send via networks, which leads to the reduction in data transmissions. Pri Remark 3. When considering the fuzzy controller ui tik = ˆsi Kis xci (tik ), it is difficult to s=1 µ maintain a synchronous premise variable among the fuzzy system, the fuzzy observer, and the fuzzy controller. For simplification of controller design, the linear controller is proposed in (12) instead of a fuzzy one.
ˆli are unavailable, the asynchronous condition µli 6= µ ˆli Remark 4. As the knowledge on µli and µ generally reduces to a linear controller instead of a fuzzy one, which declines the control performance [39]. However, when the knowledge of µli and µ ˆli is available, a corresponding fuzzy controller can be obtained, and therefore can reduce the design conservatism.
3
Main Results
A new model transformation will be proposed in this section. Then, based on the novel model, a relaxed LKF without the requirement of all positive-definite Lyapunov matrices is introduced. Combined with the SSG technique, the co-design problem of calculating the observer and controller gains, the event-triggering parameters, and the sampling period is resolved in the form of LMIs. 5
ACCEPTED MANUSCRIPT
3.1
Model Reformulation
Based on the input delay method in [40], both the measured output and the control input are reconstructed as the following delayed forms, ei (t) = xi (t) − xci (t) , yˆi (tik ) = yˆi (t − ηi (t)), u ˆi (tik ) = u ˆi (t − ηi (t)),
(14)
0 ≤ ηi (t) < hi , t ∈ [tik , tik+1 ), k ∈ N.
(15)
where ηi (t) = t − tik . It follows from the Assumptions 1-3 that In addition, we model the event-triggered counterparts as [27]
CR IP T
wyi (t) = yˆi (t − ηi (t)) − yi (t − ηi (t)),
wui (t) = u ˆi (t − ηi (t)) − ui (t − ηi (t)).
(16) (17)
Combining with (3), (7), (9), (14), (16) and (17), the augmented closed-loop control system is given by x ¯˙ i (t) = Ai (µi , µ ˆi )¯ xi (t) + Aid (¯ µi , µ ˆi )¯ xi (t − ηi (t)) N X j=1 j6=i
R3 F¯ij (µi )xj (t),
AN US
+ R1 Bi wui (t) − R2 Li (ˆ µi )wyi (t) + where
ED
M
Ai (µi ) 0 , Ai (µi , µ ˆi ) = Ai (µi , µ ˆi ) + Li (ˆ µi )Ci (ˆ µi ) Ai (ˆ µi ) − Li (ˆ µi )Ci(ˆ µi ) xi (t) Bi Ki −Bi Ki , µi , µ ¯i ) = ,x ¯i (t) = Aid (ˆ ei (t) −L (ˆ µ )C (¯ µ ) 0 i i i i I 0 I , R2 = , R3 = , Ai (µi , µ ˆi ) = Ai (µi ) − Ai (ˆ µi ), R1 = 0 I I Q g ri P µ ¯f [fiφ (xi (tik ))] Qg iφ ς ≥ 0, µ ¯fi fi (xi tik ) = 1. ¯fi fi (xi tik ) := Pri φ=1 µ i t ) f (x µ ¯ ( ) ] [ i iφ ς=1 φ=1 iφ k f =1
(18)
(19)
AC
CE
PT
Besides, the delay variations can be modeled as [41] 2 1 wdi (t) = x ¯i (t − ηi (t)) − [¯ xi (t) + x ¯i (t − hi )] , (20) hi 2 " # (1) n o wdi (t) (1) (2) where wdi (t) = , wdi (t) , wdi (t) ∈
ACCEPTED MANUSCRIPT
where Ri1 denotes the forward subsystem with the operator {wui (t) , wyi (t) , wdi (t)} → {ξui (t) , ξyi (t) , ξdi (t)}, and Ri2 denotes the feedback interconnection with the operator {ξui (t) , ξyi (t) , ξdi (t)} → {wui (t) , wyi (t) , wdi (t)}, and {∆yi , ∆ui , ∆di } denotes the uncertain operator, respectively. Based on the interconnected system given by (21) and (22), the following lemma is provided at first. Lemma 1. Given the interconnection with two subsystems Ri1 in (21) and Ri2 in (22), the operators T T (t) ξ T (t) T (t) ξui ∆i : ξi (t) 7−→ wi (t) satisfy the property k∆i k∞ ≤ 1, where ξi (t) = ξyi , di T T T T wi (t) = wyi (t) wui (t) wdi (t) , ∆i = diag{∆yi , ∆ui , ∆di }.
CR IP T
Proof. It follows from the event-triggered strategy given in (12) and (13), one has kwyi (t)k = kˆ yi (t − ηi (t)) − yi (t − ηi (t))k
≤ σyi kyi (t − ηi (t))k
1 hi (1) 1
σyi Ci (¯ µi ) xi (t) + xi (t − hi ) + wdi (t)
2 2 2 = kξyi (t)k ,
kwui (t)k = kˆ ui (t − ηi (t)) − ui (t − ηi (t))k
AN US
≤ σui kui (t − ηi (t))k
= σui Ki I −I x ¯i (t − ηi (t))
1 1 hi
I −I x ¯ (t) + x ¯ (t − h ) + w (t) = σui K i i i di
i
2 2 2 = kξui (t)k .
(23)
(24)
0
1 = 2 hi
Z t Z 0
Z t
0
−hi
Z
PT
1 h2i
ED
M
Next, considering the zero initial conditions, and by using Jensen’s inequality (Lemma A1 in the Appendix), it follows from (20) that Z t T (α)wdi (α) dα wdi
≤
CE
1 = hi
AC
1 = hi
Z
0 0
−hi Z 0 −hi Z 0
hi
Z
0
Z
β
ρi (β) ξdi (α + β) dβ
0
0
−hi
ρi (β) ξdi (α + β) dβ dα
T ρ2i (β) ξdi (α + β) ξdi (α + β) dβ dα
−hi t T ξdi (α t+β
T Z
+ β) ξdi (α + β) dα dβ
T ξdi (α) ξdi (α) dα
dβ
Z t 1 T ≤ ξ (α) ξdi (α) dα dβ hi −hi 0 di Z t T = ξdi (α) ξdi (α) dα.
(25)
0
It is not difficult to see from (23)-(25) that k∆i k∞ ≤ 1, thus completing this proof.
3.2
Stability Analysis
This section first introduces the following LKF: V (t) =
N X i=1
7
Vi (t),
(26)
ACCEPTED MANUSCRIPT
with Vi (t) = x ¯Ti (t) Pi x ¯i (t) +
Z
t
t−hi
Z
x ¯Ti (α) Qi x ¯i (α) dα + hi
0
−hi
Z
t
t+β
¯˙ i (α) dαdβ, x ¯˙ Ti (α) Zi x
(27)
where the matrices {Pi , Qi , Zi } ∈ R2nxi ×2nxi , i ∈ N are symmetric, and Pi > 0, Zi > 0. Inspired by [42], the positive definiteness of matrix Qi in (27) is not required. The following lemma is given to ensure the positive property of V (t).
CR IP T
Lemma 2. Consider the LKF in (26), the inequality V (t) ≥ k¯ x (t)k2 is satisfied, where > 0, T T ¯1 (t) x ¯T2 (t) · · · x ¯TN (t) , if there exist positive-definite symmetric matrices {Pi , Zi } ∈ x ¯ (t) = x 2n ×2n xi , and symmetric matrices Q ∈ R2nxi ×2nxi , such that for all i ∈ N the following inequalities R xi i hold: 1 −Zi hi Pi + Zi > 0. (28) ? Qi + Zi Proof. Firstly, by using Lemma A1 in the Appendix (Jensen’s inequality), one gets 0
−hi
Z
t
t+β
≥ hi = hi
x ¯˙ Ti (α) Zi x ¯˙ i (α) dαdβ Z
0
−hi 0
Z
−hi hi
Z
−1 β
Z
t
t+β
Z T ˙x ¯i (α) dα Zi
˙x ¯i (α) dα dβ
AN US
hi
Z
t
t+β
−1 [¯ xi (t) − x ¯i (t + β)]T Zi [¯ xi (t) − x ¯i (t + β)] dβ β
ED
M
1 = hi [¯ xi (t) − x ¯i (t − β)]T Zi [¯ xi (t) − x ¯i (t − β)] dβ β 0 Z hi [¯ xi (t) − x ¯i (t − β)]T Zi [¯ xi (t) − x ¯i (t − β)] dβ ≥ 0 Z t [¯ xi (t) − x ¯i (α)]T Zi [¯ xi (t) − x ¯i (α)] dα. =
(29)
t−hi
It follows from (27) and (29) that t
PT
Z
Vi (t) ≥
t−hi
x ¯i (t) x ¯i (α)
T
1 hi Pi
+ Zi −Zi ? Qi + Zi
x ¯i (t) x ¯i (α)
dα.
(30)
CE
Therefore, it is easy to see from (28) that V (t) ≥ k¯ x (t)k2 . Thus, completing this proof. Based on the interconnection model in (21) and (22), and the LKF in (26), a sufficient condition for stability of the control system in (18) is validated by the following theorem.
AC
Theorem 1. Given the large-scale fuzzy system in (3) and the decentralized event-triggering observerbased controller given by (7) and (9), the stability of the control system in (18) can be validated, if there exist positive-definite symmetric matrices {Pi , Zi , Hi3 } ∈ R2nxi ×2nxi , Hi1 ∈ Rnyi ×nyi , Hi2 ∈ Rnui ×nui , symmetric matrices Qi ∈ R2nxi ×2nxi , matrix multipliers Gi ∈ R(8nxi +nyi +nui )×2nxi , and positive scalars {σui , σyi , εij (µi ), ε0 } , εij (µi ) ≤ ε0 , such that for all i ∈ N the following matrix inequalities hold:
+ Zi −Zi > 0, ? Qi + Zi 2 ΩT (¯ 2 ΩT H Ω + Sym {G A (µ , µ Θi + σyi µi ) + σui ¯i )} Gi R3 F¯ij (µi ) i i i ˆi , µ 1 µi )Hi1 Ω1 (¯ 2 i2 2 < 0, ? −Eij (µi )
8
1 hi Pi
(31) (32)
ACCEPTED MANUSCRIPT
hi µi , µ ¯i ) 2 Adi (ˆ
CR IP T
where 2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Qi − Zi 0 0 0 , Θ = i ? ? ? −Hi1 0 0 ? ? ? ? −Hi2 0 ? ? ? ? −Hi3 ? Ω1 (¯ µi )= 0 12 Ci (¯ µi ) 0 12 Ci (¯ µi ) 0 0 0 h2i Ci (¯ µi ) 0 , hi 1 1 1 1 Ω2 = 0 2 Ki − 2 Ki 2 Ki − 2 Ki 0 0 2 Ki − h2i Ki , Ai (µi , µ ˆi , µ ¯i ) = −I Ai (µi , µ µi ,µ ¯i ) 12 Adi (ˆ µi , µ ¯i ) −R2 Li (ˆ µi ) R1 Bi ˆi ) + 21 Adi (ˆ ¯ ¯ ¯ ¯ F (µ ) · · · F (µ ) · · · F (µ ) F (µ ) = , Θ = Q − Z + ε (N − 1) R1 R1T , i1 i i i i 0 iN ij,j6=i i i(1) ij i | {z } T N −1 T T I 0 0 I R = , R = , R3 = I I , 1 2 . E (µ ) = diag {ε (µ )I · · · ε (µ )I · · · ε (µ )I} ij i i1 i i iN ij,j6=i i {z } | N −1
Proof. By taking the time derivative of V (t), one has
AN US
V˙ i (t) ≤ 2¯ xTi (t) Pi x ¯˙ i (t) + x ¯Ti (t) Qi x ¯i (t) − x ¯Ti (t − hi ) Qi x ¯i (t − hi ) Z t ¯˙ Ti (t) Zi x ¯˙ i (t) − hi + h2i x x ¯˙ Ti (α) Zi x ¯˙ i (α) dα.
(33)
(34)
t−hi
Based on Lemma A1 given in the Appendix (Jensen’s inequality), it holds that Z t T Z t Z t x ¯˙ i (α) dα −hi x ¯˙ i (α) Zi x ¯˙ i (α) dα ≤ − x ¯˙ i (α) dα Zi t−hi
t−hi
M
t−hi
T
= − [¯ xi (t) − x ¯i (t − hi )] Zi [¯ xi (t) − x ¯i (t − hi )] .
(35)
0=
N X
2χTi
i=1
PT
(t) Gi Ai (µi , µ ˆi , µ ¯i )χi (t) +
where x ¯, y¯ ∈
Rn
N X i=1
2χTi
(t) Gi R3
T (t) w T (t) w T (t) x ¯˙ Ti (t) x ¯Ti (t − hi ) wyi ¯Ti (t) x ui di
CE
where χi (t) = in (33). Note that
ED
Define the matrix multipliers Gi ∈ <(8nxi +nyi +nui )×2nxi , i ∈ N , and it follows from (20) that
and the scalar κ > 0.
AC
inequality (37), one has
i=1
2χTi
(t) Gi R3
N X
F¯ij (µi )xj (t),
(36)
j=1 j6=i
T
, and Ai (µi , µ ˆi , µ ¯i ) is defined
2¯ xT y¯ ≤ κ−1 x ¯T x ¯ + κ¯ y T y¯,
Define the scalar parameter 0 < εij (µi ) ≤ ε0 , where εij (µi ) :=
N X
N X
(37) ri P
l=1
µli εlij , i ∈ N , and by using the
F¯ij (µi )xj (t)
j=1 j6=i
≤
N X N X
T T ¯T χTi (t) Gi R3 F¯ij (µi )ε−1 ij (µi )Fij (µi )R3 Gi χi (t) +
≤
N X N X
T T ¯T χTi (t) Gi R3 F¯ij (µi )ε−1 ij (µi )Fij (µi )R3 Gi χi (t) + ε0 (N − 1)
i=1 j=1 j6=i
i=1 j=1 j6=i
9
N X N X
εij (µi )xTj (t)xj (t)
i=1 j=1 j6=i
N X i=1
xTi (t)xi (t).
(38)
,
ACCEPTED MANUSCRIPT
Define systematic positive matrices {Hi1 , Hi2 , Hi3 } , i ∈ N , and given the following index: J(t) =
=
N X
Ji (t) i=1 N Z ∞ X i=1
0
T T T ξyi (t) Hi1 ξyi (t) − wyi (t) Hi1 wyi (t) + ξui (t) Hi2 ξui (t)
T T T −wui (t) Hi2 wui (t) + ξdi (t) Hi3 ξdi (t) − wdi (t) Hi3 wdi (t) dt.
(39)
CR IP T
When considering zero initial conditions, it is easy to see that V (0) = 0 and V (∞) ≥ 0. Then, it yields from (34)-(39) that J(t) ≤ J(t) + V (∞) − V (0) N Z ∞n X T T T = V˙ i (t) + ξyi (t) Hi1 ξyi (t) − wyi (t) Hi1 wyi (t) + ξui (t) Hi2 ξui (t) ≤
0
N Z X i=1
T T T −wui (t) Hi2 wui (t) + ξdi (t) Hi3 ξdi (t) − wdi (t) Hi3 wdi (t) dt
∞
χTi (t) Σi (µi , µ ˆi , µ ¯i )χi (t) dt,
0
AN US
i=1
with
(40)
2 T 2 T Σi (µi , µ ˆi , µ ¯i ) = Θi + σyi Ω1 (¯ µi )Hi1 Ω1 (¯ µi ) + σui Ω2 Hi2 Ω2
M
+ Sym {Gi Ai (µi , µ ˆi , µ ¯i )} +
N X j=1 j6=i
T T ¯ ¯T ε−1 ij (µi )Gi R3 Fij (µi )Fij (µi )R3 Gi ,
(41)
3.3
CE
PT
ED
where all notations have been defined in (33). By using Schur complement to (32), it is straightforward to get that Σi (µi , µ ˆi , µ ¯i ) < 0, which implies V˙ (t) < 0 and J(t) < 0. By using the SSG theorem (Lemma A2 in the Appendix), the stability of system (18) can be ensured, thus completing this proof. It is noticed that the conditions given in (31) are nonlinear matrix inequalities, which brings the difficulty of calculating the numerical solution. In the followings, by virtue of some matrix inequality convexification technique, the co-design results simultaneously including the controller gains, eventtriggered parameters, and the sampling period will be provided in terms of LMIs.
Co-Design
AC
In this paper, it supposes that the matrix Bi is of full column rank, thus one gets [43] Inui Ti Bi = , 0(nxi −nui )×nui where Ti ∈ Rnxi ×nxi , i ∈ N , are nonsingular matrices. Define Gi11 Gi12 Gi1 = Ti , 0 Gi13
where Gi11 ∈ Rnui ×nui , Gi12 ∈ Rnui ×n(xi−ui) , Gi13 ∈ Rn(xi−ui) ×n(xi−ui) .
10
(42)
(43)
ACCEPTED MANUSCRIPT
Based on the expressions in (42) and (43), the coupling term Gi1 Bi Ki can be transformed into the following LMIs condition, Gi11 Gi12 Gi1 Bi Ki = Ti Bi Ki 0 Gi13 Gi11 Ki = 0(nxi −nui )×nxi ¯i K = 0(nxi −nui )×nxi = Ki .
CR IP T
(44)
It should be pointed out that when the knowledge between µli and µ ˆli are unavailable, the condition µli 6= µ ˆli degrades into a linear controller instead of a fuzzy one, which produces the design conservatism. Currently, assuming that the knowledge between µli and µ ˆli are available, that is ρli ≤
µ ˆli ≤ ρ¯li , l µi
(45)
AN US
where ρli and ρ¯li are positive constant scalars. Combining the Theorem 1 with the assumption in (45), the co-design containing the fuzzy controller gains, the sampled period, and the event-triggered parameters, is given as follows.
ED
M
Theorem 2. Given the large-scale fuzzy system in (3). A decentralized event-triggering observerbased controller given by (7) and (9) exists, such that the stability of the control system in (18) is validated, if there exist positive-definite symmetric matrices {Pi , Zi , Mi } ∈ R2nxi ×2nxi , symmetric T ,L ¯ s ∈ Rnxi ×nyi , K ¯ i ∈ Rnui ×nxi , matrices Qi ∈ R2nxi ×2nxi , matrices {Gi1 , Gi2 } ∈ Rnxi ×nxi , Xils = Xisl i and positive scalars {ε0 , i1 , i2 , hi , σi , εijl } , εisl ≤ ε0 , such that for all (l, s, f ) ∈ Li , i ∈ N the following inequalities hold: 1 −Zi hi Pi + Zi > 0, (46) ? Qi + Zi ll ρ¯li Σllf i + Xi < 0,
AC
CE
PT
ll ρli Σllf i + Xi ls sl ρ¯si Σlsf ¯li Σslf i +ρ i + Xi + Xi l slf ls sl ρsi Σlsf i + ρi Σi + Xi + Xi ls sl ρsi Σlsf ¯li Σslf i +ρ i + Xi + Xi l slf ls sl ρ¯si Σlsf i + ρi Σi + Xi + Xi 11 Xi · · · Xi1ri
where
Σlsf = i
Hi2 − Sym {Gi11 } ? ?
.. .
Xiri 1
..
. ···
.. .
Xiri ri
(47)
< 0,
(48)
< 0,
(49)
< 0,
(50)
< 0,
(51)
< 0,
(52)
> 0,
(53)
¯2 0 σui Ω h iT h i 2 Ωf Θi + σyi Hi1 Ωf1 + Sym Πls Gi R3 F¯ijl , i 1 ?
and
11
−Eijl
(54)
ACCEPTED MANUSCRIPT
2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Q − Z 0 0 0 i i , Θi = ? ? ? −H 0 0 i1 ? ? ? ? −Hi2 0 ? ? ? ? ?i −Hi3 h f f f f h 1 1 Ω1 = 0 2 Ci 0 2 Ci 0 0 0 2i Ci 0 , Ω 1 ¯ ¯ 2= 0 1 K ¯ −1K ¯ ¯ i 0 0 hi K ¯ i − hi K ¯i , − 21 K 2 i 2 i 2 Ki 2 2 Θi(1) = Qi − Zi + ε0 (N − 1) R1 R1T , T T T R1 = I 0 , R2 = 0 I , R3 = I I , T T T Gi = hGi Gi 02nxi ×(4nxii+nyi +nui ) , l l l l ¯ ¯ ¯ ¯ Fij = Fi1 · · · Fij,j6=i · · · FiN , {z } | N −1 n o l l l = diag εl I , E · · · ε I · · · ε I n n n i1 iN ij,j6 = i xi xi xi ij {z } | 0 = −Gi1 0 0
1 2 Ki 1 ¯s f − 2 Li Ci 1 2 Ki 1 ¯s f − 2 Li Ci
0
CR IP T
Gi1 Ali + 21 K i l s s ¯ Gi2 Ai − Gi2 Ai + Li Cis − 12 Cif
0 −Gi2
Gi1 Ali + 21 K i ¯s Cs − 1 Cf Gi2 Ali − Gi2 Asi + L i i 2 i
0
−Gi2 0
0
− 21 Ki 0 1 − 2 Ki 0 0
M
−Gi1
0 ¯s −L i
0 ¯s −L i 0
ED
Πlsf i
AN US
N −1
and
(55)
Gi1 Bi 0 Gi1 Bi 0 0
hi 2 Ki hi ¯ s f − 2 Li Ci hi 2 Ki hi ¯ s f − 2 Li Ci
− 12 Ki ¯ sC s Gi2 As − L
− h2i Ki 0 hi − 2 Ki 0 0
0
i
i
i
− 12 Ki ¯ sC s Gi2 As − L i
i
i
0
.
(56)
PT
Moreover, the decentralized event-triggering observer and controller gains are given by −1 ¯ ¯ Ki = G−1 i11 Ki , Lis = Gi2 Lis .
(57)
AC
CE
Proof. By applying Schur complement to (32), we have −1 −Hi2 σui Ω2 0 2 ΩT (¯ ? Θi + σyi µi ) + Sym {Gi Ai (µi , µ ˆi , µ ¯i )} Gi R3 F¯ij (µi ) < 0. 1 µi ) Hi1 Ω1 (¯ ? ? −Eij (µi )
Define Wi = diag Gi11 I6nxi +nyi +nui I(N −1)nxi , and perform a congruence transformation to (57) by Wi , one has −1 T ¯2 −Gi11 Hi2 Gi11 σui Ω 0 2 ΩT (¯ ? Θi + σyi µi ) + Sym {Gi Ai (µi , µ ¯i , µ ˆi )} Gi R3 F¯ij (µi ) < 0, (58) 1 µi ) Hi1 Ω1 (¯ ? ? −Eij (µi ) 1 ¯ ¯ 2= 0 1 K ¯ −1K ¯ ¯ i 0 0 hi K ¯ i − hi K ¯i . where Ω − 21 K 2 i 2 i 2 Ki 2 2 For simplification of controller design, the matrix multipliers Gi can be directly designated as Gi =
GTi
GTi
02nxi ×(4nxi +nyi +nui ) 12
T
,i ∈ N
(59)
ACCEPTED MANUSCRIPT
where Gi = diag{Gi1 , Gi2 }, and {Gi1 , Gi2 } ∈ Rnxi ×nxi are nonsingular matrices. Note that −1 −GTi11 Hi2 Gi11 ≤ Hi2 − GTi11 − Gi11 .
(60)
By substituting (59) into (58) and extracting the fuzzy basis functions, together with the inequality in (60), one gets ri X ri X ri X µli µ ˆsi µ ¯fi Σlsf < 0, (61) i i=1 s=1 f =1
CR IP T
which implies (58), where Σlsf is defined in (54). i In terms of the assumption in (45), and by using Lemma A2, the inequalities (46)-(53) can be derived, thus completing this proof. Besides, as a special case, i.e., the fuzzy system in (3) and the event-triggering observer-based controller in (7) and (9) share the same premise variables. It maybe exists when the measurable premise variables transmitting to the controller are not closed through the communication networks. In this case, the corresponding LMI-based co-design results are given as below.
M
i
AN US
Theorem 3. Given the large-scale fuzzy system in (3). Then, a decentralized event-triggering observer-based controller given by (7) with the premise variables fi (xi (t)) and (9) exists, such that the stability of the control system in (18) is validated, if there exist positive-definite symmetric matrices {Pi , Zi , Mi } ∈ Rn2nxi ×2nxi , symmetricomatrices Qi ∈ R2nxi ×2nxi , matrices {Gi1 , Gi2 } ∈ Rnxi ×nxi , and positive scalars ε0 , i1 , i2 , hi , σi , εlij , εlij ≤ ε0 , such that for all (l, s, f ) ∈ Li , i ∈ N , the following inequalities hold: 1 −Zi hi Pi + Zi > 0, (62) ? Qi + Zi ¯ llf < 0, 1 ≤ l ≤ ri Σ (63)
¯ lsf = Σ i
Hi2 − Sym {Gi11 } ? ?
h
2 Ωf Θi + σyi 1
iT
(64)
¯ σui Ω 0 n o h 2i ¯ lsf Gi R3 F¯ijl , Hi1 Ωf1 + Sym Π i ?
(65)
−Eijl
2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Q − Z 0 0 0 i i , Θi = ? ? ? −H 0 0 i1 ? ? ? ? −Hi2 0 ? ? ? ? ?i −Hi3 h f f f f h 1 1 Ω1 = 0 2 Ci 0 2 Ci 0 0 0 2i Ci 0 , Ω 1 ¯ ¯ 2= 0 1 K ¯ −1K ¯ ¯ i 0 0 hi K ¯ i − hi K ¯i , − 21 K 2 i 2 i 2 Ki 2 2 Θi(1) = Qi − Zi + ε0 (N − 1) R1 R1T , T T T R1 = I 0 , R2 = 0 I , R3 = I I , T T T Gi = hGi Gi 02nxi ×(4nxii+nyi +nui ) , l l l l ¯ ¯ ¯ ¯ Fij = Fi1 · · · Fij,j6=i · · · FiN , | {z } N −1 n o l l l l E = diag , ε I · · · ε I · · · ε I n n n i1 iN ij,j6 = i xi xi xi ij | {z }
AC
CE
and
PT
where
ED
¯ lsf + Σ ¯ slf < 0, 1 ≤ l < s ≤ ri Σ i i
N −1
13
(66)
ACCEPTED MANUSCRIPT
Πlsf i
−Gi1
0 = −Gi1 0 0
Gi1 Ali + 12 Ki ¯s Cs − 1 Cf L i i 2 i
0 −Gi2
Gi1 Ali + 21 Ki ¯s Cs − 1 Cf L i i 2 i
0
−Gi2
1 2 Ki ¯ sC f − 21 L i i 1 K 2 i ¯ sC f − 21 L i i
0
0
0
− 21 Ki
0 ¯s −L i 0 ¯s −L i 0
0 − 21 Ki 0 0
Gi1 Bi 0 Gi1 Bi 0 0
− 12 Ki ¯ sC s Gi2 As − L i
i i 1 − 2 Ki ¯ sC s Gi2 Asi − L i i
0
hi 2 Ki hi ¯ s f − 2 Li Ci hi 2 Ki hi ¯ s f − 2 Li Ci
0
− h2i Ki 0 − h2i Ki 0 0
.
CR IP T
and
(67)
Furthermore, the decentralized event-triggered observer and controller gains are given by −1 ¯ ¯ Ki = G−1 i11 Ki , Lis = Gi2 Lis .
(68)
AN US
When considering the unknown information with µli 6= µ ˆli , the corresponding LMI-based result can be summarized in the following theorem.
? ?
¯ σ Ω 0 h iT uih 2 i n o 2 Ωf ¯ lf Hi1 Ωf1 + Sym Π Gi R3 F¯ijl , Θi + σyi 1 i
ED
¯ lf = Σ i
Hi2 − Sym {Gi11 }
i
?
(71)
−Eijl
2 hi Zi + Hi3 Pi 0 0 0 0 ? Θi(1) Zi 0 0 0 ? ? −Q − Z 0 0 0 i i , Θi = ? ? ? −H 0 0 i1 ? ? ? ? −Hi2 0 ? ? ? ? ? −H i i3 h f f f f h 1 1 i Ω = , 0 C 0 C 0 0 0 C 0 1 2 i 2 i 2 i h h Ω 1 1 1 1 i ¯ ¯ 2= 0 ¯ − K ¯ ¯ − K ¯ 0 0 ¯i , − 2i K 2 Ki 2 i 2 Ki 2 i 2 Ki Θi(1) = Qi − Zi + ε0 (N − 1) R1 R1T , T T T R1 = I 0 , R2 = 0 I , R3 = I I , T T T Gi = hGi Gi 02nxi ×(4nxii+nyi +nui ) , l l l l ¯ ¯ ¯ ¯ Fij = Fi1 · · · Fij,j6=i · · · FiN , | {z } N −1 n o l l l = diag εl I E , · · · ε I · · · ε I n n n i1 iN ij,j6 = i xi xi xi ij | {z }
AC
CE
and
PT
where
M
Theorem 4. Consider the large-scale fuzzy system in (3). Then, a decentralized event-triggered observer-based linear controller given by (7) and (9) exists, such that the stability of the control system in (18) is validated, if there exist positive-definite symmetric matrices {Pi , Zi , Mi } ∈ R2nxi ×2nxi , symmetric matrices Qi ∈ R2nxi ×2nxi , matrices {Gi1 , Gi2 } ∈ Rnxi ×nxi , and positive scalars {ε0 , i1 , i2 , hi , σi , εlij }, εlij ≤ ε0 , such that for all (l, f ) ∈ Li , i ∈ N , the following inequalities hold: 1 −Zi hi Pi + Zi > 0, (69) ? Qi + Zi ¯ lf < 0, Σ (70)
N −1
14
(72)
ACCEPTED MANUSCRIPT
and −Gi1
Gi1 Ali + 21 Ki ¯i Cl − 1 Cf L
0
0 = −Gi1 0 0
−Gi2
i
2 i l Gi1 Ai + 12 Ki ¯i Cl − 1 Cf L i 2 i
0
−Gi2 0
0
− 21 Ki
1 2 Ki 1¯ − 2 Li Cif 1 2 Ki 1¯ − 2 Li Cif
0 − 21 Ki 0 0
0
0 ¯i −L 0 ¯i −L 0
Gi1 Bi 0 Gi1 Bi 0 0
− 12 Ki ¯ iC l Gi2 Ai − L i
− 12 Ki
¯ iC l Gi2 Ai − L i 0
hi 2 Ki hi ¯ − 2 Li Cif hi 2 Ki hi ¯ − 2 Li Cif
0
− h2i Ki 0 − h2i Ki 0 0
.
CR IP T
Πlf i
(73)
Furthermore, the decentralized event-triggering linear observer and controller gains are given by
4
Numerical Example
(74)
AN US
−1 ¯ ¯ Ki = G−1 i11 Ki , Li = Gi2 Li .
For validating the proposed method, we consider the large-scale fuzzy system with two interconnected subsystems as below: l , THEN Plant Rule Ril : IF xi1 (t) is Fi1 N P x˙ i (t) = Al xi (t) + Bi u ˆi (ti ) + F¯ l xj (t) i
k
j=1 j6=i
ij
M
y (t) = C x (t), l = {1, 2} , i = {1, 2} i i i
0.6 1 1.1 , , A21 = 0 −1.6 −1 0.8 0 1.2 0 2 1 ¯ ¯ , , F12 = F12 = 0 0 0 0 0.5 B1 = , C1 = 1 0 , 0
ED
where
0.4 0
CE
PT
A11 =
AC
for the first subsystem, and
−1.1 0.6 −1 2 = , A2 = , −0.5 0.9 −0.7 0.01 0 0.02 0 1 2 ¯ ¯ F21 = , F21 = , 0 0 0 0 0.4 B2 = , C2 = 1 0 , 0 A12
0.4 1
for the second subsystem. Here, we assume that σy1 = 0.17, σu1 = 0.11, σy2 = 0.17, σu2 = 0.19, and hi = 0.01. The normalized membership functions are given in Figure 2. For the design results given in Theorem 2, it is clear to see that the transformation matrices 2 0 2.5 0 T1 = , T2 = . 0 1 0 1 15
ACCEPTED MANUSCRIPT
1 0.8 0.6 0.4 0.2 0 −ri1
0
|xi1 (t)|
Figure 2: Membership functions
CR IP T
rule i1 rule i2
ri1
AN US
Firstly, it can be checked that the method proposed in [14] fails to find a feasible solution. By Theorem 3, however, a feasible solution exists, and the corresponding controller gains are 9.1848 8.2148 L11 = , L12 = , K1 = −9.0822 −0.4312 , 0.0004 −0.0001 8.3478 8.8277 , K2 = −12.2723 −0.4519 . , L22 = L21 = 0.0005 0.0001
PT
ED
M
Based on the above solutions, and given the initial conditions x1 (0) = [1.1, 0]T and x2 (0) = [1.3, 0]T , Figures 3 and 4 show the responses of system states and their estimations tend to zero. Figure 5 implies that the number of data transmissions decreases from 100 to 31 in the S-C channel of the subsystem 1, and from 100 to 41 in the C-A channel of the subsystem 1. Figure 6 shows that the number of data transmissions decreases from 100 to 30 in the S-C channel of the subsystem 2, and from 100 to 32 in the C-A channel of the subsystem 2. Thus validating the effectiveness and availability of the developed method.
AC
CE
1.5
x11 x12 xc11 xc12
1
0.5 0
−0.5 −1 0
1
2 3 4 5 Time in seconds Figure 3: Responses of system states and estimations for subsystem 1
16
ACCEPTED MANUSCRIPT
1.5 x21 x22 xc21 xc22
1 0.5
CR IP T
0 −0.5 −1 0
2 3 4 5 Time in seconds Figure 4: Responses of system states and estimations for subsystem 2
AN US
1
ED
M
σy1 = 0.17 σu1 = 0.11
20
40 60 80 100 Time in samples Figure 5: Data transmissions in subsystem 1
CE
PT
0
AC
σy2 = 0.17 σu2 = 0.19
0
20
40 60 80 100 Time in samples Figure 6: Data transmissions in subsystem 2
17
ACCEPTED MANUSCRIPT
5
Conclusions
Appendix
CR IP T
This paper studied the decentralized output-feedback control problem for large-scale networked fuzzy systems in the event-triggering framework. A control scheme with the two-channel event-triggering was proposed. By using a novel model transformation, and introducing an LKF without the requirement of positive definition for all Lyapunov matrices, and combining with the virtue of SSG theorem, the co-design with less conservatism was derived in the form of LMIs to simultaneously obtain the controller gains, event-triggered parameters, and the sampled period. The simulation results show that the proposed controller guarantees the stability of the resulting control system while reducing data communications in both the S-C channel and the C-A channel.
Lemma A1. [44] For any positive-definite symmetric matrix M ∈ Rn×n , and scalars d2 > d1 ≥ 0, the following inequality holds d2
x (t) dt
d1
T
M
Z
d2
d1
Z x (t) dt ≤ (d2 − d1 )
d2
d1
xT (t) M x (t) dt.
AN US
Z
Lemma A2. [39] Assume that the membership functions satisfy ρl ≤ µµˆll ≤ ρ¯l , l ∈ L := {1, 2, . . . , r} . r r P P T , such that for all (l, s) ∈ L , the µl µ ˆs Σls < 0 holds if there exist matrices Xls = Xsl Then, l=1 s=1
following inequalities hold
ρ¯l Σll + Xll < 0,
ρl Σll + Xll < 0,
M
ρ¯s Σls + ρ¯l Σsl + Xls + Xsl < 0,
ρs Σls + ρl Σsl + Xls + Xsl < 0,
ED
ρs Σls + ρ¯l Σsl + Xls + Xsl < 0,
PT
ρ¯s Σls + ρl Σsl + Xls + Xsl < 0, X11 · · · X1r .. .. > 0. .. . . . Xr1 · · ·
Xrr
CE
Acknowledgment
AC
The authors are grateful to the Editor-in-Chief, Senior Editor Prof. Michael V. Basin, Associate Editor, and anonymous reviewers for their insightful comments, which helped to improve the quality of this work.
References
[1] G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 676-697, 2006. [2] C. Han, L. Wu, P. Shi, and Q. Zeng, “On dissipativity of Takagi-Sugeno fuzzy descriptor systems with time-delay,” Journal of the Franklin Institute, vol. 349, no. 10, pp. 3170-3184, 2012. [3] L. Zhang, T. Yang, P. Shi, and M. Liu, “Stability and stabilization of a class of discrete-time fuzzy systems with semi-Markov stochastic uncertainties,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2016, DOI: 10.1109/TSMC.2016.2527627. 18
ACCEPTED MANUSCRIPT
[4] Y. Liu and S. Lee, “Stability and stabilization of Takagi-Sugeno fuzzy systems via sampleddata and state quantized controller,” IEEE Transactions on Fuzzy Systems, vol. 24, no. 3, pp. 635-644, 2016. [5] H. Shen, J. Park, and Z. Wu, “Finite-time reliable L2 -L∞ /H∞ control for Takagi-Sugeno fuzzy systems with actuator faults,” IET Control Theory & Applications, vol. 8, no. 9, pp. 688-696, 2014.
CR IP T
[6] H. Shen, J. Park, and Z. Wu, “Extended dissipative state estimation for Markov jump neural networks with unreliable links,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 2, pp. 346-358, 2017. [7] J. Qiu, G. Feng, and H. Gao, “Nonsynchronized-state estimation of multichannel networked nonlinear systems with multiple packet dropouts via T-S fuzzy-affine dynamic models,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 1, pp. 75-90, 2011. [8] Y. Liu, J. Park, B. Guo, and Y. Shu, “Further results on stabilization of Chaotic systems based on fuzzy memory sampled-data control,” IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2017.2686364.
AN US
[9] L. Zhang, Z. Ning, and P. Shi, “Input-output approach to control for fuzzy Markov jump systems with time-varying delays and uncertain packet dropout rate,” IEEE Transactions on Cybernetics, vol. 45, no. 11, pp. 2449-2460, 2015. [10] Y. Liu, B. Guo, and J. Park, “Non-fragile H∞ filtering for delayed Takagi-Sugeno fuzzy systems with randomly occurring gain variations,”Fuzzy Sets and Systems, vol. 316, pp. 99-116, 2017.
M
[11] H. Zhang, Y. Shi, and A. Mehr, “On filtering for discrete-time Takagi-Sugeno fuzzy systems,”IEEE Transactions on Fuzzy Systems, vol. 20, no. 2, pp. 396-401, 2012.
ED
[12] L. Bakule, “Decentralized control: Status and outlook,” Annual Reviews in Control, vol. 38, no. 1, pp. 71-80, 2014. [13] C. Hua and S. Ding “Decentralized networked control system design using T-S fuzzy approach,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 1, pp. 9-21, 2012.
CE
PT
[14] H. Zhang, G. Yu, C. Zhou, and C. Dang, “Delay-dependent decentralized H∞ filtering for fuzzy interconnected systems with time-varying delay based on Takagi-Sugeno fuzzy model,” IET Control and Applications, vol. 7, no. 5, pp. 720-729, 2013.
AC
[15] Z. Zhong, S. Fu, T. Hayat, F. Alsaadi, and G. Sun, “Decentralized piecewise H∞ fuzzy filtering design for discrete-time large-scale nonlinear systems with time-varying delay,” Journal of the Franklin Institute, vol. 352, no. 9, pp. 3782-3807, 2015. [16] H. Zhang, C. Dang, and J. Zhang, “Decentralized fuzzy H∞ filtering for nonlinear interconnected systems with multiple time delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 40, no. 4, pp. 1197-1203, 2010. [17] X. He, Z. Wang, X. Wang, and D. Zhou, “Networked strong tracking filtering with multiple packet dropouts: algorithms and applications” IEEE Transactions on Industrial Electronics, vol. 61, no. 3, pp. 1454-1463, 2014. [18] Y. Xu, R. Lu, H. Peng, K. Xie, and A. Xue, “Asynchronous dissipative state estimation for stochastic complex networks with quantized jumping coupling and uncertain measurements” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 2, pp. 268-277, 2017.
19
ACCEPTED MANUSCRIPT
[19] X. He, Z. Wang, L. Qin, and D. Zhou, “Active fault tolerant control for an Internet-based networked three-tank system” IEEE Transactions on Control Systems Technology, vol. 24, no. 6, pp. 2150-2157, 2016. [20] Y. Shi and B. Yu, “Output feedback stabilization of networked control systems with random delays modeled by Markov chains” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1668-1674, 2009.
CR IP T
[21] Y. Xu, R. Lu, H. Peng, and S. Xie, “Filtering for fuzzy systems with multiplicative sensor noises and multi-density quantizer” IEEE transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2017.2702119. [22] X. He, Z. Wang, Y. Liu, D. Zhou, “Least-squares fault detection and diagnosis for networked sensing systems using direct state estimation approach” IEEE Transactions on Industrial Informatics, vol. 9, no. 3, pp. 1670-1679, 2013. [23] L. Zhang, Y. Zhu, Z. Ning, and X. Yin, “Resilient estimation for networked systems with variable communication capability,” IEEE Transactions on Automatic Control, vol. 61, no. 12, pp. 41504156, 2016.
AN US
[24] Q. Lu, L. Zhang, M. Basin, and H. Tian, “Analysis and synthesis for networked control systems with uncertain rate of packet losses,” Journal of the Franklin Institute, vol. 349, no. 7, pp. 2500-2514, 2012. [25] R. Yang, G. Liu, P. Shi, C. Thomas, and M. Basin, “Predictive output feedback control for networked control systems,” IEEE Transactions on Industrial Electronics, vol. 61, no. 1, pp. 513-520, 2014.
M
[26] W. Heemels, A. Teel, N. Wouw, and D. Neˇsi´c, “Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1781-1796, 2010.
ED
[27] W. Heemels and M. Donkers, “Model-based periodic event-triggered control for linear systems,” Automatica, vol. 49, no. 3, pp. 698-711, 2013.
PT
[28] M. Donkers and W. Heemels, “Output-based event-triggered control with guaranteed L∞ -gain and improved and decentralized event-triggering,” IEEE Transactions on Automatic Control, vol. 57, no. 6, pp. 1362-1376, 2012.
CE
[29] D. Yue, E. Tian, and Q. Han, “A delay system method for designing event-triggered controllers of networked control systems,” IEEE Transactions on Automatic Control, vol. 58, no. 2, pp. 475-481, 2013.
AC
[30] D. Hristu-Varsakelis and P. Kumar, “Interrupt-based feedback control over a shared communication medium,” In Proceedings of the 2002 IEEE Conference on Decision Control, pp. 3223-3228, 2002. [31] X. Wang and M. Lemmon, “Event-triggering in distributed networked control systems,”IEEE Transactions on Automatic Control, vol. 56, no. 3, pp. 586-601, 2011. [32] P. Tabuada and X. Wang, “Preliminary results on state-triggered scheduling of stabilizing control tasks,” In Proceedings of the 2006 IEEE Conference on Decision Control, pp. 282–287, 2006. [33] J. Cheng, J. Park, H. Wang, “Event-triggered H∞ control for T-S fuzzy nonlinear systems and its application to truck-trailer system” ISA Transactions, vol. 65, pp. 62-71, 2016.
20
ACCEPTED MANUSCRIPT
[34] J. Cheng, J. Park, L. Zhang, and Y. Zhu, “An asynchronous operation approach to eventtriggered control for fuzzy Markovian Jump systems with general switching policies,” IEEE Transactions on Fuzzy systems, DOI 10.1109/TFUZZ.2016.2633325. [35] C. Peng and Q. Han, “To transmit or not to transmit: a discrete event-triggered communication scheme for networked Takagi–Sugeno fuzzy systems,” IEEE Transactions on Fuzzy systems, vol. 21, no. 1, pp. 164-170, 2013.
CR IP T
[36] X. Jia, X. Chi, Q. Han, and N. Zheng, “Event-triggered fuzzy H∞ control for a class of nonlinear networked control systems using the deviation bounds of asynchronous normalized membership functions,”Information Sciences, vol. 259, no. 20, pp. 100-117, 2014. [37] D. He, X. Jia, X. Chi, and W. Ma, “Fuzzy H∞ tracking control for a class of nonlinear networked control systems with a discrete event-triggered communication scheme,” In Proceedings of the 2013 IEEE Conference on Control and Decision, pp. 2562-2567, 2013. [38] Y. Guan, Q. Han, and C. Peng, “Event-triggered output feedback control for Takagi-Sugeno fuzzy systems,” In Proceedings of the 2013 IEEE Conference on Industrial Electronics Society, pp. 5644-5649, 2013.
AN US
[39] C. Ario and A. Sala, “Extensions to “stability analysis of fuzzy control systems subject to uncertain grades of membership”,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, no. 2, pp. 558-563, 2008. [40] E. Fridman, “A refined input delay approach to sampled-data control,” Automatica, vol. 46, no. 2, pp. 421-427, 2010.
M
[41] K. Gu, Y. Zhang, and S. Xu, “Small gain problem in coupled differential-difference equations, time-varying delays, and direct Lyapunov method,” International Journal of Robust and Nonlinear Control, vol. 21, no. 4, pp. 429-451, 2011.
ED
[42] S. Xu, J. Lam, B. Zhang, and Y. Zou, “New insight into delay-dependent stability of time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 25, no. 7, pp. 961-970, 2015.
PT
[43] J. Qiu, G. Feng, and H. Gao, “Static output feedback H∞ control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 2, pp. 245-261, 2013.
AC
CE
[44] K. Gu, “An integral inequality in the stability problem of time-delay systems,” In Proceedings of the 2000 IEEE Conference on Decision and Control, pp. 2805-2810, 2000.
21