Improved Robust Stability Criterion of Networked Control Systems with ...

Hindawi Publishing Corporation Journal of Control Science and Engineering Volume 2014, Article ID 936856, 6 pages http://dx.doi.org/10.1155/2014/936856

Research Article Improved Robust Stability Criterion of Networked Control Systems with Transmission Delays and Packet Loss Shenping Xiao, Liyan Wang, Hongbing Zeng, and Wubin Cheng School of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412008, China Correspondence should be addressed to Shenping Xiao; xsph [email protected] Received 10 August 2013; Revised 4 December 2013; Accepted 24 December 2013; Published 12 February 2014 Academic Editor: Derong Liu Copyright © 2014 Shenping Xiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The problem of stability analysis for a class of networked control systems (NCSs) with network-induced delay and packet dropout is investigated in this paper. Based on the working mechanism of zero-order holder, the closed-loop NCS is modeled as a continuoustime linear system with input delay. By introducing a novel Lyapunov-Krasovskii functional which splits both the lower and upper bounds of the delay into two subintervals, respectively, and utilizes reciprocally convex combination technique, a new stability criterion is derived in terms of linear matrix inequalities. Compared with previous results in the literature, the obtained stability criterion is less conservative. Numerical examples demonstrate the validity and feasibility of the proposed method.

1. Introduction Network control systems (NCSs) are the feedback control systems in which the control loops are closed via real-time networks [1]. NCSs have great advantages compared to the traditional point-to-point control systems, including high reliability, ease of installation and maintenance, and low cost, and they have been applied in many areas, such as computer integrated manufacturing systems, intelligent traffic systems, aircraft control, and teleoperation. However, due to the insertion of communication network in feedback control loops, time delay caused by data transmission and packet dropout in NCSs is always inevitable, which may degrade system performance or even lead to the potential system instability. Thus it is of significance to cope with the adverse influences of induced delay and packet dropout. Recently, the problem of robust stability analysis for NCSs has attained considerable attention, and a great number of research results have been reported [1–9]. In order to reduce the conservatism, Yu et al. [5] obtained the sufficient condition on the stabilization of NCSs; the admissible upper bounds of induced delay and packet dropout can be computed by using LyapunovRazumikhi function techniques and the quasiconvex optimization algorithm. The free weighting matrices method was proposed in [3] to solve the problem of network-based

control. However, using too many free weighting matrices makes the system analysis complex; what is more, some terms were neglected directly, which brings the conservative. Peng et al. [9] indicated that if more information of induced delays in NCSs were utilized, conservatism in system analysis could be reduced. Moreover, it can build a bridge to connect quality of control (QoC). In [4], a stability criterion based on maximum allowable network-induced delay rate is proposed for choosing a reasonable sampling period. Inspired by the above research results, in the present paper, the robust stabilization problem for a class of NCSs with network-induced delay and packet dropout is addressed. The NCSs are modeled as continuous-time linear systems on the basis of the input delay approach. By exploiting the information of lower and upper bounds of the delay, a new Lyapunov-Krasovskii functional is constructed. Moreover, reciprocally convex combination technique is introduced such that less conservative results are obtained. Two numerical examples are given to illustrate the validity and feasibility of the proposed results.

2. System Description In this paper, the sensor module is assumed to act in a clockdriven fashion with transmission period h; the controller

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Journal of Control Science and Engineering

and actuator modules are assumed to act in an event-driven fashion. A new packet will be used by the controller immediately after its arrival. Single packet transmission is considered, where all the data sent or received over the network are sampled at the same sampling instant and assembled together into one network packet. The plant is described by the following linear plant model proposed in [7]: 𝑥̇ (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) ,

𝑡+ ∈ [ 𝑖𝑘 ℎ + 𝜏𝑘 , 𝑖𝑘+1 ℎ + 𝜏𝑘+1 ) , 𝑘 = 0, 1, 2, 3, . . . ,

0 packet is lost,

𝑖𝑘+1 − 𝑖𝑘 = 2,

1 packet is lost,

has

0

0

𝑇

𝑟

0

(6)

Lemma 2 (see [14]). For any positive matrix 𝑅, scalars 𝑎 and 𝑏 satisfying 𝑎 > 𝑏, and 𝑎 vector function 𝑥, the following inequality holds: (𝑎 − 𝑏)2 𝑎 𝑎 𝑇 ∫ ∫ 𝑥 (𝑢) 𝑅𝑥 (𝑢) 𝑑𝑢 𝑑𝑠 2 𝑏 𝑠 𝑎

𝑎

𝑏

𝑠

𝑇

𝑎

𝑎

𝑏

𝑠

(7)

≥ (∫ ∫ 𝑥 (𝑢) 𝑑𝑢 𝑑𝑠) 𝑅 (∫ ∫ 𝑥 (𝑢) 𝑑𝑢 𝑑𝑠) .

Lemma 3 (see [15]). Let 𝐹1 , 𝐹2 , 𝐹3 , . . . 𝐹𝑁 : 𝑅𝑚 󳨃→ 𝑅 have positive values for arbitrary value of independent variable in an open subset 𝑊 of 𝑅𝑚 . The reciprocally convex combination of 𝐹𝑖 (𝑖 = 1, 2, . . . , 𝑁) in 𝑊 satisfies 𝑁 𝑁 𝑁 𝑁 1 min ∑ 𝐹𝑖 (𝑡) = ∑𝐹𝑖 (𝑡) + max ∑ ∑ 𝑊𝑖,𝑗 (𝑡) 𝑖=1 𝜂𝑖 𝑖=1 𝑖=1 𝑗=1, 𝑗 ≠ 𝑖

(8)

(3)

.. . 𝑖𝑘+1 − 𝑖𝑘 = 𝑝,

𝑟

(2)

where 𝐾 is the state feedback gain matrix, 𝜏𝑘 denotes the network-induced delay, and ℎ is the sampling period, {𝑖1 , 𝑖2 , 𝑖3 , . . .} ⊂ {1, 2, 3, . . .}; due to the introduction of logical zero-order holder, the actuator will use the latest available control input, so 𝑖𝑘+1 > 𝑖𝑘 . If {𝑖1 , 𝑖2 , 𝑖3 , . . .} = {0, 1, 2, 3, . . .}, then no packet dropout occurred in the transmission. And the numbers of consecutive packet dropouts during the time interval (𝑖𝑘 ℎ, 𝑖𝑘+1 ℎ) can be described as follows: 𝑖𝑘+1 − 𝑖𝑘 = 1,

𝑟

𝑟 (∫ 𝑤(𝑠)𝑇 𝑀𝑤 (𝑠) 𝑑𝑠) ≥ (∫ 𝑤 (𝑠) 𝑑𝑠) 𝑀 (∫ 𝑤 (𝑠) 𝑑𝑠) .

(1)

where 𝑥(𝑡) ∈ 𝑅𝑛 , 𝑢(𝑡) ∈ 𝑅𝑚 are the plant’s state and input vectors, respectively. 𝐴 and 𝐵 are known to be real constant matrices with proper dimensions. This control signal is based on the plant’s state at the instant 𝑖𝑘 ℎ. So, the control law can be described as 𝑢 (𝑡+ ) = 𝐾𝑥 (𝑖𝑘 ℎ) ,

Lemma 1 (see [13]). For any positive matrix 𝑀 > 0, scalar 𝑟 > 0, and a vector function 𝑤 : [0, 𝑟] → 𝑅𝑛 such that the 𝑟 integration ∫0 𝑤(𝑠)𝑇 𝑀𝑤(𝑠)𝑑𝑠 is well defined,

subject to 𝑝 − 1 packets are lost.

𝑁

Assume the existence of constants 𝜏𝑚 > 0, 𝜏𝑀 > 0. One

{𝜂𝑖 > 0, ∑𝜂𝑖 = 1, 𝑊𝑖,𝑗 (𝑡) : 𝑅𝑚 󳨃󳨀→ 𝑅, 𝑖=1

(9) 𝜏𝑚 ≤ 𝜏𝑘 , (𝑖𝑘+1 − 𝑖𝑘 ) ℎ + 𝜏𝑘+1 ≤ 𝜏𝑀,

𝑘 = 0, 1, 2, 3, . . . ,

(4)

where 𝜏𝑚 and 𝜏𝑀 indicate the lower and the upper bounds of the total delay involving both transmission delays and packet dropouts, respectively. From a straightforward combination of (1)–(4), the system can be rewritten as follows: 𝑥̇ (𝑡) = 𝐴𝑥 (𝑡) + 𝐴 𝑑 𝑥 (𝑡 − 𝑑 (𝑡)) , 𝑥 (𝑡) = 𝜑 (𝑡) ,

𝑡 ∈ (−𝜏𝑀, 0) ,

(5)

where 𝐴 𝑑 = 𝐵𝐾, the function 𝑑(𝑡) = 𝑡 − 𝑖𝑘 ℎ which satisfies 𝜏𝑚 ≤ 𝑑(𝑡) ≤ 𝜏𝑀 denotes the time-varying delay in the control signal, and 𝜑(𝑡) is the state’s initial function. The following technical lemmas are introduced, which are indispensable for the proof of the main result.

𝐹 (𝑡) 𝑊𝑖,𝑗 (𝑡) ] ≥ 0} . 𝑊𝑗,𝑖 (𝑡) = 𝑊𝑖,𝑗 (𝑡) , [ 𝑖 𝑊𝑖,𝑗 (𝑡) 𝐹𝑗 (𝑡)

3. Main Results Theorem 4. For a given scalar 𝜏𝑀 > 𝜏𝑚 ≥ 0, NCS (5) is asymptotically stable if there exist symmetric matrices 𝑃 = 𝑅11 𝑅12 12 [𝑃𝑖𝑗 ]4×4 > 0, 𝑄 = [ 𝑄∗11 𝑄 𝑄22 ] > 0, 𝑅 = [ ∗ 𝑅22 ] > 0, 𝑍𝑖 (𝑖 = 1, 2, 3) > 0, 𝑅𝑖 (𝑖 = 1, 2, 3) > 0, and proper dimensions matrice 𝑆12 such that

[

𝑍2 𝑆12 ] > 0, ∗ 𝑍2

Θ Θ Θ = [ 11 12 ] < 0, ∗ Θ22

(10)

Journal of Control Science and Engineering

3 Ω44 = − 𝑅22 − 𝑍2 ,

where

Θ12

𝜏𝑚2 Ψ𝑅1 ] [ 8 ] [ ] [ ] [ (𝜏 − 𝜏 )2 ] [ 𝑀 𝑚 Ψ𝑅2 ] [ ] [ 2 ] [ ] [ 2 𝜏𝑀 ] [ ] [ Ψ𝑅3 ], 8 =[ ] [ ] [ 𝜏 𝑚 ] [ Ψ𝑍1 ] [ 2 ] [ ] [ [ (𝜏𝑀 − 𝜏𝑚 ) Ψ𝑍2 ] ] [ ] [ ] [ 𝜏𝑀 ] [ Ψ𝑍3 2 ] [

Ω55 = − 𝑍1 − 𝑄11 + 𝑄22 , Ω66 = − 𝑍3 − 𝑅11 + 𝑅22 . (12) Proof. Let us choose a Lyapunov-Krasovskii functional candidate as 4

𝑉 (𝑥𝑡 ) = ∑𝑉𝑖 (𝑥𝑡 ) ,

where

Θ11 = diag {−𝑅1 −𝑅2 −𝑅3 −𝑍1 −𝑍2 −𝑍3 } , Ω11 𝑃11 𝐴 𝑑 𝑃12 −𝑃12

Θ22

[∗ [ [∗ [ [∗ [ =[∗ [ [∗ [ [∗ [

∗

[∗

Ω15

Ω17

Ω18

Ω19

𝐴𝑇𝑑 𝑃13

𝐴𝑇𝑑 𝑃14

Ω22

Ω23

Ω24

0

0

∗

Ω33

𝑆12

−𝑄𝑇12

𝐴𝑇𝑑 𝑃12

0

𝑃22

𝑃23

𝑃24

∗

∗

Ω44

0

−𝑅𝑇12

−𝑃22

−𝑃23

−𝑃24

∗

∗

∗

Ω55

0

−𝑃𝑇23

−𝑃33

−𝑃34

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

Ω66 ∗ ∗ ∗

−𝑃𝑇34 0 −𝑅1 ∗

−𝑃44 0 0 −𝑅3

Ψ = [𝐴 𝐴 𝑑 0 0 0 0 0 0 0] with 𝑇 𝑇 + 𝑃14 + 𝑃14 Ω11 = 𝐴𝑇 𝑃11 + 𝑃11 𝐴 + 𝑃13 + 𝑃13

+ 𝑅11 + 𝑄11 − 𝑍1 − 𝑍3 − 2

− (𝜏𝑀 − 𝜏𝑚 ) 𝑅2 −

2 𝜏𝑀

4

𝜏𝑚2 𝑅 4 1

𝑅3 ,

] ] ] ] ] ] ], ] ] ] ] ] ]

(11)

𝑥 (𝑡) ] [ 𝑡−𝜏𝑚 ] [ 𝑡−𝜏𝑚 ] [ ] [ ∫ [ 𝑡−𝜏𝑀 𝑥 (𝑠) 𝑑𝑠 ] [ ∫𝑡−𝜏𝑀 𝑥 (𝑠) 𝑑𝑠 ] ] [ 𝑡 ] [ 𝑡 ] 𝑃[ ], 𝑉1 (𝑥𝑡 ) = [ ] [∫ ] [∫ 𝑥 𝑑𝑠 𝑥 𝑑𝑠 (𝑠) (𝑠) ] [ 𝑡−(𝜏 /2) ] [ 𝑡−(𝜏 /2) ] [ 𝑡 𝑚 ] [ 𝑡 𝑚 ] [ ] [ 𝑥 (𝑠) 𝑑𝑠 𝑥 (𝑠) 𝑑𝑠 ∫ ∫ ] [ 𝑡−(𝜏𝑀 /2) ] [ 𝑡−(𝜏𝑀 /2) 𝑉2 (𝑥𝑡 ) 𝑇

𝑥 (𝑠) 𝑥 (𝑠) 𝜏𝑚 ] [𝑄11 𝑄12 ] [ 𝜏 ] 𝑑𝑠 =∫ [ ∗ 𝑄22 𝑥 (𝑠 − 𝑚 ) ) 𝑡−(𝜏𝑚 /2) 𝑥 (𝑠 − 2 2 𝑡

𝑇

𝑥 (𝑠) 𝑥 (𝑠) 𝜏𝑀 ] [𝑅11 𝑅12 ] [ 𝜏 ] 𝑑𝑠, [ +∫ ∗ 𝑅22 𝑥 (𝑠 − 𝑀 ) ) 𝑡−(𝜏𝑀 /2) 𝑥 (𝑠 − 2 2 𝑡

𝑉3 (𝑥𝑡 ) =

𝑡 𝜏𝑚 0 ∫ 𝑥̇𝑇 (𝑢) 𝑍1 𝑥̇ (𝑢) 𝑑𝑢 𝑑𝑠 ∫ 2 −𝜏𝑚 /2 𝑡+𝑠 −𝜏𝑚

+ (𝜏𝑀 − 𝜏𝑚 ) ∫

Ω15 = − 𝑃13 + 𝑍1 + 𝑄12 ,

−𝜏𝑀

Ω16 = 𝑅12 + 𝑍3 − 𝑃14 ,

+

𝑇 𝑇 Ω17 = 𝐴𝑇 𝑃12 + 𝑃23 + 𝑃24 + (𝜏𝑀 − 𝜏𝑚 ) 𝑅2 , 𝑇 + Ω18 = 𝐴𝑇 𝑃13 + 𝑃33 + 𝑃34

𝜏𝑚 𝑅, 2 1

Ω19 = 𝐴𝑇 𝑃14 + 𝑃34 + 𝑃44 +

𝜏𝑀 𝑅, 2 3

𝑉4 (𝑥𝑡 ) =

𝑡

∫ 𝑥̇𝑇 (𝑢) 𝑍2 𝑥̇ (𝑢) 𝑑𝑢 𝑑𝑠 𝑡+𝑠

𝑡 𝜏𝑀 0 ∫ 𝑥̇𝑇 (𝑢) 𝑍3 𝑥̇ (𝑢) 𝑑𝑢 𝑑𝑠, ∫ 2 −𝜏𝑀 /2 𝑡+𝑠

𝑡 𝑡 𝜏𝑚2 𝑡 ∫ ∫ 𝑥̇𝑇 (V) 𝑅1 𝑥̇ (V) 𝑑V 𝑑𝑢 𝑑𝑠 ∫ 8 𝑡−(𝜏𝑚 /2) 𝑠 𝑢 2

+

𝑇 Ω22 = − 2𝑍2 + 𝑆12 + 𝑆12 ,

Ω23 = 𝑍2 − 𝑆12 ,

𝑇

𝑥 (𝑡)

Ω16

−𝑃𝑇24 −𝑅2 ∗ ∗

+

(𝜏𝑀 − 𝜏𝑚 ) 𝑡−𝜏𝑚 𝑡 𝑡 𝑇 ∫ ∫ 𝑥̇ (V) 𝑅2 𝑥̇ (V) 𝑑V 𝑑𝑢 𝑑𝑠 ∫ 2 𝑡−𝜏𝑀 𝑠 𝑢 2 𝑡 𝑡 𝑡 𝜏𝑀 ∫ ∫ 𝑥̇𝑇 (V) 𝑅3 𝑥̇ (V) 𝑑V 𝑑𝑢 𝑑𝑠. ∫ 8 𝑡−(𝜏𝑀 /2) 𝑠 𝑢

𝑇 Ω24 = 𝑍2 − 𝑆12 ,

Ω33 = − 𝑄22 − 𝑍2 ,

(13)

𝑖=1

Define an extended-state vector as

(14)

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Journal of Control Science and Engineering

Δ

𝜉𝑇 (𝑥𝑡 ) = [𝑥𝑇 (𝑡) 𝑥𝑇 (𝑡 − 𝑑 (𝑡)) 𝑥𝑇 (𝑡 − 𝜏𝑚 ) 𝑥𝑇 (𝑡 − 𝜏𝑀) 𝑥𝑇 (𝑡 −

𝑡−𝜏𝑚 𝑡 𝑡 𝜏m 𝜏 ) 𝑥𝑇 (𝑡 − 𝑀 ) ∫ 𝑥𝑇 (𝑠) 𝑑𝑠 ∫ 𝑥𝑇 (𝑠) 𝑑𝑠 ∫ 𝑥T (𝑠) 𝑑𝑠] , 2 2 𝑡−(𝜏m /2) 𝑡−(𝜏𝑀 /2) 𝑡−𝜏𝑀

(15)

and then NCS (5) is simplified as

By utilizing Lemma 1, we obtain

𝑥̇ (𝑥𝑡 ) = 𝜓𝜉 (𝑥𝑡 ) .

(16)

Calculating the derivative of 𝑉1 (𝑥𝑡 ) along the trajectory of NCS (5) yields

𝜏𝑚 𝑡 𝑥̇𝑇 (𝑠) 𝑍1 𝑥̇ (𝑠) 𝑑𝑠 ∫ 2 𝑡−(𝜏𝑚 /2) 𝑇

𝑥 (𝑡) 𝑥 (𝑡) −𝑍 𝑍 𝜏𝑚 ] [ ∗ 1 −𝑍1 ] [ 𝜏 ], 1 𝑥 (𝑡 − ) 𝑥 (𝑡 − 𝑚 ) [ [ 2 ] 2 ]

≤[

𝑇

𝑥 (𝑡)

−

𝜓𝜉 (𝑡) ] [ [ ] ] [ ∫ 𝑥 (𝑡−𝜏𝑚 ) − 𝑥 (𝑡−𝜏𝑀)] [ 𝑡−𝜏𝑀 𝑥 (𝑠) 𝑑𝑠 ] [ ] ] [ [ 𝑡 ] ] 𝑃[ 𝜏𝑚 𝑉1̇ (𝑥𝑡 ) = 2[ [ ]. ] [∫ 𝑥 (𝑡− ) (𝑡)−𝑥 𝑥 𝑑𝑠 (𝑠) [ ] ] [ [ 𝑡−(𝜏 /2) 2 ] ] [ [ 𝑡 𝑚 ] [ 𝜏𝑀 ] 𝑥 (𝑡)−𝑥 (𝑡− ) 𝑥 (𝑠) 𝑑𝑠 ∫ ] 2 ] [ [ 𝑡−(𝜏𝑀 /2) (17) 𝑡−𝜏𝑚

𝑇

𝑥 (𝑡) 𝑥 (𝑡) 𝜏 ]. 𝜏𝑀 ] [−𝑍3 𝑍3 ] [ ≤[ ∗ −𝑍3 𝑥 (𝑡 − 𝑀 ) 𝑥 (𝑡 − ) 2 2 Define 𝛼 = (𝑑(𝑡) − 𝜏𝑚 )/(𝜏𝑀 − 𝜏𝑚 ), 𝛽 = (𝜏𝑀 − 𝑑(𝑡))/(𝜏𝑀 − 𝜏𝑚 ); by the reciprocally convex combination in Lemma 3, the following inequality holds:

It is easy to get 𝑇

𝑥 (𝑡) 𝑥 (𝑡) 𝑄 𝑄 𝑉2̇ (𝑥𝑡 ) = [ 𝜏𝑚 ] [ ∗11 𝑄12 ] [ 𝜏 ] 22 𝑥 (𝑡 − ) 𝑥 (𝑡 − 𝑚 ) [ [ 2 ] 2 ]

𝑇

𝜏 𝜏𝑚 𝑇 𝑥 (𝑡 − 𝑚 )] ) 𝑄 𝑄 2 ] [ 11 12 ] [ 2 − ∗ 𝑄22 ) 𝑥 (𝑡 − 𝜏 𝑥 (𝑡 − 𝜏 𝑚 ] 𝑚) ] [ [ [𝑥 (𝑡 −

𝛽 [ √ (𝑥 (𝑡 − 𝜏𝑚 ) − 𝑥 (𝑡 − 𝑑 (𝑡))) ] ] 𝑍2 𝑆12 [ 𝛼 −[ ] [∗ 𝑍 ] ] [ 𝛼 2 −√ (𝑥 (𝑡 − 𝑑 (𝑡)) − 𝑥 (𝑡 − 𝜏𝑀)) 𝛽 ] [ (21)

𝑇

𝑥 (𝑡) 𝑥 (𝑡) 𝜏𝑀 ] [𝑅11 𝑅12 ] [ 𝜏 ] +[ ∗ 𝑅22 𝑥 (𝑡 − 𝑀 ) 𝑥 (𝑡 − ) 2 2 𝜏 𝜏𝑀 𝑇 𝑥 (𝑡 − 𝑀 )] ) 𝑅 𝑅 2 ] [ 11 12 ] [ 2 . − ∗ 𝑅22 ) ) 𝑥 (𝑡 − 𝜏 𝑥 (𝑡 − 𝜏 𝑀 ] 𝑀 ] [ [ (18) [𝑥 (𝑡 −

𝛽 [ √ (𝑥 (𝑡 − 𝜏𝑚 ) − 𝑥 (𝑡 − 𝑑 (𝑡))) ] ] [ 𝛼 ×[ ] < 0. ] [ 𝛼 −√ (𝑥 (𝑡 − 𝑑 (𝑡)) − 𝑥 (𝑡 − 𝜏𝑀)) 𝛽 ] [ Due to 𝜏𝑚 ≤ 𝑑(𝑡) ≤ 𝜏𝑀, according to Lemma 1 and inequalities (21), we have

The time derivative of 𝑉3 (𝑥𝑡 ) can be represented as 𝑉3̇ (𝑥𝑡 ) = 𝑥̇𝑇 (𝑡) (( × 𝑥̇ (𝑡) − ×∫

−𝜏𝑚

−𝜏𝑀

−

𝜏𝑚2 4

2

) 𝑍1 + (𝜏𝑀 − 𝜏𝑚 ) 𝑍2 + (

2 𝜏𝑀

4

− (𝜏𝑀 − 𝜏𝑚 ) ∫ ) 𝑍3 )

𝜏𝑚 0 𝑥̇𝑇 (𝑠) 𝑍1 𝑥̇ (𝑠) 𝑑𝑠 − (𝜏𝑀 − 𝜏𝑚 ) ∫ 2 −𝜏𝑚 /2

𝑡−𝜏𝑚

𝑡−𝜏𝑀

𝑥̇𝑇 (𝑠) 𝑍2 𝑥̇ (𝑠) 𝑑𝑠 𝑡−𝜏𝑚

= − (𝜏𝑀 − 𝜏𝑚 ) ∫

𝑡−𝑑(𝑡)

𝑡−𝜏𝑀

≤−

(19)

𝑥̇𝑇 (𝑠) 𝑍2 𝑥̇ (𝑠) 𝑑𝑠

𝑡−𝑑(𝑡)

− (𝜏𝑀 − 𝜏𝑚 ) ∫

𝑥̇𝑇 (𝑠) 𝑍2 𝑥̇ (𝑠) 𝑑𝑠

𝜏𝑀 0 𝑥̇𝑇 (s) 𝑍3 𝑥̇ (𝑠) 𝑑𝑠. ∫ 2 −𝜏𝑀 /2

(20)

𝑡 𝜏 𝑥̇𝑇 (𝑠) 𝑍3 𝑥̇ (𝑠) 𝑑𝑠 − 𝑀∫ 2 𝑡−(𝜏𝑀 /2)

−

𝑥̇𝑇 (𝑠) 𝑍2 𝑥̇ (𝑠) 𝑑𝑠

𝜏𝑀 − 𝜏𝑚 𝑥 (𝑡 − 𝜏𝑚 ) 𝑇 𝑍2 −𝑍2 𝑥 (𝑡 − 𝜏𝑚 ) [ ][ ] [ ] ∗ 𝑍2 𝑥 (𝑡 − 𝑑 (𝑡)) 𝑑 (𝑡) − 𝜏𝑚 𝑥 (𝑡 − 𝑑 (𝑡)) 𝜏𝑀 − 𝜏𝑚 𝑥 (𝑡 − 𝑑 (𝑡)) 𝑇 𝑍2 −𝑍2 𝑥 (𝑡 − 𝑑 (𝑡)) [ ][ ] [ ] ∗ 𝑍2 𝑥 (𝑡 − 𝜏𝑀) 𝜏𝑀 − 𝑑 (𝑡) 𝑥 (𝑡 − 𝜏𝑀)

Journal of Control Science and Engineering

5

𝑇

𝑥 (𝑡 − 𝜏𝑚 ) − 𝑥 (𝑡 − 𝑑 (𝑡)) 𝑍 𝑆 ≤ −[ ] [ 2 12 ] ∗ 𝑍2 𝑥 (𝑡 − 𝑑 (𝑡)) − 𝑥 (𝑡 − 𝜏𝑀)

From (16)–(24), we have 𝑉̇ (𝑥𝑡 ) ≤ 𝜉𝑇 (𝑥𝑡 ) (

𝑥 (𝑡 − 𝜏𝑚 ) − 𝑥 (𝑡 − 𝑑 (𝑡)) ] ×[ 𝑥 (𝑡 − 𝑑 (𝑡)) − 𝑥 (𝑡 − 𝜏𝑀)

4

𝜏𝑚4 𝑇 (𝜏 − 𝜏 ) Ψ 𝑅1 Ψ + 𝑀 𝑚 Ψ𝑇𝑅2 Ψ 64 4

𝑇 −2𝑍2 + 𝑆12 + 𝑆12 𝑍2 − 𝑆12 𝑍2 − 𝑆12 𝑇 [ = 𝜁 (𝑥𝑡 ) ∗ −𝑍2 𝑆12 ] 𝜁 (𝑥𝑡 ) , ∗ ∗ −𝑍2 ] [

2

+

where (23)

Based on Lemma 2, an upper bound of 𝑉4̇ (𝑥𝑡 ) can be estimated as 𝑉4̇ (𝑥𝑡 ) =

𝜏𝑚4 𝑇 𝜏2 𝑥̇ (𝑡) 𝑅1 𝑥̇ (𝑡) − 𝑚 64 8 ×∫

𝑡

𝑡

∫ 𝑥̇𝑇 (𝑢) 𝑅1 𝑥̇ (𝑢) 𝑑𝑢 𝑑𝑠

4 𝜏𝑀 𝜏2 𝑥̇𝑇 (𝑡) 𝑅3 𝑥̇ (𝑡) − 𝑀 64 8

×∫

𝑡

𝑡

∫ 𝑥̇𝑇 (𝑢) 𝑅3 𝑥̇ (𝑢) 𝑑𝑢 𝑑𝑠 4

×∫

𝑡−𝜏𝑀

≤

2

(𝜏 − 𝜏 ) (𝜏M − 𝜏𝑚 ) 𝑇 𝑥̇ (𝑡) 𝑅2 𝑥̇ (𝑡) − 𝑀 𝑚 4 2 𝑡−𝜏𝑚

𝜏𝑚4

64

𝑡

Remark 7. Unlike [2],

𝑇

∫ 𝑥̇ (𝑢) 𝑅2 𝑥̇ (𝑢) 𝑑𝑢 𝑑𝑠

𝑡−𝜏

𝑀

4 𝜏𝑀

64

𝑥̇𝑇 (𝑡) 𝑅3 𝑥̇ (𝑡)

𝑡 𝜏𝑚 𝑥 (𝑠) 𝑑𝑠) 𝑥 (𝑡) − ∫ 2 𝑡−(𝜏𝑚 /2)

𝑡 𝜏𝑀 𝑥 (𝑠) 𝑑𝑠) 𝑥 (𝑡) − ∫ 2 𝑡−(𝜏𝑀 /2)

𝑇

Remark 9. The delay-departioning method in this paper reduces the conservativeness considerably, which arises from the LMI analysis of the interval [𝜏𝑚 , 𝜏𝑀]. But when the interval [𝜏𝑚 , 𝜏𝑀] is reduced, that is, when 𝜏𝑚 → 𝜏𝑀, the benefits of analysis method are shortened. The new strategy is to further make use of the information of the delay lower bound through departioning of the delay interval [0, 𝜏𝑚 ] into 𝑁 equidistant subintervals.

𝑡 𝜏𝑀 𝑥 (𝑠) 𝑑𝑠) 𝑥 (𝑡) − ∫ 2 𝑡−(𝜏𝑀 /2) 4

+

(𝜏𝑀 − 𝜏𝑚 ) 𝑇 𝑥̇ (𝑡) 𝑅2 𝑥̇ (𝑡) 4

− ((𝜏𝑀 − 𝜏𝑚 ) 𝑥 (𝑡) − ∫

𝑡−𝜏𝑚

𝑡−𝜏𝑀

× 𝑅2 ((𝜏𝑀 − 𝜏𝑚 ) 𝑥 (𝑡) − ∫

𝑡−𝜏

Remark 8. The relationship between the time-varying delay and its lower bound and upper bound is taken into account. As a result, fewer useful information is ignored in the derivative of the Lyapunov-Krasovskii functional.

𝑡 𝜏 × 𝑅1 ( 𝑚 𝑥 (𝑡) − ∫ 𝑥 (𝑠) 𝑑𝑠) 2 𝑡−(𝜏𝑚 /2)

× 𝑅3 (

𝑀

̇ conservation. In this paper, −(𝜏 − 𝜏𝑚 ) ∫𝑡−𝜏 𝑥̇𝑇 (𝑠)𝑍2 𝑥(𝑠)𝑑𝑠 is 𝑀 retained as well. Meanwhile, we deal with the integral terms of quadratic quantities by virtue of the reciprocally convex combination in Lemma 3; the obtained results are improved.

𝑇

−(

𝑡−𝜏

̇ is −(𝜏𝑀 − 𝜏𝑚 ) ∫𝑡−𝜏 𝑥̇𝑇 (𝑠)𝑍2 𝑥(𝑠)𝑑𝑠

̇ enlarged as −(𝜏𝑀 − 𝜏) ∫𝑡−𝜏 𝑥̇𝑇 (𝑠)𝑍2 𝑥(𝑠)𝑑𝑠 which may lead to

𝑠

𝑥̇𝑇 (𝑡) 𝑅1 𝑥̇ (𝑡) +

−(

̇ 𝑡 ) is negatively defined, based on If Θ < 0, then 𝑉(𝑥 the Lyapunov theory, we can conclude that NCS (5) is asymptotically stable.

Remark 6. To reduce the conservatism of stability criteria, the Lyapunov-Krasovskii functional contains some triple-integral terms in [12]. However, the new LyapunovKrasovskii functional in our paper which not only contains some triple-integral terms but also divides the lower and upper bounds of the delay into two equal segments, respectively, is proposed. The results will be less conservative.

𝑡−(𝜏𝑀 /2) 𝑠

+

2 𝜏𝑀 Ψ𝑇𝑍3 Ψ + Θ22 ) 𝜉 (𝑥𝑡 ) . 4

Remark 5. The robust stability criteria presented in Theorem 4 are for the nominal system. However, it is easy to further extend Theorem 4 to uncertain systems, where the systems matrices 𝐴 and 𝐴 𝑑 contain parameter uncertainties.

𝑡−(𝜏𝑚 /2) 𝑠

+

(25)

𝑇

+ (𝜏𝑀 − 𝜏𝑚 ) Ψ 𝑍2 Ψ

(22)

𝜁𝑇 (𝑥𝑡 ) = [𝑥𝑇 (𝑡 − 𝑑 (𝑡)) 𝑥𝑇 (𝑡 − 𝜏𝑚 ) 𝑥𝑇 (𝑡 − 𝜏𝑀)] .

4 𝜏2 𝜏𝑀 Ψ𝑇𝑅3 Ψ + 𝑚 Ψ𝑇 𝑍1 Ψ 64 4

+

𝑇

𝑥 (𝑠) 𝑑𝑠)

𝑡−𝜏𝑚

𝑡−𝜏𝑀

4. Numerical Example Example 10. Consider the following NCS [10]:

𝑥 (𝑠) 𝑑𝑠) . (24)

𝑥̇ (𝑡) = [

0 1 0 ] 𝑥 (𝑡) + [ ] 𝑢 (𝑡) . 0 −0.1 0.1

(26)

6

Journal of Control Science and Engineering Table 1: Admissible upper bound 𝜏𝑀 for various 𝜏𝑚 .

𝜏𝑚 Jiang and Han [10] Zhang et al. [8] Theorem 4

0.00 0.941 1.003 1.113

0.01 0.942 1.024 1.114

0.05 0.948 1.026 1.116

References 0.10 0.952 1.027 1.119

Table 2: Admissible upper bound 𝜏𝑀 for various 𝜏𝑚 . 𝜏𝑚 Shao [11] Sun et al. [12] Theorem 4

1 1.617 1.620 1.765

2 2.480 2.488 2.606

3 3.389 3.403 3.502

4 4.325 4.342 4.429

Assume that the state feedback gain matrix 𝐾 = [−3.75 − 11.5] when we do not consider the lower bound of the delay, that is, 𝜏𝑚 = 0.00, by applying Theorem 4, the maximum upper bound of delay obtained is 𝜏𝑀 = 1.113, while in [10], 𝜏𝑀 = 0.941, and the maximum allowable value is 𝜏𝑀 = 1.003 in [8]. A more detailed comparison for different values of 𝜏𝑚 is provided in Table 1. As shown in the table, it can be seen that our results are less conservative than the ones in [8, 10]. Example 11. Consider the NCS (5) with the following parameters [11]: 𝐴=[

0 1 ], −1 −2

𝐴 𝑑 = 𝐵𝐾 = [

0 0 ]. −1 1

(27)

Table 2 lists the maximum allowable upper bound of 𝜏𝑀 with respect to different conditions of 𝜏𝑚 along with some existing results from the literature. From Table 2, we can conclude that the criterion derived in this paper presents superior results.

5. Conclusion The problem of robust stability is addressed for a class of NCSs with network-induced delay and packet dropout. Improved and simplified stability criterion is established without involving any model transformation or free weighting matrices. The simulation results indicate that the criterion derived in this paper can exhibit better performance compared to that in the existing literature.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This work is supported by the National Nature Science Foundation of China (Grant no. 61203136 and Grant no. 61304064), the Natural Science Foundation Hunan Province of China (Grant no. 11JJ2038), and the Construct Program for the Key Discipline of electrical engineering in Hunan Province.

[1] G. C. Walsh, H. Ye, and L. G. Bushnell, “Stability analysis of networked control systems,” IEEE Transactions on Control Systems Technology, vol. 10, no. 3, pp. 438–446, 2002. [2] X. F. Jiang, Q.-L. Han, S. R. Liu, and A. K. Xue, “A new 𝐻∞ stabilization criterion for networked control systems,” IEEE Transactions on Automatic Control, vol. 53, no. 4, pp. 1025–1032, 2008. [3] H. J. Gao, T. W. Chen, and J. Lam, “A new delay system approach to network-based control,” Automatica, vol. 44, no. 1, pp. 39–52, 2008. [4] H. G. Zhang, G. T. Hui, and Y. C. Wang, “Stabilization of networked control systems with piecewise constant generalized sampled-data hold function,” International Journal of Innovative Computing, Information & Control, vol. 9, no. 3, pp. 1159–1170, 2013. [5] M. Yu, L. Wang, T. G. Chu, and F. Hao, “Stabilization of networked control systems with data packet dropout and transmission delays: continuous-time case,” European Journal of Control, vol. 11, no. 1, pp. 40–49, 2005. [6] H. Bentez-Prez and A. Bentez-Prez, “Networked control systems design considering scheduling restrictions and local faults using local state estimation,” International Journal of Innovative Computing, Information & Control, vol. 9, no. 8, pp. 3225–3239, 2013. [7] D. Yue, Q.-L. Han, and J. Lam, “Network-based robust 𝐻∞ control of systems with uncertainty,” Automatica, vol. 41, no. 6, pp. 999–1007, 2005. [8] X. X. Zhang, X. Jiang, and Q.-L. Han, “An improved stability criterion of networked control systems,” in Proceedings of the American Control Conference (ACC ’10), pp. 586–589, July 2010. [9] C. Peng, D. Yue, E. Tian, and Z. Gu, “A delay distribution based stability analysis and synthesis approach for networked control systems,” Journal of the Franklin Institute, vol. 346, no. 4, pp. 349–365, 2009. [10] X. Jiang and Q.-L. Han, “Delay-dependent robust stability for uncertain linear systems with interval time-varying delay,” Automatica, vol. 42, no. 6, pp. 1059–1065, 2006. [11] H. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009. [12] J. Sun, G. P. Liu, J. Chen, and D. Rees, “Improved delay-rangedependent stability criteria for linear systems with time-varying delays,” Automatica, vol. 46, no. 2, pp. 466–470, 2010. [13] K. Gu, “An integral inequality in the stability problem of timedelay systems,” in Proceedings of the 39th IEEE Confernce on Decision and Control, pp. 2805–2810, December 2000. [14] J. Sun, G. P. Liu, and J. Chen, “Delay-dependent stability and stabilization of neutral time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 19, no. 12, pp. 1364– 1375, 2009. [15] P. G. Park, J. W. Ko, and C. K. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, 2011.

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