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 Β© 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 [1β9]. 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
2
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]: π₯Μ (π‘) = π΄π₯ (π‘) + π΅π’ (π‘) ,
π‘+ β [ ππ β + ππ , ππ+1 β + ππ+1 ) , π = 0, 1, 2, 3, . . . ,
0 packet is lost,
ππ+1 β ππ = 2,
1 packet is lost,
has
0
0
π
π
0
(6)
Lemma 2 (see [14]). For any positive matrix π
, scalars π and π satisfying π > π, and π vector function π₯, the following inequality holds: (π β π)2 π π π β« β« π₯ (π’) π
π₯ (π’) ππ’ ππ 2 π π π
π
π
π
π
π
π
π
π
(7)
β₯ (β« β« π₯ (π’) ππ’ ππ ) π
(β« β« π₯ (π’) ππ’ ππ ) .
Lemma 3 (see [15]). Let πΉ1 , πΉ2 , πΉ3 , . . . πΉπ : π
π σ³¨β π
have positive values for arbitrary value of independent variable in an open subset π of π
π . The reciprocally convex combination of πΉπ (π = 1, 2, . . . , π) in π satisfies π π π π 1 min β πΉπ (π‘) = βπΉπ (π‘) + max β β ππ,π (π‘) π=1 ππ π=1 π=1 π=1, π =ΜΈ π
(8)
(3)
.. . ππ+1 β ππ = π,
π
(2)
where πΎ is the state feedback gain matrix, ππ denotes the network-induced delay, and β is the sampling period, {π1 , π2 , π3 , . . .} β {1, 2, 3, . . .}; due to the introduction of logical zero-order holder, the actuator will use the latest available control input, so ππ+1 > ππ . If {π1 , π2 , π3 , . . .} = {0, 1, 2, 3, . . .}, then no packet dropout occurred in the transmission. And the numbers of consecutive packet dropouts during the time interval (ππ β, ππ+1 β) can be described as follows: ππ+1 β ππ = 1,
π
π (β« π€(π )π ππ€ (π ) ππ ) β₯ (β« π€ (π ) ππ ) π (β« π€ (π ) ππ ) .
(1)
where π₯(π‘) β π
π , π’(π‘) β π
π are the plantβs state and input vectors, respectively. π΄ and π΅ are known to be real constant matrices with proper dimensions. This control signal is based on the plantβs state at the instant ππ β. So, the control law can be described as π’ (π‘+ ) = πΎπ₯ (ππ β) ,
Lemma 1 (see [13]). For any positive matrix π > 0, scalar π > 0, and a vector function π€ : [0, π] β π
π such that the π integration β«0 π€(π )π ππ€(π )ππ is well defined,
subject to π β 1 packets are lost.
π
Assume the existence of constants ππ > 0, ππ > 0. One
{ππ > 0, βππ = 1, ππ,π (π‘) : π
π σ³¨σ³¨β π
, π=1
(9) ππ β€ ππ , (ππ+1 β ππ ) β + ππ+1 β€ ππ,
π = 0, 1, 2, 3, . . . ,
(4)
where ππ and ππ 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)β(4), the system can be rewritten as follows: π₯Μ (π‘) = π΄π₯ (π‘) + π΄ π π₯ (π‘ β π (π‘)) , π₯ (π‘) = π (π‘) ,
π‘ β (βππ, 0) ,
(5)
where π΄ π = π΅πΎ, the function π(π‘) = π‘ β ππ β which satisfies ππ β€ π(π‘) β€ ππ denotes the time-varying delay in the control signal, and π(π‘) is the stateβs initial function. The following technical lemmas are introduced, which are indispensable for the proof of the main result.
πΉ (π‘) ππ,π (π‘) ] β₯ 0} . ππ,π (π‘) = ππ,π (π‘) , [ π ππ,π (π‘) πΉπ (π‘)
3. Main Results Theorem 4. For a given scalar ππ > ππ β₯ 0, NCS (5) is asymptotically stable if there exist symmetric matrices π = π
11 π
12 12 [πππ ]4Γ4 > 0, π = [ πβ11 π π22 ] > 0, π
= [ β π
22 ] > 0, ππ (π = 1, 2, 3) > 0, π
π (π = 1, 2, 3) > 0, and proper dimensions matrice π12 such that
[
π2 π12 ] > 0, β π2
Ξ Ξ Ξ = [ 11 12 ] < 0, β Ξ22
(10)
Journal of Control Science and Engineering
3 Ξ©44 = β π
22 β π2 ,
where
Ξ12
ππ2 Ξ¨π
1 ] [ 8 ] [ ] [ ] [ (π β π )2 ] [ π π Ξ¨π
2 ] [ ] [ 2 ] [ ] [ 2 ππ ] [ ] [ Ξ¨π
3 ], 8 =[ ] [ ] [ π π ] [ Ξ¨π1 ] [ 2 ] [ ] [ [ (ππ β ππ ) Ξ¨π2 ] ] [ ] [ ] [ ππ ] [ Ξ¨π3 2 ] [
Ξ©55 = β π1 β π11 + π22 , Ξ©66 = β π3 β π
11 + π
22 . (12) Proof. Let us choose a Lyapunov-Krasovskii functional candidate as 4
π (π₯π‘ ) = βππ (π₯π‘ ) ,
where
Ξ11 = diag {βπ
1 βπ
2 βπ
3 βπ1 βπ2 βπ3 } , Ξ©11 π11 π΄ π π12 βπ12
Ξ22
[β [ [β [ [β [ =[β [ [β [ [β [
β
[β
Ξ©15
Ξ©17
Ξ©18
Ξ©19
π΄ππ π13
π΄ππ π14
Ξ©22
Ξ©23
Ξ©24
0
0
β
Ξ©33
π12
βππ12
π΄ππ π12
0
π22
π23
π24
β
β
Ξ©44
0
βπ
π12
βπ22
βπ23
βπ24
β
β
β
Ξ©55
0
βππ23
βπ33
βπ34
β β β β
β β β β
β β β β
β β β β
Ξ©66 β β β
βππ34 0 βπ
1 β
βπ44 0 0 βπ
3
Ξ¨ = [π΄ π΄ π 0 0 0 0 0 0 0] with π π + π14 + π14 Ξ©11 = π΄π π11 + π11 π΄ + π13 + π13
+ π
11 + π11 β π1 β π3 β 2
β (ππ β ππ ) π
2 β
2 ππ
4
ππ2 π
4 1
π
3 ,
] ] ] ] ] ] ], ] ] ] ] ] ]
(11)
π₯ (π‘) ] [ π‘βππ ] [ π‘βππ ] [ ] [ β« [ π‘βππ π₯ (π ) ππ ] [ β«π‘βππ π₯ (π ) ππ ] ] [ π‘ ] [ π‘ ] π[ ], π1 (π₯π‘ ) = [ ] [β« ] [β« π₯ ππ π₯ ππ (π ) (π ) ] [ π‘β(π /2) ] [ π‘β(π /2) ] [ π‘ π ] [ π‘ π ] [ ] [ π₯ (π ) ππ π₯ (π ) ππ β« β« ] [ π‘β(ππ /2) ] [ π‘β(ππ /2) π2 (π₯π‘ ) π
π₯ (π ) π₯ (π ) ππ ] [π11 π12 ] [ π ] ππ =β« [ β π22 π₯ (π β π ) ) π‘β(ππ /2) π₯ (π β 2 2 π‘
π
π₯ (π ) π₯ (π ) ππ ] [π
11 π
12 ] [ π ] ππ , [ +β« β π
22 π₯ (π β π ) ) π‘β(ππ /2) π₯ (π β 2 2 π‘
π3 (π₯π‘ ) =
π‘ ππ 0 β« π₯Μπ (π’) π1 π₯Μ (π’) ππ’ ππ β« 2 βππ /2 π‘+π βππ
+ (ππ β ππ ) β«
Ξ©15 = β π13 + π1 + π12 ,
βππ
Ξ©16 = π
12 + π3 β π14 ,
+
π π Ξ©17 = π΄π π12 + π23 + π24 + (ππ β ππ ) π
2 , π + Ξ©18 = π΄π π13 + π33 + π34
ππ π
, 2 1
Ξ©19 = π΄π π14 + π34 + π44 +
ππ π
, 2 3
π4 (π₯π‘ ) =
π‘
β« π₯Μπ (π’) π2 π₯Μ (π’) ππ’ ππ π‘+π
π‘ ππ 0 β« π₯Μπ (π’) π3 π₯Μ (π’) ππ’ ππ , β« 2 βππ /2 π‘+π
π‘ π‘ ππ2 π‘ β« β« π₯Μπ (V) π
1 π₯Μ (V) πV ππ’ ππ β« 8 π‘β(ππ /2) π π’ 2
+
π Ξ©22 = β 2π2 + π12 + π12 ,
Ξ©23 = π2 β π12 ,
π
π₯ (π‘)
Ξ©16
βππ24 βπ
2 β β
+
(ππ β ππ ) π‘βππ π‘ π‘ π β« β« π₯Μ (V) π
2 π₯Μ (V) πV ππ’ ππ β« 2 π‘βππ π π’ 2 π‘ π‘ π‘ ππ β« β« π₯Μπ (V) π
3 π₯Μ (V) πV ππ’ ππ . β« 8 π‘β(ππ /2) π π’
π Ξ©24 = π2 β π12 ,
Ξ©33 = β π22 β π2 ,
(13)
π=1
Define an extended-state vector as
(14)
4
Journal of Control Science and Engineering
Ξ
ππ (π₯π‘ ) = [π₯π (π‘) π₯π (π‘ β π (π‘)) π₯π (π‘ β ππ ) π₯π (π‘ β ππ) π₯π (π‘ β
π‘βππ π‘ π‘ πm π ) π₯π (π‘ β π ) β« π₯π (π ) ππ β« π₯π (π ) ππ β« π₯T (π ) ππ ] , 2 2 π‘β(πm /2) π‘β(ππ /2) π‘βππ
(15)
and then NCS (5) is simplified as
By utilizing Lemma 1, we obtain
π₯Μ (π₯π‘ ) = ππ (π₯π‘ ) .
(16)
Calculating the derivative of π1 (π₯π‘ ) along the trajectory of NCS (5) yields
ππ π‘ π₯Μπ (π ) π1 π₯Μ (π ) ππ β« 2 π‘β(ππ /2) π
π₯ (π‘) π₯ (π‘) βπ π ππ ] [ β 1 βπ1 ] [ π ], 1 π₯ (π‘ β ) π₯ (π‘ β π ) [ [ 2 ] 2 ]
β€[
π
π₯ (π‘)
β
ππ (π‘) ] [ [ ] ] [ β« π₯ (π‘βππ ) β π₯ (π‘βππ)] [ π‘βππ π₯ (π ) ππ ] [ ] ] [ [ π‘ ] ] π[ ππ π1Μ (π₯π‘ ) = 2[ [ ]. ] [β« π₯ (π‘β ) (π‘)βπ₯ π₯ ππ (π ) [ ] ] [ [ π‘β(π /2) 2 ] ] [ [ π‘ π ] [ ππ ] π₯ (π‘)βπ₯ (π‘β ) π₯ (π ) ππ β« ] 2 ] [ [ π‘β(ππ /2) (17) π‘βππ
π
π₯ (π‘) π₯ (π‘) π ]. ππ ] [βπ3 π3 ] [ β€[ β βπ3 π₯ (π‘ β π ) π₯ (π‘ β ) 2 2 Define πΌ = (π(π‘) β ππ )/(ππ β ππ ), π½ = (ππ β π(π‘))/(ππ β ππ ); by the reciprocally convex combination in Lemma 3, the following inequality holds:
It is easy to get π
π₯ (π‘) π₯ (π‘) π π π2Μ (π₯π‘ ) = [ ππ ] [ β11 π12 ] [ π ] 22 π₯ (π‘ β ) π₯ (π‘ β π ) [ [ 2 ] 2 ]
π
π ππ π π₯ (π‘ β π )] ) π π 2 ] [ 11 12 ] [ 2 β β π22 ) π₯ (π‘ β π π₯ (π‘ β π π ] π) ] [ [ [π₯ (π‘ β
π½ [ β (π₯ (π‘ β ππ ) β π₯ (π‘ β π (π‘))) ] ] π2 π12 [ πΌ β[ ] [β π ] ] [ πΌ 2 ββ (π₯ (π‘ β π (π‘)) β π₯ (π‘ β ππ)) π½ ] [ (21)
π
π₯ (π‘) π₯ (π‘) ππ ] [π
11 π
12 ] [ π ] +[ β π
22 π₯ (π‘ β π ) π₯ (π‘ β ) 2 2 π ππ π π₯ (π‘ β π )] ) π
π
2 ] [ 11 12 ] [ 2 . β β π
22 ) ) π₯ (π‘ β π π₯ (π‘ β π π ] π ] [ [ (18) [π₯ (π‘ β
π½ [ β (π₯ (π‘ β ππ ) β π₯ (π‘ β π (π‘))) ] ] [ πΌ Γ[ ] < 0. ] [ πΌ ββ (π₯ (π‘ β π (π‘)) β π₯ (π‘ β ππ)) π½ ] [ Due to ππ β€ π(π‘) β€ ππ, according to Lemma 1 and inequalities (21), we have
The time derivative of π3 (π₯π‘ ) can be represented as π3Μ (π₯π‘ ) = π₯Μπ (π‘) (( Γ π₯Μ (π‘) β Γβ«
βππ
βππ
β
ππ2 4
2
) π1 + (ππ β ππ ) π2 + (
2 ππ
4
β (ππ β ππ ) β« ) π3 )
ππ 0 π₯Μπ (π ) π1 π₯Μ (π ) ππ β (ππ β ππ ) β« 2 βππ /2
π‘βππ
π‘βππ
π₯Μπ (π ) π2 π₯Μ (π ) ππ π‘βππ
= β (ππ β ππ ) β«
π‘βπ(π‘)
π‘βππ
β€β
(19)
π₯Μπ (π ) π2 π₯Μ (π ) ππ
π‘βπ(π‘)
β (ππ β ππ ) β«
π₯Μπ (π ) π2 π₯Μ (π ) ππ
ππ 0 π₯Μπ (s) π3 π₯Μ (π ) ππ . β« 2 βππ /2
(20)
π‘ π π₯Μπ (π ) π3 π₯Μ (π ) ππ β πβ« 2 π‘β(ππ /2)
β
π₯Μπ (π ) π2 π₯Μ (π ) ππ
ππ β ππ π₯ (π‘ β ππ ) π π2 βπ2 π₯ (π‘ β ππ ) [ ][ ] [ ] β π2 π₯ (π‘ β π (π‘)) π (π‘) β ππ π₯ (π‘ β π (π‘)) ππ β ππ π₯ (π‘ β π (π‘)) π π2 βπ2 π₯ (π‘ β π (π‘)) [ ][ ] [ ] β π2 π₯ (π‘ β ππ) ππ β π (π‘) π₯ (π‘ β ππ)
Journal of Control Science and Engineering
5
π
π₯ (π‘ β ππ ) β π₯ (π‘ β π (π‘)) π π β€ β[ ] [ 2 12 ] β π2 π₯ (π‘ β π (π‘)) β π₯ (π‘ β ππ)
From (16)β(24), we have πΜ (π₯π‘ ) β€ ππ (π₯π‘ ) (
π₯ (π‘ β ππ ) β π₯ (π‘ β π (π‘)) ] Γ[ π₯ (π‘ β π (π‘)) β π₯ (π‘ β ππ)
4
ππ4 π (π β π ) Ξ¨ π
1 Ξ¨ + π π Ξ¨ππ
2 Ξ¨ 64 4
π β2π2 + π12 + π12 π2 β π12 π2 β π12 π [ = π (π₯π‘ ) β βπ2 π12 ] π (π₯π‘ ) , β β βπ2 ] [
2
+
where (23)
Based on Lemma 2, an upper bound of π4Μ (π₯π‘ ) can be estimated as π4Μ (π₯π‘ ) =
ππ4 π π2 π₯Μ (π‘) π
1 π₯Μ (π‘) β π 64 8 Γβ«
π‘
π‘
β« π₯Μπ (π’) π
1 π₯Μ (π’) ππ’ ππ
4 ππ π2 π₯Μπ (π‘) π
3 π₯Μ (π‘) β π 64 8
Γβ«
π‘
π‘
β« π₯Μπ (π’) π
3 π₯Μ (π’) ππ’ ππ 4
Γβ«
π‘βππ
β€
2
(π β π ) (πM β ππ ) π π₯Μ (π‘) π
2 π₯Μ (π‘) β π π 4 2 π‘βππ
ππ4
64
π‘
Remark 7. Unlike [2],
π
β« π₯Μ (π’) π
2 π₯Μ (π’) ππ’ ππ
π‘βπ
π
4 ππ
64
π₯Μπ (π‘) π
3 π₯Μ (π‘)
π‘ ππ π₯ (π ) ππ ) π₯ (π‘) β β« 2 π‘β(ππ /2)
π‘ ππ π₯ (π ) ππ ) π₯ (π‘) β β« 2 π‘β(ππ /2)
π
Remark 9. The delay-departioning method in this paper reduces the conservativeness considerably, which arises from the LMI analysis of the interval [ππ , ππ]. But when the interval [ππ , ππ] is reduced, that is, when ππ β ππ, 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, ππ ] into π equidistant subintervals.
π‘ ππ π₯ (π ) ππ ) π₯ (π‘) β β« 2 π‘β(ππ /2) 4
+
(ππ β ππ ) π π₯Μ (π‘) π
2 π₯Μ (π‘) 4
β ((ππ β ππ ) π₯ (π‘) β β«
π‘βππ
π‘βππ
Γ π
2 ((ππ β ππ ) π₯ (π‘) β β«
π‘βπ
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.
π‘ π Γ π
1 ( π π₯ (π‘) β β« π₯ (π ) ππ ) 2 π‘β(ππ /2)
Γ π
3 (
π
Μ conservation. In this paper, β(π β ππ ) β«π‘βπ π₯Μπ (π )π2 π₯(π )ππ is π 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.
π
β(
π‘βπ
Μ is β(ππ β ππ ) β«π‘βπ π₯Μπ (π )π2 π₯(π )ππ
Μ enlarged as β(ππ β π) β«π‘βπ π₯Μπ (π )π2 π₯(π )ππ which may lead to
π
π₯Μπ (π‘) π
1 π₯Μ (π‘) +
β(
Μ π‘ ) is negatively defined, based on If Ξ < 0, then π(π₯ 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.
π‘β(ππ /2) π
+
2 ππ Ξ¨ππ3 Ξ¨ + Ξ22 ) π (π₯π‘ ) . 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 π΄ and π΄ π contain parameter uncertainties.
π‘β(ππ /2) π
+
(25)
π
+ (ππ β ππ ) Ξ¨ π2 Ξ¨
(22)
ππ (π₯π‘ ) = [π₯π (π‘ β π (π‘)) π₯π (π‘ β ππ ) π₯π (π‘ β ππ)] .
4 π2 ππ Ξ¨ππ
3 Ξ¨ + π Ξ¨π π1 Ξ¨ 64 4
+
π
π₯ (π ) ππ )
π‘βππ
π‘βππ
4. Numerical Example Example 10. Consider the following NCS [10]:
π₯ (π ) ππ ) . (24)
π₯Μ (π‘) = [
0 1 0 ] π₯ (π‘) + [ ] π’ (π‘) . 0 β0.1 0.1
(26)
6
Journal of Control Science and Engineering Table 1: Admissible upper bound ππ for various ππ .
ππ 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 ππ for various ππ . ππ 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 πΎ = [β3.75 β 11.5] when we do not consider the lower bound of the delay, that is, ππ = 0.00, by applying Theorem 4, the maximum upper bound of delay obtained is ππ = 1.113, while in [10], ππ = 0.941, and the maximum allowable value is ππ = 1.003 in [8]. A more detailed comparison for different values of ππ 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]: π΄=[
0 1 ], β1 β2
π΄ π = π΅πΎ = [
0 0 ]. β1 1
(27)
Table 2 lists the maximum allowable upper bound of ππ with respect to different conditions of ππ 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, βStability analysis of networked control systems,β IEEE Transactions on Control Systems Technology, vol. 10, no. 3, pp. 438β446, 2002. [2] X. F. Jiang, Q.-L. Han, S. R. Liu, and A. K. Xue, βA new π»β stabilization criterion for networked control systems,β IEEE Transactions on Automatic Control, vol. 53, no. 4, pp. 1025β1032, 2008. [3] H. J. Gao, T. W. Chen, and J. Lam, βA new delay system approach to network-based control,β Automatica, vol. 44, no. 1, pp. 39β52, 2008. [4] H. G. Zhang, G. T. Hui, and Y. C. Wang, βStabilization of networked control systems with piecewise constant generalized sampled-data hold function,β International Journal of Innovative Computing, Information & Control, vol. 9, no. 3, pp. 1159β1170, 2013. [5] M. Yu, L. Wang, T. G. Chu, and F. Hao, βStabilization of networked control systems with data packet dropout and transmission delays: continuous-time case,β European Journal of Control, vol. 11, no. 1, pp. 40β49, 2005. [6] H. Bentez-Prez and A. Bentez-Prez, βNetworked control systems design considering scheduling restrictions and local faults using local state estimation,β International Journal of Innovative Computing, Information & Control, vol. 9, no. 8, pp. 3225β3239, 2013. [7] D. Yue, Q.-L. Han, and J. Lam, βNetwork-based robust π»β control of systems with uncertainty,β Automatica, vol. 41, no. 6, pp. 999β1007, 2005. [8] X. X. Zhang, X. Jiang, and Q.-L. Han, βAn improved stability criterion of networked control systems,β in Proceedings of the American Control Conference (ACC β10), pp. 586β589, July 2010. [9] C. Peng, D. Yue, E. Tian, and Z. Gu, βA delay distribution based stability analysis and synthesis approach for networked control systems,β Journal of the Franklin Institute, vol. 346, no. 4, pp. 349β365, 2009. [10] X. Jiang and Q.-L. Han, βDelay-dependent robust stability for uncertain linear systems with interval time-varying delay,β Automatica, vol. 42, no. 6, pp. 1059β1065, 2006. [11] H. Shao, βNew delay-dependent stability criteria for systems with interval delay,β Automatica, vol. 45, no. 3, pp. 744β749, 2009. [12] J. Sun, G. P. Liu, J. Chen, and D. Rees, βImproved delay-rangedependent stability criteria for linear systems with time-varying delays,β Automatica, vol. 46, no. 2, pp. 466β470, 2010. [13] K. Gu, βAn integral inequality in the stability problem of timedelay systems,β in Proceedings of the 39th IEEE Confernce on Decision and Control, pp. 2805β2810, December 2000. [14] J. Sun, G. P. Liu, and J. Chen, βDelay-dependent stability and stabilization of neutral time-delay systems,β International Journal of Robust and Nonlinear Control, vol. 19, no. 12, pp. 1364β 1375, 2009. [15] P. G. Park, J. W. Ko, and C. K. Jeong, βReciprocally convex approach to stability of systems with time-varying delays,β Automatica, vol. 47, no. 1, pp. 235β238, 2011.
International Journal of
Rotating Machinery
Engineering Journal of
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Distributed Sensor Networks
Journal of
Sensors Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Control Science and Engineering
Advances in
Civil Engineering Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com Journal of
Journal of
Electrical and Computer Engineering
Robotics Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
VLSI Design Advances in OptoElectronics
International Journal of
Navigation and Observation Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Chemical Engineering Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Active and Passive Electronic Components
Antennas and Propagation Hindawi Publishing Corporation http://www.hindawi.com
Aerospace Engineering
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
International Journal of
International Journal of
International Journal of
Modelling & Simulation in Engineering
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Shock and Vibration Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Acoustics and Vibration Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014