Delay distribution based robust H ∞ control of networked control systems with uncertainties

Asian Journal of Control, Vol. 12, No. 1, pp. 46 57, January 2010Published online 28 October 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/asjc.156

DELAY DISTRIBUTION BASED ROBUST H∞ CONTROL OF

NETWORKED CONTROL SYSTEMS WITH UNCERTAINTIES

Chen Peng, Dong Yue, and Yu-Chu Tian

ABSTRACT

Network induced delay in networked control systems (NCS) is inher-ently non-uniformly distributed and behaves with multifractal nature. How-ever, such network characteristics have not been well considered in NCSanalysis and synthesis. Making use of the information of the statistical distri-bution of NCS network induced delay, a delay distribution based stochasticmodel is adopted to link Quality-of-Control and network Quality-of-Servicefor NCS with uncertainties. From this model together with a tighter boundingtechnology for cross terms, H∞ NCS analysis is carried out with significantlyimproved stability results. Furthermore, a memoryless H∞ controller is de-signed to stabilize the NCS and to achieve the prescribed disturbance attenu-ation level. Numerical examples are given to demonstrate the effectiveness ofthe proposed method.

Key Words: Networked control systems, H∞ control, stochastic distribution,linear matrix inequalities, uncertainties.

I. INTRODUCTION

Networked control systems (NCS) use data net-works to close both information and control loops.Significant effort is being made in the networkedcontrol community to develop general NCS networkingtechnologies, e.g. those based on Ethernet or wirelessEthernet, for control applications requiring remote anddistributed operations. Typical examples include remote

Manuscript received September 30, 2007; revised May 31,2008; accepted October 21, 2008.C. Peng and D. Yue are with the School of Electrical and Au-

tomation Engineering, Nanjing Normal University, 78 BancangStreet, Nanjing 210042, China (e-mail: [email protected],[email protected]).Y.-C. Tian (corresponding author) is with the School of

Information Technology, Queensland University of Technol-ogy, GPO Box 2434, Brisbane 4001, Australia; (e-mail:[email protected]).This work was supported in part by Natural Science Foun-

dation of China (NSFC) under grant numbers 60704024 and60774060, Australian Research Council under grant numberDP0559111), and Australian Government’s DEST (Depart-ment of Education, Science and Training) under InternationalScience Linkages (ISL) grant number CH070083.

medical treatment, urban heating systems, andwide areaplant automation over Internet Protocol (IP) networks.Due to its well-developed infrastructure, wide accep-tance, simplicity and affordability [1], IP networkinghas been promoted for networked control. Although IPnetworking is convenient for communications in large-scale and remotely distributed systems, there are stillmajor difficulties in design and implementation of NCSover IP networks [2]. Fundamentals of NCS over datacommunication networks can be found in [1, 3] and ref-erences therein for related works.

An NCS integrates information, communications,and control with control loops being closed through thenetwork. Among many problems in NCS, how to dealwith time-varying network induced delay is challeng-ing, and has been one of our recent research focuses,e.g. [2, 4–6]. Network induced delay degrades controlperformance or even causes system instability. Therehave been some reports in the open literature on dis-turbance attenuation and robust H∞ control of systemswithout transmission delay of control signals [7]. Mucheffort has also been made on this topic for NCS [3, 8, 9].By introducing some slack matrix parameters andconsidering the lower bound of network-induced de-lay, Yue et al. have studied the robust H∞ control

q 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

C. Peng et al.: Delay Distribution Based Robust H∞ 47

of uncertain NCS and obtained less conservative re-sults than previous work [8]; but the computationaldemand is high for the controller feedback gain dueto the introduced slack matrices. Under the assump-tion that the network-induced delay is less than thesampling period and satisfies Bernoulli random bi-nary distribution, Yang et al. have considered theobserver-based H∞ controller problem for NCS withrandom communication delays [3]; however, theyhave neglected the synchronization of the control sig-nals from the controller to actuator. Gao, Chen andLam have studied a special type of delay system,which contains multiple successive delay componentsin the state [10, 11]. Less conservative results canbe expected from [10, 11] because more informationon the inner delay distribution properties has beenutilized.

Although much research has been conductedin NCS analysis and synthesis, the statistical featuresof NCS communication networks have not been fullycaptured. For example, many published papers of the-oretical investigations have inherently assumed an uni-form distribution of the NCS network-induced delaybetween its lower and upper bounds. However, recentadvances have revealed that IP based NCS network-induced delay is non-uniformly distributed [1] andbehaves with a multifractal nature [6]. According toour work [12], it is our belief that the more the statis-tical features of the NCS networks are utilized in NCSanalysis and synthesis, the better the control perfor-mance obtained. The problem of H∞ robust analysisand synthesis for uncertain NCS with considerationof the stochastic distribution of the network delay re-mains open. Furthermore, simultaneous considerationof networks and control is crucial for system co-design,which links NCS quality-of-control (QoC) and NCSnetwork quality-of-service (QoS).

The overall aim of this paper is to develop a de-lay distribution-based robust H∞ control method fora class of uncertain NCS over IP networks. To makeuse of the delay characteristics of the NCS networks,we propose a new stochastic NCS model to capture thenon-uniform distribution of the network delays. Fromthis model together with a tighter bounding technology,delay dependent criteria for NCS stability and controlsynthesis are derived using a linear matrix inequality(LMI) approach. Because of the use of the delay distri-bution information, much less conservative results canbe obtained than those from existing references. Sincethe obtained conditions for the existence of admissiblecontrollers are not expressed as strict LMIs, the coneprocedure is employed to solve the non-convex feasi-bility problem.

Compared with our work in [12] which does notconsider system uncertainties, the work presented in thispaper deals with robust control with consideration ofsystem uncertainties in NCS.

Notation. N stands for positive integers, Rn denotesthe n-dimensional Euclidean space, Rn×m is the setof n ×m real matrices, I is the identity matrix ofappropriate dimensions. Matrix X>0 (respectively,X ≥ 0) for X ∈ Rn×n means that X is a real symmetricpositive definite (respectively, positive semi-definite).E{x} is the expectation of x . Pr{·} means probability.For an arbitrary matrix B and two symmetric matrices

A and C ,[A∗

BC

]denotes a symmetric matrix, where ∗

denotes the entries implied by symmetry. Ones([n,m])is an m-by-n matrix of ones.

II. NON-UNIFORM DISTRIBUTION OFNETWORK DELAY IN IP BASED

NCS

IP-based network delays display irregular be-haviour. To illustrate the characteristics of such delays,Tipsuwan et al. measured Round-Trip Time delaysfrom different Ethernet network nodes for 24 hours [1].Recently, Tian et al. have used the open source packagens2 to model and simulate typical scenarios of NCSover IP networks. For the scenarios described in [2] foran NCS over 10Mbps IP networks, the correspondinghistograms of the delays are shown in Fig. 1 [4, 12].

Figure 1 shows that the histograms skew to the left,indicating the higher probability to have delays shorter

20 40 60 80 100 120 1400

5

10

15

20

25

30

35

40

Per

cent

age

of th

e N

umbe

r of

Pac

kets

(%

)

Fig. 1. Histograms of the network delays reported in [2].


48 Asian Journal of Control, Vol. 12, No. 1, pp. 46 57, January 2010

than the median and mean delays. Actual delays canbe much longer than the median and mean, but withmuch lower probability. Generally speaking, althoughcommunication delay �(t) is time-varying, it is non-uniform. In most cases, small delays are dominant whilelarge delays are exiguous, implying that the probabilityof small delays is bigger than that of large delays. Thisphenomenon can be mathematically described as⎧⎪⎨⎪⎩

1) �0 ≤ �(t) ≤ �2<∞, ∀t ≥ 0

2) Pr[�(t) ∈ [�0, �1)] = �(t),

Pr[�(t) ∈ [�1, �2)] = 1 − �(t)

(1)

where �(t) ∈ [0, 1] is a stochastic variable, �1 is thebound which means Pr{�(t) ∈ [�0, �1)} = �(t).

It is worth mentioning that the general statisticaldistribution of the network induced delay described inEq. (1) can be regarded as one of the key characteristicsof NCS network QoS. The fundamental difficulty fromthis new description of network QoS is how to improvethe QoC under this new framework. In the followingsection, we devote to linking the QoS and QoC.

III. DELAY DISTRIBUTION BASED NCSCONTROL

Consider a class of control systems governed by

x(t) = (A + �A)x(t) + Bu(t) + B1w(t), (2)

z(t) =Cx(t), (3)

x(t) = �(t), t ∈ [−�2, �1] (4)

where x(t)∈ Rn , u(t) ∈ Rm and z(t) ∈ Rp are the statevector, control input vector and controlled output vec-tor, respectively. A, B, B1 and C are constant ma-trices with appropriate dimensions. �(t) is the initialcondition of the system. �2 and �1 are the lower andupper delay bounds, respectively. �A denotes the pa-rameter uncertainties satisfying �A=HF(t)E1, whereH and E1 are constant matrices with appropriate di-mensions, and F(t) is an unknown time-varying ma-trix, which is Lebesque measurable in t and satisfiesFT (t)F(t)≤I . w(t)∈ L2[t0,∞) denotes the externalperturbation. Throughout this paper, it is assumed thatsystem (2)–(4) is controlled over a network.

The following assumptions commonly used inNCS research in the literature are made in this work.

1. The sensors are clock-driven; and the controllersand actuators are event-driven;

2. All state variables are available for measurement;and the data is transmitted with a single-packet;and

3. Signal quantization and communication errorsover the network are not considered.

Using a similar modeling technique in [5, 8, 9], wemodel the closed-loop control system for (2)–(4) as

x(t) = (A + �A)x(t) + Bu(t) + B1w(t),

t ∈ [ikh + �ik , ik+1h + �ik+1) (5)

u(t+) =Kx(t − �ik ),

t ∈ {ikh + �ik , k = 1, 2, . . .} (6)

where u(t+) = limt→t+0 u(t), h is the sampling pe-riod, ik(k = 1, 2, 3, . . .) are some integers such that{i1, i2, i3, . . .} ⊂ {0, 1, 2, 3, . . .}, K is the state feedbackgain, the network-induced delay �ik is the time fromthe instant ikh when sensors sample from the plant tothe instant when actuators send control actions to theplant. Here, it is assumed that the control computationand other overhead delays are included in �ik [9].

In Equation (5), it is not required to have ik+1>ik .If ik+1−ik = 1, it means that there is no data packetdropout in the transmission. If ik+1>ik+1, there aredropped packets but the received packets are in or-dered sequence. If ik+1<ik , it means out-of-orderpacket arrival sequences occur; a typical scenariois that u(ik+1) =Kx(ik+1h) is implemented afteru(ikh) =Kx(ikh). Discarding the old data packet willhelp reduce network-induced delay �ik , which in turnmakes the system tolerate a larger amount of datapacket loss. Therefore, the time stamping techniqueis employed in this paper to implement the messagerejection, implying that the latest message is kept andold massages are discarded.

Define �(t) = t− ikh, t ∈ [ikh+�ik , ik+1h+�ik+1),k = 1, 2, 3, . . . , in every interval [ikh + �ik , ik+1h +�ik+1). It follows that

�ik ≤ �(t) ≤ (ik+1 − ik)h + �ik+1 . (7)

Since x(ikh) = x(t − (t − ikh)), from eqs. (6) and (7),we have

u(t+) =Kx(t − �(t)), �1 ≤ �(t) ≤ �2 (8)

where �1 = infk{�ik }, �2 = supk[(ik+1 − ik)h + �ik+1].Similar to [12, 13], define two sets �1 ={t :�(t)∈

[�0, �1)} and �2 = {t : �(t) ∈ [�1, �2)}. It is clear that�1 ∪ �2 = R+ and �1 ∩ �2 = 0. In terms of the def-initions of �1 and �2, it is seen that t ∈ �1 meansthe event �(t) ∈ [�0, �1) occurs and t ∈ �2 means the



event �(t) ∈ [�1, �2) occurs. Furthermore, we can definea stochastic variable �(t) as

�(t) ={1, t ∈ �1

0, t ∈ �2.(9)

To make use of the distribution information of theNCS network described in (1), a new feedback controllaw is proposed below to replace control law (8):⎧⎪⎨⎪⎩

u(t) = �(t)Kx(t − �1(t))

+(1 − �(t))Kx(t − �2(t)),

�1(t) ∈ [�0, �1), �2(t) ∈ [�1, �2)(10)

where⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩�1(t) = �(t)�(t),

and �(t) = 1 if �(t) ∈ [�0, �1)�2(t) = (1 − �(t))�(t),

and �(t) = 0 if �(t) ∈ [�1, �2).

(11)

Substituting (10) into (5), we obtain the followingmodel for the closed-loop control systems with the con-trolled output vector and initial conditions being givenin eqs. (3) and (4), respectively.

x(t) = (A + �A)x(t) + �(t)BKx(t − �1(t))

+(1 − �(t))BKx(t − �2(t)) + B1w(t) (12)

This equation is similar to Eq. (10) in [12] with theexception that system uncertainties are considered herein this paper.

Remark 1. Inspired by the work [3, 13–15], assumethat �(t) in (12) is a Bernoulli Distributed Sequence(BDS) with{

Pr{�(t) = 1}= E{�(t)} := �

Pr{�(t) = 0}= 1 − E{�(t)} := 1 − �(13)

where 0 ≤ � ≤ 1 is a constant. Different from the work[3, 14, 15], where BDS is employed to deal with thesystems with missing measurements, this paper utilisesBDS to describe the probability distribution of the NCSnetwork delay.

Remark 2. It is seen that system (12) makes use ofmore information about the distribution characteristicsof the NCS network induced delay. Less conservativeresults can be expected from this formulation than thosefrom the models without considering the non-uniformdelay distribution. This will be verified through numer-ical examples in Section VI.

Remark 3. When �(t)�0 or 1, model (12) is reduced to

x(t) = (A+�A)x(t)+BKx(t−�(t))+B1w(t) (14)

x(t) = �(t), t ∈ [−�2, 0] (15)

This special case implies that the statistical distributioncharacteristics of input u(t) is not considered in systemanalysis and synthesis. This case has been investigatedin [5, 16, 17]. We will show in Section VI that theirresults are the special cases of our results.

To facilitate further development, Eq. (12) can berewritten as

x(t) = �(t) + (�(t) − �)�(t) (16)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

�(t) � Ax(t) + �BKx(t − �1(t))

+(1 − �)BKx(t − �2(t))

+B1w(t) + HF(t)Ea

�(t) � BKx(t − �1(t))

−BKx(t − �2(t))

(17)

Now, let us formulate some practically computablecriteria for the asymptotic stability of the NCS describedby (12). The following definition is useful in derivingthe criteria.

Definition 1. For all admissible uncertainties, system(12) is said to be robustly asymptotically stable in themean-square with an H∞ norm bound � if the followingconditions hold:

1. For system (12) with w(t)�0, and �(t) describedin (1) and (13), the trivial solution (equilibriumpoint) is globally and asymptotically stable in themean square if the following holds:

limt−→∞ E|x(t, �(t))|2 = 0 (18)

2. Under the assumption of zero initial condition, thecontrolled output z(t) satisfies

E{‖z(t)‖2} ≤ �E{‖w(t)‖2} (19)

for any nonzero w(t)∈ L2[0, ∞) , where � is aprescribed scalar.

IV. H∞ PERFORMANCE ANALYSIS

Existing results of asymptotical stability of time-varying delay systems are delay distribution indepen-dent. This section will develop stability criteria that are



delay distribution dependent. An innovative approachwill be proposed to construct a Lyapunov functional thatutilizes the information of the delay distribution andlower bound of the delay. The main results of our NCSstability analysis are summarized below in Theorem 1.

Theorem 1. For given constants �i (i = 0, 1, 2), �>0and matrix K , if there exist matrices P , Q, R j( j = 1, . . . , 4)>0 with appropriate dimensions, scalarε>0, such that the following matrix inequality holds,then system (12) is asymptotically stable in the mean-square with an H∞ norm bound �.⎡⎢⎢⎢⎢⎣

�11 �12 �13 �14

∗ �22 0 0

∗ ∗ �33 0

∗ ∗ ∗ −ε I

⎤⎥⎥⎥⎥⎦<0 (20)

where

�11 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�11 R1 + �PBK R4 �14 PB1 PH

∗ −R1−R3 R3 0 0 0

∗ ∗ �33 0 0 0

∗ ∗ ∗ −R2 0 0

∗ ∗ ∗ ∗ −�2 I 0

∗ ∗ ∗ ∗ ∗ −ε I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

�11 = −R1 − R2 − R4 + PA + AT P + Q + CTC,

�33 = −Q − R3 − R4, �14 = R2 + (1 − �)PBK,

�12 = diag{A, �BK, 0, (1 − �)BK, B1, H}T×Ones(6, 4),

�13 = diag{0, (BK)T , 0,−(BK)T , 0, 0}Ones(6, 4),

�22 = diag

{−�−2

1 R−11 ,−

(�1 − �0

2

)−2

R−13 ,

−(

�0 + �12

)−2

R−14 ,−�−2

2 R−12

},

�14 = [εEa 0 0 0 0 0]T ,

�33 = [�(1 − �)]−1�22.

Proof. Define �= �0+�12 . Construct a Lyapunov-

Krasovskill functional candidate as

V (xt ) =5∑

i=1Vi (xt ) (21)

whereV1(xt ) = xT (t)Px(t),

V2(xt ) =∫ t

t−�xT (s)Qx(s)ds,

V3(xt ) =2∑

i=1

∫ 0

−�i

∫ t

t+sx T (v)�i Ri x(v) dv ds

V4(xt ) =∫ −�1

−�

∫ t

t+sx T (v)(� − �1)R3 x(v) dv ds

V5(xt ) =∫ 0

−�

∫ t

t+sx T (v)�R4 x(v)dvds,

R j>0, j = 1, . . . , 4. It follows from Eq. (13) that E{�−�} = 0, E{(�−�)2} = �(1−�), where �means �(t). Then,the mathematical expectation of the generator LV (xt )for the evolution of V (xt ) along the solutions of system(12) is given by

E{LV1(xt )} = 2E{xT (t)Px(t)}= 2E{xT (t)P�(t)

+(� − �)xT (t)P�(t)}= 2xT (t)P�(t) (22)

E{LV2(xt )}= xT (t)Qx(t) − xT (t − �)Qx(t − �) (23)

According to (16), we have

E

{2∑

i=1x T (t)�2i Ri x(t)

}

= E

{2∑

i=1[�(t) + (� − �)�(t)]T �2i Ri

×[�(t) + (� − �)�(t)]}

=2∑

i=1�T (t)�2i Ri�(t)

+2E(� − �)2∑

i=1�T (t)�2i Ri�(t)

+E[(� − �)2]2∑

i=1�T (t)�2i Ri�(t)

=2∑

i=1�T (t)�2i Ri�(t)

+�(1 − �)2∑

i=1�T (t)�2i Ri�(t) (24)



Applying Jessen’s inequality [18], when Ri>0, we have

−2∑

i=1

∫ t

t−�i (t)x T (v)�i Ri x(v)dv

≤2∑

i=1

[x(t)

x(t − �i (t))

]T

×[−Ri Ri

∗ −Ri

][x(t)

x(t − �i (t))

](25)

From (24) and (25), the following E{LV3(xt )} can bemore evolved.

E{LV3(xt )}

=2∑

i=1E

{x T (t)�2i Ri x(t)

−∫ t

t−�i

x T (v)�i Ri x(v)dv

}

≤2∑

i=1E

{x T (t)�2i Ri x(t)

−∫ t

t−�i (t)x T (v)�i Ri x(v)dv

}. (26)

�

The same operations can be applied to E{LV4(xt )}and E{LV5(xt )}. Then, considering (22)–(26) together,and similar to the method in [19] to deal with uncer-tainties, we have

E{LV (xt )} ≤ E{T (t)[�11 − �12�−122 �T

12

−�13�−133 �T

13](t) − z(t)T z(t)

+�2w(t)Tw(t)

+ε[F(t)Ea]T [F(t)Ea]} (27)

where T (t) =[xT (t), xT (t −�1(t)), xT (t −�), xT (t −

�2(t)), T (t), [F(t)Ea]T ], �i j (i, j = 1, 2, 3, 4) is de-fined in Theorem 1.

By Schur complement, (20) implies that

�11 − �12�−122 �T

12 − �13�−133 �T

13

+�14ε−1�T

14<0. (28)

It is clear that there exists a scalar ε>0 such that

T (t)�14ε−1�T

14(t)

−ε[F(t)Ea]T [F(t)Ea] ≥ 0 (29)

Then, combining (28) and (29), we have

T (t)[�11 − �12�−122 �T

12 − �13�−133 �T

13](t)

+ε[F(t)Ea]T [F(t)Ea] ≤ 0. (30)

With (30), (27) implies that

E{LV (xt )} ≤ E{�2w(t)Tw(t) − z(t)T z(t)} (31)

Under zero initial condition, integrating both sides of(31) from t0 to t and letting t → ∞, we have

E

{∫ ∞

t0z(s)T z(s)ds

}≤ E

{∫ ∞

t0�2w(s)Tw(s)ds

}thus E{‖z(t)‖2} ≤ �E{‖w(t)‖2}.

Next, we can prove the asymptotic stability of sys-tem (12) in mean-square. From (20) and (27), we haveE{LV (xt )}<0. It can now be concluded from Lyapunovstability theory that the dynamics of system (12) is ro-bustly asymptotically stable in the mean square. Then,by Definition 1, the result is established. This completesthe proof. �

Remark 4. Matrix inequality (20) is not an LMI dueto nonlinear terms in �22. But it is easy to transformedinto an LMI. Analysis of the transformation is omitted.

Remark 5. Model transformation is one of the sourcesof conservativeness [18]. Free weighting matrices alsointroduce conservativeness [20, 21]. Peng et al. derivedless conservative results based on an augmented matrixmethod, in which neither free weighting matrices norany model transformation have been employed [20].Using the convexity of the matrix function and employ-ing fewer unknown variables, Park et al. obtained im-proved results [22]. Compared with Moon’s inequality[23] and the slack variable matrix method [24] to dealwith cross terms in [8, 16, 25], our proof of Theorem1 uses neither model transformation nor free variablematrices. It is also worth mentioning that all vari-able matrices P and Ri in Theorem 1 appear in theLyapunov-Krasovskii functional (21); and their physi-cal interpretations are clear. Nevertheless, how to getless conservative results with reduced computationalpower demand is still challenging.



Remark 6. � reflects network QoS. The larger � is,the more network induced delays fall into the interval[�0, �1). It will be seen later that a larger � and smallerinterval [�0, �1) correspond to a larger allowable �2,which is the upper delay bound. This links the NCSQoC with the network QoS, and will help achieve anoptimized co-design of NCS networks and control tomaximize the overall performance of the NCS.

V. ROBUST H∞ CONTROLLER DESIGN

With Theorem 1, we are now ready to design thefeedback gain K to make system (12) robustly asymp-totically stable in the mean square with an H∞ normbound �. The main results of the control design are sum-marized below in Theorem 2.

Theorem 2. Given scalars �i (i = 0, 1, 2), �>0. System(12) is asymptotically stable in mean-square with feed-back gain K =VX−T and the H∞-norm constraint (19)is achieved for all nonzero (t), if there exist matri-ces Q, R j>0( j = 1, . . . , 4), a non-singular matrix X ,matrix V with appropriate dimension, scalar ε>0, suchthat the following matrix inequality holds⎡⎢⎢⎢⎢⎢⎢⎢⎣

�11 �12 �13 �14 XCT

∗ �22 0 0 0

∗ ∗ �33 0 0

∗ ∗ ∗ −ε I 0

∗ ∗ ∗ ∗ −I

⎤⎥⎥⎥⎥⎥⎥⎥⎦<0, (32)

where

�11=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�11 R1+�BV R4 �14 B1 H

∗ −R1−R3 R3 0 0 0

∗ ∗ �33 0 0 0

∗ ∗ ∗ −R2 0 0

∗ ∗ ∗ ∗ −�2 I 0

∗ ∗ ∗ ∗ ∗ −ε I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

�11 = −R1 − R2 − R4 + AXT + XAT + Q,

�14 = R2 + (1 − �)BV, �33 = − Q − R3 − R4,

�22 = diag

{−�−2

1 X R−11 X, −

(�1 − �0

2

)−2

X R−13 X,

−(�)−2X R−14 X,−�−2

2 X R−12 X

}

�12 = diag{X AT , �V T BT , 0, (1 − �)V T BT ,

BT1 , HT }Ones(6, 4),

�13 = diag{0, V T BT , 0, −V T BT , 0, 0}Ones(6, 4),�14 = [εEaX 0 0 0 0 0]T ,

�33 = [�(1 − �)]−1�22.

Proof. Pre- and post-multiply both sides of (20) withdiag(X, X, X, X, I, I, I, I, I, I, I, I, I, I, I ) and itstranspose. Defining V =KXT , X = P−1, Q =XQX,

Ri =XRi X (i = 1, · · · , 4), and applying Schur comple-ment, we have (32). This completes the proof. �

At this stage, if the above problem of (32) hasa solution, we say that there exists a feedback gainK =VX−T that guarantees the asymptotical stability ofsystem (12) in the mean square. However, the nonlin-ear equality matrix conditions X R−1

i X in (32) result indifficulties for obtaining such a solution. The followingmethod is developed to overcome the difficulties.

Define Mi , i = 1, . . . , 4, such that X R−1i X ≥ Mi ,

and define Si = M−1i and Ui = R−1

i . Replace condition(32) with the following (33) and (34)

⎡⎢⎢⎢⎢⎣�11 �12 �13 �14

∗ �22 0 0

∗ ∗ �33 0

∗ ∗ ∗ −ε I

⎤⎥⎥⎥⎥⎦< 0, (33)

[Si X

X Ui

]≥ 0, i = 1, . . . , 4 (34)

where �22 = diag{−�−21 M1, −(

�1−�02 )−2M3, −�−2M4,

−�−22 M2}, �33 = [�(1 − �)]−1�22.Using the cone complementarity approach [26],

we formulate the following nonlinear minimizationproblem with considerations of LMI conditions insteadof the original non-convex feasibility problem

Min: Trace

{XP +

4∑i=1

(Si Mi +Ui Ri )

}(35)

Subject to : (33), (34) and[P∗

IX

]>0,[

Mi I

∗ Si

]>0,

[Ri I

∗ Ui

]>0



For this optimization problem, the following algorithmis employed to obtain a suboptimal maximum of �2 forgiven �0, �1, � and �. The convergence of the algorithmis guaranteed in terms of the similar result of [26].Algorithm 1. Finding a suboptimal maximum �2 withrespect to �

1. Choose a sufficiently little initial �2>0, such thatthere exists a feasible solution to (33) and (34).Set �02 = �2.

2. Search a feasible set {X0, P0, M0i ,U0i , R0i , S0i(i = 1, . . . , 4)} satisfying LMI in (33) and (35).Set k = 0.

3. Solve the following LMI problem for the variables{X, P, Mi ,Ui , Ri , Si (i = 1, . . . , 4)}

Min : tr

(Xk P + Pk X +

4∑i=1

[Ski Mi

+MkiSi +Uki Ri + RkiUi ])

Subject to: (33) and (34).

(36)

Set Xk+1=X , Pk+1 = P , R(k+1)i = Rki , S(k+1)i = Ski ,M(k+1)i = Mik , U(k+1)i =Uki.

4. If (32) is satisfied, return to Step 2 after increasing�2 to some extent; if (32) is not satisfied withina specified number of iterations, then exit. Other-wise, set k = k + 1 and go to Step 3.

Remark 7. It is worth mentioning that there are al-ternative algorithms to solve out the non-LMIs inTheorem 3. One of such algorithms is to linearise thenonlinear item X R−1

i X by setting X = εi Ri in (32),where εi is a prescribed constant. Another algorithmis to replace the nonlinear item X R−1

i X with Ri − 2X[27]. Although the cone complementarity-based Al-gorithm 1 will increase computational complexity incomparison with the aforementioned methods, it givesless conservative results for the examples to be dis-cussed later in this paper.

So far, the controller has been designed to sat-isfy Definition 1. The results in Theorem 2 also suggestsome other types of optimization problems for controldesign, e.g. for the optimal H∞ control problem

{Minimize: �

Subject to: (33) and (34)(37)

VI. NUMERICAL EXAMPLES

This section aims to demonstrate the effectivenessand applications of the methodology developed in thispaper.

6.1 Performance analysis

Consider the following system investigated in [17]

x(t) =[0 1

0 −0.1

]x(t) +

[0

0.1

]u(t) (38)

When the effect of external perturbations on the systemis considered, (38) can be expressed as

x(t) =[0 1

0 −0.1

]x(t) +

[0

0.1

]u(t)

+[0.1

0.1

]w(t) (39)

z(t) = [0 1]x(t) (40)

Two cases are considered below for stability anal-ysis of system (39)–(40).

Case 1.1. For (38), the state feedback controllerdesigned in [17] is u(t) = [−3.75 − 11.5]x(t). It is de-signed without considering the presence of NCS net-works.

1. Without consideration of the distribution of thenetwork induced delay, the control law of [17] isemployed, i.e. u(t) = − [3.75, 11.5]x(t). Usingthe criteria from [5, 16, 17, 25, 28] and two specialcases of Theorem 1 of this paper (�0 = 0, �= 0or 1), we obtain the respective maximum allow-able transfer intervals (MATI), also called MADB[25], which guarantees the asymptotic stability ofsystem (38) controlled over a network. The re-sults are shown in Table I, which indicates that ourTheorem gives the same result as the existing best.

2. When the delay distribution is considered, thecontrol law of [12, 16] is modified to u(t) = −�(t)[3.75,−11.5]x(t − �1(t)) − (1 − �(t))[3.75,11.5]x(t − �2(t)). Our MATI results are listed inTable II. Existing methods in the open literaturecannot handle this case.

Case 1.2. Consider the H∞ performance of sys-tem (39)–(40) under the given controller and the H∞disturbance attenuation level �.



Table I. MATI results without consideration of networkcharacteristics.

Criterion [17] [28] [25] [5]MATI 0.00045 0.0538 0.7805 0.8695

Criterion Th.1† [16] Th.1‡MATI 0.8695 0.9410 0.9410

†�0 = 0, �= 0, finding the maximum �2.‡�0 = 0 and � = 1, finding the maximum �1.

Table II. MATI �2 with consideration of network charac-teristics.

�0 = 0.00 �0 = 0.02 �0 = 0.04� �1 = 0.02 �1 = 0.04 �1 = 0.06

0 0.8695 0.8695 0.86950.2 0.9721 0.9720 0.97190.4 1.1224 1.1222 1.12170.6 1.3746 1.3741 1.37320.8 1.9440 1.9429 1.94111 ∞ ∞ ∞

Table III. MATI �2 under �= 1 and with consideration ofnetwork characteristics.

�0 = 0.00 �0 = 0.02 �0 = 0.04� �1 = 0.02 �1 = 0.04 �1 = 0.06

0 0.8695 0.8695 0.86950.2 0.9710 0.9677 0.96220.4 1.1197 1.1114 1.09760.6 1.3696 1.3539 1.32730.8 1.9340 1.9042 1.85391 ∞ ∞ ∞

1. When the distribution of the network induced de-lay is not considered, i.e. � = 0, we obtain the min-imum allowable � = 1 for �1 = 0.8695.

2. When the distribution of the network induced de-lay is taken into account, we obtain the upperbound �2 under different lower bounds �0 and �1,as shown in Table III. Again, existing methods inthe open literature cannot handle this case.

From Tables I, II and III, we can draw the follow-ing general conclusions.

• When the distribution of the network induced delayis not considered, the result in this paper is as goodas the existing best in the open literature.

• It is seen from Tables II and III that when � = 0,i.e. when the distribution of the network induceddelay is considered, less conservative results canbe obtained.

• For fixed �(��0 or 1), �2 decreases as �0 or �1increases. When �0 and �1 are fixed, �2 increaseswith the increase of �. These results reflect the rela-tionship between QoS and QoC. For example, thelarger the probability of the network induced de-lay in the lower interval [�0, �1) (QoS), the largerthe allowable network induced delay upper bound�2 (QoC).• In Tables II and III, for any �0 ≤ �1 ≤ 0.8695,when � → 0 we have �2 = 0.8695. When � → 1,we have �2 → ∞, implying that most of the net-work induced delays �(t) fall into [�0, �1) and theprobability of �(t) ∈ [�1, �2) trends to zero. There-fore, when � → 1, the effect of �2 on system sta-bility is negligible.

6.2 Controller design

For controller design, we study the networked con-trol problem for an uninterruptible power system (UPS).The objective is to control the PWM inverter over anIP based NCS network such that the output AC voltageis kept at the desired value while maintaining robust-ness against load disturbances. A UPS with 1 KVA isconsidered in this example. The continuous-time modelcan be obtained from the discrete model described in[3] at half-load operating point.

x =⎡⎢⎣

0.0449 −0.093 0

0.1469 −0.0906 0

−6.6297 6.2635 −4.2872

⎤⎥⎦ x

+�A(t)x +⎡⎢⎣

0.0238

−0.0678

−3.8693

⎤⎥⎦ u(t)

+⎡⎢⎣1

0

0

⎤⎥⎦w(t) (41)

z(t) = [0.1 0 0]x(t) (42)

where ‖�A‖ ≤ 0.01.Two cases are considered below for controller de-

sign of the system.Case 2.1. Let us design an H∞ controller with or

without consideration of the IP network characteristics.Assume that the disturbance attenuation level � = 0.5.

Without consideration of the delay distributionof the NCS network (� = 0), using Theorem 2 and



Algorithm 1, we have the maximum allowable networkdelay �2 = 1.0.

When the network characteristics are considered(�= 0.5, �0 = 0, �1 = 0.2), we have the maximum al-lowable delay �2 = 1.7. The corresponding

X =⎡⎢⎣12.9 18.2 8.45

18.2 20.5 42.27

8.45 42.27 930.03

⎤⎥⎦and V =[−137.6118, 273.698,−463.8711], respec-tively.

To practically simulate the time-varying networkdelay, a stochastic delay table is established, in whichthe number of delay values �(t) ∈ [0, �1) is N1, and thenumber of delay values �(t) ∈ [�1, �2) is N2, respec-tively. For given �, it is required to have N1:N2 = � :(1− �) in order to model the distribution of the networkinduced delay.

With the delay bounds obtained above, thefeedback controller gain K can be computed asK =[−14.1371, 2.6782,−0.0432]. Furthermore, dif-ferent distributions of the network induced delay willgive different control results.

Case 2.2. Let us design the H∞ controller over anIP network described in [1] with � = 0.5, �1 = 0.0629,and �2 = 0.7562. The purpose is to minimize the H∞index, i.e. to solve the problem (37).

Solving the optimization problem (37) using theLMI Toolbox yields the the minimum allowable valueof � = 0.23 and K =[−35.2031, 1.9184, 0.0472].

However, if the network characteristics arenot considered, i.e. � = 0, we have � = 0.36 andK =[−22.1321, 2.0866, 0.0223] for �2 = 0.7562. Theresults are more conservative.

In summary, it is seen from the examples thatthe more information of the network characteristics isconsidered in NCS analysis and design, the less con-servative results can be obtained. Therefore, the delaydistribution based approach proposed in this paper iseffective to improve the NCS control performance.

VII. CONCLUSION

The disturbance attenuation problem for NCS overIP based networks has been investigated based on aLyapunov–Krasovskii functional method. A delay dis-tribution based stochastic control approach that linksQoC and QoC has been developed. Improved criteriafor H∞ performance analysis and H∞ control synthe-sis have been derived using this approach together with

a tighter bounding technology to deal with cross terms.Simulation results have been given to demonstrate theeffectiveness of the proposed approach.

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Chen Peng was born in Chinain 1972. He received his B.Sc.,M.Sc. and Ph.D. from the ChineseUniversity of Mining Technologyin 1996, 1999 and 2002 respec-tively. He was a Post-doctoral Re-search Fellow at Nanjing NormalUniversity.

From November 11 2004 toJanuary 2005, he was a Research

Associate at Hong Kong University. From July 2006 toAugust 2007, he was a Research Associate at Queens-land University of Technology. He is currently a fullProfessor at Nanjing Normal University. His researchinterests include networked control systems, and fuzzysystems and delay systems.

Dong Yue was born in 1964 inHenan, China. He received theB.S. degree in mechanical engi-neering from the Guilin Instituteof Electrical Engineering, Guilin,China, in 1985, the M.S. degreein applied mathematics from theAnhui University, Huainan, China,in 1991, and the Ph.D. degreein control engineering from the

South China University of Technology, Guangzhou,China, in 1995.

From June 1999 to September 1999, he was aresearch associate at City University of Hong Kong.From 2000 to 2001, he worked at Pohang Universityof Science and Technology as a senior scientist. FromJune to September, 2002, he was a research associateat Hong Kong University. From March 2004 to March2005, he was a visiting scholar at Central QueenslandUniversity, Australia. From January, 2007 to March,2007, he was a visiting scholar at Brunel University,



USA. He is currently a professor of control theory andcontrol engineering at Nanjing Normal University anddirector of Research Center for Information and ControlEngineering Technology. His research interests includenetworked control systems, time delay systems, robustcontrol and fuzzy control.

Yu-Chu Tian received the B.E.degree in electrical engineering(automation) fromWuhan Instituteof Engineering, China, in 1982, theM.E. degree in electrical engineer-ing (systems engineering) fromEast China University of Scienceand Technology, China, in 1987,the Ph.D. degree in Industrial Au-tomation (intelligent control and

decision) from Zhejiang University, China, in 1993,and the Ph.D. degree in Computer and Software En-gineering from the University of Sydney, Australia,in 2009.

He joined the Naval Academy of Engineering,China, in 1987 as a Lecturer in systems engineering.From 1994 to 1998, he worked at the Department ofElectrical Engineering and the Institute of IndustrialProcess Control, Zhejiang University, China, where hewas promoted to Associate Professor in 1995. From1996 to 1998, he was on leave from Zhejiang Universityto Hong Kong University of Science and Technology,

as a Research Associate working on complex systemsand process control. He joined Curtin University ofTechnology, Australia, in 1998 as a Research Fel-low, working on process systems engineering. He waselected as one of the first four Western Australia Strate-gic Research Fellows in 2000. Since 2002, he has beenwith the Faculty of Information Technology, Queens-land University of Technology, Australia, initially asa Lecturer, later as a Senior Lecturer and NetworkingTeaching Group Leader, and currently as an Asso-ciate Professor. As a visiting professor, he worked atthe Department of Computer Science, The Univer-sity of Maryland at College Park, Maryland, USA,in 2007.

Dr Tian’s research interests include complex sys-tems, systems engineering, real-time computing, em-bedded systems, robotics, computer networks, wirelesscommunications, wireless sensor/actuator networks,and control theory and engineering. He has publishedover 120 refereed journal and conference papers, andholds a patent. He has developed a large-scale net-worked control system for China’s largest and mostadvanced galvanising line, and a software system ofnetwork communications and real-time operation forthe world first, large-scale, fully-automated, multi-robot sweet corn processing line. As the Chief Investi-gator, he has won/completed 20 competitive researchgrants/projects. Dr Tian is a member of IEEE, IEAust,and IChemE. He is an Associate Editor for the Asia-Pacific Journal of Chemical Engineering.


Delay distribution based robust H ∞ control of networked control systems with uncertainties

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