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Automatica 44 (2008) 39–52www.elsevier.com/locate/automatica

A new delay system approach to network-based control�

Huijun Gaoa,∗, Tongwen Chenb, James Lamc

aSpace Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, PR ChinabDepartment of Electrical and Computer Engineering, University of Alberta, Edmonton, Alta., Canada T6G 2V4

cDepartment of Mechanical Engineering, University of Hong Kong, Hong Kong

Received 18 April 2006; received in revised form 4 December 2006; accepted 25 April 2007Available online 3 October 2007

Abstract

This paper presents a new delay system approach to network-based control. This approach is based on a new time-delay model proposedrecently, which contains multiple successive delay components in the state. Firstly, new results on stability and H∞ performance are proposedfor systems with two successive delay components, by exploiting a new Lyapunov–Krasovskii functional and by making use of novel techniquesfor time-delay systems. An illustrative example is provided to show the advantage of these results. The second part of this paper utilizes thenew model to investigate the problem of network-based control, which has emerged as a topic of significant interest in the control community.A sampled-data networked control system with simultaneous consideration of network induced delays, data packet dropouts and measurementquantization is modeled as a nonlinear time-delay system with two successive delay components in the state and, the problem of network-basedH∞ control is solved accordingly. Illustrative examples are provided to show the advantage and applicability of the developed results fornetwork-based controller design.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Linear matrix inequality (LMI); Networked control systems (NCSs); Sampled-data systems; Stability; Time-delay systems

1. Introduction

Time-delay systems, also called systems with after effector dead time, hereditary systems, equations with deviating ar-gument or differential–difference equations, have been an ac-tive research area for the last few decades. The main rea-son is that many processes include after-effect phenomena intheir inner dynamics, and engineers require models to behavemore like real processes due to the ever-increasing expecta-tions of dynamic performance. There have been a great number

� This paper was not presented at any IFAC meeting. This paper was rec-ommended for publication in revised form by Associate Editor Hitay Ozbayunder the direction of Editor Ian petersen. This work was partially supportedby Natural Sciences and Engineering Research Council of Canada, NationalNatural Science Foundation of China (60528007, 60504008), Program forNew Century Excellent Talents in University, China, an Alberta IngenuityFellowship, an Honorary Izaak Walton Killam Memorial Postdoctoral Fel-lowship, and RGC HKU 7031/06P.

∗ Corresponding author. Tel.: +86 451 86402350; fax: +86 451 86418091.E-mail addresses: [email protected] (H. Gao), [email protected] (T.

Chen), [email protected] (J. Lam).

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.04.020

of research results concerning time-delay systems scattered inthe literature. To mention a few, stability analysis is carried outin He, Wang, Lin, and Wu (2007), He, Wang, Xie, and Lin(2007b), He, Wu, She, and Liu (2004), Lin, Wang, and Lee(2006), Xia and Jia (2002); stabilizing and H∞ controllers aredesigned in Hua, Guan, and Shi (2005), and Zhang, Wu, She,and He (2005); robust filtering is addressed in Gao and Wang(2004), Liu, Sun, He, and Sun (2004), Wang and Burnham(2001), Wang, Huang, and Unbehauen (1999); and model re-duction/simplification is investigated in Gao, Lam, Wang, andXu (2004) and Xu, Lam, Huang, and Yang (2001). The impor-tance of the study on time-delay systems is further highlightedby the recent survey paper (Richard, 2003) and monographs(Gu, Kharitonov, & Chen, 2003; Niculescu, 2001).

Closely related to time-delay systems, network-based con-trol has emerged as a topic of significant interest in the controlcommunity. It is well known that in many practical systems,the physical plant, controller, sensor and actuator are diffi-cult to be located at the same place, and thus signals arerequired to be transmitted from one place to another. In mod-ern industrial systems, these components are often connected


40 H. Gao et al. / Automatica 44 (2008) 39 – 52

over network media (typically digital band-limited serial com-munication channels), giving rise to the so-called networkedcontrol systems (NCSs). Compared with traditional feedbackcontrol systems, where these components are usually connectedvia point-to-point cables, the introduction of communicationnetwork media brings great advantages, such as low cost,reduced weight and power requirements, simple installationand maintenance, and high reliability (Ishii & Francis, 2002).Therefore, NCSs receive more and more attention and becomemore and more popular in many practical applications in recentyears. Modeling, analysis and synthesis of network-based feed-back systems have been receiving increasing attention, which ishighlighted by the recent special issue edited by Antsaklis andBaillieul (2004). Among the reported results on NCSs, tomention a few, stability issue is investigated in Montestruqueand Antsaklis (2004), Walsh, Ye, and Bushnell (2002), andZhang, Branicky, and Phillips (2001), stabilizing controllersare designed in Yang, Wang, Hung, and Gani (2006), Yu,Wang, and Chu (2005a), Yu, Wang, Chu, and Hao (2005b),Zhang, Shi, Chen, and Huang (2005), Zhivoglyadov andMiddleton (2003), performance preserved control is studiedin Lian, Moyne, and Tilbury (2003), Seiler and Sengupta(2005), and Yue, Han, and Lam (2005), and moving hori-zon control is proposed in Goodwin, Haimovich, Quevedo,and Welsh (2004). In an NCS, the most significant featureis the network induced delays, which are usually causedby limited bits rate of the communication channels, by anode waiting to send out a packet via a busy channel, orby signal processing and propagation. The existence of sig-nal transmission delays generally brings negative effects onNCS stability and performance. This observation furtherenhances the importance of the study on time-delay sys-tems.

The most commonly and frequently used state-space modelto represent time-delay systems is

x(t) = Ax(t) + Adx(t − d(t)), (1)

where d(t) is a time delay in the state x(t), which is oftenassumed to be either constant, or time-varying satisfying certainconditions, e.g.,

0�d(t)� d < ∞, d(t)�� < ∞. (2)

Almost all the reported results on time-delay systems are basedon this basic mathematical model.

In a recent paper (Lam, Gao, & Wang, 2007), the followingnew model for time-delay systems is proposed:

x(t) = Ax(t) + Adx

(t −

s∑i=1

di(t)

), (3)

0�di(t)� di < ∞, di (t)��i < ∞. (4)

This model contains multiple delay components in the state,and a stability analysis result is reported in Lam et al. (2007)for systems with two successive delay components. The

introduction of this new model is motivated by the observa-tion that sometimes in practical situations, signals transmittedfrom one point to another may experience a few network seg-ments, which can possibly induce successive delays with differ-ent properties due to variable network transmission conditions,and has been clearly justified by a state-feedback remote con-trol problem. A numerical example has shown the advantageof the stability result.

The intention of Lam et al. (2007) is to expose the newmodel and to give a preliminary result on its stability analy-sis. It is worth noting that this stability condition leaves muchroom for improvement. A significant source of conservative-ness that could be further reduced lies in the calculation of thetime-derivative of the Lyapunov–Krasovskii functional. In ad-dition, only stability is analyzed, while the H∞ performancehas not been investigated, and application of this new model tothe emerging network-based control would likely yield betterperformance.

Following the work of Lam et al. (2007), it is our inten-tion in this paper to present new stability and H∞ perfor-mance conditions for systems with multiple successive delaycomponents, and apply this new model to network-based con-trol. To make our idea more lucid, we still consider the casewhere only two successive delay components appear in thestate, and the idea behind this paper can be easily extended tosystems with multiple successive delay components. New re-sults on stability and H∞ performance are proposed by ex-ploiting a new Lyapunov–Krasovskii functional and by mak-ing use of novel techniques for time-delay systems. An il-lustrative example is provided to show the significant advan-tage of the developed results. These constitute the contentsof Section 2.

In Section 3, we apply the new time-delay model to the prob-lem of network-based control. As can be seen later, a sampled-data NCS with simultaneous consideration of network induceddelays, data packet dropouts and measurement quantization canbe modeled as a nonlinear time-delay system with two succes-sive delay components in the state, which forms a solid back-ground for the new model mentioned above. Then, the H∞performance condition developed in Section 2 is exploited toinvestigate the problem of network-based H∞ control. Illus-trative examples are provided to show the advantage and appli-cability of the developed results for network-based controllerdesign.

Notation: The notation used throughout the paper is fairlystandard. The superscripts “T” and “−1” stand for matrixtransposition and matrix inverse, respectively; Rn denotesthe n-dimensional Euclidean space and the notation P > 0(�0) means that P is real symmetric and positive definite(semi-definite). In symmetric block matrices, we use an as-terisk (∗) to represent a term that is induced by symmetryand diag{. . .} stands for a block-diagonal matrix. Matrices,if their dimensions are not explicitly stated, are assumedto be compatible for algebraic operations. The space ofsquare-integrable vector functions over [0, ∞) is denoted byL2[0, ∞), and for w ={w(t)} ∈ L2[0, ∞), its norm is denotedby ‖w‖2.


H. Gao et al. / Automatica 44 (2008) 39–52 41

2. Main results on the new delay model

2.1. Stability analysis

Consider the following system with two successive delaycomponents in the state:

� : x(t) = Ax(t) + Adx(t − d1(t) − d2(t)),

x(t) = �(t), t ∈ [−d, 0]. (5)

Here x(t) ∈ Rn is the state vector; d1(t) and d2(t) represent thetwo delay components in the state and we denote d(t)=d1(t)+d2(t); A, Ad are system matrices with appropriate dimensions.It is assumed that

0�d1(t)� d1 < ∞, d1(t)��1 < ∞,

0�d2(t)� d2 < ∞, d2(t)��2 < ∞, (6)

and d = d1 + d2, � = �1 + �2. �(t) is the initial condition onthe segment [−d, 0].

The purpose of this subsection is to derive a new stabilitycondition under which system � in (5) is asymptotically sta-ble for all delays d1(t) and d2(t) satisfying (6). One possibleapproach to check the stability of this system is to simply com-bine d1(t) and d2(t) into one delay h(t) with

0�h(t)� d1 + d2 < ∞, h(t)��1 + �2 < ∞. (7)

Then, system � in (5) becomes

�1 : x(t) = Ax(t) + Adx(t − h(t)),

x(t) = �(t), t ∈ [−d, 0]. (8)

The stability of system �1 in (8) can be readily checked byusing some existing stability conditions. As discussed in Lamet al. (2007), however, since this approach does not make fulluse of the information on d1(t) and d2(t), it would be inevitablyconservative for some situations. In the following, we presenta new stability criterion.

Theorem 1. System � in (5) with delays d1(t) and d2(t) satis-fying (6) is asymptotically stable if there exist matrices P > 0,Q1 �Q2 �0, R�0, Z1 �Z2 > 0, M > 0, and S, T, U, V satis-fying[�1 + �2 + �T

2 + �3 �4∗ �5

]< 0, (9)

where

�2 = [S + V T − S U − T − U − V ],

�1 =⎡⎢⎣

�11 0 PAd 0∗ �12 0 0∗ ∗ −(1 − �)Q2 0∗ ∗ ∗ −R

⎤⎥⎦ ,

�11 = PA + ATP + Q1 + R, �12 = −(1 − �1)(Q1 − Q2),

�3 = �T31[d1Z1 + d2Z2 + dM]�31,

�31 = [A 0 Ad 0], �4 = [S T U V ],�5 = diag{−d−1

1 Z1, −d−12 Z2, −d−1Z2, −d−1M}. (10)

Proof. Define a Lyapunov–Krasovskii functional as

V (t) = V1(t) + V2(t) + V3(t) + V4(t),

V1(t) = xT(t)P x(t),

V2(t) =∫ t

t−d1(t)

xT(s)Q1x(s) ds +∫ t−d1(t)

t−d(t)

xT(s)Q2x(s) ds,

V3(t) =∫ t

t−d

xT(s)Rx(s) ds,

V4(t) =∫ 0

−d1

∫ 0

�xT(t + �)Z1x(t + �) d� d�

+∫ −d1

−d

∫ 0

�xT(t + �)Z2x(t + �) d� d�

+∫ 0

−d

∫ 0

�xT(t + �)Mx(t + �) d� d�, (11)

where P > 0, Q1 �Q2 �0, R�0, Z1 �Z2 > 0 and M > 0 arematrices to be determined. Then, along the solution of system� in (5), the time derivative of V (t) is given by

V1(t) = 2xT(t)P [Ax(t) + Adx(t − d(t))], (12)

V2(t)�xT(t)Q1x(t) − (1 − �)xT(t − d(t))Q2x(t − d(t))

− (1 − �1)xT(t − d1(t))(Q1 − Q2)x(t − d1(t)), (13)

V3(t) = xT(t)Rx(t) − xT(t − d)Rx(t − d), (14)

V4(t) = xT(t)[d1Z1 + d2Z2+dM]x(t)−∫ t

t−d1

xT(�)Z1x(�) d�

−∫ t−d1

t−d

xT(�)Z2x(�) d� −∫ t

t−d

xT(�)Mx(�) d�

� xT(t)[d1Z1+d2Z2 + dM]x(t)−∫ t

t−d

xT(�)Mx(�) d�

−∫ t

t−d1(t)

xT(�)Z1x(�) d�−∫ t−d1(t)

t−d(t)

xT(�)Z2x(�) d�

−∫ t−d(t)

t−d

xT(�)Z2x(�) d�. (15)

Note that in the above derivation, we have used the relationshipsQ1 �Q2 �0 and Z1 �Z2 > 0. By the Newton–Leibniz formula,for any appropriately dimensioned matrices S, T, U, V, we haveΥi = 0 (i = 1, . . . , 4) with

Υ1��T(t)S

(x(t) − x(t − d1(t)) −

∫ t

t−d1(t)

x(�) d�

),

Υ2��T(t)T

(x(t − d1(t)) − x(t − d(t)) −

∫ t−d1(t)

t−d(t)

x(�) d�

),



Υ3��T(t)U

(x(t − d(t)) − x(t − d) −

∫ t−d(t)

t−d

x(�) d�

),

Υ4��T(t)V

(x(t) − x(t − d) −

∫ t

t−d

x(�) d�

), (16)

where �(t) = [xT(t) xT(t − d1(t)) xT(t − d(t)) xT(t − d)]T.Then, from (11)–(16) we have

V (t)�2xT(t)P [Ax(t) + Adx(t − d(t))] + xT(t)Q1x(t)

− (1 − �1)xT(t − d1(t))(Q1 − Q2)x(t − d1(t))

− (1 − �)xT(t − d(t))Q2x(t − d(t))

+ xT(t)Rx(t) − xT(t − d)Rx(t − d)

+ xT(t)[d1Z1 + d2Z2 + dM]x(t)

−∫ t

t−d1(t)

xT(�)Z1x(�) d� −∫ t−d1(t)

t−d(t)

xT(�)Z2x(�) d�

−∫ t−d(t)

t−d

xT(�)Z2x(�) d� −∫ t

t−d

xT(�)Mx(�) d�

+ Υ1 + Υ2 + Υ3 + Υ4

��T(t)[�1 + �2 + �T2 + �3 + �6]�(t) +

10∑i=7

�i , (17)

where

�6 = d1SZ−11 ST + d2T Z−1

2 T T + dUZ−12 UT + dV M−1V T,

�7 = −∫ t

t−d1(t)

�T71Z

−11 �71 d�, �71 = ST�(t) + Z1x(�),

�8 = −∫ t−d1(t)

t−d(t)

�81Z−12 �81 d�, �81 = T T�(t) + Z2x(�),

�9 = −∫ t−d(t)

t−d

�91Z−12 �91 d�, �91 = UT�(t) + Z2x(�),

�10 = −∫ t

t−d

�T101M

−1�101 d�, �101 = V T�(t) + Mx(�).

(18)

Note that Zi > 0, i =1, 2, M > 0, thus �i , i =7, . . . , 10, are allnon-positive. By the Schur complement, (9) guarantees �1 +�2 +�T

2 +�3 +�6 < 0. Therefore, from (18) we have V (t) <−‖x(t)‖2 for a sufficiently small > 0 and x(t) �= 0, and theasymptotic stability is established (Hale & Lunel, 1993). �

Remark 1. Theorem 1 presents a new stability criterionfor system � with two successive time-varying delay com-ponents. This criterion is derived by defining the newLyapunov–Krasovskii functional in (11), which makes fulluse of the information about d1(t) and d2(t). It is also worthmentioning that some novel techniques have been exploited inthe calculation of the time derivative of V (t). On one hand, nosystem transformation has been performed to the original sys-tem and thus there is no need to seek upper bounds of the innerproduct between two vectors, which has the potential to yieldless conservative results; on the other hand, when derivingV4(t) in (15), we keep the last term

∫ t−d(t)

t−dxT(�)Z2x(�) d�,

which was often ignored in the derivation of stability conditionfor time-delay systems.

In the following, we further extend the above idea to an im-portant case. More specifically, we assume the two successivedelay components d1(t) and d2(t) have very different proper-ties in that d1(t) and d2(t) are assumed to be constant and non-differentiable, respectively. Thus the assumption in (6) reads

d1(t) ≡ d1 < ∞, 0�d2(t)� d2 < ∞. (19)

As can be seen in the next section, this case is much relatedto the model we use for network-based control. Then, we havethe following corollary.

Corollary 1. System � in (5) with delays d1(t) and d2(t) satis-fying (19) is asymptotically stable if there exist matrices P > 0,Q�0, R�0, Zi > 0, i = 1, 2, M > 0, and S, T , U, V satisfying[�1 + �2 + �T

2 + �3 �4∗ �5

]< 0, (20)

where �i , i = 2, . . . , 5, are given in (10) and

�1 =⎡⎢⎣

PA + ATP + Q + R 0 PAd 0∗ −Q 0 0∗ ∗ 0 0∗ ∗ ∗ −R

⎤⎥⎦ . (21)

Proof. Define the Lyapunov–Krasovskii functional

V (t) = V1(t) + V2(t) + V3(t) + V4(t),

where V1(t), V3(t) and V4(t) are given in (11) and

V2(t) =∫ t

t−d1

xT(s)Qx(s) ds,

with P > 0, Q�0, R�0, Zi > 0, i = 1, 2, M > 0 being matri-ces to be determined. Then, the corollary can be proved alongsimilar lines as in the proof of Theorem 1. �

2.2. H∞ performance analysis

In this subsection, we investigate the problem of H∞ per-formance analysis for systems with two successive delay com-ponents in the state. Consider the following system:

� : x(t) = Ax(t) + Adx(t − d1(t) − d2(t)) + Ew(t),

y(t) = Cx(t) + Cdx(t − d1(t) − d2(t)) + Fw(t),

x(t) = �(t), t ∈ [−d, 0]. (22)

Here x(t), �(t), d1(t) and d2(t) are the same as those in theabove subsection; w(t) ∈ Rl is the disturbance input whichbelongs to L2[0, ∞); A, Ad , E, C, Cd , F are system matriceswith appropriate dimensions. Our objective is to investigateunder what condition system � in (22) is asymptotically stablewith an H∞ disturbance attention level , that is, ‖y‖2 < ‖w‖2for all nonzero w ∈ L2[0, ∞) under zero initial condition. Wefirst consider the assumption in (6).



Theorem 2. System � in (22) with delays d1(t) and d2(t) sat-isfying (6) is asymptotically stable with an H∞ disturbanceattention level if there exist matrices P > 0, Q1 �Q2 �0,R�0, Z1 �Z2 > 0, M > 0, and S, T, U, V satisfying[�1 + �2 + �T

2 + �3 + �4 + �5 �4∗ �5

]< 0, (23)

where �4 and �5 are given in (10), and

�1 =

⎡⎢⎢⎢⎣

�11 0 PAd 0 PE

∗ �12 0 0 0∗ ∗ −(1 − �)Q2 0 0∗ ∗ ∗ −R 0∗ ∗ ∗ ∗ 0

⎤⎥⎥⎥⎦ ,

�2 = [S + V T − S U − T − U − V 0],�3 = �T

31[d1Z1 + d2Z2 + dM]�31,

�31 = [A 0 Ad 0 E], �41 = [C 0 Cd 0 F ],�4 = �T

41�41, �5 = diag{0, 0, 0, 0, −2I }, (24)

and �11, �12 are given in (10).

Proof. First, (23) implies (9), thus � in (22) with delays d1(t)

and d2(t) satisfying (6) is asymptotically stable. Now, define theLyapunov–Krasovskii functional as in (11). Then, by followingsimilar lines as in the proof of Theorem 1, along the solutionof system � in (22), the time derivative of V (t) is given by

V (t)� �T(t)[�1 + �2 + �T

2 + �3 + �6]�(t) +10∑i=7

�i ,

where �i , i = 1, 2, 3, are given in (24), �i , i = 6, . . . , 10, aregiven in (18) and �(t) = [�T(t) wT(t)]T. Thus, we have

yT(t)y(t) − 2wT(t)w(t) + V (t)

� �T(t)[�1 + �2 + �T

2 + �3 + �6 + �4 + �5]�(t)

+10∑i=7

�i . (25)

Note that Zi > 0, i = 1, 2, M > 0, thus �i , i = 7, . . . , 10, areall non-positive. Since (23) guarantees �1 + �2 + �T

2 + �3 +�6 + �4 + �5 < 0, we have

yT(t)y(t) − 2wT(t)w(t) + V (t) < 0 (26)

Table 1Calculated delay bounds for different cases

Delay bound d2 for given d1 Delay bound d1 for given d2

d1 = 1 d1 = 1.2 d1 = 1.5 d2 = 0.1 d2 = 0.2 d2 = 0.3

Theorem 1 0.512 0.406 0.283 2.300 1.779 1.453Lam et al. (2007) 0.415 0.376 0.248 2.263 1.696 1.324Wu et al. (2004), Jing et al. (2004), Fridman and Shaked (2003) 0.180 0.080 Infeasible 1.080 0.980 0.880Lee et al. (2001) Infeasible Infeasible Infeasible 0.098 Infeasible InfeasibleKim (2001) Infeasible Infeasible Infeasible 0.074 Infeasible Infeasible

for all nonzero w ∈ L2[0, ∞). Under zero initial condition, wehave V (0) = 0 and V (∞)�0. Integrating both sides of (26)yields ‖y‖2 < ‖w‖2 for all nonzero w ∈ L2[0, ∞), and theproof is completed. �

For the assumption in (19), we have the following corollary(the proof follows similar lines as in the proofs of Theorem 2and Corollary 1 and is thus omitted).

Corollary 2. System � in (22) with delays d1(t) and d2(t) sat-isfying (19) is asymptotically stable with an H∞ disturbanceattention level if there exist matrices P > 0, Q�0, R�0,Zi > 0, i = 1, 2, M > 0, and S, T, U, V satisfying[�1 + �2 + �T

2 + �3 + �4 + �5 �4∗ �5

]< 0,

where �4 and �5 are given in (10), �i , i = 2, . . . , 5, are givenin (24), and

�1 =

⎡⎢⎢⎢⎣

PA + ATP + Q + R 0 PAd 0 PE

∗ −Q 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ −R 0∗ ∗ ∗ ∗ 0

⎤⎥⎥⎥⎦ .

2.3. Illustrative example

In this subsection, we use a numerical example to illustratethe advantage of the proposed new model and the developedstability condition.

Example 1. Consider system � in (5) with the following pa-rameters, borrowed from Fridman and Shaked (2003), and Jing,Tan, and Wang (2004):

A =[−2 0

0 −0.9

], Ad =

[−1 0−1 −1

].

Suppose we know that

d1(t)�0.1, d2(t)�0.8.

Our purpose is to find the upper bound d1 of delay d1(t), or d2of d2(t), when the other is known, below which the system isasymptotically stable. By combining the two delay componentstogether, some existing stability results can be applied to thissystem. The calculation results obtained by Theorem 1 in this



paper, Theorem 1 in Lam et al. (2007), Theorem 2 in Wu,He, She, and Liu (2004), Theorem 1 in Jing et al. (2004),Theorem 3.2 in Lee, Moon, Kwon, and Lee (2001), Corollary1 in Kim (2001) and Theorem 1 in Fridman and Shaked (2003)for different cases are listed in Table 1. It can be seen from thetable that Theorem 1 in this paper yields the least conservativestability test than other approaches, showing the advantage ofthe stability result in this paper.

3. Application to network-based control

3.1. Problem formulation

In this section, we apply the results obtained above to theproblem of H∞ control for NCSs. Consider a typical NCSshown in Fig. 1. Suppose the physical plant is given by thefollowing linear system:

x(t) = Ax(t) + Bu(t) + Ew(t),

y(t) = Cx(t) + Du(t) + Fw(t). (27)

Here x(t) ∈ Rn is the state vector; u (t) ∈ Rp is the controlinput; w(t) ∈ Rl is the disturbance input which belongs toL2 [0, ∞); y(t) ∈ Rq is the output; and A, B, C, D, E, F aresystem matrices with appropriate dimensions.

In Fig. 1, it is assumed that the sampler is clock-driven,while the quantizer, controller and zero-order hold (ZOH) areevent-driven. The sampling period is assumed to be h where his a positive real constant and we denote the sampling instantof the sampler as sk , k = 1, . . . ,∞. In addition, it is assumedthat the state variable x(t) is measurable, and the measure-ments of x(t) are firstly quantized via a quantizer, and thentransmitted with a single packet. The quantizer is denoted asf (·) = [f1(·) f2(·) · · · fn(·)]T, which is assumed to be sym-metric, that is, fj (−v)=−fj (v), j =1, . . . , n. In this paper, weare interested in the logarithmic static and time-invariant quan-tizer. For each fj (·), the set of quantized levels is described by

Uj = {±u(j)i , i = 0, ±1, ±2, . . .} ∪ {0}. (28)

Network Medium

u

Controller

NetworkInduced Delay

NetworkInduced Delay

ZOH

Sampler

y

x

w

Quantizer

x

Physical Plant

Fig. 1. A typical networked control system.

According to Elia and Mitter (2001), and Fu and Xie (2005), aquantizer is called logarithmic if the set of quantized levels ischaracterized by

Uj = {±u(j)i , u

(j)i = �i

j u(j)0 , i = ±1, ±2, . . .},

∪{±u(j)0 } ∪ {0}, 0 < �j < 1, u

(j)0 > 0. (29)

Each of the quantization level u(j)i corresponds to a segment

such that the quantizer maps the whole segment to this quanti-zation level. In addition, these segments form a partition of R,that is, they are disjoint and their union equals to R. For thelogarithmic quantizer, the associated quantizer fj (·) is definedas

fj (v) =

⎧⎪⎨⎪⎩

u(j)i if

1

1 + j

u(j)i < v� 1

1 − j

u(j)i , v > 0,

0 if v = 0,

−fj (−v) if v < 0,

where

j = 1 − �j

1 + �j

. (30)

Then, at the sampling instant sk , we have

x(sk) = f (x(sk)) = [f1(x1(sk)) f2(x2(sk)) · · · fn(xn(sk))]T.

Now denote the updating instants of the ZOH as tk, k =1, . . . , ∞, and suppose that the updating signal (successfullytransmitted signal from the sampler to the controller and to theZOH) at the instant tk has experienced signal transmission de-lays �k (�k = �k + dk where �k is the delay from the quantizerto the controller and dk is the delay from the controller to theZOH. It is assumed that there is no delay between the sensorand quantizer). Therefore, the state-feedback controller takesthe following form:

u(tk) = Kf (x(tk − �k)), (31)

where K is the state-feedback control gain. Thus, consideringthe behavior of the ZOH, we have

u(t) = Kf (x(tk − �k)), tk � t < tk+1, (32)

with tk+1 being the next updating instant of the ZOH after tk .A natural assumption on the network induced delays �k can

be made as

�m ��k ��M, (33)

where �m and �M denote the minimum and the maximum de-lays, respectively. In addition, at the updating instant tk+1 thenumber of accumulated data packet dropouts since the last up-dating instant tk is denoted as �k+1. We assume that the maxi-mum number of data packet dropouts is �, that is,

�k+1 � �. (34)

Then, it can be seen from (33) and (34) that

tk+1 − tk = (�k+1 + 1)h + �k+1 − �k . (35)



Remark 2. It is worth noting that the assumption on the net-work induced delays �k made in (33) is more general than thosein Yu et al. (2005a, 2005b), and Yue et al. (2005). The maindifference lies in the lower bound we introduced. By assuming�m = 0, (33) is the same as those in Yu et al. (2005a, 2005b),and Yue et al. (2005). The introduction of the lower bound �mwill be shown later, via a numerical example, to be advanta-geous for reducing conservativeness by utilizing the idea ofsuccessive delay components developed in the above section.

Therefore, from (27)–(32) we obtain the following closed-loop system:

x(t) = Ax(t) + BKf (x(tk − �k)) + Ew(t),

y(t) = Cx(t) + DKf (x(tk − �k)) + Fw(t),

tk � t < tk+1. (36)

Remark 3. It is important to note that in (32), tk refers to theupdating instant of the ZOH. While in Yu et al. (2005a), thecontroller is expressed as

u(t) = Kx(tk), tk � t < tk+1, (37)

with tk standing for the sampling instant. It should be notedthat when the controller and actuator are event-driven, we can-not use the sampling instant to model the behavior of the ZOH.The reason is that the signal transmission delays may not nec-essarily be integer multiples of the sampling period, and thusthe ZOH may be updated between sampling instants. By usingthe updating instant in this paper, we do not need to synchro-nize the ZOH and the sampler, and thus the networked controlmodel formulated here is essentially different from that in Yuet al. (2005a) and is more general, though they appear to besimilar.

3.2. Key idea

It is noted that the closed-loop system in (36) is in the formof a sampled-data system. As the time sequence {tk} depends onboth the network induced delays and data packet dropouts, theperiod tk+1 − tk for the sampled-data system in (36) is variableand uncertain. Now, let us represent tk − �k in (36) as

tk − �k = t − �m − t + tk + �m − �k = t − �m − �(t), (38)

where

�(t) = t − tk + �k − �m. (39)

Then, from (35) we have

0��(t)��, (40)

where

� = �M − �m + (� + 1)h. (41)

By substituting (38) into (36) we obtain

x(t) = Ax(t) + BKf (x(t − �m − �(t))) + Ew(t),

y(t) = Cx(t) + DKf (x(t − �m − �(t))) + Fw(t). (42)

Remark 4. It is worth noting that in the above transformedsystem, �m is a constant delay, and �(t) is a non-differentiabletime-varying delay with bound �. Our main idea in the abovetransformation is to represent the sampled-data system in (36)as a continuous time system with two successive delay compo-nents in the state, which takes a very similar form as system �in (5) with the assumption in (19). The H∞ control problemwill be solved based on this new model.

Remark 5. If the lower bound of the network induced delaysis assumed to be zero, that is, �m = 0, (42) takes the followingform:

x(t) = Ax(t) + BKf (x(t − �(t))) + Ew(t),

z(t) = Cx(t) + DKf (x(t − �(t))) + Fw(t), (43)

with

0��(t)� �, (44)

where �=�M+(� + 1

)h. Compared with (40), the upper bound

of �(t) in (44) is increased by �m. In other words, without tak-ing the lower bound of the transmission delays into considera-tion, �m will be treated as a non-differentiable time-varying de-lay instead of a constant one when it is nonzero. Therefore, theintroduction of the lower bound �m will naturally reduce con-servativeness, which will be shown, via a numerical examplelater. However, existing results on networked control systems,such as Yu et al. (2005a, 2005b), Yue et al. (2005), and Yue,Han, and Peng (2004),did not offer to take the lower bound �minto consideration.

3.3. H∞ performance analysis

This subsection is concerned with the problem of H∞ per-formance analysis. More specifically, assuming that the matri-ces A, B, C, D, E, F in (27) and the controller gain matrix K in(31) are known, we shall study the conditions under which theclosed-loop NCS in (36) is asymptotically stable with an H∞disturbance attention level . The following theorem shows thatthe closed-loop H∞ performance can be guaranteed if thereexist some matrices satisfying certain LMIs. This theorem willplay an instrumental role in the problem of H∞ network-basedcontrol.

Before proceeding further, we give the following lemmawhich will be used later.

Lemma 1. Given appropriately dimensioned matrices �1, �2,�3, with �T

1 = �1. Then,

�1 + �3�2 + �T2 �T

3 < 0 (45)

holds if for some matrix W > 0

�1 + �3W−1�T

3 + �T2 W�2 < 0. (46)

Theorem 3. Consider the NCS in Fig. 1. Given the controllergain matrix K and a positive constant , the closed-loop systemin (36) is asymptotically stable with an H∞ disturbance atten-tion level if there exist matrices P > 0, Q�0, R�0, Zi > 0,



i = 1, 2, M > 0, S, T, U, V, and a diagonal matrix W > 0 sat-isfying⎡⎢⎢⎢⎣

�1 + �2 + �T2 + �5 + �2 �4 �3 �5 �6

∗ �8 0 0 0∗ ∗ �4 0 �7∗ ∗ ∗ −I DK

∗ ∗ ∗ ∗ −W

⎤⎥⎥⎥⎦< 0,

(47)

where �2 and �5 are given in (24), �4 is given in (10), and

�1 =

⎡⎢⎢⎢⎣

PA + ATP + Q + R 0 PBK 0 PE

∗ −Q 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ −R 0∗ ∗ ∗ ∗ 0

⎤⎥⎥⎥⎦ ,

�2 = diag{0, 0, �2W, 0, 0}, �3 = �T31�32,

�31 = [A 0 BK 0 E], �32 = [Z1 Z2 M],�4 = diag{−�−1

m Z1, −�−1Z2, −�−1M}, �7 = �T32BK ,

�5 = [C 0 KD 0 F ]T, �6 = [KTBTP 0 0 0 0]T,

�8 = diag{−�−1m Z1, −�−1Z2, −�−1Z2, −�−1M},

� = diag{ 1, . . . , n}, � = �m + �. (48)

Proof. First, considering the quantization behavior shown in(28)–(30) and according to Elia and Mitter (2001), and Fu andXie (2005), (42) can be expressed as

x(t) = Ax(t) + BK(I + �(t))(x(t − �m − �(t))) + Ew(t),

y(t) = Cx(t) + DK(I + �(t))(x(t − �m − �(t))) + Fw(t),

(49)

where

�(t) = diag{�1(t), �2(t), . . . ,�n(t)}, (50)

with

�j (t) ∈ [− j , j ], j = 1, . . . , n. (51)

By comparing system (49) and system (22) with the assumptionin (19), according to Corollary 2, we know that system (49) isasymptotically stable with an H∞ disturbance attention level if there exist matrices P > 0, Q�0, R�0, Zi > 0, i = 1, 2,M > 0, and S, T, U, V satisfying[�1 + �2 + �T

2 + �5 + �4�T4 + �2 �4

∗ �8

]< 0, (52)

where �2, �5 are given in (24), �4 is given in (10) and

�1 =

⎡⎢⎢⎢⎣

PA + ATP + Q + R 0 PBK(I + �(t)) 0 PE

∗ −Q 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ −R 0∗ ∗ ∗ ∗ 0

⎤⎥⎥⎥⎦ ,

�2 = �3[�mZ1 + �Z2 + �M]�T3 ,

�3 = [A 0 BK(I + �(t)) 0 E]T,

�4 = [C 0 DK(I + �(t)) 0 F ]T.

By the Schur complement, (52) is equivalent to⎡⎢⎣

�1 + �2 + �T2 + �5 �4 �5 �4

∗ �8 0 0∗ ∗ �4 0∗ ∗ ∗ −I

⎤⎥⎦< 0, (53)

where

�5 = [A 0 BK(I + �(t)) 0 E]T[Z1 Z2 M].Rewrite (53) in the form of (45) with

�1 =⎡⎢⎣

�1 + �2 + �T2 + �5 �4 �3 �5

∗ �8 0 0∗ ∗ �4 0∗ ∗ ∗ −I

⎤⎥⎦ ,

�2 = [�6 0 0 0], �6 = [0 0 �(t) 0 0],�3 = [�T

6 0 �T7 KTDT]T.

Then, according to Lemma 1, (53) holds if for some matrixW > 0⎡⎢⎣

�1 + �2 + �T2 + �5 �4 �3 �5

∗ �5 0 0∗ ∗ �4 0∗ ∗ ∗ −I

⎤⎥⎦

+⎡⎢⎣

�60�7DK

⎤⎥⎦W−1

⎡⎢⎣

�60�7DK

⎤⎥⎦

T

+ [�6 0 0 0]TW [�6 0 0 0] < 0. (54)

Note that W is required to be diagonal and positive definite.Then, by using a Schur complement operation and by consider-ing (51), (47) guarantees (54), and the proof is completed. �

If there is no quantizer in the NCS shown in Fig. 1, theclosed-loop system in (36) reads

x(t) = Ax(t) + BKx(tk − �k) + Ew(t),

y(t) = Cx(t) + DKx(tk − �k) + Fw(t), tk � t < tk+1. (55)

Then, we have the following corollary, which can be proved byfollowing similar lines as in the proof of Theorem 3.

Corollary 3. Consider the NCS in Fig. 1, but without the quan-tizer. Given the controller gain matrix K and a positive con-stant , the closed-loop system in (55) is asymptotically stablewith an H∞ disturbance attention level if there exist matri-ces P > 0, Q�0, R�0, Zi > 0, i = 1, 2, M > 0, and S, T, U,V satisfying⎡⎢⎣

�1 + �2 + �T2 + �5 �4 �3 �5

∗ �8 0 0∗ ∗ �4 0∗ ∗ ∗ −I

⎤⎥⎦< 0,

where �2 and �5 are given in (24), �4 is given in (10), and�i , i = 1, 3, 4, 5, 8, are given in (48).



3.4. H∞ controller design

This subsection is devoted to solving the problem of H∞controller design for NCSs.

Proposition 1. Consider the NCS in Fig. 1. Given a positiveconstant , there exists a state-feedback controller in the form of(31) such that the closed-loop system in (36) is asymptoticallystable with an H∞ disturbance attention level if there existmatrices P > 0, Q�0, R�0, Zi > 0, i = 1, 2, M > 0,K , S, T ,U , V , and a diagonal matrix W > 0 satisfying

⎡⎢⎢⎢⎢⎢⎣

�1 + �2 + �T2 + �5 �3 �5H �7 �8 �10

∗ �4 0 0 0 0∗ ∗ �6 0 �9 0∗ ∗ ∗ −I DK 0∗ ∗ ∗ ∗ −P W−1P 0∗ ∗ ∗ ∗ ∗ −�−2W

⎤⎥⎥⎥⎥⎥⎦< 0,

(56)

where �5 is given in (24) and

�1 =

⎡⎢⎢⎢⎣

AP + PAT + Q + R 0 BK 0 E

∗ −Q 0 0 0∗ ∗ 0 0 0∗ ∗ ∗ −R 0∗ ∗ ∗ ∗ 0

⎤⎥⎥⎥⎦ ,

�2 = [S + V T − S U − T − U − V 0],�3 = [S T U V ], H = [I I I ],�4 = diag{−�−1

m P Z−11 P , −�−1P Z−1

2 P ,

− �−1P Z−12 P , −�−1P M−1P },

�5 = [AP 0 BK 0 E]T, �7 = [CP 0 DK 0 F ]T,

�8 = [KTBT 0 0 0 0]T, �10 = [0 0 P 0 0]T,

�6 = diag{−�−1m Z1, −�−1Z2, −�−1M}, �9 = HTBK .

(57)

Moreover, if the above condition is feasible, a desired controllergain matrix in the form of (31) is given by

K = KP −1. (58)

Proof. From Theorem 3, we know that there exists a state-feedback controller in the form of (31) such that the closed-loopNCS in (36) is asymptotically stable with an H∞ disturbanceattention level if there exist matrices P > 0, Q�0, R�0,Zi > 0, i = 1, 2, M > 0, K, S, T, U, V, and a diagonal matrixW > 0 satisfying (47). Define

J = diag{J1, J2, J3, I, P−1},

J1 = diag{P −1, P −1, P −1, P −1, I },J2 = diag{P −1, P −1, P −1, P −1}, J3 = diag{Z−1

1 , Z−12 , M−1}.

Performing a congruence transformation to (47) by J, and aSchur complement operation to the term �2P −1WP −1 in the

(3, 3) block, together with the change of matrix variables de-fined by

P=P −1, M=M−1, W=W−1, Z1=Z−11 , Z2 = Z−1

2 ,

K = KP −1, Q = P −1QP −1, R = P −1RP −1,

[S T U V ] = J1[S T U V ]J2,

we obtain (56), and the proposition is proved. �

The condition in Proposition 1 cannot be implemented byusing standard numerical software due to the existence of theterms P Z−1

i P , P W−1P and P M−1P . By noticing Zi > 0 andM > 0, we have (Zi − P )Z−1

i (Zi − P )�0, (M − P )M−1(M −P )�0, (W − P )W−1(W − P )�0, which are equivalent to,respectively,

− P Z−1i P �Zi − 2P , −P M−1P �M − 2P ,

− P W−1P �W − 2P . (59)

By combining (56) and (59), we readily obtain the followingtheorem.

Theorem 4. Consider the NCS in Fig. 1. Given a positive con-stant , there exists a state-feedback controller in the form of(31) such that the closed-loop system in (36) is asymptoticallystable with an H∞ disturbance attention level if there existmatrices P > 0, Q�0, R�0, Zi > 0, i = 1, 2, M > 0, K , S,T , U , V , and a diagonal matrix W > 0 satisfying

⎡⎢⎢⎢⎢⎢⎢⎣

�1 + �2 + �T2 + �5 �3 �5H �7 �8 �10

∗ �4 0 0 0 0∗ ∗ �6 0 �9 0∗ ∗ ∗ −I DK 0∗ ∗ ∗ ∗ W − 2P 0∗ ∗ ∗ ∗ ∗ −�−2W

⎤⎥⎥⎥⎥⎥⎥⎦

< 0,

(60)

where �i , H are given in (57) and �4 =diag{�−1m (Z1 −2P ),

�−1(Z2 −2P ), �−1(Z2 −2P ), �−1(M −2P )}. Moreover, if theabove condition is feasible, a desired controller gain matrix inthe form of (31) is given by (58).

Remark 6. Note that (60) is an LMI not only over the matrixvariables, but also over the scalar . This implies that the scalar can be included as an optimization variable to obtain a reduc-tion of the H∞ disturbance attention level bound. Then, theminimum (in terms of the feasibility of (60)) H∞ disturbanceattention level bound with admissible controllers can be readilyfound by solving the following convex optimization problem:

Minimize subject to (60) over P > 0, Q�0, R�0, Zi > 0,i = 1, 2, M > 0, K , S, T , U ,V , and diagonal matrix W > 0.

Theorem 4 presents an LMI condition for the existence ofdesired state-feedback controllers based on the inequalities in(59). In the following, we present another approach to checkthe condition in Proposition 1.



Now introduce additional matrix variables Ni > 0, X > 0,and Y > 0 and replace (56) with

⎡⎢⎢⎢⎢⎢⎣

�1 + �2 + �T2 + �5 �3 �5H �7 �8 �10

∗ �4 0 0 0 0∗ ∗ �6 0 �9 0∗ ∗ ∗ −I DK 0∗ ∗ ∗ ∗ −Y 0∗ ∗ ∗ ∗ ∗ −�−2W

⎤⎥⎥⎥⎥⎥⎦

< 0, (61)

X − P M−1P �0, Y − P W−1P �0,

Ni − P Z−1i P �0, i = 1, 2, (62)

where

�4 = diag{−�−1m N1, −�−1N2, −�−1N2, −�−1X}.

Then, we readily obtain the following theorem.

Theorem 5. Consider the NCS in Fig. 1. Given a positive con-stant , there exists a state-feedback controller in the form of(31) such that the closed-loop system in (36) is asymptoticallystable with an H∞ disturbance attention level if there ex-ist matrices P > 0, P > 0, W > 0, Q�0, R�0, X > 0, X > 0,Y > 0, Y > 0, Ni > 0, Ni > 0, Zi > 0, Zi > 0, i = 1, 2, M > 0,K , S, T , U , V , and a diagonal matrix W > 0 satisfying (61)and[−X P

∗ −M

]�0,

[−Y P

∗ −W

]�0,[−Ni P

∗ −Zi

]�0, (63)

P P = I, XX = I, Y Y = I, WW = I ,

ZiZi = I, NiNi = I, i = 1, 2. (64)

Moreover, if the above condition is feasible, a desired controllergain matrix in the form of (31) is given by (58).

The condition presented in Theorem 5 is equivalent to thatin Proposition 1. It is noted that this condition is not a con-vex set due to the matrix equality constraints in (64). Severalapproaches have been proposed to solve such nonconvex fea-sibility problems, among which the cone complementarity lin-earization (CCL) method (El Ghaoui, Oustry, & Rami, 1997) isthe most commonly used one (for instance, the CCL algorithmhas been used for solving the controller design problems as wellas model reduction problems (Gao & Wang, 2003; Gao et al.,2004)). The basic idea in CCL algorithm is that if the LMI[

PI

IL

]�0 is feasible in the n × n matrix variables L > 0 and

P > 0, then tr(PL)�n, and tr(PL) = n if and only ifPL = I .Now using a cone complementarity approach (El Ghaoui

et al., 1997), we suggest the following nonlinear minimizationproblem involving LMI conditions instead of the original non-convex feasibility problem formulated in Theorem 5.

Problem NBCD (network-based controller design):

min tr(P P + XX + Y Y + WW +2∑

i=1

(ZiZi + NiNi))

subject to (61), (63) and[P I

I P

]�0,

[X I

I X

]�0,

[Y I

I Y

]�0,

[W I

I W

]�0,[

Zi I

I Zi

]�0,

[Ni I

I Ni

]�0, i = 1, 2.

According to El Ghaoui et al. (1997), if the solution of theabove minimization problem is 8n, that is,

min tr(P P + XX + Y Y + WW +2∑

i=1

(ZiZi + NiNi)) = 8n,

then the conditions in Theorem 5 are solvable. Algorithm 1 inEl Ghaoui et al. (1997) can be easily adapted to solve ProblemNBCD.

3.5. Illustrative examples

In this subsection, two examples are provided to illustratethe results developed above. We first use a numerical exam-ple to show the advantage by introducing the lower bound oftransmission delays. The second example is utilized to showthe applicability of the proposed controller design methods.

Example 2. Suppose the system matrices A, B, C, D, E, F in(27) and the controller gain K in (31) are given

A =[

0 1−1 −2

], B =

[01

], E =

[0.30.5

],

C = [1 0], D = 0.3, F = 0.5, K = [−1 1].The parameters for the quantizer f (·) are given by �1 =0.9 and�2 = 0.8, thus we have 1 = 0.0526 and 2 = 0.1111. It is as-sumed that the network induced delays �k satisfy �m ��k ��M,the maximum number of data packet dropouts is 2, and thesampling period is 10 ms. Our purpose is to determine the min-imum guaranteed closed-loop H∞ performances for differentvalues of lower delay bound �m.

Firstly, we assume �M = 0.4 s. When we do not consider thelower bound of the network induced delays, that is, �m = 0, byusing Theorem 3 (assuming that �m is sufficiently small), theminimum guaranteed closed-loop H∞ performance obtained ismin =3.1207. However, if we assume �m =0.1 s, the minimumguaranteed closed-loop H∞ performance obtained is min =2.8113. Secondly, we assume �M = 0.6 s. When �m = 0.3 s,we get min =6.0780. However, when �m =0.05, the conditionin Theorem 3 is found infeasible. A more detailed comparisonfor different cases is provided in Table 2, which shows thatconsidering the lower bound of the signal transmission delaygives rise to less conservative results.



Table 2Minimum feasible for different cases

�M (s) 0.4 0.6

�m (s) 0 0.2 0.05 0.2 0.3min 3.1207 2.5233 Infeasible 11.2314 6.0780

Example 3. Suppose the physical plant in Fig. 1 is the satel-lite system, considered in Biernacki, Hwang, and Battacharyya(1987). The satellite system consists of two rigid bodies joinedby a flexible link. This link is modelled as a spring with torqueconstant k and viscous damping f. Denoting the yaw angles forthe two bodies (the main body and the instrumentation mod-ule) by �1 and �2, the control torque by u(t), the moments ofinertia of the two bodies by J1 and J2, the dynamic equationsare given by

J1�1(t) + f (�1(t) − �2(t)) + k((�1(t) − �2(t))) = u(t),

J2�2(t) + f (�1(t) − �2(t)) + k((�1(t) − �2(t))) = 0.1w(t),

where w(t) denotes the disturbance. Assume the output is theangular positions �2(t). Thus, a state-space representation ofthe above equation is given by⎡⎢⎣

1 0 0 00 1 0 00 0 J1 00 0 0 J2

⎤⎥⎦⎡⎢⎣

�1(t)

�2(t)

�1(t)

�2(t)

⎤⎥⎦

=⎡⎢⎣

0 0 1 00 0 0 1

−k k −f f

k −k f −f

⎤⎥⎦⎡⎢⎣

�1(t)

�2(t)

�1(t)

�2(t)

⎤⎥⎦

+ [0 0 1 0]Tu(t) + [0 0 0 0.1]Tw(t),

y(t) = [0 1 0 0][�1(t) �2(t) �1(t) �2(t)]T.

Here we choose J1 =J2 = 1, k = 0.09 and f = 0.04 (the valuesof k and f are chosen within their respective ranges). Then, thecorresponding matrices described in Section 2 are given by

A =⎡⎢⎣

0 0 1 00 0 0 1

−0.3 0.3 −0.004 0.0040.3 −0.3 0.004 −0.004

⎤⎥⎦ ,

B =⎡⎢⎣

0010

⎤⎥⎦ , E =

⎡⎢⎣

000

0.1

⎤⎥⎦ ,

C = [0 1 0 0], D = 0, F = 0.

It is assumed that: the sampling period h=10 ms; the networkinduced delay bound in (33) are given by �m =10 ms and �M =40 ms ms; the maximum number of data packet dropouts �=2.Then, from (41) we have �=60 ms. In addition, the parametersfor the quantizer f (·) are assumed to be �1=�2 =�3=�4=0.9.

The eigenvalues of A are −0.04 + 0.4224j , −0.0400 −0.4224j , 0, 0; thus the above system is not stable. Our purpose

0 5 10 15 20 25 30-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (s)

Sta

te R

esponse

Fig. 2. State response.

is to design a state-feedback controller in the form of (31) suchthat the closed-loop system is asymptotically stable with anH∞ disturbance attention level . By using Theorem 4 (mini-mizing in (60)), we obtain the following matrices (for spaceconsideration we do not list all the obtained matrices here):

P =⎡⎢⎣

6.2203 0.2400 −0.6033 −0.61130.2400 0.6331 0.3561 −0.2321

−0.6033 0.3561 2.6993 −0.3402−0.6113 −0.2321 −0.3402 0.2044

⎤⎥⎦ ,

K = [−2.1220 0.0125 − 1.7269 0.0781].

Thus, according to (58), the gain matrix for the state-feedbackcontroller in (31) is given by

K = [−1.1789 − 1.3096 − 1.6629 − 7.3974],

and the obtained minimum guaranteed H∞ performance interms of the feasibility of (60) is ∗ = 0.7864.

We first illustrate that the closed-loop system is asymptot-ically stable under the above obtained controller. The initialcondition is assumed to be [−0.8 0.5 − 0.3 0.2]T. The stateresponses are depicted in Fig. 2, from which we can see thatall four state components converge to zero. In the simulation,the network induced delays and the data packet dropouts aregenerated randomly (meanly distributed within their ranges)according to the above assumption, and shown in Figs. 3 and4. The computed control inputs arriving at the ZOH are shownin Fig. 5, where we can see the discontinuous holding behaviorof the control inputs.

Next, we illustrate the H∞ performance of the closed-loopsystem. To this end, let us assume zero initial conditions, andselect a set of input signals as follows:

w(t) ={

sin 0.2t, 5� t �15 s,0 otherwise.

(65)



0 5 10 15 20 25 300.01

0.015

0.02

0.025

0.03

0.035

0.04

Time (s)

Netw

ork

Induced D

ela

y (

s)

Fig. 3. Network induced delays.

0 1 2 3 4 5 6 7 8 9 10

0

1

2

Time (s)

Data

Packet D

ropout

Fig. 4. Data packet dropouts.

Fig. 6 depicts the state responses. By calculation, ‖w‖2 =2.5468, ‖y‖2 = 1.3537, which yields

‖y‖2

‖w‖2= 0.5315 < ∗ = 0.7864,

showing the effectiveness of the H∞ controller design.

4. Conclusions

This paper has presented new results on stability and H∞performance for systems with two successive delay componentsin the state by exploiting a new Lyapunov–Krasovskii func-tional and by making use of novel techniques for time-delaysystems. An illustrative example is provided to show the signif-icant advantage of these results. Moreover, the proposed newresults have been utilized to investigate networked control sys-tems with simultaneous consideration of network induced de-

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (s)

Contr

ol In

put

Fig. 5. Control input signal.

0 5 10 15 20 25 30

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (s)

Sta

te R

esponse

Fig. 6. State response under (65).

lays, data packet dropouts and measurement quantization. Lessconservative and easily verifiable conditions for the existenceof admissible H∞ controllers have been obtained. Illustrativeexamples have been presented to show the advantage and appli-cability of the proposed network-based controller design meth-ods.

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Huijun Gao was born in Heilongjiang Province,China, in 1976. He received the M.S. degree inElectrical Engineering from Shenyang Univer-sity of Technology, Shengyang, China, in 2001,and the Ph.D. degree in Control Science and En-gineering from Harbin Institute of Technology,Harbin, China, in 2005. He was a Research As-sociate in the Department of Mechanical Engi-neering, University of Hong Kong, Hong Kong,from November 2003 to August 2004. FromOctober 2005 to September 2007, he carriedout his postdoctoral research in the Depart-ment of Electrical and Computer Engineering,

University of Alberta, Canada, supported by an Alberta Ingenuity Fellowshipand an Honorary Izaak Walton Killam Memorial Postdoctoral Fellowship.He joined Harbin Institute of Technology in November 2004, where he iscurrently a Professor.Dr. Gao’s research interests include network-based control, robust control/filtertheory, model reduction, time-delay systems, multidimensional systems andtheir engineering applications. He is an Associate Editor of IEEE Transac-tions on Systems, Man and Cybernetics Part B: Cybernetics, Journal of Intel-ligent and Robotics Systems, International Journal of Innovative Computing,Information and Control, and serves on the Editorial Board of InternationalJournal of Systems Science, Journal of the Franklin Institute and NonlinearDynamics and Systems Theory. He was the recipient of the University of



Alberta Dorothy J. Killam Memorial Postdoctoral Fellow Prize in 2005, andwas a co-recipient of the Outstanding Science and Technology DevelopmentAwards, from the Ministry of Machine-Building Industry of China, and fromthe Liaoning Provincial Government of China, both in 2002.

James Lam received a first class B.Sc. degreein Mechanical Engineering from the Universityof Manchester and was awarded the AshburyScholarship, the A.H. Gibson Prize and the H.Wright Baker Prize for his academic perfor-mance. From the University of Cambridge, heobtained the M.Phil. and Ph.D. degrees in con-trol engineering. He is a Scholar and Fellowof the Croucher Foundation. Dr. Lam is now aProfessor at the Department of Mechanical En-gineering, the University of Hong Kong. Priorto that, he held faculty positions at the CityUniversity of Hong Kong and the University of

Melbourne. Professor Lam is a Chartered Mathematician and Chartered Sci-entist, a Fellow of the Institute of Mathematics and its Applications, a SeniorMember of the IEEE, a Member of the IEE. He is holding a guest professor-ships at the Huazhong University of Science and Technology, South ChinaUniversity of Technology, Shandong University and Central South University.On the professional service side, Professor Lam is an Associate Editor of theAsian Journal of Control, the International Journal of Systems Science, theJournal of Sound and Vibration, the International Journal of Applied Mathe-matics and Computer Science, the IEEE Transactions on Signal Processing,the Journal of the Franklin Institute, the Conference Editorial Board of theIEEE Control Systems Society, an editorial member of the IET Control The-ory and Applications and the Open Electrical and Electronic EngineeringJournal. He has served as Editor-in-Chief of IEE Proceedings: Control The-ory and Applications and the IFAC Technical Committee on Control Design.His research interests include reduced-order modelling, delay systems,

descriptor systems, stochastic systems, multidimensional systems, robust con-trol and filtering. He has published numerous research articles in these areasand co-authored a monograph entitled Robust Control and Filtering of Sin-gular Systems (Springer, 2006).

Tongwen Chen received the B.Eng. degreein Automation and Instrumentation from Ts-inghua University (Beijing) in 1984, and theM.A.Sc. and Ph.D. degrees in Electrical En-gineering from the University of Toronto in1988 and 1991, respectively. From 1991 to1997, he was an Assistant/Associate Professorin the Department of Electrical and Com-puter Engineering at the University of Calgary,Canada. Since 1997, he has been with theDepartment of Electrical and Computer Engi-neering at the University of Alberta, Edmonton,Canada, and is presently a Professor. He held

visiting positions at the Hong Kong University of Science and Technology,Tsinghua University, Kumamoto University, and Harbin Institute of Technol-ogy Shenzhen Graduate School.His research interests include computer and network-based control systems,process control, multirate digital signal processing, and their applications toindustrial problems. He co-authored with B.A. Francis, the book OptimalSampled-Data Control Systems (Springer, 1995). Dr. Chen received a McCallaProfessorship for 2000–2001 and a Killam Professorship for 2006–2007, bothfrom the University of Alberta, and a Fellowship from the Japan Society forthe Promotion of Science for 2004. He was elected an IEEE Fellow in 2005.He has served as an Associate Editor for several international journals, in-cluding IEEE Transactions on Automatic Control, Automatica, Systems andControl Letters, and Journal of Control Science and Engineering. He is aregistered Professional Engineer in Alberta, Canada.

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