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Adaptive stabilisation of networked control systemstolerant to unknown actuator failuresA.H. Tahoun a & Hua-Jing Fang aa Department of Control Science and Engineering , Huazhong University of Science andTechnology , Wuhan 430074, ChinaPublished online: 30 Apr 2010.

To cite this article: A.H. Tahoun & Hua-Jing Fang (2011) Adaptive stabilisation of networked control systems tolerant tounknown actuator failures, International Journal of Systems Science, 42:7, 1155-1164, DOI: 10.1080/00207720903308975

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International Journal of Systems ScienceVol. 42, No. 7, July 2011, 1155–1164

Adaptive stabilisation of networked control systems tolerant to unknown actuator failures

A.H. Tahoun* and Hua-Jing Fang

Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

(Received 8 July 2008; final version received 1 September 2009)

In this article, adaptive state feedback stabilising controllers for networked adaptive control systems with unknownactuator failures are developed. The problems of networked control systems (NCSs) such as transmission delaysand data-packets dropout, induced by the insertion of data networks in the feedback adaptive control loopsare also considered. The novelty of this article consists in the combination of different aspects in NCSs: statetracking control of systems with unknown parameters, unknown actuator failures, network-induced delaysand data-packets dropout. Normalised adaptive laws are designed for updating the controller parameters.Sufficient conditions for Lyapunov stability are derived in the case of uncertainty due to actuator failures, delaysand data-packets dropout. Simulation results are given to illustrate the effectiveness of our design approach.

Keywords: Lyapunov theory-methods; networked control systems; stability; feedback control; actuator failures;delay, packet dropout; adaptive control

1. Introduction

In recent years, networked control system (NCS),system with feedback loops closed via real-time

network, has become a highly active research field.The primary advantages of an NCS are reduced systemwiring, a simpler installation and ease of systemdiagnosis and maintenance. On the other hand, factorssuch as bandwidth constraints, network-induced delaysand packet dropout, among many other peculiarities ofnetworks, often affect the performance of an NCS oreven cause instability.

The main stumbling block to be addressed whendealing with NCS is the stability of the overall systemwith network-induced delays and data packet dropout.Recently, much attention has been paid to the study ofthe stability analysis and control design of NCSs(Zhang, Branicky, and Phillips 2001; Zhang 2001;Montestruque and Antsaklis 2003; Jianyong, Shimin,andHaiqing 2004; Yue, Han, and Peng 2004; Yu,Wang,Chu, and Xie 2004; Yu, Wang, Chu, and Hao 2004;

Seiler and Sengupta 2005; Baillieul and Antsaklis 2007;Gupta, Hassibi, and Murray 2007; Hespanha,Naghshtabrizi, and Xu 2007; Hu, Bai, Shi, and Wu2007; Xiong and Lam 2007). Most of the existingmethods assume complete knowledge of the plant, whichis rarely the case in practice. Therefore, new adaptivedesigns are needed to handle uncertainties in NCSs.

The adaptive control problem of NCSs was firstdiscussed in Tahoun and Fang (2007a, b, c). In Tahoun

and Fang (2007a), the stability of adaptive control of

networked systems was considered with a network

inserted between sensors and the controller only, which

was extended in Tahoun and Fang (2007b) with the

network inserted between sensors and the controller

and between the controller and actuators, but without

delays and packet dropouts. In Tahoun and Fang

(2007c), an adaptive control model of NCSs in the

presence of time-varying network-induced delays was

proposed.Actuator failures may have significant damaging

effect on the performance of control systems and need

to be compensated, as they often cause undesired

system behaviour and sometimes lead to instability or

even catastrophic accidents. To improve system relia-

bility and performance, a controller has to be able to

accommodate those failures and compensate the effect

caused by them instantaneously and automatically

whenever the failures take place. There have been

several approaches for control of systems with actuator

failures including multiple models, switching and

tuning designs (Chen, Tao and Joshi 2001; Tao, Tang

and Joshi 2001; Tao, Joshi and Ma 2001; Chen, Tao

and Joshi 2002; Tao, Chen and Joshi 2002; Chen,

Tao and Joshi 2004; Du and Yang 2007). In this

article, we focus on direct adaptive solutions to the

actuator failure compensation problem, delays and

data-packets dropout in NCSs without explicit failure

detection.

*Corresponding author. Email: alitahoun@yahoo.com

ISSN 0020–7721 print/ISSN 1464–5319 online

� 2011 Taylor & Francis

DOI: 10.1080/00207720903308975

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An important feature of adaptive failure compen-sation is that such a design is able to adapt to changesin system failure pattern and failure values, so that inaddition to stability, asymptotic tracking of a referencesignal is ensured, despite the system and failureuncertainties.

The key design task is to find the appropriatecontroller structure and adaptive laws such that undercertain plant-model matching conditions, the adaptivecontroller can automatically adjust the remainingfunctional actuators to achieve a desired control objec-tive, despite the unknown failures of other actuators inthe controlled system (Chen, Tao, and Joshi 2004).

From the literature, it is found that the research inNCSs involves two important aspects. From theperspective of control design, the stability and theperformance of NCSs is the main issue to be consid-ered; (i.e. Zhang, Branicky, and Phillips 2001; Zhang2001; Montestruque and Antsaklis 2003; Jianyong,Shimin, and Haiqing 2004, among many others). Fromthe perspective of communication and informationtheory, the service of the network, such as schedulingstrategy, is also important when dealing with NCSs(i.e. Lee, Lee, and Lee 2005; Branicky, Phillips, andZhang 2002; Walsh and Ye 2001). The design ofcontrollers for NCSs has also been overlooked, asmany researchers start with a controller that has beendesigned, ignoring the challenges introduced by NCSsand then investigating to what extent such controllerscan guarantee stability in spite of the network. On theother hand, the results obtained for NCSs are stilllimited: most of the aforementioned results assumethat the plant is given and model parameters arecompletely available, while few articles address theanalysis and synthesis problems for NCSs whose plantparameters are partially unknown. In fact, whilecontrolling a real plant, the designer rarely knows itsparameters accurately (Narendra and Annaswamy1989). To the best of our knowledge, adaptive controlfor systems with unknown parameters in a networkenvironment has not been fully investigated, which isthe focus of this article.

This article is organised as follows: the problem isformulated in Section 2. The modelling of the delaysand data-packet dropouts is discussed in Section 3. Themain result is given in Section 4. Section 5 presents anillustrative example. Finally, we present our conclu-sions in Section 6.

2. Formulation of the problem

In the NCS shown in Figure 1, we consider two sourcesof network-induced delays: the sensor-to-controllerdelay, �sck , and the controller-to-actuator delay, �cak , in

which the compositive effect of the delay is �k.We make the following assumptions:

(A1) The sensor is clock-driven and both controller

and actuator are event driven.(A2) The data are transmitted in a single packet at

each time step.(A3) The data packets reach the controller and the

actuators by their original transmitting

sequence if they are not lost.(A4) The delay is time-varying and not known.

Remark 1: Assumption (A1) is standard in NCSs

design, assumption (A2) may need to be extended in

future work, assumption (A3) is needed to guarantee

that no more sensor updates are sent out until the

current one is received by the controller and assump-

tions (A3) and (A4) are the motivations of this article.

Also, we assume that there exists positive constant

� such that �¼ (kþ 1)h� (k� nk)hþ �kþ 1, where h is

the sampling period and nk is the number of packets

dropout at time khþ �k.In Figure 1, a class of linear time-invariant plants is

described as

_xðtÞ ¼AxðtÞþBuðtÞ,

t2 ½khþ �k,ðkþ1Þhþ �kþ1Þ, k¼ 0,1,2, . . . ,ð1Þ

where A2Rn�n and B2Rn�m are unknown constant

parameter matrices, x(t)2Rn is a state vector and

u(t)¼ [u1, . . . , um]T2Rm is a piecewise continuous

control input vector whose components may fail

during the system operation, and T denotes transpose.

The value of the input u(t) only changes at instants

(k� nk)hþ �k, and kept constant until the next control

)(ˆ tu )(tx

k(t )

Adaptivecontroller

Plant

Controller

SensorActuator

Data network

Strategy foradjusting

controller gains

sckτca

kτ

Figure 1. The block diagram of adaptive NCS.

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update is received at time (kþ 1� nkþ 1)hþ �kþ 1

(realised by a zero-order hold (ZOH)), where nk andnkþ 1 are the packets dropout at times khþ �k and(kþ 1)hþ �k, respectively. In Equation (1), the pair(A,B) is controllable and A, B are unknown matriceswith compatible dimensions.

2.1. Modelling of the actuator failures

In this article, we consider the type of actuator failuresmodelled as

uj ðtÞ ¼ �uj, t � tj, j 2 1, 2, . . . ,mf g, ð2Þ

where the constant value �uj and the failure time tj areunknown. For the control problem, it is assumed that

(A5) The system (1) is so constructed that for any upto m� 1 actuator failures, the remaining actua-tors can still achieve a desired control objective,when implemented with known parameters.

Remark 2: Assumption (A5) is a basic assumptionfor a class of actuator failure compensation (Chen,Tao, and Joshi 2001; Tao, Joshi, and Ma 2001; Chen,Tao, and Joshi 2002).

The key task is to adaptively adjust the remainingcontrols of the NCSs, shown in Figure 1, to achievesystem performance when there are up to m� 1actuator failures, without the knowledge of the systemsparameters.

As in Tao et al. (2001), in the presence of actuatorfailures, the input u(t) can be expressed as

uðtÞ¼ �ðtÞþ�½ �u��ðtÞ�,

t2 ½khþ �k,ðkþ1Þhþ �kþ1Þ, k¼ 0,1,2, . . . ,ð3Þ

where �ðtÞ is the designed control input, and

�u ¼ ½ �u1, �u2, . . . , �um�T

ð4Þ

� ¼ diagf�1, �2, . . . , �mg ð5Þ

�i ¼1 if the ith actuator fails0 otherwise

:

�ð6Þ

Given that the plant dynamics matrices (A,B) areunknown, and so are the actuator failure time tj,parameters �uj and j pattern, the control objective is todesign a state feedback control �ðtÞ such that all signalsin the closed-loop NCS are bounded and the statevector x(t) asymptotically converges to small constant.

2.2. Matching conditions and adaptive controllerstructure

To meet the adaptive control objective in the presenceof p-failed actuators, that is, uj ðtÞ ¼ �uj, j ¼ j1,

j2, . . . jp, 1 � p � m� 1, we assume that

(A6) There exist constant vectors K�1¼

½k�11,k�12, . . . ,k

�1m� 2R

n�m, k�2¼½k�21,k

�22, . . . ,k

�2m�

T2

Rm and k�3 ¼ ½k�31, k

�31, . . . , k�31�

T2Rm, such that

the following equations are satisfied:

Aþ BðI� �ÞK�T1 ¼ AþX

i 6¼j1,...jp

bik�T1i ¼ Am,

BðI� �Þk�2 ¼X

i 6¼j1,...jp

bik�2i ¼ bm,

BðI� �Þk�3 þ B� �u ¼X

i 6¼j1,...jp

bik�i þ

Xi¼j1,...jp

bj �uj ¼ 0,

ð7Þ

where Am and bm are known constant matrices, Am isthe Hurwitz the matrix satisfying PAm þ AT

mP ¼ �Q,and P and Q are symmetric and positive-definitematrices.

(A7) The sign of k�2i, sign½k�2i�, is known.

Remark 3: Assumption (A6) is the so-called matchingcondition (for more details see Chen et al. 2001, 2002),in which if A and B are known, the controller

��ðtÞ ¼ K�T1 xðtÞ þ k�3 ð8Þ

would lead to the network-free closed-loop systemfor t � 0

_xðtÞ ¼ AxðtÞ þ B��ðtÞ þ B�½ �u� ��ðtÞ�,

¼ AmxðtÞ � BðI� �ÞK�T1 xðtÞ

þ BðI� �Þ½K�T1 xðtÞ þ k�3� þ B �u�

¼ AmxðtÞ

whose state vector x(t) is exponentially stable, achiev-ing the control objective.

As A, B are unknown, the adaptive controller forthe considered NCS for t 2 ½khþ �k, ðkþ 1Þhþ �kþ1Þ,k ¼ 0, 1, 2, . . . , will be chosen as

uðtÞ ¼ �ðtÞ þ �½ �u� �ðtÞ�,

in which

�ðtÞ ¼ KT1 ðtÞ �xðtÞ þ k2ðtÞrðtÞ þ k3ðtÞ, ð9Þ

where K1ðtÞ 2 Rn�m, k2ðtÞ 2 Rm and kðtÞ 2 Rm are theadaptive estimates of K�1, k�2, k�3, respectively, and �xðtÞis the output of the sensor-controller network part.

Define the parameter errors

~k1iðtÞ ¼ k1iðtÞ � k�1i~k3iðtÞ ¼ k3iðtÞ � k�3i:

ð10Þ

for i¼ 1, . . . , m.

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2.3. Modelling of the delays and the packet dropout

Here, we present a new model of delays and packetdropout. As the controller is time varying, the network

can be considered as double switches (S1 and S2)rather than a single switch, as in Zhang (2001), Zhanget al. (2001) and Yu et al. (2004b), see Figure 2. When

the switch (S1 or S2) is closed (in position 1), a networkpacket is transmitted, whereas when it is open (inposition 2), the packet is lost and the old data packet is

used.The modelling of the packet loss between sensors

and the controller can be expressed as follows:

With no packet dropout at time khþ �sck :

�xðtÞ ¼ xðkhÞ

With one packet dropout at time khþ �sck :

�xðtÞ ¼ xððk� 1ÞhÞ

With two packets dropout at time khþ �sck :

�xðtÞ ¼ xððk� 2ÞhÞ

..

.

With ‘sck packets dropout at time khþ �sck :

�xðtÞ ¼ xððk� ‘sck ÞhÞ:

The quantity of dropped packets between sensors

and the controller is accumulated from the latest timewhen �xðtÞ has been updated.

The modelling of the packet loss between sensorsand the actuators can be expressed, with ‘cak packets

dropout at time khþ �k, as follows:

With no packet dropout between sensors andcontroller:

�ðtÞ ¼KT1 ððk�‘

cak Þhþ �

sck Þxððk�maxð0,‘sck�‘ca

kþ‘cak ,‘

cak ÞÞhÞ

þk3ððk�‘cak Þhþ �

sck Þ

With one packet dropout between sensors andcontroller:

�ðtÞ ¼KT1 ððk�‘

cak Þhþ �

sck Þxððk�maxð1,‘sck�‘ca

kþ‘cak ,‘

cak ÞÞhÞ

þk3ððk�‘cak Þhþ �

sck Þ

With two packet dropout between sensors and

controller:

�ðtÞ ¼KT1 ððk�‘

cak Þhþ �

sck Þxððk�maxð2,‘sck�‘ca

kþ‘cak ,‘

cak ÞÞhÞ

þk3ððk�‘cak Þhþ �

sck Þ

..

.

In general, the modelling of the packet loss between

sensors and the actuators with ‘sck packets dropout

between sensors and controller and ‘cak packets drop-

out between the controller and the actuators can be

summarised at time khþ �k as follows:

�ðtÞ ¼ KT1 ððk� ‘

cak Þhþ �

sck Þxððk� nkÞhÞ

þ k3ððk� ‘cak Þhþ �

sck Þ, ð11Þ

where

nk ¼ maxð‘sck , ‘sck�‘ca

kþ ‘cak , ‘

cak Þ 2 f0, 1, 2, . . .g:

Again, the quantity of dropped packets between the

controller and the actuators is accumulated from the

latest time when �ðtÞ has been updated.

Remark 4: When ‘sck ¼ ‘cak , it means that the packet is

dropped between sensors and controller and between

the controller and the actuators in the same transmis-

sion period. In this case, � (t) is replaced by

�ðtÞ ¼ kT1 ððk� ‘sck Þhþ �

sck Þxððk� ‘

sck ÞhÞ

þ k3ððk� ‘sck Þhþ �

sck Þ:

Remark 5: When ‘sck ¼ ‘cak ¼ 0, it means that no

packet is dropped or rejected in the transmission. In

this case, �(t) is replaced by

�ðtÞ ¼ kT1 ðkhþ �sck ÞxðkhÞ þ k3ðkhþ �

sck Þ:

Thus, the above model (10) of the network can be

viewed as a general form of NCS model, where the

effects of packet loss and delays are simultaneously

considered.From Equations (1) and (3), for t 2 ½khþ �k,

ðkþ 1Þhþ �kþ1Þ, k ¼ 0, 1, 2, . . . , we get

_xðtÞ ¼ AxðtÞ þ B�ðtÞ þ B�ð �u� �ðtÞÞ,

xðtÞ ¼ xð0Þ, t 2 ½0, �0Þ:

Using Equation (7), we obtain

_xðtÞ ¼ AmxðtÞ � BðI� �ÞK�T1 xðtÞ þ BðI� �Þ�ðtÞ þ B� �u:

ð12Þ

From Equations (10) and (11), it can be found that

_xðtÞ ¼AmxðtÞþBðI��Þ½KT1 ððk�‘

cak Þhþ �

sck Þxððk�nkÞhÞ

þk3ððk�‘cak Þhþ �

sck Þ�þB� �u�BðI��ÞK�T1 xðtÞ:

Plant

Adaptivecontroller

SensorActuator

)(txu (t )

u (t )

Memory sck

S1S2

12

1 2Memory cak

Figure 2. NCS setup with data packet dropout.

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Rearranging, _xðtÞ becomes

_xðtÞ¼AmxðtÞþBðI��Þ½ ~KT1 ððk�‘

cak Þhþ �

sck Þxððk�nkÞhÞ

þ ~k3ððk�‘cak Þhþ �

sck Þ�

þBðI��Þ½K�T1 xððk�nkÞhÞþk�3�

þB� �u�BðI��ÞK�T1 xðtÞ:

Using the formula

xðtÞ ¼ xððk� nkÞhÞ þ

Z t

ðk�nkÞh

_xðsÞds,

we have

_xðtÞ¼AmxðtÞþBðI��Þ½ ~KT1 ððk�‘

cak Þhþ �

sck Þxððk�nkÞhÞ

þ ~k3ððk�‘cak Þhþ �

sck Þ��BðI��ÞK�T1

Z t

ðk�nkÞh

_xðsÞds

Rearranging, we have

_xðtÞ ¼AmxðtÞþX

j6¼j1,...jp

bj ~kT1jððk�‘cak Þhþ �

sck Þxððk�nkÞhÞ

þX

j6¼j1,...jp

bj ~k3jððk�‘cak Þhþ �

sck Þ

�BðI��ÞK�T1

Z t

ðk�nkÞh

_xðsÞds ð13Þ

2.4. Normalised adaptive laws

For the considered adaptive NCS, the normalisedadaptive control law is chosen, for j ¼ 1, . . . ,m,t 2 ½khþ �k, ðkþ 1Þhþ �kþ1Þ, k ¼ 0, 1, 2 to be

_k1jðtÞ ¼ �sign½k�2j��1jxðtÞx

TðtÞPbs

,

_k3jðtÞ ¼ �sign½k�2j��2j x

TðtÞPbs

,

ð14Þ

where ¼ 1þ xððk� ‘sck ÞhÞ�� ��2þ kðk� ‘cak Þhþ �

sck

�� ��2,and �1j is an n� n symmetric positive-definite adapta-tion gain matrix, �1j4 0.0

Remark 6: The normalised adaptive law is used hereto cancel xððk� nkÞhÞ

�� �� that will appear in the upperbound � in the following theorem, in which � becamelarger.

Now, we will give our controller design method forNCS (1) with the adaptive control input (3) and (11)based on the Lyapunov stability criterion.

3. Main result

The main result of this article will be treated in thefollowing theorem.

Theorem 1: The overall adaptive NCS with a linear

time-invariant plant (1) achieving Assumptions

(A1)–(A7) and an adaptive stabiliser (3) and (11) is

globally stable if the adaptive control laws take the form

(14) and � satisfies

� ¼�P3

i¼11�iþM Ak k2þM Ak k2

P3i¼1

1�iþ �1 þ �2

for 05�5 minðQÞ, where

�1 ¼ �1 Pk k4X

j 6¼j1,...,jp

bj�� ��2 �1k k2 bmk k

2

þ �2 Pk k4X

j6¼j1,...,jp

bj�� ��2 �2k k2 bmk k

2

þ1

�4Pk k2

Xj6¼j1,...,jp

bik k2 kT1jðt� �

cak Þ þ Aþ Am

��� ���2þ �6 kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��2þM Bk k2 kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��2,�2 ¼

2

�5Pk k2

Xj6¼j1,...,jp

bik3jðt� �cak Þ þ B� �u

�� ��2þ 2�7 k3ððk� ‘

cak Þhþ �

sck Þ

�� ��2þ2�8 �u2

þ 2M k3ððk� ‘cak Þhþ �

sck Þ

�� ��2þ2M Bk k2 �u2

þ M Bk k2 kT1 ððk� ‘cak Þhþ �

sck Þ

�� ��� k3ððk� ‘

cak ÞhÞ

�� �� xððk� nkÞhÞ�� ��

þ 2M Bk k2 �u k3ððk� ‘cak ÞhÞ

�� ��þ M Bk k2 �u kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��� xððk� nkÞhÞ�� ��,

and M ¼ �3 Pk k Ak k þ Amk kð Þ2þ �4 þ �5 for positive

constants , �i, i¼ 1, 2, 3, 4, 5, 6, 7, 8.

Proof: Consider a positive-definite Lyapunov func-

tion V(t) of the form

VðtÞ ¼ V1ðtÞ þ V2ðtÞ þ V3ðtÞ, ð15Þ

where

V1ðtÞ ¼ xTðtÞPxðtÞ

þX

j6¼j1,...,jp

1

k�2j

��� ��� ~kT1jðt� �cak Þ�

�1j

~k1jðt� �cak Þ

þX

j6¼j1,...,jp

1

k�2j

��� ��� ~k23jðt� �cak Þ�

�1j ,

V2ðtÞ ¼

Z t

ðk�ikÞh

xðsÞRxðsÞds,

V3ðtÞ ¼M

Z t

t��

Z t

s

_xðvÞ _xðvÞdv ds:

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Differentiating Vi(t) i¼ 1, 2, 3, with respect to t, for

t 2 ½khþ �k, ðkþ 1Þhþ �kþ1Þ, k ¼ 0, 1, 2, . . . , we have

_V1ðtÞ ¼ _xTðtÞPxðtÞ þ xTðtÞP _xðtÞ

þX

j6¼j1;...;jp

2

k�2j

��� ���_~kT

1jðt� �cak Þ�

�1j

~k1jðt� �cak Þ

þX

j6¼j1;...;jp

1

k�2j

��� ���_~k3jðt� �

cak Þ�

�1j

~k3jðt� �cak Þ,

_V2ðtÞ ¼ xTðtÞRxðtÞ � xTððk� ikÞhÞRxððk� ikÞhÞ,

_V3ðtÞ ¼ �M _xTðtÞ _xðtÞ �M

Z t

t��

_xTðsÞ _xðsÞds

Substituting for _xðtÞ, _k1jðtÞ and _k3jðtÞ from Equations

(13) and (14), taking into account sign½k�2j�jk�2jj ¼ k�2j,

we have

_V1ðtÞ ¼ xTðtÞATmPxðtÞ þ xTðtÞPAmxðtÞ

þ 2xTðtÞPX

j6¼j1,...,jp

bj ~kT1jððk� ‘cak Þhþ �

sck Þ

� xððk� nkÞhÞ

þ 2xTðtÞPX

j6¼j1,...,jp

bj ~k3jððk� ‘cak Þhþ �

sck Þ

� 2xTðtÞPBðI� �Þk�T1

Z t

ðk�nkÞh

_xðsÞds

þ 2X

j6¼j1,...,jp

xTððk� nkÞhÞPbi ~kT1jðt� �cak Þ

� xððk� nkÞhÞ

þ 2X

j6¼j1,...,jp

xTððk� nkÞhÞPbi ~k3jðt� �cak Þ: ð16Þ

and

_V3ðtÞ ¼ �MxTðtÞATAxðtÞ þ 2�MxTðtÞATBuðtÞ

þ �MuTðtÞuðtÞBTB�M

Z t

t��

_xTðsÞ _xðsÞds:

Rearranging Equation (16) yields

_V1ðtÞ ¼�xTðtÞQxðtÞ

þ2xTðtÞPX

j6¼j1,...,jp

bj ~kT1jððk�‘cak Þhþ �

sck Þxððk�nkÞhÞ

þ2xTðtÞPX

j6¼j1,...,jp

bj ~k3jððk�‘cak Þhþ �

sck Þ

�2xTðtÞPBðI��Þk�T1

Z t

ðk�nkÞh

_xðsÞds

�2X

j6¼j1,...,jp

xTððk�nkÞhÞPbi ~kT1jðt� �cak Þxððk�nkÞhÞ

�2X

j6¼j1,...,jp

xTððk�nkÞhÞPbi ~k3jðt� �cak Þ

þ2X

j6¼j1,...,jp

xTðtÞPbi ~kT1jðt� �cak Þxððk�nkÞhÞ

�2X

j6¼j1,...,jp

xTðtÞPbi ~kT1jðt� �cak Þxððk�nkÞhÞ

þ2X

j6¼j1,...,jp

xTðtÞPbi ~k3jðt� �cak Þ

�2X

j6¼j1,...,jp

xTðtÞPbi ~k3jðt� �cak Þ: ð17Þ

Define

D ~kT1jðt� 0�cak Þ ¼~kT1jðt� �

cak Þ �

~kT1jððk� jkÞhþ �sck Þ,

D ~kT3jðt� �cak Þ ¼

~kT3jðt� �cak Þ �

~kT3jððk� jkÞhþ �sck Þ:

Then Equation (17) becomes

_V1ðtÞ ¼�xTðtÞQxðtÞ

þ2xTðtÞPX

j6¼j1,...,jp

bjD ~kT1jððk�‘cak Þhþ �

sck Þxððk�nkÞhÞ

þ2xTðtÞPX

j6¼j1,...,jp

bjD ~k3jððk�‘cak Þhþ �

sck Þ

�2xTðtÞPBðI��Þk�T1

Z t

ðk�nkÞh

_xðsÞds

�2X

j6¼j1,...,jp

Pbi ~kT1jðt� �cak Þxððk�nkÞhÞ

Z t

ðk�nkÞh

_xðsÞds

�2X

j6¼j1,...,jp

Pbi ~k3jðt� �cak Þ

Z t

ðk�nkÞh

_xðsÞds:

Using the well-known inequalities (Moon, Park, and

Kwon 2001)

�2aTb �1

�aTaþ �bTb ð18Þ

for any a, b2Rn and scalar �4 0, we haveZ khþ�k

ðk�ikÞh

_xðsÞ�� ��ds � Z t

t��

_xðsÞ�� ��ds,

t 2 ½khþ �k, ðkþ 1Þhþ �kþ1Þ, k ¼ 0, 1, 2, . . . ð19Þ

andZ t

t��

xðsÞds

� �T Z t

t��

xðsÞds

� �� �

Z t

t��

xTðsÞxðsÞds, ð20Þ

where, �¼ (1þ ik)hþ �kþ 1, _VðtÞ becomes bounded

from above as

_VðtÞ��minðQÞ xðtÞ�� ��2þX3

i¼1

�

�ixðtÞ�� ��2þ�M Ak k2 xðtÞ

�� ��2

þX8i¼6

�

�iM Ak k2 xðtÞ

�� ��2þR xðtÞ�� ��2

þ�1 Pk k2

�

Xj6¼j1,...,jp

bj�� ��2 D ~kT1jððk�‘

cak Þhþ�

sck Þ

��� ���2� xððk�nkÞhÞ�� ��2

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þ�2 Pk k2

�

Xj6¼j1,...jp

bj�� ��2 D ~k3jððk� ‘

cak Þhþ �

sck Þ

��� ���2

þ�

�4

Xj 6¼j1,...,jp

Pbi ~kT1jðt� �cak Þ

��� ���2 xððk� nkÞhÞ�� ��2

þ�

�5

Xj 6¼j1,...,jp

Pbi ~k3jðt� �cak Þ

��� ���2 � R xððk� ikÞhÞ�� ��2

þ ½�3 PBðI� �Þk�T1�� ��2þ �4 þ �5�

Z t

t��

_xðsÞ�� ��2ds

þ �6� Bk k2 kT1 ððk� ‘cak Þhþ �

sck Þ

�� ��2 xððk� nkÞhÞ�� ��2

þ �7� Bk k2 k3ððk� ‘cak Þhþ �

sck Þ

�� ��2þ�8� Bk k2 �u2

þ �M Bk k2 kT1 ððk� ‘cak Þhþ �

sck Þ

�� ��2 xððk� nkÞhÞ�� ��2

þ �M k3ððk� ‘cak Þhþ �

sck Þ

�� ��2þ�M Bk k2 �u2

þ �M Bk k2 kT1 ððk� ‘cak Þhþ �

sck Þ

�� �� k3ððk� ‘cak ÞhÞ

�� ��� xððk� nkÞhÞ�� ��þ �M Bk k2 �u kT1 ððk� ‘

cak Þhþ �

sck Þ

�� �� xððk� nkÞhÞ�� ��

þ �M Bk k2 �u k3ððk� ‘cak ÞhÞ

�� ���M

Z t

t��

_xðsÞ�� ��2ds:

From (14), where

Dk1jðt� �cak Þ�� �� � �1k k bmk k Pk k�,

Dk3jðt� �cak Þ�� �� � �2k k bmk k Pk k�,

_VðtÞ becomes

_VðtÞ � �minðQÞ þX3i¼1

�

�i

"þ �M Ak k2þ �M ATB

�� ��2

�X8i¼6

1

�iþ R

#xðtÞ�� ��2

þ �1� Pk k4X

j6¼j1,...,jp

bj�� ��2 �1k k2 bmk k

2

24� xððk� nkÞhÞ�� ��4

þ �2� Pk k4X

j 6¼j1,...,jp

bj�� ��2 �2k k2 bmk k

2

þ�

�4

Xj 6¼j1,...,jp

Pbi ~kT1jðt� �cak Þ

��� ���2þ �6� kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��2þ �M Bk k2 kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��2�Ri� xððk� nkÞhÞ�� ��2

þ�

�5

Xj 6¼j1,...,jp

Pbi ~k3jðt� �cak Þ

��� ���2þ �7� k3ððk� ‘

cak Þhþ �

sck Þ

�� ��2þ�8� �u2

þ �M k3ððk� ‘cak Þhþ �

sck Þ

�� ��2þ�M Bk k2 �u2

þ �M Bk k2 kT1 ððk� ‘cak Þhþ �

sck Þ

�� ��� k3ððk� ‘

cak ÞhÞ

�� �� xððk� nkÞhÞ�� ��

þ �M Bk k2 �u kT1 ððk� ‘cak Þhþ �

sck Þ

�� ��� xððk� nkÞhÞ�� ��þ �M Bk k2 �u k3ððk� ‘

cak ÞhÞ

�� ��þ ½�3 PBðI� �Þk�T1

�� ��2þ�4 þ �5��

Z t

t��

_xðsÞ�� ��2ds�M

Zt��

_xðsÞ�� ��2ds:

If the adaptation stopped when xððk� nkÞhÞ�� ��5 1,

4 1, and by choosing R ¼ ��1 þ ��2, where

�1 ¼�1 Pk k4X

j 6¼j1,...,jp

bj�� ��2 �1k k2 bmk k

2 xððk� nkÞhÞ�� ��4

þ �2 Pk k4X

j6¼j1,...,jp

bj�� ��2 �2k k2 bmk k

2

þ1

�4Pk k2

Xj6¼j1,...,jp

bik k2 kT1jðt� �

cak Þ þ Aþ Am

��� ���2þ �6 kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��2þM Bk k2 kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��2,�2 ¼

2

�5Pk k2

Xj 6¼j1,...,jp

bik3jðt� �cak Þ þ B� �u

�� ��2þ 2�7 k3ððk� ‘

cak Þhþ �

sck Þ

�� ��2þ2�8 �u2

þ 2M k3ððk� ‘cak Þhþ �

sck Þ

�� ��2þ2M Bk k2 �u2

þ M Bk k2 kT1 ððk� ‘cak Þhþ �

sck Þ

�� ��� k3ððk� ‘

cak ÞhÞ

�� �� xððk� nkÞhÞ�� ��

þ 2M Bk k2 �u k3ððk� ‘cak ÞhÞ

�� ��þ M Bk k2 �u kT1 ððk� ‘

cak Þhþ �

sck Þ

�� ��xððk� nkÞhÞ�� ��

and

M ¼ �3 Pk k Ak k þ Amk kð Þ2þ �4 þ �5, we get

_VðtÞ � �minðQÞ þX3i¼1

�

�i

"þ �M Ak k2

þM Ak k2X3i¼1

�

�iþ ��1 þ ��2

#xðtÞ�� ��2:

Again, by choosing

� ¼�P3

i¼11�iþM Ak k2þM Ak k2

P3i¼1

1�iþ �1 þ �2

ð21Þ

for 05�5 minðQÞ, _VðtÞ becomes

_VðtÞ � �minðQÞ þ �½ � xðtÞ�� ��2:

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Finally, we can conclude that _VðtÞ5 0, if � satisfies

(21). Therefore, x(t), k1jðtÞ, k3jðtÞ and V(t) are bounded

for all t� t0 and the over all system is globally stable.

We therefore obtained the desired result.

4. Illustrative example

Consider the following unstable system (Tao et al.

2001):

A ¼0 1 00 0 1�1 �2 �2

24

35 , B ¼

0 0 00 0 01 2 3

24

35:

Assume that the desired system parameters

Am ¼

0 1 0

0 0 1

�1 �3 �3

264

375 , bm ¼

0

0

1

264

375:

For this system, we consider the actuator failurepattern

u1ðtÞ ¼�1ðtÞ for t 2 0, 80½ Þ

1 for t 2 80,1½ Þ

�

u2ðtÞ ¼�2ðtÞ for t 2 0, 30½ Þ

1 for t 2 30,1½ Þ

�u1ðtÞ ¼ �3ðtÞ for t 2 0,1½ Þ:

Let Q ¼ 0:5I (I¼ identity matrix), A and B areunknown but only Aupp and Bupp are known,x(0)¼ [1 0.5 0.1]T, �1¼ I, �¼ 0.4, ¼ 0. 1, �i¼ 1,k1i(0)¼ [0 0 0]T and k3i(0)¼ 0. With �k¼ h and 10%packet dropped out, h is estimated online to be1.9192e� 004s. For the considered system, the systemstates for the NCS are illustrated in Figures 3–5, fromwhich we can see that at the instants of actuatorfailures the system adapts itself to compensate ortolerate the failures, as shown in the figures.

5. Conclusions

This article addressed the adaptive Stabilisation prob-lem of NCSs with delays and data-packet dropouts inthe presence of unknown actuator failures. Withoutdiscretisation of the continuous-time plant, an adaptivemodel of NCS in the presence of delays anddata-packet dropouts tolerant to actuator failureswas proposed. Sufficient conditions for Lyapunov

0 20 40 60 80 100 120 140 160 180 200−1

−0.5

0

0.5

1

1.5

Time (s)

x 1 (

t)

Figure 3. System state x1(t), one sample delay and 10%packet dropout.

0 20 40 60 80 100 120 140 160 180 200−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

x 2 (t

)

Figure 4. System state x2(t), one sample delay and 10%packet dropout.

0 20 40 60 80 100 120 140 160 180 200−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

x 3 (t

)

Figure 5. System state x3(t), one sample delay and 10%packet dropout.

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stability were developed. An illustrative example wasgiven to illustrate the effectiveness of the designapproach.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China, Grant #60574088 and #60274014.

Notes on contributors

A.H. Tahoun was born in 1976, Hereceived his BSc degree in ElectricalEngineering in 1999, and his MSc inControl Engineering in 2005 fromTanta University, Egypt. He receivedthe PhD degree in Control Theoryand Engineering from the HuazhongUniversity of Science and Technology,Wuhan, China, in 2009. He is a

Lecturer in the Department of Computer and AutomaticControl Engineering, Faculty of Engineering, TantaUniversity, Tanta, Egypt. His research interests includeadaptive control, networked control systems, fault diagnosis,robust and fault-tolerant control.

Hua-Jing Fang received his BS, MSand PhD degrees in Control Theoryand Engineering from the HuazhongUniversity of Science and Technologyin 1982, 1984 and 1991, respectively.He is working with the Department ofControl Science and Engineering,Huazhong University of Scienceand Technology, China, where he

has been a Professor since 1994, and also serves as theDirector of the Institute of Control Theory, the Vice Directorof the Center for Nonlinear and Complex Systems. He is themember of Fault Detection, Supervision and Safety ofTechnical Processes Committee and Education Committeeof the Chinese Automation Association, Associate Editorof the Journal of Control Theory and Applications. Hisresearch interests include complex networked control system,control system fault diagnosis, robust and fault-tolerantcontrol, dynamics and control of autonomous vehicle’sswarm.

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