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Fault tolerant control for switching discrete-time systems with delays: an improved conecomplementarity approachAbdellah Benzaouia a , Mustapha Ouladsine b & Bouchra Ananou ba LAEPT-URAC 28, Faculty of Science Semlalia, University Cadi Ayyad, Marrakech, Moroccob LSIS-UMR 6168, University of Paul Cézanne, Aix-Marseille, FrancePublished online: 21 Jan 2013.

To cite this article: Abdellah Benzaouia , Mustapha Ouladsine & Bouchra Ananou (2013): Fault tolerant control for switchingdiscrete-time systems with delays: an improved cone complementarity approach, International Journal of Systems Science,DOI:10.1080/00207721.2012.762561

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International Journal of Systems Science, 2013http://dx.doi.org/10.1080/00207721.2012.762561

Fault tolerant control for switching discrete-time systems with delays: an improved conecomplementarity approach

Abdellah Benzaouiaa,∗, Mustapha Ouladsineb and Bouchra Ananoub

aLAEPT-URAC 28, Faculty of Science Semlalia, University Cadi Ayyad, Marrakech, Morocco; bLSIS-UMR 6168,University of Paul Cezanne, Aix-Marseille, France

(Received 6 April 2012; final version received 27 November 2012)

In this paper, fault tolerant control problem for discrete-time switching systems with delay is studied. Sufficient conditionsof building an observer are obtained by using multiple Lyapunov function. These conditions are worked out in a new way,using cone complementarity technique, to obtain new LMIs with slack variables and multiple weighted residual matrices.The obtained results are applied on a numerical example showing fault detection, localisation of fault and reconfiguration ofthe control to maintain asymptotic stability even in the presence of a permanent sensor fault.

Keywords: switching systems; delay; stabilising control; arbitrary switching sequence; fault detection; fault tolerant control;observer; multiple Lyapunov function; LMI; cone complementarity

1. IntroductionSwitched systems are a class of hybrid systems encoun-tered in many practical situations, which involve switch-ing between several subsystems depending on various fac-tors. Generally, a switching system consists of a familyof continuous-time subsystems and a rule that supervisesthe switching between them. For example, many processesin the chemical and pharmaceutical industries operate fol-lowing batches, composed of different operations that arecarried out in sequence. This changes discontinuously thedynamics of the operation (Seborg, Edgar, and Mellichamp1989). Many other examples can be found in the automotiveindustry, in aircraft and air traffic control, and many otherfields.

Two main problems are widely studied in the literatureaccording to the classification given in Branicky (1998) andBlanchini and Savorgnan (2006): The first one, which is theone solved in this work, looks for testable conditions thatguarantee the asymptotic stability of a switching systemunder arbitrary switching rules (Daafouz, Riedinger, andIung 2002; Benzaouia, Akhrif, and Saydy 2010; Benzaouiaand Tadeo 2010; Benzaouia, Hajjaji, and Tadeo 2011; Ben-zaouia 2012), while the second is to determine a switchingsequence that renders the switched system asymptoticallystable (see Liberzon and Morse (1999) and the referencetherein). Following the first approach, Blanchini, Miani,and Mesquine (2009) investigated the problem of design-ing a switching compensator for a plant switching amongsta (finite) family of given configurations (Ai, Bi, Ci). Theextension to switching systems with delay is prompted by

∗Corresponding author. Email: benzaouia@uca.ma

the existence of transport delays in many control problemsin process control, irrigation systems, thermal systems, etc.The stabilisation problem is of interest because the exis-tence of a delay might cause instabilities (Mahmoud 2010).The stabilisation of this kind of system has been extensivelystudied in the literature.

A main problem, which is always inherent to all dynam-ical systems, is the possibility of the presence of actuatoror sensor faults. Even for linear systems, this problem hasbeen an active area of research for many years (Frank andDing 1997; Zhong, Ding, Lam, and Wang 2003; Bateman,Noura, and Ouladsine 2007). The study of this problem wasextended to switching systems (see Nouailletas, Koeing,and Mendes 2007; Wang, Wang, and Shi 2009; Belkhiat,Messai, and Manamanni 2011 and the references therein).In Wang et al. (2009), a switching discrete-time system withstate delay is considered. The design method is based onthe construction of a filter and a fault estimation. This ap-proach leads to a big number of matrices to be determined.In Benzaouia, Ouladsine, and Naamane (2012), an observeris built to detect the fault when it occurs.

In this work, the problem of fault tolerant control (FTC)is addressed for uncertain switching systems with delays.Only the case of sensor fault is addressed. It is well knownthat this problem can be solved by using a nominal con-trol law designed in the absence of any fault, associatedwith fault detection, localisation and reconfiguration tech-niques to maintain the main performances as stability of thesystem in the presence of one failing sensor. This problemobtained a great consideration during the last three decades,

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2 A. Benzaouia et al.

as described in Noura, Theilliol, Ponsart, and Chamesed-dine (2009). However, only few works were interested toswitching systems. In Zhang and Yu (2012), the problem ofFTC for switched linear systems with time-varying delaysis addressed. Only the case of actuator fault is considered.The switching sequence in this work is taken with averagedwell time. The derived conditions of asymptotic stabil-ity are obtained by using the congruence transformationtechnique as in Wang et al. (2009).

In the present work, we treat the same class of systemsas in Wang et al. (2009) but with a Luenberger observeras in Benzaouia et al. (2012) to detect sensor fault whilemaintaining system stability in its presence. The main con-tribution of this paper is to propose a new technique offault localisation and reconfiguration to preserve stabilitywhen a sensor fails temporary or permanently. The ob-tained condition of asymptotic stability in the presence offault based on the H∞ technique is then worked out ina simple way to obtain new LMIs. The proposed LMIsare different from the ones obtained in Nouailletas et al.(2007) and Benzaouia et al. (2012). The idea is to use thewell-known cone complementarity technique. However, anoptimisation problem is necessary to find the required so-lutions. To avoid the use of such optimisation problem, asimple technique proposed in Benzaouia (2012), which isactually a very common inequality found in different re-sults based on LMIs and recently used in Gao and Chen(2008), is used. Besides the introduction of slack variablesin the cone complementarity technique, the residual signalis designed with multiple matrices Vi . The applicability ofthe obtained results on a numerical example shows the use-fulness of the observer, which can work with success evenin the presence of unknown bounded permanent input andpermanent sensor fault.

This paper is organised as follows: Section 2 deals withthe problem statement, while Section 3 presents some pre-liminary results on fault detection problem. The main re-sults of this paper are developed in Section 4 together withan illustrative example.

2. Problem formulation

Consider the following delayed discrete-time switchingsystem:

x(k + 1) = Aαx(k) + Aταx(k − τ ) + Bαu(k) + Eαd(k)

y(k) = Cαx(k) + Cταx(k − τ ) + Nαd(k) + Mαf (k),

(1)

where x ∈ Rn is the state, u ∈ R

m is the control, y ∈ Rp is

the output, τ is the delay, d ∈ Rg is an external unknown

input, f ∈ Rq is the fault, and α is a switching rule that

takes its values in the finite set I := {1, . . . , N}, k ∈ Z+.Each subsystem α is called a mode. Matrices Ai ,Aτi have

the same following structure: Ai = Ai + �Ai(k),Aτi =Aτi + �Aτi(k), with norm-bounded uncertainty terms ac-cording to

[�Ai(k) �Aτi(k)] = LiW (k)[H1i H2i]. (2)

Matrices Ai,Aτi, Bi, Ei , Ci, Cτi, Ni,Mi are of appropriatesize constant known matrices. It is assumed that

• The switching system is stabilisable.• Each pair

(Ai, C

j

i

)is observable, j = 1, . . . , p,

where Cj

i stands for the matrix formed by all therows of matrix Ci except its j th row.

• Matrices Ci are of full rank.• Only one sensor is failing.• The switching rule is not known a priori but α(k) is

available at each sampling time k.• W (k)T W (k) ≤ I.

The second assumption corresponds to practical im-plementations where the switched system is supervisedby a discrete-event system or operator allowing for α(k)to be known in real time. Upon introducing the indicatorfunction,

ξ (k) = [ξ1(k), . . . , ξN (k)]T , (3)

where ξi(k) = 1 if the switching system is in mode i andξi(k) = 0 if it is in a different mode, one can write theswitching system (1) as follows:

x(k + 1) =N∑

i=1

ξi(k)[Aix(k) + Aτix(k − τ )

+Biu(k) + Eid(k)]

y(k) =N∑

i=1

ξi(k)[Cix(k) + Cτix(k − τ )

+Nid(k) + Mif (k)]. (4)

The applied control is a state feedback one. Since the statecannot be available nor the total output, the observer canthen be used. In addition to stabilise the system with freefault, the controller has to achieve asymptotic stability inclosed loop even in the presence of sensor fault occurringon sensor j and is given by

u(k) = Fij xj (k), j = 0, . . . , p. (5)

The system in closed loop is then given by

x(k + 1) =N∑

i=1

ξi(k)[Acij x(k) + Aτix(k − τ )

+Eid(k) − BiFij (x(k) − xj (k))]

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International Journal of Systems Science 3

y(k) =N∑

i=1

ξi(k)[Cix(k) + Cτix(k − τ )

+Nid(k) + Mif (k)] (6)

with Acij = Ai + BiFij . In this work, we are interested bythe synthesis of an observer for this class of systems in orderto detect a default when it occurs in the switching system.For this, consider the following switching observer:

xj (k + 1) =N∑

i=1

ξi(k)[Acij xj (k) + Aτixj (k − τ )

+Kij (yj (k) − yj (k))].

yj (k) =N∑

i=1

ξi(k)[Cj

i xj (k) + Cτj

i xj (k − τ )] (7)

j = 0, . . . , p, (8)

with Acij = Ai + BiFij , where yj stands for the vectorformed by all the components of the vector y except its j thcomponent. In this case, Kij ∈ R

n×(p−1) and yj ∈ Rp−1.

Note that

yj (k) =N∑

i=1

ξi(k)[C

j

i x(k) + Cτj

i x(k − τ )

+Nj

i d(k) + Mj

i f (k)]. (9)

In this structure, only matrices Kij and Fij are to be de-signed.

Remark 1: As assumed in the second assumption, theswitching rule is not known a priori but α(k) is available ateach sampling time k. This means that one can synchronisethe switch of the observer with the switch of the system.In this case, the problem to have the system and the ob-server evolving in different modes cannot occur as it arisesin the continuous case studied in Belkhiat et al. (2011). Itmust also be pointed out that the time delay is taken con-stant to simplify the design method, which focuses on thereconfiguration of the controller and observer technique.

The residual of the j th observer is defined as

rj (k) =N∑

i=1

ξi(k)Vij (yj (k) − yj (k)). (10)

Matrices Vij are to be computed. It is worth noting thatthe proposed structure of the residual is different from theclassical one used in Belkhiat et al. (2011) where onlyone matrix V is used. Defining the j th observer error by

ej (k) = x(k) − xj (k) leads to

ej (k + 1) =N∑

i=1

ξi(k)[(

Ai − KijCj

i

)ej (k)

+ (Aτi − KijCτj

i

)ej (k − τ )

+ (Ei − KijNj

i

)d(k) − KijM

j

i f (k)

+�Ai(k)x(k) + �Aτi(k)x(k − τ )]. (11)

Define an augmented state vector as xj (k) =[x(k)T xj (k)T ej (k)T ]T and an augmented input vector as

wk = [dT

k f Tk

]T. The corresponding dynamical system is

then derived as follows:

xj (k + 1) =N∑

i=1

ξi(k)[Acij xj (k)

+ Aτ ij xj (k − τ ) + Bijw(k)] (12)

rj (k) =N∑

i=1

ξi(k)rij (k)[Cij xj (k)

+Cτij xj (k − τ ) + Dijw(k)], (13)

with Acij = Acij + �Ai(k) and Aτij = Aτij + �Aτi(k),where

Acij =

⎡⎢⎢⎣

Ai + BiFij 0 −BiFij

0 Ai + BiFij KijCj

i

0 0 Ai − KijCj

i

⎤⎥⎥⎦ ,

Aτ ij =

⎡⎢⎢⎣

Aτi 0 0

0 Aτi KijCτj

i

0 0 Aτi − KijCτj

i

⎤⎥⎥⎦ ,

Bij =⎡⎣ Ei 0

−KijNj

i −KijMj

i

Ei − KijNj

i −KijMj

i

⎤⎦ ,

�Ai(k) =⎡⎣�Ai(k) 0 0

0 0 00 0 �Ai(k)

⎤⎦ ,

�Aτi(k) =⎡⎣�Aτi(k) 0 0

0 0 00 0 �Aτi(k)

⎤⎦ ,

Cij = [0 0 VijC

j

i

],

Cτ ij = [0 0 VijCτ

j

i

], Dij = [

VijNj

i VijMj

i

].

(14)

The uncertain terms for the augmented system canbe again developed as follows: [�Ai(k) �Aτi(k) ] =

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4 A. Benzaouia et al.

LiW [H1i H2i ], where

Li =⎡⎣Li 0 0

0 0 00 0 Li

⎤⎦ , Hsi =

⎡⎣Hsi 0 0

0 0 00 0 Hsi

⎤⎦ ,

W (k) = diag{W (k), 0,W (k)}, s = 1, 2. (15)

There are two objectives of this work:

• First, find new conditions to build an FTC able tomaintain the stability of the switching system whena sensor fails.

• Second, develop a synthesis method based on the H∞tool to compute the unknown matrices representingthe switching observer gains Kij , the residual gainsVij and the feedback control gains Fij .

One can note that the way followed in this paper isdifferent from the one used in Wang et al. (2009) wherea filter and a dynamic of the fault are completely charac-terised. With the use of a Luenberger switching observer,the number of unknown matrices is lower in our case.

3. Preliminary results

As used in the literature of fault detection (Wang et al. 2009;Belkhiat et al. 2011; Zhong et al. 2003), the identificationof the fault fk is not necessary. One can use the followingresidual criterion:

Jj

k =⎛⎝k0+k∑

s=k0

rj (s)T rj (s)

⎞⎠

1/2

, (16)

where k0 and k define the interval of the evaluation windowand k being the current sampled time. The size of thiswindow increases until it is equal to the global horizonof observation T . Then, this evaluation function can becompared to a threshold Jth to conclude if a fault occurs ornot, as follows:

Jj

k > Jth ⇒ Faults ⇒ Alarm

Jj

k ≤ Jth ⇒ No Faults.

The threshold criterion can be chosen as indicated in Wanget al. (2009) as

Jth = supd∈l2,f =0,k

J 0k , (17)

where J 0k stands for a criterion obtained with a residual

r0 computed with no sensor fault. Nevertheless, anotherevaluation function based on a past receding window can

be defined and used instead of the one of Equation (16) as

J jr =

(k∑

s=k−T

rj (s)T rj (s)

)1/2

, k > T , (18)

where T is the fixed window size and k is the currentsampled time.

Once the default is detected, the problem is how tolocate the failing sensor? According to the method of com-puting the p residuals in the absence of perturbations anduncertainties at any time k, the following two cases canarise:

• If (rj )l �= 0, l �= j = 1, . . . , p, the failing sensor isnot l,

• If (rj )l = 0, l �= j = 1, . . . , p, the failing sensor em-phasises with l,

where (rj )l stands for the lth component of vector rj .Note that one needs to compute the different residuals:the threshold residual rth computed without fault and withthe maximal allowed perturbations, the residuals rj , j =0, 1, p. The failing sensor noted j∗ can be located by thefollowing method to check for j = 1, . . . , p at any time k:

If Jj

k > Jth or maxj ‖rj (k)‖ > ‖rth(k)‖and ‖rj (k)‖ > min0≤j≤p ‖rj (k)‖,

Then sensor j is not failing, (19)

If Jj

k > Jth or maxj ‖rj (k)‖ > ‖rth(k)‖and ‖rj (k)‖ = min0≤j≤p ‖rj (k)‖,

Then sensor j∗ = j is the failing senor, (20)

where rj represents the residual without the componentj . For explaining this fact, take an example of p = 3 andassume that at time k only sensor Se3 is really failing.

(1) j = 1: Sensor Se1 is omitted, in this case the inputof the first observer are only y2, y3 and u. Sincethe sensor Se3 is failing, the corresponding residualwill be greater than the threshold, implying sensorj is not failing.

(2) j = 2: Sensor Se2 is omitted, in this case the inputof the second observer are only y1, y3 and u. Sincethe sensor Se3 is failing, the corresponding residualwill be greater than the threshold, implying sensorj is not failing.

(3) j = 3: Sensor Se3 is omitted, in this case the inputof the third observer are only y1, y2 and u. Sincethe sensor Se3 is failing, the corresponding residualwill be lower than the threshold, implying sensor j

is failing.

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International Journal of Systems Science 5

For this example, one can locate the failing sensor cor-responding to j∗ = 3. Note that the failing sensor corre-sponds to the lower norm of the corresponding residual andcan be computed by

j∗ = arg min1≤j≤p

‖rj‖. (21)

Now, the separation lemma is recalled.

Lemma 3.1 (Shi, Boukas, and Agarwal 1999): Givensymmetric matrix S and matrices L,W (k) and H of appro-priate size, then

S + LT W (k)H + HT WT (k)L < 0

holds for W (k)T W (k) ≤ I if and only if there exists a scalarε > 0 such that

S + ε−1LT L + εHT H < 0.

The technique of H∞ problem for the augmentedswitching system (12) consists of ensuring the asymptoticstability of the system for wk = 0 and x0 �= 0 while realis-ing the following condition for wk �= 0 and x0 = 0:

supwk �=0,wk∈l2[0,∞)

√rj (k)T rj (k)√

wTk wk

< γj , γj > 0. (22)

Remark 1: In the H∞ theory, the input wk is assumed tobe only bounded without any knowledge of its bound. Inour case, wk represents the fault fk and the perturbationdk . No additional assumption is required on these inputs.The idea is to reduce the impact of this exogenous input wk

on the system by reducing as far as possible the scalars γj

defined by Equation (22).

Condition (22) is realised if

J j (γ ) =Tk−1∑k=0

[rj (k)T rj (k) − γ 2

j wTk wk

]< 0. (23)

To realise condition (23), one has to use a Lyapunov func-tion Vj (x) and look for the condition realising

J j (γ ) =Tk−1∑k=0

[rj (k)T rj (k) − γ 2

j wTk wk + �Vj (x)

]−Vj (x(Tk)),

≤Tk−1∑k=0

[rj (k)T rj (k) − γ 2j wT

k wk + �Vj (x)] < 0

(24)

for any wk ∈ l2[0,∞) and with x0 = 0. It is obvious thatcondition (24) is satisfied if �Vj (x)] < 0, that is the systemis asymptotically stable for wk = 0 and x0 �= 0.

To stabilise the system even in non-fault situation, oneadd the case j = 0 corresponding to full observation andcontrol without extracting any sensor.

The first result we recall is a sufficient condition of H∞fault detection for the augmented switching system (12)presented by Wang et al. (2009).

Lemma 3.2 (Wang et al. 2009): For given scalars γj >

0, system (12) under arbitrary switching is asymptoticallystable when wk = 0 and under zero-initial conditions, theperformance index (22) is guaranteed for all wk ∈ l2[0,∞),if there exists positive definite symmetric matrices Pij andQij , i ∈ I, such that

� =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−P −1lj Acij Aτij Bi 0 0

∗ −Pij 0 0 CTij I

∗ ∗ −Qsj 0 Cτ Tij 0

∗ ∗ ∗ −γ 2j I DT

ij 0

∗ ∗ ∗ ∗ −I 0

∗ ∗ ∗ ∗ ∗ −Q−1ij

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

∀(i, l, s) ∈ I3, j = 0, . . . , p, (25)

where ∗ stands for the symmetrical term of the correspond-ing off-diagonal term.

Remark 2: Even system (12) is given with similar nota-tions as in Wang et al. (2009) to have the possibility to useLemma 3.2, the changes of variables (14) are completelydifferent from those taken in Wang et al. (2009). Further,the proof of this result is based on the use of a multipleLyapunov-Krasovskii functional given by

Vj (xk) = xj (k)T(

N∑i=1

ξi(k + 1)Pij

)xj (k)

+k−1∑

s=k−τ

xj (s)T(

N∑i=1

ξi(k)Qij

)xj (s). (26)

With these preliminary results, we are now able tosolve the main problem of designing an FTC for switch-ing discrete-time systems with delays in the presence of asensor failure, by using a switching observer.

4. Main results

The objective of this section is to work out inequality (25)to obtain an LMI enabling one to synthesise the switchingobserver together with its corresponding residual, makingpossible the design of FTCs for discrete-time switchingsystem with delay. The cone complementarity technique

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introduced by El Ghaoui, Oustry, and AitRami (1997) andused in many works as Nachidi, Benzaouia, and Tadeo(2008), Gao and Chen (2008) is also used to derive themain results of this paper.

Lemma 4.1 For given scalars γj > 0, system (12) underarbitrary switching is asymptotically stable when wk = 0and under zero-initial conditions, the performance index(22) is guaranteed for all wk ∈ l2[0,∞), if there exists pos-itive definite symmetric matrices Pij , Qij , and matrices Kij ,Fij , Vij , i ∈ I, such that

� =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−P −1lj Aij Aτij Bi 0 0 LT

i

∗ −ij i 0 CTij I 0

∗ ∗ −�ijs 0 Cτ Tij 0 0

∗ ∗ ∗ −γ 2j I DT

ij 0 0

∗ ∗ ∗ ∗ −I 0 0

∗ ∗ ∗ ∗ ∗ −Q−1ij 0

∗ ∗ ∗ ∗ ∗ ∗ −εiI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

∀(i, l, s) ∈ I3, j = 0, . . . , p, (27)

where

ij = Pij − εiHT1i H1i

�isj = Qsj − εiHT2i H2i (28)

i = εiHT1i H2i .

Proof: Using the change of variables of the uncertain termsof the augmented system (15), one can rewrite matrix � asfollows:

� = �0 + LTi W (k)Hi + H T

i W (k)Li (29)

with

�0 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−P −1lj Aij Aτij Bi 0 0

∗ −Pij 0 0 CTij I

∗ ∗ −Qsj 0 Cτ Tij 0

∗ ∗ ∗ −γ 2j I DT

ij 0

∗ ∗ ∗ ∗ −I 0

∗ ∗ ∗ ∗ ∗ −Q−1ij

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

Li = [Li 0 0 0 0 0], Hi = [0 H1i H2i 0 0 0]. (30)

Applying Lemma 3.1 for each subsystem and Schur com-plement, the inequality (27) follows readily. In addition, itwas shown in Benzaouia et al. 2009 that using the separationLemma 3.1 with εi instead of ε leads to less conservativeconditions. �

The following result is obtained by working out inequal-ity (27) by using a cone complementarity technique.

Theorem 4.2: For given scalars γj > 0, system (12) underarbitrary switching is asymptotically stable when wk = 0and under zero-initial conditions, the performance index(22) is guaranteed for all wk ∈ l2[0,∞), if there exists pos-itive definite symmetric matrices Pij , Qij , Sij and Rij , ma-trices Kij , Fij , Vij , i ∈ I, j = 0, . . . , p, such that

� =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Slj Aij Aτij Bi 0 0 LTi

∗ −ij i 0 CTij I 0

∗ ∗ −�ijs 0 Cτ Tij 0 0

∗ ∗ ∗ −γ 2j I DT

ij 0 0

∗ ∗ ∗ ∗ −I 0 0

∗ ∗ ∗ ∗ ∗ −Rij 0

∗ ∗ ∗ ∗ ∗ ∗ −εiI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

T race(PijSij ) = T race(I)

T race(RijQij ) = T race(I)

∀(i, l, s) ∈ I3, j = 0, . . . , p. (31)

Proof: Obvious.To apply this result, one uses the Schur complement to

rewrite equivalently inequalities SijPij = I as SijPij ≥ I,that is Sij − P −1

ij ≥ 0, which can be equivalently writtenas [

Sij I

I Pij

]≥ 0 (32)

[Rij I

I Qij

]≥ 0, (33)

This result presents the advantage of computing directlymatrices K

j

i , Fj

i and Vj

i . Nevertheless, an optimisationproblem must be solved to achieve PijSij � I by minimis-ing iteratively the trace of matrix PijSij extending an al-gorithm presented, for linear systems, in El Ghaoui et al.(1997) to switching systems.

This algorithm is based on a linear approximation ofT r(PijSij ) by T r(P0Sij + S0Pij ), where P0 and S0 are par-ticular solutions of the LMI constraints (31)–(33).

(Pb.1) :

{min(Sij ,Pij ,Qij ,Rij ) θ

s.t. (31), (32), (33),

where θ is a function with multiple objective

θ = β Trace(PijSij ) + (1 − β) Trace(QijRij ), 0 < β < 1.

In Gao and Chen (2008), a different nonlinear optimisationproblem involving LMI conditions is proposed to solve theobtained cone complementarity conditions. To avoid theoptimisation problem, which may be less conservative buthard to implement, a commonly used technique in the liter-ature of LMIs, applied in Benzaouia (2012) for stabilising

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International Journal of Systems Science 7

saturated uncertain switching systems, can be used in thiscontext. The following result deals with this technique.

Corollary 4.3: For a given scalar γj > 0, system (12)under arbitrary switching is asymptotically stable whenwk = 0 and under zero-initial conditions, guarantees theperformance index (22) for all wk ∈ l2[0,∞), if there ex-ists positive definite symmetric matrices Pij , Qij , Sij andRij and matrices Xij , Zij , Kij , Fij , Vij , i ∈ I, j = 0, . . . , p

such that

� =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Slj Aij Aτij Bi 0 0 LTi

∗ −ij i 0 CTij I 0

∗ ∗ −�ijs 0 Cτ Tij 0 0

∗ ∗ ∗ −γ 2j I DT

ij 0 0

∗ ∗ ∗ ∗ −I 0 0

∗ ∗ ∗ ∗ ∗ −Rij 0

∗ ∗ ∗ ∗ ∗ ∗ −εiI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(34)

[−Xij − XTij + Pij −Xij − XT

ij + I

∗ −Xij − XTij + Sij

]< 0 (35)

[−Zij − ZTij + Qij −Zij − ZT

ij + I

∗ −Zij − ZTij + Rij

]< 0, (36)

∀(i, l, s) ∈ I3, j = 0, . . . , p. (37)

Proof: The idea is to realise inequalities −P −1ij < −Sij and

−Q−1ij < −Rij . The first inequality is equivalent by Schur

complement to

[−P −1ij −I

∗ −S−1ij

]< 0. (38)

By pre-multiplying and post-multiplying this inequality bydiag

{XT

ij , XTij

}, where Xij is any non-singular matrix, it

follows equivalently:

[−XTijP

−1ij Xij −XT

ijXij

∗ −XTijS

−1ij Xij

]< 0. (39)

The following development

(XT

ij − Pij

)P −1

ij (Xij − Pij )

= XTijP

−1ij Xij − Xij − XT

ij + Pij > 0, (40)

leads to

− XTijP

−1ij Xij < −Xij − XT

ij + Pij (41)

− XTijXij < −Xij − XT

ij + I. (42)

Replacing Equations (41) and (42), one obtains Equation(35). A similar development is used to obtain Equation(36). Inequalities −P −1

ij < −Sij and −Q−1ij < −Rij are

then used to bound inequality (25) to obtain LMIs (34)–(36). �

Remark 1: It is worth noting that the proposed techniqueto avoid the optimisation problem given by Equations (35)and (36), is based on the idea, which is actually a very com-mon inequality found in different results based on LMIs andrecently used in Gao and Chen (2008), to bound terms oftype PM−1P by N such that N − PM−1P ≤ 0. The opti-misation problem, which may be less conservative, is hardto implement in our actual context of multiple Lyapunovfunction with matrices Pij , Sij , Rij ,Qij .

Now that all the gain observers Kij and the gain con-trollers Fij are computed offline, the procedure to be fol-lowed to stabilise the switching system despite the presenceof a failing sensor is very simple. It consists, for the failingdetected sensor by the technique given by Equation (20),of choosing controller gains Gij∗, i = 1, . . . , N . This sim-plicity is due to the proposed method based on one sensorisolation. Note also that in this case, we only need p ob-servability assumption for (Ai, C

j

i ), j = 1, . . . , p for eachmode.

Once all the gains Kij , Fij are computed offline forj = 0, . . . , p, the initial applied control is one of the no-fault obtained with j = 0. Then, an algorithm of detectionand localisation based on the technique described above islaunched to stand guard on the eventual occurrence of afault. The algorithm will allow to locate the failing sen-sor j∗ and act the corresponding set of controllers andobservers gain with a simple correspondence Kij∗, Fij∗.Of course, the algorithm of surveillance is working in realtime.

Example 4.4: : Consider the following numerical exam-

ple similar to the one studied in Wang et al. (2009) withmodified default matrices. The used matrices are dividedby 10 to be closed to the perturbation matrices. Besides, theperturbation dk is taken 100 times the one of Wang et al.(2009).

Mode 1:

A1 =[

0.2 −0.10 0.4

], Aτ1 =

[0.1 00.1 0.3

],

B1 =[

0.1 0.30.2 0.1

], E1 =

[0.2 00 0.1

],

C1 =[

0.1 00 1

], Cτ1 =

[0 0.1

0.1 0

],

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8 A. Benzaouia et al.

L1 =[

0.010.1

], H11 = [0.1 0.01], H21 = [0.01 0.01],

N1 =[

1.1 00 1

], M1 =

[3 00 2

].

Mode 2:

A2 =[

0.4 0.10.1 0.3

], Aτ2 =

[0.1 00.2 0.1

],

B2 =[

0.3 0.40.2 0.1

], E2 =

[0.2 00 0.6

],

C2 =[

0 0.10.1 0

], Cτ2 =

[0.1 00 0.1

],

L2 =[

0.10.1

], H12 = [0.1 0.1], H22 = [0.1 0.1],

N2 =[

1.2 00 1

], M2 =

[4 00 3

].

For this simple example, LMIs (34) are of size 42, whichpresents a big computation complexity with existing soft-wares. Using Scilab 5, the LMIs (34)–(36) are feasible forγ = 0.6. It is worth noting that each time LMIs (34)–(36)are feasible, the corresponding ones with common matricesPi = P and Qi = Q are also feasible for this example. Theobtained observer and residual gains are given by:

For j = 0:

K10 =[

0.0099905 −0.00544970.0016062 0.0289834

],

K20 =[

0.0182866 0.00133780.0063633 0.0612819

],

V10 = 0.1314996; V20 = 0.1436017.

F10 =[

0.1667700 −1.1868018−0.4974717 0.5791786

],

F20 =[

0.1598631 −1.7259354−0.5036928 1.3249117

].

For j = 1:

K11 =[−0.0053866

0.0257757

], K21 =

[0.00394110.0532697

],

V11 = 0.1410116; V12 = 0.1717656.

F11 =[

0.1120497 −1.4429344−0.4720887 0.6527653

],

F21 =[

0.0953194 −1.7100553−0.5286821 1.2418702

].

Figure 1. The evolution of the switching rule α.

Figure 2. The evolution of the residuals rj (k) and the thresholdresidual rth in the absence of sensor fault: rth in red, r0 in blue, r1

in green and r2 in magenta.

For j = 2:

K12 =[

0.00989940.0014567

], K22 =

[0.01813950.0072554

],

V12 = 0.1410116; V22 = 0.1717656.

F12 =[

0.1656434 −1.0828299−0.4937376 0.5406356

],

F22 =[

0.0009425 −1.8701884−0.3970519 1.4456168

].

The fault is generated from k = 150 with a unit magnitude.The bounded unknown input is a permanent random signalof maximal magnitude 0.2. Three cases of sensor fault aretested separately: free faults, sensor 1 failing and sensor 2failing.

Figure 1 presents the switching sequences α(k), whileFigure 2 plots the evolution of the different residuals:the threshold residual rth computed with free fault in thepresence of the maximal allowed perturbation signal d(k).Figures 2–4 plot the residuals corresponding to rj , j =0, 1, 2: with free fault, senor 1 failing and sensor 2 failing.These figures confirm that when a sensor fault occurs, thecorresponding residual rj computed without using the j thoutput has the minimum value amongst all the others.

To show the interest of the designed fault tolerant con-troller, Figure 5 plots the evolution of the states and ob-servers in the presence of a fault occurring at k = 150

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International Journal of Systems Science 9

Figure 3. The evolution of the residuals of fault rj (k) in thepresence of a fault on sensor 1.

Figure 4. The evolution of the residuals of fault rj (k) in thepresence of a fault on sensor 2.

Figure 5. The evolution of the states (in blue) and observers (inred) in the presence of a fault on sensor 2 without reconfigurationof the controllers.

Figure 6. The evolution of the states (in blue) and observers (inred) in the presence of a fault on sensor 2 with reconfiguration ofthe controller and observer gains.

without using any reconfiguration of the controllers. Thatis, the computed controller for free fault is still used evena sensor fault occurs. In consequence, the performances ofthe system are not guaranteed in the presence of one sensorfailing. However, Figure 6 plots the evolution of the statesand observers in the presence of a fault occurring at k = 150while using reconfiguration of the controller and observergains. In consequence, the asymptotic stability of the sys-tem is preserved even in the presence of one sensor failing,which shows the usefulness of the proposed method.

5. Conclusion

In this paper, sufficient conditions for building a feed-back FTC based on a Luenberger observer for discrete-timeswitching systems with delay are presented. The obtainedconditions are worked out by using the cone complemen-tarity technique to obtain new LMIs. Besides the introduc-tion of slack variables Xi, Zi to realise −P −1

i < −Si and−Q−1

i < −Ri , the residual signal is designed with multi-ple matrices Vi . The observer is then used in fault detectionand localisation. In particular, the proposed technique al-lows to detect, localising and reconfigure the controllers toachieve asymptotic stability during the presence of the fail-ing sensor. Besides, the observer works with success even inpresence of uncertainty and permanent bounded unknowninput. A numerical example is studied to illustrate the ap-plicability of the obtained results. As the aim of this work

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10 A. Benzaouia et al.

is to propose new techniques to design FTC by fault detec-tion and reconfiguration, the study is limited, at this level,to discrete-time switching systems with fixed delay andwithout uncertainties on matrices Bi . These results can beextended to continuous-time switched systems by using thesame technique developed in Wang, Shi, and Wang (2010)and to discrete-time switching systems with time-varyingdelay as studied in Zhang and Yu (2012). Other new ideasdeveloped in Shaker and Wisniewski (2012) and Sun, Liu,Wang, and Ress (2012) have to be explored to improve theobtained conditions.

Notes on contributorsAbdellah Benzaouia was born in Attaouia(Marrakech) in 1954. He received the de-gree of Electrical Engineering at the Mo-hammedia school (Rabat) in 1979 andthe Doctorat (PhD) at the University CadiAyyad in 1988. He is actually a Professor atthe University of Cadi Ayyad (Marrakech)where he is also head of the laboratory of re-search LAEPT, CNRST laboratory. His re-

search interests are mainly constrained control, robust control,pole assignment, systems with Markovian jumping parameters,hybrid systems and fuzzy systems. He collaborates with manyteams in France, Canada, Spain and Italy. He recently published abook on Saturated Switching Systems with Springer.

Mustapha Ouladsine received his Ph.D. de-gree in Nancy, 1993 in the estimation andidentification of nonlinear systems. In 2001,he joined the laboratory of information sci-ences and systems in Marseille in France,which he is heading since 2006. His re-search interests are the nonlinear estimationand identification, diagnosis and prognosisof failures in complex systems and their ap-

plications in the field of transport. He has published numeroustechnical articles in the field.

Bouchra Ananou received her Ph.D. degreein 2003 in Marseille ‘by Electron Param-agnetic Resonance Study of Volcanic Sed-iments Trapp of Ethiopia’. In 2005, shejoined the laboratory of information sci-ences and systems in Marseille, France. Herresearch interests are the diagnosis of com-plex systems by data analysis and their ap-plications to production systems. She has

published numerous technical articles in the field.

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