fault diagnosis in a networked control …matwbn.icm.edu.pl/ksiazki/amc/amc24/amc2449.pdf · fault...

Int. J. Appl. Math. Comput. Sci., 2014, Vol. 24, No. 4, 809–820DOI: 10.2478/amcs-2014-0060

FAULT DIAGNOSIS IN A NETWORKED CONTROL SYSTEMUNDER COMMUNICATION CONSTRAINTS:

A QUADROTOR APPLICATION

KARIM CHABIR, MOHAMED AMINE SID, DOMINIQUE SAUTER

CRAN CNRS UMR 7039University of Lorraine, BP239, 54506 Vandoeuvre Cedex, France

e-mail: karim.chabir,[email protected]

This paper considers the problem of attitude sensor fault diagnosis in a quadrotor helicopter. The proposed approach iscomposed of two stages. The first one is the modelling of the system attitude dynamics taking into account the inducedcommunication constraints. Then a robust fault detection and evaluation scheme is proposed using a post-filter designedunder a particular design objective. This approach is compared with previous results based on the standard Kalman filterand gives better results for sensor fault diagnosis.

Keywords: networked control systems, transmission delays, robust residual generation, adaptive residual evaluation.

1. Introduction

Unmanned aerial vehicles (UAVs) have received a greatdeal of attention during the last few years due totheir high performance in several applications suchas search and critical missions, surveillance tasks,geographic studies and various military and securityapplications. As an example of UAV systems, thequadrotor helicopter is a relatively simple, affordableand easy-to-fly system, and thus it has been widelyused to develop, implement and test-fly methods incontrol, fault diagnosis, fault tolerant control as wellas multi-agent based technologies in formation flight.Navigation and guidance algorithms may be embeddedon the on-board flight microcomputer/micro-controller, orwith the remote interference of ground wirelesses/wiredcontrollers. In our setting the quadrotor is controlled overa real time communication network with time-varyingdelays and therefore is considered a networked controlsystem (NCS). In general, the NCS is composed of alarge number of interconnected devices (system nodes)that exchange data through a communication network.

Recent research on NCSs has received considerableattention in the automatic control community (Niculescu,2001; Tipsuwan and Chow, 2003; Mirkin and Palmor,2005; Hespanha et al., 2007; Richard, 2003; Fang et al.,2007). The major focus of the research activities areon system performance analysis regarding the technical

properties of the network and on controller designschemes for NCSs (Xia et al., 2011; Bemporad et al.,2010).

However, the introduction of communicationnetworks in the control loops makes the analysisand synthesis of NCSs a highly complex task(Morawski and Zajaczkowski, 2010). There are severalnetwork-induced effects that arise when dealing withthe NCS, such as time-delays (Niculescu, 2001; Nilssonet al., 1998; Pan et al., 2006; Schöllig et al., 2007; Yiet al., 2007; Zhang et al., 2005), packet losses(Xiong and Lam, 2007; Sahebsara et al., 2007; Yuet al., 2004; Georges et al., 2011) and quantizationproblems (Goodwin et al., 2004; Montestruque andAntsaklis, 2007; Fang et al., 2007). Because of theinherent complexity of such systems, the control issuesof NCSs have attracted attention of many researchers,particularly taking into account network-induced effects.A typical application of these systems ranges over variousfields, such as automotive engineering, mobile robotics,or advanced aircraft.

Fault diagnosis is one of the most important researchfields in modern control theory (Frank and Ding, 1997;Gertler, 1998; Isermann, 2005; Stoustrup and Zhou, 2008;Basseville and Nikiforov, 1993). However, the study offault detection (FD) of the NCS is a new research topicthat has been receiving more attention in the last few years

karim.chabir, [email protected]

810 K. Chabir et al.

(Simon et al., 2013). For instance, the results of Sauter andBoukhobza (2006), Llanos et al. (2007) and Chabir et al.(2008; 2009; 2010) focus on networked-induced delays.The problem studied by Zhang et al. (2004) and Wanget al. (2009) is the analysis and design of FD systemsin case of missing measurements. Fault detectability andisolability in NCSs have been discussed by Sauter et al.(2009), Chabir et al. (2009) or Sid et al. (2012). The faulttolerant structure is studied by Patton et al. (2007), Jainet al. (2012) and Jamouli et al. (2012). A method based onthe reference governor can be found in the work of Weberet al. (2012).

Delays are known to drastically degrade theperformance of control systems. For this reason, manyworks aim at reducing the effects of induced networkdelays on NCSs (Tipsuwan and Chow, 2003; Yu et al.,2004; Li et al., 2006). In the majority of the studiesconcerning NCSs, the delay is classified according to itsnature either as deterministic or stochastic delay. It canalso be classified as long or short delay, according toits duration. The delay is said to be short if its durationis less than one sampling period and long otherwise(Hu and Zhu, 2003; Lincoln and Bernhardsson, 2000).Generally, the dynamics of the delay corresponding to thecharacterization of the network are not taken into account.Thus, one interesting approach is to estimate the delay,in order to generate an optimal control, as well as robustalgorithms of faults detection that take into account thenetwork characteristics.

For dealing with the short delay effect, many workshave been proposed in the literature. For instance, Sauteret al. (2009) formulate the delay effect as an unknowninput with a variable distribution matrix by using Taylorapproximation. The same approximated model is used byYe and Ding (2004) for the generation of a time varyingparity space based fault indicator. Stochastic delay canbe modelled by a Markov chain (Yi et al., 2007; Zhanget al., 2005). Sauter et al. (2009) use a fault isolation filterfor monitoring a system under Markovian short delays.The proposed filter parameters are designed using linearmatrix inequalities (LMIs) (Boyd et al., 1994). Zhenget al. (2003) propose a reduced order fault detection filterfor improving the robustness to constant long delay andreduce the complexity of the design problem.

Wang et al. (2006) set forth a method for faultdetection in NCSs under stochastic and probably longduration delay. The model given by Ray and Halevi (1988)as well as Hu and Zhu (2003) is adopted for the design.However, this model can be seen as an extension of theunidimensional Taylor approximation given by Ye andDing (2004) for the multidimensional case. Wang et al.(2008) consider mixed delay composed of a constant partand random part. The delay effect is approximated bypolytopic uncertainties and uses the “reference model"fault detection technique (Ding, 2008) for the design

of observer based fault detection. A majority of faultdetection approaches of NCSs that exist in the literatureare model based (Sauter et al., 2013). However, artificialintelligence methods are considered less suitable for realtime implementation (Rahmani et al., 2008).

The objective in this study is the diagnosisof quadrotor attitude sensors fault under variabletransmission delay. First, an attitude dynamic modeltaking into account variable transmission delay ispresented. Then we propose a robust residual generationand evaluation scheme using a post-filter that verifies aparticular design objective. This approach is comparedwith previous results based on the standard Kalman filterand gives better results for sensors fault diagnosis.

The rest of the paper is organized as follows. InSection 2, the quadrotor helicopter attitude dynamicsmodelled and then controlled using the LQR approach.Section 3 presents the first main result of this paper,which is related to the modelling of networked controlsystems. Finally, Section 4 we present our second mainresult concerned with residual generation and evaluationusing an adaptive threshold. Simulation results are givenin Section 5 and the paper is concluded in Section 6.

2. Description of the quadrotor helicopterdynamics

The mini-helicopter under study has four fixed-pitchrotors mounted at the four ends of a simple cross frame,cf. Fig. 1. The attitude is modelled with the Euler-anglerepresentation, which provides an easier expression forthe linearised model. Moreover, this representation ismore intuitive. The inertial measurement unit model isgiven with the quaternion representation of the attitude.This choice is governed by the implementation of theattitude observer, which will be easier with the quaternionparametrization of the attitude.

Fig. 1. Quadrotor mini-helicopter.

Fault diagnosis in a networked control system under communication constraints. . . 811

2.1. Quadrotor model. The quadrotor is a small aerialvehicle controlled by the rotational speed of four blades,driven by four electric motors. A quadrotor is consideredto be a VTOL (vertical take off and landing) vehicleable to hover. Two frames are considered to describe thedynamic equations: the inertial frame N(exn , eyn , ezn)and the body frame B(exb

, eyb, ezb

) attached to the UAVwith its origin at the centre of mass of the vehicle.

The quadrotor orientation can be parametrized bythree rotation angles with respect to frame N : yaw (Ψ),pitch (Θ) and roll (Φ). ω ∈ R

3 is the angular velocity ofthe quadrotor relative to N expressed in B. The quadrotoris controlled by independently varying the rotationalspeed ωmi, i = 1, . . . , 4, of each electric motor. The forcefi and the relative torque Qi produced by motor i areproportional to ωmi,

fi = bω2mi, (1)

Qi = kω2mi, (2)

where k > 0, b > 0 are two parameters depending on thedensity of air, the radius, the shape, the pitch angle of theblade and other factors. The three torques that constitutethe control vector for the quadrotor are expressed in frameB as

τφa = d(f2 − f4), (3)

τθa = d(f1 − f3), (4)

τψa = Q1 +Q3 −Q2 −Q4, (5)

where d represents the distance from one rotor to thecentre of mass of the quadrotor. From the Newton–Eulerapproach, the kinematic and dynamic equations of thequadrotor are given by

(φ, θ, ψ) = Mω, (6)

Fig. 2. Quadrator mini-helicopter configuration: the iner-tial frame N(exn , eyn , ezn) and the body frameB(exb , eyb , ezb).

If ω = −ω × Ifω + τa +Ga, (7)

where If ∈ R3×3 represents the constant inertial matrix

expressed in B (i.e. If = diag(Ifx, Ify, Ifz)) and × in(7) denotes the cross product. The matrix M is given by

M =

⎡⎣1 tg θ sinφ tg θ cosφ0 cosφ − sinφ0 sinφ

cos θcosφcos θ

⎤⎦⎡⎣ωxωyωz

⎤⎦ , (8)

where ωx, ωy, ωz are the three measurements from tri-axerate gyros. Due to the rotation combination of thequadrotor four rotors, the gyroscopic torquesGa are givenas follows:

Ga =4∑i=1

Ir(ω × ezb)(−1)i+1ωmi, (9)

where Ir is the inertia of the so-called rotor (composed ofthe motor rotor itself, the shape and the gears). A linearcontrol law that stabilizes the system described by thenon-linear model ((6) and (7)) around the hover conditionsis designed.

Note that non-linearities are second order. Therefore,it is reasonable to consider a linear approximation. From(6) and (7) and under the hover condition (φ ≈ θ ≈ ψ ≈0), we can write

(φ, θ, ψ)T = (ω1, ω2, ω3)T . (10)

Then the dynamical model is obtained in terms of Eulerangles,

φ = θψIfy − IfzIfx

+τφaIfx

, (11)

θ = φψIfz − IfxIfy

+τθaIfy

, (12)

ψ = φθIfx − Ify

Ifz+τψaIfz

. (13)

The gyroscopic torques Ga are not considered for thedesign of the control law. However, they are taken intoaccount in simulations in order to analyse the robustnessfeatures.

2.2. Attitude control. In this section, the linearisedmodel of (6) and (7) is first derived. Then a control law isbriefly summarized. Note that this paper is not dedicatedto the design of a particular control law (Tayebi andMcGilvray, 2006; Castellanos et al., 2005). Therefore, weuse a simple LQ controller for stabilizing the quadrotorsystem. For the state feedback we use an estimated stateprovided by an extended Kalman filter that estimates boththe system state and the network-induced delay.

The linear dynamics of the system described beforeare given by the following state space model:

xT = (φ, φ, θ, θ, ψ, ψ). (14)


The system (12) around the hover conditions is given by

xt = Axt +But, (15)

where

A =

⎡⎣A0 0 00 A0 00 0 A0

⎤⎦ , B =

⎡⎣Bx 0 00 By 00 0 Bz

⎤⎦ ,

A0 =[0 10 0

], Bi =

[01Ifi

].

The attitude stabilization problem consists in drivingthe quadrotor attitude from any initial condition to adesired constant orientation and maintaining it thereafter.As a consequence, the angular velocity vector is alsobrought to zero and remains null once the desired attitudeis reached, x → 0, t → ∞. The discrete linear controlleris given by

ukh = −Lxkh. (16)

The control is designed to minimize to following objectivefunction:

J =N−1∑k=0

[xTkQdxk + uTkRduk] + xTNQ0xN , (17)

where

Φ = eAh, Γ =

(k+1)h∫

kh

eAsB ds

Qd =

(k+1)h∫

kh

ΦT (s)QΦ(s) ds

and

Rd =

(k+1)h∫

kh

(ΓT (s)QΓ(s) +R) ds

where matrices Q, R are symmetric and positive definite.Furthermore, the following assumptions are taken

into account.

Assumption 1. The full state vector is available (anglesand angular velocities). In practice, these state variablesare obtained from the measurements of rate gyros,accelerometers and magnetometers by using a dedicatedattitude observer (Castellanos et al., 2005).

Assumption 2. A periodic sampling is used.

Assumption 3. The control signals remain constantbetween two updates.

Proposition 1. Consider the quadrotor rotational dyna-mics described by (12). Then, the discrete control is givenby

ukh =[τφa,kh τθa,kh τψa,kh

]= −Lxkh. (18)

2.3. Control simulation. The weighting matrices Qand R are chosen in order to obtain a suitable transientresponse, and only feasible control signals are applied tothe actuators. Then for a sampling time h = 0.01 s thematrix gain which minimizes (17) and locally stabilizesthe quadrotor at x = 0 is given by

L =

⎡⎢⎢⎢⎢⎢⎢⎣

0.0352 0 00.0284 0 0

0 0.0352 00 0.0284 00 0 0.03520 0 0.0284

⎤⎥⎥⎥⎥⎥⎥⎦.

Here we simply present some results of the drone attitudesimulation with a variable step response (Fig. 3) and theLQ controller signal (Fig. 4).

3. Model for NCSs under the fault effectand communication delay

Induced time delays in networked controlled systems canbecome a source of instability and degradation of controlperformance (Yi et al., 2007; Zhang et al., 2004; Xiongand Lam, 2007; Sahebsara et al., 2007). When the systemis controlled over a network, we have to take into accountthe sensor to controller delays and controller to actuatordelays. Note that delays, in general, cannot be consideredconstant and known. Network-induced delays may vary,depending on the network traffic, medium access protocoland the hardware.

Assumption 4. For data acquisition it is supposed thatthe sensor is time-driven and the sampling period isdenoted by h. Both the controller and the actuator are

0 500 1000 1500 2000 2500

20

0

20

Rol

l °

time instants

drone attitude

Output signal desired trajectery

0 500 1000 1500 2000 2500

20

0

20

Pitc

h °

time instants

0 500 1000 1500 2000 2500

20

0

20

time instants

Yaw

°

Fig. 3. Quadrotor attitude (ϕ, θ, ψ) and reference.


event-driven. This signifies that calculation of the newcontrol or actuator signal is started as soon as the newcontrol or actuator information arrives, as illustrated inFig. 5

Assumption 5. Unknown time-varying network-induceddelay at time step k is denoted by τk, and τk =τsck + τcak is smaller than one sampling period τk ≤h, τsck and τcak are the sensor-to-controller delay andthe controller-to-actuator delay, respectively. There isno packet dropout in the networks. Thus, the controlinput (zero-order hold assumed) over a sampling interval[kh, (k + 1)h] is

ut =

uk−1, t ∈ [kh, kh+ τk],uk, t ∈ [kh+ τk, (k + 1)h].

(19)

Let us first assume that the residual generation andevaluation algorithms are executed instantaneously atevery sampling period k. Based on this assumption, if thecontrol input is kept constant over each sampling intervalh and if we consider that fault inputs have slow dynamics,the discrete time system can be described by

xk+1 = Φxk + Γ0,τk

uk + Γ1,τkuk−1,

yk = Cxk + vk.(20)

From

Γ0,τk=

h−τk∫

0

eAsB ds,

Γ1,τk=

h∫

h−τk

eAsB ds

0 500 1000 1500 2000 2500

1

0

1

tau

Rol

l °

time instants

control signal

0 500 1000 1500 2000 2500

1

0

1

tau

Pitc

h °

time instants

0 500 1000 1500 2000 2500

1

0

1

time instants

tau

Yaw

°

Fig. 4. Control signal.

it follows that

Γ =

h∫

0

eAsB ds = Γ0,τk+ Γ1,τk

and hence Γ0,τk= Γ − Γ1,τk

.In accordance with the properties of the definite

integral, if we introduce the control increment Δuk, theplant (20) with an unknown disturbance vector and a faultvector, which must be detected, is described by

⎧⎨⎩

xk+1 = Φxk + Γuk + Γ1,τkΔuk

+ Ξxdk + Ψxfk,yk = Cxk + Ξydk + Ψyfk,

(21)

where fk ∈ Rq is the fault vector and dk ∈ R

p is the noisevector.

Remark 1. Adding the sensor fault effect in both processand observation equations (21) is for the generalisation ofthe study. In our application which considers only sensorfaults we take Ψx = 0.

The matrix A is called diagonalizable if there existsan invertible matrix P such that

A = PΛP−1 = Pdiag(λ1, λ2, . . . , λn)P−1, (22)

where λ1, λ2, . . . , λn are the eigenvalues of the matrix A.Then we can write

eAt = I +At+ · · · + 1nAntn (23)

= PP−1 + PΛP−1t+ · · · + 1n

(PΛP−1)ntn (24)

= P (I + Λt+ · · · + 1n

Λntn)P−1 (25)

= PeΛtP−1. (26)

Fig. 5. Timing diagram for data communication.


From Eqn. (21), we can write

Γ1,τkΔuk

=

h∫

h−τk

P eΛ sP−1B ds Δuk (27)

= P

h∫

h−τk

eΛs dsP−1BΔuk (28)

= P

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h∫h−τk

eλ1s ds 0 · · · 0

0. . .

. . ....

.... . .

. . . 0

0 · · · 0h∫

h−τk

eλns ds

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

× P−1B Δuk (29)

= P

⎡⎢⎢⎢⎢⎣

1λ1eλ1 h 0 · · · 0

0. . .

. . ....

.... . .

. . . 00 · · · 0 1

λneλn h

⎤⎥⎥⎥⎥⎦P−1B Δuk

− P

⎡⎢⎢⎢⎢⎣

1λ1eλ1 (h−τk) 0 · · · 0

0. . .

. . ....

.... . .

. . . 00 · · · 0 1

λneλn (h−τk)

⎤⎥⎥⎥⎥⎦

(30)

× P−1B Δuk (31)

= ΓΔ Δuk − P diag[

1λ1, . . . ,

1λn

]diag(βk)

×

⎡⎢⎢⎢⎣

eλ1(h−τk)

eλ2(h−τk)

...eλn(h−τk)

⎤⎥⎥⎥⎦ (32)

= ΓΔ Δuk − ΓΔ,k

⎡⎢⎢⎢⎣

eλ1(h−τk)

eλ2(h−τk)

...eλn(h−τk)

⎤⎥⎥⎥⎦

= ΓΔ Δuk − ΓΔ,k dτk(33)

with

βk = [β1k β

2k . . . β

nk ]T = P−1BΔuk ∈ R

n,1,

diag(βk) =

⎡⎢⎢⎢⎢⎣

β1k 0 · · · 0

0 β2k

. . ....

.... . .

. . . 00 · · · 0 βnk

⎤⎥⎥⎥⎥⎦,

ΓΔ = P

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1λ1eλ1h 0 · · · 0

0. . .

. . ....

.... . .

. . . 0

0 · · · 01λneλnh

⎤⎥⎥⎥⎥⎥⎥⎥⎦,

ΓΔ,k = P diag[

1λ1

1λ2

· · · 1λn

]diag(βk).

According to (32), the model of (21) can also be rewrittenas ⎧⎨

⎩xk+1 = Φxk + Γuk + ΓΔ Δuk − ΓΔkdτk

+ Ξxdk + Ψxfk,yk = Cxk + Ξydk + Ψyfk

(34)by defining

Γak =[

ΓΓΔ

], uak =

[uk

Δuk

], Ξax,k =

[Ξx

−ΓΔk

],

Ξay =[Ξy0

], dak =

[dkdτ,k

],

which results inxk+1 = Φxk + Γaku

ak + Ξax,kd

ak + Ψxfk,

yk = Cxk + Ξaydak + Ψyfk.(35)

In a practical situation, ensuring the residualgenerator robustness against unknown input disturbancesis considered the main issue of FDI algorithm design.In the case of structured types of uncertainties, theexisting literature provides a wide variety of solutions forachieving robustness (see, for instance, Chen and Patton,1999; Ding, 2008). In the next section, FDI is revisited,considering network-induced effects. Model based faultdetection relies on the generation of a residual signalwhich is sensitive to failures and able to decouple faultsfrom other unknown disturbance inputs. The design mustensure that residuals are closed to zero in a failure-freecase while clearly deviating from zero in the presence offaults. In a first attempt, the idea is to consider a residualgenerator based on the following state observer:

xk+1 = Φxk + Γaku

ak + L(yk − yk),

yk = Cxk.(36)

The residual generator is given by

rk = T (yk − yk), (37)

where L and T are matrices designed to fulfil faultdetection and isolation requirements. From (36) and (37),it results that the estimation error εk = xk−xk propagatesas follows:

εk+1 = (Φ − LC)εk + (Ξax,k − LΞay)dak

+ (Ψx − LΨy)fk. (38)


The observer gain L is designed to stabilize the matrix(Φ − LC).

After the application of the Z-transformation, weobtain the following transfer function model:

zk = T (C(zI − Φ + LC)−1(Ξax,k − LΞay) + Ξay)dak

+ T (C(zI − Φ + LC)−1(Ψx − LΨy) + Ψy)fk.(39)

The matrix parameters T and L are determined to verifythe following requirements:

• asymptotic stability under fault free conditions, i.e.,fk = 0,

• minimization of disturbance effects,

• maximization of fault effects.

Perfect fault detection means the total decoupling ofthe residual signal from unknown inputs. This can bedescribed by

T (C(zI − Φ + LC)−1(Ξax,k − LΞay) + Ξay)dak = 0,

T (C(zI − Φ + LC)−1(Ψx − LΨy) + Ψy)fk = 0.(40)

Actually, there are various approaches (Gertler,1998; Chen and Patton, 1999; Frank and Ding, 1997;Ding, 2008) to the design for the gain matrices L andT . Therefore, developing a new technique does not makethe main objective of this paper. In the remainder of thepaper, we suppose that the system is controlled over acommunication network. Thus, we take in considerationthe sensor to controller delay and the controller toactuator delay in our design. For the illustration of FDIperformance degradation under a delay constraint, weperform a simulation using the system described by (15)as a plant model. It is supposed that the FD system basedon standard Kalman filtering is connected to the plant viaa network.

For this simulation, the network delay is supposedto be a Gaussian variable, the fault associated to the firstattitude sensor (φ: Roll) occurs at time instant k = 1000and the fault associated to the second attitude sensor (ψ:Yaw) occurs at time instant k = 1500. The result shownin Fig. 7 does not allow us to distinguish between the faultand the network variable delay effects. Hence, it appearsthat the robustness of the fault diagnosis system againstnetwork-induced delays depends on the amplitude of theunknown term ΓΔ,k , dτ,k.

It is clear that any robust design has to decoupleor at least minimize the effect of delay on the residual.This problem is equivalent to fault detection under theeffect of Gaussian noise and unknown inputs at thesame time (Darouach et al., 2003). The delay effect can

be considered an unknown input with a time-varyingdistribution matrix ΓΔ,k. In the sequel, we use a robustfilter for detection of faults that may occur in the quadrotorsystem.

4. Robust residual generation andevaluation

The objective of fault diagnosis is to perform two maindecision tasks (Frank and Ding, 1997): fault detection,consisting in deciding whether or not a fault has occurred,and fault isolation, consisting in deciding which elementof the system has failed. The general procedure comprisesthe following two steps:

• Residual generation: the process of associating,with the pair model–observation, features that allowevaluating the difference with respect to normaloperating conditions.

• Residual evaluation: the process of comparingresiduals with some predefined thresholds accordingto a test and at a stage where symptoms are produced.

This implies designing residuals that are close to zeroin fault-free situations while clearly deviating from zeroin the presence of faults, and possess the ability todiscriminate between all possible modes of faults, whichexplains the use of the term isolation. Therefore, theobjective here is to design a residual generator similar tothe one given by Eqn. (37) with the additional proprietyof robustness against network delay effects. Severalapproaches have been proposed in the literature (Wanget al., 2009; Sauter et al., 2009; Chabir et al., 2008).

0 500 1000 1500 2000 25001

0

1

Rol

l °

time instants

Residuals generation

0 500 1000 1500 2000 25001

0

1

Pitc

h °

time instants

0 500 1000 1500 2000 25001

0

1

time instants

Yaw

°

Fig. 6. Residuals generation by the standard Kalman filter (Cha-bir et al., 2010).


4.1. Residual generation. A solution of the abovementioned problem towards the design of an observerbased residual generator will be derived. First, let usdefine the vector

zk =[xkek

]. (41)

The overall system dynamics, which include the plant andthe residual generator, can be expressed as

zk+1 = Azk + Buak + Ξk,xdak + Ψxfkrk = T Czk + TΞayd

ak + TΨyfk,

(42)

where

A =[Φ 00 Φ −KkC

],

C =[0 C

], C =

[Γak0

],

Ξk,x =[

Ξak,xΞak,x − LΞay

], Ψx =

[Ψx

Ψx − LΨy

].

It is assumed that the plant is mean square stable,since the observer gain matrix L has no influenceon the system in (42). The overall system dynamics(plant + residual generator) is mean square stable. Thepost-filter T and the observer gain matrix L are the designparameters for the residual generator. The main objectiveof the design of the residual generator is to improvethe sensitivity of the FD system to faults while keepingrobustness against disturbances. Thus, the selection ofthe design parameters L and T can be formulated as thefollowing optimization problem:

supJ = supL,T

∥∥Grfz∥∥−

‖Grdz ‖∞, (43)

where

Grdz = T C(zI − A+ LC

)−1 Ξk,x + TΞay, (44)

Grfz = T C(zI − A+ LC

)−1 Ψx + TΨy. (45)

4.2. Residual evaluation. The second step of thefault detection procedure is to evaluate the residual.Residual evaluation is an important step of model basedFD approach (see, for instance, Ding, 2008). This stepincludes the calculation of the residual evaluation functionand determination of the detection threshold. The decisionfor successful fault detection is based on the comparisonbetween the results obtained from the residual evaluationfunction and the determined threshold. The followingresidual evaluation function is proposed:

Jek = ‖rk‖2,N

=

√√√√(

1N

N∑i=1

rk−i

)T (1N

N∑i=1

rk−i

),

(46)

where N is the length of the evaluation window. Thevariance of the residual signal can be expressed as

σrk = E((rk − rk)

T (rk − rk)). (47)

Under the assumption that the unknown input and controlinput are L2 bounded, the following theorem can be given.

Theorem 1. Given the system (15) and the constants γ1 >0, γ2 > 0, the following equation holds true:

σrk = E((rk − rk)

T (rk − rk))

< γ1

k∑j=0

(vTj vj + ΔuTj Δuj

)(48)

+ γ2

(vTk vk + ΔuTkΔuk

)

if there exist P > 0 such that

⎡⎢⎢⎣

−P PA PB Ξk,x( ) −P 0 0( ) ( ) −I 0( ) ( ) ( ) −I

⎤⎥⎥⎦ < 0, (49)

[ −P C( ) −γ1I

]< 0, (50)

[ −I Ψy

( ) −γ2I

]< 0, (51)

where the symbols ( ) denote the symmetric terms

Ξk,x =[

Ξax,kΞax,k − LΞay

], Ξax,k = [Ξx,−ΓΔ,k],

and ΓΔ,k is calculated for Δu = max(Δu).

Proof. Define the following Lyapunov function candidate:

Vk = zTk Pzk

with P > 0 and V0 = 0. This equation satisfies

EVk+1 − EVk <k−1∑j=0

(daTj daj ).

It follows that

EVk = EzTk P zk= zTk P zk + E

(zk − zk)T (zk − zk)

= zTk P zk + trace(Pσz),

where σz = E (zk − zk) .


By evaluating Evk+1, we get

EVk+1 = EzTk+1 P zk+1= E(Azk + Buak + Ξk,xdak

)T× P

(Azk + Buak + Ξk,xdak

)= E

[zk uak dak

]M1,0

[zk uak dak

]T+[zk uak dak

]M1,1

[zk uak dak

]

+ trace(σzMA),

where

M1,0 =[AT0 BT0 ΞT0

]TP[A0 B0 Ξ0

],

M1,1 =l∑i=1

(σ2i

⎡⎣

0BTi0

⎤⎦P [

0 Bi 0] ),

MA = ATp PAp,

such that

Ap =[

Φ 00 Φ − ρ1LC

], Bp =

[Γapρ2Γap

]

Cp =[

0 ρ1C],

where p ∈ 0, 1 and M1 = M1,0 +M1,1.Suppose that

M1 <

⎡⎣P 0 00 I 00 0 I

⎤⎦ . (52)

Then⎡⎣

AT0(B0 + σ1B1)T

ΞTx,0

⎤⎦ (P−1)−1

[A0 (B0 + σ1B1) Ξx,0

]

−⎡⎣P 0 00 I 00 0 I

⎤⎦ < 0 (53)

and hence

MA < P. (54)

By using the Schur complement, we get⎡⎢⎢⎣

−P−1 A0 (B0 + σ1B1) Ξx,0( ) −P 0 0( ) ( ) −I 0( ) ( ) ( ) −I

⎤⎥⎥⎦ < 0.

Equivalently,⎡⎢⎢⎣

−P PA0 P (B0 + σ1B1) P Ξx,0( ) −P 0 0( ) ( ) −I 0( ) ( ) ( ) −I

⎤⎥⎥⎦ < 0.

Note that the LMI given in (53) implies (54). From(53), (54), it is evident that

EVk+1 < EVk + (da Tk dak + ua Tk uak). (55)

This leads to

zTk P zk + trace(P σz) < k−1∑j=0

(daTj daj + ua Tj uaj ).

Now, from (48), we get

σrk = E(rk+1 − rk+1)T (rk+1 − rk+1)

=[zk uak dak

]M1

⎡⎣zkuakdak

⎤⎦+ trace(σzMc),

where Mc = (ρ1C)T (ρ1C) = ρ21C

TC.If

σrk < Vk ⇔ trace(σzMc) < zTk P zk + trace(P σz),

then it is evident that

σrk < γ1E k−1∑j=0

(daTj daj + ua Tj uaj

+ γ2EdaTk dak + γ2Eua Tk uak

Using the Schur complement, we get

Mc < γ1P ⇔ 1γ1ρ21C

TC − P < 0,

M1 < γ2 ⇔ 1γ2ρ21C

TC − P < 0.

This concludes the proof.

Note that Δuk is set to the allowed upper bound ofthe control input max(Δuk). The threshold can be set as

J thk =√αNβ, (56)

where

β = supσrk

= γ1(δd,2 +k∑j=0

(ΔuTj Δuj)) (57)

+ γ2

(δd,∞ + ΔuTkΔuk

)(58)

such that

δd,2 ≥k∑j=0

(vTj vj), δd,∞ ≥ vTk vk

are the L2, and L∞ norm of the unknown input,respectively. The parameter 0 < αN < 1 is a constant


value that depends on the length of the evaluation windowN . The constants γ1 and γ2 are parameters that representthe bounds on the variance of the residual signal. Notethat since the residual signal is a white noise process,the threshold will depend on the statistical part of it(which means the variance of the residual signal). Afterthe determination of a threshold, a decision whether afault occurs has to be made. The decision logic for theFD system can be defined as follows: Jek > J thk ⇒ fault,Jek < J thk ⇒ no fault. The threshold J thk is adaptive andis influenced by Δuk, which has to be calculated online.The simulations in the next section are performed in orderto validate the results of the proposed residual evaluator.

5. Simulation

The upper bounds on the unknown inputs are δd,2 = 0.15and δd,∞ = 0.28. The length of the evaluation windowis set to 50 and αN is set to 0.3. The parameters ofthe threshold (bounds on the variance of residual) arecomputed as γ1 = 0.0058 and γ2 = 0.05. The thresholdis then to be determined (adaptively) on-line during thesimulation. From the result shown in Fig. 7, it is clearthat the adaptive threshold allows fault detection and thelikelihood of the false alarm rate is drastically minimized.

6. Conclusion

In this paper we deal with the residual generation andevaluation issue within the framework of networkedcontrol systems. The problems addressed in this paperinclude (i) robustness against network delays as well asnoise and (ii) reducing the false alarm rate. In this context,a quadrotor attitude sensor fault is detected by a post-filter

0 500 1000 1500 2000 25000

0.2

0.4

Rol

l °

time instants

Residuals evolution

evaluated residual adaptive threshold

0 500 1000 1500 2000 25000

0.05

0.1

Pitc

h °

time instants

0 500 1000 1500 2000 25000

0.2

0.4

time instants

Yaw

°

Fig. 7. Evaluated residual.

and compared with an adaptive threshold that considersthe variation of control inputs as well as unknown inputs.The problem of threshold design is established in termsof linear matrix inequalities. Validation results show theeffectiveness of the obtained results.

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Karim Chabir received his Master’s in automatic and intelligent tech-niques in 2006 from the National Engineering School of Gabes (Tunisia)and a Ph.D. in automatic control from Henri Poincaré University (Fran-ce) and the University of Gabes in 2011. His research works were carriedout at the Research Centre for Automatic Control of Nancy (CRAN) andat the Research Unit of Modelling, Analysis and Control Systems of theNational Engineering School of Gabes. He was a member of the depen-dability and system diagnosis group (SURFDIAG). His current researchinterests are focused on model-based fault diagnosis and fault tolerantwith emphasis on networked control systems. He was a secondary scho-ol teacher of Gabes from 2003 to 2007, where he was also an assistantprofessor in the Faculty of Science of Gabes from 2007 to 2011. He hasbeen with the Temporary Teaching and Research (ATER) at the Facultyof Science and Technology of Nancy since 2011.

Mohamed Amine Sid was born in 1986 in Algeria. He received hisMaster’s degree in automatic control from the Department of ElectricalEngineering at Sétif University, Algeria. Since 2010, he has been wor-king towards a Ph.D. in automatic and signal processing at the ResearchCentre for Automatic Control of Nancy (CRAN, CNRS). He is curren-tly an associate professor at Setif University, Algeria. His main researchinterests are in networked control systems and fault detection.

Dominique Sauter received the D.Sc. degree (1991) from Henri Poin-carè University, Nancy 1 (now the University of Lorraine), France. Since1993 he has been a full professor at this university, where he teaches au-tomatic control. He was the head of the Electrical Engineering Depart-ment for four years, and now he is a vice-dean of the Faculty of Scienceand Technology. He is a member of the Research Center for AutomaticControl of Nancy (CRAN) associated with the French National Centerfor Scientific Research (CNRS). He is also a member of the French-German Institute for Automatic Control and Robotics (IAR), where hehas chaired a working group on intelligent control and fault diagnosis.His current research interests are focused on model-based fault diagnosisand fault tolerant control with emphasis on networked control systems.The results of his research works are published in over 50 articles injournals and book contributions as well as 150 conference papers.

Received: 15 July 2013Revised: 19 December 2013Re-revised: 25 February 2014

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