Abstract—This paper deals with observer-based actuator fault detection, isolation and estimation (FDIE) schemes for a class of linear systems with external disturbances. To isolate and estimate the possible actuator faults, we first relate each possibility to a faulty model, then for each possible faulty model, we design a corresponding Sliding Mode Observer (SMO), the residuals are defined as the norm of the magnitudes of the output estimation errors resulting from all possible faulty models. It is proved that, for the faulty model corresponding to the faulty actuator, the designed SMO can ensure the related output estimation error converges to a sufficiently small neighborhood. It is also shown that, for all other possible faulty models, the related output estimation errors are much larger. Finally, a Research Civil Aircraft Model (RCAM) is used to illustrate the design procedures and the effectiveness of the proposed actuator FDIE method.

I. INTRODUCTION CTUATOR, sensor or process faults may drastically change system behavior, they may cause the plant state

variables to deviate beyond acceptable limits, ranging from performance degradation to instability. To improve plant efficiency, maintainability and reliability can be achieved and accomplished by the design and application of fault detection and isolation (FDI) technique, which involves early detection and isolation of system faults, fault accommodation schemes or system reconfiguration, to avoid unexpected system deviations and to achieve the system goal in spite of the faults. During the past two decades, the design and analysis of such

strategies has received considerable attention and fruitful results can be found in several excellent survey papers [1]-[4] and books [5]-[7]. Amongst various FDI methods, the observer-based FDI technique is one of the most extensively studied schemes. In short, the plant output is compared with the output of an observer designed from a model of the system, and the discrepancy is processed to form a residual, which decides whether a fault occurs or not. In this way, the general procedure for using an observer for fault diagnosis consists of three main steps:

Jingjin Liu is with the School of Automation Engineering, Nanjing

University of Aeronautics and Astronautics, Nanjing, JS210016 China. (e-mail: ljjpaper@yahoo.com.cn).

Bin Jiang is with the School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, JS210016 China. (e-mail: binjiang@nuaa.edu.cn).

Youmin Zhang is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec H3G 1M8, Canada. (e-mail: ymzhang@encs.concordia.ca).

1. Estimating the output of the system by using an appropriate structure of the observer. 2. Comparing the estimated and the measured outputs, i.e.

generating the so-called residuals. 3. Analyzing the residuals and deciding if a fault occurred or

not. Generally speaking, two observer-based FDI schemes have been proposed: One is called dedicated observer scheme. In this scheme, to isolate one fault among N possible faults, N observers are designed to generate N residuals and the i th residual is designed only sensitive to the i th fault but decoupled from all other faults, where Ni ≤≤1 . The scheme can only be designed for some special cases and can only be used to detect and isolate one single fault. The other one is called generalized observer scheme proposed by Frank [1]. Similarly, N observers are also designed to generate N residuals. The idea here is to make the i th residual sensitive to all faults except the i th one. Once such residuals can be designed, the decision for fault isolation becomes straightforward. In this paper, we adopt the latter one.

An abundance of research work on FDI has concentrated on linear systems and only limited results for linear uncertain systems have been reported, but the model of the system for which the observer is designed will not be a perfect representation of the system. Consequently, the development of FDI schemes which are robust to model uncertainties has been the focus of a great deal of emerging research and received more and more attention recently. In particular, thanks to its robustness to system uncertainties and external disturbances, sliding mode observer (SMO) based FDI method, which provides robustness to parametric variations in the plant or system unknown uncertainties, has witnessed large amount of theoretical and application results, see [8]-[12].

In [10]-[12], the robustness properties of sliding mode observers were exploited to generate residuals. The sliding mode observer under consideration feeds-back the output estimation error through a discontinuously switched term which induces a sliding motion in the state estimation error space. Besides generating residuals, sliding mode observer-based strategy can also be used to reconstruct the fault signals [10-12], i.e. magnitude information about the faults can be obtained. Therefore, the use of SMO has two advantages: One is that it can deal with any types of bounded actuator faults (constant and inconstant faults); the other is that it can provide a method to estimate the faults.

Sliding Mode Observer-Based Fault Detection and Isolation in Flight Control Systems

Jingjin Liu, Bin Jiang and Youmin Zhang

A

16th IEEE International Conference on Control ApplicationsPart of IEEE Multi-conference on Systems and ControlSingapore, 1-3 October 2007

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In this paper, we make further discussion on the actuator fault diagnosis problems for linear systems with external disturbances based on [12]. We wish to not only detect and isolate the faults, but also estimate the magnitude of faults. After formulating our fault diagnosis problem for a class of linear uncertain systems and defining all possible fault models, we will design a bank of SMOs based on the faulty models. For the designed SMOs, a sufficient condition is derived based on Lyapunov stability theory. Finally, we apply our proposed actuator FDI method to a Research Civil Aircraft Model (RCAM), and simulation results demonstrate the efficiency and feasibility of the proposed scheme.

II. SYSTEM DESCRIPTION AND PRELIMINARIES This section introduces the preliminaries necessary for the

work presented in this paper. Consider initially the linear uncertain system described by the following model:

Cxy

tfuBAxx a

=Δ+++= )()(&

(1)

where nRx ∈ is the state vector, pRy ∈ is the output

vector, and mRu ∈ is the input vector. The matrix A , B , C are all of appropriate dimensions. af denotes system

actuator fault, and )(tΔ represents unknown external disturbance.

We assume that only actuator faults occur and no sensor faults are involved. For simplicity, we consider the case that only one single actuator is faulty at one time. The actuator fault diagnosis problem is formulated as: with the output y available, we can design an observer-based scheme to isolate the single faulty actuator after the fault has been detected and to estimate the faults.

To solve the problem, we need to design a bank of SMOs. To design SMOs with desired fault actuator isolation and fault estimation properties, we need the assumptions as follows: Assumption A1: Matrix B is of full column rank. Assumption A2: There exist appropriate matrices L , F and positive definite matrices P , Q :

TT

T

FCPBQLCAPPLCA

=

−=−+− )()( (2)

Assumption A3: There exists a constant W such that

Wt ≤Δ )( , i.e. )(tΔ is norm bounded. Meanwhile, the

system actuator fault is bounded, to be exactly, there also

exists a constant M , Mfa ≤ , and actuator fault is much

larger than system external disturbance, namely MW << . Remark 1: A 1 can be replaced by a weaker assumption that

),( CA is detectable. A 3 describes the types of system disturbance and actuator faults we can deal with. Basically, it allows any types of bounded actuator faults (including constant faults and inconstant faults).

III. SLIDING MODE OBSERVERS DESIGN AND ACTUATOR FAULT ISOLATION AND ESTIMATION

In this section, we first define all possible faulty models, and then we propose SMO designs based on the faulty models, meanwhile, some properties of the SMOs are derived and proved. Moreover, a sufficient condition for actuator fault detection and isolation based on the designed SMOs is finally deduced and a fault estimation method is also given.

A. Fault Models Once any actuator of the system is faulty, the resulting

system from (1) is called faulty model in our paper. Since we have m actuators and we assume only single fault will occur, therefore, we have m possible faulty models in total.

When the thl − actuator is faulty, the faulty models can be described as follows:

Cxy

tfububAxx alll

m

ljjj

=

Δ++++= ∑≠

)()(& (3)

where )( 1 mbbB L= , alf is the thl − actuator fault, ju is

the desired control when the thj − actuator is healthy. lu is

the control corresponding to the thl − actuator.

Remark 2: When all actuators are healthy, the true system is definitely different from any of the above m fault models. When thl − actuator is faulty, then the true system reduces to the thl − faulty model, while different from any other models.

B. Sliding Mode Observers Design and Their Properties In the last subsection, we’ve defined m possible faulty

models. The measured output is the output of the above faulty model.

According to A2 and using SMO design technique based on [12], we can design a bank of sliding mode observers for all possible faulty models as below:

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mieFeF

xCybBuyyLxAx

yii

yiii

ii

iiiii

≤≤−=

=++−−=

1

ˆˆ)ˆ(ˆˆ

ρμ

μ&

(4)

where iF is the thi − row of F and the switching gains of

the designed SMO: ρ is chosen such that M≥ρ .

yxCe iyi −= ˆ . L is chosen such that LCA − is a Hurwitz

matrix. Remark 3: It follows from (4) that we have to solve the matrix inequality and equality for P , L and F in (2). It would be very difficult if we try to solve it directly. Fortunately, it is equivalent to solve the following LMIs for appropriate P , L and F .

YPLYCYCPAPA

FCPBFCPBYCYCPAPA

TTT

T

TTTTT

1,0

00

−−=<+++

≤⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+++

(5)

Therefore, we can find the solutions by using Matlab® LMI

toolbox. The designed SMOs have the following properties: Theorem 1: Under the above mentioned assumptions A1~ A3, when the l-th actuator is faulty, i.e. li = , we define le

as lll xxe −= ˆ , if )(/2 min QWPel λ⋅> holds, then the

uniformly ultimately bounded stability of system state estimation error is guaranteed. When li ≠ , denote allii fbbD −= μ , if WD >> , then we

have li ee >> , hence, we can use the designed a bank of

sliding mode observers (4) to detect and isolate the faulty actuator. Proof: For li = , denote LCAA −=0 . According to (3)

and (4), we have )()(0 tfbeAe alllll Δ−−+= μ& (6)

We choose the following Lyapunov function:

lT

l PeeV = (7)

[ ] )8(2)(

2)(

)(2)(2

)(2)(2)(

min

2min

00

WPeQe

WPeeQ

MPbetPeQee

fPbetPeePAPAe

ePePeeV

ll

ll

lT

lllT

l

alllT

lT

llTT

l

lT

llT

l

⋅−−=

⋅⋅+−≤

−−Δ⋅⋅+−≤

−+Δ−+=

+=

λ

λ

ρ

μ

&&&

If )(/2 min QWPel λ⋅> , then 0<V& and le converges to a

small neighborhood:

{ })(/2)( min QWPeeU ll λ⋅≤= (9)

Thus, we can say that the system state estimation error is

uniformly ultimately bounded stable.

Remark 4: We can choose an appropriate observer gain matrix L such that we have proper neighborhood )(eU , hence, we can guarantee that the residual of the observer

yxCtr il −= ˆ)( is sufficiently small.

For li ≠ , similarly, it follows from (3) and (4) that: )(0 tfbbeAe alliiii Δ−−+= μ& (10)

Because matrix B is of full column rank (assumption A1), we know that ib and lb are independent. When it comes to

system external disturbance )(tΔ , if WfbbD allii >>−= μ

holds, we certainly have Wtfbb allii >>Δ−− )(μ

(since Wt ≤Δ )( ), therefore, compare (6) with (10), we can

say li ee >> , the proof method was the same as above

mentioned. ¶ Given all above statements, conclusion can be easily made.

So that we can use the designed sliding mode observers to implement actuator fault detection and isolation effectively.

C. Actuator Fault Detection, Isolation and Estimation We define the residuals of SMO (4) as:

miyxCtetr iyii ≤≤−== 1,ˆ)()( (11)

According to Theorem 1, when the l-th actuator is faulty, the residuals )(trl must tend to be extremely small; while for any

li ≠ , basically, )()( tetr yii = does not equal to zero on

any small time intervals, and we have )()( trtr li >> . Consequently, the faulty actuator can be detected and isolated successfully by monitoring the residual signal )(tri . That is to say, our designed bank of SMOs scheme for actuator fault detection and isolation is effective and feasible. However, to realize these SMOs in practice, we may face the chattering problem that they may exhibit. To reduce the chattering in SMO application, we suggest using the following bank of modified SMOs:

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mieF

eFbBuyyLxAx

ii

iii

iiiii

≤≤+

−=

++−−=

1,

)ˆ(ˆˆ

δρμ

μ (12)

where δ is a quite small constant.

We need to choose a suitable threshold such that )(trl would tend to be extremely small when the l-th actuator

is faulty, while other residuals litri ≠),( are not equal to zero

on any small time intervals and much greater than )(trl . Thus the modified SMOs (12) can not only diminish the chattering problem in practice, but also realize fault diagnosis successfully.

Once the l-th actuator is detected and identified as the

faulty one, we can say that )()( tetr yll = must tend to be

extremely small. If we further assume )(tel& is also very

small, based on (6), we can give a method to estimate the actuator fault as follows:

δρ

+−=

yll

yllal eF

eFf̂ (13)

where alf̂ is the estimation of the actuator fault alf .

IV. SIMULATION ON A RESEARCH CIVIL AIRCRAFT MODEL This section involves following works: A Research Civil

Aircraft Model (RCAM) is presented first. Then banks of desired SMOs are designed for the actuator fault detection and estimation. Finally, simulation results are demonstrated to prove the effectiveness of the method.

A. The Research Civil Aircraft Model The Research Civil Aircraft Model (RCAM) is originally a

nonlinear aircraft model with a 6 degree-of-freedom, both the initial purpose for developing this model and the detailed description of the model can be found in [13]. By trimming the nonlinear model and linearizing it at a specific aircraft mass, center of gravity location, aircraft altitude and air speed, a linear RCAM longitudinal model was developed in [14].

Here, we will add some bounded external disturbances (e.g. wind disturbances) to the system and use this linear uncertain RCAM to test the effectiveness and feasibility of our actuator fault diagnosis scheme.

The linear uncertain RCAM model has two inputs, five outputs, and five states. It can be given as follows:

CxytBuAxx

=Δ++= )(&

(14)

where

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−−−

−−−

−−

=

099.003.097.7900670.0220.0770.0360.770074.0028.0780.9190.2000000.10016.00098.0

A ,

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

=

000480.6620.19180.00058.044.2

B ,

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−−−=

10000099.0028.097.7900029.099.0000068.0023.0078.088.700001

C

We choose the system unknown disturbances as: ]0000)sin(2.0[)( tt =Δ Thus we know that 2.0)( ≤Δ t , i.e. 2.0=W .

The details about the definition of system inputs, states and outputs are listed below: The inputs are defined as:

(rad)deflectionthrottle

(rad)deflectiontailplane

th

t

−−

δδ (15)

The states are defined as:

)()()(

)()(

1

1

1

maircraftofaltitudezsmaxiszaircraftinvelocitywsmaxisxaircraftinvelocityu

radanglepitchsradratepitchq

−⋅−

⋅−

−⋅−

−

−

−

θ (16)

The measured outputs are defined as:

)()(

)(

)()(

1

1

1

maircraftofaltitudezsmaxiszearthinvelocityw

smaxisstabilityinairspeedV

gfactorloadbodyverticalnsradratepitchq

E

A

z

−⋅−

⋅−

−⋅−

−

−

−

(17)

For further reference about the design of the desired controller, please refer to [13].

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Here, we assume there are totally two actuators in the system, and only one single actuator is faulty at one time. The assumed system actuator fault can be described as:

⎪⎩

⎪⎨

⎧≤<

≤<−=

otherst

ttfa

01054511

1 , 02 =af (18)

It follows from (18) that we choose af as an incipient fault,

and Mfa ≤ , 4=M . Meanwhile, WM >> is satisfied. Since we only have two actuators here, our actuator fault

isolation problem reduces to identify which of the two actuators is faulty. There are only two possible faulty models, so we can design two modified SMOs corresponding to the two actuators as given below:

111

111

11111 )ˆ(ˆˆ

δρμ

μ

+−=

++−−=

y

y

eFeF

bBuyyLxAx&

(19)

and

222

222

22222 )ˆ(ˆˆ

δρμ

μ

+−=

++−−=

y

y

eFeF

bBuyyLxAx&

(20)

where 1

0 )( −−= CAAL , positive definite matrices P and Q

are chosen as 5IP = , 52IQ = . 0A is chosen as 50 IA −= . The

switching gain ρ is chosen as 20=ρ , such that M≥ρ .

1δ and 2δ are chosen as small positive constants

as 121 == δδ . We define the residuals of the SMOs respectively as

yxCtetr y −== 111 ˆ)()( , yxCtetr y −== 222 ˆ)()( .

B. Simulation Results If the first actuator is faulty, (19) is the true system model

in practice, otherwise, (20) will be the true system model. Thus, the fault diagnosis can be implemented in this way: If

)(1 tr is extremely small, while )(2 tr does not equal to zero on

any small time intervals and )()( 12 trtr >> , then the first

actuator is faulty and the fault can be estimated by 1μ . On the other hand, if )(2 tr is extremely small, while )(1 tr does not

equal to zero on any small time intervals and )()( 21 trtr >> , then the second actuator is faulty and the fault can be estimated by 2μ .

In the following simulation, we assume that the first actuator has been faulty and fault is an incipient fault which is described as (18).

Fig. 1 Actuator faults as describe in (18).

Fig. 2 System unknown external disturbances: [ ]0000)sin(2.0)( tt =Δ .

Fig. 3 Comparison of )(1 tr and )(2 tr .

Fig. 4 Comparison of actuator fault and fault estimation value. The blue

solid line represents the actuator fault, the red dotted line represents the fault estimation 1μ .

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In Figs. 3 and 4, we plot the residuals corresponding to the two actuators respectively and the estimation of fault. From Fig. 3, one can see that )(1 tr is always small, while )(2 tr is far from zero as time t increases and therefore )()( 12 trtr >> . This observation shows that the system actuator faults can be detected and isolated successfully and effectively. From Figure 4, we can clearly draw a conclusion that the actuator faults can be estimated accurately and promptly (within one second) as t increases.

Fig. 5 Block diagram of overall FDI system structure

Last but not least, to clearly demonstrate and illustrate the

function and placement of SMO-based FDI scheme in an overall active fault-tolerant flight control system, the block diagram which illustrates the overall structure of a typical SMO-based FDI control system is shown in Fig. 5. As can be seen from Fig. 5, the SMOs are the FDI module in active fault-tolerant control system. In the FDI module, actuator faults in the flight control systems should be detected and isolated as quickly as possible, and the actuator faults in the fight control system should also be estimated promptly by the reconfiguration mechanism. Therefore, the desired stability of our control system can be guaranteed and maintained.

V. CONCLUSIONS AND FUTURE WORK This paper proposes a kind of sliding mode observer based

actuator fault detection, isolation and estimation schemes for a class of linear uncertain systems with unknown external disturbance. To detect, isolate and estimate the possible actuator faults, we design a bank of Sliding Mode Observers (SMOs) corresponding to the system actuator respectively, the residuals are defined as the norm of the magnitudes of the output estimation errors resulting from all possible faulty models. In addition, a sufficient condition for actuator fault isolation is derived. A Research Civil Aircraft Model (RCAM) is used to illustrate the design procedures and the efficiency of the proposed actuator FDI method.

Simulation results showed that the scheme can detect and isolate the system actuator faults successfully and effectively, and estimate the corresponding actuator faults accurately and

promptly after a short transient period. Thus, the information about the faults can be obtained efficiently.

Further extensions of this scheme to nonlinear systems with partially known states are our future research topics.

ACKNOWLEDGMENT This research was supported by National Natural Science

Foundation of China (60574083).

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