Abstract—Today the problem of fault detection and diagnosis (FDD) is considered as an important and essential counterpart of control engineering systems. Because of importance and existence of faults that don't have a known structure in control system, i.e., fault occurred because of tangle of complex factors, In this paper a Lipschitz nonlinear system with unmeasured states and unknown faults is considered and a novel FDD architecture for it is presented. A neuro/fuzzy model consisting of few locally linear models (LLMs) with on-line updated centers and width vectors is used to approximate the model of the fault. A nonlinear observer is used to estimate the states of the system that are inputs to LLMs. The stability analysis of system is carried out via Lyapunov theory, from which the parameter updating rules are derived. At the end of this paper some numerical simulation is given to show the effectiveness of the method.

I. INTRODUCTION

AULT detection and diagnosis, i.e., monitoring

system to observe the time, situation and significance of the fault may occur in control system, is considered as a crucial counterpart of engineering system because of high reliability demand in control systems. Observer based fault detection is used in many studies [1], [ 2]. The fundamental idea behind it lies in output estimation using some nonlinear observer. The error of estimation is used as residual in this case. Then the weighted residual is used to generate fault function. A Luenberger-like observer can be used to estimate output in a deterministic framework [3], [ 4]. In situations where deterministic information of the output can’t be measured such as chemical engineering processes [5], neural network based observers that measure system output information based on stochastic value rather than deterministic value [6]-[8] can be useful.

On the other hand, since most of the faults occurring in a dynamic system have unknown structures, model based FDD approaches, which can be considered as the major group of proposed FDD design schemes will not be effective [9]. Therefore fault approximation approaches with the ability of learning and adaptation is used [10], [ 11].

Farzad Soltanian is Control eng. M.sc. student at Sahand University of Technology (SUT), Tabriz, Iran. (e-mail: f_soltanian@ sut.ac.ir).

Ahmad Akbari alvanagh is with Sahand University of Technology (SUT), Tabriz, Iran as Assistant professor of Electrical Engineering Department. (e-mail: a.akbari@sut.ac.ir).

Mohammad Javad Khosrowjerdi is with Sahand University of Technology (SUT), Tabriz, Iran as associate professor of Electrical Engineering Department. (e-mail: khosrowjerdi@sut.ac.ir).

Therefore, in this paper an adaptive neuro/fuzzy model, with the description, detailed in section IV, is used for estimation of the fault, the inputs of this LLMs are system states, since the system states aren’t measurable a nonlinear observer is used to estimate system states. Then stability analysis is given via Lyapunov theory and parameter updating rules are derived. C. –S. Liu and et.al used a RBF neural network to approximate the model of unknown fault (2007). This paper is a development of adaptive RBF FD. In this paper a neuro/fuzzy model consisting of few locally linear models (LLMs) with on-line updated centers and width vectors is used to approximate the model of the fault. If all coefficients of the Weight vector, except the constant part, equaled to zero then adaptive RBF equation obtains.

This paper is organized as follows. In section II, a general nonlinear system is presented. In section III, a Lipschitz observer is designed to estimate system states. In section IV, LLMs based model is introduced for detection and diagnosis of fault and stability analysis is considered, and parameter updating rules are obtained from it. Finally in section V, the numerical example is presented to illustrate the effectiveness of the proposed method.

II. REPRESENTATION OF SYSTEM Consider a nonlinear system which has the following

structure with unknown fault:

( ) ( ) ( ),x A t T f x

y C

x x ux

φ β= + + −

= (1)

Where nx ∈ is the unmeasured state vector, mu ∈ is the input vector, and py ∈ is the output

vector ( ) : n nxφ → is the known vector field that represents the nonlinear dynamics of system and

( ) : n nf x → is the unknown vector field that represents the nonlinear fault function of the system. C is known matrices. The matrix function of ( )t Tβ − represents the time profile of fault, where denotes the fault occurrence time. ( )t Tβ − Has a diagonal form as:

( ) ( )( )

1,...,it T diag t T

i nβ β− = −

= (2)

Adaptive Locally-linear-models-based Fault Detection and Diagnosis for Unmeasured States and Unknown Faults

Farzad Soltanian, Ahmad Akbari alvanagh, Mohammad Javad Khosrowjerdi

F

2011 2nd International Conference on Control, Instrumentation and Automation (ICCIA)

978-1-4673-1690-3/12/$31.00©2011 IEEE 507

Where iβ denotes the function of affecting fault on ith state equation. Without loss of generality, it is supposed that the fault is a function of system states only. If the fault is a function of inputs as well, this method will also be effective. It is sufficient to suppose that input vector of LLMs is a function of inputs and states, instead of system states only.

III. DESIGNING OBSERVER To generate and estimate the system states that are inputs

of fault diagnosis LLMs, nonlinear observers such as Lipschitz [2], [12], Luenberger like [13] and so on can be used.

In this section a nonlinear Lipschitz observer is designed for the system before occurring fault to estimate the system states.

The system equation is rewritten as follows:

( , )x Ax x u

y Cx

φ= +

= (3)

The nonlinear Lipschitz observer has the following form: ˆ ( ) ( )ˆ ˆ ˆ,ˆ ˆx A G y

C

x x u yy x

φ= + + −

= (4)

Defining ˆe x x= − , from (3) and (4), the following state error equation is obtained:

1 ˆ( ) ( )e Ae x x= + Φ − Φ (5)

Where 1A A GC= − is designed as a Hurwitz matrix. The Lipschitz system satisfies the following condition:

ˆ ˆ( ) ( )x x x xγΦ − Φ ≤ − (6)

Lipschitz theorem: for nonlinear system (3), the nonlinear observer (4) where 1A , C is observable, error equation (5) is asymptotically stable [12, 14], if equation (6), in which γ is bounded by

min max( ) 2 ( )Q Pγ λ λ< (7)

Where 0TP P= > and Q are positive definite

matrices given by 0TA P PA Q+ = − > , is satisfied. Proof: The Lyapunov candidate can be selected as

follows:

1TV e Pe= (8)

By differentiation we obtain:

1

1

1

ˆ[ ] 2 [ ( ) ( )]ˆ2 [ ( ) ( )]

T T

T T T

T T

V e Pe e Pe

V e A P PA e e P x x

V e Qe e P x x

= +

= + + Φ − Φ

= − + Φ − Φ (9)

If Lyapunov condition is asserted on system, the equation

can be reformed as follows:

1

2min

2min

2

( ) 2

[ ( ) 2 ]

T T T

T T

V e Qe e P e

Q e e P e

Q P e

γ

λ γ

λ γ

≤ − + ≤

− + ≤

− − (10)

Which implies that if min ( ) 2Q Pλ γ> is Chosen, then

1 0V < .

IV. FAULT APPROXIMATOR A Locally linear NeuroFuzzy model (LLNFM) is used in

this study to model the fault. A diagram of typical LLNFM with 'm LLM and n input is shown in Fig. 1 where each LLM has the following structure [17]

0 1 1ˆ ...i i i in ny w w x w x= + + + (11) Corresponding to each LLM there exist an activation

function which specifies validity degree of that LLM for different inputs.

Fig. 1- Diagram of LLNFM with m′ LLMs and n inputs

Universally LLMs is used in many engineering processes

where system structure is known[15], or faults of the system are known such as Cement rotary kiln [16]; in this situation at first system must be trained by data that are available, then primary model for fault is constructed, in the second step by asserting infinite data on fault estimated model, its fitness is tested; finally the validation of LLMs is considered, if validation is suitable training is stopped, else estimation model must be modified. In this situation by training of network, inner model for fault is constructed and the model is modified by minimization of error between desired output and its estimated value, LLM estimation Algorithm for this situation can be such as follow:

508

1-If number of LLM isn’t available, system estimated by one LLM, where has a following structure:

0 1 1ˆ ... n ny w w x w x= + + + (12) Where denotes output estimated value and number of

input to LLM, respectively; Obviously In this situation validation function is equal to 1.

Otherwise, if number of LLM be available, by partitioning input space, system can be estimated by LLM, where denotes the number of LLM. And validation function is constructed for these LLMs, such as follow:

'

'

0 1 1

1

1

ˆ ...

( ) 1

ˆ ( )

i i i in n

m

ii

m

i ii

y w w x w x

x

y y x

=

=

= + + +

Θ =

= Θ

(13)

Where 1

( ) 1m

ii

x′

=

Θ = , represented validity function.

2-Then, the following cost function is constructed:

2

1

ˆ( ) ( )m

i i ii

J y y x′

=

= − Θ (14)

Where ˆ, iy y are stand for desired and estimated value respectively.

3-If the cost function is smaller than desired error; then training stopped. Otherwise, available input space should be divided to 2 partitions and assigned LLM to them. Then validity function reconstructed for them. Finally the worst one that has a maximum cost functions of estimation funded.

4- Go to step3 since error be smaller than it’s desired.

In situation that fault structure is unknown; algorithm can be modified as follow:

If fault occurred, from (1), state equation has a following structure:

( ) ( ),x A f x

y C

x x ux

φ= + +

= ( )T t≤ (15)

The out pout ( )ˆf x is used to approximate ( )f x , where

1( )ˆ ˆ ˆ ˆ( ) ( )

m

i ii

f x wv x x′

=

= Θ (16)

B is unknown matrices, that is designed such that satisfies condition of theorem1, 1 ( 1)ˆ : n n n n

iw+ +× → is weight

function, where i denote the i th locally linear models (LLMs) are used in neuro/fuzzy model,

1ˆ( ) : n nv x +→ is function of input vector where

1ˆ ˆ ˆ( ) [1, ,..., ]Tnv x x x= and ˆ( )i xΘ ∈ denote validity function of LLM s, where:

1

ˆ( )ˆ( )ˆ( )

ii m

ii

xx

x

μ

μ′

=

Θ = (17)

Where:

1 1

1

ˆ ˆˆ( ) (exp( ))...(exp( ))i n ini

i in

x d x dxμσ σ− −= denote

activation function of ith LLM, pd ∈ where

'1[ ,..., ]T Tm

d d d= and p nm′= , [ ]1 . . .i nd d d= ,pσ ∈ , where 1[ ,..., ]T T

pσ σ σ= and [ ]1 ,. . .,i nσ σ σ= .

By using LLM to estimate fault function, Equation (15) can be rewritten as follow:

1( ) ( ), ( ) ( ) ,

m

i ii

A B

y C

x x x u wv x x t T

x

φ ε′

=

= + + +

=

Θ ≥ (18)

Where ε is the approximation error of LLMs. Therefore observer with fault has the following structure:

1( ) ( ) ( )ˆ ˆ ˆ ˆ ˆ ˆ ˆ, ( ) ( )

ˆ ˆ

m

i ii

A B G y

C

x x x u wv x x y

y x

φ′

=

= + + + −

=

Θ (19)

Assumption1: Exist an ideal weight matrix ˆiw , such as

i iw w≤ and δ δ≤ for all , 1,...,ndx A i m′∈ ⊂ = ,

where iw andδ are positive constant and dA is compact set.

Assumption2: suppose ( )f x f≤ for

all ndx A∈ ⊂ , where f is positive constant.

Assumption 3: system states are bounded and fault occurs

after estimation of system states, because estimated states are considered as inputs of fault approximator.

The error equation is obtained as follows:

1

1

ˆˆ ˆ ˆ ˆ ˆ( ) ( ) ( ( ) ( , , )1

ˆ( ( ) ( , , ))) ( ) ( )1

ˆˆ ˆ ˆ ˆ( ( ) ( , , ) ( ) ( , , )))1

i i

i i

i i i i

me Ae x x B wv x x d

im

wv x x d Ae x xim

B wv x x d wv x x di

σ

σ δ

σ σ δ

′= + Φ − Φ + Θ −

=′

Θ − = + Φ − Φ +=

′Θ − Θ −

=

(20)

The following relation is defined:

509

ˆ

ˆ( , , ) ( , , )

i i i

i i i

i i

w w wv v v

x d x dσ σ

= −= −

Θ = Θ − Θ

(21) By substituting eq. (21) in (20) the error eq. is obtained as

1ˆˆ ˆ ˆ ˆ ˆ( ) ( ) ( ( ) ( , , )

1

ˆ( ( ) ( , , ))) ( ( ) )1 1

i i

i i i

me Ae x x B wv x x d

im m

wv x x d B w v xii i

σ

σ δ

′= +Φ −Φ + Θ −

=′ ′

Θ + Θ −= =

(22)

By expanding Taylor’s series of ˆ( , , )i x d σΘ at ˆ ˆ( , )d σ The following formula is obtained:

ˆ ˆˆˆ ˆ ˆ( , , ) ( , , ) (.)

iii i iidx d x d d hoσσ σ σΘ =Θ −Θ −Θ +

(23)

Where ˆ ˆ|i i i

ii d d d

idδδ =

ΘΘ = , ˆ ˆ|i i i

ii

iσ σ σ

δδσ =ΘΘ =

and ˆ ˆ,d d d σ σ σ= − = − then by using above relation error equation can be rewritten as follow:

1

ˆ ˆ

ˆ ˆ

1

ˆˆ ˆ ˆ( , ) ( , ) ( ( , , )1

ˆ ˆ ˆ ˆ( ) ( ( ) ) (1 1

( ))1 1 1

ˆˆ ˆ ˆ( , ) ( , ) ( ( , , )1

ˆ ˆ(

ii

ii

i i

i i iid

i i i i iid

i

i

me Ae x u x u B wv x d

im mwv x wv x

i im m m

B wv wv wvi i i

mAe x u x u B wv x dii

wv x

σ

σ

σ

δ

σ

′= +Φ −Φ + Θ +

=′ ′

Θ + Θ − += =

′ ′ ′Θ + Θ − Θ

= = =′

= +Φ −Φ + Θ +=

ˆ ˆˆ ˆ) ( ( ) ) ( )1 1

iii iid

m mwv x

i iσ δ

′ ′Θ + Θ − +Λ

= =

(24)

Where:

ˆ ˆ

1

(1 1

ˆˆ ˆ ˆ( ( , , ) ( , , )))1

iii i ii d

m

i i i ii

m mB wv wvi i

mw v v x d v x d

i

σ

σ σ′

=

′ ′Λ = Θ + Θ −

= =′

Θ − Θ − Θ=

Lemma1: if the following parameter updating equations are employed, then will be bounded:

22 ˆ

ˆ ˆ ˆ( ) (1 ),ˆ ˆ ˆ, 0

ˆ ˆ( )

iT T Tw y i i

i i iT T T

W y i

wL e x v w

w wwL e x v else

βγ β

γ

>− Θ + −

= >− Θ

(25)

2

2

ˆ

ˆ

ˆˆˆ ˆ (1 ),ˆ ˆ ˆ, 0

ˆ ˆ ,

i

i

i dT T Td i y d iid

ii i

T T Td i yid

dv w L e d

d d dv w L e else

βγ β

γ

>− Θ + −

= >− Θ

(26)

ˆ

ˆ

ˆ

ˆ ˆˆ ˆ ˆ , 0( ) (1 ),ˆ ˆ ˆ ˆ ˆ ˆ( ) (1 ), , 0

ˆ ˆ( ) ,

i

i

i

T Ti iy i i

T Ti y i i i i

T Ty i

e Lwv

e Lwvelsee Lwv

σ σ

σ σ

σ σ

σ κ σγ κ σσ γ σ σ σ

γ

< <− Θ + −

= − Θ −Ω+ > Ω >

− Θ

(27)

Theorm1: for the nonlinear system (19), if parameter updating rules of (25), (26), (27) are used and L satisfies

TPB C L= , then error equation (24) will be bounded. Proof: The Lyapunov candidate can be selected as

follows:

21 1

1

1 1 1( ) ( )2 2 2

1 ( )2

n nT T T

i i i ii iW d

nTi i

i

V e Pe tr w w tr d d

trσ

γ γ

σ σγ

= =

=

= + +

+ (28)

By differentiation, following equation is obtained:

'

'

2

1

1

ˆ

1 ˆ[ ( , ) ( , )]2

1 1 ˆˆ( ) ( (( ) ( )

1 1ˆ ˆ( ))) [ ( , ) ( , )]2

1ˆ ˆ ˆ( ) ( (( ( ) )

1 ˆˆ ˆ( )i

T T

mT T T

i ii W d

T T T

mT T T T T

i y i ii W

T T T T Ti yid

d

V e Qe e P x u x u

e P tr w w d d

e Qe e P x u x u

e P tr w L e x v w

d v w L e d

σ

δγ γ

σ σγ

δγ

σγ

=

=

= − + Φ −Φ +

Λ + + + +

= − + Φ −Φ

+ Λ + + Θ + +

Θ + + ˆ1ˆ ˆ ˆ(( ) ))

i

T Ty ie Lwv σ

σ

σγ

Θ +

(29)

By using assumption 3, expanding ˆ( , , )i x d σΘ at ˆ ˆ,d σ , and equation 22, the following equation obtained; for more detail consult with [3], [18].

510

2

2min

2min

1 ˆ[ ( , ) ( , ) ( )]212

1 ( )2

1( ( )2

T T

T T T T

T T T

V e Qe e P x u x u

e Qe e P e e P

Q e e P e e P

Pe Q P

e

δ

γ σ

λ γ σ

σλ γ

≤ − + Φ − Φ − Λ + ≤

− + + Λ + ≤

− + + Λ + =

Λ +− − +

(30)

If max

min max

( )1( ( ) ( ))2

Pe

Q P

λ δ

λ γλ

Λ +>

− is chosen, then

according to theorem 1, 2 0V < and also ,d σ are bounded.

V. SIMULATION EXAMPLE In order to show effectiveness of this method, following example is investigated. Example: a nonlinear system by following structure is given:

( ) ( ) ( ),A t T f X

y C

x x x ux

φ β= + + −

=

Where ( )f X is unknown. Choosing the parameters of the system as:

[ ]

1 2 1 2

2 1

12 8, 1 1

5 5

8 0.1 sin( )( )

3 40.5sin(0.2 )

A C

x x x xx

x x

u t

φ

=−

=−

− +=

−=

For simulation, following parameters are designed:

[ ]8 228.4999 190.4166, 6 5 ,

5 107.8133 89.8444

25 2.66 1000 80,

2.66 50 80 1000

G L B

P Q

= = =

= =

Initial values that asserted on system are:

[ ] [ ]

'

0.4,0.4 , 0.4,0.6 , 1, 20.9, 0.1, 10, 0.1, 0.1

10;

i i

w d

d i

mσ

σκ γ γ γ

∈ − ∈ =Ω = = = = =

=

,

.

Fig. 2 shows system states and their estimation without fault. In Fig. 3 nonlinear fault and states estimation errors, where fault is a function of system states only showed. Obviously from Fig. 3,4 can be concluded that this approach is suitable for detection and diagnosis of external faults that have a function of time instead of system states, too.

Fig. 2. state and estimation state before occurring fault

Fig. 3. fault and estimated fault, state error estimation,

511

where

2

2 1cos ( )( )

0

x xf x =

Fig. 4. fault and estimated fault, state error estimation, where

[ ( 5( 1)) ( 5)] ( 5), 10( )

( ) 50 1000

r t i r t i u t it

f x i=

− − − − −− ≥

∞

=

Fig. 5. fault and estimated fault, state error estimation,

where 0.1

( )0

f x =

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