# Nonlinear Observer-Based Fault Detection

**ABSTRACT** This communication deals with the problem of designing a nonlinear observer in order to achieve fault detection and localization for a wide class of nonlinear systems subjected to bounded nonlinearities. A dedicated nonlinear observer scheme (DNOS) for fault detection and identification in reconstructible systems is proposed. INTRODUCTION State observation of nonlinear dynamical systems is becoming a growing topic of investigation in the specialized literature. The reconstruction of the time behaviour of state variables remains a major problem both in control theory and process diagnosis. Researchers attention is being particularly focused on the design of adaptive observers for on-line process state estimation. There is increasing awareness that to ensure robustness in performance requires simpler and stable adaptive observer schemes. Linear systems have received considerable attention (Luenberger, 1966), (O'Reilly, 1983) leading to several stable adaptive observer systems (Kreisselm...

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**ABSTRACT:**This paper propose a robust nonlinear unknown input observer "this an extension of the Luenberger observer in unknown inputs" based on first order Taylor expansion. The observer is characterized by its simplicity in the mathematical development can also attack a large class of nonlinear systems without go through a canonical transformation. A systematic method for calculating the gain of the observer is presented (11). The necessary and sufficient conditions for the existence of the observer are given. A numerical example is given to illustrate the attractiveness and the simplicity of the new design procedure. I. INTRODUCTION A fault tolerant system is able of maintaining stability and a degree of performance in the presence of disturbances. These systems are generally classified into two approaches: Passive Fault-Tolerant Control Systems and Active Fault-Tolerant Control Systems. In our case we will consider the Passive Fault-Tolerant Control Systems.01/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper provides some new developments in the design of robust local unknown input observers for nonlinear systems based on first order Taylor approximation. The necessary and sufficient conditions for the existence of the observer are given. A numerical example is given to illustrate the attractiveness and the simplicity of the new design procedure.01/2011; - SourceAvailable from: vbn.aau.dk

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Nonlinear observer-based fault detection

Kondo ADJALLAH, Didier MAQUIN, José RAGOT

Institut National Polytechnique de Lorraine

Centre de Recherche en Automatique de Nancy - CNRS UA 821

BP 3 - 2, Avenue de la Forêt-de-Haye

France - 54 516 Vandoeuvre Cedex

Abstract - This communication

problem of designing

order to achieve fault detection and localization

for a wide class of nonlinear systems subjected

to bounded nonlinearities. A dedicated nonlinear

observer scheme (DNOS) for fault detection and

identification in reconstructible

proposed.

deals with the

a nonlinear observer i n

systems i s

INTRODUCTION

State observation of nonlinear dynamical systems is

becoming a growing topic of investigation in the

specialized literature. The reconstruction of the time

behaviour of state variables remains a major problem both

in control theory and process diagnosis. Researchers

attention is being particularly focused on the design of

adaptive observers for on-line process state estimation.

There is increasing awareness that to ensure robustness in

performance requires simpler and stable adaptive observer

schemes. Linear systems have received considerable

attention (Luenberger, 1966), (O'Reilly, 1983) leading to

several stable adaptive observer systems (Kreisselmeier,

1977, 1979), (Magni and Mouyon, 1991). Linear observer

systems involving unknown inputs have also been

developed and analyzed (Viswanadham and Srichander,

1987). Nevertheless, the design of asymptotically stable

observers remains a hard task in the nonlinear case, even

when the nonlinearities are fully known.

Several observer design approaches have been proposed in

recent years for nonlinear systems. Walcott and Zak

(1986) and (1987) for instance, proposed a new type of

observer for systems subjected to bounded nonlinearities

or uncertainties. This type of observer does not require

exact knowledge of the system nonlinearities. Bastin and

Gevers (1988) also

observer/identifier, for SISO

systems, capable of state estimation and parameter

adaptation. Marino (1990) also developed the same idea

and proposed a simpler observer restricted however to a

class of systems with constant unknown parameters. As a

main result, the construction of an observer may be

performed by finding suitable state space and output

changes of coordinates to transform the nonlinear system

into an observable form from which can be derived an

observer with linear dynamics. Xia and Gao (1988) and

Krener and Isidori (1983) have given Lie algebraic and

rank conditions for nonlinear process observability and

observer existence purpose.

described an adaptive

observable nonlinear

Simultaneously, the techniques of fault detection and

isolation (FDI) are increasingly discussed in both research

and applications. It is based on the use of analytical

redundancy. The general procedure first generates the so-

called residuals (i.e., faults accentuated functions) before

proceeding to fault detection and isolation (i.e.,

determination of its location, duration, type, magnitude,

source). The use of observers is among the well-

established concepts (Patton et al, 1989) for linear

systems. For the general case of nonlinear systems

(Hengy and Frank, 1986), (Seliger and Frank, 1991) little

has so far been achieved in the development of associated

FDI observers. Solutions so far proposed are difficult to

apply in real situations.

This note is organized into four sections. The first

presents the observer design for nonlinear systems. The

second considers dedicated observers while the third deals

with the analysis of generated residuals. The last section

is devoted to computational aspects and uses the

numerical results of a synchronous machine to highlight

the strategy employed to isolate faults.

1. NONLINEAR OBSERVER DESIGN

We shall consider nonlinear systems of the form:

˙ x(t) = f x(t), u(t)

(

y(t) = Cx(t)

x ∈ Rn, u ∈ Rm and y ∈ Rp

), x(0) = x0

(1)

It is assumed that for any input u(t) and initial state x0,

the corresponding state trajectory x(t) is defined for all t

and that f is continuously differentiable. Proceeding by

analogy to the classical observer design approach in the

linear case, we seek an observer of the following form:

ˆ˙x = f ˆ x(t), u(t)

(

ˆ y = Cˆ x(t)

)+ g y(t)

()− g ˆ y(t)

(), ˆ x(0) = ˆ x0

, (2)

where the analytical function g: Rp → Rn is to be

determined. The state and output errors are respectively

defined by:

e(t) = x(t)−ˆ x(t)

ε(t) = y(t)−ˆ y(t)

(3)

For a sake of simplicity, the time variable will be now

omitted. The dynamic of estimation error e(t) is then:

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˙ e = f(x, u)− f(ˆ x, u)− g(y)+ g(ˆ y)

(4)

Assuming

asymptotically to the state of the system, one can

consider the state error (equation (3)) in the neighbourhood

of zero. This allows the use of a first order Taylor

expansion of the function f:

that the observer state converges

f(x, u) = f(ˆ x + e, u)

= f(ˆ x, u) + Dˆ x( f )e

(5a)

where Dˆ x is a differential operator defined by:

Dˆ x( f ) =∂f(x, u)

∂xT

x=ˆ x

Similarly, for g:

g(y) = g(ˆ y)+ Dy(g)Ce

(5b)

with:

Dy(g) =∂g(y)

∂yT

y=ˆ y

Consequently, the dynamic of the estimation error may be

rewritten:

[

˙ e = Dˆ x( f )− Dy(g)C

]e

(6)

For state reconstruction, the idea is to select a continuous

mapping g(y) so that ˆ x(t) becomes a state estimator of

the process under consideration. If the pair Dˆ x( f ), C

is observable at any time t, then a matrix Dy(g) must be

determined so that (6) has arbitrary stable poles at each

operating point parametrized by the control u. The natural

question that arises is when does the function g exist ?

Only sufficient condition may be found (Misawa and

Hedrick, 1989). A particular structure of the observer is

proposed in order to simplify the calculation of this

mapping:

{}

ˆ˙x = f ˆ x, u

(

ˆ y = Cˆ x

)+ R ˆ x, u

() y −ˆ y

(), ˆ x(0) = ˆ x0

(7)

The state error is then solution of the equation:

˙ e = f(x, u)− f(ˆ x, u)− R ˆ x, u

() y −ˆ y

()

(8)

The matricial function R(ˆ x, u) is chosen so that the

state error e(t) asymptotically decreases and approaches

zero as t tends to infinity. The error e(t) is then considered

to be in the neighbourhood of zero. By using (5a) and

(5b), a first order Taylor expansion of the function f(x, u)

in the neighbourhood of the estimated state trajectory

ˆ x(t) is substituted in (8), that gives:

˙ e = Dˆ x( f )− R(ˆ x, u)C

[]e

(9)

So, let us consider the quadratic Lyapunov function:

V(e) =1

2eTPe

(10)

where P is a positive definite matrix. We require time

derivative of V(e) to be negative:

˙V(e) = eTP˙ e

˙V(e) = eTP Dˆ x( f )− R(ˆ x, u)C

[]e

(11)

This condition ensures that e decreases exponentially to

zero (Ogata 1970, Corless 1988). For a particular

structure of the function f, Tsinias (1989) proposed an

algorithm for determining the gain R(ˆ x, u) based on the

assumption that Ker C

( )

{

generalization of this algorithm which is based on a

sequential determination of P and R(ˆ x, u) avoiding the

nonlinear coupling appearing in (8). The algorithm

comprises two steps, the first one being devoted to the

determination of P and the second one to the

determination of R(ˆ x, u) using the previous value of P.

} ≠ {0}. We propose here a

Step 1:

is reduced to:

If e ∈ Ker C

{

( )

}− 0 { }, then the equation (11)

˙V(e) = eTPDˆ x( f )e

(12)

The problem is to find a matrix P which ensures the

condition:

˙V(e) = eTPDˆ x( f )e < 0

(13)

Solving inequation (13) yields a value for P. For that

purpose, assuming

Ker C

( )

{

transformation:

}≠ 0 { }, one can use the

e = Ke

(14)

where K is right orthogonal to C. Substituting (14) into

(13) gives:

˙V(e) = e

TKTPDˆ x( f )Ke

(15)

where the dimension of e is less than those of e. Making

˙V(e) negative, by majorization techniques, gives the

matrix P (see the example of section 4).

Step 2:

chosen as an identity matrix, we now allow e ∈ Rn and

try to determine R(ˆ x, u) which verifies the following

inequality:

If step 1 produces a suitable P, else P can be

˙V(e) = eTP Dˆ x( f )− R(ˆ x, u)C

[]e < 0

(16)

A sufficient condition to fulfill this inequality is that the

matrix Dˆ x( f )− R(ˆ x, u)C be negative semidefinite.

Page 3

This is achieved by using first the following structure

proposed for R(ˆ x, u):

R(ˆ x, u) = P−1F(ˆ x, u)CTQ

(17)

where F(ˆ x, u) and Q are respectively n and p

dimensional square matrices

Substituting equation (17) in equation (16) gives:

to be determined.

˙V(e) = eTPDˆ x( f )e − eTF(ˆ x, u)CTQCe < 0 (18)

A map F(ˆ x, u) which satisfies the inequality (18) is a

positive definite one defined for all t such that:

eTPDx( f )e < eTF(ˆ x, u)e

(19)

Secondly, assuming that such a map exists, then using

equation (18), one can write:

˙V(e) ≤ eTPDˆ x( f )e − eTF(ˆ x, u)Dˆ x(h)TQDˆ x(h)e (20)

According to inequality (19) we say that a sufficient

condition for satisfying the Lyapunov stability condition

(13) can be summarized as follows: find a (p, p) matrix Q

such that the map CTQC − I

[

All positive defined matrix F(x, u) that verifies the

inequality

] be positive semidefinite.

PDx( f ) < F(x, u)

fulfill the inequality (19). The matricial norm is those

induced by the Euclidean vector norm . . We then

propose for F(x, u) the following map:

F(x, u) = diag φi(x, u)

()

where diag defines a diagonal matrix in which the diagonal

elements are defined by:

φi(x, u) =1

2

αij(x, u) + αji(x, u)

j=1

n

∑

where αij are the elements of PDx( f ).

To summarize, the existence of P, verifying inequation

(13), and those of Q verifying (20) are the two conditions

needed to design the state observer (2) which is described

by:

ˆ˙x = f(ˆ x, u)+ P−1F(ˆ x, u)CTQ(y − Cˆ x)

The block diagram of the resulting nonlinear observer is

shown in figure 1 where the time invariant matrix

R(ˆ x, u) has to be determined using the preceding

algorithm (equation 16). This will be illustrated by an

example in the last section of the paper.

f

u(t)

y(t)

x(t)

x(t)

^

OBSERVER

PLANT

ε(t)

∫

h

fh

∫

R

Figure 1: nonlinear observer structure

This nonlinear observer method can be extended to

systems with nonlinear output (Adjallah, 1993).

2. DEDICATED NON LINEAR OBSERVER SCHEME (DNOS)

The basic idea of this approach is to reconstruct the state

and output of the process under consideration and then

analyze the output estimation error. It is worthwhile

recalling here the differential equation governing the

dynamics of the state estimation error:

˙ e = Dˆ x( f )− R(ˆ x, u)C

[]e

In the presence of process or sensor faults, equation (1)

may be modified as follows:

˙ x = f(x, u)+ δ, x0= x(0)

y = Cx +ζ

δ ∈Rn

ζ ∈Rp

(21)

where δ(t) and ζ(t) represent process and sensor faults

respectively. In this case the dynamic of the state

estimation error is given by:

˙ e = Dˆ x( f )− R(ˆ x, u)C

[]e − R(x, u)δ +ζ

(22)

Since the output estimation error ε(t) = Ce(t) is a

function of δ(t) and ζ(t), it can be used as a residual for

indicating that a fault has occurred. It is clear from

equation (22) that the output estimation error is affected

by the faults. The system represented by this equation is

asymptotically stable, since the stability conditions of the

observers are fulfilled. In the ensuing development, we

shall limit our attention only to sensor and actuator

faults. Generally, fault detection is achieved by comparing

the residuals (normalized by their variance) to a specified

threshold. To be more precise, the observer has to be

designed to facilitate faults isolation. A well-known

approach for sensor fault isolation based on dedicated

observers scheme (Clark, 1978) or generalized observer

scheme introduced by Frank (1987) to increase robustness

of such observer-based FDI scheme, may be extended here

to the nonlinear case. Each observer is driven by the input

vector u and the output of a set of dedicated sensors. The

complete output (or a part of it, for systems that are not

completely observable) is estimated and the corresponding

residuals are generated and analyzed (figure 2). In this way,

Fang (1993) proposed a new method for robust residuals

for failure detection and localization.

Page 4

Process

Observer 1

Observer p

u(t)

y(t)

Logic unit

for fault

detection

and

isolation

alarms

Figure 2: observer scheme for residual generation and fault

isolation

A specific number of faults can thus be detected and

isolated when a set of observers designed with different

outputs combination of the process is used (Ge and Fang

(1989)).

4. EXAMPLE

We consider here the nonlinear model of a synchronous

machine (Mukhopadhyay, 1972) governed by the

following differential equations, in which ζ is a vector of

sensors faults:

˙ x1

˙ x2

˙ x3

=

x2

−A1x2− A2x3sin(x1) − 0.5B2sin(2x1) + B1u1

u2− D1x3+ D2cos(x1)

x1

x2

x3

y1

y2

=

1

0

0

1

0

0

+

ζ1

ζ2

The f derivative with respect to x is:

Dx( f ) =

010

−A2x3cos(x1)

− B2cos(2x1)

−D2sin(x1)

−A1

0

−A2cos(x1)

−D1

A1 = 0.2703,

B2 = - 48.04,

A2 = 12.01,

D1 = 0.3222,

B1 = 39.19,

D2 = 1.9

Figure 3 shows the input signal u(t) and the two output

signals y1(t) and y2(t). In this example, a bias of 0.4

magnitude of output signals is simulated between 0.4 sec.

and 0.8 sec. on the first sensor and between 1.2 sec. and

1.6 sec. on the second.

210

0,8

1,0

1,2

1,4

1,6

1,8

u2

u1

Figure 3a: input signal u(t) (t: sec.)

u1 is the percent of variation of the mecanical input power

while u2 is the percent of variation of control field voltage.

210

-3

-2

-1

0

1

2

3

y1

y2

Figure 3b: outputs y1(t) and y2(t) (t: sec.)

Keeping to the proposed method, the first step is to find a

positive definite matrix such that for e ∈ Ker(C)

inequality (13) holds. Solving this inequality leads to

determine the matrix P. As suggested in step one, we find

a matrix K which column describes the null space of C:

{}, the

K = 001

()T

We then propose a diagonal structure and positive definite

matrix for P and choose:

P = diag(p1, p2, p3)

eTKTPDˆ x( f )Ke = −D1p3e2< 0 for any real and positve

value of p3.

We can then calculate F(x, u) = diag(φi), (i = 1, 2, 3):

φ1(x, u) =1

2

p1− p2A2x3cos(x1)+ B2cos(2x1)

()

+

p3D2sin(x1)

φ2(x, u) =1

2

p1− p2A2x3cos(x1)+ B2cos(2x1)

(

+

2p2A1

)

+

p2A2cos(x1)

φ3(x, u) =1

2

p3D2sin(x1) + p2A2cos(x1) + 2p2A1

[]

The next step is to find a matrix Q rending CTQC − I

positive semidefinite. We find:

[]

Q =

2

0

0

2

The observer is then described by the equations:

ˆ˙x = f(ˆ x, u)+ P−1F(ˆ x, u)CTQ(y − Cˆ x)

Here, our aim is to detect sensor faults. We then calculate

the output residuals r(t) defined as r(t) = y(t)−ˆ y(t).

Figures 4a and 4b show respectively the first and the

second residuals: after a transient due to arbitrary initial

conditions applied to the observer, the residuals are centred

at the origin in the absence of fault. The faults are

simultaneously accentuated in both residuals making

impossible to know which of the sensors is faulty.

Page 5

210

-1

0

1

Figure 4a (Nonlinear observer scheme results): residual

r01.

210

-1

0

1

Figure 4b (Nonlinear observer scheme results): residual

r02.

Now we will design two observers, each one fed by the

input u and an output yj (j = 1, 2). As the system is

observable in both cases, the outputs may be

reconstructed and the residuals generated. For the design

purpose, we consider the output y1 and the observer fed by

y1 = C1x,

C1= 1 0

(

0

)

in the case of DNOS1 and for DNOS2 the output y2:

y2 = C2x,

C2= 01 0

()

A fine reconstruction of the states would result in the fact

that the jth residual of the ith observer rij (i ≠ j) will be

sensitive to faults of all sensors while rii will be sensitive

to faults on the ith sensor only.

Nonlinear observer dedicated to y1

For e ∈ Ker(C1)

{}, a matrix K1 such that (K1C = 0) is:

K =

0

0

1

0

0

1

T

and a matrix P which verifies the inequality (13) can be:

P = P1 = P =

74.9 −0.1 17.6

−0.11.5

17.6 −0.1

−0.1

6.3

One can then calculate F(x, u) and find Q. The DNOS1

has the following form:

ˆ˙x = f(ˆ x, u)+ P−1F(ˆ x, u)CTQ(y1− C1ˆ x)

with Q = 0.5. Comparing figures 5a and 5b, on the time

interval from 1.2 to 1.6 seconds reveals that the observer

input is fault free. That means the system's state is

correctly estimated. It also means ˆ y1 and ˆ y2 are correctly

reconstructed, with only residual r12 remaining fault

sensitive on the second sensor. Comparison on the time

interval from 0.4 to 0.8 seconds shows that r11 and r12 are

simultaneously accentuated by faults on the observer

input y1. r11 can be used to isolate faults on the first

sensor.

210

-0,5

0,0

0,5

Figure 5a (Dedicated nonlinear observer scheme 1):

residual r11.

210

-1

0

1

2

3

Figure 5b (Dedicated nonlinear observer scheme 1):

residual r12.

Nonlinear observer dedicated to y2

In this case, when e ∈ Ker(C2)

{}, we have

K =

1

0

0

0

0

1

T

and we propose the matrices P = P1 and Q = 0.025. The

DNOS2 has the following form:

ˆ˙x = f(ˆ x, u)+ P−1F(ˆ x, u)CTQ(y2 − C2ˆ x)

The second output helps to reconstruct the system state

with dedicated nonlinear observer scheme. Results are

interpreted in analogous fashion as in the preceding case:

the first residual is sensitive to faults due to both sensors

while the second is sensitive only to fault due to the

second sensor. Figures 6a and 6b show respectively the

residual r21, and the residual r22, with faults simulated on

the first and the second sensors.

We conclude that the observer controlled by the input u

and the output of all the sensors able the detection of

sensor faults but not their localization. Localization of

faults necessitates the use of dedicated observers which

yield faults decoupled residuals with particular geometric

Page 6

fault direction.

210

-0,5

0,0

0,5

Figure 6a (Dedicated nonlinear observer scheme 2):

residual r21.

210

-0,5

0,0

0,5

Figure 6b (Dedicated nonlinear observer scheme 2):

residual r22.

CONCLUSION

In this paper, we have discussed the analytical redundancy

approach to FDI in nonlinear dynamic systems. An

observer design method with good fault detection

properties was presented. Simulation and experimental

results were used to illustrate the application of the

dedicated nonlinear observer scheme to the isolation of

sensor faults. Contrary to linearized systems, the resulting

nonlinear observer is a solution to one of the aspect of

robustness problems with respect to the nonlinear

systems operating point. It embraces a very large class of

nonlinear systems including bilinear systems.

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