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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