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Fault estimation in a class of first order nonlinear systems

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Reformulated principle of fault estimation design for one class of first order continuous-time nonlinear system is treated in this paper, where a neural network is regarded as model-free fault approximator. The problem addressed is presented as approach based on sliding mode methodology with combination of radial basis function neural network to design robust nonlinear fault estimation. The method utilizes Lyapunov function and the steepest descent rule to guarantee the convergence of the estimation error asymptotically. Simulation results show the feasibility of the proposed approach.
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Fault Estimation in a Class of First Order
Nonlinear Systems
R. Fónod and D. Gontkoviˇ
c
Technical University of Košice, Faculty of Electrical Engineering and Informatics
Department of Cybernetics and Artificial Intelligence
Košice, Slovakia
e-mail: robert.fonod@student.tuke.sk, daniel.gontkovic@tuke.sk
Abstract—Reformulated principle of fault estimation de-
sign for one class of first order continuous-time nonlinear
system is treated in this paper, where a neural network is
regarded as model-free fault approximator. The problem
addressed is presented as approach based on sliding mode
methodology with combination of radial basis function
neural network to design robust nonlinear fault estimation.
The method utilizes Lyapunov function and the steepest
descent rule to guarantee the convergence of the estimation
error asymptotically. Simulation results show the feasibility
of the proposed approach.
Index Terms — Sliding mode state estimator, radial ba-
sis function neural network, Lyapunov function, steepest
descent rule, linear matrix inequalities.
I. INTRODUCTION
The theory of sliding mode has emerged as a method
capable of use in given robust control systems, as well
as state estimation [15], [19]. The main advantage of
sliding mode in state estimation exists in the sliding
where the state error trajectories are constrained on
the predetermined equivalent system. The sliding mode
observer design can be then divided into two phases.
First, in the reaching phase, give an error switching law
such that the estimated error trajectory is trapped on a
switching surface and remain on it thereafter. And second,
in the sliding phase, determine the switching surface such
that the error dynamics in the sliding mode have good
performance.
It should be pointed out that the robustness of a
variable structure resides in its sliding phase, but not
in its reaching phase. There is no easy way to shape
the error dynamics of the reaching phase. In fact, the
error dynamic is not completely robust over all time. In
addition, if the system suffers from uncertainties or faults,
then the error behavior in the sliding mode is not only
governed by the switching surface but also determined
by faults. In this case, the state estimator stability may
not be assured [4], [11]. However, in many cases sliding
mode control is supposed [16], [17], [18], and different
observer structures used to estimate sensor, actuator, and
system faults [3], [7], [10].
Recently, neural networks has been successfully used
in model-free fault estimation in nonlinear systems. The
known supports of these methods give their ability to
good performance for approximation of nonlinear func-
tions. Combining sliding mode observer technology with
neural networks to design fault estimation structures offer
new opportunity in design [6].
Radial basis function (RBF) neural networks seems to
be one of the neural networks with high approximation
and regularization capability [2], where the essential
phenomenological rationale for the use of RBF rest in the
realm of the purpose of feed-forward networks and fea-
ture extraction possibility. Gaussian RBF are employed
most frequently, since it is bounded, strictly positive and
continuous on IR, and the optimization of RBF networks
provides better approximation and interpolation capabil-
ity as compared to the sigmoid functions. However, the
performance of RBF neural networks depends on the
number of neurons in hidden units, and on methods used
for determining the output neuron weights, where training
of a RBF neural network is, in general, a challenging
nonlinear optimization problem.
The paper is focused on robust system fault estimation
and isolation (FDI) approach for one class of nonlinear
single input/output (SISO) systems. The FDI scheme is
based on sliding mode neural observer [9], which is
robust against system uncertainty, and fault estimation be
realized using the sliding boundary size. When a system
fault is occurred the estimate part in the observer for
faults is enabled. When a RBF neural network is used
to approximate a system fault the fault estimation can be
formulated in a simpler way.
The main contribution of the paper is to present a
reformulated design method for continuous-time non-
linear SISO systems to solve system fault estimation
problem using a sliding mode observer. Used structure
is motivated by the need for robustness in model-free
fault estimation. Under defined conditions, the stability of
the observer is assured, where the state observer error is
bounded. In comparison with methods based on analytical
models (see e.g. [5], [8], [13], [14]) design conditions for
specified system class are here derived using radial basis
function neural network.
II. PROBLEM DESCRIPTION
The system under consideration is a first order SISO
nonlinear dynamic systems, which is given as
˙q(t)=aq(t)+bu(t)+f(q(t)) (1)
y(t)=q(t)(2)
SAMI 2011 • 9th IEEE International Symposium on Applied Machine Intelligence and Informatics • January 27-29, 2011 • Smolenice, Slovakia
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where q(t)IR,u(t)IR,andy(t)IR are the state,
input and output variables, respectively, and scalars a
IR,bIR. A system fault is modelled by an unknown
additive function f(q(t)) IR.
The problem of interest is to design sliding-mode
observer based estimation of system faults.
III. OBSERVER STRUCTURE
Since the calculation of f(q(t)) may be not accurate
enough, in order to rectify this kind of errors as well as
to reduce the number of sensors a sliding mode observer
can be used to estimate f(q(t)), i.e. the sliding mode
observer of system (1), (2) can be designed e.g. as
˙qe(t)=aqe(t)+bu(t)+j(y(t)ye(t)) + s(t)(3)
ye(t)=qe(t)(4)
where qe(t)IR is the state observer variable, ye(t)IR
is the estimated output, and jis an observer gain. The
term s(t)IR, which represents the unknown f(q(t)),
is defined as a switching function of the tracking error of
the observer, i.e.
s(t)=sign(e(t)) (5)
where e(t)is the observer estimation error defined as
e(t)=q(t)qe(t)(6)
and is a constant factor. The constant has to be chosen
in such way that sliding mode is enforced in the manifold
νe=q(t)qe(t)=0. Once sliding mode is enforced, the
differential equations (1) and (3) have the same solution
and the terms f(q(t)) and s(t)have to be equivalent.
During application a low pass filter has to be used to gain
the average value s(t)of the discontinuous time function
(5).
Model-free sliding mode observer does not require
the model of f(q(t)), but it cannot arrive finite time
convergence, the sliding mode gain should be bigger than
an upper bound, and incomplete information about the
rest nominal parameters causes chattering phenomenon. It
seems to be natural to construct its estimate f(q(t)|w(t))
depending on a parameter w(t), which can be adjusted
online by means of an updating law. In the next, a neural
network is used to approximate f(q(t)) and construct a
neural observer of the form
˙qe(t)=aqe(t)+bu(t)+
+j(y(t)ye(t)) + fa(qe(t)|w
a(t)) + (1r(t)) s(t)(7)
ye(t)=qe(t)(8)
fa(qe(t)|w
a(t)) = wT
a(t)ϕ(qe(t)) (9)
Here s(t)IR is an external feed-forward compensation
signal and r(t)is a switch function. The switch function
switch between the neural estimator and the sliding mode
observer in the dependence on the output error (6), i.e.
r(t)=1if e2(t)δ
0if e2(t)(10)
where δis known upper bound of the neural modelling
error.
If r(t)=1, the observer is pure neural observer, and
(7), (8) becomes
˙qe(t)=aqe(t)+bu(t)+
+j(y(t)ye(t)) + fa(qe(t)|w
a(t)) (11)
ye(t)=qe(t)(12)
If after time t0e2(t),i.e.r(t)=0then wT
a(t)is a
constant vector wT
a=wT
a(t0), and the observer (7), (8)
becomes pure sliding mode observer
˙qe(t)=aqe(t)+bu(t)+
+j(y(t)ye(t)) + fa(qe(t)|w
a)+s(t)(13)
ye(t)=qe(t)(14)
IV. RBF NEURAL NETWORK
The model-free fault estimation employed in the next
use a radial basis function neural network (RBFNN) to
approximate an unknown fault combining with the sliding
mode observer.
Supposing that there are preceptive field units in
the neural network hidden layer than the output of the
RBFNN is
fa(qe(t)) =
p
h=1
wh(t)ϕh(qe(t)) = wT
a(t)ϕ(qe(t))
(15)
ϕh(qe(t)) = exp
(qe(t)ch)2
σ2
h
(16)
wT
a(t)=w1(t)w2(t)··· wp(t)(17)
ϕT(qe(t))=ϕ1(qe(t)) ϕ2(qe(t)) ··· ϕp(qe(t))
(18)
where ch,σhare the center and width (spread factor)
of the neural cell of the h-th neuron in the hidden layer,
respectively, and w
h(t)IR is the weight connecting the
h-th hidden layer neuron and the network output.
Thus, the optimal weight values of RBF neural network
can be considered as follows
w=arg min
waΩ
fsup
qeSq
|f(qe(t)|wa)f(qe(t))|(19)
where
Ωf={wa:wa≤δw}(20)
is a valid field of wa(t),δwis a parameter, and SqIR
is a variable space of the state observer variable.
Because radial basis function ϕ(qe(t)) satisfies Lips-
chitz condition
ϕ(q(t)) ϕ(qe(t))2κϕ(q(t)qe(t))2(21)
where κϕcan be selected by users, then according to the
Stone-Weierstrass theorem, the smooth function f(q(t))
can be written as
f(q(t)) = wT ϕ(q(t)) + ε(t)(22)
R. Fónod and D. Gontkovic • Fault Estimation in a Class of First Order Nonlinear Systems
- 318 -
where wT is a fixed weight matrix of the neural network,
and ε(t)is is the smallest approximation (modelling)
error satisfying condition
ε2(t)δ(23)
Then, according to (15) is
f(q(t)) fa(qe(t)|wa(t)) = ν(t)=
=wT ϕ(q(t)) + ε(t)wT
a(t)ϕ(qe(t)) =
=wT (ϕ(q(t))ϕ(qe(t))) + ε(t)+
+(wT wT
a(t))ϕ(qe(t))
(24)
where with notations
wT
Δϕ(t)=wT(ϕ(q(t))ϕ(qe(t))) (25)
wT
e(t)=wT wT
a(t)(26)
it yields
ν(t)=ε(t)+wT
Δϕ(t)+wT
e(t)ϕ(qe(t)) (27)
V. SLIDING MODE NEURAL OBSERVER STABILITY
The neural sliding mode observer design method given
by [9] is represented as follows. Assuming that the sliding
mode neural observer switches between two models then
if r(t)=1then (1), (11), (6), and (27) implies
˙e(t)=aq(t)+bu(t)aqe(t)bu(t)
j(y(t)ye(t)) + f(q(t))fa(qe(t)|w
a(t)) =
=(aj)e(t)+ν(t)
(28)
˙e(t)=
=(aj)e(t)+ε(t)+wT
Δϕ(t)+wT
e(t)ϕ(qe(t)) (29)
respectively, where
Δϕ(t)=ϕ(qe(t)+e(t)) ϕ(qe(t)) (30)
Since neural networks have to be discrete updated and
e(i)be used in the updating law, then
we(i+1)=we(i)r(i)μ(i)ϕ(qe(i))e(i)=
=we(i)μ(i)ϕ(qe(i))e(i)(31)
where 0(i)<1.
Defining Lyapunov function candidate as
v(we(i)) = we(i)2=trace we(i)wT
e(i)(32)
then
Δv(we(i)) = we(i+1)2−we(i)2=
=we(i)μ(i)ϕ(qe(i))e(i)2−we(i)2=
=μ2(i)ϕ(qe(i))2e(i)2
2μ(i)we(i)ϕ(qe(i))e(i)
(33)
respectively. Inserting (29) it yields [12]
2μ(i)we(i)ϕ(qe(i))≥
2μ(i)wT
e(i)ϕ(qe(i))=
=2μ(i)˙e(i)(aj)e(i)ε(i)wT
Δϕ(i)≥
μ(i)2e(i)wT
Δϕ(i)−
2e(i)(aj)e(i)2e(ie(i)−2e(i)ε(i)
(34)
and it is obvious that (21) implies
2e(i)wT
Δϕ(i)≤e(i)2+wTΔϕ(i)2
(1 + κϕwT)e2(i)(35)
Since for a smooth stable system and any i>0yields
˙e(i)
e(i)i(36)
then
2e(ie(i)≥−ie(i)2=ie2(i)(37)
and
2e(i)(aj)e(i)=2(aj)e2(i)0(38)
Because it yields, too
2e(i)ε(i)≥−e(i)2−ε(i)2=−e(i)2δ(39)
then
2μ(i)wT
e(i)ϕ(qe(i))≥−μ(i)δ+
+μ(i)(1+ κϕwT−i2(aj)1)e2(i)(40)
Subsequently, (33) implies
Δv(we(i)) μ(i)δ
μ(i)μ(i)ϕ(qe(i))2+
+κϕwT−2(aj)ie2(i)<0(41)
Setting
μ(i)= κϕwT−2(aj)i
1+ϕ(qe(i))2(42)
where μi>0is the learning rate, (33) takes the form
Δv(we(i)) ≤−μ(i)e2(i)+μ(i)δ<0(43)
Since in this case e2(i)δ,thenΔv(we(i)) <0,
v(we(i)) is bounded, and also we(i)is bounded.
Here the initial condition wTcan be selected as
κϕwT<1+i+2(aj)(44)
which gives
0
ϕwT−i2(aj)<1(45)
and it is evident that (42) implies 0(i)<1.
If at time t=t0be r(t)=0the weights become
constants, the observer (7), (8) becomes pure sliding
mode taking form (13), (14). The error dynamics obtained
from (1), (2), (13), and (14) is
˙e(t)=aq(t)+bu(t)aqe(t)bu(t)s(t)
j(y(t)ye(t)) + f(q(t))fa(qe(t)|w
a)=
=(aj)e(t)+Δf(t)s(t)
(46)
ey(t)=e(t)(47)
where Δf(t)is neural modelling error when the weight
of the neural networks is fixed as w
a.
Defining the Lyapunov function of the form
v(e(t)) = 1
2pe2(t)(48)
SAMI 2011 • 9th IEEE International Symposium on Applied Machine Intelligence and Informatics • January 27-29, 2011 • Smolenice, Slovakia
- 319 -
0 5 10 15
100
0
100
200
300
400
500
600
time [s]
velocity (RPM)
Reference
Response
Fig. 1. Responses of the controlled system
with p>0IR, and taking the time derivative of v(e(t))
results in
˙v(e(t)) = pe(te(t)=e(t)p(aq(t)+bu(t)+Δf(t)
s(t)aqe(t)bu(t)j(q(t)qe(t)) (49)
˙v(e(t)) = e(t)p(aee(t)s(t)+Δf(t)) <0(50)
where
ae=aj(51)
Supposing Δf(t)=s(t)and setting pj =1(50) implies
p(aj)=pa 1<0(52)
Thus, for any psatisfying (52) be j=p1. To proceed
Δf(t)=s(t), a function
s(e(t)) = p1
e(t)
e(t)e(t)pΔf(t)=
=(e(t),Δf(t),p)p1
e(t)
e(t)
(53)
be introduced conditioned by e(t)=0,ands(e(t)) = 0
if e(t)=0. This function characterizes the sliding mode
in state estimation, and it is evident that with a positive
psatisfying (52) the observer error is bounded.
The sliding mode neural observer requires two design
parameters: switch constant δand the upper bound Δ
of neural modelling error Δf(t)when start the sliding
mode compensation. Here δdecide when neural network
learning is stopped and sliding mode observer is started.
The bigger δis, the shorter training time the neural
observer has and since the neural modelling error is
bigger, so Δshould be bigger. Usually Δ, because δ
corresponds to the modelling error with optimal synaptic
weight, while Δcorresponds to the modelling error when
e2(t). Note, this parameters are user-defined.
VI. ILLUSTRATIVE EXAMPLE
Considering the synchronous reluctance motor drive
system [4] described by the equation
˙q(t)=aq(t)+bu(t)+f(q(t))
where q(t)=ω(t)is velocity, the input control u(t)is
the electromagnetic torque, J=0.00076 kgm2is the
0 5 10 15
100
0
100
200
300
400
500
600
time [s]
velocity (RPM)
Real output
Estimated output
Fig. 2. The corresponding observer output response
inertia moment of rotor, m=0.00012 Nms/rad is the
viscous friction coefficient, respectively, and
a=m
J,b=1
J
A simple feedback controller has been introduced to
stabilize the nominal system for demonstration purposes
with the state control law of the form [12]
u(t)=kq(t)+r(t)+ 1
TIt
0
(ωr(τ)ω(τ))dτ
where k=0.0075,n=0.0076,TI=1.5sec.
In this simulation was tested different numbers of
hidden nodes with the nest result that after the hidden
nodes number is more than 5 the estimation accuracy be
not improved a lot.
There is consider no parameter variations except sys-
tem fault at t=5sec. The simulation results depicted
in Figs. 1 and 2 represent the system velocity response
and its estimation, and show that the proposed control
approach is effective. Figs. 3 and 4 provide the fault
estimation and synaptic weight adaptation obtained using
the sliding mode neural observer.
VII. CONCLUDING REMARKS
This paper explores use of a sliding mode scheme
for fault estimation in nonlinear system of the first
order. Modified design method is presented where system
fault estimation is achieved by the RBF neural network
inclusion into the sliding mode observer. The stability
analysis is given using the Lyapunov method and LMI
observer condition to garanty observer error convergence,
where LMI procedure give desirable observer dynamics.
It is demonstrated that the sliding mode observer can
be used to detection and estimate system faults. The
ability to estimate the faults directly is very desirable for
fault detection, and preferred to handle problem of fault
occurrence and faults behavior.
ACKNOWLEDGEMENTS
The work presented in this paper was supported by
VEGA, Grant Agency of Ministry of Education and
Academy of Science of Slovak Republic under Grant No.
1/0256/11. This support is very gratefully acknowledged.
R. Fónod and D. Gontkovic • Fault Estimation in a Class of First Order Nonlinear Systems
- 320 -
0 5 10 15
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
time [s]
fault estimation
Real fault
Estimated fault
Fig. 3. System fault and its estimation
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SAMI 2011 • 9th IEEE International Symposium on Applied Machine Intelligence and Informatics • January 27-29, 2011 • Smolenice, Slovakia
- 321 -
... The main contribution of the paper is to adapt the work presented in [8], [9] to actuator faults, acting on the system in additive form. Moreover, the idea is extended to a class of continuous-time linear Multi-Input Multi-Output (MIMO) systems, using a bank of observers. ...
... If rank(B) = r, then for each actuator fault f i a (q(t)), it is necessary to construct an own observer of the structure (7), (8), i.e. if all actuator faults are monitored, then a bank of r observers (see Fig. 1) have to be designed. In the next, for the sake of simplicity, the index i will be omitted, and the following equivalences are hold ...
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Variable structure systems consist of a set of continuous subsystems together with suitable switching logic. Advantageous properties result from changing structures according to this switching logic. Design and analysis for this class of systems are surveyed.
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In this paper, a sliding mode observer scheme of sensor fault diagnosis is proposed for a class of time delay nonlinear systems with input uncertainty based on neural network. The sensor fault and the system input uncertainty are assumed to be unknown but bounded. The radial basis function (RBF) neural network is used to approximate the sensor fault. Based on the output of the RBF neural network, the sliding mode observer is presented. Using the Lyapunov method, a criterion for stability is given in terms of matrix inequality. Finally, an example is given for illustrating the availability of the fault diagnosis based on the proposed sliding mode observer.