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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,b∈IR. 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)) + (1−r(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)<δ (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)<δ,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

wa∈Ω

fsup

qe∈Sq

|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 Sq∈IR

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))2≤κϕ(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)) = wT ϕ(q(t)) + ε(t)(22)

R. Fónod and D. Gontkovic • Fault Estimation in a Class of First Order Nonlinear Systems

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where wT 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)=

=wT ϕ(q(t)) + ε(t)−wT

a(t)ϕ(qe(t)) =

=wT (ϕ(q(t))−ϕ(qe(t))) + ε(t)+

+(wT −wT

a(t))ϕ(qe(t))

(24)

where with notations

wT

Δϕ(t)=wT(ϕ(q(t))−ϕ(qe(t))) (25)

wT

e(t)=wT −wT

a(t)(26)

it yields

ν(t)=ε(t)+wT

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)) =

=(a−j)e(t)+ν(t)

(28)

˙e(t)=

=(a−j)e(t)+ε(t)+w∗T

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<μ(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)−(a−j)e(i)−ε(i)−w∗T

Δϕ(i)≥

≥μ(i)2e(i)w∗T

Δϕ(i)−

−2e(i)(a−j)e(i)−2e(i)˙e(i)−2e(i)ε(i)

(34)

and it is obvious that (21) implies

2e(i)w∗T

Δϕ(i)≤e(i)2+w∗TΔϕ(i)2≤

≤(1 + κϕw∗T)e2(i)(35)

Since for a smooth stable system and any i>0yields

˙e(i)

e(i)≤i(36)

then

−2e(i)˙e(i)≥−ie(i)2=−ie2(i)(37)

and

−2e(i)(a−j)e(i)=−2(a−j)e2(i)≥0(38)

Because it yields, too

−2e(i)ε(i)≥−e(i)2−ε(i)2=−e(i)2−δ(39)

then

2μ(i)wT

e(i)ϕ(qe(i))≥−μ(i)δ+

+μ(i)(1+ κϕw∗T−i−2(a−j)−1)e2(i)(40)

Subsequently, (33) implies

Δv(we(i)) ≤μ(i)δ−

−μ(i)−μ(i)ϕ(qe(i))2+

+κϕw∗T−2(a−j)−ie2(i)<0(41)

Setting

μ(i)= κϕw∗T−2(a−j)−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 w∗Tcan be selected as

κϕw∗T<1+i+2(a−j)(44)

which gives

0<κ

ϕw∗T−i−2(a−j)<1(45)

and it is evident that (42) implies 0<μ(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)=

=(a−j)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

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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>0∈IR, and taking the time derivative of v(e(t))

results in

˙v(e(t)) = pe(t)˙e(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=a−j(51)

Supposing Δf(t)=s(t)and setting pj =1(50) implies

p(a−j)=pa −1<0(52)

Thus, for any psatisfying (52) be j=p−1. To proceed

Δf(t)=s(t), a function

s(e(t)) = p−1

e(t)

e(t)e(t)pΔf(t)=

=(e(t),Δf(t),p)p−1

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>δ, because δ

corresponds to the modelling error with optimal synaptic

weight, while Δcorresponds to the modelling error when

e2(t)<δ. 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)+nωr(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

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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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