Adaptive Fault Detection and Isolation for DC Motor Input and

Sensors

Nikita Kolesnik

a

, Alexey Margun

b

, Artem Kremlev

c

and Andrei Zhivitskii

1d

Control Systems and Robotics Dept., ITMO University Saint Petersburg, Russia

Keywords: Fault Detection, Fault Isolation, DC Motor, Identification, Adaptive System.

Abstract: The paper is devoted to the development of an adaptive approach to the fault detection and isolation of input

and sensor failures of armature-controlled direct current motors. The proposed detection method is based on

the full state Luenberger observer. Isolation scheme uses the directional residual set and relationships between

fault directions and residual vector. Adaptability is provided by dynamic regressor extension and mixing

approach for online estimation of parameters. Proposed scheme allows to isolate following faults:

unaccounted load acting on the rotor, input voltage disturbance, failures of velocity and current sensors.

Simulation results confirm performance of the proposed approach.

1 INTRODUCTION

The development of technologies leads to use of

process automation systems in various fields of

human activity: industrial manufacturing,

autonomous cars and aircrafts, HVAC, etc. These

systems typically have a complex structure that

includes interconnected sensors, actuators and

passive elements. Failures of system parts may cause

sufficient consequences. Therefore, timely fault

detection and isolation is of particular importance,

especially for safety-critical systems. Such function

allows to increase reliability, perform predictive

maintenance, effective reconfiguration and quick

failure elimination. According to Wunnenberg (1990)

faults can be classified as follows: com- ponent fault

(deviation of a plant parameters from its nominal

values); sensor fault (sensor measurement doesn’t

corresponds to real physical value); actuator fault

(deviation of control signals from the desired values).

The most common methods of fault detection are

observer based approaches, parity relations,

parameter identification based algorithms and

machine learning approaches (Chen & Patton, 1999).

Parity relation methods rely on hardware or temporal

redundancy. Hardware duplication is effective and

a

https://orcid.org/0000-0002-8630-4202

b

https://orcid.org/0000-0002-5333-0594

c

https://orcid.org/0000-0002-7024-3126

d

https://orcid.org/0000-0002-0632-778X

does not require system model but demands

additional financial costs for adding sensor and

maintenance (Ray & Luck, 1991). Another drawback

of a sensor duplication is a confines imposed by

technological restrictions. Temporal redundancy

requires accurate plant model, but doesn’t need

additional hardware devices. Both approaches have a

good performance for sensor faults detection in linear

systems and are applicable for DC motors.

Observer based approaches use difference

between estimated and measured state variables

(residual) for fault detection. Isolation problem can be

solved with structured and directional residual sets or

fault detection filters (Patton & Chen, 1997). The

structured set method is based on synthesis of specific

residual generator sensitive for only one or all-but-

one corresponding fault. The main idea of directional

generators set is changing of residual signals in only

one direction that corresponds to the specific fault in

a residual space. Fault direction filters approaches use

special procedures of observer synthesis to make it

sensitive to the specific failures. Mentioned above

methods are effective for actuator and sensor fault

detection and isolation (Chung & Speyer, 1998).

Problem of robustness with respect to the

parametric uncertainties, disturbances, noises and

Kolesnik, N., Margun, A., Kremlev, A. and Zhivitskii, A.

Adaptive Fault Detection and Isolation for DC Motor Input and Sensors.

DOI: 10.5220/0011336700003271

In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 703-710

ISBN: 978-989-758-585-2; ISSN: 2184-2809

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

703

additive nonlinearities can be solved with the use of

unknown input observers (Chen, Patton & Zhang).

However, the synthesis procedure for these

algorithms has solution only for a class of linear

systems with sufficient limitations on plant matrices.

Model of DC motor with measured velocity or current

doesn’t satisfy the necessary conditions

(Wunnenberg, 1990).

Identification approaches use online parameters

estimation algorithms (for example, gradient descent

or least squares). These methods provide detection

and isolation of component faults on the base of

deviation between nominal and estimated parameters

(Isermann, 1997).

Last researches propose to use artificial

intelligence and neural networks to detect and isolate

faults. In (Santos et all, 2018) the fault detection and

classification schemes are proposed. The fault is

detected by a classical Luenberger observer. The

classification is based on a representation which

combines the subctrative clustering algorithm with an

adaptation of particle swarm clustering. DC motor

fault detection, isolation and identification based on a

neural networks approach is presented in (Adouni,

Abid & Sbita, 2016). However, this method requires

a lot of computational power and don’t guarantee

results. Experimental researches of fault detection

and diagnosis methods for DC motor drives are

analyzed in (Isermann, 2006).

This paper is devoted to the actuator and sensor

adaptive fault detection and isolation for armature

controlled direct current motors. Unknown input

observers don’t exist in the cases of sensor and

actuator faults occurring in mechanical and electrical

parts (equations for its synthesis have no solution

(Wunnenberg, 1990)). Proposed research describes

easy for computation and application method of fault

detection and isolation. The Motor is assumed to be

equipped with a velocity and current sensor. Fault

detection is based on full order state observer.

Isolation algorithm is provided by online parameters

identification with the use of dynamic regressor

extension and mixing. The proposed approach is an

adaptive extension of previous authors’ research

(Margun, Kremlev & Vlasov, 2021; Nguev, Vlasov,

Margun & Kirsanova, 2021;Margun,) where DREM

is used for components fault detection and isolation.

Simulation results confirm performance of the

proposed approach.

The paper is organized as follows. Section II

describes a mathematical model of the motor under

faults and problem statement. General detection and

isolation scheme, algorithms of observers calculation

and residual directions are shown in Section III.

Simulation results are shown in Section V.

2 PROBLEM STATEMENT

Consider a model of DC motor. Its dynamic is

described by equations

,

,

b

f

di

LRiuE

dt

JMM

ω

+=−

=−

(1)

where L is an inductance, R is an armature

resistance, i is a current, u is an input voltage, E

b

is a

back electromagnetic force, ω is a rotor angular

velocity, J is a rotor and load inertia, M is a motor

torque, M

f

is a friction momentum,

,

,

,

bb

m

ff

Ek

M

ki

Mk

φ

ω

φ

ω

=

=

=

(2)

where k

b

, k

m

and k

f

are constants,

φ

is a magnetic

flux assumed to be constant.

If motor is equipped with a velocity and current

sensor then the dynamic in state space representation

takes the form:

,

,

x

Ax Bu

yCx

=+

=

(3)

where

12

34

0

10

,,,

1

01

f

m

b

k

k

aa

JJ

xA BC

aa

i

k

R

L

LL

φ

ω

φ

−

== = ==

−−

Consider the model (3) under following faults:

external torque applied to the rotor (this failure can be

caused by wear-out of bearing or any unaccounted

load); input voltage disturbanc; velocity sensor fault;

current sensor fault. Torque and voltage are classified

as actuator faults because of they are directly acting

on state vector derivatives.

A motor model under the above faults is described

by equations

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

704

(4)

where f

a1

is a torque fault, f

a2

is a voltage fault, f

s1

is a velocity sensor fault, f

s2

is a current sensor fault

signals assumed to be unknown.

The goal of the research is to develop an adaptive

scheme for actuator and sensor faults detection and

isolation that remains operability under uncertain or

non-stationary parameters. First, consider the case of

known motor parameters. Next, an adaptive

modification is proposed for the case of parametric

uncertainties.

3 FAULT DETECTION AND

ISOLATION SCHEME

The basis of the proposed approach is the use of bank

of full order Luenberger state observers for fault

detection (Clark, 1979):

(5)

where is an estimate of state vector, =

an observer design matrix specific for i-th

fault.

Residual signal is chosen as the difference

between sensor measurements () (4) and observer

output () (5):

(6)

where () is a state estimation error.

The dynamic of considered faults residual is

described by equations:

(7)

where vector

defines fault direction in two-

dimensional residual space,

is an i-th fault signal.

It is necessary to develop a K synthesis algorithm

for each considered faults. The matrix should satisfy

following condition to provide isolability property

(Chen & Patton, 1999):

1)

;()

=1 to provide

unidirectional residual for faults in residual

space;

2) (A − KC) should be stable to provide stability

of the observer;

3) all vectors

should be linearly independent

for faults separability.

Additionally, mutual faults are separable if above

conditions are satisfied for all occurred failures.

Hence, it is necessary to design observer synthesis

algorithm for each of failures and develop an isolation

scheme on the base of residual signals (fig. 1). Let’s

analyze motor behavior under actuator and sensor

faults.

Figure 1: Fault detection and isolation scheme.

3.1 Torque Fault Detection

Some unaccounted force acting on the rotation of

mechanical parts causes torque fault. Error dynamic

takes the form

(8)

where

is an external force momentum acting

on rotor.

Consider condition 1:

(9)

It holds if we choose

=

:

(10)

1

2

1

2

1

0

,

1

0

10

,

01

a

a

s

s

f

J

xAxBu

f

L

f

yCx

f

=++

=+

ˆˆ ˆ

(),

ˆˆ

,

x

Ax Bu K y y

yCx

=++ −

=

ˆˆ ˆ

() ,r yyCxx CeyCx=−= − = =−

(),

,

ii

eAKCelf

rCe

=− +

=

[]

11

11

() ,

1

00,

aa

T

a

eAKCelf

ll

J

=− +

==

[]

111

11

331

1

()

;( ) 1

0( )

aa

akl

rank l A KC l rank

J

akl

−

−= =

−

111

1

()

1

00

akl

rank

J

−

=

Adaptive Fault Detection and Isolation for DC Motor Input and Sensors

705

Consider condition 2. Characteristic polynomial

of error model (8) takes the form:

(11)

where s is a complex variable,

(12)

Characteristic polynomial doesn’t depend on

,

because of

=

and

=

. So we

can define

=0. One can choose positive n, m to

provide desired observer behaviour and complete K

calculation by solution of equations (12) with respect

to the k

1

, k

4

with known k

2

, k

3

and n, m chosen by the

designer.

Condition 3 is satisfied because we have only one

fault direction vector.

3.2 Voltage Fault Detection

A voltage fault occurs due to some failure in

electronic circuits and disturbances of input voltage

(for example, the crash of the transistor in motor

driver or influence of powerful non-stationary

external magnetic field). Error dynamic takes the

form

(13)

where f

a2

is an additive voltage applied to the

motor input.

Consider condition 1:

(14)

It holds if we choose k

2

= a

2

:

(15)

Consider condition 2. Characteristic polynomial

of error model (13) is the same as in for force fault

case (12). It doesn’t depend on k

3

because all terms

with k

3

are rejected due to k

2

= a

2

. So we can define

k

3

= 0. In the same way as in previous subsection one

can choose positive n, m to provide desired observer

behaviour by pole placement procedure and complete

K calculation by solution of equations (12) with

respect to the k

1

, k

4

.

Condition 3 holds because residual directions l

a1

and l

a2

are orthogonal.

3.3 Velocity Sensor Fault Detection

This fault occurs due to mechanical or electronic

failure in velocity sensor or its data channels.

Multiply or stuck measurement value are the most

common types of the failures. Taking into account

(4), error dynamic takes the form

(16)

where f

s1

is velocity sensor fault signal.

Residual directions l

a1

and l

a2

are basis vectors in

two dimensional residual space. Therefore, it is

impossible to build linearly independent l

s1

with

respect to l

a1

and l

a2

in the same time. Let us choose

the following residual direction

=

=

11

. If the residual signal is along this direction,

then this fault will be more likely. Moreover, we can

provide mutual isolability with one of actuator faults.

Consider condition 1:

(17)

It is impossible to design such k

2

and k

4

that the

columns will be linearly dependent like in actuator

faults case because one of observer poles will be

equal to zero. Let rows will be linearly dependent to

satisfy the condition. Therefore:

(18)

One can find k

2

as a solution of (18):

(19)

The last coefficient k

4

is chosen to satisfy

condition 2 with use of characteristic polynomial

(12).

It is impossible to provide condition 3 for all

simultaneous faults, but proposed scheme allows to

isolate a velocity sensor fault with one of actuator

faults.

2

1

det( ( )) ,

a

p

sI A KC s ms n=−−=++

114 4

14 14 14 14 23 23 32 23

,

.

mk a a k

nkaaakkakkkakakaa

=−−+

=−+−−++−

[]

22

22

() ,

1

00,

aa

T

a

eAKCelf

ll

L

=− +

==

[]

222

22

442

0( )

;( ) 1

1

()

aa

akl

rank l A KC l rank

akl

L

−

−= =

−

442

00

1

1

()

rank

akl

L

=

−

11

1

1

3

() ,

1

,

0

s

s

s

eAKCelf

k

lKC

k

=− +

==

[]

1111223

11

3331443

()( )

;( ) 1

()()

ss

kakkakk

rank l A KC l rank

kakkakk

−+−

−= =

−+−

111 2 23

1

3331443

()( )

()()

akk a kk

k

kakkakk

−+−

=

−+−

22

113 2 3 31 413 13 4

2

2

3

akk ak ak akk kkk

k

k

+−− +

=

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

706

3.4 Current Sensor Fault Detection

Reasons for current sensor faults are the same as for

velocity sensor. Error dynamic is described by

equations

(20)

where f

s2

is a velocity sensor fault signal. Choose fault

direction as

=

=

21

. This direction

is isolable from one of previous faults (condition 3 is

partially satisfied). Consider condition 1:

(21)

Similarly to previous subsection:

(22)

Therefore, k

3

is a solution of (22):

(23)

The last coefficient k

1

is chosen to satisfy

condition 2 with use of the characteristic polynomial

(12).

3.5 Fault Isolation Scheme

Faults directions in residual space are illustrated in

figure 2.

Figure 2: Fault directions in residual space.

However, it is impossible to provide explicit

separation of all simultaneous faults because two of

its directions define basis of two dimensional residual

space. But we can propose a scheme that allows faults

detection and isolation with the use of directional

relationship similarly to (Chen et all, 1996).

Introduce directional relationship Z between

residual vector r and fault direction l

i

2

2

.

T

i

i

T

i

lr

С

lr

=

(24)

Coefficient Z

i

denotes normalized value of

residual projection on the i-th fault direction. If Z

i

>

Z

j

then fault i is more likely then j. The most likely

fault corresponds to max(

), i = {

,

,

,

}.

Robustness with respect to the noises can be

provided by use of threshold:

,

0,

,

,

r if r Threshold

r

if r Threshold

≥

=

<

(25)

Problem of insensitivity to parametric

uncertainties can be overcome with the use of

identification algorithms. However, observer matrix

becomes depending on estimates of plant parameters

to perform all necessary detection conditions.

It should be noted, condition 3 is not satisfied for

all possible mutual faults. Hence, proposed scheme

may lead to isolation errors in cases of multiple faults.

For example, two simultaneous actuator failures can

cause increasing of the residual vector in sensor fault

direction. However, such situation is unlikely in

practice and detection algorithm remains its

performance.

4 ADAPTIVE MODIFICATION

Combine proposed method with the method of

dynamic regressor extension and mixing (DREM)

(Aranovskiy, Bobtsov, Ortega & Pyrkin, 2016;

Aranovskiy, Belov, Ortega, Barabanov & Bobtsov,

2019) for online estimation of parameters to ensure

FDI operability under uncertainties and non-

stationarity.

4.1 Plant Parameterization

It is necessary to transform (3) to the autoregressive

model for use of DREM. Rewrite plant (3) in transfer

function representation. Transfer functions with

current and velocity outputs take the form:

12

2

34

() ,

i

s

Ws

ss

ωω

ωω

+

=

++

(26)

5

2

34

() ,Ws

ss

ω

ω

ωω

=

++

(27)

22

2

2

4

() ,

0

,

1

s

s

s

eAKCelf

k

lKC

k

=− +

==

[]

2112224

22

4332444

()( )

;( ) 1

()()

ss

kakkakk

rank l A KC l rank

kakkakk

−+−

−= =

−+−

2112224

4332444

()( )

()()

kakkakk

kakkakk

−+−

=

−+−

22

32 424 124 24 124

3

2

2

ak akk akk ak kkk

k

k

+−−+

=

Adaptive Fault Detection and Isolation for DC Motor Input and Sensors

707

where

1

1

,

L

ω

=

2

,

f

k

L

J

ω

=

3

,

f

Lk RJ

LJ

ω

+

=

4

,

fbm

Rk k k

LJ

φ

ω

+

=

5

.

m

k

k

L

J

φ

=

There are unmeasured derivatives of i(t) and ω(t)

that prevents transformation to the autoregressive

model.

Rewrite (26), (27) as differential equations:

2

341 2

2

341 2

() () () () (),

() () () () (),

s

it sit it sut ut

s

tsititsutut

ωωω ω

ωω ωω ω

=− − + +

=− − + +

(28)

Apply second order stable linear filter with

characteristic polynomial

()=

+2+1 to

the left and right parts of (28) according to Ioannou &

Sun (2012):

2

3

41 2

2

3

41 2

,

() () () () ()

,

() () () () ()

s

ss

iiiuu

sssss

s

s

s

iiuu

s

sss s

ω

ωω ω

ωω ω

ω

ω

=− − + +

ΛΛΛΛΛ

=− − + +

ΛΛΛΛΛ

(29)

Coefficients λ

0

, λ

1

do not affect convergence time,

but appropriate choice allows to filter measurement

noises.

Equations (29) can be represented in desired form

with measured signals:

,

,

T

fi i i

T

f

yq

yq

ωωω

η

η

=

=

(30)

where

2

10 0

22

1

2

,,

fi f

ss

yiy

ss ss

ω

ω

λλ λλ

==

++ ++

110

2

0

2

1

;;

T

i

s

qii

ss ss

λλ λλ

=− −

++ ++

01

22

10

1

;

s

uu

ss ss

λλ λλ

−=

++ ++

11 1 1

1234

;;; ,

ii i i

qqqq

=

[

]

[

]

1234 1234

;;; ;;; ,

T

i

ηω ηωω η ηω

η

==

(31)

10 1

22

0

1

(); ;()

T

s

qt

ss s

t

s

ω

ωω

λλ λλ

=− −

++ ++

10

11 1

123

2

1

() ; ; ,ut q q q

ss

ωω ω

λλ

=

++

[

]

[

]

534 534

;; ;; .

T

ω

ηηηωωω

η

==

and

η

i

,η

ω

are transfer function unknown parameters

vectors to be identified.

4.2 Identification Algorithm

Let us use DREM method for DC motor parameters

online identification (Aranovskiy et all, 2016). This

approach provides independent estimation of plant

parameters and convergence speed tuning.

According to Margun et all (2021) and Nguev et

all (2021), apply different stable linear filters to (30)

2

22

33 44

22 3

2

3

22

,

,,

,,

TT

fi

TT

fi

T

f

ii

fi

T

ff

ii

yyqq

ss

yq yq

yq yq

ωωωω

η

η

ηη

λ

η

λλ

λ

η

==

++

==

=

=

=

(32)

where λ

i

are unique positive constants. Algorithm

for λ

i

selection doesn’t exists. However, there are

following heuristics can be used: large differences

between filter’s parameters decrease convergence

time; too large or small λ

i

can sufficiently increase

computation complexity; it is good practice to choose

filters parameters that are ten times different.

Obtain an extended system which includes (30)

and (32) in matrix representation:

,,

iii

YQY Q

ωωω

η

η

==

(33)

where

=

;

;

;

,

=

;

;

Multiply both equations (33) from the left by

adjoint matrix of

Q

i

,Q

ω

respectively. Yields

{

}

{}

,

,

iii

diag

diag

ωωω

ξϕ

η

ξϕ

η

=

=

(34)

where

φ

i

and

φ

ω

are determinants of matrices

Q

i

and

Q

ω

respectively.

Multiplication of (33) by adjoint matrix provides

independent regressors for parameters estimation

(one separate regressor for each unknown plant

parameter). This allows to design an independent

scalar identification algorithm for each parameter

similar to the classical gradient descent approach with

simplified tuning and fast convergence:

()

2

,

ˆˆ

nn nnn т

ηξϕϕηγ

−=

(35)

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

708

where

γ

n

>0 is a design parameter that allow to tune

the convergence speed, ̂

is an estimate of the

corresponding parameter n. Separate parameters

identification and convergence speed tuning are main

advantages of DREM. One parameter change doesn’t

influence on others parameters estimates. This fact

provides the robustness of fault isolation in

comparison with gradient descent and least squares

approaches.

4.3 Adaptive Fault Detection and

Isolation

To ensure the adaptability of fault isolation scheme it

is necessary to combine it with a parameters

identification algorithm. It should be noted, that the

observer matrices depend on the identifier outputs.

This leads to the fact that during transient processes

the values of the observer matrices will have

significant errors in comparison with the desired

values. This may lead to false faults detections. We

need to update values of observers matrices only after

the end of identification algorithm transients to

overcome this drawback. It can be performed with the

use of sliding window:

(

̅

(

)

̂

(

)

)

∀,

(36)

where ̅ =

(

)

/ is a mean value on

period

(

;

)

,

is a threshold value.

Moreover, detection scheme should be insensitive

to the noises, small disturbances and deviations. This

problem can be solved with a residual deadzone

condition:

∆

=0

(37)

5 SIMULATION RESULTS

Consider motor with following plant

Observers matrices

K

a1

,K

a2

,K

s1

,K

s2

are updated

after parameters estimation transient time. Their

initial values can be calculated for nominal plant.

Characteristic polynomial for plants (26), (27)

parameterization is ()=

+2+1.

Parameters of DREM filters are chosen as follows:

for current output transfer function

=0.1,

=1,

= 10; for velocity output transfer function

=

0.1,

=1. Threshold values are

= 0.05,

=0.1, all values equal to one, =

10

(

0.1

)

+sin

(

)

to satisfy the persistent

excitation condition.

Identification algorithm results are shown figure

3. Parameters estimate converges to the true value of

six seconds. Observers matrices tend to the following

values:

Figure 4 illustrates fault isolation algorithm

output during external momentum acting on rotor

shaft from 10 to 15 seconds. Fault signal is a constant

that increases velocity. The largest value of

directional relationships

C

a1

corresponds to the

failure.

Figure 3: Plant parameters identification.

Figure 4: Directional relationships with force momentum

fault from 10 to 15 seconds.

The case of voltage fault is illustrated in figure 5.

The fault signal is an additional voltage applied to the

motor input. The directional relationship allows to

isolate this fault.

The case of sensor fault is illustrated in figure 6.

The fault is a current sensor zero shift that may be

caused by corruption of information bites. Proposed

scheme allows to isolate this sensor fault.

Figure 5: Directional relationships with voltage fault from

10 to 15 seconds.

Adaptive Fault Detection and Isolation for DC Motor Input and Sensors

709

Figure 6: Directional relationships with current sensor fault

since 10 seconds.

6 CONCLUSION

Actuator and sensor adaptive fault detection and

isolation scheme for direct current motor is proposed.

The motor is assumed to be equipped with velocity

and current sensors. Detection algorithm is observer

based. Isolation scheme uses directional relationship

between residual and fault directions.

Adaptability is provided by DREM approach.

Proposed solution allows to isolate torque fault, input

voltage fault, velocity sensor fault and current sensor

fault. Simplicity of observers and residual generators

synthesis and its trivial computation are advantages

of the scheme.

Robustness with respect to the noise is obtained

by use of threshold. Insensitivity to uncertainties is

provided by the DREM approach and switching

techniques for the tracking of estimation end.

Simulation results confirm the effectiveness of the

proposed approach.

ACKNOWLEDGEMENTS

The work is supported by the Russian Science

Foundation grant (project 19-19-00403).

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