LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Actuator Fault Detection in Nonlinear Systems
Using Neural Networks
Rastko Selmic, Ph.D.
Department of Electrical Engineering and Institute for
Micromanufacturing
Louisiana Tech University
Ruston, LA 71272, USA
Email: rselmic@latech.edu
Web: http://www2.latech.edu/~rselmic/
1
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Contents
Introduction
Problem Formulation
Actuator Fault Detection, Fault Dynamics, and Fault
Detectability
Two cases considered:
- State feedback
- Output feedback
Simulation Results
Conclusion
Other projects, ideas, etc.
2
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Introduction
Collaborative work with Marios Polycarpou and Thomas
Parisini
An actuator fault identification in unknown, input-affine,
nonlinear systems using neural networks is presented
Two cases are considered: state feedback and output
feedback case
Neural net tuning algorithms and identifier have been
developed using the Lyapunov approach
A rigorous detectability condition is given for actuator faults
relating the actuator desired input signal and neural netbased observer sensitivity
Simulation results are presented to illustrate the
detectability criteria and fault detection in nonlinear
systems.
3
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Questions to be Answered
 What kind of actuator faults can be detected?
 Under what conditions faults are detectable using NN
identifiers?
 If faults are not presently detectable, how identifier
parameters need to be adjusted in order to detect the
faults?
4
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Problem Formulation
 Consider a nonlinear, input-affine system given by
x  f ( x)  g ( x)u  d (t )
y  h( x )
where x  [ x1 , x2 ,, xn ]T is a system state and f, g:  n   n , h:  n  
are unknown, smooth functions, u is the input control signal, y is the
output, and d represents a system disturbance.
 It is assumed that the disturbance is bounded such that
d (t )  DB
 It is assumed first that the full state measurement is available.
5
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Problem Formulation
 The actuator of the nonlinear system generates the control signal u
based on desired actuator signal v.
 We consider the Gang Tao’s actuator fault model given by
u (t )  v(t )   (u  v(t )) ,
where the fault value of the actuator input is u .
  0 models the actuator without a fault.
  1 models the actuator with a fault.
 The actuator fault value can be considered as a time-varying function
u (t )  v(t )   (u (t )  v(t ))
 Combined, the nonlinear system and the actuator fault model is
x  f ( x)  g ( x)v(t )   (u (t )  v(t ))  d (t )
6
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Case I: State Feedback
 A NN system observer is given by
xˆ   Axˆ  Ax  Wˆ fT  f ( x)  Wˆ gT  g ( x)v(t ) ,
where the matrix A is Hurwitz and two NNs are used to approximate
nonlinear functions f(x) and g(x)
f ( x)  W fT  f ( x)   f ( x) ,
g ( x)  WgT  g ( x)   g ( x) .
W f , Wg are some ideal target NN weights, and  f (x ) ,  g (x) are NN
approximation errors bounded by  fM and  gM .
 An observer error given by
e  x  xˆ
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LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
A NN System Observer
u
v
Nonlinear System with Actuator Failure
Model
 1
u x  f ( x)  g ( x)u  d (t )
 0
x
e

NN OBSERVER
x̂
Figure 1. NN system observer – fault identifier.
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LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
NN Tuning Law
 A nonlinear system and observer dynamics are given by
~
~
e   Ae  W fT  f ( x)   f ( x)  WgT  g ( x)v(t )   g ( x)v  d (t )
where W~f  W f  Wˆ f , W~g  Wg  Wˆ g .
Theorem: Stable NN Observer Tuning Law (Lewis’ NN Tuning)
Given the nonlinear system and the NN observer, let the estimated NN
weights be provided by the NN tuning algorithm
Wˆ f  C f  f ( x)eT  kC f e Wˆ f
Wˆ  C  ( x)v(t )eT  kC e Wˆ
g
g
g
g
g
with any constant matrices C f  C f  0 , C g  C g  0 , and a small design
parameter k. Then the state observer error e and the NN weight estimation
~ ~
errors W f , Wg are uniformly ultimately bounded. The convergence region
for the state observer error e can be reduced by increasing the gain A.
T
T
9
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Stability Analysis
 Stability proof is carried out using the Lyapunov function
1
1 ~
1 ~
1 ~
1 ~
L  eT e  tr W fT C f W f  tr WgT C g Wg
2
2
2




 The bound on the tracking error is given by
1 2 1 2
W fM  WgM   fM   gMVB  DB
4
e 4
B
min
where min is a minimum eigenvalue of A, and VB is a bound on a control
signal. We assume that the control signal is bounded (the system has
been stabilized).
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LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
The State Observer Error
 Modified closed-loop error dynamics due to actuator fault is given by
~
~
e1   Ae1  W fT1 f ( x )   f ( x )  WgT1 g ( x )v(t )
  g ( x )v(t )  d (t )  g ( x )(u (t )  v(t ))
where x denotes the state x when there is a fault in the actuator.
 It is important to consider the difference in behavior of the state
observer error when the actuator is healthy and when there is an
actuator fault.
 This difference has its own dynamics - fault dynamics.
11
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Dynamics of a Fault
 Fault, as a process, has its own dynamics. The fault dynamics is
given by
~
~
~
e~   Ae~  W fT  f ( x)  W fT1 f ( x )   f ( x)   f ( x )  WgT  g ( x)v(t )
~
 WgT1 g ( x )v(t )   g ( x)v(t )   g ( x )v(t )  g ( x )(u (t )  v(t ))
where e~  e  e1
 Equivalently, the dynamics is given by
~ ~
e~  h(e~,W fT ,W fT1 , x, x , t )  g ( x )(u (t )  v(t )) ,
where the function h( ) is given by
~ ~
~
~
h(e~,W fT ,W fT1 , x, x , t )   Ae~  W fT  f ( x)  W fT1 f ( x )   f ( x)
~
~
  f ( x )  WgT  g ( x)v(t )  WgT1 g ( x )v(t )   g ( x)v(t )   g ( x )v(t )
12
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Detectability of Actuator Faults
Theorem: Detectability of Actuator Faults
Given a NN observer from Theorem 1, a fault in the system actuator can be
detected after time interval T if the following condition is satisfied
t 0 T
 g ( x )(u ( )  v( ))d  2eB  T C1
t0
~
~
where C1  eB max  2W fB  2WgB  2 M VB  1.
 The above condition provides rigorous justification for an intuitive
concept of fault detectability which says that for the fault to be detected
there should be “enough” difference between the desired input signal
and failed actuator values.
 The result also relates the time before fault has been detected and the
NN estimator parameters.
13
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Case II: Output Feedback
Consider a single-input single-output nonlinear system in a normal form
given by
x  Ax  b f ( x)  g ( x)u  d (t )
y  hT x
where x  [ x1 , x2 ,, xn ]T denotes the state vector, f, g: n   are
unknown smooth functions, u is the input control signal, y is the output, and
d represents a system disturbance. We assume that the disturbance is
bounded, that is d (t )  DB .
0
0

A
0

0
1 0  0
1 
0 
0 1  0

 

 , h  .
0 
0 0  1
 
0
0 0  0
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LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
A NN Observer
 A NN system observer is given by


xˆ  Axˆ  b Wˆ fT  f ( xˆ )  Wˆ gT  g ( xˆ )v(t )  r0 (t )  L( y  hT xˆ )
yˆ  hT xˆ
where x̂ is an estimate of the state vector x and ŷ is an estimate of the
output y. The observer gain matrix L is chosen so that As  A  Lh T is
stable.
 Two NNs are used to approximate the nonlinear functions f(x) and g(x):
f ( x)  W fT  f ( x)   f ( x)
g ( x)  WgT  g ( x)   g ( x)
where W f , Wg are some ideal target NN weights, and  f (x) ,  g (x) are
NN approximation errors.
15
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
A NN Observer
Nonlinear System with Actuator Failure
u
v
 1
 0
u
NONLINEAR
SYSTEM
y
~
y

ŷ
NN OBSERVER
Figure 2. NN system observer – fault identifier.
16
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
NN Observer Tuning Law
Theorem: Stable NN Observer Tuning Law (Lewis’ NN Tuning). Given
the nonlinear system and the NN observer, choose the robustifying term as
~
y
ro  kr ~
y
with a scalar k r . Let the estimated NN weights be provided by the NN tuning
algorithm
Wˆ f  C f  f ( xˆ ) ~
y  kC f ~
y Wˆ f
Wˆ  C  ( xˆ )v(t ) ~
y  kC ~
y Wˆ
g
g
g
g
g
with any constant, symmetric matrices C f  C f T  0 , Cg  Cg T  0 , and a
x and the
suitably small design parameter k. Then the state observer error ~
~ ~
NN weight estimation errors W f , Wg are uniformly ultimately bounded.
Proof: Select the Lyapunov function as
1 T ~ 1 ~ T 1 ~
1 ~
~
L ~
x Px  tr W f C f W f  tr WgT C g 1Wg
2
2
2




17
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Dynamics of a Fault
 The fault dynamics is given by
~
~
ex  As ex  b{W fT  f ( xˆ )  WgT  g ( xˆ )v(t )  c1 (t )  r0 (t )}
~T
~T
 b{W fF
 fF ( xˆ F )  WgF
 gF ( xˆ F )v(t )  c1F (t )  r0 F (t )
 g ( xF )(u (t )  v(t ))}
e y  hT e x
where ex  ~
y~
yF . The system can then be represented in
x ~
xF and e y  ~
the following form:
~ ~T ~T ~T
ex  As ex   (W fT ,W fF
,Wg ,WgF , x, xˆ , t )  bg ( xF )(u (t )  v(t ))
e y  hT e x
18
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Detectability of Actuator Faults
Lemma. Given the NN observer and its NN parameters, the fault
detectability condition is given by the following inequality
t 0 T
t 0 T
t0
t0
T
 bWˆ g  g (u ( )  v( ))d  (WgM   Ng )  (u ( )  v( )) d 
exp( As T )(2 ~
xB  2C2T )
19
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Detectability of Actuator Faults
The result relates observer parameters, i.e. NN weights,
with fault detectability and the actuator control signal
It also shows when actuator faults can not be detected or
what needs to be done with NN observer to improve
sensitivity.
20
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Simulation Example
 Consider the second order nonlinear system given by
x1  x2
x2  5 x13  2 x2  ( x1  1)u (t )
y  x1
 Fault model is given by
u (t )  v(t )   (u (t )  v(t ))
 Two NNs are used for the system observer. Both NNs have 2, 30, and
2 neurons at the input, hidden, and output layers, respectively.
 Standard sigmoid activation function is used.
 For both NNs, the first-layer weights are uniformly randomly distributed
between -1 and 1.
21
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Simulation Example
 The threshold weights for the first layer are uniformly randomly
distributed between -20 and 20.
 The second layer weights W are initialized to zero for both neural
networks.
 Neural network tuning parameters are chosen as k=0.0001, Cf=20,
Cg=20.
 The state observer matrix is A=diag{30,30}.
 An ideal control signal v(t) is given by
v(t )  10 sin(t )
22
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Simulation Example
State observer errors e1(t), e2(t)
Norm of state observer error
0.4
1.5
0.2
0
1
-0.2
-0.4
0.5
-0.6
-0.8
-1
0
5
10
15
time (sec)
System state observer errors e1(t) (full line)
and e2(t) (dotted line).
0
0
5
10
15
time (sec)
Norm of the error e(t).
23
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Simulation Example
 NN weights bounds and approximation error bounds are estimated as
W fM  WgM  3 ,  fM   gM  0.1. Actual simulation or experimental results
can also be used to estimate the above parameters.
 The error bound is estimated as eB  0.2 .
 Assume that there is a fault at t=5sec where u  18 .
24
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Simulation Example
State observer errors e1(t), e2(t)
Norm of state observer error
0.5
1.5
0
1
-0.5
0.5
-1
0
5
10
15
time (sec)
Actuator fault at t=5sec; system state
observer errors e1(t) (full line) and e2(t)
(dotted line).
0
0
5
10
15
time (sec)
Actuator fault at t=5sec; norm of the error
e(t).
25
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Conclusions
 It is shown how neural net-based system can be used for
actuator fault detection in unknown, nonlinear, input-affine
systems.
 Stable neural net tuning laws are given and estimate on the
state observer error is provided using Lyapunov approach.
 Sufficient conditions for actuator fault detectability are
presented.
 An open research problem is to combine active fault
detection methods in case detectability conditions are not
satisfied.
26
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Intelligent Sensors and Actuators Group
Research interests:
 Wireless sensor networks for chemical agents monitoring
 Suboptimal coverage control missions in mobile sensor
networks.
 Intelligent actuator control using neural networks
 Actuators and sensors failure detection and compensation
 Intelligent wireless sensor networks
Group members: Dr. Rastko Selmic, 3 Ph.D. students,
4 M.S. students, and 2 undergraduate students.
The group has two laboratories with several control
system setups, sensors, wireless sensor nodes, two
mobile robots, 11 PC computers.
27
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Intelligent Sensors and Actuators Laboratory
The newest lab in EE – 11
PC computers, 8 control
system experimental
setups, sensors, wireless
sensor nodes, two mobile
robots, 2 oscilloscopes.
28
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Smart Actuator Control Using IEEE 1451 Standard
Develop a smart actuator control that is
compatible with IEEE 1451 standard for
smart transducers.
The concept allows for intelligent control
based on data and metadata gathered by
the network of smart sensors.
Control action depends on sensor data
and information stored in TEDS and
HEDS.
Smart Actuator
STIM
Network
(TCP/IP)
NCAP
Memory
TEDS, HEDS
Actuator
Power
Module
29
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Testbed Development – Chemical Agent
Monitoring
 Developed a chemical
sensor board for WSN
applications based on
Xbow motes.
 Sensor nodes monitor for
carbon monoxide (CO),
nitrogen dioxide (NO2),
and methane (CH4).
 Research problem: a
suboptimal sensor network
coverage of the area of
interest while providing
quasi real-time tracking
and monitoring of the
focus area observation
space.
Link
Local
Computer
Remote Sensor Nodes
RS-232
Remote
Computer
Base Station
Radio Link
30
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Simulation Tool for Coverage Control in Mobile
Sensor Networks
Simulation tool is needed to experiment with variety of algorithms
for sensor node deployment under localization and network
connectivity conditions.
Development based on C (optimization, network conditions) and
C++ (GUI).
C language chosen so simulation can be ported to High
Performance Computing machines in case it is needed for very
large networks.
Examples of different scenarios in sensor network coverage
control: uniform coverage, focused coverage, balanced coverage
control of sensor nodes.
31
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
32
LOUISIANA TECH UNIVERSITY
Department of Electrical Engineering and Institute for Micromanufacturing
INTRODUCTION
PROBLEM FORMULATION
STATE FEEDBACK
OUTPUT FEEDBACK
OTHER PROJECTS
Thank you! Any questions?
33