L OUISIANA T ECH U NIVERSITY Department of Electrical Engineering and Institute for Micromanufacturing INTRODUCTION PROBLEM FORMULATION STATE FEEDBACK

LOUISIANA TECH UNIVERSITYDepartment of Electrical Engineering and Institute for Micromanufacturing

INTRODUCTION PROBLEM FORMULATION STATE FEEDBACK OUTPUT FEEDBACK OTHER PROJECTS

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Actuator Fault Detection in Nonlinear Systems Using Neural Networks

Rastko Selmic, Ph.D.

Department of Electrical Engineering and Institute for Micromanufacturing

Louisiana Tech UniversityRuston, LA 71272, USA

Email: [email protected]: http://www2.latech.edu/~rselmic/



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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.



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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 net-based observer sensitivity

Simulation results are presented to illustrate the detectability criteria and fault detection in nonlinear systems.



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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?

Questions to be Answered



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Problem Formulation

Consider a nonlinear, input-affine system given by

)(

)()()(

xhy

tduxgxfx

where T

nxxxx ],,,[ 21 is a system state and f, g: nn , 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 BDtd )(

It is assumed first that the full state measurement is available.



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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 ))(()()( tvutvtu ,

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 ))()(()()( tvtutvtu Combined, the nonlinear system and the actuator fault model is )())()(()()()( tdtvtutvxgxfx

Problem Formulation



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A NN system observer is given by )()(ˆ)(ˆˆˆ tvxWxWAxxAx g

T

gf

T

f ,

where the matrix A is Hurwitz and two NNs are used to approximate nonlinear functions f(x) and g(x) )()()( xxWxf ff

T

f ,

)()()( xxWxg gg

T

g .

fW , gW are some ideal target NN weights, and )(xf , )(xg are NN

approximation errors bounded by fM and gM .

An observer error given by

xxe ˆ

Case I: State Feedback



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)()()( tduxgxfx

NN OBSERVER

v

u

01 u x

x

e

Nonlinear System with Actuator Failure Model

Figure 1. NN system observer – fault identifier.

A NN System Observer



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NN Tuning Law

A nonlinear system and observer dynamics are given by )()()()(

~)()(

~tdvxtvxWxxWAee gg

T

gff

T

f

where fff WWW ˆ~ ,

ggg WWW ˆ~ .

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

ff

T

fff WekCexCW ˆ)(ˆ

gg

T

ggg WekCetvxCW ˆ)()(ˆ

with any constant matrices 0 T

ff CC , 0 T

gg CC , and a small design

parameter k. Then the state observer error e and the NN weight estimation errors fW

~ , gW~ are uniformly ultimately bounded. The convergence region

for the state observer error e can be reduced by increasing the gain A.



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Stability Analysis

Stability proof is carried out using the Lyapunov function

gg

T

gff

T

f

T WCWtrWCWtreeL~~

2

1~~2

1

2

1 11

The bound on the tracking error is given by

min

22

41

41

BBgMfMgMfM

B

DVWWe

where min is a minimum eigenvalue of A, and BV is a bound on a control signal. We assume that the control signal is bounded (the system has been stabilized).



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The State Observer Error

Modified closed-loop error dynamics due to actuator fault is given by

))()()(()()()(

)()(~

)()(~

1111

tvtuxgtdtvx

tvxWxxWAee

g

g

T

gff

T

f

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.



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Dynamics of a Fault

Fault, as a process, has its own dynamics. The fault dynamics is given by

))()()(()()()()()()(

~)()(

~)()()(

~)(

~~~

1

1

tvtuxgtvxtvxtvxW

tvxWxxxWxWeAe

ggg

T

g

g

T

gfff

T

ff

T

f

where 1

~ eee Equivalently, the dynamics is given by ))()()((),,,

~,

~,~(~

1 tvtuxgtxxWWehe T

f

T

f ,

where the function h( ) is given by

)()()()()()(

~)()(

~)(

)()(~

)(~~),,,

~,

~,~(

1

11

tvxtvxtvxWtvxWx

xxWxWeAtxxWWeh

ggg

T

gg

T

gf

ff

T

ff

T

f

T

f

T

f



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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

1

0

0

2))()()(( CTedvuxg B

Tt

t

where 12~

2~

2max1 BMgBfBB VWWeC .

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.



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Case II: Output Feedback

Consider a single-input single-output nonlinear system in a normal form given by

xhy

tduxgxfbAxxT

)()()(

where Tnxxxx ],,,[ 21 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 BDtd )( .

0000

1000

0100

0010

A ,

0

0

0

1

h .



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A NN Observer A NN system observer is given by

xhy

xhyLtrtvxWxWbxAx

T

Tg

Tgf

Tf

ˆˆ

)ˆ()()()ˆ(ˆ)ˆ(ˆˆˆ 0

where x is an estimate of the state vector x and y is an estimate of the

output y. The observer gain matrix L is chosen so that Ts LhAA is

stable. Two NNs are used to approximate the nonlinear functions f(x) and g(x):

)()()( xxWxf ffTf

)()()( xxWxg ggTg

where fW , gW are some ideal target NN weights, and )(xf , )(xg are

NN approximation errors.



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A NN Observer

NONLINEAR SYSTEM

NN OBSERVER

v

u

01 u y

y

Nonlinear System with Actuator Failure

y~

Figure 2. NN system observer – fault identifier.



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NN Observer Tuning LawTheorem: Stable NN Observer Tuning Law (Lewis’ NN Tuning). Given the nonlinear system and the NN observer, choose the robustifying term as

y

ykr ro ~

~

with a scalar rk . Let the estimated NN weights be provided by the NN tuning algorithm

fffff WykCyxCW ˆ~~)ˆ(ˆ

ggggg WykCytvxCW ˆ~~)()ˆ(ˆ

with any constant, symmetric matrices 0 Tff CC , 0 T

gg CC , and a

suitably small design parameter k. Then the state observer error x~ and the NN weight estimation errors fW

~ , gW~ are uniformly ultimately bounded.

Proof: Select the Lyapunov function as

ggTgff

Tf

T WCWtrWCWtrxPxL~~

2

1~~2

1~~2

1 11



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Dynamics of a Fault

The fault dynamics is given by

xT

y

F

FFFgFTgFFfF

TfF

gTgf

Tfxsx

ehe

tvtuxg

trtctvxWxWb

trtctvxWxWbeAe

))}()()((

)()()()ˆ(~

)ˆ(~

{

)}()()()ˆ(~

)ˆ(~

{

01

01

where Fx xxe ~~ and Fy yye ~~ . The system can then be represented in

the following form:

xT

y

FTgF

Tg

TfF

Tfxsx

ehe

tvtuxbgtxxWWWWeAe

))()()((),ˆ,,~

,~

,~

,~

(



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Lemma. Given the NN observer and its NN parameters, the fault detectability condition is given by the following inequality

)2~2)(exp(

))()(()())()((ˆ

2

0

0

0

0

TCxT

dvuWdvuWb

BAs

Tt

tNggM

Tt

tg

Tg




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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.



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Simulation Example

Consider the second order nonlinear system given by

1

12

3

12

21

)()1(25

xy

tuxxxx

xx

Fault model is given by ))()(()()( tvtutvtu 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.



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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 )sin(10)( ttv

Simulation Example



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System state observer errors e1(t) (full line) and e2(t) (dotted line).

0 5 10 15-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

time (sec)

State observer errors e1(t), e

2(t)

0 5 10 150

0.5

1

1.5

time (sec)

Norm of state observer error

Norm of the error e(t).

Simulation Example



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NN weights bounds and approximation error bounds are estimated as 3 gMfM WW , 1.0 gMfM . Actual simulation or experimental results

can also be used to estimate the above parameters. The error bound is estimated as 2.0Be . Assume that there is a fault at t=5sec where 18u .

Simulation Example



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Actuator fault at t=5sec; system state observer errors e1(t) (full line) and e2(t)

(dotted line).

Actuator fault at t=5sec; norm of the error e(t).

0 5 10 15-1

-0.5

0

0.5

time (sec)

State observer errors e1(t), e

2(t)

0 5 10 150

0.5

1

1.5

time (sec)

Norm of state observer error

Simulation Example



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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.



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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.



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Intelligent Sensors and Actuators LaboratoryThe newest lab in EE – 11 PC computers, 8 control system experimental setups, sensors, wireless sensor nodes, two mobile robots, 2 oscilloscopes.



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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.

NCAP TEDS, HEDS

Smart Actuator

PowerModule

Actuator

Memory

Network (TCP/IP)

STIM

NCAP TEDS, HEDS

Smart Actuator

PowerModule

Actuator

Memory

Network (TCP/IP)

STIM



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Testbed Development – Chemical Agent Monitoring

Base Station

Remote Computer

Local Computer

Link

Remote Sensor Nodes

Radio Link

RS-232

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.



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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.



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Thank you! Any questions?

Documents

L OUISIANA T ECH U NIVERSITY Department of Electrical Engineering and Institute for Micromanufacturing INTRODUCTION PROBLEM FORMULATION STATE FEEDBACK