FRACTIONAL ORDER FAULT TOLERANT CONTROLLER FOR

AUV

Dhananjay Talange¹ and Sneha Joshi²

1, 2 Department of Electrical Engineering, Pune University, Pune, India

s.d.joshi@hotmail.com

dbt.elec@coep.ac.in

Abstract— Here a innovative approach of Auto tuned fractional order control is proposed for fault tolerant control of Autonomous

Underwater Vehicle (AUV). From fault detection and reconfiguration, fractional order controller is designed. Reconfiguration is designed

using Eigen structure assignment technique. The proposed approach is tested on REMUS-100 AUV model. Variable step input is used as a

reference input.

Keywords— Fractional order, AUV, fault tolerant control

I. INTRODUCTION

Autonomous Underwater Vehicle (AUV) has been a subject of

research and development in exploring unknown marine

environment and carrying out different military missions.

AUV is complex, nonlinear system due to hydrodynamic

uncertainties involved in it. AUV maneuvering and control has

become a crucial task. The control system design for AUV has

become very challenging as it is not possible to correct

failures manually. Various advanced control systems are

available. Literature survey was carried out and different

control systems currently under use were studied. They are

PID, Neural network control, sliding mode control, fuzzy logic

control and optimal control. Different tuning techniques are

used apart from manual control. Robotics is a notable example

of this tuning technique. In majority cases robotic systems are

governed such that their behavior obeys a defined motion.

However during their operations it is conceivable that fault

may occur and system may malfunction. Thus it is essential to

take appropriate action to detect the fault and reconfigure the

same automatically during operation of AUV.

While on mission AUVs require periodic maintenance still

thrusters or sensors may fail due to uncertain hydrodynamic

forces in the ocean. If actuator fails there is a partial or total

loss of control action of AUV. To overcome failure of actuator

duplicating actuator in the system is one solution. But for

AUVs it is not advisable due to its high size and price hence

fault tolerance has to be achieved through existing actuators

only. Hence in fault tolerant control (FTC) system the

actuators are expected to overwork and maintain specified

system performance within tolerance limit. Dynamic model in

degraded performance is used as reference, to avoid saturation

of remaining actuator new command input is set for controller.

Actuators if work beyond their capacity may lead to

saturation. In AUVs usually there are four actuators two for

vertical and two for horizontal movements.

The proposed work concentrates on the vertical motion

considering faults on the vertical thruster. The method of

accommodation when fault occur consists of mainly three

steps: fault identification, fault reconfiguration. An algorithm

is necessary to make the behavior of the AUV within tolerable

limits.

II. MODELLING OF AUV ACTUATOR UNDER FAULT

2.1. Modeling of Actuator Fault

Consider normal system with differential equation shown

below.

(1)

(2)

(3)

The equivalent discrete time representation can be :

(4)

(5)

(6)

F= , G=

) B, H=C, , T sampling

period. represents modeling uncertainties and represents noise. To model actuator faults, we can write:

(7)

is post fault input. is control effectiveness factor.

i=1……Ɩ, , where

(8)

Indicates that actuator is healthy and is = Ɩ means

total failure, in between value shows partial loss of actuator. The total model can be represented as:

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(9)

(10)

(11)

(12)

Above model is model for fault diagnosis. The actuator

fault if occurs at instant the becomes non-zero.

2.2. FTC Design Objectives

While designing FTC the system performance under

dynamic and under steady state conditions and under normal

and under faults must be studied. Under normal condition

system must perform as per specifications. Under faults system

should survive with specified acceptable performance.

In AUV while fault occurs it is necessary to maintain its

stability. Other objectives are battery life, AUV

maneuverability and approach safely at the surface. The

prominent factor is stability. In actual AUV control when any

of the actuator gives failure system becomes handicapped; to

avoid this actuator redundancy [4] is one solution. But

practically in AUV, size and weight restrictions are the

limitations on redundancy. Hence in initial actuator failure,

actuator may cause further damage of rest of the system. In

AUV, degraded performance lowers speed and depth in case of

failure of vertical thrusters. In this article we propose suitable

reference model to propose strategy for new reference input

adjustment so that the system will not deteriorate under fault

conditions. Here after fault occurs Fractional order controller is

auto tuned for the degraded model.

III. PROPOSED FTC FOR AUV DEPTH SYSTEM

The structure of FTC [5] for AUV is as shown in figure 1. It

consists command input management, reference fault model,

control reconfiguration. Before any fault occurs system is

under normal condition, model is represented as normal

model. When actuator fault occurs performance of system

decreases, degrades and is called fault model. Controller is

Auto tuned FOPID controller is reconfigured for fault model

as reference. To prevent saturation of healthy actuator

command input is readjusted from new model. [6] To design

FTC, active model after fault occurrence has to be obtained.

Parameter estimation and state space model is to be provided.

The reconfiguration is designed using Eigen structure

assignment. After reconfiguration, FOPID designed according

to it.

Fig.1 Fault Tolerant System with FOPID

IV. REFERENCE MODEL AFTER OCCURRENCE OF FAULT

4.1 Reference Model after Occurrence of Fault

Desired reference model without fault can be given as:

(13)

(14)

Performance of the system gets reduced. Hence Eigen values

of degraded fault model will be:

Ψ = diag [ ] this is degradation matrix as per desired reference model. From this new can be obtained.

4.2 Command Input after Occurrence of Fault

When fault occurs in one actuator other may go into saturation

to avoid it command input is adjusted to appropriate value

during control reconfiguration. We simulated a model using

changed command input. Command input under normal

condition is .When fault occurs in actuator there is reduction

in control. At time t suppose a fault occurs the system

degrades in performance hence control input signal has to be

reduced to protect actuator from saturation.

Command input of faulty actuator should be less than the command input on other actuators. Hence the command input should be reconfigured so that control distribution will be proper. Weighing matrix is used to assign proper weights for redistribution of available control among all the actuators.

Closed loop control signal is and command input is under steady state. At steady state the command input can be given as:

is steady state closed loop control input for normal condition. W is weighing matrix. Degraded performance of system at steady state is improved by , but immediately after changing command input the transition occurs may cause

saturation of actuator. Hence gain of controller should be properly selected.

FOC

Reconfiguration

Fault

Detection

AUV Vertical Actuator

Sensor

Command I/P Adjustment as per

Ref Model after

Fault

Degraded Ref

Model under

Fault

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4.3 Design of Model after Reconstructing Command Input

During normal condition state space model is presented as:

k< t1 is under normal condition

, k>t2 at faulty condition of

actuator.

Once a fault occurs new controller gains are set as per the

Fault reference model here we are using FOPID controller and

is auto tuned for new gains after occurrence of fault. There are

two models one is desired model and other is degraded in

performance model after actuator fault.

V. INTRODUCTION TO FRACTIONAL ORDER PID CONTROLLER

As per the basics of fractional calculus [7] three definitions

are used for fractional differentegral i.e. Grunwald Letnikov

(GL), Riemann-Liouville (RL) and the Caputo.

The GL definition is given by (15)

[ / ]( ) ( 1) ( )lim

00

t a h rjrD f t h f t jh

a t jjh

(15)

RL Definition is given by (6) 1

( ) 1 / ( ) / ( ) / ( )nn t r n

D f t n r d dt f taa t

(16)

(n -1) < r < n and () is Gamma function.

The Caputo definition is given by (17) :

1( ) 1/ ( ) ( ) / ( )

t n r nD f t r n f t daa t

(17)

( n -1) < r < n

The state space form of fractional order system is expressed

in Laplace Transform and the transfer function is as follows

(18):

G(s) = 1

01..... 1 0na S a S a Sn

(18)

The most common form of fractional order PID controller [8]

is in the PI D form. The order of integrator is and that

of differentiator is and both are real numbers. The transfer

function of such PID controller is (19):

( ) 1( )

( )P I D

U sG s k k k s

E s s

( , 0)

(19)

E(s) is error, U(s) is controller output and G(s) is transfer

function of controller. In time domain it can be expressed as:

(20)

( ) ( ) ( ) ( )P DIu t k e t k D e t k D e t

(20)

With , =1 the FOPID controller will behave as classical

PID controller. For dynamic systems FOPID enhances control

performance. The band limit of FOPID is important. The finite

dimensional approximation should be done by selecting

proper frequency range. Though theoretical fractional order

system is with infinite memory practically, finite memory

approximation is required. Different methods like Oustaloup’s

method are used for finite memory approximation. [9][10]

Here FOPID is tuned as per the fault model as reference. The

reconfiguration is done using Eigen structure design. λ, μ, P,I

& D are varied as per reconfigured model.

5.1 Auto tuned FOPID

The conventional PID controller is used in many control

applications due to its simple structure in spite of its time

consuming tuning process. Tuning requires complete

knowledge about the system like order of model, dead time,

settling time. The other alternative is auto tuning. If the phase

of the open loop system is flat around cross over frequency

then controller is robust for gain variations. Actual

implementation of such controllers is complicated hence auto

tuning method is used in PID or FOPID controller. Auto

tuning of controller can be formulated as :[11]

(21)

This controller has two different parts as:

(22)

(23)

Eq.(22) corresponds to fractional order PI controller and

eq.(23) is for fractional order PD controller. Here fractional

order PI controller is used to cancel the slope of the phase

response at and around gain cross over frequency and

phase margin . So the phase curve is flattened in open loop

phase response of the system. This enhances the robustness of

the system. Following are the few steps for auto tuned FOPID

controller.

By fixing and for the system the resulting

pairs of frequency and phase are obtained for n

iterations. The slope of system is obtained from it.

λ and are obtained from slope of the system. the

gain is obtained for flat frequency response.

From flat gain, parameters of fractional order PD

controller are obtained.

Very small value of μ is fixed and then x and are

obtained.

If x is –ve then value of μ is increased till x becomes

+ve.

This value of μ ensures flatness of phase curve.

Therefore values of λ and μ are obtained and FOPID is fixed.

With a switch method auto tuning process is continued and is

implemented as shown in Fig. 1

VI. SIMULATION ON AUV DEPTH MODEL

6.1 Depth System of AUV

The linear zed state space model of depth system of AUV can be represented as: [12]

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x Ax Bu , The state and control vector are:

[ ] , [ ]T T

sx w q z u

In depth system four variables are involved w as heave

velocity, pitch rate q , pitch angle and depth z . Control

variable is deflection angle of stern, s .Here we have referred

REMUS-100 AUV model for simulations by neglecting w.

and is as under:

3 2 1

6.406. .

0.82 0.69T F

s s s

(21)

6.2 Design of Reference Model & Command Input

Referring design considerations in section IV and if the weighing matrix is selected as:

The parameters of the system normal and the degraded reference models and their Eigen values [13] are shown in Table I.

Table 1. Parameters of Model

Normal model gives performance as per specifications. In degraded fault model the model is active model under actuator fault condition of AUV.

Fig.2 Closed Loop Depth system without Fault

Here we have applied unit step input . From Figure 2 ,4

and Figure 5, 8 it is seen that output is tracking in closed

loop response as well in normal and degraded fault

model. Actuator fault in controlled system is as shown in

Figure 9 in simulink. From Figure 7 it is seen that

without reconfiguration the system output does not track

the input in degraded model. From figure 3 it is seen that

fault occurs output tend to reduce in closed loop system

and in degraded model it tend to oscillate (Figure 7).

The system output recovers back after reconfiguration of

controller [14] as well reduction/ modification in control

input.

Fig.3 Closed Loop Depth Model at Fault

Fig.4 Closed Loop Response after Reconfiguration

Fig.7 step Response for Fault Model without

Reconfiguration

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time in Sec

Depth

step I/P

Depth O/P

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

Time

Depth

step I/P

Depth O/P

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

Time

Depth

step i/p

depth o/p

Step Response

2 3 4 5 6 7 8 9 10-200

-100

0

100

200

300

400

500

Time

Depth

Model A B Eigen values of

A

Open Loop Model

Closed

loop

Model

without

fault

Reconfi

gure

model

after

fault

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Fig.8 Fault model after Reconfiguration

Fig.5 Reconfigured Response for Model

From the above simulation results it is seen that the response

of the system before occurrence of fault and in degraded

reference model the response after reducing command in put ,

output tries to track the input satisfactorily. The performance

parameters are shown in TABLE II.

Table II. Performance Parameters

Name of the

parameter

Closed Loop

System

Normal

system

Degraded

system

Rise Time (sec) 2.67 11.1 32.1

Settling Time (sec) 31.8 46.6 133

Overshoot 14.5% 9.9% 4.28%

Peak 1.14 1.16 1.08

When fault occurs the system performance degrades but using

degraded fault model and reduction in command input, the

system response approaches within limit using FOPID

controller.

VII. CONCLUSION

In this paper FTC is structured based on degraded fault

model. FTC is based on two concepts which are discussed in

this paper one is based on reconfiguration of controller and

other is on fractional order control. Two reference models are

discussed in this paper one for normal performance and other

for fault or degraded one. Keeping degraded performance as

reference, reconfiguration of fractional order controller is done

and actuator saturation is prevented and also performance of

AUV is tried to maintain.

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0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Depth

Step Response

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

1.2

1.4

time in sec

Depth

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Institute of Technology and the Woods

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DOI:10.1016/J.ARCONTROL.2008.03.008

Computational Science and Systems Engineering

ISBN: 978-1-61804-362-7 292