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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:
Computational Science and Systems Engineering
ISBN: 978-1-61804-362-7 287
(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
Computational Science and Systems Engineering
ISBN: 978-1-61804-362-7 288
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]
Computational Science and Systems Engineering
ISBN: 978-1-61804-362-7 289
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
Computational Science and Systems Engineering
ISBN: 978-1-61804-362-7 290
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.
REFERENCES:
[1] Fossen, Thor I. "Guidance and Control of Ocean
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[2] Jin-Kyu Choi and Hayato Kond, "On Fault-tolerant
control of a hovering AUV with four
horizontal and two vertical thrusters",OCEANS 2010
IEEE - Sydney, vol. 1, May 2010.
[3] Gilbert, E. G., and Tan, K. T. (1995) Discrete-time
reference governors and the nonlinear control of
systems with state and control constraints
International Journal of Robust and Nonlinear
Control, 5, 5(Aug. 1995), 487–504.
[4] Jiang, J., and Zhao, Q. (2000) Design of reliable
control systems possessing actuator redundancies.
Journal of Guidance, Control,and Dynamics, 23, 4
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[5] M. Blanke, R. Izadi-Zamanabadi, S.A. Bøgh and C.P.
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[8] I. Podlubny , Fractional Order systems and PIαDλ
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[9] R. Caponetto, L. Fortuna, and D. Porto, “A new
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[10] Chunna Zhao, Dingy¨u Xue and Yangquan Chen, “A
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[11] A.Monje,B.M.Vinagre,Vicente Feliu,Y.Chen,
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[12] Timothy Prestero, “Verification of a Six-Degree of
Freedom Simulation Model for the REMUS
Autonomous Underwater Vehicle”, Massachusetts
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
Computational Science and Systems Engineering
ISBN: 978-1-61804-362-7 291
Institute of Technology and the Woods
HoleOceanographic Institution September
2001,M.S.Thesis.
[13] Jiang, J. (1994) Design of reconfigurable control
systems using eigen structure
assignment.International Journal of Control, 59,2
(Feb. 1994),395–410.
[14] Zhang, Y. and J. Jiang, 2008. Bibliographical review
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DOI:10.1016/J.ARCONTROL.2008.03.008
Computational Science and Systems Engineering
ISBN: 978-1-61804-362-7 292
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