Adaptive Output Feedback Control Based on DRFNN

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Authors Accepted Manuscript

Adaptive output feedback control based on DRFNN

for AUV

Li-Jun Zhang, Xue Qi, Yong-Jie Pang

PII: S0029-8018(09)00084-5

DOI: doi:10.1016/j.oceaneng.2009.03.011

Reference: OE 1636

To appear in: Ocean Engineering

Received date: 19 February 2009

Accepted date: 29 March 2009

Cite this article as: Li-Jun Zhang, Xue Qi and Yong-Jie Pang, Adaptive out-

put feedback control based on DRFNN for AUV, Ocean Engineering (2009),

doi:10.1016/j.oceaneng.2009.03.011

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Adaptive output feedback control based on DRFNN for AUV

Li-Jun Zhang a , Xue Qib , Yong-Jie Pang c

a, b College of Automation, Harbin Engineering University, Harbin Heilongjiang 150001,China

c College of Shipbuilding Engineering, Harbin Engineering University, Harbin Heilongjiang 150001,China

Abstract

The tracking control problem of AUV in six degrees of freedom (DOF) is addressed in this paper. In general, the

velocities of the vehicles are very difficult to be accurately measured, which causes full state feedback scheme is

not feasible. Hence, an adaptive output feedback controller based on dynamic recurrent fuzzy neural network

(DRFNN) is proposed, in which the location information is only needed for controller design. The DRFNN is used

to online estimate the dynamic uncertain nonlinear mapping. Compared to the conventional neural network,

DRFNN can clearly improve the tracking performance of AUV due to its less inputs and stronger memory features.

The restricting condition for the estimation of the external disturbances and network's approximation errors, which

is often given in the existing literatures, is broken in this paper. The stability analysis is given by Lyapunov

theorem. Simulations illustrate the effectiveness of the proposed control scheme.

Keywords: Autonomous underwater vehicles; Adaptive control; Output feedback; Dynamic recrrent fuzzy neural network

Introduction

Due to the highly nonlinear dynamics of marine vehicles and the significant environment

disturbances, it is a great challenge to control marine vehicles, such as ships, submarines, remotely

operated vehicles and autonomous underwater vehicles. In AUV systems, linear translational


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velocities such as surge, sway and heave velocities are usually difficult to be obtained from

location measurements. And these measurements are usually corrupted by noises, so velocities

obtained by numerical location differentiation are not feasible.

Some adaptive control algorithms for AUVs tracking and dynamic positioning problems are

proposed in [1-5], where all of the given control algorithms are needed to know both locations and

velocities in 6 DOF, which cause more difficulties in practical operations. In order to solve the

problem of unmeasurable velocities, paper [6] provides an observer-based neural network adaptive

controller, in which the velocities are obtained by state observer. In [7], a full state observer is

designed to estimate the velocities of AUV moving in shallow water area, and an output feedback

controller is provided by using backstepping techniques to complete the tracking mission.

However, backstepping techniques are required to know the accurate modeling of AUV, which is

too much difficult for AUV with nonlinear structure and lots of uncertain parameters. In addition,

backstepping techniques also rise complexity of controllers structure.

In order to deal with the uncertain nonlinear parts in the AUV's dynamics, many researchers

concentrated their interests on the applications of neural networks to the AUV's control problems.

In [8], [9], and [19], a single hidden layered neural network is used to approximate the smooth

uncertainties of the vehicles' dynamics, where inputs of the networks are all the states of AUV.

This probably increases the number of hidden neurons and weighting parameters, and further slow

down the response velocity of the system. DRFNN is a dynamic recurrent multilayered

connectionist network with the memory elements and feedback loop. Compared to the traditional

fuzzy neural network (TFNN), DRFNN not only constructs a set of fuzzy rules and realizes fuzzy

inference, but also adds feedback connections to TFNN. Since a recurrent neuron has an internal


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feedback loop, it captures the dynamic response of a system, thus the number of inputs of DRFNN

are less than that of TFNN, this can simplify the network structure and speed up the response

velocity. DRFNN has been used to approximate the nonlinearity of the dynamics systems in

[10-12]. In this paper, we will use DRFNN to estimate the unknown parts of AUVs dynamic

modeling.

In practice, network's reconstruction error is not avoided in estimating the unkown nonlinear

part of dynamics, and the external disturbance will always exist, therefore, robustness has become

one of the most important issues in the control problems. Many researchers have studied the

applications of robust control methodology to the motion control of underwater vehicles in

[13-15]. In these literatures, the bounds of uncertainties are assumed to be known in advance. This

is a somewhat strong restricting condition in many practical applications because of the high

nonlinear and unpredictable operating environments of the vehicles. This kind of restrictions will

be relaxed in this paper.

Motivated by the results of the above mentioned papers, an adaptive output feedback controller

based on DRFNN for AUV is proposed to solve the trajectory tracking problem. The proposed

controller does not depend on the accurate modeling of AUV. The structure of the controller is

divided into three parts: dynamic compensator, adaptive control item and robust control item.

Dynamic compensator is used to stabilize the linear part of tracking error dynamics. Adaptive

control item is based on DRFNN which is used to approach the nonlinear dynamics of AUV.

According to the dynamic recurrenct features of DRFNN, the issues of the unmeasurable

velocities of AUV is solved without increasing the number of network nodes, which differs from

conventional neural networks. Robust control item is provided to compensate the unbounded


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estimated error of DRFNN and external disturbances. Motivated by [20], an observer is designed

to estimate the derivative of tracking error which will be used in the designed adaptive updating

laws. The stability analysis will be given by Lyapunov theorem. In order to demonstrate the

effectiveness of proposed control scheme, certain simulation studies on an AUV are presented.

The tracking performance of the horizontal location, the depth location and the yaw angle are

considered in the simulation studies.

The remainder of this paper is organized as follows: AUV's dynamics equation is derived in

Section 2. In Section 3, a DRFNN-adaptive output feedback controller for trajectory tracking is

presented. Section 4 contains the stability analysis. In order to demonstrate the effectiveness of the

proposed control scheme, certain simulation studies are presented in Section 5. Finally, we make a

brief conclusion on the paper in Section 6.

2. Problem formulation

The mathematical model of an AUV in 6 DOF can be described as

d

J

M C D g

yout

K K Q

Q Q Q Q Q K W W

K

(1)

where the positive definite inertia matrix RB AM M M includes the inertia RBM of the

vehicle as a rigid body and the added inertiaA

M due to the acceleration of the wave. The matrix

6 6u

C RQ groups the Coriolis and centripetal forces, this matrix is skewsymmetrical. The

hydrodynamic damping term 6 6uD RQ takes into account the dissipation of energy due to the

friction exerted by the fluid surrounding AUV. The vector 6g RK is the combined

gravitational and buoyancy forces in the body-fixed frame.d

W is the external disturbances.


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> @T

x y zK I T M denotes the vehicle location and orientation in the earth-fixed frame.

> @T

u v w p q r Q is the vector of vehicles velocities expressed in the body-fixed frame.

6RW is the input of the system and the vector of the forces and moments on AUV induced by

the thrusters and fins. K is the output of the system. J K is the kinematic transformation

matrix expressing the transformation from the body-fixed frame to earth-fixed frame.

1

2

0

0

JJ

J

KK

K

1

c c s c c s s s s c c s

J s c c c s s s c s s s c

s c s c c

M T M T M T I M I M I T

K M T M T I T M M I T M I

T T I T I

2

1

0

0

s t c t

J c s

s c c c

I T I T

K I I

I T I T

where sin , cos s c and tan t .

For a given desired trajectorydK which satisfies > @, ,

T

d d dK K K is bounded, there exists a

maping

1

2

d

dJ

[ K K

[ K Q K

that transforms system (1) into the tracking error system

1 2

1

2 1 2 1

1

,

df J M

hout

[ [

[ [ [ [ W W

[

(2)

where

11 2 1 1 2, ( ) df J J[ [ [ [ [ K

1 1 1

1 1 2 1 2( ( ) ( ) d dJ M C J J[ [ [ K [ [ K

1 11 2 1 2 1( ) ( ) ) d dD J J g[ [ K [ [ K [ dK

is a nonlinear uncertain function in which C Q Q and D Q Q contain velocities interaction in


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6 DOF.1[ is the output tracking error.

The control objective of this paper is to design controller W only using the output tracking

error1[ such that the solutions of system (2) and the other signals are all uniformly ultimately

bounded (UUB).

3. Controller design

From the tracking error system (2), if 1 2,f [ [ and dW are exactly known, the controller

will be designed as follows

1 11 1 1 2 1, dMJ u f J M W [ [ [ [ W (3)

where 1u is chosen as the output of a dynamic compensator to stabilize the following system at

point 0,0

1 2

2 1

u

[ [

[(4)

The structure of the dynamic compensator is shown in Fig. 1, i.e.

1

1 1

c d c d

d c d

A B

u C D

[ [ [

[ [ (5)

where c R[ , , , ,d d d d A B C D are designed parameters such that the closed-loop system

1 2

2 1

1

d c d

c d c d

C D

A B

[ [

[ [ [

[ [ [

(6)

is asymptotically stable (in [18]).

However, in fact, 1 2,f [ [ is not known exactly. We might want to use a neural network to

approximate 1 2,f [ [ in controller (3) which compensates for its effects. This will be

introduced in following part.


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3.1 DRFNN architecture

In this paper, DRFNN is used to compensate the unknown nonlinear function 1 2,f [ [ online.

A schematic diagram of DRFNNs stucture with n inputs, 1, , "im i n term nodes for each

input variable, single output is shown in Fig.2.

Layer 1 accepts input variables and each node for each input variable. The nodes in this layer

only transmit input variables1

, , "T

N N N

nx x x to the next layer directly.

Layer 2 represents the terms of the respective linguistic variables and is used to calculate

Gaussian membership value, i.e.

2

exp

N

i ijj

i

ij

x xP

V (7)

whereijx and ijV are the center and width of the Gaussian function respectively.

Nodes at layer 3 represent fuzzy rules. Layer 3 forms the fuzzy rule base. Links before layer 3

represent the preconditions of the rules, and the links after layer 3 represent the consequences of

the rule nodes, i.e.

1 ,1

m

a

ja c j

j

s a w k zD (8)

, 1 c j jk kD D (9)

0.5

1

1

a s ak f s a

eP

D (10)

At time k, every rule node a kD not only contains 11 1 " niia N N

n nz k x k x kP P , but

also contains recurrent rule nodes at time 1k .

Layer 4 realizes normalized unit for rules at layer 3, i.e.

1

aa m

i

i

DD

D

1,2, , "a m (11)


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where m denotes rule number.

Layer 5 is the output layer. Each node is for actual output to the pumped out this system. The

links between layer 4 and layer 5 are connected by the weighting value aN

y , i.e.

1

1

ma

N aNa

N m

a

a

y

y F x

D

D

(12)

Define the parameter vector of DRFNN as follows

1 T

T T T T

Nx w yE V

Since 1 2,f [ [ is a function about 1[ and 2[ . 2[ is not measurable directly, so it can't be

the input vector of DRFNN. However, there exists derivative relationship between1[ and 2[ .

For the reason that DRFNN has memory elements and inner feedback loop which can reflect

dynamic relationship between1[ and 2[ , choose 1[ as the input vector is reasonable.

Due to the approximation capacities of neural networks for nonlinear mappings, we have

following lemma.

Lemma 1 ([16]). Given*

0!H , there exists a set of bounded parameter vector E , such that

1 2,f [ [ can be approximated over a compact set : by a DRFNN with the input vector 1[ ,

i.e.

1 2 1, f F[ [ [ E H *H H (13)

for all 1 2, :[ [

Here, we replace 1 2,f [ [ in (3) with 2u , i.e.

2 1 1 2 , u F f[ E [ [ (14)

where 1 T

T T T T

Nx w yE V is the estimation of E . 1 6R[ .

With equations (13) and (14), neural network's reconstruction error is written as follows:


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1 2 2, f u[ [ 1 1 F F[ E [ E H

1 2

w

w D

TF

E E

[ EE E H

E (15)

where Taylor expansion is used.

E E E 2

2lim 0of

D

t

E

E

Equation (15) will be used in the immediate adaptive control design and stability analysis.

3.2 Robust controller

In literatures [13-15], the neural network's reconstruction error and external disturbances are all

bounded by known constants. This is a strong restricting condition for AUV dynamics, and it will

be broken in following part.

Assumption 1. Assume

11 2 2 1, dd d df u J M c[ [ [ W M (16)

with dc is an unknown constant and dM is a known bounded nonnegative function for

1 2, :[ [ . Clearly, Assumption 1 relaxes the restricting conditions in [13-15], which is very

valuable for the control design of AUV since its complex nonlinearity.

Here, we replace 11

dJ M[ W in (3) with 3u . The robust controller 3u satisfies

3 d du c M (17)

dc is the estimation of dc defined in (16).

The controller in (3) is rewritten as

1 1 1 2 3 MJ u u uW [ (18)


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3.3 Observer and adaptive laws design

In order to obtain 2u and 3u , the adaptive laws of parametersE and

dc will need the error

vector > @1 2T

c[ [ [ in which 2[ is not acquirable directly. So the observer will be first

designed to estimate the state 2[ , and then the adaptive laws of E and dc will be derived from

Lyapunov-based method.

Define

> @1 2T

c[ [ [ [

0 1 0

00

d d

d d

A D CB A

> @0 1 0T

b

From (2), (4) and (5), the error dynamics can be written as

11 2 2 3 1, dA b f u u J M[ [ [ [ [ W (19)

Define a new vector

> @1T

cg [ [

where g is measurable. With this definition, the error dynamics can be rewritten as

11 2 2 3 1,

dA b f u u J M

g C

[ [ [ [ [ W

[ (20)

where

1 0 0

0 0 1

C

Clearly ,A C is observable. With the measurable vector g as the input signal, the stucture

of a linear observer is shown as following


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A Kg

g C

[ [

[ (21)

where g g g is estimated error, K is a gain matrix and is chosen such that A KC is

asymptotically stable.

Define

[ [ [ A A KC

Then, the observer error dynamics can be written as

11 2 2 3 1,

dE AE b f u u J M[ [ [ W (22)

By (22) , we can obtain the estimate of 2[ .

Based on Lyapunov theorem, the adaptive law of the parameter vector for DRFNN can be

designed as follows:

1 0 2

w

w

TF PbE EE E

[ EE J [ O E E

E(23)

in which0E may be used to include a best guess of some E , where E is an ideal parameter

vector to estimate 1 2,f [ [ . 0!EO , EJ is adaptive gain matrix. P is the positive definite

solution of the Lyapunov equation

TA P PA Q (24)

for some , 0 !TP P Q .

In the similar idea, dc is updated by the following adaptive law:

0 [ 2 ]

d d

T

d c d c d d c Pb c cJ M [ O (25)

Ior some 0!dc

O , 0dc is a best guess of dc , and dc is an ideal quantity.dc

J is an adaptive

gain.


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4. Stability analysis

Define

d d dc c c

Choose Lyapunov function candidate as

1 11 1

2 2

d

T T T

d c dV P P c cE[ [ [ [ E J E J (26)

where P is the positive definite solution of the Lyapunov equation

TA P PA Q

in which

, 0 ! TP P Q

Here we assume that

2

min ! Q Pb PbO (27)

2

2!dc d

O M (28)

where min QO is the minimum eigenvalue of matrix Q .

Theorem 1. For system (1) and (2), we take observer dynamics (21), the controller (5), (14), (17)

and (18), adaptive updated laws (23) and (25), then all the states in the closed-loop system and the

control signals are uniformly ultimately bounded.

Proof. Consider the Lyapunov function candidate

1 11 1

2 2

d

T T T

d c dV P P c cE[ [ [ [ E J E J

The derivative of V along (19) and (22) will be

1 2 2 32 ( , T T TV Q Q Pb f u u[ [ [ [ [ [ [ 11 )

dJ M[ W 1 22 ( , T Pb Pb f[ [ [

12 3 1 ) du u J M [ W

1 1 d

T

d c dc cEE J E J


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T TV Q Q[ [ [ [ 1 2 1 3 12 ( )w

w

DT T

d

FPb u J M

E E

[ E[ E E H [ W

E

11 2 2 3 12 , T dPb Pb f u u J M[ [ [ [ W 1 1

d

T

d c dc cEE J E J

From (23), it follows

2 13 12 DT T T dV Q Q Pb u J M [ [ [ [ [ E H [ W

11 2 2 3 12 , T dPb Pb f u u J M[ [ [ [ W 10 dT

d c dc cEO E E E J

22

min mind V Q QO [ O [

1

2

w

w

T Td d d d

FPb c c

E E

[ E[ M M E

E

2 ( ) T d d dPb Pb c c[ M M 10 dT

d c dc cEO E E E J

22

min mind V Q QO [ O [ 2

T

d dPb c[ M 1

2

w

wT T

FPb

E E

[ E[ E

E

2 ( ) T d dPb Pb c[ M 10 dT

d c dc cEO E E E J (29)

Put (17) and (25) into (29), we have

22


w

wT FPb[ E

E2 ( ) T d dPb Pb c[ M

0 0 dT

c d d d c c cEO E E E O

Since

222 22 ( ) d T d d d d Pb Pb c Pb Pb c[ M [ M

0 0 T TE EO E E E O E E E E 2 2

02 2

d E EO O

E E E

0 0 d dc d d d c d d d d c c c c c c cO O22

2 2d d d

c c

d d doc c cO O

we find

22


w

wT FPb[ E

E

2 222 2

2 d dPb Pb c

EOE [ M

2 2 2

0 02 2 2

d dc c

d d dc c cE

O OOE E

Since


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22

2

22

2

w

ww d

w

T

T

FPb

FPb

E

E

[O E

E [ EE O

we have

222

min min[ ]d V Q Q Pb PbO [ O [ 2 2

2

d

c

d dcO

M2 2

0 02 2

dc

d dc cE

OOE E

22

2w

w

T FPb

E

[E

O

d VO U (30)

where O and U are positive constants defined by

2

minmin 2

max max

min{ , , 2 }

d dc c d

Q Pb PbQ

P P

OOO J O M

O O

22

2 2

0 0

2

2 2

w

w d

T

c

d d

FPb

c cE

E

[OO E

U E EO

Inequality (30) satisfies

0 0

d d

tV V e OU U

O O(31)

When oft ,V is bounded byU

O. Moreover, if 1 dVJ [ , then

1

1lim of

dt

U[ J

O(32)

That is, [ , [ , E and dc are uniformly ultimately bounded. ,

5. Simulation analyses

To illusrate that the approach is applicable to the AUV systems, simulation study is carried out

based on omni-directional intelligent navigator [17]. Due to the complicated motion system of

AUV, make the following assumption to simplify the problem: if AUV is moving with a changing


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yaw angle, its underwater depth will keep in constant; If AUV is moving with a changing depth,

its yaw angle will keep in constant. In this part, we will show the trajectory tracking of surge,

sway, yaw angle and depth respectively. The AUV dynamics is described by equation (1). The

desired velocities and positions are generated by reference trajectory generators. The initial

parameters of DRFNN are random. In the simulation studies, the external disturbance is taken as

10 ( ) d randW Q with > @1,1 rand , and the corresponding restricting functions are chosen

as 1dM . Dynamic compensator parameters are designed as

5, 1, 30, 8 d d d d A B C D

Observer gain matrix

1 1

0.2 5

0.1 1

K

Other design parameters are selected as

1, 0.02, E E

J O02, 0.25, 10 d dc c dcJ O

In the following figures, solid lines are the trajectory of AUV, and dot lines are the desired

trajectory.

Fig.3 and Fig.4 show the trajectory tracking of surge with DRFNN and with TFNN respectively.

The input variables of TFNN contain the measurable state and the estimated states of the AUV

system. This will let the estimated error into the neural network, and then increase the output error

of the TFNN. So its trajectory tracking performance is worse than that of the trajectory tracking

with DRFNN.

Fig.5 and Fig.6 show the trajectory tracking of sway with DRFNN and tracking error

respectively.


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Fig.7 and Fig.8 show the trajectory tracking of depth and yaw angle with DRFNN respectively.

At the beginn ing, the output of the controller is almost provided by dynamic compensator. With

the learning of DRFNN parameters, after a few minutes the output of the controller is almost

provided by DRFNN. From these simulation figures, we can see that excellent tracking of position

is obtained by the controller proposed.

6 Conclusion

For the tracking control of AUV, an adaptive output feedback controller based on DRFNN was

proposed and shown to track the desired trajectory in the earth-fixed frame. DRFNN was chosen

to compensate the nonlinear uncertain part of the AUV dynamics online. Based on the advantage

of DRFNN, the measurable state variable is only chosen as the input of the neural network, it

decreased the output error of the neural network. This was helpful for improving the accuracy of

trajectory tracking. A focus of this paper was taken on an attempt to break the traditional

restricting condition, which is usually added to the AUV's external disturbances. Here, DRFNN's

error and the external disturbances was bounded by a unknown constant, and certain adaptation

scheme for this constant was introduced through Lyapunov-based method. A linear observer was

proposed to estimate the tracking error which was used in updating adaptive and robust laws. The

stability of the controller was analysed by Lyapunov theorem. Simulation results showed that the

output feedback controller performs well with stability and robustness.

References

[1] Jian-Cheng Yu, Ai-Qun Zhang, Xiao-Hui Wang, Li-Juan Su, Direct adaptive control of underwater vehicles

based on fuzzy neural networks, ACTA Automatica Sinica. 33 (2007) 840-846.


18/28

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pted

man

uscript

[2] Yoerger D R, Slotine J J E, Robust trajectory control of underwater vehicles J, IEEE Journal of

Oceanic Engineering 10 (1985) 462-470.

[3] Fossen T I, Sagatun S I, Adaptive control of nonlinear systems: a case study of underwater robotic

systems J, Journal of Robotic Systems 8 (1991) 393-412.

[4] Corradini M L, Orlando G, A discrete adaptive variable structure controller for MIMO systems and

its application to an underwater ROV J, IEEE Transactions on Control Systems Technology 5 (1997)

349-359.

[5] Antonelli G, Chiaverini S, Sarkar N, et al, Adaptive control of an autonomous underwater vehicle:

experimental results on ODIN J, IEEE Transactions on Control Systems Technology 9 (2001)

756-765.

[6] Jian-Shu Gao, Zhi-Wei Xing, Hong-Bo Zhang, Observer-based neural network adaptive control of

underwater vehicles, ROBOT 26 (2004) 515-518.

[7] Shu-Yong Liu, Dan-Wei Wang, Eng Kee Poh, Chin Swee Chia, Nonlinear output feedback

controller design for tracking control of ODIN in wave disturbance condition, OCEANS. (2005)

1808-1810.

[8] Ji-Hong Li, Pan Mook Lee, A neural network adaptive controller design for free-pitch-angle diving

behavior of an autonomous underwater vehicle, Robotics and Autonomous System 52 (2005)

132-147.

[9] J.H.Li, P.M.Lee, S.J.Lee, Neural net based nonlinear adaptive control for autonomous underwater

vehicle, Proceedings of the IEEE International Conferenceon Robotics and Automation, Washington

DC, USA, (2002) 1075-1080.

[10]You-Wang Zhang, Direct adaptive dynamic recurrent fuzzy neural network control and its


19/28

Acce

pted

man

uscript

application. Information and Control 37(3), 269-274.

[11]Liu-Ping Mao, Yao-Nan Wang, Wei Sun, Yu-Xin Dai, An adaptive control using recurrent fuzzy

neural network, ACTA Electronica Sinica 34 (2008) 2285-2287.

[12]You-Wang Zhang, Rong-Zhu Wang, Design of indirect adaptive dynamic recurrent fuzzy neural

network controller, J.CENT.SOUTH UNIV.(NATURAL SCIENCE) 35 (2004) 253-257.

[13]P.Kiriazov, E.Kreuzer, F.C.Pinto, Robust feedback stabilization of underwater robotic vehicle, Rob.

Autonom. Syst.21 (1997) 415-423.

[14]P.M.Lee, S.W.Hong, Y.K.Lim, C.M.Lee, B.H.Jeon, J.W.Park, Discrete-time quasi-sliding mode

control of an autonomous underwater vehicle, IEEEJ. Ocean. Eng.24 (1999) 388-395.

[15]A.J.Healey, D.Lienard, Multivariable sliding mode control for autonomous diving and steering of

unmanned underwater vehicles, IEEE J. Ocean. Eng.18 (1993) 327-339.

[16]Naira Hovakimyan, Flavio Nardi, Nakwan Kim, Adaptive output feedback control of uncertain

nonlinear systems using single-hidden-layer neural networks, IEEE Transacyions on neural networks

13 (2002) 1420-1431.

[17]Gianluca Antonelli, Stefano Chiaverini, Nilanjan Sarkar, Michael West, Adaptive control of an

autonomous underwater vehicle: experimental results on ODIN, IEEE TRANSACTIONS ON

CONTROL SYSTEMS TECHNOLOGY 9 (2001) 756-765.

[18]Frederick M Brasch, JR., James B.Pearson, Pole placement using dynamic compensators, IEEE

Trans. Automat.Contr.15 (1970) 34-43.

[19]A.J.Calise, N.Hovakimyan, M.Idan, Adaptive output feedback control of nonlinear systems using

neural networks, Automatica 8 (2001) 1201-1211.


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[20]N.Hovakimyan, F.Nardi, A.Calise, A novel observer based adaptive output feedback approach for

control of uncertain systems, IEEE Trans. Automat. Contr. 147 (2002) 1310-1314.


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Fig.1. Controller architecture

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Fig.2. DRFNN architecture

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Fig.3. Trajectory tracking of surge with DRFNN

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Fig.4. Trajectory tracking of surge with TFNN

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Fig.5. Trajectory tracking of sway with DRFNN

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Fig.6. Trajectory tracking error of sway with DRFNN

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Fig.7. Trajectory tracking of yaw angle with DRFNN

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Fig.8. Trajectory tracking of depth with DRFNN

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Adaptive Output Feedback Control Based on DRFNN