International Journal of Scientific & Engineering Research, Volume 2, Issue 5, May-2011 1

ISSN 2229-5518

A Robust Hoo Speed Tracking Controller for Underwater Vehicles via Particle Swarm Optimization

Mohammad Pourmahmood Aghababa, Mohammd Esmaeel Akbari

Abstract— This paper presents an Hoo controller designing method for robust speed tracking of underwater vehicles, using Particle Swarm Optimization (PSO). Nonlinearity mapping of the underwater vehicles model to a nominal linear model, by employing a linear controller for a nonlinear model, is one of the main contributions of this paper. For reaching the linear Hoo controller, the nonlinear models linearized around an operating point. Both nonlinear and linearized models are discussed. A brief explanation of Hoo synthesis is given. Also frequency dependent weighting functions are used for penalizing tracking errors, setpoint commands and measured outputs noises using PSO. Obtained controller is reduced order to achieve a lower order controller. After simulating the reduced order Hoo controller it is embedded into the nonlinear model. By nonlinear simulations, robustness and efficient performance of the Hoo controller is shown. Control efforts of actuators revealed no saturation, therefore it is feasible to implement.

Index Terms— Hoo controller, underwater vehicles, particle swarm optimization, robustness.

—————————— • ——————————

1 INTRODUCTION

n the past two decades, underwater vehicles have be- come an intense area of oceanic researches because of their emerging applications, such as scientific inspection of deep sea, exploitation of underwater resources, long range survey, oceanographic mapping, underwater pipe- lines tracking and so on. Developing a control system that can achieve the aforementioned goal is challenging for a variety of reasons such as: the nonlinearity of the dynam- ics, the multivariable character of the vehicle with coupl- ing among different channels, the consistent amount of uncertainty due to the lack of precise knowledge of hy- drodynamic drag coefficients and evaluation of external
disturbance due to environmental interaction.
Several control techniques have been proposed in lite-
rature to deal with uncertainty. Sliding mode controller
for trajectory control of underwater vehicles, neglecting
the cross coupling terms, is proposed in [2]. Multivariable sliding mode control for diving, steering and speed con- trol of underwater vehicles with decoupled design is used in [3].
An Hoo autopilot for subzero II that had two sub- controllers, the longitudinal controller for the forward
speed and depth, and the lateral controller for the head- ing angle is presented in [4]. A reduced order Hoo control

————————————————

e Mohammad Pourmahmood Aghababa, Department of Electrical Engineer- ing, Ahar Branch, Islamic Azad University, Ahar,Iran;

m-Pourmahmood@iau-Ahar.ac.ir

e Mohammd Esmaeel AkbariDepartment of Electrical Engineering, Ahar

Branch, Islamic Azad University, Ahar, Iran; m-Akbari@iau-Ahar.ac.ir
that had three SISO decoupled controllers for the for- ward speed, heading angle and depth control was ap- plied to subzero III in [5]. A time delay control law for robust trajectory control of underwater vehicles is pro- posed in [6].
In this paper, designing of an Hoo controller for robust speed tracking is the major purpose. Position control can-
not be performed without suitable speed tracking. Here, both linear and angular speeds are considered to be con- trolled. Using Particle Swarm Optimization (PSO), weighting functions, that capture the disturbance charac- teristics and performance requirements are selected to take advantage of Hoo design algorithm.
For designing the speed controller, the nonlinear mod- el is linearzed around an operating point. Afterwards, parameters changing mapped to the linear model as un-
certainties. Weighted noises are also added to measured outputs. Then weighting functions for setpoint com- mands and tracking errors are obtained. It is assumed that all states can be measured by sensors, so that the state estimator is not necessary. After designing the Hoo controller and order reduction, it is embedded to full non- linear model. Tracking robustness and efficiency of the proposed controller is shown by nonlinear simulation. The required control efforts of thrusters are possible to realize.
The proposed method has the following characteris-
tics: a) the problem of speed tracking is considered as a new work, b) designed controller is MIMO (Multi Input- Multi Output) without neglecting the cross coupling terms, c) the Hoo controller is designed by using PSO, and d) the linear controller robustness is shown when it is embedded to the nonlinear model.

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The rest of this paper is preceded as follows. Section 2 presents the nonlinear and linearized motion equations of underwater vehicles. In Section 3, a general scheme of Hoo control synthesis is firstly explained. Then, the main pro- cedure of PSO method is given. In Section 4, an Hoo con- troller is designed for speed tracking aim of the underwa-
originates from the medium. This so called added mass is accounted for by the matrix MA.
2) Coriolis and Centripetal forces, C(v): For matrix
C(x), a similar discourse can be held. Both the coriolis and
centripetal forces are forces that are proportional to mass
and inertia. Hence, the matrix consists of two matrices:

T

ter vehicles and numerical simulations are performed.

C (v) CRB (v) + CA (v) CRB -CRB

(4)
Finally, some conclusions are given in section 5.

2 MOTION EQUATIONS AND DYNAMICS OF

UNDERWATER VEHICLES

2.1 The Nonlinear Model of Underwater Vehicles Throughout the marine robotics literature a vehicle’s six degrees of freedom dynamic equations are expressed as

[1]:
where CRB represents forces and moments due to the
mass and physical characteristics of the craft, CA(x) incor- porates the terms originating from the added mass.
3) Damping terms, D(v): In the damping matrix, D(x), four terms are combined [1]:
D(x)=Dp+ Ds(x) + Dw + DM(x) (5) where Dp is the potential damping, Ds(x) is linear and quadratic skin friction, Dw is wave drift damping and DM(x) is damping due to vortex shedding.

Mv + C (v)v + D(v)v + g (11 ) 1

(1)
Potential damping is introduced due to forces on the

11 J (11 )v

J (11 ) diag{J1 (11 ), J 2 (11 ) }

(2)
body when the latter is forced to oscillate. Skin friction
where s(.)=sin(.), c(.)=cos(.), t(.)=tan(.), 11 is the position and orientation of the vehicle in the Earth fixed frame,

E R6X1 , v is linear and angular velocity of the vehicle in

the body fixed frame, E R6X1 , M is the inertia matrix in- cluding added mass, E R6X6 , C(v) is a matrix consisting Coriolis and centripetal terms, E R6X6 , D(v) is a matrix
effects can be shown to constitute both a linear and a qu-
adratic term. Wave drift damping only plays a major role
at the surface where it can be interpreted as added resis-
tance due to incoming waves. Damping due to vortex
shedding is a result of the non-conservative nature of a
moving system in water with respect to energy. The visc-
consisting damping or drag terms, E R6X6 ,

g (11 )

is the
ous damping force due to this phenomenon is a function
vector of restoring forces and moments due to gravity and buoyancy, E R6X1 , and 1 is the vector of forces and
of the relative velocity of the craft, its physical characteris-
tics and the density and viscosity of the water.
moments of propulsion,

E R6X1 .The matrix

J (11 )

con-

4) Gravitation and Buoyancy, g (11 ) : This term models

verts velocity in a body fixed frame, v, to velocity in an earth fixed frame,11 , as shown in Fig. 1. In fact J1 (11 ) and J2 (11 ) convert linear and angular velocities in a body fixed frame, v, to velocities in an earth fixed frame,11 ,

respectively. A detailed derivation of these nonlinear eq- uations of motion can be found in [1]. Below a small
the restoring forces which result from gravitation and
buoyancy.
5) Thruster model, 1 : Usually, propellers are used as propulsion devices for underwater vehicles. The load torque Q from the propeller, and the thrust force T, are then usually written as [1]:



4

summary of the modeled phenomena is given.

Q pD 5

KQ (J 0 ) n n ,

T D

KT (J 0

) n n

(6)

Fig. 1. Inertial and body coordinate frames


where n is rotational velocity of the thruster, is the mass
density of water, D is the diameter of the propeller, KQ and KT are the torque and the thrust coefficients of the propeller, and J0 is the advance ratio.
In this paper, the thrusters are assumed to be driven by
DC motors. DC motors are usually controlled by velocity
feedback. It is assumed that six propellers are erected in six freedom degrees. Therefore, ni will be the physical input related to thruster number i. It can be also shown that an algebraic relation exists between the thrust of
propeller i and the physical input. Therefore, the thrust will be chosen as input in the model ui = Ti.

2.2 The Linearzed Model for Underwater Vehicles The nonlinear speed system of the underwater vehicles can be described in state space form by defining a six di- mensional state vector x=(u, v, w, p, q, r) as follows.

x

f ( x) + Bu,

u 1

(7)
1) Mass and Inertia, M: In matrix M, two inertial com-

f ( x) M -1 (-C ( x) - D( x) - g (11)), B -M -1

(8)
ponents are accounted for [1],
M=MRB+MA, M=MT, M>0 (3)
The rigid body inertial matrix, MRB, represents the mass and inertia terms due to the mass and other physical
characteristics of the craft. However in a dense medium such as water, a considerable contribution to the mass
For a linear controller design, it is necessary to extract the linearzed model from the nonlinear model around a representative operating point. In this paper, the nominal value of rotational speed of the propellers is considered
100 rpm. Using this assumption, the operating point is

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obtained:
x0=(1, 1, 1, 1, 1, 1 ) (9) The linearized model is:
put/output transfer functions such that when ||Ted||oo <
1, the relationship between e and d is suitable.
Without lack of generality, a mathematical overview of

x A x + B u, y

T

C x

Hoo synthesis is as follows. Figure 4 shows a standard feedback system, where w is the input vector of exogen-

x [u, v, w, p, q, r ] , u

y [u, v, w, p, q, r]T

[11 , 12 , 13 , 14 , 15 , 16 ]

(10)
ous signals, e is the output vector of errors to be reduced, y is the vector of measurements that are available for
where A and B are 6× 6 matrices and C is a 6×1 vector,

1i i=1, 2, …, 6 are the propeller forces, explained in the previous section. [u, v, w] and [p, q, r] are the linear and angular speeds of the underwater vehicle in a body fixed coordinate system, respectively.

The step response of linearized model is shown in Fig.
2. As seen in this figure, the step response is not tracked
and system modes are not decoupled.
feedback and u is the vector of external force signals. Let
Tew denote the closed loop transfer matrix from w to e. The Hoo synthesis problem is to find, among all controllers
that yield a stable closed loop system, a controller K that
minimizes the infinity norm ||Ted||oo. Throughout this paper we assume that all states are available for mea-

surement, that is, y equals the internal state of the genera- lized plant P.

u1 u2

4

2

0

4

2

0


2

1

0


4

2

0

4

2







0




u3 u4 u5 u6





Fig. 4. Standard feedback configuration





Suppose that a state space realization for P can be writ- ten as





2 x C1 x + B1w + B2u

(11)

1 e C1 x + D12u, y x

(12)

0

0 20 40

0 20 40 0 20 40

0 20 40

0 20 40

0 20 40

and assume that (A, B2) is stabilizable, D12 has indepen-

(sec)

Fig. 2. Step response of open loop linear model

dent columns and the system with input u and output e has no zeros on the imaginary axis.
Theorem. Suppose y > 0 is a given positive number.

3 METHODOLOGIES

Let

H (y)

\ H11
LH 21

H12 l

22 J

denote the Hamiltonian matrix

3.1 Hoo Synthesis Approach

Figure 3 shows a tracking problem, with disturbance re- jection, measurement noise, and control input signal limi-
with entries
H11=A-B2(D12TD12)-1D TC1

12

H12= y -2 B1B1T-B2(D12TD12)-1B2T

TD )-1 D

T)C

tations. K is a controller to be designed, G is the system,
H21=-C1T(I- D12 (D12 12

12 1

as nonlinearity uncertainties modeled, to be controlled and Wnoise, Wcmd and Wperf are weighting functions for
H22=-AT+C1TD12 (D12TD12)-1 B2T (13)
Then, there exists a stabilizing controller K such that
sensor noises, setpoint commands and tracking errors,
||Ted||oo < y if and only if: i)

H (y) has no eigenvalues

respectively. A reasonable design objective would be to
on the imaginary axis and there exist a basis for the spec-
design K to keep tracking errors and control input signal
tral subspace X- H (y) of

H (y)

of the form [X1T, X2T]
small for all reasonable reference commands, sensor nois- es, and external force disturbances.
Hence, a natural performance objective is the closed loop gain from exogenous influences (reference com- mands, sensor noise, and external force disturbances) to regulated variables (tracking errors and control input signal). Specifically, let T denote the closed loop mapping from the outside influences to the regulated variables. Good performance is associated with T being small. The mathematical objective of Hoo control is to make the closed loop MIMO transfer function Ted to satisfy ||Ted||oo<1. The weighting functions are used to scale the in-
where X1 and X2 are square matrices of appropriate di-

Fig. 3. Generalized Performance Block Diagram

mensions and X1 is invertible. ii) X( y )=X2X1-1 is positive

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semi definite. In this case, one such controller is K(s) = F, where
represents cognition, or the private thinking of the par- ticle when comparing its current position to its own best.

TD )-1[D

TC +B

X( y )] (14)
F=-(D12 12

12 1 2T

The third term in equation (15), on the other hand,
Existence and computation of X( y ) is a standard ma-
trix algebra problem that can be solved using a standard technique for solving Riccati equations based on the real Schur decomposition [9].

3.2 Particle Swarm Optimization

A particle swarm optimizer is a population based stochas- tic optimization algorithm modeled after the simulation of the social behavior of bird flocks. PSO is similar to ge- netic algorithm (GA) in the sense that both approaches are population-based and each individual has a fitness function. Furthermore, the adjustments of the individuals in PSO are relatively similar to the arithmetic crossover operator used in GA. However, PSO is influenced by the simulation of social behavior rather than the survival of the fittest. Another major difference is that, in PSO each individual benefits from its history whereas no such me- chanism exists in GA. In a PSO system, a swarm of indi- viduals (called particles) fly through the search space. Each particle represents a candidate solution to the opti- mization problem. The position of a particle is influenced by the best position visited by itself (i.e. its own expe- rience) and the position of the best particle in its neigh- borhood. When the neighborhood of a particle is the en- tire swarm, the best position in the neighborhood is re- ferred to as the global best particle and the resulting algo- rithm is referred to as a gbest PSO. When smaller neigh- borhoods are used, the algorithm is generally referred to as a lbest PSO. The performance of each particle (i.e. how much close the particle is to the global optimum) is meas- ured using a fitness function that varies depending on the optimization problem.
The global optimizing model proposed by Shi and Eber- hart [7] is as follows:
represents the social collaboration among the particles, which compares a particle’s current position to that of the best particle [8]. Also, to control the change of particles’ velocities, upper and lower bounds for velocity change is limited to a user-specified value of Vmax. Once the new position of a particle is calculated using equation (16), the particle, then, flies towards it [7]. As such, the main pa- rameters used in the PSO technique are: the population size (number of birds); number of generation cycles; the maximum change of a particle velocity Vmax and w. Generally, the basic PSO procedure works as follows: the process is initialized with a group of random particles (solutions). The ith particle is represented by its position as a point in search space. Throughout the process, each par- ticle moves about the cost surface with a velocity. Then the particles update their velocities and positions based on the best solutions. This process continues until stop condition(s) is satisfied (e.g. a sufficiently good solution has been found or the maximum number of iterations has been reached).

4 HCONTROLLER DESIGNING PROCEDURE

4.1 Weight Selection and Building Model

Uncertainty

To take advantage of Hoo design algorithm, we formu- late the design as a closed loop gain minimization prob- lem. So we select weighting functions that capture the disturbance characteristics and performance requirements to help normalize the corresponding frequency depen- dent gain constraints.
Wcmd is included in Hoo control problems that require tracking of a reference command. Wcmd shapes the norma-
lized reference command signals into the reference sig-

v w X v

+ RAND X c X (P

- x ) + rand X c X

nals that we expect to occur. It describes the magnitude

i +1 i

(G best - x i )

1 best i

2

(15)
and the frequency dependence of the reference com- mands generated by the normalized reference signal. Ref-

x i+1

x i + vi+1

(16)
erence commands for underwater vehicles linear and an- gular speeds are usually flat. This means that underwater
where vi is the velocity of particle i, xi is the particle posi-
tion, w is the inertial weight. c1 and c2 are the positive
vehicle speed does not change frequently and has no high oscillations. Therefore, Wcmd, is selected equal to
constant parameters, Rand and rand are the random func- tions in the range [0,1], Pbest is the best position of the ith

Wcmd

A

s + B

(17)
particle and Gbest is the best position among all particles in the swarm.
The inertia weight term, w, serves as a memory of pre- vious velocities. The inertia weight controls the impact of the previous velocity: a large inertia weight favors explo- ration, while a small inertia weight favors exploitation [7]. As such, global search starts with a large weight and then decreases with time to favor local search over global search [7].
It is noted that the second term in equation (15)
where A and B are two constants that are determined
using PSO.
Wperf weights the difference between the response of the closed loop system and the ideal model. Often we
might want accurate matching of the ideal model at low frequencies and require less accurate matching at higher frequencies, in which case Wperf is flat at low frequencies, rolls off at first or second order, and flattens out at a small, nonzero value at high frequencies. Therefore, the error weights penalize setpoint tracking errors on u, v, w, p, q and r. Hence, Wperf is considered as follows, for all of

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them. 6 6

W perf

C + E

s + D

(18)

min F ( P)

Q , R

22[(1 - e-a

i 1 j 1

)X (T + T )

as A and B; C, D and E are three constants that are found

+ e- a (M

pij

+ Essij )]

(21)
using PSO.
Wnoise represents frequency domain models of sensor noise. Each sensor measurement feedback to the control-
ler has some noise, which is often higher in one frequency range than another. The weighting function for the sen- sors would be small at low frequencies, gradually in- crease in magnitude as a first order or second order sys- tem, and level out at high frequencies. Then a high pass filter is selected for weighting functions of measured states.
is used for evaluating the Hoo controller performance. where Mpij is the maximum overshoot, Tsij is the settling time, Trij is the rise time and Esij is the integral absolute error of step response (i, j=1, 2, …, 6). Note that desired steady state of diagonal modes of the system (i.e. i=j) is 1 while for non-diagonal modes (i.e. i#j) it is desired to be 0.

a E [0, 4] is the weighting factor. The optimum selec- tion of a depends on the designer’s requirement and the

characteristics of the plant under control. We can set a to

Wnoise

Fs

s + G

(19)
be smaller than 0.7 to reduce the overshoot and steady-
state error. On the other hand, we can set a to be larger
where F and G constants are found by PSO.
To complete the uncertainty model, changing of the
underwater vehicle speeds due to vehicle parameters
changing, that can be produced by hydrodynamic drag
coefficients and propellers rotational speeds and external
disturbance, should be considered in controller designing
procedure. In this paper it performed by evaluating the underwater vehicle nonlinear behavior, when the men- tioned vehicle parameters are changed reasonably, and mapping it to the linear model. Therefore, we will build
an uncertainty model that matches our estimate of uncer- tainty in the physical system as closely as possible. Be- cause the amount of the model uncertainty or variability typically depends on frequency, our uncertainty model involves frequency-dependent weighting functions to normalize modeling errors across frequency. The follow- ing frequency dependent weighting function for both linear and angular speeds is chosen.
than 0.7 to reduce the rise time and settling time. If a is
set to 0.7, then all performance criteria (i.e. overshoot, rise
time, settling time, and steady-state error) will have the
same worth.
The minimization process is performed using PSO al- gorithm. Step response of the plant is used to compute four performance criteria Mp, Ess, Tr and Ts in the time domain. At first, the lower and upper bounds of the pa-
rameters are specified. Then a population of particles and a velocity vector are initialized, randomly in the specified range. Each particle represents a solution (i.e. weighting functions parameters P) that its performance criterion should be evaluated. This work is performed by compu- ting Mp, Ess, Tr, and Ts using the step response of the plant, iteratively. Then, by using the four computed pa- rameters, the performance criterion is evaluated for each particle according. Then using equations (15) and (16) the next likely better particles (solutions) are determined.
This process is repeated until a stopping condition is sa-

Wuncertaint y

Hs + I

s + J

(20)
tisfied. In this stage, the particle corresponding to Gbest is

where H, I and J constants are computed by PSO.

4.2 HController Design and Simulation results

Now that all plant components, as illustrated in Figure 3, are described and nonlinearity uncertainties and the weighting functions are constructed. We can design a desired Hoo controller. By using sysic function of MATLAB Robust Control Toolbox, the weighted uncertain model is built. Nonlinearity uncertainties are modeled by using ultidyn function.
the optimal vector P. The optimal P is obtained as P=[0.15,
1.23, 98.47, 0.95, 0.11, 0.2, 1.51, 5.73, 1.29, 10.33].

After constructing the weights and the weighted plant, we have recast the control problem as a closed loop gain minimization. A gain minimizing controller for the uncer- tain plant can be computed by using hinfsyn function. By using this function, the desired H= controller (K in Figure

3) is obtained. The obtained controller has 12 inputs (plant noisy outputs and weighted setpoint commands), 6 outputs for control forces of the plant, and 18 states, with

The weighting functions unknown parameters are

nominal performance y min

0.65 . For model order re-

computed using PSO. Therefore, minimizing a cost func- tion, determining the vector P=[A, B, C, D, E, F, G, H, I, J] is the main purpose. For doing this, a performance index as a cost function- that should be minimized- must be selected. The performance criterion is defined based on some typical desired output specifications in the time domain such as overshoot Mp, rise time Tr, settling time Ts, and steady-state error Ess. Therefore, in this paper, a time domain performance criterion defined by

duction, modred and balreal commands are used. Small

Henkel singular values indicate that the associated states are weakly coupled. With discarding these negligible Henkel singular values, the controller order is reduced to

11. Figure 5 shows the reduced order H= controller beha- vior (step response), when it is engaged with the linear plant. As illustrated in this figure, the H= controller can control the vehicle to track the desired speeds, efficiently.

To assess the behavior of the designed Hoo controller, it is embedded to the full nonlinear model of the underwa-

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ter vehicle as described in section 2.1 to form a closed loop system. Simulations are implemented in MATLAB Simulink. By the step response, the speed tracking quality is examined. Figure 6 shows the robust behavior of the designed controller against the nonlinearity of the nonli- near model. As shown in this figure, when a step is simul- taneously commanded to the actuators, the proposed Hoo

Step Response

small. This means that the designed controller can behave robustly against to the nonlinearities.

Figures 7 and 8 show the control efforts of the Hoo con- troller with the linearized and the nonlinear models, re- spectively. As illustrated in the figures, the control effort of actuators reveals no saturation and so it is feasible to implement.

2

1

0

2

1

0


1

0.5

0

1

0.5

0

1

0.5

0

2

1

From: u1


From: u2


From: u3


From: u4


From: u5


From: u6














Fig. 8. Control efforts of the Hoo controller embedded into the nonlinear plant, (time unit is 0.1s)






0

0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5

Time (sec)

Fig. 5. Step response of linear model with reduced order Hoo controller

5 CONCLUSIONS

A robust H= controller for underwater vehicles speed

Respons e of u

2

1

0

0 10 20 30 40 50

0.1 (sec ) Response of w

2

1

0

0 10 20 30 40

0.1 (sec ) Respons e of q

2

1

0

0 10 20 30

0.1 (sec )

Respons e of v

2

1

0

0 10 20 30

0.1 (sec ) Respons e of p

2

1

0

0 10 20 30

0.1 (sec ) Response of r

2

1

0

0 10 20 30 40

0.1 (sec )

tracking is introduced, in this paper. Nonlinearity of nonlinear model is mapped onto the nominal linear model as uncertainties. Using frequency dependent weighting functions that are determined by PSO, track- ing errors and noise errors are eliminated, robustly. The designed controller order is reduced. Using nonlinear simulations, robust behavior of the proposed controller is shown. The actuator control efforts were at the suita- ble rang for implementation. The future work can focus on control of underwater vehicles using nonlinear me- thods hybrid with intelligent techniques.

Fig. 6. Step response of the reduced order Hoo controller integrated in the nonlinear model

controller can follow the signal with small errors. Fur- thermore, steady state and amplitude errors are desirably

Fig. 7. Control efforts of the Hoo controller embedded into the linear plant, (time unit is 0.1s)

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[3] A.J. Healey and D. Lienard, “Multivariable Sliding Mode Con-

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1993, pp. 327–339.

[4] Z. Feng, and R. Allen, “Hoo Autopilot Design for an Autonom- ous Underwater Vehicle”, Proceedings of the 2002 IEEE CCA/CACSD, 2002, pp. 350 -354.

[5] Z. Feng, and R. Allen, “Reduced Order Hoo Control of an Auto- nomous Underwater Vehicle”, Journal of Control Engineering

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[7] Y. Shi, and A. R. Eberhart, “Modified Particle Swarm Optimiz- er”, Proceedings of the IEEE conference on evolutionary com- putation. Piscataway, NJ: IEEE Press, 1998, pp. 69–73

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ISSN 2229-5518

[8] J. Kennedy and R. Eberhart, "Particle Swann Optimization," In Proceedings of IEEE International Conference on Neural Net­ works, Perth, Australia, vol.4, 1995, pp.1942-1948.

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