Proceedings of 2008 IEEE International Conference onMechatronics and Automation

TB1-4

978-1-4244-2632-4/08/$25.00 ©2008 IEEE

Abstract—Most designs for heading controller adopted Taylor linearization technique and neglected its nonlinear factors. For this reason, the accuracy of heading control for Autonomous Underwater Vehicle (AUV) was lower. In order to solve this problem and enhance the accuracy, robustness and stability for heading control system, a nonlinear model of AUV in horizontal plane was established, which include uncertainties. Based on linearization via state feedback, a linear model of AUV was obtained at the operating point, and a robust heading controller was designed with closed-loop gain shaping algorithm. The computer simulation was used to demonstrate the robust controller designed via state feedback, has a better performance than the controller, which based on a linear model using Taylor linearization technique.

I. INTRODUCTION

UV has been a subject of research and development, particularly in exploring unknown marine environment, completing the specific underwater

operation and carrying out submarine military missions. Although studies have been made of AUV over the past thirty years, still AUV technology limitations remain, as in[1], [2]. AUV is a typical system which is characterized by high nonlinearities, couplings and uncertainties. In general, the motion of AUV changes obviously when it navigates under the influence of ocean currents and ocean waves. Then the hydrodynamic effect becomes more complicated, and the nonlinearity of its motion becomes more notable. Therefore, the nonlinear dynamics of motion model for AUV can not be ignored, as in [3].

However, the most heading controller of AUV is designed based on linear motion model, which linearized by Taylor series expansion and neglected of the high nonlinearities for local linearization. Thus, this method will result in low precision and poor stable for AUV heading control system, as in [4], [5]. Ref. [6], [7], [8], [9], linearization via state feedback is the application of differential geometry. It is different from the Taylor series expansion for local

Manuscript received March 10, 2008. This research work was supported from postdoctoral science research developmental foundation of Heilongjiang province, No.LHK-04010.

Xiang-qin Cheng is with the Automatic Department, Harbin Engineering University, Harbin, 150001, China

(e-mail: chengxiangqin1983@163.com). Doctor candidate. Zhe-ping Yan is with the Automatic Department, Harbin Engineering

University, Harbin, 150001, China (e-mail: yanzheping @ hrbeu.edu.cn). Doctor tutor. Xin-qian Bian is with the Automatic Department, Harbin Engineering

University, Harbin, 150001, China (e-mail: bianxinqian@hrbeu.edu.cn). Doctor tutor. Jia-jia Zhou is with the Automatic Department, Harbin Engineering

University, Harbin, 150001, China (e-mail: choujiaa123@163.com). Doctor candidate.

linearization. It does not neglect any nonlinear factor in the process of linearization. Therefore, this method is not only kind of precision, but also applicable to the entire region. In order to enhance the accuracy, robustness and stability for AUV heading control, the motion model of AUV in horizontal plane is linearized base on the method of linearization via state feedback, then the robust heading controller is designed by the closed loop gain shaping algorithm.

The paper is organized as follows: Section II provides a dynamics model of AUV in horizontal plane. A linear dynamics model of AUV for heading control is obtained at the model of operating point, based on linearization via state feedback in section III. And a robust heading controller is designed with closed-loop gain shaping in section IV. In section V, under different simulation environment, the heading control curves of AUV are obtained at different navigation speed. The final section contains the conclusion and further works.

II. DYNAMIC MODEL OF AUV IN HORIZONTAL PLANE

A. Reference Frames Dynamic modeling for AUV is a complicated problem of

dynamics, whose research should be not in Onboard Coordinate, but in an inertial coordinate. This paper selects Geodetic Coordinate E��� as inertial coordinate. The origin point E can be any point on the ground or surface of sea. The positive axis E� points to the earth’s core. The axis E� and E� are orthogonal. E� points to anywhere. E��� is a right handed coordinate. The north-east-depth coordinate is adopted in this paper.

Fig. 1 Reference frames

The Onboard Coordinate OXYZ, which is not an inertial coordinate, is also defined. Axis OX is parallel to the datum plane of the vehicle and points to the head of AUV. Axis OYis parallel to the datum plane and points to starboard of AUV. Axis OZ points to the bottom of the vehicle. The relationship of these two reference frames is shown in Fig. 1.

Application of Linearization Via State Feedback to Heading Control for Autonomous Underwater Vehicle

Xiang-qin CHENG, Zhe-ping YAN, Xin-qian BIAN and Jia-jia ZHOU

A

477

B. Modeling Heading Control System for AUV In general, the motion of AUV that neglected of the

rolling movement and the influence of coupling betweenhorizontal plane and vertical plane, can be simplified as the two planes movement campaign, respectively. Consequently, it is useful and convenient for researching on the heading maintenance and control for AUV, as in [10].

The dynamic model can be obtained by Kinetic Research. According to fluid dynamics, Momentum Theorem and Moment of Momentum theorem, the dynamic equations for AUV heading control system are the equations (1)-(3):

3 2 2 2 21 1 12 2 2

4 2 3 2 2 ' 21 1 12 2 2

[ ]

r r

u uu vv

rr vr r prop

m u vrL X u L X u L X v

L X r L X vr L u X Xδ δ

ρ ρ ρρ ρ ρ δ

−′ ′ ′= + + +′ ′+ + +�

�� (1)

1 4 ' ' ' '| |2

1 3 ' ' ' ' '| |2

1 2 ' ' '| |2

1 2 ' 22

[ ]

[ | |]

[ | |]

[ | |]

( ) cos sinr

r pq qr r r

v ur wp vq v r

uv vw v v

r

m v ur wp

L Y r Y pq Y qr Y r r

L Y v Y ur Y wp Y vq Y v r

L Y uv Y vw Y v v

W B L Y uδ

ρ

ρ

ρ

θ ϕ ρ δ

+ −

= + + + +

+ + + + +

+ + +

− +

�

�

��

� (2)

1 5 ' ' ' '| |2

1 4 ' ' ' '| |2

1 13 ' ' ' 3 ' 2| |2 2

( )

[ | | ]

[ | | ]

[ | | ]r

z y x

r pq r r qr

v vq wq v r

v v v vw r

I r I I pq

L N r N pq N r r N qr

L N v N vq N wq N v r

L N uv N v v N vw L N uδ

ρ

ρ

ρ ρ δ

+ −

= + + + +

+ + + +

+ + +

�

�

�

�

� (3)

The attitudes of the vehicle in the inertial frame ψ and θare computed by (4):

( sin cos ) / cosq rψ ψ ψ θ= +� . (4) Where, L , m , xI , yI , zI respectively are length, quality, and moment of inertia along coordinates; u , v , w are translational motions along OXYZ; p , q , r are rotational motions (angular velocity) about OXYZ. ϕ , θ , ψ , are the attitude angles; 'X , 'Y , 'N are hydrodynamic coefficients;

rδ is the rudder angle; ρ is the density of seawater, W , Brespectively are gravity and buoyancy of AUV.

Assume that 0u is the navigation speed of AUV, and the center of gravity on the origin of the coordinate system. Meanwhile, w , p , q are small amounts and hardly influence the heading control system. Therefore, the influence of vertical movement and roll motion could be neglected. However, when AUV has a strong maneuver, the speed v and angular velocity r change in the broader context. Thus, the changes of velocity and the impact of nonlinear hydrodynamics should be considered. Then, the heading control system can be

3 ' 4 '1 12 2

2 2 4 2 21 1 10 0 02 2 2

3 2 21 10 0 02 2

( ) ( )

( ) ( ) ( )

( ) ( )

v r

uvv v r r

ur r r v

m L Y v L Y r

L Y u v L Y u r L Y u v

L Y u mu r L Y u dδ

ρ ρρ ρ ρ

ρ ρ δ

− + −

′ ′ ′= + + +

′ ′− + +

� �� �

(5)

4 5 '1 12 2

3 ' 2 5 ' 21 12 2

3 41 10 02 2

3 2102

( ) ( )

( ) ( )

( ) ( )

( )

v z r

v v r r

uv ur

prop r r r

L N v I L N r

L N v L N r

L N u v L N u r

N L N u dδ

ρ ρρ ρ

ρ ρρ δ

′− + −

= + +

′ ′+ +

′+ +

� �� �

(6)

r dψψ = +� (7) Where vd , rd , dψ include the error, uncertainty and outside interference, which are produced by linear approximation and bounded toward the system input.

Defining the heading instruction is a constant, rψ , and 0rψ =� , heading error is eψ .

Then e rψ ψ ψ= − (8) ( )r

ed r d

dt ψψ ψψ ψ−= = − = − −� � (9)

Dynamic model for AUV heading control in horizontal plane are:

2 2 211 12 13 0 14 0 11 0

2 2 221 22 23 0 24 0 21 0

0e

r r

v

r dr a v a r a u v a u r d b uv a v a r a u v a u r d b u

ψψδ

− −� � � � � �� � � � � �= + + + + +� � � � � �� � � � � �+ + + +� � � � � �

���

(10)

Where: 2 4 2 31 1 1 1

| | | |2 2 2 23 5 3 41 1 1 1

| | | |2 2 2 2

v v r r uv ur

v v r r uv ur

L Y L Y L Y L Y mP

L N L N L N L Nρ ρ ρ ρρ ρ ρ ρ

′ ′ ′ ′� �−= � �′ ′ ′ ′� �� �

,

212

312

r

r

L YQ

L Nδ

δ

ρρ

′� �= � �′� �

,3 ' 4 '1 1

2 24 5 '1 1

2 2

v r

v z r

m L Y L YH

L N I L Nρ ρ

ρ ρ� �− −

= � �′− −� �

� �

� �

,

11 12 13 14 1

21 22 23 24

a a a aA H P

a a a a−� �

= =� �� �

, [ ] 111 21

TTB b b H Q−= = .

III. DESIGNING HEADING CONTROL SYSTEM BASED ONLINEARIZATION VIA STATE FEEDBACK

A. Linearization via State Feedback Linearization via state feedback has proved to be a

powerful tool in control of nonlinear systems, as in [11]. It is a branch of geometry of theoretical system for nonlinear system, the meaning of which is: through state feedback control and nonlinear coordinates transform, the nonlinear system is transformed into linear system which has a good dynamic quality, stability and fully capable of controlling.

Consider an affine nonlinear dynamic system: ( ) ( )x f x g x u= +� (11)

( )y h x= (12)

Where, f , g : n nR R→ , h : n lR R→ are smooth functions on the state space nR , u , ly R∈ are the system input and output, respectively.

It is known that if a real-valued function ( )xλ can be found and a change of coordinates

1 1 1( ) ( ( ) ( ) ( ))Tx x x xφ φ φΦ = � (13)

478

by1

1

( ) ( )( ) ( ),2i f i

x xx L x i n

φ λφ φ −

== ≤ ≤

(14)

is possible. Meanwhile the following conditions must be satisfied:

1( ) 0 , 2 ,

( ) 0.g i

g n

L x i nL xφφ

− = ≤ ≤

≠ (15)

Then by setting ( )xξ = Φ , the system (11) can be transformed into

1 2

2 3

1

,

,,

,

( ) ( ) .n n

n f n g nL x L x u

ξ ξξ ξ

ξ ξξ φ φ

−

=

=

=

= +

�

�

��

�

(16)

Let n K vξ ξ= +� , a feedback control law can be ( ) ( )

( )f n

g n

L x K x vu

L xφ

φ− + Φ +

= (17)

Where, 0 1 1( )nK k k k −= � is the coefficient row vector of a Hurwitz polynomial (later in this paper, K as well as

0 1nk k −� have the same meaning as here). In (14)–(17), the terms

11

11

( ) ( ),

( ) ( ).

if i

ig i

L x f xx

L x g xx

φφ

φφ

−−

−−

∂=

∂∂

=∂

(18)

are called Lie derivatives of 1( )i xφ − along ( )f x and ( )g x , respectively.

It has been proved that (15) is equivalent to 1

1

( ) 0, 1 1,

( ) 0.

if

nf

ad g

ad g

L x i n

L x

λ

λ

−

−

= ≤ ≤ −

≠ (19)

Where0

1

1

( ) ( ),

( ) [ , ]

( ) ( ),

( ) [ , ]( ), 0,1, 2, , 1.

f

f

k kf f

ad g x g x

ad g x f gg ff x g xx x

ad g x f ad g x k n−

=

=

∂ ∂= −∂ ∂

= = −�

(20)

are called Lie products. Equation (19) can be written as

2( ) ( ) ( ) ( ) 0.nf f

x g x ad g x ad g xx

λ −∂ � � =� �∂� (21)

If equation (21) is solvable in a neighborhood of 0x (i.e., there exists a function ( )xλ to make the system (11) have relative degree n at 0x ), the well-known conditions (sufficient and necessary) for the solution of the state space exact linearization problem are the following.

1). The matrix [ 0( )g x 0( )fad g x � 2 0( )nfad g x− ] has the

rank n.2). The distribution span { g 1

fad g � 2nfad g− } is

involutive in a neighborhood of 0x .The first condition guarantees that the equation (21) has a

nontrivial solution, it is said that, the differential quotient ( ) /x xλ∂ ∂ is a basis of the one-dimensional codistribution

around 0x . And the second condition is an integrability condition, or called involutivity condition. It means that there exists a real-valued function ( )xλ whose differential quotient ( ) /x xλ∂ ∂ spans the codistribution.

The above feedback linearization is an exact feedback linearization. The transformation of a nonlinear system into a linear one involves solving the first-order partial differential equation (21), which normally is a quite difficult problem. And another technique is input–output feedback linearization. If the system (11) has relative degree n with the output h(x) (or say the output h(x) is a function ( )xλ ), then the coordinates can be

1 21

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

Tn

n Tf f

x x x xh x L h x L h xφ φ φ

−

Φ =

=

��

(22)

A feedback control law can be given as 1

0 11

( ) ( ) ( )( )

n nf n f

ng f

L h x v k h x k L h xu

L L h x

−−

−

− + + + +=

� (23)

For many systems, (19) can only be satisfied till the order r in a neighborhood of 0x (r < n).That means

1

0

( ) 0, 2 1,

( ) 0.g i

g r

L x i r

L x

φ

φ− = ≤ ≤ −

≠ (24)

Then the system is called to have relative degree r. In this case the system can be divided into two parts: the linearizable part and the zero dynamic part. When the zero dynamic part is asymptotically stable, the change of coordinates for the linearizable part can be

1 21

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

Tr

r Tf f

x x x xh x L h x L h xφ φ φ

−

Φ =

=

��

(25)

And a feedback control law can be completed by 1

0 11

( ) ( ) ( )( )

r rf r f

rg f

L h x v k h x k L h xu

L L h x

−−

−

− + + + +=

� (26)

Where 0 1nk k −� are the coefficients of an r order Hurwitz polynomial.

B. Linearize AUV Heading Control System A feedback control law for AUV heading control system

should be given based on Linearization via state feedback in this section.

In equation (10), set the system input ru δ= and output ( , , )e ey h r vψ ψ= = .

Let

479

2 211 12 13 0 14 0

2 221 22 23 0 24 0

( , , )e

rf r v a v a r a u v a u r

a v a r a u v a u rψ

−� �� �= + + +� �� �+ + +� �

(27)

2 211 0 21 0( , , ) 0

T

eg r v b u b uψ � �= � � (28) then, the AUV heading control system can be transformed into

[ ] ( , , ) ( , , )( , , )

Te e e

e e

r v f r v g r v uy h r vψ ψ ψ

ψ ψ= +

= =

� � � (29)

By the method of linearization via state feedback, the AUV heading control system (29) can be linearized as following:

1). System for Relative Degree: Set k = 0, then

[ ] 211 0

221 0

( , , )( , , ) ( , , )( , , )

01 0 0 0

eg e eT

e

h r vL h r v g r vr v

b ub u

ψψ ψψ

∂=∂

� �� �= =� �� �� �

(30)

According to the formula (24), let k = 1,

[ ] 2 211 12 13 0 14 0

2 221 22 23 0 24 0

( , , )( , , ) ( , , )( , , )

1 0 0

ef e eT

e

h r vL h r v f r vr v

ra v a r a u v a u ra v a r a u v a u r

r

ψψ ψψ

∂=∂

−� �� �= + + +� �� �+ + +� �

= −

(31)

[ ] 2 211 0 11 0

221 0

( , , )( , , ) (( , , ))

( , , )0

0 1 0 .

f eg f e eT

e

L h r vL L h r v g r v

r v

b u b ub u

ψψ ψ

ψ∂

=∂

� �� �= =� �� �� �

(32)

For this heading control system, 211 0 0b u ≠ .So the relative

degree k = 2. 2). Coordinate ransformation: In order to change the

coordinate, set 1( , , )e er vφ ψ ψ= , 2 ( , , )e r v rφ ψ = . Because of the relative degree k = 2, we need another function

3 ( , , )e r vφ ψ , which must be satisfied:

33

( , , )( , , ) ( , , ) 0( , , )

eg e eT

e

r vL r v g r vr v

φ ψφ ψ ψψ

∂= =∂

(33)

By computing, we get the function as 2 2

3 21 0 11 0( , , )e r v b u r b u vφ ψ = + (34) by setting ( )xξ = Φ , the heading control system can be transformed into

1 2

2

3

1

,

( ) ( ) ,

( ),.

b a u

y

ξ ξξ ξ ξξ η ξ

ξ

=

= +

==

���

(35)

Where, 1 2

11 0( ) ( ( ))g fa L L h b uξ φ ξ−= = (36)

21

2

11 11

11

2 1

212 2 212 14

11 3 32 2 411 00

12 21 14 212 3 13 0 22 2

110

( ) ( ( ))

( )

2( ) ( ) ,

fb L h

a b a aab ub b u

a b a ba u

bb u

ξ φ ξ

ξ ξ ξ

ξ ξ ξ

−=

= + + + −

+ −

(37)

21 21

11

11

21

21

11

13

3 212 22 2 2

11 21 21 11 0 2211

2 212 21 22 14 213 24 0 32 2

11 110

214 3

13 21 23 11 24 21 0 211

212 22 21

2 3211

( ) ( ( ))

( )

1( ) ( )

( )

2 2( ) .

fL

a b a ba b a b u

bb

a b a a b a ub bb u

a ba b a b a b u

b

a b a bbb

η ξ φ φ ξ

ξ

ξ ξ

ξ

ξ ξ

−=

= + + + +

+ + + +

+ − − −

+

(38)

3). State Feedback: By setting 1 ( ( ) )( )

u b ua

ξξ

∗= − + (39)

the system (35) can be transformed into

1 2

2

3

1

,

,

( ),.

u

y

ξ ξξξ η ξ

ξ

∗

=

=

==

���

(40)

The heading control system could be partial feedback linearized due to the relative degree k = 2(k < n). Consulting some parameters and hydrodynamic coefficients of AUV, we can get

0.8436 1.3145 0.0167 0.25261.0498 0.6152 0.0812 0.5550

A− − −� �= � �− − −� �

(41)

0.06130.0296

B−� �= � �

� � (42)

According to the formulae (36) and (37), we can get the coefficients for ( )a ξ and ( )b ξ . Then the state feedback control law for AUV heading control system can be can be written as

480

2

2 232 4

0 0

2 3 0 2 320 0

1 ( ( ) )( )

1 349.8( 0.53710.0613

20.7091 4.12070.1387 )

u b ua

u u

u uu u

ξξ

ξ ξ

ξ ξ ξ ξ

∗

∗

= − +

= − − + −

+ − +

(43)

The next step is to design the new input u∗ based on closed-loop gain shaping algorithm.

IV. DESIGNING THE CONTROLLER BY CLOSED-LOOP GAIN SHAPING ALGORITHM

Ref.[12], it is shown that the closed-loop gain shaping algorithm is significance of the project. The slope (high-frequency asymptotic lines slope) of the closed loop transfer function is 40(dB/dec), and the sensitivity function curve can structured the spectrum curve of the inertial system which has the largest singular value 1 (that is T=1/( T1s+1)2 ) . Compar-ed with the typical second-order oscillation link, T has an equivalent damping as 1, which ensures that the spectrum can not have peak.

By the controlled object G and the bandwidth frequency of the closed-loop system, the coefficient row vector of controller can be given as

11

1

2 ( 1)2

K TG T s s=

+ (44)

then u Kξ∗ = − , the state feedback control law can be completed by

[ ]

2

2 23 2 32 4 2

0 0 0

0 2 3 1 2 30

1 349.8 20.7091( 0.53710.0613

4.12070.1387 ).

uu u u

u ku

ξ ξ ξ ξ

ξ ξ ξ ξ ξ

= − − + − +

− +(45)

Where 1 eξ ψ= , 2 rξ = , 2 23 21 0 11 0b u r b u vξ = + , eψ , r , v could

be measured by the Octans (attitude sensor). Thus, 1ξ , 2ξ , 3ξcan be computed.

Let ru δ= , then the formula (45) can be written as

[ ]

2 2 2 221 0 11 02 4

0 0

2 221 0 11 0 02

0

2 221 0 11 0 3

0

1 349.8( 0.5371 ( )0.0613

20.7091 ( ) 0.1387

4.1207 ( ) )

r

e

r b u r b u vu u

r b u r b u v u ru

b u r b u v k ru

δ

ψ ξ

= − − + + −

+ + −

+ +

(46)

The formula (46) is the heading controller for AUV at the operating point.

V. SIMULATION FOR AUV HEADING CONTROL SYSTEM

Dynamic model of AUV in horizontal plane is used in the design progress for heading controller. In order to verify the controller is useful and validity on the AUV 6-DOF

mathematical model, as well as demonstrate the robust controller has improve the dynamic characteristics and robustness, the disturb model of ocean waves are established. Figs. 2 and 3 show the responses of the wave exciting force and moment when AUV navigating under the conditions of slight sea.

0 3 0 6 0 9 0 1 2 0-1 2 0 0

- 9 0 0

- 6 0 0

- 3 0 0

0

3 0 0

Forc

e(N

)

T im e ( s )

Fig. 2 Responses of the wave exciting force

0 3 0 6 0 9 0 1 2 0- 2 0 0

0

2 0 0

4 0 0

6 0 0

Mom

ent(N

m)

T im e (s )

Fig. 3 Responses of the wave exciting moment

0 3 0 6 0 9 0 1 2 0

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

Hea

ding

(Deg

ree)

T im e( s)

u = 2 k n u = 4 k n u = 6 k n

Fig. 4 Response of headingAUV navigates at the speed of 2.0 kn, 4.0 kn, 6.0 kn,

respectively. Meanwhile, the coefficient row vectors can be computed based on closed-loop gain shaping algorithm (44) at the different speed: K2.0=[2.0316 8.6438 0.0753], K4.0=[1.5326 6.1683 0.0429] , K6.0=[1.3497 5.0346 0.0377] .

By the feedback control law (46), the heading control curves obtained at the different speed of AUV under the conditions of slight sea.

At first, it is known that a heading controller is designed based on Taylor linearization technique and neglected its nonlinear factors in [4]. Fig.4 and 5 show the responses of heading and rudder, respectively. Before the time 30(s), the heading instruction is 0 (degree), after the time 30(s), the heading instruction is 60 (degree). Where the solid lines, dash lines and dash-dot lines represent the speed of AUV at 2.0 kn, 4.0 kn, 6.0 kn, respectively.

481

0 30 60 90 120

-10

0

10

20

30

40

Rud

der(

Deg

ree)

Time(s)

u=2 kn u=4 kn u=6 kn

Fig. 5 Response of rudder Then the simulation results under the heading controller,

which designed based on linearization via state feedback in this paper, are presented as Fig.6 and 7. We can see that the heading error and stabilizing time for heading control system are reduced by the method in Fig.6, compared to the Fig.4.

0 30 60 90 120

0

10

20

30

40

50

60

70

Hea

ding

(Deg

ree)

T ime(s)

u=2 kn u=4 kn u=6 kn

Fig. 6 Response of heading

0 30 60 90 12 0

-10

0

10

20

30

40

Rud

der(

Deg

ree)

T ime(s)

u=2 kn u=4 kn u=6 kn

Fig. 7 Response of rudder In order to get a better comparison, Table 1 gives an

overview of the simulations. Where 0u means the navigation speed of AUV , tr means rise time, ts means stabilizing time, σ means the maximum overshoot.

TABLE I Overview of the simulations

u0(m/s)

Taylor linearization technique

Linearization via state feedback

tr(s) ts(s) σ (%) tr(s) ts(s) σ (%) 2.0 16 40 6.7 18 27 4.5 4.0 12 16 1.8 11 13.5 1.5 6.0 11.5 15 1.1 12 12 0.1

From the results it can be learned that the controller designed based on linearization via state feedback can stabilize the system more rapidly than the controller based on Taylor linearization technique. Changing the stabilizing time ts from 40 to 27, 16 to 13.5,15 to 12 (sec) at the speed of

2.0, 4.0, 6.0(m/s), respectively. Meanwhile, it also reduces the maximum overshoot from 6.7 to 4.5, 1.8 to 1.5, 1.1 to 0.1(%), respectively. So the results proof that the controller designed based on linearization via state feedback can improve the dynamic characteristics and robustness of the AUV heading control system in a certain extent.

VI. CONCLUSION

In this paper the design of a control law for AUV heading control system based on linearization via state feedback is introduced. The dynamic model of AUV has the nonlinear terms, which can’t be ignored by Taylor series expansion. Simulations show that the proposed techniques can reduce the negative affects of the nonlinearities, and the controller based on it can improve the dynamic characteristics and robustness for the AUV heading control system in a certain extent under the influences of ocean waves. However, the techniques have to require a known system model and analytically get a nonlinear transformation. For a complex system, when the model and parameters are unknown or unclear, and a nonlinear transformation is difficult to find, the method can be used to get a better approximate solution.

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