Robust adaptive path following for a nonlinear third

2010 7th International Multi-Conference on Systems, Signals and Devices

Robust adaptive path following for a nonlinear third order Nomoto's ship model M. TAKTAK-MEZIOUl, J. GHOMMAMl, N. DERBELI 1 Intelligent

Control Design and Optimization of Complex Systems Unit, ENIS, Tunisia

manel.taktak©yahoo.com, jghommam©gmail.com, n.derbel©enis.rnu.tn Abstract-In this paper, an adaptive backstepping method is used for a ship course tracking control. This has been done by applying a reference signal derived from a LOS algorithm. The ship model is described by a third order nonlinear model whose parameters are unknown. The control design uses estimate values of the unknown parameters of the system. Then, adaptive laws of the estimation of these values have been proposed. The stability of the controlled system has been ensured by the use of a Lyapunov function. Simulation results show the effectiveness of the proposed approach and the designed controller can be used to the ship course tracking with good performances. Key-words- Adaptive backstepping control, nonlinear con­ trol, adaptative control, Line Of Sight.

and Fossen [4], [7] and calculations have been developed in [11]. This shows that it is an effective method for marine vehicle navigation and this is thanks to its simplicity, its low cost and flexibility to sudden changes in the desired specifications for the followed path. The reminder of this work is organized as follows. In section 2, the formulation of the problem is presented. In section 3, we will present the LOS algorithm for the path following problem. In section 4, the adaptive control based on backstepping method is described. Simulation results illustrating the effectiveness of the proposed method and the convergence to the desired path are given in section 5.

1. INTRODUCTION

II. PROBLEM FORMULATION

Scientific development in various areas of science and engineering techniques pushed automatic research to study more and more non-stationary systems including variable parameters. Because of using a command with fixed parameters can lead in some works to a failure and cannot maintain or achieve "good" performance indices that are increasingly required in several industrial applications. In this paper, we focus on this paper on- the control law with parameters that can be adjusted during on line (i.e. use an Adaptive command structure), and this using the a well known technique of adaptive backstepping [1]. The adaptive version of the backstepping technique includes the ability to summarize the controllers for a large class of nonlinear systems with known structure and uncertain parameters [8]. Although the robustness of the control law has been intensively studied in the case of linear systems [3], many other robust controllers were involved in the case of nonlinear systems [13]. In this paper, we are studying the control problem of a third order Nomoto model. The study unfolds in two steps: first, an adaptive backstepping procedure is applied to an uncertain strict feedback form which presents the model of the ship, to track a given reference asymptotically and estimate the uncertain parameters which characterize the ship's model. Second, the reference signal is supplied along with the new position to be attained by the ship. To ensure this reference, a LOS algorithm is therefore presented. The main idea of the LOS algorithm is to give an intuitive understanding of the behavior of the ship: This consists on steering the actual ship's heading (i.e. 'lj;(t)) to a desired angle called 'lj;zos (t). The guidance by the LOS algorithm has been also treated by Healey, Lienard

A ship model described by the third order Nomoto model is given by [7]:

978-1-4244-7534-6/1 0/$26.00 ©2010 IEEE

1 H �(t) Tl T2

(

�2 ( T3J(t)

T

{

) +

c5(t) (1)

)

where 'lj;(t) is the ship course and c5(t) is the rudder angle. Based on [13], we propose the following change of variables:

Xl(t) X2(t) X3(t) X4(t)

= = = =

'lj;(t) �(t) ;(;(t) � cc5(t) c5(t)

with the initial conditions Xl(O) �cc5(O).

=

X2(O)

(2)

=

0 and X3(O)

=

As a result, the system can be described by the following state equations:

(3)

Assumptions a, b, d and e are unknown variable parameters in the design taken as:

(4)

2010 7th International Multi-Conference on Systems, Signals and Devices

and H [X2(t)] x�(t) + X2(t) is a nonlinear function and u(t) is the controlling input. =

Ship

I +

I

I

X,Y

Ylos �Yk-1 Xlos �Xk-1

l

u(t)

The desired path consists on the collection of way-points in a table of points, but the LOS position is now located on the segment linking the previous and the current point, given respectively by Pk-1 and Pk. Referring to [ 12], two equations are used to calculate the LOS position denoted Plos which are: Yk �Yk-1 Xk �Xk-1

( 6)

I

(7)

'l/Jlos

Fig.

1.

I I

I

LOS

where the subscript k is relative to the current way-point, while k �1 is relative to the previous way-point. To compute the LOS angle 'lj;los (t), we use the following expression:

I

'lj;los (t)

Block diagram of the controlled system

Control objective: The task consists now on designing the control law u(t) such that for a given 'lj;los (t), we obtain lim I'lj;(t) � 'lj;los (t) 1

t->oo

=

where ii, b, d and e, the estimate values of a, b, d, and are bounded.

e,

IV. CONTROL DESIGN

Step 1: Using ( 2 ) , in this step new state variables are introduced which are defined by the following equations X1(t) � 'lj;d(t) X2(t) �(Y1(zd X3(t) � (Y2(Zl, Z2) X4(t) � (Y3(Zl, Z2, Z3)

III. L OS ALGORITHM The principle of the LOS algorithm is an intuitive understanding of the behavior of a ship [ 6]. The principle is that, if a vessel may maintain its ship course 'lj;(t) aligned with the angle named 'lj;los (t), then the convergence to the desired position is achieved. In short, the geometric exercise to manoeuvre the vessel can be re-formulated as follows: force the position p of the ship to converge to a desired path by forcing its angle 'lj;(t) to converge to 'lj;los (t) (5).

(8)

where x and Y define the current position of the ship. The atan2 function ensures that 'lj;los (t) E [�7f, 7f].

(5)

0

arg [(Xlos � x) + j (Ylos �y )] arctan 2 (Ylos �Y, Xlos �x)

(9) (10) (11) ( 12 )

where 'lj;d(t) is the desired ship course, and (Y1(Zl), (Y2(Zl, Z2), and (Y3(Zl, Z2, Z3) are stabilizing functions. The derivative of the first equation in the sub-system ( 2 ) is given by: (13) The role of (Y1(zd is to stabilize equation (13). This has been done by considering the first Lyapunov function (14) Its derivative, with respect to time, is given by:

Plos circle of acceptance for k

In order to ensure the convergence to 0 of the error Zl, (Y1(zd takes the following form:

( 16) Finally, the derivative of V1(Zl) is expressed by: (17 ) Substituting ( 16) into ( 13 ) gives:

Fig.

2.

Principle of the manoeuvre by the

LOS

algorithm

(18)

[11]

where k1

>

0 is a tuned parameter.

2010 7th International Multi-Conference on Systems, Signals and Devices

Step 2: In this step, the derivative of the second equation (10) takes the form

Substituting and (31) into (30) gives:

. V3

(19)

2 2 2 -k1 Z1 - k2Z2 - k3Z3 + z3Z4 l 1 -a ,I a + X3(t)Z3 - b ,i b + H (X3(t)) Z3 (32 )

(20)

In order to ensure the stability of the system, and th� negativity of V3, expressions of the adaptive laws a et b take the following forms:

where a2(zl, Z2) is the second stabilizing function The derivative of a1(Zl) is

=

(

)

)

(

The second Lyapunov function is defined by

V2(Zl, Z2)

=

1 V1(Zl) + Z22 "2

(33) (21)

and its derivative takes the form:

V1(Zl) + Z2Z2

V2

-k1 Zr + Z2 (Zl + a2- (h(Zl)) + Z2Z3

(22)

In (22), we made the following substitution to ensure the negativity of V2(Zl, Z2):

(23 )

( 34 ) Step4: In the final step, our aim is to determine expres­ sions of adaptive laws d et �, and then, the expression of the adaptive control u(t). In fact, the global Lyapunov function is given by 1 2 1 -2 1 2 V3 + eZ4 + '3-1d + '4-1e"2 "2 "2 The derivative of the fourth state variable is

V4

=

( 36 )

In fact, the obtained expression of a2(zl, Z2) is

a2(Zl, Z2)

=

-k2Z2- Zl + (h(zd

(24 )

Finally, V2(Zl, Z2) is defined by

(25 ) Substituting (24) into (19), we obtain

(26 ) where k2 > 0 is a tuned parameter chosen later. Step 3: The derivative of the equation (11) is given by b H (X2(t))- aX3(t) + Z4

+ a3(zl, z2, Z3)- Oc2(Zl, Z2)

Substituting ( 36 ) into the expression of V4(Zl, Z2, Z3, Z4), we obtain V3 +eZ4z4-131dd-'41�e 3 . ki zT -131dd-'41�e i=l ( 37 ) +Z4 [Z3- dX4(t) -eOc3 + u(t)]

- z:=

From (37), we pick the control law u(t) as u(t)

(27)

(28 )

The third stabilizing function a3(zl, Z2) which ensures the stability of the system, has therefore the following form:

Oc2(Zl, Z2) + iiX3(t) + bH (X2(t)) -Z2- k3Z3

a

-,1X3(t)Z3 b� -,2H (X2(t)) Z3 =

=

e

V2 + Z3Z3-'l1aa -li1bb l 2 2 aa- -'2- bA b + Z3Z4 -k1 Zl - k2Z2 -'h-1� +Z3 (Z2- b H (X2(t))- aX3(t) + a3- Oc2) (30)

=

( 38 )

( 39 )

=

where 11 and 12 are constant. The derivative of V3 is given by:

a3

-k4z4- Z3 + dX4(t) + �Oc3

� -,3x4(t)z4

-Zl- k2z2 + O:l(Zl)

V3

{

=

with adaptive laws given by

where the time derivative of a2(zl, Z2) takes the form

3 (kr - I)Zl- (k1 + k2)Z2 + 1/J� \ t)

( 35 )

( 31 )

=

-,4Oc3z4

Theorem 1: Consider the nonlinear model of the ship given by equation (1), with uncertain parameters a, b, d and e, having the bounds (4). Applying the control law u(t) given by (38) and the update laws (39) of estimated param­ eters ii, b, d and e, states (Xl, X2, X3, X4) are asymptotically stable and (ii, b, d, e) are bounded. Proof: Substituting in ( 37 ) the control law u(t) given by ( 38 ) and ( 39 ) , we can write: 4 1 2 1 ki zi - d 13 d + X4(t)Z4 - e(14 � + Oc3Z4) V4 i=l 4 ki zi2 (40) i=l =

z:=

(

)

- z:=

Then, it is obvious that V4 < o. Thus, we conclude, using Barbalat's lemma [5], that (Xl, X2, X3, X4) and (ii, b, d, e) are bounded, and therefore, Zl, Z2, Z3 and Z4 converge to o.

2010 7th International Multi-Conference on Systems, Signals and Devices

V. SIMULATION RESULTS In this section, we consider numerical simulations to show the effectiveness of the proposed controllers. Con­ troller gains and adaptive gains are chosen as: kl = 0. 51, k2 = 0.4, k3 = 0. 5, k4 = 0.4, '11 = 0.4, '12 = 0.2, '13 = 0.25, and '14 = 0.3. Initial conditions are [Xl(O) X2(0) X3(0) X4(0)]= [� 000]. In figure 3, we can see a good convergence of the ship course 1/;(t) to the angle 1/;zos (t) for which is obvious that the error (1/; (t) - 1/;zos (t)) converges to O. This allows us to say that the principle of the LOS algorithm is successfully achieved for this system. In figure 6, we observe the convergence of the error variables to 0, and figures 5 and 7 show that state variables and estimate parameters are bounded. In figure 8, it is clear that estimation errors are also bounded. The control input u(t), in figure 4, shows a good convergence to the origin yielding 1/;( t) = 1/;zos (t). Figure 9 shows that the vessel, in the plan (x, y) realizes a good tracking.

"

"

1.5

0.5

r

0.5 -0.5 -0.5

0

100

200 timers)

300

400

"

0

100

200 timers)

300

400

300

400

300

400

300

400

"

1.5

0.5

0.5

-0.5

-0.5 -,

-,

-, 100

0

200 time[s)

300

400

-1.5

200 timers)

100

0

8

Fig. 5.

6

1.5

States Xl, X2, x3and X4

4

2

0

� �

�

�

"

"

-0.2 0.5

-0.4 -0.5

-1

�

-0.6

0:-------:c,oo�--=2OO�--:::300::---C400 timers)

-0.8

�

�

3.

l

I

100

200 timers)

300

-1

(a)'!f;(t) and '!f;los(t), (b ) Error ('!f;(t) - '!f;los(t))

-0.5 0

100

200 timers)

300

400

-1

0.5

0.2

I

-0.4

0

100

Fig.

-5

0�---;500:----;'=OO---";!;50:---='OO;---c:!'50-:::300=---='50�---:!400 timejs]

Fig.

4.

Control input u(t)

l

I

II

I

-0.5

-0.6

-,

200 timers] "

-0.2

-,

100

0

"

0.4

-0.8

I ,

I

-0.5

0.6

Fig.

l

0.5

200 timers]

6.

300

400

-1

0

100

Error variables Zl, Z2, Z3 and Z4

200 timers]

2010 7th International Multi-Conference on Systems, Signals and Devices

VI. CONCLUSION 0.01,---_-_-_----,

0.05,---_-_-_----,

-0.01

�

�

-0.05

e

-0.02

_O.'

-0.03 -0.15

-0.04

L--,""00--200�-300=-----,J4OO

-o·05 0

-0.2

time[s)

--,J L-=----=o------,-,_ , 300 100 400 0

.

::s]

0.1 0.08

1.5

0.06 0.04

:;;:

0.02

0.5

-0.02 -0.04

0

100

300

200

timers)

400

-

O.5

0�-,:-::oo----;;:200=--,30--:!.4OO time[s]

In this paper, we have considered an uncertain model of a nonlinear third order Nomoto's ship, for which, we applied the adaptive backstepping technique to regulate the actual orientation of the ship, with a desired angle provided by a LOS algorithm. This technique has been applied providing an effective tool, for nonlinear systems, allowing to built directly and systematically a control law that ensures the stability of the closed loop system. Simulation results show the effectiveness of the proposed approach. Further, research directions will deal with NN algorithms combined with the backstepping procedure to address the trajectory tracking of such model. RE FERENCES [lJ Astrom K.J and Wittenmark B., Adaptative Control.

Reading, MA: AddisonWesley, 1989

Fig.

7.

Estimation parameters

art), b(t), d(t)

and

[2J Brockett R. W., Asymptotic Stability and Feedback

e(t)

0.2 0.1

0.15

0.08 "",-.

..!!.

0.1

e

0.06

0.05

0.04 0.02 100

300

200

tirne[s]

400

-0.05

0

100

time[s]

200

300

400

100

time[sj

200

300

400

1.8

0.05

1.6 1.4

:;;:

"

� 1.2

0

0.8 0 .•

-0.05

0.4 -0.1

0

Fig.

100

8.

300

200

time[s]

Parameter errors

400

0.2

0

art), b(t), d(t)

1200

1000

800

Fig.

9.

The ship tracking

and

e(t)

Stabilization, in Differential Geometric Control Theory . Birkhiiuser, Boston, USA, 181�191, 1983 [3J Feurer A and Morse A.S., Adaptive Control of Single­ Input, Single-Ouput Linear Systems. IEEE Trans. on Automatic Control. 23 ( 4 ) : 557�569, 1978 [4J Fossen T.I., Marine Control Systems. Guidance, Navi­ gation, and Control of Ships, Rigs and Underwater Ve­ hicles. Trondheim, Norway: Marine cybernetics, 2002 [5J Fernando A.C.C., Fontes and Lalo Magni. A General­ ization of Barbalat's Lemma with Applications to Ro­ bust Model Predictive Control, Proc. of the MTNS'04 Mathematical Theory of Networks and Systems, Louvain, Belgium, 2004 [6J Ghommam J., Mnif F. and Derbel N., Path Following for Underactured Marine Craft using Line Of Sight, Int. Conf. on Machine Intelligent, Tozeur-Tunisia, 2005 [7J Healey A.J and Lienard D., Multivariable Sliding­ Mode Control for Autonomous Diving and Steering of Unmanned Underwater Vehicles. IEEE Journal of Oceanic Engineering, 18 ( 3 ) : 327�339, 1993 [8J Kanellakopoulos 1., Kokotovic P.V. and Morse A.S., Systernatic Design of Adaptive Controllers for Feedback Linearizable Systems. IEEE Trans on Automatic Con­ trol. Vol.36 ( 1l ) , 1241-1253, 1991 [9J Kokotovic P. and Arcak M. Constructive Non Linear Control: A Historical Perspective. Automatica. 37 ( 5 ) : 637�662, 2001 [1OJ Krstic M., Kanellakolulos 1. and Kokotovic P.V. Non Linear and Adaptive Control Design. New York, Wiley, 1995 [11J Morten B., Nonlinear Maneuvering Control of Under­ actuated Ships. Master of Science Thesis, Norway Uni­ versity of Science and Technology, Departement of En­ gineering Cybernetics, Trondheim, Norway, 2003 [12J Paquin L.-N., Application du Backstepping a une Colonne de Flottation. Memoire, Departement de Mines et Metallurgie, Faculte des Sciences et De Genie, Uni­ versite Laval, 2000 [13J Witkowska A., Tomera M. and Smierschalski R., A BacksteppingApproch to Ship Course Control. Dans Int. J. Appl. Math. Comput. Sciences., 17 ( 1 ) : 73�85, 2007