Applicability of Sliding Mode Control in Autopilot Design for Ship ...

IEEE International Conference on Recent Advances and Innovations in Engineering (ICRAIE-2014), May 09-11, 2014, Jaipur, India

Applicability of Sliding Mode Control in Autopilot Design for Ship Motion Control Swarup Das Member IEEE Faculty of Navy Military Institute of Training Girinagar, Pune - 41lO25 Email: sdas [email protected] Abstract-The ship autopilot design is an active area of research and application of optimum controllers leads to saving of time, fuel cost, manpower and provides comforts to the crew as well as longevity to the machinery. However practical implication poses various challenges since a ship is always under unknown parametric uncertainties which are of structured as well as unstructured nature. In this paper a simple yet effective sliding mode controller is designed wherein signum function of the controller is replaced with a smooth switching function within a boundary layer in the vicinity of switching surface to reduce chattering phenomenon. Third order yaw rudder model has been used based on Nomoto's second order yaw rate rudder model. Numerical analysis carried out and the results are presented to show the robustness of the controller. Finally its applicability onboard a naval ship is discussed. Keywords- autopilot; signum/unction; I. INTRODUCTION Control system design has mammoth influence in the marine field in diverse applications like depth control of submarine, course keeping, roll stabilisation, diving etc. Advent of sophisticated controllers for autopilot design contributes for the ship to carry out tasks with reduced manpower, sufficient reliability, and adequate economy and with optimum performance [1]. Over the last two decades, considerable amount of research effort has been seen in the autopilot design of controller for course keeping, course changing as well as ship trekking. Transformation from purely mechanical mode to microprocessor based design of controller led to reduction of journey time in bad weather thereby saving cost, providing comfort to the crew. Thus, design of simple autopilot is required which can adapt itself to the parametric uncertainties due to external disturbances and model inaccuracies. Course keeping and course changing for a ship during maneuver is achieved by control of movement of rudder [2]. The purpose of the autopilot controller design is to control the movement of the rudder to maintain and change the ships course as per desired trajectory. A surface vessel at sea in always under effect of varying parameters which are difficult to estimate accurately due to various environmental disturbances caused by wind, tides, waves, underwater sea currents, change of depth under keel varying over a wide ranges depending on locations on earth and climatic situation depending on time. Measuring these [978-1-4799 -4040-01 14/$3l.00 © 2014 IEEE]

disturbances accurately and getting a precise model is almost impossible. The performance is also affected by ship sailing order such as trim, loading, ballast speed etc. as they modify the hydrodynamic coefficients and finally affecting the parameters of the ship model. Moreover, the dynamic of ship varies widely with the wide ranges of ships which are of various classes depending on the type of tasks they are expected to perform and this has a large role to play in the implementation of autopilot. Various robust control theories have been considered for autopilot design of ship. Although the PID controller is simple, reliable, and easy to construct, they need weather adjustment for different sea condition which require an undesirable work for operator. The state feedback approach gives good result but necessitates the need of precise modeling information. In this paper, Sliding mode Controller based on design of simple sliding surface is proposed which is robust and does not require a prior knowledge of unknown parameters and has the advantages of parametric invariance. In addition, the controller can be used onboard for any class of ship irrespective of different system parameters. Sliding mode control has been used in various fields and has shown extra ordinary advantages over conventional control strategies[12]. In our work, a linearized third order yaw-rudder model based on Nomoto's second order model is presented on horizontal plane. The design of sliding mode controller involves designing a stable surface which is one order less than the order of the system followed by designing controller. The robust exact differentiators, also known as Levant differentiators have been used as observer to find unknown states. The principle objective of the controller is to improve the ship's performance in terms of effectiveness and robustness irrespective of the sailing condition. The rest of the paper is organized as follows: In section II, the mathematical model of the ship under the influence of hydrodynamic effects is explained. Section III deals with the concept of sliding mode control which is divided in to two parts: design of switching surface and design of sliding mode controller. Simulation results showing the application of the autopilot design and analysis are presented in section IV followed by conclusion in section V. II.

MATHEMATICAL MODELLING


Intricate dynamics of ship maneuvering is fully described by coupled non linear differential equations. Conversely, for the design of ship steering autopilot, a simple model with average predictive capability is usually preferred. Also the use of sliding mode controller does not require precise modeling. A three DOF plane motion including surge, sway and yaw motion is considered satisfactory, however roll motion can't be neglected and hence a 4 DOF including surge, sway, yaw and roll motion is used to describe motion of a ship. Fig. 1 represents the Standard Notation followed by the Society of Naval Architects and Marine Engineers notation convention (SNAME. 1950) applied on the ship motion (6 DOF). The equation of motion describing the dynamics of the ship motion obtained from Newton's Law are given by [3]

{I

l:.. .F orces= L: lUoments

For simplicity it is assumed that the only external force and moment are caused by a single rudder angle noted by , where: [Yf N 0 K 0]

M .v+

[Y N 0 K 0]

T

(3)

fl."-) =Yruu+ Y�ii+ YrT+ Y;;T -+ Y� � +Yp. P + Y�p + Y.s5

J-=� =Kpp + Kpp - mifJMq,+ l 1�=:;j; = JIlI'I- + NrT +N

+ NrlP

Npp + N'U ,+ NiIi· + Yao

dt

IIJ

l' uir + K1"'r +.r ... r + KtfD

(1)

d(Iy;)

=

With the assumption of constant forward speed Uo and and upon linearization with respect to straight line and decoupling surge equation, following linear sway-yaw-roll equation is achieved

d( m u ) dt

T

(4)

where Yvv, Yvv, denote the hydrodynamic coefficients; for instance,

(5) indicates the derivative of the sway force Y to the sway speed V evaluated at the reference condition; m is the mass of the ship; and g is the acceleration due to gravity. lxx, and Izz, are the moment of inertia about the x-axis and z- axis respectively. GM is the metacentric height, which indicates the restoring capability of a ship in rolling motion[4 ]. Fig.

I.

Applying Laplace to (4) and rearranging them we get Standard notation and sign conventions for ship motion description

(SNAME, 1950)

The notations as per SNAME are described bellow in tabular form[12]. The motion of the ship including surge, sway, heave, roll, yaw and pitch is described as follows: TABLE I

al1J

bl4> Cl � '

=

=

COORIDANTE FRAME VESSEL PARAMETER

11.1

- Ii) - } v = Yp 2 + y� + Yq'I

uo� v� w

ranslation vel in body fixed frame [m/s1

a2

orce components along body axes [kg]

a3 =

r.0tational vel in body fixed frame [rad/s] [Moments components along body axes [kg.m]

a

..

+ yO' + muo

=Y 0. The function f(t) is assumed to be a Lebesgue - measurable function, the unknown sampling noise f(t)- fO(t) is assumed bounded. The task is to find real-time estimations of fO, DO, . . . .,fOn using only values of f and the number L. The estimations are to be exact in the absence of noises when f(t) = £D(t). Then the nth-order differentiator has the outputs xi where i = 0, 1, . . . . .. , n defined recursively mentioned as follows:[9]

Xl

l

=

-Alx - f (tl l 'l! sign(x - f t}) + ign(x - f(tn

(26)

CD fd

NUMERICAL SIMULATION AND ANALYSIS

Numerical simulations are performed in simulink environment of Matlab software. In the mathematical model it is assumed that only yaw angle is available. The velocity and

X2

Fig. (2) shows the schematic or graphical representation of the equation used to derive Levant Differentiated value. The same logical sequence is used in simulink environment of Matlab to estimate x2 and x3 i.e. velocity and acceleration form xl.

for (J = 0. The value of S should be selected to trade off the requirement to maintain an ideal performance with that of ensuring a smooth control action. IV.

using

A. A. Robust Exact Differentiator

For the simulation purpose, gain K has been taken as 50, and the control law which is of discontinuous nature becomes u(t) = -50sgn(S)+l.99 x2 +2.S6x3

acceleration are Differentiators[ S].

o.

J

Fig. 2. Schematic chart in simulink showing calculation involved Levant DifTerentiator


B. Results of Simulation The simulation was carried out with a step response as disturbance signal and the results of simulations are shown bellow. Both the Fig. (3) & Fig. (4) are results of sliding mode control but in Fig. (4 ) modified sliding surface has been used with boundary in the neighborhood of sliding surface. The difference in the control action in both case is very much evident which clearly indicates use of smooth switching function reduces chattering.

avoid chat-tering and yet allow the controller to maintain its trajectories. The controller will be prohibited rom giving any signal if the desired course is within a boundary, in a way, chattering will be nullified completely. Moment the trajectory goes beyond the minimum set boundary, SMC will be implemented and the course will be corrected. Therefore, this controller can be used with minor modification and little bit of zig zag movement.

Rudder Control loop

Fig. 5. Block diagraqrn of autopilot controller for Ships

V. CONCLUSION

�

"I!

...

-

""

..

...

...

..

-

Fig. 3. Perfonnance of the System with a step response

C. Analysis In theory, though the trajectories slide along the switching function, but in practice, there is formulation of high frequency switching which occurs in the vicinity of the switching surface due to the introduction of signum function in the control algorithm. This is called chattering and happens due to mechanical limitation in any system. This 'zig-zag' curved phenomenon is the cause of several factors such as low control accuracy, heat loss and high wear in instruments as well as instability. A typical block diagram of autopilot controller for course keeping and maneuverability is explained by Fig.(5). Servo

In this paper, a methodology for constructing autopilot for ship based on sliding mode control is described. This controller is very simple in nature compared to modern so phisticated and intelligent controller yet proves to be very effective. Third order yaw-rudder model derived based on Nomoto's second order yaw rate to rudder angle model has been used for the purpose of design of controller. Yaw rate r and r is calculated from yaw angle by using robust exact differentiator. The numerical simulation has been carried out in simulink environment. A smooth switching function is used in stead of signum function inside the boundary layer which is in the vicinity of switching surface. Thereby controller ensures the region inside boundary remains attractive and at the same time chattering is reduced. The accuracy of controller and robustness along with the frequency of control action is decided by the thickness of the boundary.

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Fig. 4. Performance of the System with a step response with modified sliding surface

mechanism of steering machine and rudder movement are restricted by Slew Rate Limitation (SRL) and Saturation effect. These restrictions wont allow sliding mode controller to be successfully implemented on board any naval ships. However, an ad hoc strategy an be devised to completely

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[9]

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