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Submarine Depth Control

NENAD POPOVICH M.Sc. (Eng), SUDEEP LELE, NIRAJ GARIMELLA

School of Engineering

Faculty of Design and Creative Technology

Auckland University of Technology

31-33 Symonds Street, Auckland

NEW ZEALAND

Abstract: -In this paper the mathematical model of the closed loop system: submarine, steering gear

and controller is defined. A preliminary stability analysis has been performed. Several different control

algorithms are investigated for the “real world” conditions and for various environmental regimes. A

nonlinear model of steering gear for overcoming a “derivative” kick was included. Simulink has been

used to test the controller performance and results are given. A disturbance model is proposed and its

effect on the control system is outlined.

Key-Words: -Submarine, Depth control, Non-linear system, Modelling, Simulink.

1 Introduction There are not many papers dealing with the

control problems of the submarine (most

submarines are used for navy and military

purposes, i.e. classified).

Finding a suitable transfer function to describe

the depth-dynamics of the submarine is a

challenging task. Once a suitable transfer

function for the submarine and its steering gear

were obtained, the analytical methods were

used for analysing the system as well as for

designing suitable parameters of the controller.

Several non-linear characteristics also have to

be included in the model using Simulink.

2 Mathematical model Fig.1 shows a submarine dynamic reference

frame [1]. The kinetic equations are defined in

terms of the body-fixed velocities:

surge (u), sway (v), heave (w), roll (p), pitch

(q) and yaw (r).

Fig.1 Submarine dynamic reference frame

An appropriate Simulink model block diagram

for the closed loop system: controller - steering

gear – submarine is shown on Fig.2.

In this model non-linear nature of the steering

gear is also included, as well as some

disturbances caused by waves and sea current.

Fig.2 Submarine depth control system

Proceedings of the 3rd WSEAS/IASME Int. Conf. on Electroscience & Technology For Naval Engineering, Greece, July 14-16, 2006 (pp1-5)

3 Stability analysis The submarine itself is a stable system

(excluding some small range), as seen on Fig.3.

Fig.3 Root locus of submarine

Unfortunately, when the steering gear is

included, a system: steering gear – submarine,

becomes unstable for the whole range of the

gain constants (see Fig.4 where two poles are

always in the right hand side of the “s-plane”).

Fig.4 – Root locus of submarine and

steering gear

Of course, the submarine can not be controlled

by itself and has to be driven by steering gear.

That means a proper control action has to be

chosen and the proper controller parameters

have to be selected to achieve stable system

with the desired dynamic behaviour.

The first Ziegler-Nichols tuning method for the

controller’s parameters was not an option as it

requires no integrators in the system transfer

function (the submarine transfer function itself

has an “integrator” [6]), as well as the open

loop response is expected to be in “S-shape”.

The second Ziegler-Nichols tuning method

insists that just changing the values of the

proportional controller (Kp) in the closed-loop

system response should result in sustained

oscillations, i.e. to find the ultimate (or critical)

proportional gain. And again, in our case it was

not an option, because system is unstable for

the whole range of gain constants (as stated

earlier). The Routh stability criteria gives the

same results [2].

4 Defining controller The principle: “keep the controller as simple as

possible” [3] will lead us to use PD or PI

controller (because the simplest P controller

does not lead to stable system). If a satisfactory

response can not be achieved with those two

types then use PID controller.

In many papers [4], [5] dealing with the marine

vehicle control systems, there are

recommendations for using PD controller if the

steady state error is not a dominant criteria, or

if you already have at least one integrator in the

controlled system (as in our case). In addition,

if the system is unstable itself the use of “D”

component is essential [7]. Generally, an “I”

controller component will slow down the

system dynamics, and destabilise a system,

which is not the best choice for our (from the

“start”, i.e. even without controller) unstable

system. On the other hand, a “D” controller

component will speed up a system, stabilise it,

but cause a considerable amount of lag in the

system response. In the case of step function

for the input, i.e. a rapid change from zero to

the desired depth value, (even for small depth

value) the “D” component will react “too” fast

to this change. It will cause the effect called the

“derivative kick”, where a huge spike of a

relatively large magnitude (14*1013

) is

noticeable (see Fig.5 in Appendix).

Of course, this will be not good for our steering

gear. One way of avoiding a possible damage

of the steering gear is to introduce some non-

linear elements (saturation type) in front of it.

The value of “D” component is good to

maintain at a lower level at most times, because

increasing the value of that component

complicates an already existing problem.

All above mentioned are just recommendations

for the selections of the controller and its


parameters. Just recommendations, nothing

else, especially for our “non-typical”, unstable

and nonlinear control system!

It seems that only simulation can give us right

selection of the controller type and its

“optimal” parameters.

4.1 Linear model Assume a linear system, i.e. no saturations of

the steering gear, relatively small step function

as an input, i.e. a small change in depth, a deep

water conditions, calm sea and without

disturbances: waves, winds or sea current.

Based on that assumption model, several

“educational” trials have been performed in

order to find the most suitable controller type

and its “optimal”, parameters.

The best choice for the controller type is PD

which gives us a satisfactory response:

relatively small overshoot and settling time, as

well as no steady state error.

The recommended values are shown below in

Table 1. The response can be found on Fig.6 (in

Appendix). Response, in PID case is shown on

Fig.7 (in Appendix).

PD Controller PID Controller

Step

Response

Kp = 2 Kd = 1.5 Kp= 2 Ki = 0.66

Kd = 1.5

Table 1 – Recommended controller values

4.2 Non-linear model As stated before the main reason for

implementing the nonlinear components is to

curb the effect caused by the derivative

component i.e. “derivative kick”. In addition,

those nonlinearities have a positive effect on

stability.

The block diagram shown on Fig.2 will help to

understand that concept. The first saturation

block is used as a rudder limiter or a stern plane

limiter. This block helps to limit the movement

of the rudder and protect the steering gear from

damage.

The second saturation block would act as a

rudder/stern plane rate limiter. That means it

limits a fast change in the rudder movement.

When selecting PD controller the values shown

in Table 2 help to maintain the submarine

dynamics at safe level. The responses based on

those values are shown on Fig.8 (in Appendix).

It can be seen that response has a very small

overshoot (less than 1%, almost negligible).

The response for the recommended limits (see

Table 2) in the case of PID controller is shown

in Fig.9 (in Appendix). The overshoot is bigger

than in Fig.8. That verifies our selection of PD

controller.

However, if gradual descent for the submarine

is demanded i.e. a ramp function as an input is

required, then PID controller is a better

selection in terms of zero steady state error.

Further, in the case of shallow water even small

overshoot is not desirable, and one of the

operational criteria has to be a non-oscillatory

response.

PD

Controller

PID

Controller

Step Response Kp = 2

Kd = 1.5

Kp = 2

Ki = 0.66

Kd = 1.5

Stern-plane

limiter

-0.9 to 0.9

(Suggested

range: +0.2

to +1)

-0.5 to 0.5

(Suggested

range: +0.23

to +1)

Stern-plane

rate limiter

-0.8 to 0.8

(Suggested

range: +0.3

to +1)

-0.85 to 0.85

(Suggested

range: +0.6 to

+1)

Table 2 – Recommended values for saturation

blocks

5 Disturbances If the disturbances are present: wind for the

surface (navigation) operation, waves for the

periscope depth (regime) or sea current for the

whole operational spectrum (mode, regimes)

then a different model has to be considered

which will include some or all about mentioned

disturbances. One such model is seen on Fig.2.


Several simulations (trials) have been

performed in the presence of the disturbances.

In the case of PD controller, the response with

some steady-state error is shown on Fig.10 (in

Appendix).

6 Conclusion

Mathematical model of the submarine depth

control system is defined. The analytical and

graphical methods are used to determine

stability of the system. Ziegler-Nichols tuning

methods were unable to find initial controller

parameters due to instability for that unusual

(specific) control system. Non-linear model has

been presented and the simulation by Simulink

has been performed. The results for several

types of controllers in the different operational

modes are given. The presence of the

disturbances is pointed out.

7 Appendix

Fig.5 Derivative kick

Fig.6 Response with PD controller

Fig.7 Response with PID controller

Fig.8 Response with PD controller and

saturation blocks

Fig.9 Response with PID controller and

saturation blocks

Fig.10 Response with PD controller, saturation

blocks and disturbances


References:

[1] Euan MCGookin, Reconfigurable Sliding

Mode Control for Submarine Manoeuvring

[internet] IEEE, 2001. Available from:

http://ieeexplore.ieee.org/ie15/7644/20885/

00968102.pdf.

[2] Ogata, Katsuhiko. Modern Control

Engineering. Prentice Hall, 4th edition,

2002.

[3] CTMS- Control Tutorial for MATLAB and

Simulink, http://wolfman.eos.uoguelph.ca/

~jzelek/matlab/ctms/pid/pid.htm.

[4] Fossen Thor. I. Guidance and Control of

Ocean Vehicles. John Wiley, 1999.

[5] Hausen Anca Daniela, Predictive control

and Identification Applications to steering

dynamics, PhD, 1996.

[6] Klee, Harold. Continuous System

Simulation II. [Internet] University of

Central Florida, Class Notes. Retrieved

on 1st July 2005 from the World Wide Web:

http://classes.cecs.ucf.edu/eel5891/klee/

[7] Nenad Popovich, Synthesis of the ship

automatic steering control system, AMSE

Press, Vol. 4A, p 141-148, 1988.


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