IMPROVING “SIMUSUB” CODE

MANOEUVRABILITY OPTIMISATION FOR THE NAVANTIA S-80 SUBMARINE PROGRAMME

4. IMPROVING “SIMUSUB” CODE

Warship 2008: Naval Submarines 9, Glasgow, UK

The “SIMUSUB” code was designed in order to analyse qualities of primary interest for operation submarines in the coupled horizontal and vertical planes:

The ability to hold a straight course with minimum amount of rudder activity

The ability to turn tightly

The ability to initiate and change course or change depth quickly

So, in order to answer these questions, the simulator must have the possibility to analyse in a programmed way different standard manoeuvres in the horizontal and vertical plane such as zig-zag tests, turning circles, meander tests, stopping tests depth and heading changes, overweight’s reaction, etc… In a second phase of development of the code, and taking into account the importance of the emergency reaction against rudder jamming or uncontrolled flooding, the code was implemented to simulate the auto-pilot reaction including the normal and emergency blowing of its ballast tanks.

The general non-linear DTNSRDC equations of motion were implemented as the mathematical representation of the six degrees of freedom for the physical S-80 Class behaviour. In the final part of the article, figures related with the structure and presentations of this code are shown.

Figure 10 shows the algorithm block diagram to integrate the named non linear equation.

Figure 11 shows the Autopilot control algorithm block diagram, and some details about the main block are given in the subsequent paragraphs.

Figure 13 shows the algorithm block diagram for blowing and some detail about physical assumption for the non-linear-related integration.

Figure 14 shows a view of the interface running in a PC- station of the current “SIMUSUB” code.

Warship 2008: Naval Submarines 9, Glasgow, UK

The “hydrodynamic derivatives” were calculated using dedicated software by the Navantia Cartagena Preliminary Ship Design Dept. as:

m`: 1,71*10-2, kx: 0.030, ky: 1.047, kz: 0.849, k`y:

0,732, k`z: 0,932, I`y: 8,70*10-4, I`z: 8,73*10-4, X`u`: - 5.15*10-4, Y`v`: -1.80*10-2, Z`w`: -1.46*10-2, M`q`: - 6.37*10-4, N`r`: -8.14*10-4, M`w`: -1.40*10-3, N`v`:

6.80*10-4, Y`r`: -8.30*10-4, Z`q`: -7.00*10-5, Z`w:- 2.46*10-2, M`w: 6.26*10-3, Z`q: -7,60*10-3, M`q: - 3,46*10-3, Y`v: -5,79*10-2, N`v: -2,30*10-2, m`-Y`v:

1,70*10-2, N’r: -7.44*10-3, Z`Gb: -3,19*10-3, M`Gb:

6,38*10-4, Z`Gs: -6,25*10-3, M`Gs: -2,45*10-3, Y`Gr:

9,03*10-3, N`Gr: -4,06*10-3 where the linear terms Z'w, Z`q, M'q, Y'v, Y'r, N'r are assumed as zero.

Coefficients are only representative of the SSK forward speed manoeuvres when avoiding detachment phenomena under high rudders angles.

4.2 MATHEMATICAL MODEL

Taking as a starting point the general nonlinear DTNSRCDC submarine equations of motion (see [1,3]) and making the basic assumptions: (a) constant surge velocity, (b) small variations in pitch and yaw Euler angles, and (c) some particular geometrical hypotheses on the submarine, we obtain a mathematical linear model in the form

) ( ) ( ) ( ' ) 1

( x t Ax t Bu t

where t is the time variable, A is a 7x7 matrix and B is a 7x3 matrix. The coefficients of these matrices include hydrodynamics derivatives for the SSK model and depend on surge velocity and the properties of the underwater vehicle.

Precisely, A a ¹b, B a ¹c, where

¸¸

¨¨

1 0 0 0

0 0 0

0 1 0 0

0 0 0

0 0 180 ' ' ' 0 0 0

0 0 ' '

' 0 0 0

0 0 0 0

1 0 0

0 0 0 0

180 0 ' ' '

0 0 0 0

180 0 ' ' '

2 2

S S

M L k m M

L Z Z

m N L

k m N

Y L Y m

q yy w

q w

r zz v

r v

¸¸

¨¨

0 180 0

0 0 0 0

0 0

1 0 0 0 0

180 0 2

'180 '

0 0 0

0 180 0

' ' ' 0 0 0

0 0

0 0 0 1 0

0 0

0 0 180 0 ' '

0 0

0 0 180 0 ' ' '

0 4 0 0

0 0

S S U

S S S

u B Z W L Z M u

L M u

m u L Z Z u N u

L N u

m u L Y Y u

B G q

w q w v

v r v

and

0 0

' 180 ' 180

0 0

0 180 0

0 0

180 '

2 0 2

0 2

0 2 0

¸¸

¨¨

L M u L M u

L Z u L Z u L N u

L Y u

b s

S S

G G

For a detailed description of the constants appearing in these matrices we refer to [1]. The state variable is

) , , , , , , ( )

( t v r w q z x

x \ T

where v = sway velocity, r = yaw rate, ȥ= yaw Euler angle, w = component of velocity in z-direction, q = pitch rate, ș = pitch angle, and z = depth. The control variable is

) , , ( )

(t _r _s _b

u G G G

withįr = deflection of rudder, įs = deflection of aft plane, andįb = deflection of the forward plane.

The controllability problem we address in this work follows. For a fixed final time T and given an initial state x₀⁷ and a final state x_T⁷, we wonder if there exists a control variable u = u(t) ,0 t T, such that the solution of (1) is driven from x₀ to x_T at time T, i.e., x(0)

= x₀ and x(T) = x_T.

By using controllability theory one deduces that this problem has a positive answer. Then, the question of numerically computing the control u(t) is in order. Since the linear model (1) is mainly designed to simulate changes of depth, we will focus only on the control of this manoeuvrability. In the next section, we prove that the control term u can be easily computed from the Gramiam matrix that comes from the controllability theory. Then, we show a numerical experiment to compare these results with the more classical approach based on a Linear Quadratic Regulator (LQR).

5. CONTROLLABILITY VERSUS LQR As is well-known (see [5, p. 737]), the linear system (1) is exactly controllable at time T if and only if the controllability matrix

Q_C = [B AB A²B ... A⁶B]

has maximal range. In our situation, this is so. Moreover, the control u(t) is explicitly given by

>

⁽ ⁾

@

)

(t B^*e ^*⁽ ⁾ P T ¹ x e x₀

u ^A ^T^t _T ^AT

where A^*, B^* are the transpose of A and B, respectively, e^AT is the exponential matrix, and [P(T)]^-1 is the inverse of the Gramiam matrix

Warship 2008: Naval Submarines 9, Glasgow, UK

> @

³

dt t QCu t u t x QS t x u J

* *

) ( ) ( ) ( ) ( ) (

³

^TêÂT^BBêÂ^T^dt

P 0

* .

)

( ^*

Once the control u is determined, the state x(t) is obtained in the closed form

³

t s t A

Atx e Bu sds

e t x

0 ) (

0 ( ) .

) (

On the other hand, a LQ controller is designed by solving the optimal control problem

Minimize in u:

subject to the state equation (1). Here QS > 0 and QC 0 are two weighting matrices and x(t) x(t)x_T.The optimal feedback control u (t) has the form

⁽⁾

)

(t Lxt x_T

u 0dtdT

for an appropriate 3x7 diagonal matrix L and being x_T= (0,0,0,0,0,0,z_f) the final state.

Next, we show some numerical results obtained by implementing both approaches in ‘Matlab’ for a depth change of 10m in a time T=200s.

Figure 8: Depth change.

Figure 9: Controls.

6. CONCLUSIONS

With respect to the first part of the paper the main conclusions which can be underlined are:

(1) A first analysis of the mission scenario is a useful guidance to obtain a successful set of control planes for a submarine.

(2) The design decisions shall be quantified initially by using a set of appropriated coefficients related to the manoeuvrability characteristics.

(3) The utilization of a tool like ‘SIMUSUB’ code, simulating the submarine motions in real time, including the recovery manoeuvres and a control system is essential for assessing the design prior to the validation process.

Regarding the second part the conclusions are:

In addition to the classical LQ control strategy, a new approach based on controllability theory has been implemented for the automatic simulation of a depth change manoeuvrability for an SSK submarine. The main differences between both methodologies are as follows:

(4) The controls obtained from the LQ controller appear in a feedback form. So, from a practical point of view it is necessary to complete the control system with a suitable Kalman filter to correct the data of the state provided by the sensors of the submarine. On the contrary, the controls obtained from the controllability theory do not require the use of those because they are found in an explicit form.

(5) No constraints on the controls are imposed in the controllability strategy. This may lead in some cases, for instance for short times, to some unrealistic results with sharp changes of controls and states. With the LQ controller, these sharp changes may be corrected by

Warship 2008: Naval Submarines 9, Glasgow, UK

using appropriate weights in the associated cost. This, however, requires a post-processing work. A similar strategy could be applied to the controllability approach.

For instance, a minimum time interval for which controllability is physically admissible may be easily calculated.

a) Concerning the accuracy of reaching the final state, it is evident that the best strategy is controllability theory (see Figure 8). Nevertheless, the LQ controller can be also designed to improve this property by choosing an appropriate cost functional.

b) As for the optimality of controls, we notice that the controls obtained from the controllability theory are optimal in the L² (0,T;³) norm (see Figure 9). This norm can be considered as a measure of the manoeuvrability energy usage. In this sense, controllability provides the optimal energy usage strategy, so, the minimum energy cost in hydraulics and most reduced acoustic impact if directly linked.

c) Finally, it is important to emphasize that the results in this work easily extend to the case of yawing manoeuvrabilities.

d) As indicated in the abstract, the present work is only a preliminary study on this topic. Many interesting open questions have emerged and scheduled to be analysed.

7. ACKNOWLEDGMENTS

Work supported by Navantia S.A. (special thanks to Dr.

Carlos Merino, Engineering Vicepresident and Dr.

Remigio Díez, Cartagena Engineering Direction Delegate). The third and last part of the document is devoted to develop the works supported by Francisco Periago and Alberto Murillo, Applied Mathematics Professors in the Polytechnic University of Cartagena, UPCt, and Isabel Ainhoa Nieto, Combat System Engineer by UPCt.

8. REFERENCES

1. FELMAN, J., “Revised standard submarine equations of motion”. Report DTNSRDC/SPD- 0393-09, David W. Taylor Naval Ship Research and Development Center, Washington D.C., 1979.

2. UCL, “Submarine Design Procedure.

Manoeuvrability Lecture”, University Collage London, Revised 2002.

3. FERNÁNDEZ-CARA, E. and ZUAZUA E.,

“Control theory: History, mathematical achievements and perspectives”, Bol. Soc. Esp.

Mat. Apl. 2), pp. 79-140, 2003.

4. FOSSEN, T. I., “Guidance and control of ocean vehicles”, John Wiley and sons, 1994.

5. OGATA, K., “Ingeniería de control moderna”, Prentice Hall, 1998.

9. AUTHORS BIOGRAPHY

Jesús Pascual holds the current position of Cartagena Engineering Direction Deputy Delegate and Principal Naval Architect for the S-80 Submarine.

Javier García holds the current position of Cartagena Engineering Delegation General Design Manager. He is responsible for Hydrodynamics and Signatures Management of the S-80 Project.

Domingo Pardo holds the current position of Combat System Engineer at Cartagena Engineering Delegation.

Warship 2008: Naval Submarines 9, Glasgow, UK

[M][dV/dt] = [J][V] + [K][ G ]

*[M] ^-

Dalam dokumen WARSHIP 2008: NAVAL SUBMARINES 9 (Halaman 147-151)

MANOEUVRABILITY OPTIMISATION FOR THE NAVANTIA S-80 SUBMARINE PROGRAMME

4. IMPROVING “SIMUSUB” CODE

) , , , , , , ( )

( t v r w q z x

x \ T

>

@

> @

³

³

³

[M][dV/dt] = [J][V] + [K][ G ]

*[M] -

*[M] ^-