Modelling of Ship s Motion Using Artificial Neural Networks

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1 Modelling of Ship s Motion Using Artificial Neural Networks BOGDAN ZAK, ZYGMUN KIOWSKI, JOZEF MALECKI Department of Mechanical and Electrical Engineering Polish Naval Academy Gdynia, ul. Smidowicza 69 POLAND Bzak@amw.gdynia.pl Abstract: - In the paper using of the artificial neural networks for determination of coefficients of state equations of ship s motion in a horizontal plane is presented. he recurrent optimisation network is used to identify parameters of the ship s dynamics. A structure and the operating principle of the network and results of computer simulation of ship s motion along a desired trajectory are described. Values of state variables generated by a differential linear model and a neural model are inserted. Key-Word: - Ship, Neural Network, Modelling, Identification, Control 1. Introduction A problem of mathematical modelling of dynamic objects is one of the most important task in during designing of control systems. heoretical and practical researches were focused on problems of mathematical modelling and especially on methods of parametric identification were developed. hose methods were very efficient for both linear statically and dynamical objects but not for non-linear ones. In recent years the neural networks have been he results of research were many methods which allow to model the linear both static and dynamic object but non-linear object because of its difficulty were modelling only by approximation methods. In this case to model the non-linear object in last year more often is use the neural network. he attractive of use the neural network in modelling is approximation of any curve and to tune the structure basely on experimental and another data.. Mathematical model of ship s dynamic Dynamical equations of ship s motion regarded as control object can be formulated as follow [3]:. ( x = Ax( + Bu( + C( x. (1) where: u( control vector, x( state vector, A state matrix, B control matrix, C(x, vector of external disturbances. he equation (1) in discrete form can be written as: X [( k + 1) ] = Φ( ) X ( k ) + G( ) u( k ) + N ( ) C( x, () where: Φ() state matrix, G() control matrix, X(k)- state vector, N() disturbance matrix, u(k) control signal, but this value are defined as: 3 n 3 n Φ ( ) = I + A + A + A A (3)! 3! n! where: I identity matrix, 3 n + 1 G( ) = IB + AB + A B A B,! 3! ( n + 1)! N ( ) = I + A + A! A 3! n (4) n + 1, ( n + 1)! n (5) Sequence of control signal u(k) must take into consideration the physical limit of maximal values of stern plane saturation and stern angel speed u k 35. (6) ( ) 3. A Neural network for modelling ship s dynamics A task of the neural network is to calculated the values of matrix Φ() and G(), in such way that the neural model of ships will behave as real ships described by equation (1). he identification is

2 made by minimisation the energetic function F(φ,g) which is defined for this model as below: F(φ,g) = 0.5 ε Τ ε F(φ,g) = ε 1 + ε + ε 3 + ε 4 + ε 5 + ε 6 + ε 7, (7) he total error ε is a difference between values of the state vectors x o - generated by analytical model and state vector x m - generated by neural model, for every step of an iteration: ε = x o - x m = [ε 1, ε, ε 3, ε 4, ε 5, ε 6, ε 7 ] Τ (8) After substituting equation () and (8) the energetic function F(φ,g) can be written as follows: F(φ,g) = 0.5 ( x o - φx - gu) ( x o - φx - gu) (9) A gradient method is used to minimise this function. After calculation the partial derivative of F(φ,g) by φ and g, and equal them to zero we have the mathematical prescription which is a rule of teaching the neural network: φ g ij [( k + 1) ] = φij ( k ) Uε ix j ( k ) [( k + 1) ] = g ( k ) Uε u( k ) ij ij i (10) where U is the coefficient of teaching from partition of 0 to 1. his coefficient is defined experimental. he equation (10) describe the method of modification of values of coefficients φ ij and g i that guaranty the minimisation of energetic function in every step of iteration. A such method assures that parameters of neural model will approach to the parameters of identified object. In the figure the elements of Hopfield s recurrent neural network are presented. he basic elements are: block of data selection, block of preliminary solution, block of improvement of solution, block of comparison vector and block of parameters modification. A block of data selection is analyses state vectors and eliminates the vectors which are restricted, because this vectors has the false information which not permit to precision identification of matrix Φ, G.. PARAMEERS MODIFICAION φ i g PRELIMINARY SOLUION DAA COLLECION x o ε IMPROVEMEN OF SOLUION x m COMPARISON VECORS x o i x m. x o BLOCK OF DAA SELECION MAHEMAICAL MODEL OF SHIP Φ, G x o [(k+1)] Fig. Hopfield s recurrent network u(k ) EACHING ALGORYHM NEURAL MODEL φ, g x m. [(k+1)] NEURAL NEWORK Fig.1 he teaching of neural network for of ship s move. ε Block of preliminary solution calculates values of φ, g of the neural model s approximation. It is realised by analytic solution of matrix equations based on selected status vectors from the block of data selection. Receive in this way neural model is improved. Block of improvement of solution consists of 8 neurones which works on loop back. he results operation this block are new parameters of the status vector x m. he parameters are new input data for the neural network for next iteration. In a block of parameters modification take effect modification importance network φ ij and g i, according to the rule of teaching. Values of ε is the basic modification importance.

3 Block of comparison vectors calculate total error ε between the state vectors x o - generated by analytical model and state vector x m - generated by neural model, for every step of an iteration. 4. he results of model research For a ship described in equations () was generated 500-elements set of state vectors being input data for neural network. Verification of the neural model for ship standard manoeuvring trials was performed; for Kempf manoeuvres zig-zag test and spiral test. he testes were made for the neural and mathematical models of minesweeper for speed 15 knots. he trajectories and coordinates vector states for both models for Kempf test are presented in Fig.3 to Fig.10 without disturbances and Fig.11 to Fig.18 with disturbances (black colour mathematical model, red colour neural model). Fig 5. Speed of ship for Kempf test Fig.6 Engine s rototional speed for Kempf test Fig.3 rajectory of ship for Kempf test Fig.4 Course of schip for Kempf test Fig.7 Speed turn of ship for Kempf test.

4 Fig.11 rajectory of ship for Kempf test Fig.8 Drift angel of ship for Kempf test Fig. 1 Course of ship for Kempf test Fig.9 Angel put out of steer for Kempf test Fig.13 Speed of ship for Kempf test Fig.10 Speed put out of steer for Kempf test

5 Fig.14 Engine s rototional speed for Kempf test Fig.17 Angel put out of steer for Kempf test Fig.15 Speed turn of ship for Kempf test. Fig.18 Speed put out of steer for Kempf test Fig.16 Drift angel of ship for Kempf test Fig.19 rajectory for spiral test without disturbances.

6 5. Conclusion he results of simulation show a big efficacy of the presented neural network for modelling of ship s dynamics. he relative error between state vectors generated by ship and its neural model is smaller then 1%. In presented method exist the strong dependence between preciseness neural model and the value coefficient teaching U, that s why important problem is strategy selection and modification value coefficient U in the iteration process Fig.0, rajectory for spiral test with disturbances. In Fig.19 to fig. are presented trajectories for spiral test. he test were made for speed 15 knots for without and with disturbance. In two first figures lay the rudder was 35 o, and in the next 0 o. References: [1] W.. Miller, R.S. Sutton, P.J Werbos, Neural Networks for Control, he MI Press, 1990 [] K.S. Narendra, K Parthsarathy, Identification and Control of Dynamical Systems Using Neural Networks, IEEE rans Neural Networks, Vol.1,No.1,1990, pp [3] J.Garus, J.Ma³ecki, B. ak, Application of Network Hopfield s for Modelling of Motion Ship, XIII Polish Automation Conference, Poland, Opole, Vol., 1999, pp , (in polish). [4] J. urada, Introduction to Artificial Neural Systems, West Publishing Company, USA 199. [5] J.Hertz, A.Krogh, R.G.Palmer, Introduction to he heory of Neural Computation, Addision-Wesley Publishing Company, Inc Fig.1 rajectory for spiral test without disturbances. Fig., rajectory for spiral test with disturbances.

Selected method of artificial intelligence in modelling safe movement of ships

Safety and Security Engineering II 391 Selected method of artificial intelligence in modelling safe movement of ships J. Malecki Faculty of Mechanical and Electrical Engineering, Naval University of Gdynia,