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The availability of adequate system models to reproduce, as faithfully as possible, the actual behaviour of the experimental systems is of key importance. In marine systems, the changing environmental conditions and the complexity of the infrastructure needed to carry out experimental tests call for mathematical models for accurate simulations. There exist a wide number of techniques to define mathematical models from experimental data. Support Vector Machines (SVMs) have shown a great performance in pattern recognition and classification research areas having an inherent potential ability for linear and nonlinear system identification. In this paper, this ability is demonstrated through the identification of the Nomoto second-order ship model with real experimental data obtained from a zig-zag manoeuvre made by a scale ship. The mathematical model of the ship is identified using Least Squares Support Vector Machines (LS-SVMs) for regression by analysing the rudder angle, surge and sway speed, and yaw rate. The coefficients of the Nomoto model are obtained with a linear kernel function. The model obtained is validated through experimental tests that illustrate the potential of SVM for system identification.

Nowadays, the breakthroughs in computer science and the many research works on control engineering and autonomous vehicles call for the availability of adequate mathematical models with which new designs and ideas can be tested in simulation to predict the behaviour of the real systems with high accuracy. Furthermore, due to the high cost of practical implementations and tests, it is of higher importance to have available tools and methods to compute these mathematical models for simulation purposes. In this sense, system identification is one of the highlights among the research topics in engineering and one of the most important stages in control of autonomous vehicles. In the present work, we will focus on the computation of a mathematical model that describes the dynamical behaviour of a surface marine vessel by means of experimental data. This model is computed with a Support Vector Machines (SVMs) technique, the Least Squares Support Vector Machines (LS-SVMs). SVM has shown a great performance in pattern recognition and classification research areas, and it has a potential ability for linear and nonlinear system identification that will be shown throughout this work.

The literature on linear and nonlinear system identification is extensive and covers many areas of engineering research. For a short survey on some essential features in the identification area and a classification of methods the reader is referred to [

In surface and underwater marine robotics, the systems and vehicles employed can have a high degree of complexity. Moreover, the changing environmental conditions and the complexity of the infrastructure needed to carry out experimental tests call for mathematical models for accurate simulations. System identification of marine vehicles starts in the 70s with the works of [

Despite this, in most cases and practical situations, it is possible to employ a simple model to reproduce the dynamics of a vessel in standard operational conditions. For this reason, in this paper the Nomoto second-order model is used, [

Artificial-neural-networks-based techniques have been employed for system identification such as multilayer perceptron (MLP); see, for example, [

In support vector regression (SVR), a nonlinear model is represented by an expansion in terms of nonlinear mappings of the model input, so that they define a feature space that may be of infinite dimension. A convenient feature in SVR is that the optimal model complexity is obtained as part of the solution and does not have to be determined separately. For a survey and a tutorial on support vector regression, the reader is referred to [

An inadequate model identification may yield large prediction errors. It is important to remark that even though the problem of system identification is of great importance and the use of SVM has experienced a great boost and diversification in the last few years, not many results are available on the topic of system identification with SVM for regression yet. Exceptions include a series of interesting results such as the work of [

In this work, a variation of SVM, the Least Squares Support Vector Machines (LS-SVMs) [

With respect to the identification of marine vehicles by using SVM, we can find two interesting references, [

The key contributions of the present paper are twofold: (i) the mathematical model of a surface marine vessel is obtained with the LS-SVM technique from experimental data collected from a 20/20 degree zig-zag manoeuvre; and (ii) in striking contrast to what is customary in the literature, the prediction ability of the mathematical model computed with LS-SVM regression is tested on an open air environment with the real ship. This allows us to explicitly address the connection between the mathematical model identified and the ship, so that the model can be used to design control strategies to be implemented on the real vehicle and to predict its behaviour.

The document is organized as follows. In Section

In this section, we briefly introduce LS-SVM for system identification. As mentioned in Section

The use of LS-SVM for regression is very similar to its use for classification. Following the notation and definitions in [

The above primal problem cannot be solved when

In marine systems, as in a multitude of engineering areas, an adequate mathematical model is essential to simulate and predict the behaviour of a real system with the best possible accuracy. If reliable mathematical models are available and they predict the behaviour of the real systems accurately, then control actions can be planned and tested in simulation avoiding the costly, in time and money, practical tests.

A reliable mathematical ship model, like the Abkowitz model [

For the above reason, simpler vehicle models are commonly used so that, although without modelling all the dynamic features of the vehicle, each of these models reproduces the real behaviour of ships with large accuracy in most of practical scenarios. For the problem at hand, we consider a constant surge speed and we determine a mathematical model that defines the ship steering equations. Among these kinds of models, we can find the Nomoto models [

Given the linear steering dynamics:

Body (O) and earth-fixed (

Following the procedure of [

From (

In addition, we can express the sway velocity in a similar manner:

The main advantage of the Nomoto second-order model is its simplicity and that the hydrodynamic derivatives do not have to be computed explicitly it is possible to define the parameters of the Nomoto model directly from the experimental data. Therefore, we can compute in a fast and simple manner a ship model for control purposes because it is not the aim of this work to know explicitly the hydrodynamic derivatives.

To proceed with the reconstruction of the ship manoeuvring model, the continuous equations of motion, (

The vectors (

The vehicle used for the experimental tests is a scale model of an operational vessel, in a

Main parameters and dimensions of the vessel and the scale ship model.

Parameter | Vessel | Scale ship model |
---|---|---|

Length between perpendiculars ( |
74.400 m | 4.389 m |

Maximum beam ( |
14.200 m | 0.838 m |

Mean depth to the top deck ( |
9.050 m | 0.534 m |

Design draught ( |
6.300 m | 0.372 m |

Scale ship model used in the experimental tests.

The scale ship model, henceforth referred to as the ship, is equipped with an electric motor to control the propeller turning speed and a servo to control the rudder angle. The desired rudder angle and surge speed are commanded through a Wi-Fi connection between the ship and the control station at land.

The training data used for system identification are obtained from a 20/20 degree zig-zag manoeuvre with a sample time of 0.2 seconds and a nominal surge speed of 1 m/s. The zig-zag manoeuvre was executed for 100 seconds; so a set of 500 samples is trained. In Figure

20/20 degree zig-zag manoeuvre. Yaw angle (solid line) and rudder angle (dashed line).

Using the data shown in Figure

Figure

Yaw rate obtained in the zig-zag manoeuvre with the ship (solid line) and in simulation (dotted line).

In the same way, in Figure

Sway velocity measured in the zig-zag manoeuvre with the ship (solid line) and in simulation (dotted line).

Once the Nomoto model is well defined and it fits well with the training data, as shown in Figures

In Figures

Yaw rate obtained in two consecutive turning manoeuvres with the ship (solid line) and in simulation (dotted line).

Sway velocity obtained in two consecutive turning manoeuvres with the ship (solid line) and in simulation (dotted line).

We can notice in Figure

In Figures

Error in the identification of the yaw rate, that is, difference between the yaw rate obtained in simulation and in the real setup.

Error in the identification of the sway velocity, that is, difference between the sway velocity obtained in simulation and in the real setup.

Therefore, it is clear that the mathematical model defined for a surface marine vehicle with LS-SVM provides a satisfactory result which predicts with large accuracy the dynamics of the experimental system.

Now, it is interesting to compare the result of LS-SVM identification with the result that can be obtained by standard identification methods. The latter model is computed using the identification toolbox of MATLAB called Ident; for the documentation of this toolbox, the reader is referred to [

In Figures

Comparison between the experimental yaw rate (solid line), the yaw rate of the LS-SVM identification model (dotted line), and the one of the identification model obtained with the Ident tool of MATLAB (dashed line) for a 20/20 degree zig-zag manoeuvre.

Comparison between the experimental sway velocity (solid line), the sway velocity of the LS-SVM identification model (dotted line), and the one of the identification model obtained with the Ident tool of MATLAB (dashed line) for a 20/20 degree zig-zag manoeuvre.

The main difference between both methods lies in the important fact that the SVM approach can deal with a large number of variables and few data, as shown in [

In this work, the Nomoto second-order model of a ship has been determined through real experimental data obtained from a zig-zag manoeuvre test. This ship model has been identified by analysing the rudder angle, surge and sway speeds, and yaw rate obtained from experimental tests. This identification has been done with a Least Squares Support Vector Machines algorithm. The LS-SVM technique has been widely used for pattern recognition and classification with very good results, and in this work it has been shown that this technique has a high potential to be used in system identification. The model obtained with LS-SVM has been validated through experimental tests showing that the behaviour of the mathematical model is very similar to that of the real ship. Furthermore, the model obtained is suitable to be used for testing control algorithms that can be implemented on the ship. It has been proved that LS-SVM is a powerful tool for system identification that can deal with relative few experimental data and low time cost.

Future work will aim at (i) extending the methodology developed to deal with more complex ship models in which the hydrodynamic coefficients can be well defined and to deal with models whose structure is not known in advance, and (ii) studying the performance of control algorithms for path following and tracking with the ship model defined in comparison with the results obtained for the real vehicle.

The authors wish to thank the Spanish Ministry of Science and Innovation (MICINN) for support under Project DPI2009-14552-C02-02.