3.3 Identification using filtering techniques
As already mentioned, a simulator similar to SIMSUP has been developed in MATLAB environment and coupled to the second order filter DD2. The basic functioning of the filtering technique is explained in figures 9 and 10; the filter uses all the time history of the experimental manoeuvres (thus differentiating from the previous method that utilizes only main parameters) estimating in real time the states by means of a predictor-corrector algorithm, considering uncertainties in both measurements and process.
Fig.9 Scheme of Identification procedure with filters
Fig.10 Scheme of filters functioning
As can be seen in figure 9, input values to the filter are the time histories of the measured variables (e.g. absolute velocity U, heading y, rate of turn r and the trajectory coordinates x and y) and of system input (rudder angle d, propeller RPM is in this case not considered since the model runs with constant RPM for the sake of simplicity).
The functioning of filters, schematized in figure 10, in general is structured as follows:
- the "predictor" part, using the state equation of the system and starting from values of state variables and their associated error covariance at a certain time i, predicts the new values at time i+1 and projects also the error covariance considering the process error;
- from the state variables values of the system outputs (if different) at time i+1 are evaluated in order to be compared to measured values;
- the predicted values are then "corrected" using the measured values at time i+1 (if existing), considering both the uncertainties of measured data (measurement error) and of the predicted data caused by process error, evaluating new state variables values;
- the procedure is repeated at each time step examining the complete time history from experimental data.
In the case of system identification, filters are used incrementing artificially the number of state variables considering the unknown parameters (the hydrodynamic coefficients) as state variables and filtering them together with the other parameters. Since the value of the hydrodynamic coefficients should be almost constant during the manoeuvre, the new state equations which are added to the explicit form of (2) are simply a series of equations like the following one for Yv, given as an example:
∂Yv/∂t = 0
It has to be noted that, in order to obtain the final values using this technique, the filtering procedure has been applied iteratively more than one time to the same time history, starting each time with the final identified value from the previous filtering until convergence.
As done for the first procedure used, a first set of runs have been performed starting from simulated data, in order to evaluate the feasibility of the method. As a first step, the full set of coefficients established with the sensitivity analysis was identified starting from a simulated 10°/10°ZigZag manoeuvre without any addition of disturbances.
Even in this favourable condition the contemporary identification of the whole set of coefficients created some problems, with cancellation effects between different coefficients; it was decided, therefore, to analyse just the five linear coefficients, in analogy with the procedure adopted using the optimization technique. In this case the coefficients are identified rather easily and without any problem of cancellation using the input data from the ZigZag manoeuvre; as an example, in figure 11 the values computed at different steps for Yv coefficient are reported (similar results were obtained for the other coefficients investigated).
Identification of Yv coefficient from 10°/10° ZigZag manoeuvre without disturbances
It has to be noted, however, that the condition assumed is clearly an ideal case, because in a real situation the time histories are likely to be affected by measurement noise and the mathematical model used is just an approximation of the real system, while in this case mathematical model and system are coincident. Before starting with the identification of coefficients from experimental data, therefore, an intermediate step was made analysing the same simulated data with the addition of a Gaussian noise in order to simulate at least measurement error. Moreover, since the majority of data available from sea trials do not comprehend the two velocities of sway and surge but just the resultant, the two velocities have been combined and the input data corresponds to figure 9 (absolute velocity, rate of turn, heading and ship coordinates during the manoeuvre).
Under these conditions, the identification becomes more difficult, with a loss of accuracy; it was possible, anyway, to identify with a reasonable accordance the target values, as can be seen in figure 12 where the identification steps for the Yv coefficient are reported again as example. It has to be noted, moreover, that, contrarily to what was done for the optimization procedure, Turning Circle and ZigZag manoeuvre were not analyzed contemporarily up to now. From the analysis of the Turning Circle manoeuvre alone, anyway, it was not possible to obtain good results, as explained in the following.
Identification of Yv coefficient from 10°/10°ZigZag manoeuvre with Gaussian noise
Once the procedure was established and tested, it was applied to the same experimental ZigZag manoeuvres used with the optimization method. As anticipated before, it was rapidly clear that, in this case, the Turning Circle manoeuvre is useless. All the identification runs performed using just the input data from Turning Circle did not converge to any final value, with the coefficients "drifting" indefinitely in pairs; this result is again in accordance to what was found in , and it is probably due to the low amount of information on the system provided by a manoeuvre which is for a large part in a steady state. On the contrary the ZigZag manoeuvre gave much better results, probably because of its transient nature, therefore in the following the Identification from ZigZag manoeuvre is described.
Using the procedure described above, the values of the coefficients were identified with a reasonable agreement (table 2, column named "Filters"), with an average error of 10%, similar to the one obtained using the optimization procedure.
In the following figures 13 and 14, the experimental Turning Circle and ZigZag manoeuvres and the simulated ones using the coefficients identified are compared.
Fig.13 35°Turning Circle - Comparison of results
Fig.14 10°/10°ZigZag - Comparison of results
As it can be seen the ZigZag manoeuvre and the first part of the Turning Circle manoeuvre are simulated with a good accuracy. On the contrary, the second part of the Turning Circle manoeuvre is estimated worse, probably because the coefficients are estimated from a manoeuvre with a transient nature, while this second part is in a steady state.
Moreover the non-linear coefficients were not object of this study since their overall influence was judged negligible, however the influence on this specific part of the manoeuvre is probably more marked. It can be noted that similar result was obtained with the previous procedure.