T

static model

Kss

^ss

Table 2. An overview of equations for parameter calculation based on open loop step responses

Table 2. An overview of equations for parameter calculation based on open loop step responses

The open-loop identification method was applied to VideoRay Automarine AUV yaw degree of freedom, Stipanov et al. (2007). The case vehicle was developed at the University of Zagreb, Faculty of Electrical Engineering and Computer Science, Laboratory for Underwater Systems and Technologies. The identification results are shown in Fig. 11 and the identified inertia and drag are shown in Table 3. Blue dots show the experimental data, green line shows the fitted nonlinear drag curve while the green line shows the fitted linear drag curve. It is obvious that nonlinear mathematical model describes this vehicle better. This fact was used in determining yaw inertia parameter (see Fig. 11b). Validation results are shown in Fig. 11c where blue line shows the experimental data, red line the linear model and green line the nonlinear model.

Parameter a

1.018 1.257

Table 3. Identified nonlinear model parameters for VideoRay Automarine Module 4.3 Zig-zag method applied to underwater vehicles

In determining marine surface vehicles' dynamic behavior, zig-zag maneuvers are widely accepted. Zig-zag maneuver is used for designing ship autopilots, i.e. determining yaw motion of a surface vessel, Lopez et al. (2004). The maneuver which is usually run for ships consists of the following steps, while the ship is sailing in advance at a predetermined speed:

• turn the rudder at the maximum speed to the starboard side at 10° (20°)

• when ships course changes by 10deg (20°) from the initial course, turn the rudder to the opposite side (port) at 10deg (20°). After a while, the ship will turn to port.

• when ship course changes by 10deg (20°) from the initial course on the opposite side, turn the rudder again to the starboard side at 10 deg (20°)

The heading and the rudder position should be recorded all the time during the experiment. This algorithm can be simulated as shown in Fig. 12. The response of the zig-zag maneuver are shown in Fig. 13.

The initial assumption for this method is that yaw motion can be described using a simple Nomoto model given with j = where 8 is rudder deflection, ^ heading and K and T

parameters which are to be determined. The same model can be applied to underwater vehicles yaw model, + = N - in this case the exciting force is yaw moment. The unknown parameters can be determined by pure integration of the Nomoto model, López et al. (2004).

Fig. 12. Simulation scheme for the zig-zag manoeuvre

Fig. 13. The zig-zag experiment: integration area for determining a) drag and b) inertia at + /3Lr = N aftdt+pLfrdt = fNdt

If the integration is performed between the first two time instances when extreme headings appear (the yaw rate at these points equals zero), equation (25) is obtained.

rf2 Ndt Jti

If (24) is integrated between two consequent zero crossing point of the heading response, equation (26) is obtained. In this case, yaw rate value at the zero crossing points is needed.

rft4wdt

It is clear that in order to get the two parameters, integration of the control input has to be performed. In Fig. 13 the shaded areas are to be integrated in order to determine inertia and linear drag.

This procedure is practical if linear Nomoto model describes the vessel's dynamic properly. However, if nonlinear terms in the drag appear, the procedure cannot be used. If there is external disturbance present, the Nomoto model should be modified (which makes the procedure much more complex) otherwise the results will be false.

4.4 Identification by use of self-oscillations (IS-O)

The concept of identification by use of self-oscillations is similar to the zig-zag procedure. It was more then 20 years ago when Astrom & Hagglund (1984) derived a so called ATV (autotuning variation) method used for system identification, that is simple and appropriate for in situ identification. The method is based on using a relay-feedback to bring the system to self-oscillations. Then Luyben (1987) used this method in chemical industry to identify a transfer function of extremely nonlinear systems (distillation columns). Since then, inducing self-oscillations proved to be a great tool for controller tuning in processes and for process identification, see Li et al. (1991) and Chang & Shen (1992), especially in pharmaceutical industry.

The IS-O method is based on forcing a system into self-oscillations using the same scheme as shown in Fig. 12. In most cases, these oscillations are induced by introducing a relay with hysteresis, but it should be noted that other nonlinear elements can cause the same behavior, Vukic et al. (2003).

Unlike the zig-zag experiment, the IS-O procedure is not based on solving the differential equation which describes the process, but uses magnitudes and frequencies of the induced oscillations to determine system parameters. Having Fig. 12 in mind, the relation between the magnitudes and frequencies of self-oscillations and system parameters can be found through the Goldfarb principle, given with (27) where GN(Xm) is the describing function of the nonlinear element (relay with hysteresis), Xm is the magnitude of oscillations at the input of the nonlinear element and Gp(y'a>) is the process frequency characteristic.

Equation (27) can be graphically interpreted as finding intersection points between Nyquist frequency characteristic of the LTI process and an inverse negative describing function of the nonlinear element, see Vukic et al. (2003) and Miskovic et al. (2007b). The describing function of the relay with hysteresis is given with (28) and (29) where xa is half the width of the hysteresis, and C is the relay output.

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