If the system dynamics can be described by using equation (6), i.e. if there is a bias term, than the self-oscillations will not be symmetric. This is the case with heave DOF where there is almost always a difference between weight and buoyancy of the vehicle. If TH represents the time when relay output is in "high" position, and TL represents the time when relay output is in "low" position, TH will differ from TL. This implies that equations (30), (31), and (32) are not valid. However, based on times TH and TL the bias term S can be determined, using (6) - this way the bias can be compensated for within the controller.
This equation can be applied to a general process of n-th order which includes a constant
dtn >'">x(.t)>8) = TW, see Miskovic et al. (2008) for details.
The main assumptions that are posed on the self-oscillation method are that the oscillations are symmetric and that higher-order harmonics are negligible in comparison to the first, dominant harmonic. Since these two assumptions are never completely fulfilled in real systems, this method always introduces a slight error in the estimation of the parameters. A detailed analysis on the error which occurs in the application of the method for yaw identification can be found in Miskovic et al. (2007c). In the same paper, it is shown that the error will be small if the ratio between the established oscillations and the width of the relay with hysteresis is about 1.5.
The IS-O method was applied to the VideoRay Automarine AUV. The oscillation parameters are shown in Table 4. Since the experiment was performed in a laboratory pool external disturbance is negligible - parameters TH and TL are practically the same. The identified parameters of the nonlinear model are shown in Table 5. For more details, the reader is referred to Miskovic et al. (2007c).
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