Parameter identification based on unity feedback responses for steering control in model experiments

Abstract

Nomoto’s linear model is always utilized in model-based steering control design for model experiments to analyze the steering dynamics and optimize the controllers using numerical tools. The effective identification of parameters for the free running model is important for reducing the time cost of model experiments. In this paper, a parameter identification method based on closed-loop responses using unity feedback is discussed. Unity feedback can transform the original dynamics of the model into the familiar 2nd-order mechanical mass-damper-spring system, which is convenient for analyzing the system. Although the identification of parameters based on closed-loop responses is a common technique in control engineering, few approaches using the free running model have been applied in ocean engineering. Given this motivations, parameter identification methods based on unity feedback responses are evaluated through numerical simulations and model experiments. From the results of this investigation, it is clear that noise in measurements and the initial yaw rate of the running model have detrimental impacts on the identified values.

Appendix A. straight-line stability criterion

When the system approaches critical stability, $\zeta$ goes to 0. Therefore, the mechanical characteristics in the case of $\zeta =0$ presents the stability criteria. As $\zeta$ can be expressed as Eq. 19, when $\zeta$ goes to 0:

$$\begin{aligned} TK = \frac{1}{4} \frac{(\ln |p_{0}|)^{2}+\pi ^{2}}{(\ln |p_{0}|)^{2}} \rightarrow \infty \end{aligned}$$

(A.1)

Because $\ln |p_{0}|$ goes to 0 to satisfy Eq. A.1, $|p_{0}|$ moves to 1. Therefore, $|p_{0}|=1$ can be considered the stability criterion in the unity feedback response.

On the other hand, in the frequency domain, the poles of the system in Eq. 4 are popular indexes for stability.

$$\begin{aligned} \sigma _{1,2} = \frac{-1 {\pm } \sqrt{1-4TK}}{2T} \end{aligned}$$

(A.2)

If $(1-4TK)<0$, the real part of the poles is $-1/(2T)$. The sign of T is an important index for the stability. In the case of $(1-4TK)>0$, because of Eq. A.1, TK is positive. Therefore, $0<\sqrt{1-4TK}<1$ can be satisfied. The real part of the system is expressed by writing:

$$\begin{aligned} \sigma _{1}= & {} \frac{-1 + \sqrt{1-4TK}}{2T}< 0,\nonumber \\ \sigma _{2}= & {} \frac{-1 - \sqrt{1-4TK}}{2T} < -\frac{1}{T} \end{aligned}$$

(A.3)

As $\sigma _{2}$ plays a significant role in the criterion, the sign of T is also important for the stability criterion in this case. The Routh-Hurwitz criterion based on Eq. 1 also arrives at the same conclusion.

The criterion for the closed-loop system is summarized in Table 6. The criterion for the open-loop system, which is called the straight-line stability in ship technology, is also shown. In the case of the open-loop system, the criterion in the time domain is not revealed by the analytical approach. Therefore, special manoeuvering tests, i.e., Diedonné’s spiral or Bech’s reverse spiral manoeuvre tests, are used to check whether the ship exhibits straight-line stability. The criterion for the poles in the open-loop system is same as that in the closed-loop system. Therefore, if the straight-line stability for the free-running model is investigated, the stability of the closed-loop system needs to be surveyed only indirectly. The first peak ($|p_{0}|$) in the unity feedback response can be also used as the criterion for investigating the straight-line stability.

Table 6 Comparison of the stability criterion between the open-loop and closed-loop system for the steering dynamics