Controlling period-doubling route to chaos phenomena of roll oscillations of a biased ship in regular sea waves

Abstract

Dynamics, control, and stability of roll oscillations of a biased ship in regular sea waves are investigated. The ship under roll oscillation is modelled as the classical Helmholtz–Duffing oscillator with strongly nonlinear asymmetric restoring moment characteristics. The incremental harmonic balance method, amended with a pseudo-arc-length continuation approach, is employed to obtain the uncontrolled and controlled frequency responses. The primary and subharmonic responses of the uncontrolled system are examined through the period-doubling route to chaos path. The same ship roll model is then investigated under state feedback control with time delay. In the control scheme, a moving weight is actuated by the delay controller to generate the anti-rolling moment. The stability of periodic responses is studied using the semi-discretization method with extended Floquet theory. Bifurcation points on periodic response branches are identified from the eigenvalue study. The effect of gains and time delay in the feedback loop in controlling the primary and subharmonic responses is investigated. Several chaotic responses obtained through period-doubling route to chaos paths are successfully controlled. Stable, periodic, and steady-state solutions obtained from the IHB method are verified by numerical integration of the equation of motion as and when applicable. Solutions are aided with phase portrait, Poincaré map, time history, and Fourier spectrum for better clarity. It is shown that appropriate selections of control parameters can effectively reduce the roll amplitude to a great extent with the improved measure of stability.

Appendix A: root locus plots

Close-loop transfer function from Eq. (2) is written as

$$ \frac{G\left( s \right)}{{1 + G\left( s \right)H\left( s \right)}} = \frac{1}{{s^{2} + 1 - \left( {g_{d} + sg_{v} } \right)e^{{ - s\tau_{d} }} }}. $$

(A.1)

The close-loop control system is shown in Fig. 26.

Fig. 26

Close-loop control system

Since $G(s)=1/({s}^{2}+1)$, the open loop transfer function is expressed by

$$ G\left( s \right)H\left( s \right) = - \frac{1}{{s^{2} + 1}}\left( {g_{d} + sg_{v} } \right)e^{{ - s\tau_{d} }} . $$

(A.2)

Since

$$ e^{{ - s\tau_{d} }} \approx \frac{{1 - s\tau_{d} }}{{1 + s\tau_{d} }}, $$

(A.3)

$$ G\left( s \right)H\left( s \right) = g_{d} \frac{{\left\{ {\left( {g_{v} /g_{d} } \right)\tau_{d} } \right\}s^{2} + \left\{ {\tau_{d} - \left( {g_{v} /g_{d} } \right)} \right\}s - 1}}{{\tau_{d} s^{3} + s^{2} + \tau_{d} s + 1}}, $$

(A.4)

$$ = \left( { - g_{v} } \right)\frac{{\left( { - \tau_{d} } \right)s^{2} + \left\{ {1 - \left( {g_{d} /g_{v} } \right)\tau_{d} } \right\}s + \left( {g_{d} /g_{v} } \right)}}{{\tau_{d} s^{3} + 2s^{2} + \tau_{d} s + 1}}. $$

(A.5)

For ${g}_{v}=0$ (or ${g}_{v}/{g}_{d}=0$), and ${\tau }_{d}=0.1$, the root locus plot (refer Eq. A.4) of positive displacement feedback is shown in Fig. 27.

Fig. 27

Root locus plot under positive displacement feedback with time delay τd = 0.1

For ${g}_{d}=0$ (or ${g}_{d}/{g}_{v}=0),$ and ${\tau }_{d}=0.1$, the root locus plot (refer Eq. A.5) of the negative velocity feedback is shown in Fig. 28.

Fig. 28

Root locus plot under negative displacement feedback with time delay τd = 0.1