State feedback control of surge oscillations of two-point mooring system

Abstract

Stability analysis of surge oscillations of two-point mooring system under state feedback control with time-delay is investigated. The two-point mooring system is harmonically excited and essentially represents a strongly nonlinear Duffing oscillator. In this paper, a frequency domain based method viz. incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary and higher order subharmonic responses which may be present in such system. The IHBC is then reformulated in a manner to treat two-point mooring system under state feedback control with time-delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses for primary and higher order subharmonic oscillations is obtained by Floquet’s theory using Hsu’ scheme; whereas the stability of controlled responses is obtained by applying semi-discretization method for delay differential equation. The study focussed on the controlling primary, higher order subharmonics and chaotic responses by considering appropriate feedback gains and delay by way of (i) appreciable reduction of primary, subharmonic responses, (ii) exclusion of all higher order subharmonics 2T, 3T, 5T and 9T (1/n subharmonics or period-n solutions), and (iii) reduction of the extent of domain of all instability phenomena represented by various type of bifurcation of solutions, jump phenomena, chaotic responses etc. In the study, negative velocity feedback is observed to be much effective than state feedback for better controlling of surge oscillation of two-point mooring system. Also, the effect of larger gain values is investigated by an extensive parametric study for vibration control with different delay values.