Rolling-optimized model predictive vibration controller for offshore platforms subjected to random waves and winds under uncertain sensing delay

Adaptive neural network H∞$H_\infty$ control for offshore platform with input delay and nonlinearity

“…in which c 1 (t ) and k 1 (t ) are stiffness and damping coefficients of platform, c 2 (t ) and k 2 (t ) are stiffness and damping of AMD, respectively. The details of stiffness and damping are given by [36] c…”

“…The time series of Wind velocity is supposed to be extracted from spectra, and then equivalent wind loads are to be generated. According to Davenport wind spectrum [36], a typical form of fluctuating wind velocity is defined as…”

“…When ocean waves are relatively high, the wind-induced external loads are considered additionally. Around 16% of the hydrodynamic-aerodynamic mixed loads are aerodynamic excitation, which is usually neglected in a calm sea situation [36]. Davenport spectrum (7) with reference exponential mean wind velocity is set to depict the wind field.…”

“…The details of stiffness and damping are given by [36] $$$$\begin{equation} \begin{aligned} {c_{1}}(t)=2{m_{1}}{\xi }_{1}(t)\iota _{1}(t) ,\ {k_{1}}(t)={m_{1}}\iota _{1}^{2}(t), \end{aligned} \end{equation}$$$$in which $${\xi }_{1}(t)={\xi }_{1}+{\hat{\xi }_{1}}\Delta {\tilde{\xi }_{1}}(t)$$, ξ 1 is the nominal value of damping ratio, satisfying $$|\Delta {\tilde{\xi }_{1}}(t)|\le 1, \forall t\ge 0$$, where $${\hat{\xi }_{1}}$$ is the maximum perturbation; $$\iota _{1}(t)$$ is the natural frequency, satisfying $${\iota }_{1}(t)={\iota }_{1}+{\hat{\iota }_{1}}\Delta {\tilde{\iota }_{1}}(t)$$, where ι 1 is the nominal value of the damping ratio, $${\hat{\iota }_{1}}$$ is the maximum perturbation, …”

“…According to Davenport wind spectrum [36], a typical form of fluctuating wind velocity is defined as $$$$\begin{equation} {\bar{\omega }}{S_\mu ({\bar{\omega }})}=\frac{4{k_\mu }{\mu _0}^2{\bar{x}}^2}{ \left(1+{\bar{x}}^2 \right)^{\frac{4}{3}}},\ {\bar{x}}=\frac{1200{\bar{\omega }}}{\mu _0}, \end{equation}$$$$where $${\bar{\omega }}$$ is the frequency of fluctuating wind, μ 0 is the average velocity, which is the reference velocity at height z , $$k_\mu$$ is the roughness exponent of sea surface (A‐type). Define $$\mu (t)$$ as wind velocity, containing mean wind velocity with reference velocity μ 0 , and fluctuating wind velocity with fluctuating wind frequency defined by (7) under a wind velocity function $$\mu (t)=f_{\mu }(\mu _0,{\bar{\omega }})$$ [36], the wind force $${ \bar{f}}_{j}(t)$$ of $$\bar{p}(z,t)$$ on the target platform is …”