## 1. Introduction

The offshore platforms provide the basic support for marine operation, such as the oil exploration, drilling operations and transportation. A considerable amount of theoretical and experimental research effort has been aimed at improving the control performance of offshore platforms, including passive and active control [

1,

2]. Passive control is the classic technology to enhance the safety by using excessive construction materials to guarantee the stability of the offshore structures. Nevertheless, passive systems have limitations in improving control performance, even with the huge energy cost. In comparison, active control technologies have great potential to meet the optimal performance requirements with low consumption requirement [

3]. Recently, various kinds of active schemes have been derived and employed for offshore platforms, e.g., using the concepts of non-fragile control [

4], sliding mode theory [

5] and

$H\infty $ control [

6].

With the development of advanced sensor, actuator and digital control technology, offshore platforms under digital control system have been attracting extensive attention in the field of controller design [

7,

8]. Offshore platforms generally involve sophistication of the superstructure in the deep water, in which various inevitable factors directly affect offshore platforms in ocean environment, especially vibration caused by external wave force [

9]. In order to describe and identify the characteristic of wave force, various idealized spectra are employed to describe the empirical relationship that defines the distribution of energy with frequency within the ocean [

10], such as the Pierson–Moskowitz spectrum [

11] and Joint North Sea Wave Project (JONSWAP) spectrum [

12]. Compared to the Pierson–Moskowitz spectrum, the JONSWAP spectrum is effectively a fetch-limited version to analyze the wave force in the wave-frequency domain [

13]. Many control schemes were proposed to attenuate the wave-induced vibration thereby ensuring the safety and satisfying the performance requirements. For example, the wave vibration problem was formulated as an optimal tracking control problem and a corresponding tracking control scheme was proposed to reduce the displacement and velocity of offshore platform in [

14]; a network-based state feedback control scheme was designed for steel offshore platform based on the stability criterion in [

15] with small control consumption; a non-fragile sampled-data controller was proposed in [

4], thereby reducing the oscillation amplitudes of the offshore platform; an event-triggered controller was derived from the Lyapunov–Krasovskii function approach to guarantee the stability of offshore structure in [

16]. It should be noted that the aforementioned works are with huge energy consumption. Meanwhile, the optimal vibration control theories provide efficient methods to improve the control performance of control systems with smaller control consumption. Therefore, one target of this paper is to apply the optimal control technology to the offshore platform under digital control system.

In general, offshore platforms have their inevitable nonlinear characteristics caused by rigid structure, multiplying springs and dampers, flexibility and complexity structures. Unfortunately, the unsafely behaviors and destabilization of offshore platforms mainly result from nonlinear dynamics. Many researchers have engaged in the efforts to the implementation of control schemes for nonlinear offshore platforms, e.g., a robust mixed control method for wave-excited offshore jacket platforms is proposed in [

17] to minimize the upper bound of the performance measure on platform dynamics satisfying some norm bound constraint simultaneously; a novel sliding mode control scheme is proposed by using information about mixed current and delayed states in [

18]; by combining a sliding mode control technique, the adaptive control algorithm and wavelet support vector machine, an adaptive integral sliding mode control to handle the nonlinear behavior of the offshore platform in [

19]. Meanwhile, the optimal vibration control theories provide efficient methods to improve the control performance of control systems with smaller control consumption [

20,

21,

22,

23]. However, optimal control for a nonlinear discrete system will lead to a Hamilton–Jacobi–Bellman (HJB) equation with no exact analytical solution except [

24]. In order to obtaining approximate solutions to the HJB equation, many approaches have been developed, such as the power series approximation [

25], the successive Galerkin approach [

26], and approximating sequence of Riccati equations approach [

27]. However, it still is difficult to seek the solution under the external disturbance and nonlinear dynamics. This is the motivation of this paper.

The irregular wave disturbance attenuation problem for a nonlinear offshore platform under digital control system is investigated in this paper. An approximation of optimal wave disturbances attenuation controller (AOWDAC) is proposed to compensate the external irregular wave force and nonlinear dynamics. First, the discrete nonlinear jacket-type offshore platform is established under a digital control system, in which the external wave force acting on the offshore platform is viewed as the output of an external system. In order to minimize the average quadratic performance index and energy consumption, AOWDAC is developed to obtain the approximation solution of a derived nonlinear two-point-boundary-value (TPBV) problem caused from the nonlinear offshore structure, which is made up of disturbance compensation of wave force, feedback items of offshore platforms and the vector sequences of nonlinear compensation. Based on the value of performance index in the process of iteration, the feasibility of AOWDAC is realized. By analyzing the displacement and velocity of a jacket-type offshore platform located in Bohai Bay, the effectiveness of developed AOWDAC is proved compared with an optimal feedback and feedfoward vibration controller (OFFVC).

The rest of this paper is as follows. The irregular wave attenuation problem for a nonlinear offshore platform under a digital control system is formulated in

Section 2. The main results are shown in

Section 3, in which the AOWDAC is designed to compensate the irregular wave force and nonlinear dynamics, and the feasibility of AOWDAC is realized by using the designed iterative algorithm. In

Section 4, a jacket-type platform located in Bohai Bay is introduced to prove the effective of the designed AOWDAC and algorithm. Concluding remarks are drawn in

Section 5.

## 5. Conclusions

A digital-control-based AOWDAC was developed for a nonlinear jacket-type offshore platform under external irregular wave disturbances, which consists of feedback items of offshore platform state, feedfoward item for attenuating the external wave disturbance, and compensation sequences for responding to the nonlinear dynamic of offshore platforms. First, based on the JONSWAP wave spectrum and linearized wave theory, the wave forces were formulated as the output of an exosystem. Meanwhile, a discrete model of a nonlinear jacket-type offshore platform was established. After that, by solving an introduced nonlinear TPBV problem, the digital-control-based AOWDAC was designed based on the Riccati equation, Stein equation and two vector sequences. By designing an iterative algorithm, the feasibility of digital-control-based AOWDAC was realized based on the performance index in each iteration. Applying the digital-control-based AOWDAC to an offshore platform and comparing with OFFVC, the displacement and velocity of offshore platforms were reduced under proposed digital-control-based AOWDAC significantly, and the energy consumption of digital-control-based AOWDAC requires a smaller control force than other typical control scheme.

Due to the limitations of actuators, sensors and communication networks, one aspect of our future work will focus on the vibration control problem for the networked nonlinear offshore platform with delays and faults in actuator and measurements. Meanwhile, the predictive wave disturbance attenuation problem for offshore platforms is another aspect of our future work based on the wave feedforward information measured from advanced sensor technology.