#### Abstract

This paper presents a robust sampled-data control for vibration attenuation of offshore platforms with missing measurements subject to external wave force. It is well known that the phenomena of missing measurements are unavoidable due to various reasons, such as sensor aging and sensor temporal failure, which may degrade the control system performance or even cause instability. For active vibration control of offshore platforms with sampling measurements, to deal with the missing measurements, a robust sampled-data control method is proposed in this paper. The robust sampled-data state feedback controller is obtained in terms of the solvability of certain linear matrix inequalities (LMIs). Finally, simulations on an offshore platform are exploited to demonstrate the effectiveness of the proposed method. The simulation results show that the designed robust sampled-data control scheme is effective to attenuate the external wave force in the presence of missing measurements.

#### 1. Introduction

In the modern world, the offshore platforms, especially the oil and gas production platforms, play a more and more important role. The offshore platforms generally undergo various disturbances coming from the hostile environment that they are located in, such as wave, wind, ice, and earthquake, and the self-excited nonlinear hydrodynamic force [1, 2]. These external loads inevitably induce large continuous vibrations which make the offshore platform deformation, fatigue damage, and even unsafety. To prevent fatigue damage and ensure safety and production efficiency, the vibrations of the offshore platforms should be limited.

Up to now, a large number of researches have been made to improve the control performance of the system via active control scheme. For example, for offshore steel jacket platforms with an active tuned mass damper (TMD) mechanism, the multiloop feedback design method [1], the nonlinear control scheme, and the robust state feedback control scheme [2] have been developed to reduce the internal oscillation amplitudes of the offshore platforms. By using an active mass damper (AMD), some optimal control based schemes have been applied to improve the performance of the jacket platforms [3–6]. Recently, the dynamic output feedback control scheme [7] and the integral sliding mode control method [8, 9] have been presented to improve the performance of the offshore platforms. More recently, a delay-dependent state feedback controller has been developed to stabilize the offshore platforms subject to self-excited hydrodynamic force and actuator time-delays [10]. In [11], by artificially introducing a proper time-delay into control channel, a delayed controller is designed to attenuate the wave-induced vibration of the offshore platform and thereby improve the control performance of the system. It is indicated that the aforementioned active control schemes are effective ways to deal with the vibration problem of offshore platforms subject to nonlinear wave force.

It is well known that computers are usually used as digital controllers to control continuous-time systems in modern control systems [12] with the rapid progress of computer and digital technologies. With recent focus on wireless monitoring and control of offshore platforms [13–18] based on networked control technique [19], studying sampled-data control problem for offshore platforms is becoming significant, and the vibration control methods adopted in [1–11] are no longer applicable to offshore platforms with sampling measurements. On the other hand, because of various reasons, such as sensor aging and sensor temporal failure, the phenomena of missing measurements are unavoidable, which may degrade the control system performance or even cause instability [20–26]. Thus, it is necessary to consider the missing measurements encountered in practical issues for vibration control. However, to the best of our knowledge, no results have been made on sampled-data control for vibration attenuation of offshore platforms in presence of missing measurements, which motivates our work in this paper.

This paper is concerned with the robust sampled-data controller design for vibration attenuation of offshore platforms subject to missing measurements and external wave force. The issue of vibration attenuation is transformed into an disturbance attenuation problem of the system. A Lyapunov functional approach is used to solve the disturbance attenuation problem, and the controller design is formulated in terms of linear matrix inequalities (LMIs). To validate the effectiveness of the proposed approach, the designed controllers are applied to reduce the vibration of an offshore platform. Simulation results show good vibration attenuation performance in spite of involving missing measurements.

The rest of this paper is organized as follows. In Section 2, the description of an offshore platform with an AMD mechanism is given first. Then, the formulation of robust sampled-data control problem with missing measurements is presented. In Section 3, the robust sampled-data controller design problem involving missing measurements is solved. An illustrative example is given in Section 4, and we conclude the paper in Section 5.

*Notation.* The notation used in the paper is fairly standard. The superscript “” stands for matrix transposition; denotes the space of real numbers; denotes the -dimensional Euclidean space; the notation (≥0) means that the matrix is real symmetric and positive definite (semidefinite); and represent the identity matrix and zero matrix of appropriate dimensions, respectively; is the infinitesimal operator; and stands for a block-diagonal matrix. In addition, means expectation of the stochastic variable . The space of square-integrable vector functions over is denoted by and for , its norm is given by . For simplicity, the symmetric term in a symmetric matrix is denoted by “.”

#### 2. Problem Formulation

In this section, a dynamic model of an idealized two-degree-of-freedom system with an AMD mechanism will be presented and the active vibration attenuation problem for the system with sampling measurements will be formulated by applying a state feedback control scheme.

Consider an offshore platform shown in Figure 1 [11]; the dynamic equations of the offshore platform can be described as where and are displacements of the deck motion of the offshore platform and the AMD, respectively; , , and are the modal mass, natural frequency, and damping ratio of the offshore platform, respectively; , , and are the mass, natural frequency, and damping ratio of the AMD, respectively; is the active control of the system; is the external wave force acting on the offshore platform and can be numerically calculated by referring to [6].

Define the following state variables: and let

Then, the dynamic model of the offshore platform (1) can be written as where

To guarantee the performance index from the wave force to the control output to be realized with the specified requirement, the displacement response and the velocity response of the offshore platform are chosen as the control output. In this situation, the control output equation is given as where

In practice, for offshore platform system, only sampled measurements of state variables are available at discrete instants of time. Without loss of generality, it is assumed that the state variables of the active vibration system are measured at time instants ; that is, only is available for interval , where and is the maximum sampling interval. We are interested in designing a state feedback controller, which may contain missing measurements described by where is the controller gain matrix to be designed and is the stochastic variable having the probabilistic density function on the interval with mathematical expectation and variance ; that is, where and are known positive scalars.

*Remark 1. *In real systems, due to various reasons, such as sensor aging and sensor temporal failure, the measurements missing at one moment might be partial and therefore the missing probability cannot be simply described by 0 or 1 [24]. It is easy to see that the widely used Bernoulli distribution is included here as a special case.

According to (4) and (8), the closed-loop system is given by
It should be noticed that the closed-loop system (10) is actually a stochastic system, since it contains the stochastic quantity . Therefore, in the sequel, we will use the notion of stochastic stability in the mean-square sense.

Here, the issue of vibration attenuation for the offshore platform can be transformed into an disturbance attenuation problem of system (10). The objective is to determine the controller (8) such that(1)the closed-loop system (10) with is asymptotically mean-square stable;(2)under zero initial condition, the performance
of the closed-loop system (10) is guaranteed for nonzero and a prescribed .

To obtain the main results, the following lemma is needed.

Lemma 2 (Schur complement [24]). *Given constant matrices , and , where and , then if and only if
*

#### 3. Main Results

In this section, we will solve the problem of robust sampled-data controller design for systems (4) and (6) with missing measurements by the recently developed input delay approach [27–29]. The key idea behind this approach is that we represent the sampling instant as where . Then, from (8) and (13), we obtain where is a discrete-time control signal and the time-varying delay is piece-wise linear and satisfies and for . By using the input delay approach, the closed-loop system (10) can be transformed into a time-delay system as follows:

In the following, we will determine the controller gain matrix such that system (15) is asymptotically mean-square stable and satisfies the disturbance attenuation in (11).

*Remark 3. *Recently, the vibration control problem for offshore platforms with constant input delay has been addressed in [10, 11]. It is worth pointing out that the transformed system in our problem contains nondifferentiable time-varying delay in the states, which hinders the results in [10, 11] to be directly applied to the problem considered here.

##### 3.1. Performance Analysis

The following theorem provides a sufficient condition for system (15) to be asymptotically mean-square stable and satisfy the disturbance attenuation in (11).

Theorem 4. *Given scalars , , and controller gain matrix , the closed-loop system (15) with is asymptotically mean-square stable and the disturbance attenuation in (11) is satisfied for the wave force and a prescribed , if there exist matrices , , , , , , and with appropriate dimensions such that the following LMIs hold:
**
where
*

*Proof. *Considering a Lyapunov-Krasovskii functional as follows:
where and are matrices to be determined.

The infinitesimal operator of is defined as [26]
Combining (19) and (20) yields
where
where .

From (16), we have
Combining (21)–(24), one yields
where

By Lemma 2, it can be obtained from (17) that
It follows from (25)–(27) that

When , it is easy to get from (28) that , which means that system (15) with is asymptotically mean-square stable.

On the other hand, integrating both sides of (28) from 0 to and noting the fact that under zero initial condition and , we obtain
which indicates that the disturbance attenuation (11) is guaranteed.

This completes the proof.

##### 3.2. Sampled-Data Controller Design

In this section, the controller design problem is solved in the following and the controller gain matrix is given in terms of a solution to LMIs.

Theorem 5. *Given scalars , and , the closed-loop system (15) with is asymptotically mean-square stable and the disturbance attenuation in (11) is satisfied for the wave force and a prescribed , if there exist matrices , , , , , , , and with appropriate dimensions such that the following LMIs hold:
**
where
**Moreover, if inequalities (30) have a feasible solution, then the controller gain matrix in (8) is given by
*

*Proof. *By pre- and postmultiplying (16) by and pre- and postmultiplying (17) by , (16) and (17) can be converted into the following equivalent inequalities:
where

Noting that , one has
By letting , , and , we obtain that (33) hold if (30) is satisfied. Then, by Theorem 4, we can conclude that the theorem is true. The proof is completed.

#### 4. Simulation Results

In this section, the parameters of an offshore platform and the wave are given first. Then, the proposed robust sampled-data control (RSDHC) scheme will be applied to control the offshore platform. In order to verify the feasibility and effectiveness of the proposed scheme, simulation results of the numerical example are presented. Besides, the effect of missing signal on the control for the offshore platform will be investigated.

##### 4.1. The Parameters of the Offshore Platform and the External Wave Force

In Figure 1, the masses, natural frequencies, and the damping ratios of the offshore platform and the AMD and other related parameters of the offshore platform are the same as those in [11], which are listed in Table 1, where denotes the length of the offshore platform and represents the diameter of the cylinder. Based on the settings in Table 1, the matrices , , and in (3) can be calculated as

To investigate the effectiveness of the proposed control scheme, we need to compute the values of wave force, which can be solved by referring to [6]. The parameters regarding the external wave are taken from [6] and listed in Table 2, where is the significant wave height, is the water depth, is the peak frequency, and are the drag and inertia coefficients, respectively, is the peakedness coefficient, and is the fluid density. Based on the settings in Table 2, the power spectrum density (PSD) of wave elevation is shown in Figure 2. The shape function is given as [6] where is the vertical coordinate with the origin at the sea floor.

Then, one can compute the wave force according to [6], which is shown in Figure 3.

Due to the irregular nature of the wave force adopted in this paper, we investigate both the peak values and the root mean square (RMS) values of displacement and velocity of the offshore platform and the control force [11]. The same terms , , and are used to represent the peak values of displacement, velocity of the offshore platform, and the required control force, respectively. , , and denote the RMS values of displacement, velocity of the offshore platform, and the control force, respectively, where with as a given measurement period.

##### 4.2. Simulation Results of RSDHC for Offshore Platform with Missing Measurements

In this section, simulation results of RSDHC for offshore platform with missing measurements are presented. Let , and let the probabilistic density function of in be described by from which the expectation and the standard deviation can be easily calculated as and . Then, by Theorem 5, the controller gain matrix in (8) can be obtained as

We give the displacement, velocity, and acceleration curves of the offshore platform system without control and with RSDHC in Figures 4, 5, and 6, respectively. The corresponding control curves are presented in Figure 7. When the RSDHC is used to control the offshore platform, the peak values of displacement and velocity of the offshore platform are reduced from 0.1264 m and 0.2419 m/s to 0.0719 m and 0.1149 m/s, respectively. The maximum control force required is about . The RMS values of displacement and the velocity are reduced from 0.0424 m and 0.0826 m/s to 0.0223 m and 0.0365 m/s, respectively, and the RMS value of control force is about .

The peak and RMS values of displacement, velocity of the offshore platform under no control and RSDHC, and the corresponding control force for different values of are listed in Table 3, where and . Based on Table 3, one can yield the average reduction of peak and RMS values of the offshore platform under RSDHC with missing measurements, which are presented in Table 4.

From Figures 4–6 and Tables 3-4, it can be found that the proposed RSDHC scheme is effective. The peak and RMS values of the displacement are reduced by about 43 percent and 47 percent of the ones without control, respectively, and the peak and RMS values of the velocity are reduced by 52 percent and 56 percent of the ones without control, respectively. From Figure 6, we can see that the acceleration of offshore platform has been decreased significantly.

##### 4.3. Effects of the Missing Signal on RSDHC for Offshore Platform

Now, we turn to investigate the effects of the artificially introduced missing signal on robust sampled-data control for offshore structure. First, we give four missing signals with different probabilistic density functions, and their expectations and standard deviations are computed, which are listed in Table 5. For the cases of the different introduced missing signals, the peak and RMS values of displacement, velocity of the offshore platform, and the control force are computed and listed in Table 6, where . Based on Table 6, one can yield the reduction of peak and RMS values of the offshore platform under RSDHC, which are presented in Table 7.

From Tables 6 and 7, it can be seen that the proposed RSDHC scheme is effective to control the offshore platform, even though the expectation and standard deviation of the missing signal are 0.25 and 0.34, respectively. In addition, we can find that the peak and RMS values reduction of displacement and velocity become smaller with the decrease of and .

#### 5. Conclusions

In this paper, we have developed the problem of vibration control for offshore platforms with missing measurements subject to external wave. A robust sampled-data control scheme has been proposed. It is found from the simulation results that the designed scheme is effective to attenuate the external wave force and thereby improve the control performance of the offshore platform. In our further work, we will extend the proposed results to the case of multiple missing measurements of sensors. Also, we can investigate the sampled-data control for vibration attenuation of offshore platforms subject to multiple external forces. In addition, we may consider the sampled-data control for offshore platforms by using the finite frequency approach [30, 31].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work has been supported by the National Natural Science Foundation of China under Grant no. 61273120 and by Norwegian Centre for Offshore Wind Energy (NORCOWE) under Grant no. 193821/S60 from Research Council of Norway (RCN). NORCOWE is a consortium with partners from industry and science, hosted by Christian Michelsen Research.