# Dynamic Response of DP Offshore Platform-Riser Multi-Body System Based on UKF-PID Control

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. PID Control Method

_{p}is the proportional gain, K

_{i}is the integral gain, and K

_{d}is the differential gain.

#### 2.2. Unscented Kalman Filter Algorithm and Its Principle

_{xx}, respectively. Then, a set of sigma points is selected, and the nonlinear transformation is applied to the sigma points of each sampling data to obtain the point set $\overline{y}$ and P

_{yy}for the nonlinear transformation.

_{xx}, a sigma point set {X

_{i}} under a sampling strategy (i = 0, 1, 2, …, 2n), and the corresponding weights W

_{i}

^{m}and W

_{i}

^{c}are selected, where i is the number of sigma points, W

_{i}

^{m}is the weight used for mean weighting, and W

_{i}

^{c}is the weight used for covariance weighting. Without proportional correction, W

_{i}

^{m}= W

_{i}

^{c}= W

_{i}.

_{i}of 2n + 1 sampling points is calculated:

_{i}is the weight of the ith sigma point.

_{yy}of the output variable y are obtained.

_{i}

^{x}is the column vector composed of the n dimensions of Xia; is the column vector from n + 1 to n + q dimensions of X

_{i}

^{a}; and X

_{i}

^{w}is the column vector from n + q + 1 to n + q + m dimensions of X

_{i}

^{a}. The sampling points will then be transferred using the system’s state equation:

_{i}

^{m}and W

_{i}

^{c}, is used to calculate the predicted mean $\widehat{x}(k+1|k)$ and covariance $P(k+1|k)$.

#### 2.3. State Estimation and Filtering Model of the Dynamic Positioning Offshore Platform under UKF

**η**= [x,y,Ψ]

^{T}; and (ii) the local coordinate system O

_{0}-x

_{0}y

_{0}z

_{0}of the offshore platform, which is fixed on the offshore platform and moves with it and its coordinate origin is usually set at the center of gravity of the offshore platform to describe its velocity vector v = [u,v,r]

^{T}. Two coordinate systems are shown in Figure 1.

**M**is the inertial force matrix of the offshore platform with additional mass and

**D**is the linear damping matrix of the offshore platform,

**u**is the control force matrix of the platform,

**b**is a deviation term describing low-frequency environmental disturbances and un-modeled dynamics, and

**w**

_{1}and

**w**

_{2}are random processes of zero mean white noise.

**R**(Ψ) is the rotation matrix of an offshore platform and can be defined as follows:

**η**

_{p}, and the motion transformational relation between the parallel coordinate system and the earth coordinate system can be expressed as:

**x**= [

_{L}**η**,

_{p}^{T}**v**]

^{T}is the state variable,

**y**

_{L}is the controlled output variable,

**w**

_{L}is the disturbance vector including the deviation term

**b**, and

**n**

_{L}is the measurement of Gauss white noise. The matrices in Equation (28) are defined as follows:

**ξ**can be written as:

**w**zero mean white noise process,

_{3}**η**= [x

_{w}_{w}, y

_{w}, Ψ

_{w}]

^{T}, is the high-frequency motion vector of waves. The matrices in Equations (30) and (31) are defined as follows:

_{0}

_{i}is the peak frequency of the wave and ζ

_{i}is the relative damping coefficient, which is usually taken from 0.05 to 0.2.

**x**and the noise vector

**w**can be written as the following, respectively:

**y**=

**η**+

**η**+

_{w}**n**is the measurement model, including the high- and low-frequency motion of the offshore platform, and

**n**is the measurement noise of the system. Based on the state estimation and filtering model of the dynamic positioning offshore platform, Equation (36) can be rewritten as a time-discrete form:

**t**is the sampling time;

**x**is the state vector of the system at K time step; and

_{k}**y**is the state measurement of the system at K time step. The system noise

_{k}**w**is Gaussian white noise with a nonzero mean and its variance is

_{k}**Q**; the measurement noise

_{k}**n**is a white Gauss noise independent of system noise and its variance is

_{k}**R**.

_{k}_{i}

^{m}and W

_{i}

^{c}(i = 0,1, …., 2n) represent the weights of sigma point mean and variance, respectively;

**n**is the dimensionality of the system; λ = [α

^{2}(n + k) − n] is the scale factor; α determines the distribution of sigma points around the mean point, usually between 10

^{−4}and 1; k is usually taken as 0; and β determines the distribution state of prior state estimation. For a Gaussian distribution, β = 2 is the best.

**K**, state estimation value, and state error covariance matrix

_{k}**P**.

_{k}**Q**and R simultaneously, it is simple to produce the divergence of filtering results when both the process noise

**Q**and the measurement noise

**R**are unknown. The direct distinction between the state transition and the actual process is represented by

**Q**, which is the covariance of the process excitation noise.

**Q**is frequently problematic since the process cannot be observed directly. In engineering issues, the choice of process noise is sometimes a trade-off based on experience between convergence rate and steady-state accuracy. When the noise value in the process is high, the filtering convergence speed is rapid but the stability is weak, and vice versa. In the target tracking and positioning problem, on the one hand, we want to track the response fast when the target is navigating, so we should pick a bigger process noise, but when the target is moving smoothly, the filtering error will be greater. On the other hand, if we choose a lower process noise to enhance the steady-state estimate accuracy, the tracking error will increase abruptly owing to the underestimation of the maneuverability of the target. The offshore platform in this research belongs to the position-holding model rather than the target’s large-scale maneuvering model because it is positioned in a specific spatial coordinate location based on specific operating characteristics. Therefore, a relatively small process noise covariance

**Q**could be set to reduce the influence of model error under the premise of fast convergence speed after repeated debugging. For

**R**, both too-large values and too-small values will make the filtering effect worse. The smaller the value of

**R**, the faster the convergence will be. Therefore, considering the characteristics of

**Q**and

**R**, in order to improve the accuracy and ensure that the filtering results do not diverge,

**Q**and

**R**are generally adjusted and updated in inverse proportion. In this work,

**Q**is considered as 0.01, and

**R**is taken as 10 after repeated debugging and weighing.

**x**and

_{0}**P**are often uncertain or unknown, so they can only be assumed. The problem of filter stability is to investigate the effect of filter initial value selection on filter stability; that is, as filter time increases, the state estimation value

_{0}**x**and the state error covariance matrix

_{k}**P**become increasingly independent of the initial estimation values

_{k}**x**and

_{0}**P**, respectively. According to the researcher’s hypothesis, if

_{0}**P**cannot be determined precisely, we can utilize its higher, more conservative value by selecting a range of probable values. The actual filtering error matrix can fulfill the requirements as long as the estimate error covariance matrix derived in the filtering computation meets the requirements. Therefore, if the system model is assumed to be accurate, only the effects of the initial state value

_{0}**x**, initial covariance value

_{0}**P**, process noise

_{0}**Q**, and measurement noise

**R**are considered. Since the UKF algorithm has strong robustness, it is insensitive to filtering parameters, especially for more complex nonlinear systems. Therefore, the setting of

**P**is relatively loose. In general, as long as it is greater than 0, it can converge and have no effect on the filtering effect. As a result, when there is little confidence in the starting value of the parameters, picking the bigger one can increase the convergence speed and accuracy of parameter identification. Under normal circumstances,

_{0}**P**= k ×

_{0}**I**, k can be taken as a large number. In this paper, the k of

_{3×3}**P**is taken as 100, and

_{0}**x**is the positioning target position of the offshore platform (Target X = +30 m, Target Y = −20 m, Target Yaw = 90°).

_{0}#### 2.4. Calling up the Unscented Kalman Filter in Python

#### 2.5. Modeling of DP Offshore Platform-Riser Multi-Body System Based on UKF-PID Control with Python Language Embedded in OrcaFlex

_{x,y}is the restoring resultant force that contains the longitudinal restoring force F

_{x}of the vessel’s surge motion and the lateral restoring force F

_{y}of the vessel’s sway motion; M

_{z}is the restoring moment of the ship’s yaw motion; e

_{x}is the difference between the surge and the target value, e

_{y}is the difference between the sway and the target value, and e

_{θ}is the difference between the vessel’s yaw angle and the target value; K

_{p}is the proportional gain, K

_{i}is the integral gain, and K

_{d}is the differential gain; and a

_{w}is the wind angle, v

_{w}is the wind speed, and F

_{w}is the force or moment of the wind acting on the vessel. In this paper, the required restoring force of the platform is 1000 kN for every 1 m of displacement, and the required restoring moment of the platform is 1000 kN. m for every 1° of deflection in the yaw direction. The proportional coefficient of force K

_{pf}is 10, the proportional coefficient of moment K

_{pm}is 8, K

_{i}is 0.02, and K

_{d}is 5. Through the cut-and-trial method, the three coefficients of the PID control are adjusted and calculated. The specific process is: when the initial offset of the platform is large, the proportional gain coefficient K

_{p}is mainly adjusted, and then the integral gain coefficient K

_{i}is adjusted after the offset of the platform reaches a certain degree. After the two coefficients are adjusted to a certain extent, the two coefficients are kept unchanged within a certain range, and then the differential gain coefficient K

_{d}is slightly adjusted by using the control variable method until the positioning effect reaches a satisfactory level.

## 3. Validation

^{8}Pa, the top tension is 2500 kN, and the bottom tension is 1200 kN (in this paper, the same effect can be achieved directly by applying the corresponding pretension at the top and bottom in the initial stage of simulation). The riser is divided into 10 segment units.

_{C}= 1 m/s, and it is assumed that the current velocity at the middle layer is 75% of the surface current velocity, the current velocity on the seabed is 15% of the surface current velocity, and the current direction is 30°. The profile of the current in the water depth direction is shown in Figure 2. The significant wave height is 7 m, the peak period is 14 s, and the direction is 20°. The average wind speed is 15 m/s and the wind direction is 20°.

## 4. Results and Discussion

#### 4.1. Model Establishment

^{3}. The parameters of the slip joint are as follows: the outer diameter is 0.65 m, the inner diameter is 0.6 m, the bending stiffness is 508,944 kN·m

^{2}, the axial stiffness is 1 kN, Poisson’s ratio is 0.3, the torsional stiffness is 391,500 kN·m

^{2}, and the linear density is 0.385 t/m. The environmental parameters are as follows: the seawater density is 1025 kg/m

^{3}; the water depth is 500 m; the seabed is flat; the JONSWAP spectrum is chosen for the modeling; the peak enhancement factor γ is 4.7934; a is 0.0086, which is calculated by the specified H

_{s}and T

_{z}; σ

_{1}is 0.07 and σ

_{2}is 0.09; the significant wave height H

_{s}is 7 m; the zero crossing period T

_{z}is 9 s; the wave direction is 270°; the current is 0.5 m/s; and the direction of the current is the same as the wave direction. The offshore platform has eight pontoons, the draught of the offshore platform is 24.38 m, the displacement is 10,000 t, and the moments of inertia are I

_{x}= 500,000 t.m

^{2}, I

_{y}= 7,000,000 t.m

^{2}, I

_{z}= 7,000,000 t.m

^{2}, respectively. The positioning coordinates of the offshore platform are (target X = +30 m, target Y = −20 m), target heading = 90°. The schematic diagram of the model is shown in Figure 9.

#### 4.2. Calculation Analysis

#### 4.2.1. Effective Tension of Riser under the UKF-PID and Single PID

#### 4.2.2. Bending Moment and Curvature of Riser under UKF-PID and Single PID

#### 4.2.3. Rotation Angle of Riser under the Action of UKF-PID and Single PID

#### 4.2.4. Six Degrees of Freedom of the Offshore Platform under the Action of UKF-PID and Single PID

## 5. Conclusions

- (1)
- The effect of the UKF-PID dynamic positioning system on the variation of the effective tension of the riser is not significant, which is fully reflected in the fluctuation, coordination, and synchronism of the variation. Compared with the multi-body system controlled by a single PID static positioning system, the bending moment of the platform riser becomes larger when the positioning system is changed. With the addition of the unscented Kalman filter, the strong nonlinearity of riser bending change in the whole system is better reflected. This is helpful for the visual capture of this nonlinearity in engineering practice and the investigation of potential safety hazards and measures to improve safety.
- (2)
- Compared with the dynamic positioning system under the control of a single PID, the bending moment of the riser in the UKF-PID dynamic positioning system and the transmission of bending along the length of the riser will change. This change is mainly reflected in a relatively larger bending moment. The bending moment of the riser at a certain position will be more severe, but the relative hysteresis of the bending moment and curvature still exists. In addition, the hysteresis of load and strain transfer will be further enhanced at some locations. In this case, the overall synchronization and coordination of riser curvature changes along the length direction will be reduced.
- (3)
- The proposed control approach of this paper improves the nonlinearity of the three-degrees-of-freedom translational movements of the offshore platform and modifies the energy distribution in three translation motions of the offshore platform. Under UKF-PID control, the overall motion nonlinearity of the offshore platform-riser multi-body system has been significantly improved. The enhancement of nonlinearity makes the system more sensitive to the change of its own motion response when overcoming the change of external environmental load, which also leads to the increase in the Ry and Rz angles of the riser.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**(

**a**) Top riser angle versus offset of offshore platform. (

**b**) Bottom riser angle versus offset of offshore platform[52].

**Figure 8.**Bending moment and curvature of the riser under the action of UKF-PID and single PID [55].

**Figure 9.**Model of the rigid–flexible fluid multi-body system with a dynamic positioning system of UKF-PID.

**Figure 10.**Dynamic response of the effective tension of the riser under the action of UKF-PID and single PID.

**Figure 12.**Dynamic response of the rotation angle of the riser under the action of UKF-PID and single PID.

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**MDPI and ACS Style**

Zhang, D.; Zhao, B.; Bai, Y.; Zhu, K.
Dynamic Response of DP Offshore Platform-Riser Multi-Body System Based on UKF-PID Control. *J. Mar. Sci. Eng.* **2022**, *10*, 1596.
https://doi.org/10.3390/jmse10111596

**AMA Style**

Zhang D, Zhao B, Bai Y, Zhu K.
Dynamic Response of DP Offshore Platform-Riser Multi-Body System Based on UKF-PID Control. *Journal of Marine Science and Engineering*. 2022; 10(11):1596.
https://doi.org/10.3390/jmse10111596

**Chicago/Turabian Style**

Zhang, Dapeng, Bowen Zhao, Yong Bai, and Keqiang Zhu.
2022. "Dynamic Response of DP Offshore Platform-Riser Multi-Body System Based on UKF-PID Control" *Journal of Marine Science and Engineering* 10, no. 11: 1596.
https://doi.org/10.3390/jmse10111596