Self-Adaptive Fuzzy Control Approach for Jack-up Rig Jacking System Based on Particle Swarm Optimization

Many offshore projects, such as offshore oil and gas exploration and offshore wind farm development, have required controlling the position of elevating the hull in a stable and balanced manner in recent years. The Jack-up Rig (JuR) jacking control systems are a revolutionary innovation that is already being employed in offshore drilling and other maritime structures. The system is utilized automatically to control and stabilize the position of the JuR during sea state disturbances to keep platforms from being displaced. As a result, developing an improved control theory for the Jacking system (JS) of a JuR is very important. In this paper, first, we investigate how to adapt to the effects of external forces and hydrodynamic amplification using a particle swarm optimization strategy based on a fuzzy controller. Then, using the fuzzy controller as the foundation, we proposed the PSO - Self Adaptive Fuzzy Controller (PSO-SAFC) guaranteed by Lyapunov criteria and compares it with other fuzzy-based algorithms to show the benefit of adaptively and stability of the system. To verify the proposed algorithm, embedded STM32F746VG central processor used to build the algorithm and run Simulink models on Matlab, the collected data on the positions of the legs and the tilt compared it to the setting command values to move the Rig body and provided control signals to the Drive system. Eventually, the advantages of the proposed approach are demonstrated by the simulation and experiment outcomes.


I. INTRODUCTION
A. BACKGROUND Offshore oil and gas exploration, drilling, and work-over have all been done on jack-up drilling platforms for decades. In the jacking operation of a JuR, there are numerous approaches [1], [2], [3], [4] for ensuring stable control and reducing influencing forces (JuR) to the process. Even if each technique has its own set of advantages, the study of advanced control algorithms pushes researchers to develop theory and The associate editor coordinating the review of this manuscript and approving it for publication was Yanli Xu . practice solutions to help the system work more consistently and securely control system and has many more operations than before.
The JuR is a kind of movable platform consisting of a large floating hull outfitted with multiple detachable legs, usually three or four, capable of elevating the hull above the sea's surface [3], [4]. The 3-legs JUR 3D-designed model shown in Fig. 1, for example, are triangular barges that have been completely fitted during the drilling process and are motivated by three truss legs, the other shapes are utilized less frequently due to their complexity in structure and organization. JuR is also known for developing the jacking control system, which has three, four, six, and eight movable legs that can extend upward or below the hull. However, according to recent studies on the laboring environment, the JuR can drill in waters up to 350 feet deep, but for deep water less than 600 feet deep, the overall economics and operational efficiency of deep-water JuRs are typically more favorable than shallow water semisubmersibles [5], [6], [7], [8]. In the conception of jack-up and semi-submersible rigs, the JuR will always be preferred over a semi-submersible rig providing both are capable of digging the well.
Many different types of offshore platforms have been developed depending on a variety of variables, including technical and scientific advancements, economics, the need to use deeper natural resources [9], [10], and disturbance limits [11]. Offshore solutions for the challenges include developing more high-speed technology and conducting more strong study efforts [12], [13], [14], whereas they need more costly expenditures and system design adjustments. Meanwhile, these solutions may be switched by content economically and task methodologies that can be maintained constant at the same time for cost savings, while still utilizing the same system architecture and meeting standards and laws. Offshore engineering has recently received extensive attention from intellectual and industrial researchers to assist service providers and technology suppliers satisfy the high demand requirements for an offshore platform [15], [16]. Just developed systems must eventually assure that human, marine, and coastal ecosystems are safe, while also maintaining the significant time and resource expenditures made.

B. LITERATURE REVIEW
As can be seen, there are many challenges related to the JuR model, methodologies, and applications, as well as financial concerns, but they all revolve around two questions: 1) which system metrics should be improved (quality, unit cost, scheduling, hourly rate, safety, communication, and technology) and 2) which solutions should be used to ensure JuRs working in a rough environment while keeping operation and maintenance costs as low as possible [17]. Jacking systems (JSs) were designed as vertical transit vehicles to make it easier for the Rig body to go from one level to another by lowering the time it takes to move and the amount of physical energy in the sea. The system is synchronously controlled by the drives attached to the Rig's legs, much the same as heavy lifting equipment. It consists of sets of motors installed together to drive coaxially while simultaneously moving the whole Rig's body in various condition operating modes. In reality, in the field of marine science, various control systems have been investigated to improve system quality [18], assure safety in difficult situations [19], and save energy [20]. Because of the similarities in nature, we can apply a lot of new ideas to the JS if we consider it as an elevator system. Only one word of caution: the Rig is huge, has high inertia, and operates in harsher conditions than land-based elevators. Accordingly, with such qualities in mind, this work focuses on challenges relating to fuzzy logic adaptive control, in which the Particle Swarm Optimization (PSO) algorithm is used as a form of artificial intelligence to improve the automatic control adaption capability of fuzzy controllers.

1) SELF-ADAPTIVE FUZZY CONTROL (SAFC)
Nonlinear adaptive control has been studied for several years to deal with the difficulty of uncertainties and disruptions [18], [21], [22], [23]. An improved evolutionary algorithm based on fuzzy logic which is a dynamic multi-objective evolutionary method based on fuzzy logic is described in [24]. When an environmental change happens, it is critical to provide a real and truthful reaction to the change to enhance the variety of options. Robust adaptive control is presented as the best approach for reducing the influence of error and uncertainty disturbance and noise. Therefore, the efficiency of a robust adaptive control employing a bounded estimates scheme of the disturbance by using Genetic algorithm (GA), and the tracking performance with robustness is maintained [18]. In conventional control theory, Lyapunov-based approaches have long been utilized to build controllers that assure the system's closed-loop stability. Tracking the reference input takes a large amount of time using a Lyapunov-based approach, and the inherent approximation error leads the system to have larger integral absolute errors [25]. Thus, adaptive control Lyapunov-based approach appreciate on JS control because this system controls the slow speed with large inertia. Moreover, several existing adaptive fuzzy controllers [26], [27] have a key limitation in that they were designed for systems with limitless actuators. As a result, it can be determined that the SAFC based on Lyapunov criteria always maintains stability under actual conditions and assumptions, and thus the Adaptive fuzzy control (AFC) based on Lyapunov is regarded as viable when the control object is JS.

2) PARTICLE SWARM OPTIMIZATION (PSO)
The optimization of fuzzy systems has received a bit of attention in the recent literature. Optimization algorithms can tackle nonlinear, well-constrained, and even NP-hard problems, making them a powerful tool. Genetic Algorithms [18], Ant Colony Optimization (ACO) algorithms [28], Cerebellar Model Articulation Controller (CMAC) [29], and PSO [21], [28], for example, are among the most widely used optimization approaches. Especially, PSO is a novel evolutionary optimization approach that is rapid, easy, and likely to be utilized for searching a large search space for the optimal answer. Unlike other heuristic strategies, it features a versatile and well-balanced method to boost global and local exploration skills. Furthermore, specifying the target function and placing indefinite boundaries on the optimized parameters is sufficient, and PSO has proven to be successful in many applications when combined with fuzzy logic or artificial intelligence.

3) HYBRID CONTROL APPROACH
Hybrid Control approach focused on a combination of a Lyapunov-theory-based technique and a PSO based optimization approach for developing stable adaptive fuzzy controllers [30], [31], [32] which a potential research direction. The goal is to create a SAFC by optimizing both its structure and parameters such that the controller can ensure necessary stability while also providing sufficient performance. To establish stable Fuzzy control (FC) and assure faster convergence, a Lyapunov-based technique is used in conjunction with PSO. Therefore, we may predict that combining PSO and SAFC approaches will result in a hybrid algorithm that is more efficient.

C. CONTRIBUTION
To overcome the aforementioned problems, we present a novel model of the PSO -Self adaptive fuzzy controller (PSO-SAFC) to cope with position control for a JS of the JuR. The contributions of the paper are presented as follows: 1. The PSO-SAFC model is proposed with the goal of adapting to external forces, hydrodynamic amplification, and mechanical errors. The PSO algorithm is used to construct the optimal coefficient λ i to enhance the membership function of the fuzzy controller used to increase control quality in a time-varying environment, which is represented in a novel algorithm.
2. A simulation experiment with two case studies compares the FC, AFC, FPID, and PSO-SAFC guaranteed by Lyapunov criteria to demonstrate the benefit of the recommended approach under the influence of uncertain factors. 3. The control algorithm is embedded in the STM32F746-VG central processor with Matlab software, which gathers data on leg locations and rig body tilt and compares it to command values to move the Rig body and generate control signals to the Drive system. Thus, the advantages of the proposed approach are demonstrated by the simulation and experiment outcomes.
The rest of the paper is organized as follows. Section II presents a problem description and motivation. Section III presents the proposed a SAFC by PSO, a Lyapunov based technique is used in conjunction with PSO such that the controller can ensure necessary stability, while in Section IV, simulation and experimental case studies for both conventional FPID, FC, AFC and proposed PSO-SAFC are demonstrated. Section V concludes the paper. Figure 1 depicts the hull's motions as well as the relationship of forces. The hull vibrates vertically and horizontally around the X-and Y-axes, moving up and down in the Z-direction. The hull is driven by electric motors positioned on three points on the hull that correspond to the position of the legs in elevating mode. In working conditions, the motors on each position have always had the identical properties and are synchronously controlled at the same speed. Figure 2 shows a schematic of the JS.

A. DYNAMIC EQUATIONS OF JACKING SYSTEM
The elevating mechanism is used to move the hull to a fixed position. According to the overall diagram in Fig. 2, the force elements pushing on the JuR's hull are determined in the system of differential equations defining the kinematics as follows [33]: where M = n i=1 m i is total weight of the JuR; C, K are damping and stiffness of the system, and τ (t) = τ m (t)+τ d (t), τ m (t) = n i=1 τ i (t) is overall torque of motors. In addition, wave, wind, current, and external noise components are included in the total of the influences of external and internal disturbances. The following is the definition of the equation:

B. DYNAMIC AMPLIFICATION FACTOR IN TRANSFER FUNCTION OF KINEMATICS OF JS
In general, there are always factors that create fluctuations in actuality. In which, deviations owing to oscillations relating to Dynamic Amplification Factor (DAF) values would impact the actual position in control process of the JS [34]. Assumption 1: The JS of the JuR is an Single Degree of Freedom (SDOF) system in elevating mode.
Then, the displacement DAF of a damped JS subjected to a step load is consequently calculated as the ratio of the dynamic response to the static response. There are always factors that create fluctuations in actuality. As a result, deviations owing to oscillations relating to DAF values would impact the actual position of the hull.
The dynamic response of damped JS can be determined based on the Duhamel's integral as where ζ = (0.05 − 0.2) is the damping ratio of the system. Define DAF as the ratio of the dynamic response to the static response, the displacement DAF of an damped SDOF system subjected to a step load is therefore [35]: Remark 1: Following Society of Naval Architects and Marine Engineers [34], in case of the Rigs have Mass > 2500Ton we have conditions ζ < 0.07 and t > 0.124.
Assumption 2: In the case of Assumption 1 is satisfied, the DAF is bounded in range of 0 < DAF < 2.
In this study, using Matlab to calculate the DAF values that satisfy the conditions in Remark 1 along with Assumption 1 and Assumption 2, we get the results as shown in Table 3, in which ω n = πn/30 = 48, 984(rad/s). Therefore, the selected optimum value of DAF is 0.6284 satisfied conditions in Remark 1. Thus, this value will be used to calculated a transfer function of kinematics of JS.
For convenience and simplicity in embedded control and simulation, we use the method of decoupling system to separate the system into subsystems with the same transfer function. Then, substituting DAF = 0.6284 into an eq. (3), we have . Therefore, eq. (1) is rewritten as Using Laplace transform to eq. (6) with initial conditions The displacement of the hull is calculated by Substituting eq. (8) into eq. (7), we have with k e = 1/R is the inverted factor in rotary -translational drive (including the number of motors and transmission coefficients). Finally, the equation describing the transfer function of kinematics of JS is written as:

III. ANALYSIS NEGATIVELY FACTORS EFFECTING ON CONTROL PROCESS A. EXTERNAL FACTORS
When the sea state is rough, the JuR is normally not allowed to operate. As a result of the influence of environmental force vectors such as wave, wind, and current force, termed F envi by Fossen et al. [36], and Dang et al. [18], JS operations are only possible when the sea conditions are calm.
Waves are the most significant indication of the effect on offshore constructions. In six degrees of freedom, waves force an offshore structure to surge, pitch, roll, sway, heave, and yaw. The F wave wave force equation for the control issue [36] is as follows: with φ qr ∈ (0−2π) is the phase angle of wave components. S is the wave spectrum, ψ r is the direction wave. The ω and ψ is the harmonic amplitudes of wave frequency ω q . The k q = 2π/λ q is the number of waves, in which λ q is the wavelength.
Wind loads are calculated using one of four methods: wind tunneling, numerical modeling, formulae based on experience, and field testing. The wind force is estimated using the formula [36] with a calm sea state and the system functioning in the condition that the wind frequency and direction are simulated as slowly changing values.
where g R is wind direction, V R is wind speed. A T and A L are the transverse and lateral projected areas, represented by C X and C Y . Even though the current velocity fluctuates with time, space, depth, and vortex vibration, the current is considered stable. We need to compute the velocity and velocity profile of the flow on the water if we ignore the following elements in the calculation procedure for offshore works. The current velocity is below where u c and v c are current velocity compositions, and ψ L and ψ H are angular compositions impacted by high and low frequency values, respectively. In this study, the simulation would be used to investigate the effects of environmental factors while referencing the parameter Table 5 [21] under suitable weather situations.

B. MECHANICAL ERRORS
The rack-gear system that raises and lowers the JuR is a cultivated elastic mechanical system that produces vibration and noise. The following sorts of defects can occur in a simple gear system: contact position error, bearing assembly error, axial error, pitch error, and gear radial error.

2) COMPOSITION OF GEOMETRIC ERRORS
If several geometric faults impact a part, the total error components are overlaid by each error, therefore where: dx ij , d y ij , d z ij , δx ij , δy ij , δz ij are the jth error of the ith part, respectively. The displacement errors corresponding to the three rotation errors of the same portion under overlapping are given by x 0i , y 0i , z 0i , ∂ 0i , β 0i , γ 0i .

3) AXIAL RUNOUT
Axial runout can be caused by the clearance between the inner bore and the shaft, resulting in transmission problems. Furthermore, a specific amount of clearance is allocated in the design of gear trains for ease of assembly, or clearance fit is employed in each mating surface. The clearance of the reserved assembly and the clearance fit produce the axial runout. The addition of reserved clearance improves the accuracy of the transmission error estimate. As a consequence, the clearance-induced rotation angle error is given by: where θ Jx and θ Jy are the clearance-induced rotation errors of the gear axis around the x-and y-axes, respectively. θ Jx = J R and θ Jy = J R may be used to calculate θ Jx and θ Jy . The dividing circle radius of the gear is R, and the maximum clearance value or clearance after assembly adjustment required by design is J . The phase angle of eccentric assembly, γ , is a uniform distribution in the range of [0, 2π].

4) TOOTH PITCH ERRORS
The common error factors contribute to tooth pitch mistakes: the tangential error of the dividing circle, as well as changes in tooth pitch due to gear runout and tooth profile form failures. It indicates the angle variation error of the gear in turning a pitch angle (360 0 /z), which might damage the gear transmission's stability, and its value is −f pt ≤ f pt ≤ f pt . The tooth pitch's linear error is converted to an angular error on the gear dividing circle, and Monte Carlo is used to simulate the error fluctuation on each tooth pitch of the gear. As a result, the tooth pitch error's angular error p k may be determined. Consequently, the angular error p k due to the tooth pitch error can be calculated as follows: where f pt (µm) represents the limit deviation of individual tooth pitch; R = mz/2(mm) is the theoretical dividing circle radius of the gear, m refers to the module (mm), and z is the number of teeth. In general, the accuracy of the rig's raising and lowering actions is determined by the geometry of the gear and rack system. When the teeth are geometrically defective, the gear and rack system's performance suffers. However, defects in gear and rack transmission systems include radial misalignment, axial misalignment, tooth profile misalignment, pitch, and other unexplained flaws, among others. The major source of the system's model error is this factor. Uncertainty factors induced by this parameter enhance control complexity while simultaneously promoting the system's self-adaptability. This inaccuracy is treated as a random effect during simulation and experimentation as part of the suggested approach toward a robust adaptive for the mechanical system. Assumption 3: On the gear dividing circle, the linear error of the tooth pitch is bound and converted to an angular error.
Assumption 4: Assume that a maximum value of the error for control and normally states is comprehended or may be determined for each state.
Remark 2 The error fluctuation on each tooth pitch of the gear is simulated using Monte Carlo, which utilizes a normal distribution (a normal distribution indicates that outcomes are evenly distributed around the mean). The position error bound is a basic limit of position accuracy for ultrawide bandwidth systems working in allowed conditions.

IV. PARTICLE SWARM OPTIMIZATION SELF-ADAPTIVE FUZZY CONTROL APPROACH A. FUZZY CONTROLLER
In this study, the authors apply a fuzzy model including a double-input signal e, de/dt, and a single-output signal u [18]. The fuzzy model consists of a set of rules and their consequences, which are often assumed to be linear functions of the inputs. If-Then rules are represented as a rule notation form within B i as follows: . .and χ n is L i n then u is B i (19) where L 1 i , L 2 i , ..L n i ∈ R h and B i ∈ R h are indicates the fuzzy sets of output signal and input signals [38]. By using the singleton fuzzifier, the Max-Prod inference rule and the center averaged defuzzifier, the output signal u is defined as for m is the amount of fuzzy rules, µ L i j (χ j ) indicates the Membership Functions (MFs). We get as the fuzzy parameter vector. Define the fuzzy basis functions with φ is represented as follows: and define φ(χ) =[φ 1 (χ), φ 2 (χ), . . . ,φ m (χ)] T as the fuzzy basis function vector. Thus, the fuzzy output is able to be expressed as the linearization parametric form Lemma 1 [39]: For any given real continuous function ϕ(χ) on the compact set ⊂ R n and arbitrary ξ > 0, there exists a fuzzy system u(χ) in the form of an equation as According to Lemma 1, the nonlinear function ϕ(χ) can be approximated as Thus therein, Z and , respectively, are compact regions for χ and δ. Hence, define the minimum approximation error as

B. ADAPTIVE FUZZY CONTROLLER
We establish a AFC structure on m.file format with the MFs flexible value to improve the structure of controller. Therefore, the fuzzy modulator has a single-output u P (χ) and a double-input e(t), de/dt. The fuzzy output performance is represented as Following eq. (28), the AFC response can be described as

C. PROPOSED PSO-SELF ADAPTIVE FUZZY CONTROLLER
We apply the PSO algorithm to calibrate the parameter of MFs because it's easily implemented and computationally efficient. In this study, the PSO algorithm is used to find the optimal parameter controllers for operating the JuR nonlinear object in conditions of Assumptions 1, 2, 3, and 4. The determination of optimal coefficient λ i based on PSO algorithm [40], to deal with the improving control quality in the time-varying environment is represented in Algorithm 1. The PSO algorithm completes the correction of fuzzy set values with coefficient λ i to achieve the optimal parameters.
Where, y d is the desired position, y is the JuR response, u P (χ) is the input control to JuR model, β is a constant and is a small constant. The second term of eq. (36) is to obtain the minimum input control u P (χ), and controlled by β.

Algorithm 1 The PSO algorithm calibrates the MFs values [21]
Initialization: for each particle i(1 ≤ i ≤ s) do Randomly initialize the position λ p i (k) Randomly initialize the velocity v i (k) Initialize calibration coefficient λ i (λ 1 , λ 2 , λ 3 , λ 4 ) end for for each iteration k do (a) Each particle λ p i (k) update the values of λ p i,j (k) and λ Gb i,j (k) where (d) Apply newly coefficient λ (e) A suitable point to start is to compare the obtained value to the termination condition. If the convergence criteria do not fulfill the criterion, raise k and return to step 1. (a).

end for
The search process of the values λ i (λ 1 , λ 2 , λ 3 , λ 4 ) carries out in 4-dimensional space. The limited search range will lead to locally convergent values (express at Definition 1). A large search space will achieve a globally optimal solution (such as Definition 2), but the convergence time will be slow. Definitions 1 and 2 are defined as follows: Definition 1 [41]: Local convergenceλ ∈ D of a fitness function f : D ← is the input element with f (λ) > f (λ) for ∀λ in the surroundings ofλ. Definê Definition 2 [41]: Global convergence λ * ∈ D of a fitness function f : D ← is the input element with f (λ * ) > f (λ) for ∀λ. We get 86070 VOLUME 10, 2022 The fuzzy control gives different results for nonlinear systems like the JuR model. So if we choose the Lyapunov func-tionV (χ) only has a Negative value, and the approximate of the nonlinear components are not satisfied. Thus, our method is to combine the parameters of the adaptation and the fuzzy controller so that the PSO searching obtains the negative derivative of Lyapunov functionV de (χ) [41], that iṡ where σ > 0, that will give an exponential convergence rate.
The new performance index criteria can be rewritten by therein, ξ v represents the erroneous of the desired derivative Lyapunov function, which is defined as The goal of the PSO-SAFC solution is to bring ξ v to zero, that is,V (χ) ≤V de (χ). Minimum the erroneous of the desired derivative Lyapunov function ξ v depends on the SAFC coefficient δ T λ (express at Eq. (27)) under working conditions in Remarks 1, 2, and Assumption 2. So the approximation error ξ v is always initial within the limit range [32], satisfy Based on the stable condition (at Eq. (39)),V de (χ) is less than −σ V (χ), soV (χ) can be converted as follows [41] The PSO-SAFC solution ensures self-adaptive by δ * T λ coefficient within a limited range (Eq. (42)), and process control under stable conditions (Eq. (39)) through the Lyapunov evaluation constraint (Eq. (27)). Thereby, the proposed PSO-SAFC guarantees that the JuR closed-loop control system is stable and enables tracking of the control goal. Table 4 presents the parameter dimensions of the TamDao05 JuR model which is scaled down by the ratio of 1:100. The parameters of the environmental impact are given in the Table 5. The purpose of this work is to use simulation and experimental contents to make it easier to design, build, and operate embedded systems using Matlab. We begin by constructing the JuR legs, which will be powered by DC motor pluses through the shaft system as in Eqs (6) to (10). We used the Matlab 2019b software to simulate the JuR model under the same climatic conditions and with the same parameters.  The fuzzy's structure is based on the m.file matlab code, which helps the PSO searching algorithm easier to identify the convergence value. These optimal coefficients aim to adjust the MFs value of the fuzzy controller to fulfill the control requirement.

2) FUZZY-PID CONTROLLER DESIGN
In this subsection, the authors will compare the JuR positioning response of the proposed PSO-SAFC controller in this paper with that of the FPID controller as follows:   control cycle according to the erroneous amplitude. The Proportional Integral Derivative (PID) coefficient is updated by: The updated coefficients u f u f ( k p ), u f ( k i ), u f ( k d ) are defined by fuzzy system. The notation of MFs for NB is Large Negative, for NBB is Near Big Negative, for NS is Small Negative, for NSS is Near Small Negative, for ZE is Zero, for PSS is Near Small Positive, for PS is Small Positive, for PBB is Near Big Positive, and for PB is Large Positive. We use a fuzzy system with a double-input e(t) = [−0.5 ÷ 0.5], de/d(t) = [−10 ÷ 10], and a triple-output k p = [60 ÷ 100], k i = [0 ÷ 1], and k d = [0 ÷ 50] to calibrate the coefficients K p (s), K i (s), and K d (s) [21].

3) PSO-SAFC CONTROLLER DESIGN
To fairly compare the simulation results obtained, PSO-SAFC and other solutions apply the same number of training parameters and vessel structure modelling. The performance of PSO-SAFC is compared with that of the FPID [35] for JS control. In this study, we use the PSO searching parameters as same as Table 6 for implementing the proposed system. Then, the searching process of λ i optimal coefficient is randomly generated. Thus, 20 iteration were applied with random initial models so that every iteration process could be able to wear a  different position to search the JuR control. Through searching process, the best convergence result is 8.97409 × 10 −4 which corresponds to λ i (λ 1 , λ 2 , λ 3 , λ 4 ) values. In addition, the convergence performance of PSO-SAFC is described in detail by Fig. 3. The best fitness value and the mean fitness value are getting closer and obtaining better performance.

4) NUMERICAL SIMULATIONS
Comparison of simulation results obtained when using FPID and PSO-SAFC controllers are presented as shown in Fig. 4, and Fig. 5. Figures 4(a), and Fig. 5(a) show the actual position response of the JuR under the impact of environmental forces when applying the FPID approach (red line) and the PSO-SAFC approach (blue line). Figures 4(b), and Fig. 5(b) are the results obtained when the force is applied. Besides that, Fig. 4(c), and Fig. 5(c) express the environmental force acting on the truss. The environmental forces τ d , act on the kinetic model of JuR (expressed by eq.(1)) and the thrusters dynamic τ i , cause the JuR response erroneous. The PSO-SAFC approach (presented in section III) is conducted on TamDao05 JuR model in two cases study.
Case study 1: The simulation results that apply the FPID and PSO-SAFC controllers are pointed out in Fig. 4 with the 86072 VOLUME 10, 2022  Fig. 4(a). Table 7 presents the findings of the comparisons made between the suggested method and those of Fuzzy controller [42] and FPID controller [21], and Adaptive Fuzzy Controller (AFC) [19]. Maximum overshoot, PSO-SAFC, AFC, FPID, and Fuzzy values are 3.5 cm, 5 cm, 5.5 cm, and 6.5 cm, respectively, in the details. The Fuzzy, FPID, and AFC all have reaction times that are slower than the proposed PSO-SAFC, which has a response time of about 4.5 seconds, 4 seconds, and 2 seconds. The proposed solution with the self-adaptive coefficient δ i has reduced the amplitude of fluctuations and overshoots caused by the impact of the environment, thereby making the JuR response in a low erroneous case. Besides, the optimal correction coefficient λ i helps the proposed solution to provide a better response than the others. Additionally, PSO-SAFC has a substantially smaller fluctuation amplitude than the competition.
Case study 2: The JuR model moved to the desired position (30 cm) by using the FPID and PSO-SAFC controllers under the influence of the environment in this example, the rig body is moved to the predetermined position (30 cm) while counteracting external forces, and then they keep lifting it until it reaches 60 cm from the 30 cm reference position at the 25th-second. In the first scenario, the identical PSO-SAFC  controller parameters are utilized. According to Fig. 5(a), the PSO-SAFC controller has a roughly smaller overshoot amplitude than the FPID controller. Therefore, the comparison findings in Table 8 demonstrate that the overshoot of PSO-SAFC is around 4 cm, 2.5 cm, and 1.5 cm less than those of the Fuzzy, the FPID, and the AFC. The fluctuation value using the PSO-SAFC is lower than the 0.015 cm value using the AFC. Changing the reference position in combination with environmental influences will make the JuR position erroneously higher than case 1. The results using the PSO-SAFC solution still maintain the JuR position in the stable domain with self-correction coefficient δ i and guarantee the response quality with the optimal coefficient λ i . The comparative findings therefore show that the PSO-SAFC solution meets the technical criteria in a good manner. It is clear that the proposed PSO-SAFC controller can keep the JuR response quality stable even when the set-point conditions change.

B. ANALYSIS OF PROPOSED CONTROL APPROACH BY EXPERIMENT STUDIES
In this study, with parameter of JuR model dimensions in Table 4, the main parameters of the UT-JuR 01 are presented in Table 9 To control the DC motor system for lifting the JuR, the authors use Matlab software to implement the proposed realtime PSO-SAFC and real-time FPID controllers into the main processor STM32F746NG. Six DC motors are used to move the JS up and down on three movable legs and to the required position. Therefore, the computer receives the experimental model's response signal through the UART standard and syn- thesizes it into charts for analysis. Last but not least, Fig. 6 depicts the JuR experiment model (named UT-JuR 01). The following is the operating principle: • The distance sensors (Encoder) locate the JuR body, and the IMU sensor block detects the JuR's inclination. The feedback signal is then sent to the STM32F746NG central processor via the ADC converter circuit.
• To determine the value of the control signal, the real-time embedded controller compares the values of the desired signal and the feedback signal.
• To raise and lower the JuR to the necessary position, the driver circuit converts the control signal into a PWM pulse that controls the speed and direction of the DC motor. The experiment study is conducted in two examples in this section to verify the efficacy and adaptability of the proposed PSO-SAFC approach against changing environmental conditions. Figures 8 and 9 depict the JuR's actual 3-leg reaction while using the FPID embedded technique (red line) and the PSO-SAFC embedded approach, respectively, during the lifting of the rig body (blue line).

1) EXPERIMENT STUDIES IN UT-JuR 01 a: EXPERIMENT CASE STUDY 1
In this scenario, the real-time embedded controllers for the FPID and PSO-SAFC force the JuR's experiment model to the desired position of 30 cm from the reference position of 00 cm and keep it balanced for 580 seconds (Fig. 8). Moreover, adopting the PSO-SAFC approach as opposed to the FPID at 1.35 cm, the Rig legs' oscillation amplitude is less. Additionally, the proposed PSO-SAFC solution and the FPID solution have reaction times of 120 and 135 seconds, respectively. Consequently, the suggested solution is precise, extremely effective in the actual process, and satisfies the quality standards.
Then real-time embedded controllers continue to lift the experiment model to position 60 cm from the reference of 30 cm at the 300th-second.

b: EXPERIMENT CASE STUDY 2
In this case, the real-time embedded controllers for the FPID and PSO-SAFC force the JuR model to the designated point (30 cm) in approximately 280 seconds. The experiment model is then raised by embedded controls until it is 60 cm from the set point of 30 cm at the 300th-second. Figure 9 displays the outcomes of experimental case 2. The reaction time for the PSO-SAFC scenario is 120 seconds, which is quicker than the FPID solution's 155 seconds response time. The response amplitude that implements the suggested PSO-SAFC solution changes gradually over time rather than all at once. When utilizing the PSO-SAFC controller, the JuR legs' oscillation amplitude is 1.55 cm less than comparing with the FPID controller. Thus, it has been demonstrated that the proposed control approach works well under realworld circumstances. To adequately illustrate the proposed approach, the JuR experimental model has to be tested extensively in various working modes, extensively in various working modes.

2) EVALUATE SOME EXPERIMENTAL RESULTS
Following is a summary of the key aspects based on analysis of two experimental cases, as shown in Figures 8 and 9.
• Although the PSO-SAFC response is more consistent than FPID, the results are not significantly different because of the high sensor noise, the identical mechanical components on the hardware, and the reaction delay of an actuator.
• When both controllers are based on fuzzy technology, they maintain the feature of fuzzy sets, which is sharp impulse responses (because the membership function is a built-in isosceles triangle. This response characteristic can be improved when the number of fuzzy laws increases. As there are more fuzzy regulations, this response characteristic can be enhanced. In this study, we only employed 23 membership functions integrating 86074 VOLUME 10, 2022  35 rules for each JuR leg controller due to the slow processing speed of the central controller. • The PSO-SAFC controller's position response on three JuR legs is more even than the FPID's. Although there is some deviation, the overshoot amplitude remains within the permitted range. The realistic outcomes demonstrate that PSO-SAFC is superior to other approache. In light of the additional experimental findings listed in Table 10, which were deemed satisfactory, we proceed with the relevant observations: • In all 3 experiment cases in Table 10, the legs position oscillation amplitudes yielded with the established quality of the achieved request(< 6%).
• The are no hard fluctuations within the response amplitude changes steadily over time.
• In experimental conditions, the shortest reaction time needed to attain the equilibrium value is within the permitted range (< 5.5 seconds). The experimental model of the JuR needs to be tested more in different working modes in order to effectively demonstrate the proposed solution.

VI. CONCLUSION
During sea state disturbances, the system is used automatically to regulate and stabilize the position of the JuR to prevent platforms from being shifted. In this paper, establishing a better control theory for a Jack-up Rig's jacking system is crucial.First, we use a PSO technique based on a fuzzy controller to examine how to adapt to the impacts of external forces, mechanical errors and hydrodynamic amplification. The Self Adaptive Fuzzy Particle Swarm Optimization controller is then compared to a Fuzzy Proportional Integral Derivative controller that meets Lyapunov requirements. By simulation on Matlab, the optimal coefficients λ i (1.279 × 10 18 , 1.828 × 10 18 , 1.225 × 10 18 , 8.734 × 10 17 ) are selected from PSO process, the results of two case studies on Table 7 and Table 8 indicating and demonstrating the performance of the proposed method. Next, the control algorithm is embedded in the STM32F746VG central processor, which gathers data on leg locations and JuR body tilt and compares it to command values to move the rig body and provide control signals to the Motor drive. Finally, the experiment results indicate the benefits of the proposed PSO-SAFC controller.  LE ANH-HOANG HO received the master's degree in automation and control engineering from the Ho Chi Minh City University of Transport, Vietnam, in 2018. He is currently a Lecturer with Van Hien University, Vietnam, and a Researcher at Artificial Intelligent in Transportation (AIT), Ho Chi Minh City University of Transport. His research interests include control theory, automation, the Internet of Things, and artificial intelligence.
VAN-VANG LE was born in Haiphong, Vietnam, in 1965. He received the Ph.D. degree in machinery and equipment in Russia, in 2010. He is currently working as the Vice President of the Ho Chi Minh City University of Transport, Vietnam, where he is also a Researcher at Artificial Intelligent in Transportation (AIT). His current research interests include maritime technology, underwater vehicles, ship machinery and equipment, and offshore engineering. VOLUME 10, 2022