# Design of Active Disturbance Rejection Controller for Dynamic Positioning Based on Improved Particle Swarm Optimization

Authors: Jia Hu; Wentao Chen
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Intelligent control methods play an important role in the dynamic positioning system (DPS). To improve the control accuracy of dynamic positioning systems, an improved active disturbance rejection controller (IADRC) is designed in this study, which can optimize the steady-state performance of the system and improve the tracking accuracy of the system. For nonlinear active disturbance rejection controllers, their internal parameters are complex and numerous, with difficult settings. Proper parameters cannot be found with the trial and error method, and traditional optimization algorithms showcase some problems, such as slow convergence speed, leading to frequent failures in local optimal solutions. An optimized particle swarm optimization (IPSO) algorithm is applied to the IADRC parameter setting to boost the global search ability and the local development function. Simulation analyses demonstrate that, compared with other intelligent control methods, the IPSO-based IADRC dynamic positioning system has advantages such as fast response speed and strong anti-interference ability.


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## Body

## 1. Introduction

Dynamic positioning means that the ship can resist the disturbance of wind, wave, current, and other environmental interference through its propeller. As a closed-loop control system to control ships’ positions and heading, the ship dynamic positioning system is mainly composed of four subsystems: control subsystem, thrust distribution subsystem, propulsion subsystem, and measurement subsystem. With many advantages, such as good flexibility, high positioning accuracy, and resistance to water depth, DPS is widely implemented in offshore oil drilling platforms, salvage vessels, drilling vessels, and other equipment [1, 2]. The first DPS was developed in the 1960s, with a single-input single-output proportional-integral-derivative (PID) controller designed for controlling the horizontal motion of a plane. Subsequently, known as the second-generation product of DPS, the control technology based on modern control theories was applied to DPS. Since then, backstepping designs, fuzzy controllers, and nonlinear PID control have been implemented [3–6].With the rapid development of the shipping industry, higher requirements are imposed on ships’ safety, economy, and intelligent control. In the face of a complex and ever-changing ocean environment, ships entail high accuracy of dynamic positioning systems [7, 8]. To meet the requirements of high-precision dynamic positioning systems for special ships, many control methods have been proposed over recent years, such as fuzzy control, adaptive control, fine anti-interference control, and iterative synovial control [9]. Most of such control methods rely on the accuracy of the model itself, while the ship model is often a simplified model only used in research. As a result, the robustness and adaptability of these control methods will be reduced when they are applied in the ship field.Active disturbance rejection controller (ADRC) was first put forward by Han in 1998 [10]. Nowadays, the ADRC has been investigated and applied in many areas, including aircraft flight control, robot control, and ship control, with a lot of results achieved. As a model-independent control algorithm, it does not require high precision for the mathematical models of the controlled objects; therefore, many scholars have implemented it in dynamic positioning systems [11–13]. The ADRC was first presented as a nonlinear structure, and then, the nonlinear ADRC was parameterized and modified to linear ADRC with linearized gains by Zheng and Gao [14]. A comparison study [15] was conducted between two active rejection disturbance controllers in terms of transient characteristics and robustness. And it was further found that NLADRC offers better dynamic behavior than LADRC, but the latter possess better robustness characteristics than the former [16]. The IADRC based on the improved dragonfly algorithm was introduced to fast steering mirrors, and this control law can meet the requirements of FSM tracking accuracy [12].Applying ADRC control to a dynamic positioning system, Ye et al. [17] compared its advantages and disadvantages with those of proportional-integral-derivative (PID) control. Their simulation experiments were carried out in an ideal environment, without considering the interference of the environment. Xiong et al. [18] analyzed the control characteristics of ADRC under strong interference and proved that it can deliver better dynamic performance than PID control. Chen et al. [19] improved the nonlinear FAL function to make an extended state observer with better anti-interference performance, but it has a shortcoming: the parameters in the tracking differentiator (TD) and the extended state observer (ESO) are adjusted according to an empirical formula. Yang et al. [20] applied a linear/nonlinear active disturbance rejection switching strategy to ship dynamic positioning systems by integrating the advantages of both, but the switching strategy and conditions were not mature enough. Wu et al. [21] introduced an artificial bee colony algorithm into nonlinear ADRC to optimize the design of complex parameters; however, this algorithm shows a slow convergence speed and is easy to fall into local optimal results. Wang et al. [22] proposed a hybrid particle swarm optimization algorithm in combination with adaptive inertial weight. To solve the complex nonlinear system and model uncertainty problem, a cloud-model intelligent control algorithm was introduced to ship dynamic positioning systems [8]. To solve the parameter setting problem of cloud-model controllers, adaptive particle swarm optimization was made for parameter setting, with a better control effect obtained [23].At present, some special ships are required to have high control accuracy for such environmental disturbances as winds and waves. This study takes a test ship as the research object. The main objective is to use the proposed intelligent control law to keep the position and heading of a vessel at a set value while keeping high control precision under main environments. Dynamic positioning systems based on IADRC are investigated in this study, which makes the following main contributions:(1)
Based on the advantages and disadvantages of the basic particle swarm optimization algorithm and grey wolf optimization, an improved particle swarm optimization (IPSO) is proposed. Adopting a dynamic adaptive inertial weightω, this study avoids the complex process of particle swarm algorithms in constructing the adaptation function and addresses the problem that the traditional PSO is more likely to fall into local extreme value in the early stage and converge at a low speed in the later stage.(2)
The improved particle swarm algorithm is used to optimize the parameters, and the improved active disturbance rejection controller is applied to the DPS. Then, an ADRC-based DPS control simulation is built in Simulink, verifying that the improved active disturbance rejection controller can deliver a better control effect on dynamic positioning control.


## 2. Mathematical Model

This section will introduce the mathematical model of ships. To describe a ship’s motion, the geodetic coordinate system and ship coordinate system are established, with the center of gravity of the ship taken as the origin of the ship’s coordinate system. The coordinate system is shown in Figure1, where u,v, and r represent the forward speed, sway speed, and yaw speed, respectively. E−xy is the geodetic coordinate system, where E is the origin and Ex represents the north direction [5, 24].Figure 1
Diagram of the plane movement of a ship.The transformation formula between two coordinate systems is as follows:(1)η˙=Rψv,where η=xyψT indicates the position of the ship in geodetic coordinates, v=μνrT indicates the velocity vector of the ship in shipboard coordinates, Rψ indicates the transformation matrix.(2)Rψ=cosψ−sinψ0sinψcosψ0001.There are certain relationship equations between forces and movements. The equation can be expressed as(3)Mν˙+Dν=τ⋅τ=τT+τω,M=m−Xu˙000m−Yν˙mxG−Yr0mxGIzz−Nr˙,D=−Xu000−Yν−Yr0xG−Nr,where τT=XTYTNT indicates propulsion; τω=XωYωNω indicates the environmental interference force; M indicates the mass matrix; D indicates the damping matrix.The model of environmental disturbance is described in(4)XW=Fecosβe−ψYW=Fesinβe−ψNW=lxsinβe−ψ−lycosβe−ψ,where Fe indicates the constant value, βe indicates the average direction of disturbance change, and lx,ly indicates the position of the interference force acting on the ship [15].


## 3. The Principle of ADRC

The ADRC consists of three components: the tracking differentiator (TD), the extended state observer (ESO), and the nonlinear state error feedback (NLSEF) [10]. But this method does not depend on a specific model. Its structure is illustrated in Figure 2. TD is to arrange the transition process for the attitude angle command and obtain its differential signal. The core of ADRC is the extended state observer (ESO), which can treat the external and internal disturbance of the system as a whole. A special feedback mechanism is used to make the observed value follow the actual value. The complete algorithm of the ADRC is as follows [10, 25].Figure 2
The principle of ADRC.The formula for tracking differentiators is depicted by(5)x1k+1=x1k+hx2k,x2k+1=hfhanx1−v0,x2,r,h+x2k,where v0 indicates the input signal, x1k denotes the tracking input signal v0, x2k denotes the tracking differential v0, and r is the speed factor, which represents the speed of system tracking. h is the sampling step length.The formula offhanx1,x2,r,h can be expressed as follows:(6)d=rh,d0=hd,y=x1+hx2,a0=d2+8ry,a=x2+sgnya0−d2，y>d0x2+yh,y≪d0,fhan=−rsgna,a>d−ard,a≪d.The ESO formula can be expressed as follows [10]:(7)ε1=z1k−yk,z1k+1=z1k+hz2k−β01ε1,z2k+1=z2k+hz3k−β02falε1,α1,δ+buk,z3k+1=z3k−hβ03falε1,α2,δ,where 0<α2<α1<1, α1=0.5, and α2=0.25 can be taken in general, and α1 and α2 are empirical values. β01,β02, and β03 are parameters to be set. b is a compensating factor [16]. The fal formula is given as follows:(8)falε,α,δ=εαsgnε,ε>δεδ−α,ε<δ,where ε indicates the input error; 0<α2<α1<1, α1=0.5, and α2=0.25 can be taken in general, and α1 and α2 are empirical values.The state error feedback law is given by(9)E1=v1k−z1kE2=v2k−z2ku0=β1fale1,α1,δ+β2fale2,α2,δuk=u0−z3kb0,where E1 and E2 indicate the state error; β1 and β2 indicate the feedback control gain; u0 indicates the extracted signal of nonlinear combination; z3k/b0 indicates the disturbance compensation quantity; and uk indicates the output of ADRC.


### 3.1. Improved NLSEF Design

The improved nonlinear error state feedback law is based on the idea of PID, and the adjustable parameters are analogous to the proportional gain and differential gain in classical PID control [26]. Figure 3 illustrates the structure of IADRC.Figure 3
The structure of IADRC.The model of INLSEF is expressed as follows:(10)e1=v1k−z1k,e2=v2k−z2k,e3=∫e1dt,u0=β1fale1,a1,δ1+β2fale2,a2,δ2+β3fale3,a3,δ3,uk=u0−z3kb0,where e1, e2, and e3 indicate the state error; β1, β2, and β3 indicate the feedback control gain; uk indicates the output of IADRC; z3k/b indicates the disturbance compensation quantity; u0 is the extracted signal of nonlinear combination; δ1, δ2, and δ3 indicate the filtering factors.According to the improved nonlinear error state feedback law, the nonlinear factora represents the nonlinear length of the fall function, and it can be adjusted by following the regulations [26]:As for the proportional-like elementβ1fale1,a1,δ1, a large gain is used for small errors and a small gain is used for large errors. a1 generally takes a value between 0 and 1.As for the integral-like gain elementβ3fale3,a3,δ3, it requires −1<a3<1, and the smaller the value of a3, the faster the response.For differential-like gain elementβ2fale2,a2,δ2, it only needs to play a small role when the system stands in an approximately steady state. Therefore, a2 in fall function takes a value greater than 1.


## 3.1. Improved NLSEF Design

The improved nonlinear error state feedback law is based on the idea of PID, and the adjustable parameters are analogous to the proportional gain and differential gain in classical PID control [26]. Figure 3 illustrates the structure of IADRC.Figure 3
The structure of IADRC.The model of INLSEF is expressed as follows:(10)e1=v1k−z1k,e2=v2k−z2k,e3=∫e1dt,u0=β1fale1,a1,δ1+β2fale2,a2,δ2+β3fale3,a3,δ3,uk=u0−z3kb0,where e1, e2, and e3 indicate the state error; β1, β2, and β3 indicate the feedback control gain; uk indicates the output of IADRC; z3k/b indicates the disturbance compensation quantity; u0 is the extracted signal of nonlinear combination; δ1, δ2, and δ3 indicate the filtering factors.According to the improved nonlinear error state feedback law, the nonlinear factora represents the nonlinear length of the fall function, and it can be adjusted by following the regulations [26]:As for the proportional-like elementβ1fale1,a1,δ1, a large gain is used for small errors and a small gain is used for large errors. a1 generally takes a value between 0 and 1.As for the integral-like gain elementβ3fale3,a3,δ3, it requires −1<a3<1, and the smaller the value of a3, the faster the response.For differential-like gain elementβ2fale2,a2,δ2, it only needs to play a small role when the system stands in an approximately steady state. Therefore, a2 in fall function takes a value greater than 1.


## 4. Improved PSO Optimization Algorithm

As for the intelligent optimization algorithm, the particle swarm optimization algorithm’s core idea is to find the optimal solution through mutual cooperation and information sharing between individuals in the population [26]. The particle velocity and position update formula are expressed as follows:(11)vk+1=ωvk+c1pbest−xk+c2gbest−xk,xk+1=xk+vk+1,where ω indicates inertia weight, which is 0.9; pbest indicates the location of the individually optimal solution; gbest indicates the position of the optimal solution in the entire population; c1 and c2 indicate learning factors; vk indicates the velocity vector of the particle; xk indicates the current position of the particle.At the beginning of iteration, a large inertia weight should be chosen for the particle, so as to expand the search range and reduce the possibility of falling into local optimal. At the end of iteration, a small inertia weight should be chosen for the particle, so as to accelerate the convergence of the algorithm. The methods for adjusting inertia weightω have been studied, and some scholars find that the PSO algorithm would converge rapidly when ωstart=0.4 and ωend=0.9 [26]. Therefore, this study proposes a method to calculate the inertia weight by using an improved decreasing exponential function, which combines the merits of nonlinear function and inertia weight PSO algorithm. The formula is shown as follows:(12)ωt=ωstart−ωendtan0.8751−ttmaxk+wend,where ωstart=0.4 and ωend=0.9; t and tmax denote the current and the maximum iteration times of the algorithm, respectively.In order to obtain the global optimal solution more efficiently, the paper introduces a dynamic method to assign values toc1 and c2. At the beginning of the search, the meaning of individual information is greater than global information, so the value of c1 is greater than c2. At the end of the search, the value of c2 is greater than the value of c1, and the indicate learning factors are obtained as follows:(13)c1=2sin2π21−ttmax,c2=sin2πt2tmax.In order to verify the validity and superiority of IPSO, this study uses three different test functions to test PSO, GWO, and IPSO. The population size of these algorithms is 40, and the number of iterations is 200. Each algorithm is randomly tested for 30 times, and the average and minimum values of the four algorithms are recorded. The details of the test functions are shown in Table1.Table 1
The test functions.
Function nameEquationIntervalOptimal valueBealeFx=1.5=x1+x1x22+2.25−x+x1x222+2.625−x1+x1x232[−4.5 4.5]0RastrigionFx=10d∑i=1dxi2−10cos2πxi[−5.12,5.120SchafferFx=0.5+sin2x12−x22−0.5/1+0.001x12+x222[−100,100]0As can be seen in Table2, compared with PSO, IPSO increases the degree of data development, speeds up the convergence of the algorithm, and is conductive to finding the global optimal solution of the test functions.Table 2
The results of test functions.
Function nameTheoretical minimumAverage valueOptimal valuePSOGWOIPSOPSOGWOIPSOBeale04.17e-065.79e-073.54e-088.97e-177.81e-200Rastrigion03.75e-043.02e-37.09e-058.07e-151.25e-120Schaffer01.13e-024.90e-32.56e-057.85e-063.43e-082.54e-11


### 4.1. Fitness Function

To limit the time for the system to reach a steady state and diminish overshoots, this study chooses the ITAE index to evaluate the dynamic performance of the system.(14)J=∫0ttetdt.Since three active disturbance rejection controllers are used in this study to control each of the three degrees of freedom of a ship, the objective function is selected as follows:(15)J=ω1∫0tte1tdt+ω2∫0tte2tdt+ω3∫0tte3tdt,where e1t, e2t, and e3t are errors of the three directions, respectively; t is the simulation time; ω1, ω2, and ω3 represent the weights of the three directions, respectively.The improved PSO algorithm flow is as follows:Step 1: Initialize the population size and the maximum iteration numbers; set the upper and lower limits of particle position and velocity; and generate the initial particle velocity and position randomly.Step 2: Replace the six parameters to be optimized in IADRC with particles generated by IPSO.Step 3: Run the model for simulation, and calculate the fitness of particles.Step 4: Update the individual and global extreme values of the particle swarm.Step 5: Determine whether the global extreme value is less than the target value. If it meets the requirement, skip to Step 8; otherwise, run Step 6.Step 6: Judge whether the number of iterations reaches the maximum. If so, skip to Step 8; otherwise, run Step 7.Step 7: Update the inertia weightW and return the number of iterationsStep 8: The algorithm ends and prints g_best.The flow chart of the improved particle swarm optimization is shown in Figure4.Figure 4
Flow chart of particle swarm optimization.


## 4.1. Fitness Function

To limit the time for the system to reach a steady state and diminish overshoots, this study chooses the ITAE index to evaluate the dynamic performance of the system.(14)J=∫0ttetdt.Since three active disturbance rejection controllers are used in this study to control each of the three degrees of freedom of a ship, the objective function is selected as follows:(15)J=ω1∫0tte1tdt+ω2∫0tte2tdt+ω3∫0tte3tdt,where e1t, e2t, and e3t are errors of the three directions, respectively; t is the simulation time; ω1, ω2, and ω3 represent the weights of the three directions, respectively.The improved PSO algorithm flow is as follows:Step 1: Initialize the population size and the maximum iteration numbers; set the upper and lower limits of particle position and velocity; and generate the initial particle velocity and position randomly.Step 2: Replace the six parameters to be optimized in IADRC with particles generated by IPSO.Step 3: Run the model for simulation, and calculate the fitness of particles.Step 4: Update the individual and global extreme values of the particle swarm.Step 5: Determine whether the global extreme value is less than the target value. If it meets the requirement, skip to Step 8; otherwise, run Step 6.Step 6: Judge whether the number of iterations reaches the maximum. If so, skip to Step 8; otherwise, run Step 7.Step 7: Update the inertia weightW and return the number of iterationsStep 8: The algorithm ends and prints g_best.The flow chart of the improved particle swarm optimization is shown in Figure4.Figure 4
Flow chart of particle swarm optimization.


## 5. Simulation and Result Analysis

In this section, a dynamic positioning experiment ship is adopted as the simulation object to implement simulations, so as to demonstrate the effectiveness of the proposed control law. According to the ship mathematical model deduced above, a simulation model is built up in MATLAB to verify the control effect of IPSO-IADRC. The structure diagram of the simulation system is shown in Figure5, and the specific parameters of the ship are listed in Table 3.Figure 5
Settings of IADRC parameters based on IPSO.Table 3
Ship parameters.
NameParameterUnitShip length99mShip width15.20mDraught6mLength design waterline88.00mFull load displacement4,642.88tDimensionless inertia matrixM and damping matrix D are given as follows:(16)M=0.92700001.7502−0.17540−0.17540.1578,D=0.05580000.1183−0.04240−0.01510.0506.Assume that the ship’s starting position isη=0,0,0T and the target location is 40,50,10T. In the optimization process, the number of particle populations is set to 20, and the number of interactions is set to 50.The traditional ADRC, PSO-ADRC, and IPSO-ADRC are compared in the performance analyses of the ship’s dynamic positioning. The simulation time is set to 100 s. The simulation parameters adjusted by IPSO are shown in Tables4–6.Table 4
Optimal parameters of IADRC (surge).
ESO.ValueNLSEFValueTDValueβ0189.3647β166.5359h0.001β02976.5212β296.5135r50β03303.0062β31.2568b02α10.75α10.5α21.25α20.25α3−0.25δ10.025δ30.1δ20.025Table 5
Optimal parameters of IADRC (sway).
ESO.ValueNLSEFValueTDValueβ01769.6302β1655.3659h0.001β0231.0268β2497.0025r50β030.3595β332.3841b02α10.75α10.5α21.25α20.25α3−0.25δ10.025δ30.1δ20.025Table 6
Optimal parameters of IADRC (yaw).
ESO.ValueNLSEFValueTDValueβ0113.5410β1511.2582h0.001β02153.5745β239.1210r50β03852.2404β3358.2030b00.0015α10.75α10.5α21.25α20.25α3−0.25δ11δ30.1δ21


### 5.1. Case 1: In Calm Water

The initial states are chosen as [0, 0, 0]. The desired positions and heading are chosen as [40, 50, 10]. When no interference is added, the simulation results are shown in Figures6–8. We can see that PSO-IADRC, IPSO-IADRC, and GWO-IADRC have similar control effects. The ship can approach to the desired position and heading in approximately 30 s.Figure 6
The response curve in surge direction.Figure 7
The response curve in sway direction.Figure 8
The response curve in yaw direction.


### 5.2. Case 2: Under Disturbance Conditions

The parameters of the external slow disturbance simulated for formula (4) are set as Fe=1000, βe=120sin0.5t, and lx,ly=20,5. The simulations apply disturbances in 30–35 seconds. After the interferences are added, the simulation results are shown in Figures 9–11. Figure 12 shows the motions of the ship driven by the designed controllers, and the performance indicators are shown in Table 7.Figure 9
The response curve in surge direction (with interference).Figure 10
The response curve in sway direction (with interference).Figure 11
The response curve in yaw direction (with interference).Figure 12
The ship position response curve (with interference).Table 7
Performance indicators of controllers (with interference).
ControllerAdjustment time (s)ITAEOvershoot (m)PSO-IADRC16.86401.2958.98IPSO-IADRC13.03130.0231.32GWO-IADRC15.23389.8795.93Based on the above data, it is easy to conclude that all the control methods can achieve the goal. Especially, the IPSO-IADRC method not only delivers a shorter adjustment time but also provides a smaller value of ITAE. These findings demonstrate that the IPSO-IADRC-based DP controller is able to boost the accuracy and robustness of the control system effectively and can deliver better dynamic performance than the PSO-IADRC controller. The maximum position error of the proposed method is 1.32. But, for PSO-IADRC and GWO-IADRC, the maximum position errors are 8.98 and 5.93, respectively.


### 5.3. Case 3: Change Ship Model

In order to test the control system based on IPSO-IADRC, the parameters of the ship mathematical model are changed. The inertia matrix and damping matrix are obtained as shown in (7).The simulation results are depicted in Figure13, which shows that the proposed control system can force the ship to converge to the desired position. Although the ship parameters are changed, the response curve of the controlled variables could slightly change. The results prove the good robustness of IPSO-IADRC.(17)M=1.08520002.0575−0.40870−0.40870.2153,D=0.08650000.07620.0151000.01510.0031.Figure 13
System respond curve.


## 5.1. Case 1: In Calm Water

The initial states are chosen as [0, 0, 0]. The desired positions and heading are chosen as [40, 50, 10]. When no interference is added, the simulation results are shown in Figures6–8. We can see that PSO-IADRC, IPSO-IADRC, and GWO-IADRC have similar control effects. The ship can approach to the desired position and heading in approximately 30 s.Figure 6
The response curve in surge direction.Figure 7
The response curve in sway direction.Figure 8
The response curve in yaw direction.


## 5.2. Case 2: Under Disturbance Conditions

The parameters of the external slow disturbance simulated for formula (4) are set as Fe=1000, βe=120sin0.5t, and lx,ly=20,5. The simulations apply disturbances in 30–35 seconds. After the interferences are added, the simulation results are shown in Figures 9–11. Figure 12 shows the motions of the ship driven by the designed controllers, and the performance indicators are shown in Table 7.Figure 9
The response curve in surge direction (with interference).Figure 10
The response curve in sway direction (with interference).Figure 11
The response curve in yaw direction (with interference).Figure 12
The ship position response curve (with interference).Table 7
Performance indicators of controllers (with interference).
ControllerAdjustment time (s)ITAEOvershoot (m)PSO-IADRC16.86401.2958.98IPSO-IADRC13.03130.0231.32GWO-IADRC15.23389.8795.93Based on the above data, it is easy to conclude that all the control methods can achieve the goal. Especially, the IPSO-IADRC method not only delivers a shorter adjustment time but also provides a smaller value of ITAE. These findings demonstrate that the IPSO-IADRC-based DP controller is able to boost the accuracy and robustness of the control system effectively and can deliver better dynamic performance than the PSO-IADRC controller. The maximum position error of the proposed method is 1.32. But, for PSO-IADRC and GWO-IADRC, the maximum position errors are 8.98 and 5.93, respectively.


## 5.3. Case 3: Change Ship Model

In order to test the control system based on IPSO-IADRC, the parameters of the ship mathematical model are changed. The inertia matrix and damping matrix are obtained as shown in (7).The simulation results are depicted in Figure13, which shows that the proposed control system can force the ship to converge to the desired position. Although the ship parameters are changed, the response curve of the controlled variables could slightly change. The results prove the good robustness of IPSO-IADRC.(17)M=1.08520002.0575−0.40870−0.40870.2153,D=0.08650000.07620.0151000.01510.0031.Figure 13
System respond curve.


## 6. Conclusion

An IPSO-based IADRC system is designed in this study. The proposed IPSO can overcome the defects of traditional algorithms by adjusting parameters. The IPSO-IADRC control strategy is compared with the other two strategies. The simulation results demonstrate that the dynamic positioning system based on IPSO-IADRC can deliver faster response speeds, better anti-interference capabilities, and better dynamic characteristics than the dynamic positioning systems based on PSO-IADRC and traditional IADRC. It can provide higher control accuracy when facing environmental disturbances. The main contributions of this study are summarized as follows.(1)
In theory, an improved NLSEF strategy is proposed, which introduces an integral-like gain parameter to increase the type of the first-stage system, provides the system with better steady-state performance, and boosts the tracking accuracy of the system(2)
In practice, the proposed method can meet the requirements of ship dynamic positioning control systems for rapidity and high accuracy.



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*Source: 5991281-2022-04-20.xml*