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.
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 weight , 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 Figure 1, where , and represent the forward speed, sway speed, and yaw speed, respectively. is the geodetic coordinate system, where is the origin and represents the north direction [5, 24].
The transformation formula between two coordinate systems is as follows:where indicates the position of the ship in geodetic coordinates, indicates the velocity vector of the ship in shipboard coordinates, indicates the transformation matrix.
There are certain relationship equations between forces and movements. The equation can be expressed aswhere indicates propulsion; indicates the environmental interference force; M indicates the mass matrix; D indicates the damping matrix.
The model of environmental disturbance is described in where indicates the constant value, indicates the average direction of disturbance change, and 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].
The formula for tracking differentiators is depicted bywhere indicates the input signal, denotes the tracking input signal , denotes the tracking differential , and is the speed factor, which represents the speed of system tracking. is the sampling step length.
The formula of can be expressed as follows:
The ESO formula can be expressed as follows [10]:where , , and can be taken in general, and and are empirical values. , and are parameters to be set. is a compensating factor [16]. The formula is given as follows:where indicates the input error; , , and can be taken in general, and and are empirical values.
The state error feedback law is given bywhere and indicate the state error; and indicate the feedback control gain; indicates the extracted signal of nonlinear combination; indicates the disturbance compensation quantity; and 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.
The model of INLSEF is expressed as follows:where , , and indicate the state error; , , and indicate the feedback control gain; indicates the output of IADRC; indicates the disturbance compensation quantity; is the extracted signal of nonlinear combination; , , and indicate the filtering factors.
According to the improved nonlinear error state feedback law, the nonlinear factor represents the nonlinear length of the fall function, and it can be adjusted by following the regulations [26]:
As for the proportional-like element , a large gain is used for small errors and a small gain is used for large errors. generally takes a value between 0 and 1.
As for the integral-like gain element , it requires , and the smaller the value of , the faster the response.
For differential-like gain element , it only needs to play a small role when the system stands in an approximately steady state. Therefore, 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:where indicates inertia weight, which is 0.9; indicates the location of the individually optimal solution; indicates the position of the optimal solution in the entire population; and indicate learning factors; indicates the velocity vector of the particle; 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 weight have been studied, and some scholars find that the PSO algorithm would converge rapidly when and [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:where and ; t and 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 to and . At the beginning of the search, the meaning of individual information is greater than global information, so the value of is greater than . At the end of the search, the value of is greater than the value of , and the indicate learning factors are obtained as follows:
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 Table 1.
As can be seen in Table 2, 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.
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.
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:where , , and are errors of the three directions, respectively; t is the simulation time; , , and 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 weight and return the number of iterations Step 8: The algorithm ends and prints g_best.
The flow chart of the improved particle swarm optimization is shown in Figure 4.
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 Figure 5, and the specific parameters of the ship are listed in Table 3.
Dimensionless inertia matrix M and damping matrix D are given as follows:
Assume that the ship’s starting position is and the target location is . 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 Tables 4–6.
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 Figures 6–8. 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.
5.2. Case 2: Under Disturbance Conditions
The parameters of the external slow disturbance simulated for formula (4) are set as , , and . 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.
Based 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 Figure 13, 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.
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.
Data Availability
The data used in this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no potential conflicts of interest.
Acknowledgments
This work was supported by the Science and Technology Commission of Shanghai Municipality and Shanghai Engineering Research Center of Ship Intelligent Maintenance and Energy Efficiency under Grant 20DZ2252300.