Fractional-Order Controller for Course-Keeping of Underactuated Surface Vessels Based on Frequency Domain Specification and Improved Particle Swarm Optimization Algorithm

Abstract

In this paper, a new fractional-order (FO) PI^λD^µ controller is designed with the desired gain and phase margin for the automatic rudder of underactuated surface vessels (USVs). The integral order λ and the differential order μ are introduced in the controller, and the two additional adjustable factors make the FO PI^λD^µ controller have better accuracy and robustness. Simulations are carried out for comparison with a ship’s digital PID autopilot. The results show that the FO PI^λD^µ controller has the advantages of a small overshoot, short adjustment time, and precise control. Due to the uncertainty of the model parameters of USVs and two extra parameters, it is difficult to compute the parameters of an FO PI^λD^µ controller. Secondly, this paper proposes a novel particle swarm optimization (PSO) algorithm for dynamic adjustment of the FO PI^λD^µ controller parameters. By dynamically changing the learning factor, the particles carefully search in their own neighborhoods at the early stage of the algorithm to prevent them from missing the global optimum and converging on the local optimum, while at the later stage of evolution, the particles converge on the global optimal solution quickly and accurately to speed up PSO convergence. Finally, comparative experiments of four different controllers under different sailing conditions are carried out, and the results show that the FO PI^λD^µ controller based on the IPSO algorithm has the advantages of a small overshoot, short adjustment time, precise control, and strong anti-disturbance control.

1. Introduction

In recent years, the maneuverability of underactuated surface vessels (USVs) has become a hot topic of ship control [1,2,3,4,5]. An underactuated system is one in which the number of degrees of freedom of the system is greater than the number of control inputs. An underactuated system refers to a situation where the dimension of the system control input vector space is smaller than the dimension of the system’s generalized coordinate vector space [6]. In the study of ship maneuverability, the effectiveness of ship course control has a direct impact on ship maneuverability and the safety of navigation at sea. For vessels at sea, to keep the ship on the expected course, the rudder must be controlled by a control system to correct for the yaw caused by disturbances. Digital PID rudders are widely used as course controllers for ships at sea, but they are too sensitive to disturbances such as wind and waves, which can lead to frequent steering of the rudder. A great deal of research has been carried out on automatic rudder steering for ships in order to achieve satisfactory maneuvering performance, and many results have been obtained [7,8,9,10,11,12]. In [7], Le proposed to combine fuzzy logic control with linear control theory for the design of a ship autopilot controller and used the controller for ship course control, achieving good control results. In [8], optimal control based on the linear quadratic optimal was proposed and applied to a ship’s autopilot. In [9], the PSO method for an optimal autopilot PID controller was designed. In [10], the H∞ controller was solved by the “2-Riccati” equation, and it was used for the rudder and flap of a ship‘s course under system uncertainty. In [11], a new robust adaptive nonlinear feedback algorithm based on error-driven function was proposed, and it was used for ship course-keeping. In [12], an indirect adaptive robust controller was proposed. First, the parameter estimation law under system uncertainty was studied for a ship model. Secondly, an indirect adaptive robust controller was applied for a ship’s heading.

With the continuous improvement of precision control theory, fractional order calculus theory [13,14,15] and fractional order PI^λD^µ controllers [16,17,18] have been proposed one after another. In 1999, Podlubny proposed fractional order PI^λD^µ controllers on the basis of fractional order calculus theory [16]. Compared with the traditional PID controller, the introduction of integral order λ and differential order μ makes the PID controller have two more control parameters, expanding the original “point” control to “surface” control, further improving the control accuracy and making the controller more flexible [19,20,21,22,23,24,25,26,27]. In [19], a two-degrees-of-freedom planar robot was controlled by an FO PID controller tuned with particle swarm optimization (PSO) and a genetic algorithm (GA). In [20], an FO PI^λD^µ controller was applied to the speed control of subway trains, and optimization studies were carried out to make the train achieve better speed control and stability. In [21], a new FO PI^λD^µ controller was proposed, and a structural transformation method was designed to obtain the controller parameters, which were used to improve the anti-interference of the controller. In [22], a special case of fractional-order systems was first investigated, and on this basis, the FOPD controller and the FOPID controller were designed, which were used in the control of fractional-order systems to achieve a closed-loop system with a monotonic amplitude-frequency response. In [23], for a three-link rigid manipulator system with coupled multiple-input multiple-output characteristics and high nonlinearity, a fractional-order self-tuned fuzzy PID controller was proposed, which was applied to a robotic manipulator system and achieved good control results. In [24], an adaptive fractional order PID controller was proposed, which was applied to a photovoltaic grid system. In [25], an FOPID controller was proposed in a magnetic ball suspension system, and a modified opposition-based learning technique and the hunger games search algorithm were used to tune the parameters of the FOPID controller. In [26], an FO PI^λD^µ controller was applied to an automatic voltage regulator (AVR), and a self-regulated off-line optimal tuning method based on the gradient-based optimization algorithm was proposed to tune the five parameters of the FO PI^λD^µ controller. Davut Izci proposed an FOPID controller for a buck converter system and hybrid Lévy flight distribution and simulated annealing algorithms (LFDSAs), which involved a balanced structure in terms of explorative and exploitative phases being used to tune the parameters of the FOPID controller [27]. In addition, some new algorithms were also proposed to optimize the PID [28,29,30,31,32,33,34].

In this paper, an FO PI^λD^µ controller is applied to a ship’s course control. A modified Oustaloup approximation method is adopted. Given the values of the gain margin A_m and phase margin ϕ_m, the parameters of the FO PI^λD^µ controller are calculated. The effectiveness of the algorithm is demonstrated by simulations comparing the FO PI^λD^µ controller with a PID controller. However, as the FO PI^λD^µ controller is designed for ship course control at a rated speed, if the ship speed changes dynamically, the non-linear model of the ship will also change accordingly, which will result in the original controller parameters no longer meeting the needs of the ship’s course control and needing to be recalculated. Therefore, this paper proposes an improved particle swarm algorithm (IPSO) for dynamic adjustment of the FO PI^λD^µ controller parameters. By dynamically changing the learning factor, the particles carefully search in their own neighborhoods at the early stage of the algorithm to prevent them from skipping the global optimum and converging on a local optimum, while at the later stage of evolution, the particles converge on the global optimal solution quickly and accurately to improve the convergence speed. Comparative experiments are carried out without wind and wave disturbances and under wind and wave disturbances. The results show that the FO PI^λD^µ controller based on the IPSO algorithm has the advantages of a small overshoot, short adjustment time, precise control, and strong anti-disturbance control.

2. USV Model

2.1. Ship Motion Model

The movement of a ship has six degrees of freedom (DOF), which represent the position and direction of the ship’s movement, and therefore six separate coordinates are required to describe these six degrees of freedom. (x, y, z) represents the position of a ship in the inertial frame, and the first-order differentiation of (x, y, z) represents the translational motion along x, y, and z. The first three DOF are surge, sway, and heave. (φ, θ, ψ) represents the direction of a ship, and the first-order differentiation of (φ, θ, ψ) represents the rotation of the ship. The last three DOF are roll, pitch, and yaw [35]. The ship motion description is shown in Figure 1.

Ship maneuvering usually considers only the movement of the ship in the horizontal plane. For most ship movements and their control problems, it is possible to ignore heave, pitch, and roll motions, and thus only the sway, surge, and yaw are discussed. Therefore, the ship motion can be regarded as a plane motion, and the plane motion has only three degrees of freedom, which further simplifies the research object. The maneuvering mathematical model of a USV [36] can be written as

$$\left\{\begin{array}{l}\dot{x}=u cos\psi -v sin\psi \\ \dot{y}=u sin\psi +v cos\psi \\ \dot{\psi }=r\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}\dot{u}=\frac{m_{22}}{m_{11}}vr-\frac{d_{11}}{m_{11}}u+\frac{1}{m_{11}}{\tau }_u\\ \dot{v}=-\frac{m_{11}}{m_{22}}ur-\frac{d_{22}}{m_{22}}v\\ \dot{r}=\frac{m_{11}-m_{22}}{m_{33}}uv-\frac{d_{33}}{m_{33}}r+\frac{1}{m_{33}}{\tau }_r\end{array}\right.$$

(2)

where x and y denote the surge and sway, ψ denotes the yaw angle, u denotes the surge velocity, v denotes the sway velocity, r denotes the yaw velocity, m₁₁, m₂₂, and m₃₃ are uncertainty parameters representing the intrinsic and additional masses, d₁₁, d₂₂, and d₃₃ are also uncertainty parameters, d₁₁ is the surge hydrodynamic damping factor, d₂₂ is the sway hydrodynamic damping factor, and d₃₃ is the yaw angle hydrodynamic damping factor. For simplicity, the high-order nonlinear damping term is ignored. τ_u is the longitudinal force of the propeller, and τ_r is the torque of the propeller.

2.2. Nonlinear Ship Model

The nonlinear ship model is a dynamic system. The rudder angle δ is the input, and the yaw angle ψ is the output. By capturing the main threads of the ship’s dynamics from δ → $\dot{\psi }$ → ψ, the resulting differential equation retains the nonlinear influences and even allows the effects of wind and wave disturbances to be converted into a disturbed rudder angle δ_D as an input signal that enters the ship’s model together with the actual rudder angle δ. The responding nonlinear ship model is shown in Figure 2.

The Norrbin nonlinear model for course control [37] can be derived from the ship motion maneuvering model in cases where steering is not very frequent:

$$\stackrel{‥}{\psi }=-a_1\dot{\psi }-a_2{\dot{\psi }}^3+b\delta$$

(3)

where ψ and δ are the yaw angle and rudder angle. a₁ = αb, a₂ = βb, and b = K/T, where K, T, α, and β are all related to the navigation speed and their values have a large impact on the accuracy of the simulation model.

3. FO PI^λD^µ Controller

Fractional order (FO) calculus is essentially an arbitrary-order calculus whose order can be real or even complex, which is a further extension of integer order calculus. To improve the accuracy of the PID controller, the differential order µ and the integral order λ are extended to the real domain to obtain the FO PI^λD^µ controller, whose differential equation [16] is expressed as

$$u\left(t\right)=K_{p}e\left(t\right) + K_i{D}_t^{-\lambda }e\left(t\right) + K_d{D}_t^{\mu }e\left(t\right)$$

(4)

where $D_t^{\mu }$ ≡ ${}_a{}^{c}D_t^{\alpha }$ is defined by Caputo and λ > 0 and µ > 0 are any real number and the order of the FO controller. We can then find the Laplace transform of the FO calculus defined by Caputo:

$$L\left\{{}_a{}^{c}D_t^{\alpha }f\left(t\right)\right\}=s^{\alpha }F\left(s\right)-{{\displaystyle \sum }}_{k=0}^{n-1}{s}^{a-k-1}{f}^{\left(k\right)}\left(0\right)$$

(5)

From Equations (4) and (5), the FO PI^λD^µ controller transfer function can be obtained:

$$G_c\left(s\right)=K_p+\frac{K_i}{s^{\lambda }}+K_d{s}^{\mu }$$

(6)

Since the differential µ and the integral order λ are arbitrary positive real numbers, the PI controller is λ = 1 and µ = 0, the PD controller is λ = 0 and µ = 1, and the PID controller is λ = 1 and µ = 1.

Since λ and µ are allowed to take non-integer values, the adjustment range of the controller parameters is changed from the “point” adjustment of the PID controller in Figure 3a to the “plane” adjustment in Figure 3b, thus enabling the FO PI^λD^µ controller to control the controlled object more flexibly and to obtain better control results.

In order to better study the FO PI^λD^µ controllers, researchers wish to extend the design methods of integer-order controllers to FO controllers, which requires approximating the FO calculus to integer-order calculus. This paper uses an improved Oustaloup approximation for the FO calculus operator $s^{\alpha }$ to convert it to integer-order calculus [38]. We set the order to $N$, where (${\omega }_b$,${\omega }_h$) is the fitting frequency range and $s^{\alpha }$ is transformed into an integer-order transfer function as follows:

$$s^{\alpha }={\left(\frac{d{\omega }_b}{b}\right)}^{\alpha }\left(\frac{d{s}^2+bs{\omega }_h}{d\left(1-\alpha \right)s^2+bs{\omega }_h+d\alpha }\right){\left[\frac{1+\frac{s}{d{\omega }_b/b}}{1+\frac{s}{d{\omega }_h/b}}\right]}^{\alpha }$$

(7)

where $0<\alpha <1$, $b>0$, $d>0$, and $d=j\omega$. We express the FO part of Equation (7) (i.e., $K\left(s\right)$) as a rational transfer function in the form of zeros and poles:

$$K(s)=\underset{N\to \infty }{{\displaystyle \lim }}{K}_N(s)=\underset{N\to \infty }{{\displaystyle \lim }}{\displaystyle \prod _{k=-N}^N\frac{1+s/{\omega }_k^{\prime }}{1+s/{\omega }_k}}$$

(8)

The $k$ zero pole is

$${\omega }_k^{\prime }={\left(\frac{b}{d}\right)}^{\frac{2k-\alpha }{2N+1}}{\omega }_h{}^{\frac{N+k+\frac{1}{2}\left(1-\alpha \right)}{2N+1}}{\omega }_b{}^{\frac{N-k+\frac{1}{2}\left(1+\alpha \right)}{2N+1}}$$

(9)

$${\omega }_k={\left(\frac{b}{d}\right)}^{\frac{2k+\alpha }{2N+1}}{\omega }_h{}^{\frac{N+k+\frac{1}{2}\left(1+\alpha \right)}{2N+1}}{\omega }_b{}^{\frac{N-k+\frac{1}{2}\left(1-\alpha \right)}{2N+1}}$$

(10)

An FO calculus operator is obtained by constructing a continuous rational transfer function model:

$$G\left(s\right)=K\left(\frac{d{s}^2+b{\omega }_{h}s}{d\left(1-\alpha \right)s^2+b{\omega }_{h}s+d\alpha }\right){{\displaystyle \prod }}_{k=-N}^N\frac{1+s/{\omega }_k^{\prime }}{1+s/{\omega }_k}$$

(11)

where $K={({\omega }_b{\omega }_h)}^{\alpha }$.

4. The FO PI^λD^µ Controller of a Ship’s Course

4.1. FO PI^λD^µ Controller Design

The working principle of the FO PI^λD^µ controller system of a ship’s course is shown in Figure 4.

In Figure 4, the actual course is determined with the compass and compared with the given course to generate the course deviation, which enters the FO PI^λD^µ controller. It then calculates the rudder angle signal to be generated, and the rudder is turned to the required angle when the rudder receives the command signal to make the ship start to change its course. The rudder then returns to its zero position, completing the course revision.

In this paper, given the values of the gain margin $A_m$ and phase margin ${\varphi }_m$, the parameters of the FO PI^λD^µ controller parameters can be calculated [39]. Starting from the basic definitions of $A_m$ and ${\varphi }_m$, the dynamically controlled object $G_p\left(s\right)$ and the controller $G_c\left(s\right)$ should satisfy the following relationships:

$${\varphi }_m=arg\left[G_c\left(j{\omega }_g\right)G_p\left(j{\omega }_g\right)\right] + \pi$$

(12)

$$A_m=\frac{1}{\left|G_c\left(j{\omega }_p\right)G_p\left(j{\omega }_p\right)\right|}$$

(13)

where ${\omega }_g$ meets

$$\left|G_c\left(j{\omega }_g\right)G_p\left(j{\omega }_g\right)\right| =1$$

(14)

and where ${\omega }_p$ meets

$$arg\left[G_c\left(j{\omega }_p\right)G_p\left(j{\omega }_p\right)\right] =-\pi$$

(15)

By substituting $G_c\left(s\right)$ with Equation (6), the following relationship is obtained:

$$K_p+K_i\frac{cos\frac{\pi \lambda }{2}}{{\omega }_p^{\lambda }}+K_{d}cos\frac{\pi \mu }{2}{\omega }_p^{\mu }=R_{mp}$$

(16)

$$K_p+K_i\frac{cos\frac{\pi \lambda }{2}}{{\omega }_g^{\lambda }}+K_{d}cos\frac{\pi \mu }{2}{\omega }_g^{\mu }=R_{mg}$$

(17)

$$-K_i\frac{sin\frac{\pi \lambda }{2}}{{\omega }_p^{\lambda }}+K_{d}sin\frac{\pi \mu }{2}{\omega }_p^{\mu }=I_{mp}$$

(18)

$$K_i\frac{sin\frac{\pi \lambda }{2}}{{\omega }_g^{\lambda }}+K_{d}sin\frac{\pi \mu }{2}{\omega }_g^{\mu }=I_{mg}$$

(19)

where

$$-\frac{1}{A_m{G}_p\left(j{\omega }_p\right)}=R_{mp}+j{I}_{mp}$$

(20)

$$\frac{-cos{\phi }_m-jsin{\phi }_m}{G_p\left(j{\omega }_g\right)}=R_{mg}+j{I}_{mg}$$

(21)

When the controller is designed, as the controlled object $G_p\left(s\right)$ and $A_m$ and ${\varphi }_m$ are known from Equations (16)–(19), the error squared minimization equation is introduced into the parameter calculation:

$$J={{\displaystyle \int }}_0^{\infty }{e}^2\left(t\right)dt$$

(22)

Here, ${\omega }_p$, ${\omega }_g$, $\lambda$, and $\mu$ need to meet the following constraints:

$$\begin{gathered} \left({\omega }_g^{\lambda +\mu }-{\omega }_p^{\lambda +\mu }\right)\left(R_{mp}-R_{mg}\right)+\left({\omega }_p^{\lambda +\mu }{I}_{mp}+{\omega }_g^{\lambda +\mu }{I}_{mg}\right)cot\left(\frac{\pi \mu }{2}\right)+ \\ \left({\omega }_p^{\lambda +\mu }{I}_{mg}+{\omega }_g^{\lambda +\mu }{I}_{mp}\right)cot\left(\frac{\pi \lambda }{2}\right)-\left({\omega }_p^{\lambda }{\omega }_g^{\mu }{I}_{mp}+{\omega }_g^{\lambda }{\omega }_p^{\mu }{I}_{mg}\right)\left(cot\frac{\pi \lambda }{2}+cot\frac{\pi \mu }{2}\right)=0 \end{gathered}$$

(23)

If ${\omega }_p$, ${\omega }_g$, $\lambda$, and $\mu$ are known, then the parameters $K_p$, $K_i$, and $K_d$ are expressed as follows:

$$\begin{gathered} K_p=\left[{\omega }_p^{\lambda }{R}_{mp}-{\omega }_g^{\lambda }{R}_{mg}-\left({\omega }_p^{\lambda }{I}_{mp}- \\ {\omega }_g^{\lambda }{I}_{mg}\right)cot\left(\frac{\pi \mu }{2}\right)\right]/\left({\omega }_p^{\lambda }-{\omega }_g^{\lambda }\right) \end{gathered}$$

(24)

$$K_i=\frac{{\omega }_p^{\lambda }{\omega }_g^{\lambda }\left({\omega }_g^{\mu }{I}_{mp}-{\omega }_p^{\mu }{I}_{mg}\right)}{\left({\omega }_p^{\lambda +\mu }-{\omega }_g^{\lambda +\mu }\right)sin\left(\frac{\pi \lambda }{2}\right)}$$

(25)

$$K_d=\frac{{\omega }_p^{\lambda }{I}_{mp}-{\omega }_g^{\lambda }{I}_{mg}}{\left({\omega }_p^{\lambda +\mu }-{\omega }_g^{\lambda +\mu }\right)sin\left(\frac{\pi \mu }{2}\right)}$$

(26)

4.2. Autotuning of the FO PI^λD^µ Controller Based on the IPSO Algorithm

However, as the FO PI^λD^µ controller is designed for ship course control at a rated speed, if the ship speed changes dynamically, the non-linear model of the ship will also change accordingly, which will result in the original controller parameters no longer meeting the needs of the ship’s course control and needing to be recalculated. Therefore, many novel optimization algorithms [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54] are used to tune the parameters of the controller. This paper proposes an IPSO algorithm for dynamic adjustment of the FO PI^λD^µ controller parameters.

The PSO algorithm was originally designed to optimize the unpredictable movement of simulated flocks of birds. It is an iterative random search algorithm that is easy to implement, is robust, has good parallel processing capabilities, and is likely to obtain a global optimal solution [55,56]. However, in the later stages of optimization, the convergence time is long, the accuracy of the optimization is not high, and it easily falls into local extremes [57,58,59]. The mathematical description of the PSO algorithm [55,56] is as follows. We set the population size to $N$. At the iteration time, the position vector of each particle in the $D$-dimensional space is expressed as ${\overline{x}}_i\left(t\right)=\left(x_i^1,x_i^2,\dots ,x_i^d,\dots ,x_i^D\right)$. The velocity of each particle in the $D$-dimensional space is ${\overline{v}}_i\left(t\right)=\left(v_i^1,v_i^2,\dots ,v_i^d,\dots ,v_i^D\right)$. The optimal position for the particle $i$ to search thus far is $\overline{p_i}\left(t\right)=\left(p_i^1,p_i^2,\dots ,p_i^d,\dots ,p_i^D\right)$. The optimal position for all of the particles searched thus far is $\overline{p_g}\left(t\right)=\left(p_g^1,p_g^2,\dots ,p_g^d,\dots ,p_g^D\right)$. The velocity ${\overline{v}}_i\left(t\right)$ and the position ${\overline{x}}_i\left(t\right)$ of particle $i$ are adjusted at time $t+1$ as follows:

$$\begin{gathered} {\overline{v}}_i\left(t+1\right) = \omega {\overline{v}}_i\left(t\right) + c_1{r}_1\left({\overline{p}}_i\left(t\right) - {\overline{x}}_i\left(t\right)\right) + \\ {c}_2{r}_2\left({\overline{p}}_g\left(t\right) - {\overline{x}}_i\left(t\right)\right) \end{gathered}$$

(27)

$${\overline{x}}_i\left(t+1\right) = {\overline{x}}_i\left(t\right) + {\overline{v}}_i\left(t+1\right)$$

(28)

$$\left\{\begin{array}{c}{v}_i^d=v_{max},if v_i^d>v_{max} \\ v_i^d=-v_{max},if v_i^d<-v_{max}\end{array}\right.$$

(29)

where $i=\left[1,N\right]$, $\omega$ is the inertia weight and non-negative, learning factors $c_1$ and $c_2$ are non-negative, and $r_1$ and $r_2$ are pseudo-random numbers that obey a uniform distribution on $\left[0, 1\right]$. $v_i^d\in \left[-v_{max},v_{max}\right]$, where $v_{max}$ is a constant. If the search space is in $\left[-x_{max},x_{max}\right]$, one can set $v_{max}=k{v}_{max}$, where $0.1\le k\le 1.0$.

From Equation (27), it can be seen that in the PSO algorithm, the particle seeking performance is related to the learning factors $c_1$ and $c_2$, which are used to regulate the maximum step size in the direction of motion of the particles $\overline{p_i}\left(t\right)$ and $\overline{p_g}\left(t\right)$ respectively. A small learning factor can make the particles oscillate in the region away from the target, while a large learning factor can make the particles move rapidly toward the target region or even move away from the target. The learning factors $c_1$ and $c_2$ reflect the exchange of information between particle swarms and determine the influence of a single particle and swarm experience on the particle trajectories. The effect of the values of $c_1$ and $c_2$ on the search for an optimum is as follows:

• If c₁ and c₂ are both set to 0, the particle flight velocity will remain constant, making it impossible to search the entire space;
• If c₁ = 0, the particle loses its cognitive ability and easily falls into a local optimum solution;
• If c₂ = 0, which leads to a lack of communication and cooperation between particles, the algorithm will fall into a local minimum, and the probability of obtaining an optimal solution will become smaller;
• Smaller values for both c₁ and c₂ result in particles flying away from the target area and oscillating in that area;
• With larger values for both c₁ and c₂, particles fly faster toward the target area, and this may also cause particles to fly away from the target area.

In general, when using PSO algorithms to solve engineering optimization problems, the goal is to allow the particle in the swarm to perform optimization in the entire search space at the early stage to avoid the problem of lost local optima. In the later stages of the search, the goal is to increase the convergence accuracy and algorithm speed, which can lead to finding the global optimal solution quickly and efficiently. Therefore, the disadvantages of traditional methods can be overcome by introducing the dynamic accelerated learning factors $c_1$ and $c_2$ in the PSO algorithms. The maximum initial value of the learning factor is set to 2, while $c_1$ and $c_2$ are constructed as functions (i.e., monotonically decreasing and monotonically increasing functions, respectively) with the expressions shown in Equations (30) and (31):

$$c_1\left(t\right)=2\left(1-e^{\left(-t/t_{max}\right)}\right)$$

(30)

$$c_2\left(t\right)=2e^{\left(t-t_{max}\right)/t_{max}}$$

(31)

Therefore, the particle swarm velocity update equation is

$$\begin{gathered} {\overline{v}}_i\left(t+1\right) = \omega {\overline{v}}_i\left(t\right) + c_1\left(t\right)r_1\left({\overline{p}}_i\left(t\right) - {\overline{x}}_i\left(t\right)\right) \\ + c_2\left(t\right)r_2\left({\overline{p}}_g\left(t\right) - {\overline{x}}_i\left(t\right)\right) \end{gathered}$$

(32)

In the early stages of evolution, if $c_1$ takes a larger value and $c_2$ takes a smaller value, this will allow the particles to move throughout the search space. In the later stages, if $c_1$ takes a smaller value and $c_2$ takes a larger value, this will increase the convergence rate toward the optimal solution.

In this paper, two classical benchmark functions (Griewank and Schaffer) are selected to conduct comparative simulations on the PSO algorithm [31] and the IPSO algorithm with the dynamic acceleration learning factor, which are used to verify the correctness and superiority of the IPSO algorithm.

The Griewank function is as follows:

$$minf\left(x_i\right) ={{\displaystyle \sum }}_{i=1}^N\frac{x_i^2}{4000}-{{\displaystyle \prod }}_{i=1}^{N}cos\left(\frac{x_i}{\sqrt{i}}\right) + 1$$

(33)

where $x_i\in \left[-600,600\right]$. The function has a local minimal value, the number of which is related to the dimensions of the problem, with a global minimum of 0, and $\left(x_1,x_{2,}\dots ,x_n\right) = \left(0,0,\dots ,0\right)$ can be obtained. The function is typically a nonlinear multi-modal function with a wide search space and is often considered a complex multi-modal problem that is difficult for optimization algorithms to handle.

The Schaffer function is as follows:

$$minf\left(x_1,x_2\right)=0.5+\frac{{\left(sin\sqrt{x_{1+}^2{x}_2^2}\right)}^2-0.5}{{\left[1+0.01\left(x_{1+}^2{x}_2^2\right)\right]}^2}$$

(34)

where $x_1,x_2\in \left[-10,10\right]$. The function is a two-dimensional complex function with an infinite number of minimal value points, achieving a minimum value of 0 at (0,0). Since the function has a strongly oscillatory nature, it is difficult to find a globally optimal solution. The curves of the Griewank and Schaffer functions when their independent variables are two-dimensional are shown in Figure 5.

In this paper, the Griewank and Schaffer functions are used to test the performance of the IPSO algorithm. The parameter settings are shown in

Table 1. Parameter settings for different PSO algorithms.

Algorithm	Particles m	Weight ω	Learning Factors c1, c2	Maximum Speed Vmax	Maximum Iterations Tmax
PSO	20	1	2, 2	1	100
IPSO	20	1	Equations (30) and (31)	1	100

, and the test results are shown in Figure 6.

From the simulations, it can be seen that the test results using the Griewank function show that the PSO and IPSO algorithms could converge to the optimal solution, but the number of iterations used to find the optimal IPSO algorithm was significantly smaller than that of the PSO algorithm. The results of the Schaffer function test show that the PSO and IPSO algorithms could converge to the optimal solution, but the number of iterations used to find the optimal solution was significantly smaller for the IPSO algorithm than for the PSO algorithm. Therefore, it is reasonable to believe that the addition of dynamic learning factors to the PSO algorithm in the process of finding the best search faster and with higher convergence accuracy and higher stability to solve the PSO algorithm easily fell into the local optimum problem.

In this part, the IPSO algorithm is used to autotune the five parameters ($K_p$, $K_i$, $K_d$, $\lambda$, and $\mu$) of the FO PI^λD^µ controller in Equation (6), which is used for course-keeping of the USV. The working principal diagram and model of the FO PI^λD^µ controller with the IPSO algorithm are shown in Figure 7 and Figure 8.

The IPSO algorithm is used to autotune the five parameters ($K_p$, $K_i$, $K_d$, $\lambda$, and $\mu$) of the FO PI^λD^µ control, which is a five-dimensional solution space. Considering the system overshoot and the square integral of the control loop error, the equation of the optimization objective function is as follows:

$$J={\varpi }_1{{\displaystyle \int }}_0^{+\infty }{e}^2\left(t\right)dt+{\varpi }_2\sigma$$

(35)

where $e\left(t\right)$ is the systematic error, $\sigma$ is the system overshoot, and ${\varpi }_1$ and ${\varpi }_2$ are weights.

5. Simulations

5.1. FO $P{I}^{\lambda }{D}^{\mu }$ Controller

The research object used the actual ship data of COSCO’s large container vessel of 5446 TEU, whose parameters are shown in

Table 2. Structure data of 5446 TEU container vessel.

Structure	Value	Structure	Value
length of ship	280 m	length of two columns	267 m
width of ship	39.8 m	distance of gravity center	2.64 m
area of Rudder	61.0 m2	square coefficient	0.67
no-load weight	3.5453 M tons	full load weight	6.5531 M tons
draft	12.532 m	draft with fully load	14.023 m

. Let the ship’s rated speed be V = 19.8 knots = 10.186 m/s (1 knot = 1.852 km/s) and the ship’s nonlinear course model adopt the Norrbin nonlinear model. Since the ship could be regarded as a large mass object in movement, the ship’s course control could be regarded as a kind of slow turning movement when the steering was not very frequent, so Equation (3) could be simplified. The simplified equation [37] is shown here:

$$T\stackrel{·}{\psi }+\psi =K\delta$$

(36)

According to the calculation, the model coefficients K = 0.1955 and T = 255.8837 were obtained. A_m = 1.3, ∅_m = π/4, λ = 0.1, and μ = 0.85 for the FO PI^λD^µ controller were determined by the system characteristics and practical experience. Then, K_p = 1.4516, K_i = 1.1355, and K_d = 95.0636 were calculated from Equations (22)–(26), respectively. The fitting frequency of the modified Oustaloup filtering method (ω_b, ω_h) chose [0.001, 1000] and the order N = 5. According to the design method of the PID ship autopilot [37], the K_p, K_i, and K_d are shown here:

$$K_p=\frac{T{\omega }_n^2}{K},K_i=\frac{T{\omega }_n^3}{10K},K_d=\frac{2T\epsilon {\omega }_n-1}{K}$$

(37)

where ${\omega }_n$ is the system’s natural frequency and $\epsilon$ is the system’s relative attenuation coefficient. Usually, one would have 0.8 ≤ ε ≤ 1.0, ω_n = 0.06, and ε = 0.9. From Equation (37), we could obtain K_p = 4.712, K_i = 0.028, and K_d = 136.243. The FO PI^λD^µ controller was applied to the ship’s course, where the maximum rudder angle was restricted from −35° to +35°, and the desired course angle was 30° when time was 0–1000 s. The FO PI^λD^µ controller and PID controller were compared. The comparison results can be seen in Figure 9, Figure 10 and Figure 11 and

Table 3. Performance of PID and FO PI^λD^µ controller.

Controller Type	Adjust Time ts	Rise Time tr	Overshoot M	Ess
FO PIλDµ	114	75	4.42%	0.002
PID controller	519	71	14.25%	0.007

.

In the next step, we set the ship’s course angle to change in the following order: (1) 0 ≤ t ≤ 1000 while setting the desired course angle to ψ_r(t) = 20° and (2) 1000 ≤ t ≤ 2000, while setting the desired course angle to ψ_r(t) = 30°. The course-tracking curve and rudder angle curve with the FO PI^λD^µ controller and the PID controller are shown in Figure 12 and Figure 13.

By comparing the simulation experiments, it can be seen that the FO PI^λD^µ controller had a significantly better control effect than the PID automatic rudder controller proposed in [37]. The FO PI^λD^µ controller had the advantages of a small overshoot, short adjustment time, and accurate control.

5.2. FO PI^λD^µ Controller Based on the IPSO Algorithm

Since the ship’s nonlinear model parameters changed with the speed, the calculated FO PI^λD^µ controller parameters could not meet the dynamic changes of the model’s parameters. The corresponding nonlinear model’s parameter values for ship course control at different speeds are shown in

Table 4. Corresponding model parameters under different speeds.

V (knots)	K	T	α	β
27.1	0.2676	186.9556	10.4915	7.5343
26.5	0.2617	191.1885	10.7291	8.0578
24.5	0.2419	206.7958	11.6049	10.1966
22.5	0.2222	225.1776	12.6366	13.1644
19.8	0.1955	255.8837	14.3601	19.3172

.

In the next stage, the objective was to check the self-adjustment of the controller based on IPSO. The simulation used the Norrbin nonlinear model in Equation (3). The five parameters of the FO PI^λD^µ controller were adjusted using the IPSO algorithm for the ship’s course, and the effectiveness of the algorithm was verified by simulation experiments. Following the design step of the FO PI^λD^µ controller, the fitting frequency of the Oustaloup filtering method (${\omega }_b$, ${\omega }_k$) was [0.001,1000] and $N=5$. The population size was set to 50. The maximum number of the birds’ steps was 50. The particle dimensions were 5, meansing [K_p, K_i, K_d, λ, µ], and the initial values of c₁ and c₂ were set at 2. The inertia weight was set to 0.9. Four different control strategies were used for the ship’s course, which were the PID controller [37], the PID controller based on the PSO algorithm [9], the FO PI^λD^µ controller based on the GA algorithm [19], and the FO PI^λD^µ controller based on the IPSO algorithm. The ship’s speed increased from V = 19.8 knots to V = 24.5 knots at a constant speed, the maximum rudder angle was restricted from −35° to +35°, and the desired course angle was 30° when the time was 0–1000 s. The comparison results can be seen in Figure 14 and Figure 15 and

Table 5. Performance of four different controllers.

Controller Type	Adjust Time ts	Rise Time tr	Overshoot M	Ess
IPSO-FOPID (proposed)	103	103	0.03%	0.005
PSO-PID	416	67	31.16%	0.024
PID	519	71	14.25%	0.007
GA-FOPID	163	65	9.53%	0.086

.

Meanwhile, wind and wave disturbances have been the main causes of yawing of ships when they are sailing at sea. In this design, wind and wave disturbances were introduced into the system, which verified the robustness. The disturbances used white noise to drive a typical second-order oscillation. The wave model [60] obtained under the effect of a force-six wind is that of Equation (38), and the simulation curve is shown in Figure 16.

$$h(s)=\frac{0.4198s}{s^2+0.3638s+0.3675}$$

(38)

Four different control strategies were used for a ship’s course under wind and waves, and the results can be seen in Figure 17 and Figure 18.

It can be seen from Figure 17 and Figure 18 that the FO PI^λD^µ controller based on the IPSO algorithm had a significantly better control effect than the automatic rudder PID controller, the PID controller based on the PSO algorithm, and the FO PI^λD^µ controller based on a GA algorithm. The FO PI^λD^µ controller based on the IPSO algorithm had the advantages of a small overshoot, short adjustment time, accurate control, and strong wind and wave resistance.

6. Conclusions

This paper proposed two methods for an FO PI^λD^µ controller for USV course control. First, this paper proposed an FO PI^λD^µ controller based on frequency domain specification, which used a desired gain margin and phase margin to meet the performance requirements of the ship’s course control. The simulation results showed that the FO PI^λD^µ controller had the advantages of a small overshoot, short adjustment time, and precise control over the automatic rudder PID controller. However, if the ship’s speed changed, the ship’s nonlinear course model would change. The parameters of the FO PI^λD^µ controller no longer met the needs of course control. Secondly, the FO PI^λD^µ controller based on the IPSO algorithm was proposed for USV course control. To address the influence of the learning factor on the search speed of the particles in the solution space, this paper constructed a particle swarm optimization algorithm with a time-varying learning factor. In the initial stage of optimization, the particles had a stronger self-learning ability and weaker social learning ability to strengthen the global search ability. In the later stage, the particles had a stronger social learning ability and weaker self-learning ability to facilitate convergence at the global optimum. Comparative experiments were carried out without wind and wave disturbances and with wind and wave disturbances. The results show that the FO PI^λD^µ controller based on the IPSO algorithm had the advantages of a small overshoot, short adjustment time, precise control, and strong anti-disturbance control. The controller output was subjected to wind and wave interference, and the control object itself changed due to certain factors. It could also overcome these effects and had a strong robustness. The control performance was better than that of the three other controllers, and it was easy to modify the existing PID-type automatic rudder while having a high promotion value.

In light of the effectiveness of the FO PI^λD^µ controller based on the IPSO algorithm, more improvements need to be studied in future. An interesting aspect for future study is how to combine an FO PI^λD^µ controller with other intelligent algorithms (e.g., neural networks, Lévy flight distribution, the Nelder–Mead algorithm, and fuzzy control) and apply them to a ship’s course control to further verify the superiority of the FO PI^λD^µ controller in ship course control. Another aspect which needs to be studied involves the use of FO PI^λD^µ controllers for a ship’s trajectory control.