Design of Anti-Swing PID Controller for Bridge Crane Based on PSO and SA Algorithm

Abstract

Since the swing of the lifting load and the positioning of the trolley during the operation of a bridge crane seriously affect the safety and reliability of its work, we have not only designed Proportional Integral Derivative (PID) controllers for the anti-swing and positioning control but also proposed a hybrid Particle Swarm Optimization (PSO) and Simulated Annealing (SA) algorithm to optimize the gains of the controllers. In updating the PSO algorithm, a nonlinear adaptive method is utilized to update the inertia weight and learning coefficients, and the SA algorithm is also integrated when the PSO algorithm is searching for a global optimal solution, to reduce the probability of falling into the local optimal solution. The simulation results demonstrate that the PSO–SA algorithm proposed in this paper is prone to be a more effective method in searching for the optimal parameters for the controllers, compared with three other algorithms. As shown by the experimental results, the swing angle stabilization time of the novel algorithm is 6.9 s, while the values of the other algorithms range from 10.3 to 13.1 s under a common working condition. Simultaneously, the maximum swing angle of the novel algorithm is 7.8°, which is also better than the other algorithms.

1. Introduction

A bridge crane is a piece of lifting equipment used in workshops, warehouses, and stockyards to lift materials, playing a significant role in current national economic production, and is a crucial component of modern industrial production for the mechanization and automation of the production process. Steel wire rope and pulley blocks are essential crane components, which make the crane swing easily while lifting heavy objects, due to inertia and other external factors. This residual swing may make it difficult to position heavy objects precisely and creates potential collision risks. The anti-swing control and automatic positioning technology of the crane spreader are the key factors to enhance the crane’s automation level, aiming at enhancing the crane’s operation efficiency and reducing the operator’s labor intensity.

For the anti-swing control and positioning control of the bridge crane, many control methods have been proposed, which can be categorized as either closed-loop control or open-loop control. Open-loop control provides no data feedback in real time to the controlled object. Garrido et al. [1], Alghanim et al. [2], Cao et al. [3], and Maghsoudi et al. [4] adopted input shaping technology. Although shaping technology has achieved a certain effect in the anti-sway control of cranes, its control object is best determined using model parameters, so the effect is not ideal when encountering external interference. Chen et al. [5] used the model predictive control (MPC) method to control a crane to operate according to the reference trajectory, which was also an open-loop control method. Therefore, there is sensitivity to system parameter changes and external interferences as well as difficulty in successfully suppressing the crane’s swing.

Regarding closed-loop control, many studies have been conducted in recent years. The parallel neural network and sliding mode control are primarily utilized to control the bridge crane’s position and swing angle [6,7]. With the input shaping method, the control effect is superior. Ranjbari et al. [8] utilized fuzzy control to control the bridge crane’s swing angle, compared it with the optimal controller (LQR), and found the proposed method to be effective. Zhai et al. [9] used an adaptive fuzzy controller to solve the swing problem of an offshore crane. The controller ensured the relative stability of the cargo with the ship deck when lifting and significantly reduced the swing angle of the cargo. Sun et al. [7] used the sliding mode control method to simulate and optimize the anti-swing of the crane, and ADAMS software was also used to simulate the control effect in real-time. The adaptive coupling control method was utilized to solve the anti-swing and positioning problems of the crane under lifting/lowering and uncertain parameter conditions, and the load weight was determined [10]. The optimal control strategy was used to estimate the parameters of the dynamic model of the crane online, while the PSO algorithm was employed to solve the time-constrained optimization problem [11]. Cao et al. [12] used a combination of the auto disturbance rejection control (ADRC) method and the sliding mode control method to realize anti-sway control of large lifting equipment. The simulation results showed that the method achieved good control performance. Sliding mode control, neural network control, and fuzzy control have been applied to the anti-sway control of bridge cranes and achieved satisfactory results. However, some methods depend excessively on model parameters and are challenging to implement in engineering. Therefore, a control method with a simple structure, strong adaptability, and excellent control effect is needed to meet the project’s requirements. PID control is widely used in industry due to its simple design, reliable performance, and high stability. The selection of the PID controller’s three control parameters (proportional KP, integral Ki, and differential Kd) significantly affects the controller’s stability and response characteristics. Therefore, it is essential to design an efficient PID parameters optimization method. In recent years, scholars have studied many intelligent optimization methods for PID parameters. Issa et al. [13] adopted arithmetic optimization algorithm (AOA) to determine the optimal value of PID parameters, combining the AOA algorithm with the Harris Hawk Optimization (HHO) algorithm, avoiding falling into the local lower limit of AOA–HHO by adding disturbance and mutation factors. Du et al. [14] used the improved Artificial Bee Colony (ABC) algorithm to optimize the PID controller parameters. The algorithm introduced the best food source and modified the probability of the food source. The optimized PID controller has achieved good results in controlling a radar servo system. Chen et al. [15] used an improved Ant Colony Optimization (ACO) algorithm to optimize the PID control parameters. The algorithm proposed a nonlinear incremental evaporation rate and improved the method of pheromone incremental updating. Its improved PID controller achieved a good control effect in the thermal system. Liu et al. [16] used the improved quantum bacteria foraging (QBFO) algorithm to optimize the parameters of the PID controller. The algorithm kept the quantum rotation angle updated. At the same time, it improved the probability amplitude operator, to improve the diversity of the quantum bacteria population, and enhanced the search ability of the algorithm for global optimization.

Since introduced in 1995 [17], the PSO algorithm has been utilized in many fields. Its algorithm is simple, and its convergence rate is rapid, so it has been used to solve numerous optimization problems. However, the PSO algorithm is prone to fall into a local optimal solution; therefore, improving the PSO algorithm has become the focus of research. Mao et al. [18] improved the inertia weight and learning factor of the algorithm using a dynamic response mechanism. Huang et al. [19] optimized particle topology using the dual inheritance framework of the cultural algorithm. Zhang et al. [20] proposed a new PSO algorithm, based on clustering and ring topology, to solve the multi-objective optimization problem. Yang et al. [21] developed an adaptive decreasing function to optimize the moment of inertia. Chen et al. [22] automatically adjusted the inertia weight using the triangle probability density function. In addition, researchers attempted to optimize the PSO algorithm by combining it with other optimization algorithms. Vijayakumar et al. [23] combined the Ant Colony algorithm with the PSO algorithm to increase the convergence rate of the algorithm. Gmili et al. [24] combined the Cuckoo algorithm and PSO algorithm to improve the PSO algorithm’s local optimization capability. Ma et al. [25] incorporated crossover and mutation operations into the PSO algorithm, thereby integrating the concept of the Genetic algorithm and enhancing the PSO algorithm’s optimization capability.

Based on these studies, the PSO algorithm uses a fixed weight coefficient value while searching for optimization, ignoring the influence of the inertia weight coefficient on global optimization and local optimization at different stages of iteration. The learning factor also uses a fixed coefficient value and fails to consider the different influences of the learning factors on individual experience and group experience, which makes it easy for the algorithm to fall into a local optimal solution.

To solve the shortcomings of the PSO algorithm itself, we propose a nonlinear adaptive method to optimize the weight coefficient and learning factors. At the same time, we integrate the jumping idea of the SA algorithm and the PSO algorithm to reduce the probability of falling into the local optimal solution. The new algorithm is applied to the optimization process of the PID controller parameters of a bridge crane. Through the analysis of the simulation results, the control system optimized by the PSO–SA algorithm shows better control performance compared with the PSO and SA algorithms and the CPM method, and the system has good robustness and adaptability with external disturbances.

The remainder of the article is organized as shown in Figure 1 below. In Section 2, the dynamic model of the bridge crane is stabled by the Lagrange equation. In Section 3, the bridge crane’s PID controllers are designed. In Section 4, the proposed algorithm is introduced. In Section 5, the proposed methods are utilized for simulation and analysis. In Section 6, we discuss the control effects of different algorithms in relation to the parameter change of a bridge crane and disturbance. In the last section, the conclusions are given.

2. Mathematical Modeling of Bridge Crane

The Lagrange equation [6,8,26] has been used to model bridge cranes by many researchers. In the modeling process, the swing behavior of the lifting load can be modeled as a single pendulum [27,28] or a double pendulum [26,29]. The nonlinear and robust coupling dynamic model of the bridge crane can be obtained by using two equivalent methods. In this paper, the lifting load is compared to a simple pendulum, and the dynamic model of the crane is established using the Lagrange equation.

The working conditions of the cranes in actual work are complex, and they will also be disturbed by external interference, so the physical system of the crane should be simplified to some extent. The movement of the lifting load is the integrated movement of the cart and trolley in the horizontal directions, and its effect on the vibration of the lifting load is also a result of the integrated movement in the two directions. Therefore, when simplifying the crane model, we only consider the movement of the model in the two-dimensional plane and regard the crane trolley and the lifting load connected by the steel wire rope as a single pendulum model, as shown in Figure 2.

At the same time, the following assumptions are made for the model.

(1)

The movement of the crane trolley and the load being lifted are considered particle motion, and the wind resistance is disregarded during movement.

(2)

The quality of the steel wire rope, its elasticity, and the friction between the steel wire rope and the trolley drum are not considered.

(3)

The nonlinear influence of the trolley motors, couplings, reducers, and other transmission mechanisms are also not considered, and the driving force of the motors is controllable and can drive the trolley directly.

It is specified that the direction of the trolley speed is the positive X direction (horizontal to the right) of the system, while the downward direction along the mass center of the trolley is the positive Y direction. Figure 2 shows the diagram of the bridge crane, where $M$, $m$, $F$,$x$, $\theta$, $g$, $\mu$, and $F_l$ are trolley mass, payload mass, driving force horizontal position of the trolley, the rope length, swing angle, gravitational acceleration, damping friction, and tension of the rope, respectively.

Newton’s mechanics are used to analyze the system’s force, and the Lagrange equation is used to establish the system’s nonlinear motion equations. The following are the motion equations for the bridge crane system.

$$\{\begin{array}{c}\left(M+m\right)\ddot{x}+m\ddot{l}sin\theta +2m\dot{l}\dot{\theta }cos\theta +\\ ml\ddot{\theta }cos\theta -m\dot{l}{\dot{\theta }}^{2}sin\theta +\mu \dot{x}=F\\ 2\dot{l}\dot{\theta }+l\ddot{\theta }+\ddot{x}cos\theta +gsin\theta =0\\ m\ddot{l}+m\ddot{x}sin\theta -ml{\dot{\theta }}^2-mgcos\theta =F_l\end{array}$$

(1)

The differential equations are simplified for the convenience of the research. The anti-swing control of the lifting weight is investigated under a fixed rope length in a working condition, so $l$ does not change, and the swing angle of the lifting weight generally changes about 0° around the Y-axis. Thus, it can be approximated that sin $\theta =\theta$, cos $\theta =1$ and $\dot{l }=\ddot{ l}=0 , \dot{\theta }=0$. The tension of steel wire rope $F_l$ can be regarded as a constant value, so the simplified differential equation of the system is as follows:

$$\{\begin{array}{c}\left(M+m\right)\ddot{x}+ml\ddot{\theta }+\mu \dot{x}=F\\ l\ddot{\theta }+\ddot{x}+g\theta =0\end{array}$$

(2)

Laplace transformation acts on both sides of Equation (2) [30].

$$\{\begin{array}{c}F\left(s\right)=\left(M+m\right)s^{2}X\left(s\right)+ml{s}^2\theta \left(s\right)+\mu sX\left(s\right)\\ 0=l{s}^2\theta \left(s\right)+s^{2}X\left(s\right)+g\theta \left(s\right)\end{array}$$

(3)

According to Equation (3), the transfer functions of trolley displacement and load swing angle can be obtained as follows:

$$\{\begin{array}{c}\frac{X\left(s\right)}{F\left(s\right)}=\frac{l{s}^2+g}{Ml{s}^4+\mu l{s}^3+\left(M+m\right)g{s}^2+\mu gs}\\ \frac{\theta \left(s\right)}{F\left(s\right)}=\frac{s^2}{Ml{s}^4+\mu l{s}^3+\left(M+m\right)g{s}^2+\mu gs}\end{array}$$

(4)

3. PID Controller Design

In industrial control, the classical PID controller is the most widely used and prevalent [8]. The PID controller outputs the control quantity through a linear combination of the proportional link, integral link, and differential link. Its input signal $e\left(t\right)$ is the deviation between the given signal $i\left(t\right)$ and the feedback signal $y\left(t\right)$, and the output signal $u\left(t\right)$ is the control quantity. The relationship between the two signals above is shown in the following formula:

$$u\left(t\right)=K_p[e\left(t\right)+\frac{1}{T_i}{{\displaystyle \int }}_0^{t}e\left(t\right)dt+T_d\frac{de\left(t\right)}{dt}]$$

(5)

where $K_p$ represents the proportional coefficient, $u\left(t\right)$ is the output of control quantity, and $T_i and T_d$ are the constant of integral time and the constant of differential time, respectively.

Their transfer function is as follows:

$$G\left(s\right)=K_p+K_i\frac{1}{s}+K_d$$

(6)

where $K_p$ represents the scale factor,$K_i$ is the integral coefficient, and $K_d$ represents the differential coefficient.

The aforementioned three parameters will have a direct impact on the system’s control effect; therefore, the PID control parameters’ tuning requires suitable techniques to identify the three optimal coefficients. The three parameters have the following operational effects on the control system.

K_P is used to proportionally reflect the error and immediately generate the controller signal, K_i is used to eliminate static error, and K_d is used to reflect the error’s changing trend and generate an efficient early correction signal.

The anti-swing control system of the bridge crane is a system with a single input and multiple outputs; its outputs are the position of the trolley and the swing angle of the lifting load. Therefore, the system employs a double-loop feedback control model, and PID controllers are used for trolley positioning control and swing control. The control architecture is shown in Figure 3.

4. Proposed Methods

In this section, the intelligent and classical control methods of PSO, SA, collaborative PSO–SA, and the classical CPM method are outlined for application in bridge crane control.

4.1. PSO Algorithm

PSO is an evolutionary computing tool that was proposed by Eber Hart and Kennedy in 1995 [17]. By simulating the coordination relationship between group movement and individual movement during the movements of birds, we can locate the birds’ final footholds and determine the optimal values. The potential solution to the optimization problem is abstracted as a bird particle without volume and mass in the search space [31]. Each particle has its position and velocity vector. The particle updates its position and velocity through constant motion, and the fitness value is calculated by the fitness function, which updates the optimal position for each individual particle. By comparing the individual optimal positions of each particle in the particle swarm, it is possible to determine the population optimal value, which is the global optimal value for the optimization problem [32]. Each particle in the colony will update the direction of its velocity vector, based on its own optimal position and the optimal position of the colony as a whole, and then move at the updated velocity. The speed and position of particles can be updated according to the following formulas:

$$\left\{\begin{array}{c}\begin{array}{c}{v}_i\left(k+1\right)=\omega v_i\left(k\right)+C_1{r}_1\left[P_i\left(k\right)-x_i\left(t\right)\right]\\ +C_2{r}_2\left[P_g\left(k\right)-x_i\left(t\right)\right]\\ x_i\left(k+1\right)=x_i\left(k\right)+v_i\left(k+1\right) \end{array}\end{array}\right\}$$

(7)

where $v_i\left(k\right)$ represents the velocity of the i-th particle in the k-th iteration calculation, $x_i\left(k\right)$ is the position of the i-th particle during the k-th iteration, $\omega$ is the inertia weight, $C_1$ and $C_2$ are learning coefficients, $r_1$ and $r_2$ are random variables generated from a uniform distribution in [0,1], and $P_i\left(k\right)$ and $P_g\left(k\right)$ represent the optimal position found by a single particle and the whole particle swarm, respectively, up to now [24].

The advantages of the PSO algorithm are that the optimized function is not required to have mathematical properties such as differentiability or differentiability, and the convergence rate is fast. However, the disadvantage is also evident. While it is simple to search for the best value within a local region, it is easy to fall into the local optimal solution.

4.2. SA Algorithm

The SA algorithm, which is based on the annealing process of high-temperature metal solids, is an optimization algorithm that adopts the random search method to obtain the global optimal solution [33]. Annealing is a process of heat treatment from high to low temperature. After the metal solid has been heated to a high temperature, the particles within it are in a state of disorder and free motion. In the process of temperature decreasing, the particles inside the solid are in different states. Particles tend to become steadier and more organized over time and, ultimately, remain in a state of orderly arrangement. The SA algorithm starts from a high initial temperature, and the solution approaches stability gradually, as the temperature parameter decreases, but the stable solution may be the local optimal solution. At this point, the SA algorithm jumps out of the local optimal solution with a certain probability, which increases the likelihood of searching for the global optimal solution, to a certain degree.

In the SA algorithm, the solution to the optimization problem and its corresponding function value are regarded as the state of the object and the energy function in physical annealing, respectively, and the optimal solution is its lowest energy state [34]. The algorithm accepts the deteriorating solution, with a certain probability “P”, in the process of searching a solution, and its judgment is based on the Metropolis criterion. The formula of probability “P” is as follows:

$$P=\{\begin{array}{c}1 dy<0\\ \exp \left(-\frac{dy}{T}\right) dy\ge 0\end{array}$$

(8)

where $dy=y\left(x_2\right)-y\left(x_1\right)$, which represents the cost function difference between the new solution and the current solution. If $dy<0$, the new solution is accepted with probability 1; if $dy\ge 0$, the new solution is accepted with probability $\exp \left(-\frac{dy}{T}\right)$.

$T$ represents the annealing temperature, and its value relates to the initial annealing temperature $T\mathrm{o}$ and the change law of temperature. The relationship between $T_{k+1}$ and $T_k$ is $T_{k+1}=\alpha T_k$, where $\alpha$ is generally between 0.8 and 1. In this way, the temperature is relatively high during the initial phase of cooling down, and the probability of an acceptable deterioration solution is rather high; consequently, it is possible to escape the local optimal solution [35].

4.3. Proposed PSO–SA Algorithm

The basic PSO algorithm can ensure that the particles converge to a stable position by adjusting the parameters. Still, it cannot guarantee that the position is the global optimal solution. The likelihood of falling into a local optimal solution will increase as the complexity of the optimization problems (including the size and dimension of the search space) increase. To overcome the aforementioned issues, the nonlinear adaptive method is incorporated into the fundamental PSO algorithm, and the SA algorithm concept is also introduced to establish a fusion PSO–SA algorithm in this paper. The specific improvements are described below.

(1)

Nonlinear Adaptive Inertia Weight ω

Inertia weight ω is directly related to the convergence rate, and large ω can enhance the global search ability of the algorithm and help to detect new favorable areas. Instead, with a small ω, the search process will focus more on the local area. The best way is that the algorithm has a strong global search capability at the start of the iteration and a focus on local search at the end. Therefore, with the fixed constant of the basic Particle Swarm Optimization algorithm ω, in contrast, time-varying ω is more conducive to solving the optimization problem. In this paper, a nonlinear adaptive method is proposed; ω can be expressed by the following formula:

$$\omega ={\omega }_1-\left({\omega }_1-{\omega }_2\right){\left(\frac{iter}{Maxiter}\right)}^2$$

(9)

where ${\omega }_1$ represents the upper limit of $\mathsf{\omega }$’s value, ${\omega }_2$ represents the lower limit of $\omega$’s value, iter is the current generation selection times, and Maxiter is the maximum generation selection times.

(2)

Asynchronous Change Method of Learning Coefficients $c_1$ and $c_2$

c₁ and c₂ are coefficients used to determine the influence of individual and group experiences on particle trajectories. If c₁ and c₂ are inappropriate, the optimization may not converge or falls into the local optimal solution. So, a nonlinear adaptive change strategy is introduced into these learning coefficients. In the early stage of iteration, c₁ should be large, and c₂ should be small, so particles can learn more from the self-optimization process and less from the social-optimization process, which is conducive to strengthening the global search ability. Similarly, in the final stage of the search, small c₁ and large c₂ should be employed, so particles can learn more from the social optimization process and less from the self-optimization process, which is conducive to global convergence. The asynchronous change equations of the learning coefficients are as follows:

$$\{\begin{array}{c}{c}_1=c_{11}-\left(c_{11}-c_{12}\right)exp\left(\frac{iter}{Maxiter}-1\right)\\ c_2=c_{21}-\left(c_{21}-c_{22}\right)exp\left(1-\frac{iter}{Maxiter}\right)\end{array}$$

(10)

where $c_{11}$ represents the upper limit of $c_1$’s value, $c_{12}$ represents the lower limit of $c_1$’s value, $c_{21}$ represents the upper limit of $c_2$’s value, and $c_{22}$ represents the lower limit of $c_2$’s value;

(3)

Introducing SA into the Process of the Optimization for PSO

Using the PSO algorithm cannot solve the optimization problem only. Therefore, the SA algorithm is used to optimize the PSO algorithm, so that the PSO algorithm may jump out of the local optimal value and locate the global optimal solution. The process steps of the SA algorithm to optimize the PSO algorithm are as follows.

Step 1. Set the parameters, including the maximum number of iterations, the range of inertia weight, the range of position and velocity, initial temperature, the coefficient of annealing, the number of particles, and the range of learning factor; initialize the position and velocity of each particle at random; and determine the range of PID parameters.

Step 2. Update the position and velocity; calculate the fitness $y\left(x_i\left(k\right)\right)$ of the particles for each iteration; and record the optimal fitness of the current particle as $p_{ibest}\left(k\right)$ and the optimal fitness of all particles as $p_{gbest}\left(k\right)$.

Step 3. Incorporate the SA algorithm; begin calculating the fitness of the current particle’s optimal value and the fitness of the current global optimal value; and determine if the current optimal value should be accepted as the global optimal value based on Equation (8) (Metropolis criterion). If accepted, the current optimal value is recorded, and the original $p_{gbest}\left(k\right)$ is replaced by $p_{gbest}^{\text{'}}\left(k\right)$, which is taken as the global optimal value for the next iteration.

Step 4. Update the speed and the positions of all particles, according to Equation (7), and then the speed update formula becomes as follows:

$$v_i\left(k+1\right)=\omega v_i\left(k\right)+c_1{r}_1\left[P_i\left(k\right)-x_i\left(k\right)\right]+c_2{r}_2\left[p_{gbest}^{\text{'}}\left(k\right)-x_i\left(k\right)\right]$$

(11)

Continue to calculate the fitness of each particle, and then conduct the operation of SA.

Step 5. When the algorithm reaches the stop condition or the maximum number of iterations, the optimization stops. Otherwise, return to Step 2 to continue optimizing.

The proposed new algorithm idea is shown in Figure 4.

The PSO algorithm is the core idea of the PSO–SA algorithm. However, the PSO–SA algorithm should adopt the SA algorithm’s idea, when determining the global optimal solution each iteration. The PSO–SA algorithm improves not only the global optimization capability of PSO but also its local optimization capability. It can effectively locate the local optimal solution and obtain the global optimal solution. To evaluate the efficacy of the proposed PSO–SA method, it will be compared with the intelligent PSO and SA methods in the following section, as they are used to achieve the positioning control and anti-swing control of a bridge crane.

5. Simulation and Analysis

5.1. Optimization Methods Based on SA, PSO, and PSO–SA Intelligent Algorithm

The intelligent algorithm is linked to the Simulink model of the PID control system of the bridge, and the initial particles generated by the intelligent algorithm are assigned to the parameters of the two PID controllers. The Simulink model of the PID controllers is then simulated and analyzed, and the system’s dynamic performance indices are obtained. The dynamic performance indexes of the system are passed to the intelligent algorithm program, as the adaptive values of the intelligent algorithm, in this iteration. The adaptive values determine whether the algorithm’s termination conditions have been met. If so, the optimal solution is returned. Otherwise, the iterative calculation continues, and the preceding parameter setting steps are repeated, until the optimal PID controller parameters have been determined.

In this study, the integral absolute error (IAE) is selected as the system performance evaluation metric. Given that there are two outputs and two PID controllers, the expression for the fitness function is as follows:

$$IAE={{\displaystyle \int }}_0^T\left|E\left(t\right)\right|dt$$

(12)

$$G={{\displaystyle \int }}_0^T\left|E_1\left(t\right)\right|dt+{{\displaystyle \int }}_0^T\left|E_2\left(t\right)\right|dt$$

(13)

where $E\left(t\right)$ is the difference between the output and input of the control system, $E_1\left(t\right)$ and $E_2\left(t\right)$ are the errors of the position controller and the anti-swing controller, respectively, T represents the time of the simulation, and G is the sum of the integrals of the two errors and also the fitness function of all the algorithms.

5.2. Simulation and Analysis

In this section, the two PID controllers’ parameters are set using the PSO–SA, PSO, SA, and CPM methods, to verify the proposed algorithm’s optimization effect. MATLAB is used to establish the simulation control system for each optimization method, and the control performances obtained by each parameter setting method are compared and analyzed under a common condition.

By manually adjusting the parameters of the PID controller repeatedly in advance, the change range of each parameter is roughly determined, which can significantly reduce the range of PID parameter optimization. Through the repeated debugging and operation of all algorithms, the values of each parameter are shown in

Table 1. Parameter values of all algorithms.

Symbol	PSO	SA	PSO–SA	Parameter
Nmax	10	/	10	Population size
maxiter	100	100	100	Maximum iterations
Lx	/	10	/	Markov chain length
[vmin,vmax]	[−1,1]	/	[−1,1]	Velocity range
ω	0.8	/	/	inertia weight
ω1	/	/	0.8	Maximum inertia weight
ω2	/	/	0.4	Minimum inertia weight
c1	2	/	/	Learning coefficient 1
c2	2	/	/	Learning coefficient 2
c11	/	/	4	Maximum value of c1
c12	/	/	2	Minimum value of c1
c21	/	/	4	Maximum value of c2
c22	/	/	2	Minimum value of c2
T0	/	100	100	Initial temperature
Tf	/	0.07	/	Final temperature
α	/	0.93	0.93	Annealing coefficient
[X1min,X1max]	[200,500]	[200,500]	[200,500]	Kp search range (PID)
[X2min,X2max]	[0,0.5]	[0,0.5]	[0,0.5]	Ki search range (PID)
[X3min,X3max]	[500,1000]	[500,1000]	[500,1000]	Kd search range (PID)
[X4min,X4max]	[200,600]	[200,600]	[200,600]	Kp search range (PD)
[X5min,X5max]	[50,200]	[50,200]	[50,200]	Kd search range (PD)

.

The iteration curves of the three intelligent algorithms are shown in Figure 5 and Figure 6. The performance index functions of the PSO–SA algorithm, SA algorithm, and PSO algorithm are 11.91, 12.97, and 13.36, respectively. The value of the performance index function of the PSO–SA algorithm is the smallest of the three, and the number of iterations is also the smallest, which is 58 times.

To evaluate the control performance of the PID controllers of the bridge crane, the expected signal of the crane trolley [x] position is set to a step signal of 5 m, and the swing angle [θ] of the crane load is fixed to 0°. The parameters of the crane are the rope length of 6 m, the lifting weight of 500 kg, and the trolley weight of 300 kg, which are the same as the actual crane. The output of the PID controllers’ parameters, calculated by the PSO, PSO–SA, SA, and CPM methods, can be seen in

Table 2. PID parameters obtained with PSO, SA, PSO–SA, and CPM methods for the bridge crane.

Bridge Crane Output	Optimal PID Gains
	PSO			SA
	Kp	Ki	Kd	Kp	Ki	Kd
x	298.82	0.02	662.54	420.63	0.21	848.92
θ	453.22	0	134.25	556.21	0	118.62
	PSO–SA			CPM
	Kp	Ki	Kd	Kp	Ki	Kd
x	351.28	0.12	912.58	552.31	0.11	961.27
θ	402.35	0	76.37	481.23	0	148.57

. The output of the trolley displacement [x] and lifting load swing angle [θ] obtained by these methods are shown in Figure 7 and Figure 8, respectively.

Table 3. Performances obtained using PSO, SA, PSO–SA, and CPM methods for the bridge crane.

Output	Performances	Methods
		CPM	PSO	SA	PSO–SA
x	TS (s)	10.23	10.33	8.47	6.32
	MP (%)	10.96	8.80	8.65	0.26
	IAE	13.02	12.89	12.48	11.53
θ	TS (s)	12.31	13.12	10.32	6.87
	θmax (°)	11.36	8.17	9.49	7.87
	IAE	0.67	0.47	0.49	0.37

shows the response performance values of the bridge crane, including stability time (T_S(s)), the maximum overshoot of trolley displacement M_P(%), the maximum swing-angle of the lifting load (θ_max(°)), and the errors mentioned above (IAE).

From Figure 7 and Figure 8, we can see that the maximum overshoot of the swing angle of the PID control system optimized by the PSO–SA algorithm is 7.87°, under a common working condition, while the maximum overshoots of the other control systems optimized by the CPM, PSO, and SA algorithms are 11.36°, 8.17°, and 9.49°, respectively. It can be seen that the PID control system optimized by the PSO–SA algorithm can achieve a better anti-swing control effect for lifting weights. The maximum overshoot of the control system optimized by PSO–SA is 0.26% for the positioning of the crane trolley, while the maximum overshoots of the other control systems optimized by the CPM, PSO, and SA algorithms are 10.96%, 8.80%, and 8.65%, respectively. It can be seen that PID control system optimized by the PSO–SA algorithm shows better control performance for the control of positioning. From the experimental simulation, the response of the control system optimized by the PSO–SA algorithm shows good performance in terms of overshoot, stability time, and error data, which shows that the control system optimized by the PSO–SA algorithm has good performance in the positioning control and anti-swing control of the bridge crane.

6. Robustness Analysis

6.1. Experimental Simulation for Bridge Crane without Interference

The previous section demonstrates that the PID control gains of the bridge crane’s trolley positioning and load swing angle are calculated using the PSO, PSO–SA, SA, and CPM methods, under a common working condition, given that the rope length is 6 m and the lifting weight is 500 kg. However, the actual rope length of the bridge crane is not fixed during operation; rather, it corresponds to other rope lengths under different working conditions, so the lifting objects vary within the range of the rated load. Therefore, in order to verify the adaptability and robustness of the PID gains obtained by each optimization method, combined simulations are conducted for various rope lengths and lifting weights, and simulations are conducted of additional working conditions, as shown in

Table 4. Parameters of the crane under other three working conditions.

Working Condition	Rope Length (m)	Payload Weight (kg)	Trolley Weight (kg)
1	9	800	400
2	6	800	400
3	9	500	400

.

Comparing Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, it can be seen that the overshoot of the control system gains obtained by PSO–SA in the trolley positioning control under the three working conditions are about 7.18%, 4.62%, and 3.7%, which are the smallest values in each group. The trolley stabilization time and the values of IAE in the position control are also the smallest in each group. From

Table 5. Performances obtained using PSO, SA, PSO–SA, and CPM methods under working condition 1.

Output	Performances	Methods
		CPM	PSO	SA	PSO–SA
x	TS (s)	17.10	17.43	15.03	10.70
	MP (%)	17.22	15.84	15.02	7.18
	IAE	23.25	17.77	19.52	15.66
θ	TS (s)	20.30	15.81	17.88	12.92
	θmax (°)	8.31	6.03	7.02	5.84
	IAE	0.637	0.41	0.49	0.37

,

Table 6. Performances obtained using PSO, SA, PSO–SA, and CPM methods under working condition 2.

Output	Performances	Methods
		CPM	PSO	SA	PSO–SA
x	TS (s)	12.63	17.17	14.63	11.17
	MP (%)	14.54	13.84	12.02	4.62
	IAE	17.12	21.23	18.45	15.53
θ	TS (s)	15.75	15.61	17.24	12.62
	θmax (°)	9.83	6.96	8.25	6.86
	IAE	0.52	0.41	0.52	0.39

and

Table 7. Performances obtained using PSO, SA, PSO–SA, and CPM methods under working condition 3.

Output	Performances	Methods
		CPM	PSO	SA	PSO–SA
x	TS (s)	19.10	14.83	15.58	12.16
	MP (%)	13.92	11.62	11.48	3.70
	IAE	24.26	18.35	18.87	16.32
θ	TS (s)	24.66	16.09	15.47	14.59
	θmax (°)	9.21	6.96	7.95	6.62
	IAE	0.73	0.42	0.43	0.39

reveal that the maximum swing angles of the control system gains obtained by the PSO–SA algorithm are about 5.84°, 6.86°, and 6.62°, which are also the smallest values in each group, and the swing angle’s stability time and the corresponding IAE values are also extremely small in each group. This demonstrates that, compared to the SA, PSO, and CPM algorithms, the PID gains obtained by the PSO–SA algorithm have better adaptability and can achieve satisfactory control effect under new working conditions.

6.2. Experimental Simulation for Bridge Crane with Interference

This simulation experiment simulates the impact of instantaneous wind and environmental noise when the bridge crane is working outside. When the trolley is about to reach the target position, and the swing angle of the lifting weight is about to reach zero, a pulse signal and white noise are superimposed on the feedback signal to simulate the effect of the wind and the environmental noise, and the controllers readjust the system until the position of the trolley and the swing angle of the lifting weight are close to the expected value. From this, the robustness of the controllers is tested. By comparing the overshoot and setting time of the control optimization algorithms, the optimization effect of each algorithm can be determined. To comprehensively reflect the advantages and disadvantages of various control optimization algorithms, the physical parameters of the four common crane working conditions are shown in

Table 8. Parameters of crane under four working conditions.

Working Condition	Rope Length (m)	Payload Weight (kg)	Trolley Weight (kg)
1	6	500	300
2	9	800	400
3	6	800	400
4	9	500	400

.

Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 shows that when the crane operates under the four common parameters, the maximum overshoot of the swing angle of the PID control system optimized by the PSO–SA algorithm is 2.45° when affected by a crosswind and environmental noise, while the maximum overshoots of the other control systems optimized by the CPM, PSO, and SA algorithms are 3.78°, 3.65°, and 3.68°, respectively. It can be seen that in the presence of external interference, the PID control system optimized by the PSO–SA algorithm can achieve a good anti-swing control effect for lifting weights. At the same time, the maximum overshoot of the control system optimized by PSO–SA is 0.38 m for crane trolley positioning, while the maximum overshoots of the other control systems optimized by the CPM, PSO, and SA algorithms are 0.47 m, 0.55 m, and 0.46 m, respectively. It can be seen that in the presence of external interference, the PID control system optimized by the PSO–SA algorithm shows better control performance.

From the above analysis, it can be seen that the PID controller optimized by the PSO–SA algorithm has a good adaptability and can achieve a good control effect in the chosen working conditions. This also reflects that the PSO–SA algorithm has better performance than the PSO algorithm, SA algorithm, and CPM method in PID parameter optimization, and the PID parameters optimized by PSO–SA have better adaptability.

7. Conclusions

In this study, the PSO algorithm is improved, and the SA algorithm is integrated into a single-input double-output PID controller system, to control the lifting anti-sway and trolley positioning of a bridge crane. A nonlinear adaptive method is proposed to define the weight coefficients and learning factors. When the improved PSO algorithm is searching for the global optimal solution, the jumping idea of the SA algorithm is integrated, which can enhance the global optimization ability of the algorithm.

The experimental simulation results demonstrate that, compared to the other three methods, the proposed PSO–SA method provides the best comprehensive effect for the bridge crane’s trolley positioning control and the lifting load swing angle control. Its swing angle stabilization time is about 6.9 s, and the values optimized by the other algorithms are between 10.3 and 13.1 s. The maximum swing angle is 7.8°, which is also the smallest of these algorithms. The maximum swing angle of the lifting load is 2.45°, under the working condition with interference, which is almost 30% less than that of the other three algorithms. The maximum overshoot of trolley positioning is 0.38 m, which is about 25% less than that of the other three algorithms.

The PSO–SA algorithm can be applied to lifting equipment using single-input and double-output PID controllers, such as a gantry crane, a quayside crane, and other lifting equipment that needs to have anti-swing and positioning functions.