1. Introduction

2. Proposed Optimization

2.2. Classical chaotic mappings initialize populations

The traditional ALO algorithm employed random initialisation of the population, which tends to produce an inhomogeneous distribution of the initialised population and is not conducive to covering further areas in the search space. By taking a chaotic initialisation method to initialise the population, the coverage as well as the diversity can be increased. At present, the most commonly used methods of chaotic initialisation are logistic mapping, sine mapping,tent mapping and so on.

$$\mathrm{Logistic}\mathrm{map}:x_{i+1}=4r{x}_i\left(1-x_i\right)$$

(5)

$$\mathrm{Sin}\mathrm{e}\mathrm{map}:x_{i+1}=rsin\left(\pi x_i\right)$$

(6)

$$\mathrm{Tent}\mathrm{map}:x_{i+1}=\left\{\begin{array}{cc}2r{x}_i\hfill & \mathrm{if}{x}_i<0.5\hfill \\ 2r\left(1-x_i\right)\hfill & \mathrm{if}{x}_i\ge 0.5\hfill \end{array}\right.$$

(7)

Combining a diversity of low-dimensional chaos to form a new composite chaotic system. This type of chaotic system can effectively overcome the shortcomings of low-dimensional chaos, and is less complex and can be realised more conveniently than high-dimensional chaos.Tent-Logistic-Cosine chaos mapping utilises the existing Logistic, Sine and Tent maps as seed maps [14] as in Equation(8).

$$x_{i+1}=\left\{\begin{array}{cc}cos\left(\pi \left(2r{x}_i+4(1-r)x_i\left(1-x_i\right)-0.5\right)\right)\hfill & \mathrm{for}{x}_i<0.5;\hfill \\ cos\left(\pi \left(2r\left(1-x_i\right)+4(1-r)x_i\left(1-x_i\right)-0.5\right)\right)\hfill & \mathrm{for}{x}_i\ge 0.5,\hfill \end{array}\right.$$

(8)

The distribution (left) and histogram (right) of the 1000 generations of Tent-Logistic-Cosine chaos mapping were generated on MATLAB as shown in Figure 2

Figure 1. (a)Chaotic sequence distribution diagram. (b) Histogram of distribution of chaotic sequence.

Figure 1. (a)Chaotic sequence distribution diagram. (b) Histogram of distribution of chaotic sequence.

The above figure shows that the sequence generated by the iteration of Tent-Logistic-Cosine chaotic mapping is not distributed uniformly in the phase space, meanwhile its parameter space in the chaotic state is narrower, and the chaotic effect is less effective.

2.3. Improved ALO

To address the above problems of traditional ALO, this paper improves the algorithm by introducing improved composite chaotic mapping and Elite Opposition-Based learning to initiate the population and increase the diversity of the population.

2.3.1. Tent-Logistic-Cotangent composite chaotic mapping

To avoid the existence of small circles and unstable points in the chaotic mapping, the periodicity, ergodicity and regularity of the chaotic variables are maintained.In this paper, we present a new chaotic mapping, named Tent-Logistic-Cotangent composite mapping, with the mathematical expression as in Equation (9).

$$x_{i+1}=\left\{\begin{array}{cc}mod(\left(\pi \left(rcot\left(4\pi x_i\right)+4(1-r)x_i\left(1-x_i\right)-0.2\right)\right),1)\hfill & \mathrm{for}{x}_i<0.5;\hfill \\ mod(\left(\pi \left((1-r)cot\left(4\pi x_i\right)+4(1-r)x_i\left(1-x_i\right)-0.15\right)\right),1)\hfill & \mathrm{for}{x}_i\ge 0.5,\hfill \end{array}\right.$$

(9)

In contrast to Equation (8),the distribution (left) and histogram (right) of the 1000 generations of Tent-Logistic-Cotangent chaos mapping were also generated on MATLAB as shown in Figure 2.

Figure 2. (a)Tent-Logistic-Cotangent chaotic sequence distribution diagram. (b) Histogram of distribution of chaotic sequence.

Figure 2. (a)Tent-Logistic-Cotangent chaotic sequence distribution diagram. (b) Histogram of distribution of chaotic sequence.

Comparing Figure 1 and Figure 2, the proposed Tent-Logistic-Cotangent composite chaotic mapping iteration produces sequences that are more uniformly distributed in the phase space with excellent ergodicity.

2.3.2. Elite Opposition-Based learning

Opposition-Based Learning (OBL) is a machine intelligence strategy proposed by Tizhoosh [15] in 2005, and is used effectively to improve the quality of the solution by applying it to initialize the population. Elite Opposition-Based learning (EOBL) can improve on the basis of Opposition-Based Learning [16] , which extracts the elite individuals in the population in order to calculate the reverse individuals, and then select the candidate solutions with superior fitness values from the candidates.

Due to the great chance and blindness of the solutions generated by chaotic mapping, the generated solutions are subjected to Elite Opposition-Based learning to save convergence time and improve the quality of the solutions.It has been shown that real-world engineering problems have been solved by Elite Opposition-Based learning [17] .

Thus the elite backward learning based update initialization of the formulation as in Equation (10).

$$X_{i,j}^{t+2}=k\ast (lb+ub)-X_{i,j}^{t+1}$$

(10)

where $k\in [0,1]$. $lb$ is the lower bound of the solution space. $ub$ is the upper bound of the solution space.

2.3.3. Introduction of Cosine Factor Strategy

Traditional ALO algorithms later stage ant individuals perform random wandering towards ill-adapted ant lions, which can lead to weak iterative results and local optimum stagnation problems.Therefore, the cosine factor is introduced into the elitist update formula for adjusting the search direction to be closer to the direction of the optimization goal, and the algorithm is more exploratory, better avoids falling into the local optimum, and approaches the global optimum solution faster during the search process.Inspired by the cultural-particle swarm optimization algorithm (C-PSO) [18] , the algorithm uses adaptive inertia weights to balance global and local search capabilities.

Based on this, this paper designs a cosine factor for a given parameter can effectively reduce the situation of falling into the local optimum, and improve the search ability of local development and global,as in equations (11)-(12).

$$\theta \left(k\right)={\omega }_{min}\ast (1-cosh)+{\omega }_{max}\ast cosh$$

(11)

$$\phi \left(k\right)=r\ast cosh+{\omega }_{max}\ast cosh$$

(12)

$$Ant{P}_i^t={\theta }^2\left(k\right)R_A^t+{\phi }^2\left(k\right)R_E^t$$

(13)

where $h=\pi t/2t_{max}$. $t_{max}$represents the maximum number of iterations. ${\omega }_{max}=0.5$,${\omega }_{min}=0.1$.

3. Performance Analysis on Benchmark Functions

4. Performance Analysis of High-speed 3D printing model for Temperature Control

4.2. Controller Design

The quality of the parameters in the PID controller will greatly affect the quality of the controller. In this paper, we design an optimal PID controller based on the improved Ant-Lion optimization algorithm, with the aim of finding a set of optimal parameters in the solution space $K_p$, $K_d$,$K_i$, so that the system meets the control requirements and performs well. In designing the OCALO algorithm PID controller, the objective function should be set in accordance with the performance index of the control system and obtain satisfactory dynamic characteristics of the iterative process, and the evaluation function T is set as shown in Equation (16).

$$T={\int }_0^{\infty }\left(w_1\left|e\left(t\right)\right|+w_2{u}^2\left(t\right)\right)\mathrm{d}t$$

(16)

where $e\left(t\right)$is the error of the output value with respect to the input value. $u\left(t\right)$ is the control value of the control margin. $w_1$ is the weight value, which takes the value [0, 1], and usually $w_1$ = 0.999. $w_2$ is the weight value, which takes the value [0, 1], and usually $w_2$ = 0.001. In addition, a restriction is needed to prevent the overshooting, i.e., when the overshooting occurs, an additional overshooting term is introduced in the objective function, and the overshooting term $w_3\left|e\left(t\right)\right|$is introduced in the objective function Q as shown in Equation (17).

$$Q={\int }_0^{\infty }\left(w_1\left|e\left(t\right)\right|+w_2{u}^2\left(t\right)+w_3\left|e\left(t\right)\right|\right)\mathrm{d}t$$

(17)

Where $w_3$ is the weight value,$w_3$>>$w_1$,in general, $w_3$=100. The simulation model is built in simulink according to Equation (15). The improved ant-lion optimization algorithm is used to design the PID controller, setting the parameter ranges $\mathrm{Kp},\mathrm{Kd},\mathrm{Ki}\in [0,100000]$, and Equation (17) as the fitness function, and the goal of the optimization search is to find a set of PID values that minimise the Q error by correcting the three parameters through the OCALO algorithm. The block diagram of the PID controller design based on the OCALO algorithm is shown in Figure 9.

Figure 8. Block diagram of optimized PID controller design based on OCALO algorithm.

Figure 8. Block diagram of optimized PID controller design based on OCALO algorithm.

4.3. System Simulation and Results Analysis

In this paper, only OCALO algorithm is listed with PSO algorithm, ALO algorithm and IALO algorithm, because WALO algorithm and GALO algorithm optimize the PID of the model temperature system with a very slow convergence speed, so they do not draw their curves on the graph. From Figure 9, it is easy to see that the OCALO algorithm has a faster response, the least amount of overshooting and the fastest response when it optimizes the PID of the temperature system.

Figure 9. Partial algorithmic PID controller for temperature systems.

Figure 9. Partial algorithmic PID controller for temperature systems.

According to Table 8, through the peak amplitude, peak time, rise time, Settling time four aspects of the analysis, OCALO algorithm peak amplitude is the smallest indicates that the system in the process of PID rectification compared to other algorithms overshoot the smallest amount of the system is the most stable; the peak time as well as the shortest rise time indicates that the process of PID rectification compared to other algorithms with the shortest response time, the fastest response time; stabilisation time compared to other algorithms is the shortest, so that can be to a certain extent to save the high-speed printing time costs.

Table 9. Simulation results of PID controller.

Table 9. Simulation results of PID controller.

Variable	OCALO	PSO	ALO	IALO
Peak amplitude	1.59774	5.7589	14.3121	4.0686
Peak time	321.4505	385.3550	469.4281	451.4830
Rise time	272.0981	328.0076	407.3485	379.7559
Settling time	351.3188	860.1506	619.9532	1040.6734

The results show that OCALO has stronger robust search ability and the convergence speed can meet the real-time requirements. After establishing the FDM 3D printing nozzle model and determining its parameters, the OCALO-PID controller is designed, and through experimental simulation, the control curves of the OCALO-PID controller and the classical PID controller are compared under different control requirements, and it is concluded that the overall control performance of the OCALO-PID controller is better.

4.4. Interference Test

In performing high-speed 3D printing, the 3D printing nozzle in the face of a sharp turn of the printing task, due to the rapid rotation will cause the 3D printing nozzle vibration, so that has been stabilised at the target temperature of the 3D printing nozzle temperature system sudden load, resulting in temperature instability, affecting the quality of the print, this paper adds a perturbation at 520s, the OCALO algorithm with the classical algorithm(left) and the other improvements of the ALO algorithm(right) comparison for example Figure 11.

Analyzing Figure 10, compare the time for various controllers to recover to within 2error bars after being disturbed by a 0.1 step signal at 520 s, as shown in Table 10. Comparing the immunity of each algorithm in simulation.Determine the immunity to interference in the simulation by both the maximum peak value and the time required for recovery.

Figure 10. (a)Comparison of classical algorithms after sudden loading. (b) Comparison of Improved Algorithms after sudden loading.

Figure 10. (a)Comparison of classical algorithms after sudden loading. (b) Comparison of Improved Algorithms after sudden loading.

After the perturbation at 520 s, OCALO-PID has the smallest maximum peak, PSO-PID and ALO-PID have already experienced oscillatory phenomena, and all other algorithms have larger peaks than OCALO-PID. the amplitudes of PSO-PID and ALO-PID are still outside of the2 % error bars at 1000 s, and the shortest time used by OCALO-PID is 225.1 s , 22.4 s faster than GALO-PID, 19.1 s faster than IALO-PID, and 17.8 s faster than WALO-PID, and the difference in the recovery time is smaller compared to SSA-PID, but it is also 10.8 s faster, verifying that the OCALO-PID controllers are more resistant to disturbances than the other controllers.The OCALO-PID controller shows good performance in simulation, but it needs to be validated by applying it to an actual high-speed 3D printing temperature system.

5. Performance Analysis on Physical Platform Validation

5.1. Experiment platform construction

In order to verify the control effect of OCALO-PID controller on the temperature system of high-speed FDM 3D printing, this paper builds an experimental platform as in Figure 11, and the upper computer is the web-side interface based on Klipper firmware. The real-time temperature monitoring module of the Klipper firmware provides feedback on the heating and vibration cancellation of the FDM 3D printing nozzle.

Figure 11. Setting up the experimental environment.

Figure 11. Setting up the experimental environment.

5.3. Actual Interference Test

In real high-speed 3D printing application environments, different perturbations are often encountered,the cause of the perturbations may be due to human control requirements, or due to fast stepping movements, or may be indeterminate. Therefore, perturbation testing of algorithms is needed as a way to verify that the robustness of the algorithm meets the requirements. Perturbation of all algorithms.Add artificial perturbations in real environment 3D printing run 250 s. Figure 13 shows the perturbation test plot for all algorithms.

Table 12. Performance index.

Table 12. Performance index.

Algorithm	Steady-State Error (◦C)	Recovery Time (s)
PSO-PID	30.4	-
GALO-PID	18.5	376
WALO-PID	26.7	369
IALO-PID	10.4	370
ALO-PID	9.3	352
PID	9.6	340
OCALO-PID	6.5	261

Through the perturbation test, the OCALO-PID controller restores the steady state after 261 s, which is 115 s, 108 s, 109 s, 91 s, 79 s, and 131.4 s less than PSO-PID and GALO-PID, respectively, The steady state value of the OCALO-PID controller is 216.5 °C, and the error between setpoint and 121 °C is only 6.5 °C. The recovery time is much shorter than that of the PSO-PID and GALO-PID. The steady-state value of the OCALO-PID controller is 216.5 °C, and the error between the setpoint and 121 °C is only 6.5 °C, which is 30.4 °C, 18.5 °C, 26.7 °C, 10.4 °C, 9.3 °C, and 9.6 °C lower than that of the PSO-PID and GALO-PID, respectively, The steady state errors of PSO-PID,WALO-PID, IALO-PID, ALO-PID and PID are smaller, respectively. Therefore, the OCALO-PID controller has excellent performance to meet the robustness of high-speed 3D printing.

6. Conclusions

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