## 1 Introduction  
In control manufacturing, standard PID control algorithms are frequently utilized as a control strategy, providing the benefits of unequivocal simplicity and excellent performance [1], among others. PID control algorithms, alongside traditional methods, are mainly used for control-parameter optimization [2, 3]. However, the PID control algorithm has several drawbacks, including difficulty in parameter setting and susceptibility to disturbance. More PID tuning algorithms, such as neural networks [4], machine learning [5, 6], and heuristic algorithms [7, 8], should be developed for use in actual control processes to address these issues.  
However, as controlled systems become increasingly complex, there is a growing need for control strategies that can handle time-varying behavior and time delays. The complexity of the controlled system contributes to the frequently subpar control effects of a PID controller [9, 10]. At present, many intelligent algorithms have been developed. Systems with time-varying and hysteresis delays can benefit significantly from these approaches [11, 12]. By manually modifying the control rules, the expert PID control algorithm can be applied to specific simulation systems [13]. Fuzzy PID control can be included in the controller using a lookup table, making it simpler to obtain better control effects [14, 15, 16, 17, 18, 19]. However,the expert PID and fuzzy PID controllers always have subpar timing precision and limited anti-interference capabilities. Due to the benefits of rapid adjustment velocity and excellent accuracy [20; 21], the neural network PID control approaches have been applied in numerous complicated systems [22; 23; 24]. In 2022. [PERSON] et al. [25] used the PSO-BPNN-PID algorithm to control the nutrient solution, and the simulation results demonstrated that the reaction time and accuracy of the control system were superior to those of the conventional PID control method. The same year, a BPNN-PID controller was developed using a Xilinx field-programmable gate array (FPGA) technology in [26]. The findings revealed that the suggested system exhibited fast convergence and dependable performance. To better regulate the delay system in 2023, a particle swarm optimization-radial basis function (PSO-RBF) is utilized in place of a traditional PID controller [27]. In general, BP neural network research has thus shown positive results.  
In addition, to achieve effective controller performance, these parameters of the BPNN-PID controller must be properly optimized. Moreover, heuristic methods are typically used in the process of parameter optimization [28; 29; 30; 31; 32]. In 2018, the BP neural network was optimized using a mind evolutionary algorithm to predict wave heights; the model achieved great accuracy and a fast running time [33]. In 2021, using an algorithm known as the water cycle method, [PERSON] et al. [34] improved the BP neural network model for landslide prediction. In 2022, it was proposed that the Boost circuit's dynamic and generally pro capabilities be improved by combining a genetic algorithm with BP neural network PID control (GA-BP-PID) [35]. Therefore, an ALGWO-BPNN control model is proposed in this paper as a better way to control time-varying and time delay systems. The primary contributions of the proposed controller are as follows:  
1. The GWO technique is used to propose and optimize a PID control model based on a BPNN. Considering that the GWO algorithm is not necessarily the best for solving complex problems, a stochastic adjustment convergence factor formula is proposed, and the beta and delta are adjusted in combination with a differential evolution (DE). With the enhanced GWO method, the connection weight of the BPNN achieved a verifiably better control effect.
2. Several benchmark functions are selected to analyze ALGWO.
3. Some activation functions for the output layer of the BPNN model are analyzed and compared.
4. The ALGWO-BPNN model is simulated and compared with the conventional PID controller.