# Modeling and Multi-Objective Optimization Design of High-Speed on/off Valve System
## 1 Introduction  
Traditional electro-hydraulic servo systems typically utilize servo or proportional valves as control components. These valves offer numerous advantages, including excellent accuracy, low hysteresis, and minimal dead zone, which contribute to improved control function. However, servo valves also suffer from certain drawbacks, such as high cost, poor efficiency, and reduced reliability due to their sensitivity to the contamination of transmission media [1]. Consequently, the replacement of servo valves with a combination of high-speed on/off valves (HSV) in digital hydraulic systems has emerged as a growing trend [2]. HSVs constantly operate in fully open or fully closed states, effectively discretizing continuous flow and serving as the fundamental components of digital hydraulic systems [3]. Over the past years, digital hydraulic components have obtained extensive traction in the hydraulic systems field due to their ability to be directly controlled by pulse width modulation signals [4]. The integration of digital hydraulic systems can significantly enhance overall efficiency, making HSVs a prominent research focus in the hydraulic domain [5, 6]. Nevertheless, further advancements are required for HSVs to fulfill the demands of digital hydraulic servo systems, particularly in terms of achieving quicker response times, smaller physical dimensions, and higher flow capacities. In the research on the optimal design of HSV, [PERSON] et al. [7] introduced positive and negative pulse voltage control to the coil of screw-in cartridge valves (SCV), enabling SCVs to replace HSVs in certain application areas without changing their structure. [PERSON] et al. [8] utilized current feedback analysis based on critical switching currents to estimatethe operating state of HSVs, thereby improving their dynamic performance. [PERSON] et al. [9] designed a novel flat-plate HSV spool structure and established a flow field model for the flat-plate spool, enhancing HSVs' flow field stability and flow rate. [PERSON] et al. [10] proposed a conical armature structure design and a dual-duty cycle pulse width modulation control method to improve HSVs' steady-state and transient characteristics. [PERSON] et al. [11] established an HSV parameter analysis model based on the electromagnetic mechanism of solenoid valves to improve HSVs' dynamic response performance under varying operating conditions. [PERSON] et al. [12] introduced a double magnetic circuit-driven fast on/off valve (DAFV) with a multi-stage excitation control algorithm (MSEC), improving the stability of the HSV response time. [PERSON] et al. [13] developed a pre-excitation control algorithm that allows the coil current to quickly maintain a desired state, thereby enhancing the dynamic performance of HSVs. These studies demonstrate that the primary goal of HSV optimization design is to improve its dynamic characteristics, specifically the response time for valve working. The interrelations among various design parameters of HSVs necessitate a comprehensive consideration of each relevant factor during the design process to achieve optimal performance [14]. This complexity gives rise to a multi-objective optimization problem (MOO). Optimization algorithms are commonly utilized to address complex MOO problems, with metaheuristic methods being widely employed in the field of engineering. Metaheuristic algorithms are algorithmic frameworks that operate independently of the specific problem and draw inspiration from natural phenomena, biological behavior, or even mathematics [15]. Metaheuristic methods offer advantages such as randomness, ease of implementation, and the ability to handle black box functions, enabling them to effectively solve a wide range of complex engineering problems [16; 17]. Among the numerous metaheuristic algorithms, the artificial fish swarm algorithm (AFSA) has recently garnered significant attention in the literature [18].