## 1 Introduction  
Bridges are recognized as critical components of global infrastructure, serving as vital links in geographical and economic lifelines. The role they play in facilitating social and economic development is of paramount importance. However, their safety and stability are significantly threatened by seismic activity [1]. Severe earthquakes have been shown not only to cause physical damage to bridges but also to lead to structural failure, resulting in traffic disruptions, risks to human life, and considerable economic losses. Therefore, enhancing the seismic performance of bridges has been identified as a key research focus within civil engineering. Traditionally, seismic design for bridges has relied on increasing structural stiffness and load-bearing capacity to resist seismic forces. However, limitations in this approach have been observed, as increasing seismic forces leads to heightened internal stresses, requiring more complex designs and escalating costs. In response to these challenges, base isolation technology has been introduced as a viable solution [2]. By incorporating an isolation layer between the bridge structure and its foundation, the transmission of seismic forces to the superstructure is reduced, minimizing structural deformation and damage [3, 4]. Base isolation systems are generally classified into passive and active systems, each with specific characteristics and application scenarios [5]. Passive isolation systems are designed to function without external energy, reducing the transmission of seismic energy to the superstructure through the installation of flexible isolation devices between the bridge and the foundation of the bridge [6; 7]. Devices commonly used in passive isolation systems include lead rubber bearings, friction pendulum bearings, and high-damping rubber bearings [8]. These devices work by absorbing seismic energy via the displacement of the isolation layer, thereby reducing the vibration amplitude of the superstructure. The simplicity and cost-effectiveness of passive isolation systems have allowed them to be widely adopted in various bridge structures, with minimal maintenance required. However, since design parameters such as stiffness and damping are predetermined during construction, passive isolation systems cannot adapt during seismic events [9]. When subjected to seismic motions of varying frequency and intensity, passive systems may not respond optimally, thus limiting their effectiveness [10]. This limitation has prompted a growing interest in more adaptive isolation systems that adjust to changing seismic conditions. To address these shortcomings, active isolation technology has been increasingly investigated [11]. Unlike passive systems, active isolation systems incorporate sensors, controllers, and actuators to monitor seismic activity and the structural response in real time. Through control algorithms, counteracting forces are applied, reducing the seismic impact on the bridge structure. The flexibility of active isolation allows for the real-time adjustment of system parameters during seismic events, improving seismic performance [12]. The core of the active isolation system lies in the design of its control strategy, with Proportional-Integral-Derivative (PID) controllers being commonly employed due to their simplicity and stability. In such systems, proportional, integral, and derivative parameters are adjusted to provide real-time control over the structure's dynamic response, enabling rapid adaptation to changing seismic conditions [13]. In response to seismic accelerations or displacements detected by sensors, the PID controller calculates and applies the necessary control forces, reducing the structural vibration response [14; 15]. Although effective in practice, optimizing PID controller parameters to handle complex seismic environments remains a significant challenge [16].