## 1 Introduction  
Cantilever beams are a fundamental element in structural engineering, widely used in various applications, including civil engineering structures, aircraft, and machinery. These beams, fixed at one end and free at the other, are prone to vibrations when subjected to external forces. Such vibrations can lead to structural fatigue and failure if not adequately controlled. To address this, smart materials like piezoelectric transducers (PZTs) are employed as sensors and actuators to monitor and mitigate vibrations. PZT materials have the ability to convert mechanical energy into electrical energy and vice versa, making them ideal for vibration control in smart structures. A smart structure is a structural system that can sense and respond to its environment. It is typically made up of a conventional structure (such as a beam, plate, or shell) with embedded sensors and actuators. The sensors measure the state of the structure, such as its strain or displacement. The actuators then take action to control the structure's response, such as by changing its shape or stiffness. Piezoelectric zirconate titanate (PZT) materials are a type of smart material that can be used to make sensors and actuators for smart structures. They generate an electric charge when they are deformed, while deformed when an electric field is applied to them [1]. This property makes them ideal for sensing and controlling the vibrations of structures. Engineering structures can be made vibration-free by using these smart materials. Engineering applications for piezoelectric smart structures are quite promising [2]. Vibration is a widespread problem in many engineering structures, such as buildings and bridges. It can cause damage to the structure, reduce its lifespan, and make it uncomfortable or even dangerous to use. Structural vibration control has always received a lot of research interest from the civil and control engineering community [2]. Several studies have been conducted to control and reduce the vibration of structures, and various control technologies have been tested and applied to solve this issue. Piezoelectric made of lead zirconate titanate (PZT) is a type of smart material that can be used to control vibrations in structures. PZIs produce an electric charge when they are deformed, or conversely, deform when an electric field is applied to them [3]. This property can be used to generate forces or moments that can be used to counteract the vibrations of a structure. The application of deep learning to control systems has gained significant traction, offering new methods for enhancing the precision and adaptability of these systems. In the domain of vibration control for cantilever beams, deep learning techniques such as multi-layer neural networks and Recurrent Neural Networks (RNNs) present promising alternatives to traditional control strategies. These methods are highly effective at detecting complex patterns in data, which can lead to more accurate and efficient control of smart structures. The ability of deep learning models to refine control strategies beyond what is possible with conventional methods positions them as a cutting-edge solution in this domain. Proportional-integral--derivative (PID) controllers have long been the industry standard for control systems due to their straightforward design and proven effectiveness. However, the challenge of tuning PID controllers to consistently achieve optimal performance across diverse operating conditions remains a significant limitation. Traditional tuning techniques, though widely used, may not fully exploit the capabilities of PID controllers, particularly in complex and variable environments. The integration of deep learning into the PID tuning process offers a novel solution to this challenge. By employing multi-layer neural networks, the PID parameters--proportional, integral, and derivative gains--can be optimized, resulting in a more responsive and effective control system.