## 1 Introduction  
### Motivation  
Proportional-integral-derivative (PID) controllers are widely recognized as a fundamental control strategy in industrial and academic applications. Their simplicity, ease of implementation, and effectiveness in a wide range of systems make them a very important tool of control engineering [1]. Despite their strengths, traditional PID controllers often exhibit limitations when applied to systems requiring precise position control, particularly in scenarios involving stringent transient performance requirements such as overshoot reduction and fast settling times [2]. Several modifications to the classical PID structure have been proposed to address these challenges. These include adaptive tuning methods [3], nonlinear PID strategies [4], and hybrid control schemes [5]. However, these approaches often require complex calculations or introduce new challenges such as increased sensitivity to noise or system uncertainties. The hybrid system for trajectory restoration combines the advantages of continuous and discrete dynamics, allowing a versatile and robust control strategy for discontinuous changes in reference signals. The proposed method is particularly suitable for applications such as robotic manipulators and servomechanisms [6]. In [7], a hybrid control strategy combining model predictive control (MPC) with PID controllers was introduced to improve the performance of supercritical power units. This approach demonstrated significant improvements in response time and energy savings compared to traditional PID methods. In [8], a fuzzy gain programmed PID controller was developed to improve power quality in grid-connected hybrid solar PV and PEM fuel cell systems. The proposed controller effectively reduced total harmonic distortion, demonstrating its potential for improving energy efficiency. In [9], a hybrid optimization approach for energy control in electric vehicle controllers regulating three-phase induction motors is derived. This method improved the energy efficiency and system performance. Recent developments reflect the increasing integration of intelligent and hybrid methodologies within PID control architectures. In [10], a neuro-fuzzy and robust PID controller is integrated via active filters to enhance performance under uncertain conditions; the neuro-fuzzy component manages transient behavior, while the robust PID ensures steady-state accuracy. In [11], a neural-augmented PID controller based on deterministic learning is applied to robotic manipulators, guaranteeing the stability and convergence of tracking errors under partial excitation conditions. A hybrid controller combining PID/fractional-order PID with neural network tuning via a zebra optimization algorithm is proposed in [12], yielding enhanced robustness and energy efficiency on a two-link robot system. Reference [13] introduces a 3 DOF neuro-fuzzy PID strategy for automatic generation control in hydrogen-integrated multi-area power systems, effectively handling nonlinearities and improving frequency regulation. In [14], a hybrid adaptive PID scheme combining LMS and backpropagation enables real-time gain adjustment, demonstrating high adaptability and robustness, and serving as a foundation for nonlinear extensions. A hybrid control strategy blending PID and model predictive control (MPC) using a fractional-order RAM model is proposed in [15], where MPC refines the PID reference trajectory to enhance overall system response. The work in [16] presents a hybrid PID framework for automated guided vehicles (AGVs), using genetic algorithms (GAs) to optimize controller gains and support vector regression (SVR) to reduce computational complexity. A boundary logic-based hybrid controller combining sliding mode and PID control for underactuated nonlinear cascade systems is proposed in [17], improving performance on complex dynamic tasks.