## 1 Introduction
One of the first attempts to obtain an analytical PIDA controller design procedure for a third-order system [19; 20] neglected the effects of derivative filters required in the implementation of the controller transfer function. To determine the four parameters of the ideal series controller, four equations were solved using the root locus method. The proposed solution was based on the choice of a pair of complex conjugate poles of the closed loop, a real pole with the same real part and a real pole shifted to higher frequencies (resulting in negligible dynamics). The approach has been verified with different types of third-order plants and using different performance metrics such as percent overshoot, settling time and peak time. The setpoint responses of this approach typically exhibitedovershoot, even without considering control signal limitations. In ref. [21], this approach was further modified by [PERSON]'s method to reduce overshoot and provide robust stability. In ref. [23], the PIDD\({}^{2}\) (i.e., PIDA) controller is presented as an alternative to fractional-order (FO) PID controllers (which are implemented by higher-order filters). The optimal setting of the four controller parameters for the automatic voltage regulator (AVR) of a synchronous generator was realized by minimizing the integral of the time-weighted absolute error (ITAE) using particle swarm optimization. Among the main implementation problems was the use of filters required to obtain the derivative and the second derivative of the control error signal. The use of PIDA control was extended with the aim of balancing the differences between the generated power and the constantly changing load demand in a power system from the AVR to the load frequency control (LFC). To achieve robust and reliable performance, the work proposed to optimize the parameters of the ideal parallel PIDA controller with soft computing techniques using ITAE criteria without considering the necessary filters and control signal constraints. The work [30] continues with the application of artificial intelligence methods. An optimal design of a PIDA controller for a higher-order nonlinear time-delay system using a modified butterfly optimization algorithm is proposed to achieve an optimal control response complemented by a mutuality system. The goal is to reduce overshoot due to large disturbances but the focus is on the computational aspects of the optimization. Filter and signals have not been investigated. Another application of PIDA control with derivative filter was developed by [31] to improve the flight stability of a quad copter in a noisy environment. It is associated with a heuristic genetic filter design that addresses the challenges of nonlinearity, uncertainties, coupling and measurement noise that cause instability in flight and in automatic hover. The tuning of the proposed parallel PIDA controller with multiple inputs and outputs is performed by a stochastic dual simplex algorithm. Despite the fact that the controller's filter has a significant impact on the closed-loop stability [34], the filters required for a reliable implementation of PIDA controllers in practice have not yet been studied in detail. For this reason, ref. [28] considered designing a PIDA controller using an interactive tool for designing a parallel PIDA controller with minimum order filters in the derivative and acceleration terms. However, such an approach unnecessarily increased the number of controller parameters and did not allow the design of a controller for signal limitations. Furthermore, similarly to in [35], with two additional filter parameters, the complexity of controller tuning increases. The authors of [25] recognized the importance of controller filters in terms of circuit robustness and considered the time constant of the first-order filter as a tuning parameter in their design of an IMC-PIDA controller for an interconnected time-delayed power system. In Ref.