## 1 Introduction
[26], an optimal low-pass filter is combined with each derivative term of an IMC-PIDA (PIDD2) controller applied to a dual rotor aerodynamic system. The parameters of the PIDA controller for controlling the output voltage of a non-ideal dc-dc converter were obtained using a modified Gray-Wolf optimizer [27]. In ref. [32] the authors deal with the detailed modeling and design of load frequency controllers (LFCs) of single and multi-area power systems using PID controllers and PID controllers with serial compensators. These controllers, referred to as PIDC, can be considered equivalent to PID controllers with higher-order derivatives (PIDA controllers are equivalent to PIDC1 control). The explicit tuning formulas for the different turbine types are characterized by the use of two free parameters that affect the performance and robustness of the system. They define a single and a triple real pole of a closed-loop control system and the corresponding poles and zeros of the controller under study. The integral performance measures IAE and ITAE, the total variation TV used to evaluate the control effort, and the sensitivity indices derived by simulations and calculations (maximum sensitivity \(M_{s}\), maximum complementary sensitivity \(M_{p}\) and measurement noise sensitivity \(M_{n}\)) are analyzed in terms of these tuning parameters. The parallel PIDD2 controller combined with a second-order binomial filter [33] for the automatic voltage regulator (AVR) of a synchronous generator is designed by minimizing the load disturbance response while considering the constraints on the sensitivity functions: \(M_{s}\), \(M_{p}\) and \(M_{n}\). The five controller parameters are obtained by iteratively solving the optimization problem for given sensitivities \(M_{s}\) and \(M_{n}\) and for a fixed value of controller zeros damping factor \(\xi\). The robustness index \(M_{p}\) is then calculated using the auxiliary parameter \(\xi\). Although several limitations of the internal signals are mentioned in the description of the AVR control process, the experiments performed were linear. Several approaches are based on setting the multiple real dominant poles [14; 36]. Other solutions [37; 38] propose conditions for the maximum decay rate. The paper shows that the formulation of conditions for the closed-loop poles can be avoided by considering the performance portrait method. The method can be advantageous both with regard to the transcendental nature of a closed circuit with an infinite number of poles and also with regard to the nonlinear character due the control signal limitations. In this context, the paper is organized as follows. Section 2 summarizes the main results of the design of an ideal PDA controller for the IPDT system using the multiple real dominant poles method, its extension by a prefilter and low-pass implementation filters. Section 3 extends the problem to the series PIDA controller with a simple disturbance-observer-based disturbance reconstruction and a compensation of the disturbance by a positive feedback from the controller output. This section also includes a brief analysis of the problems caused by saturation of the control signal. In Section 4, the optimal tuning of the PIDA controller is performed using the performance portrait method and illustrated by an example of controlling a thermal process. Absolute stability of the loops with modified PIDA controllers are proven by the circle criterion in Section 5. The main results of the work are discussed in Section 6, emphasizing the superior properties obtained by reliable controller tuning and in the saturating control. A paper summary and future developments are included in the conclusions.