## 1 Introduction  
Automatic control is the most basic task used in the industry. To obtain a satisfactory system response, many control structures have been developed. Because complex control systems are not trivial to analyze, selecting the control structure parameters does not seem trivial. Therefore, automatic tuning procedures based on nature-inspired optimization algorithms have attracted significant attention in recent years. Such auto-tuning allows the user to obtain optimal system responses, in terms of a defined objective function, regardless of the control system complexity. The proper formulation of the objective function is crucial for the proper optimization process. In general, higher values of control structure proportional gains allow for the achievement of better dynamics. On the other hand, the limitation of the physical system's accuracy is related to delays, quantization, measurement noises, or unmodeled non-linearities during control structure design, which limits the possible dynamic of the control system. The objective function requires expert knowledge of the control system to maximize the system's dynamics and still provide a chattering-free control signal. However, formulating the objective function seems less difficult than selecting the optimal control structure parameters. It should be noted that automatic selection of control structure parameters was intensively researched in recent years. The following control structures were considered: Proportional-integral-derivative (PID) controllers [1, 2, 3, 4, 5, 6], two degree-of-freedom PID (2 DoF PID) [7], state-feedback controllers [8, 9, 10, 11], multi-resonance controllers [12], neural network-based controllers [13], fuzzy logic controllers [14], and event-triggered controllers [15]. Moreover, selecting nature-inspired optimization algorithms is not crucial for properly operating automatic tuning. Due to the no free lunch theorem, there is no best algorithm to solve all optimization problems. In the case of global optimization algorithms, which are nature-inspired, every algorithm can solve the same problem. The difference is related to the optimization time required to find the global optimum. Therefore, the selection of the optimization algorithm can be interpreted as user preferences. The most commonly used in the literature for automatic tuning are particle swarm optimization [2; 16; 17], artificial bee colony algorithm [10; 12], and grey wolf optimizer [13; 18; 19]. The simple objective function with the integral time absolute error (ITAE) indicator was used in [17; 20], while the particle swarm optimization and Java optimization algorithms were used to solve the problem, respectively. Robust two-degree-of-freedom state-feedback PI-controller tuned by the particle swarm optimization algorithm for an automatic voltage regulation system was proposed in [21]. The objective function was not presented, except to mention that it utilizes the output and control signal. In [22], a nature-inspired optimization algorithm has been employed to obtain the weighting matrices values needed to calculate discrete state-feedback speed controllers for permanent magnet synchronous motor drive. This approach selects eight controller parameters using one weighting factor in the objective function. The objective function was based on minimizing the speed and current error in the \(d\)-axis of the rotational coordinate system. The penalty function was related to changes in the control signal, and this part has an additional coefficient that the user should select. In [23], the particle swarm optimization algorithm is used to optimize the linear quadratic regulator for a voltage source inverter with an LC output filter. The proposed objective function is based on the ISE indicator and the penalty for control signal chattering. The 15-D problem has been reduced to the selection of a single penalty coefficient. The design process of the adaptive neural controller implemented for a non-linear object using grey wolf optimizer was presented in [19].