## 1 Introduction  
In spite of innovations in predictive and advanced control techniques, PID controllers are widely used in industries. Due to the introduced extensions and optimal tuning, they can meet even the highest requirements of the user. These controllers can use their inherent capabilities according to the dynamics of the processes and the requirements of the control systems. Depending on the control objective and the tools used, they can be classified into several groups based on response characteristics, ensuring the required stability margin, minimizing an appropriate performance criterion. Methods making use of the internal model control approach or the direct synthesis method should also be included in this list. The descriptions of such methods are available in the literature. Probably the best-known method, developed by [PERSON] and [PERSON] [1], was directed at the adequate stability margin and maintained similar properties via set-point tracking and disturbance rejection. A % decay ratio is considered to be a design criterion for this method. Some tuning methods are based on the process reaction curve [2; 3; 4]. [PERSON] et al. [5] and [PERSON] and [PERSON] [6] discussed tuning based on performance minimization criteria. The development of an internal model controller (IMC) by [PERSON] and [PERSON] [7] proved the way for the formulation of PID controller settings that would be the equivalent of an adequate first- or second-order IMC controller [5; 8; 9; 10]. When a control system is primarily intended to follow set-point changes, it is often expected to reach new set-points without overshooting and preferably in the shortest possible time. The requirements formulated above also take into account the economic aspect of the control process, as their required control costs are low, as opposed to time-optimal control systems. Several methods of tuning PI/PID controllers have been developed with the goal of set-point tracking with the minimum settling time and without overshooting. The first to be mentioned is the method proposed by [PERSON], [PERSON], and [PERSON] (CHR) [11], in which proportional, integral, and derivative values of the controller settings depend on the parameters of the First-Order Plus Dead-Time (FOPDT) model. Tuning methods based on heuristic methods can be very helpful in the controller design process. Controller parameters similar to the CHR method can be determined on the basis of the so-called nomograms, which graphically represent the controller parameter dependencies, such as a function of the ratio of the time delay (\(\theta\)) and the time constant (\(T\)). A certain inconvenience in using nomograms is the logarithmic scale of both the axes of these plots. By their nature, however, these methods are only approximate. A clearer approach to this issue is provided by the direct synthesis method. If the expected requirements illustrating the control objective are reflected in the form of the desired closed-loop transfer function, it is sufficient to know only the model in order to design the controller. The use of this method for tuning controllers was presented by [PERSON] [12], [PERSON] and [PERSON] [13], and [PERSON] and [PERSON] [14]. In this case, too, a significant difficulty in determining the correct regulator settings is the time delay occurring in the control process. The lack of an analytical method for tuning controllers of time-delayed processes allowed for determining such regulator settings, which would ensure the desired but also achievable response of the system. This, in turn, generated motivation to undertake research in this area of knowledge. The paper is organized as follows: The next chapter, which is Section 2, describes the controller design method using direct synthesis for typical inertial objects, including inertia without delay as an important reference point for further arguments. The derivation and the description of the proposed method are included in Section 3. The simulation results are given in Section 4.