## 1 Introduction  
Some processes, such as distillation columns and chemical reactors [1], aircraft pitch control, power supply systems with hydro turbine [2], and boost converters [3], have an initial response to an input step change in the opposite direction to that of its new steady-state value. This represents a strong limitation for the use of robust control methods based on high-gain controllers [4]. The control of these processes is influenced by two parallel competing dynamics working against each other [5, 6]. In the simple case with two counteracting first-order subsystems with gains \(K_{1},K_{2}\) and time constants \(T_{1},T_{2}\),  
\[S_{0}(s)=\frac{K_{1}}{T_{1}s+1}-\frac{K_{2}}{T_{2}s+1}=\frac{(K_{1}T_{2}-K_{2}T _{1})s+K_{1}-K_{2}}{(T_{1}s+1)(T_{2}s+1)} \tag{1}\]  
the system \(S_{0}(s)\) has a non-minimum phase (unstable) zero for \((K_{1}-K_{2})(K_{1}T_{2}-K_{2}T_{1})<0\). With \(K=K_{1}-K_{2}\), \(T_{1}=T,T_{2}=aT_{1}\) and \(bT=-(K_{1}T_{2}-K_{2}T_{1})/(K_{1}-K_{2})\), it can be rewritten as  
\[S_{0}(s)=\frac{K(-bT_{5}+1)}{(Ts+1)(aT_{5}+1)} \tag{2}\]  
If system (2) is extended by a dead time \(T_{d}\), which represents the sum of the remaining shorter loop delays, the system becomes a second-order plus unstable zero (SOPUZ)  
\[S(s)=\frac{Y(s)}{U(s)}=S_{0}(s)e^{-T_{d}s} \tag{3}\]The presence of an unstable zero causes problems in model-based control design approaches based on process inversion such as the disturbance observer-based (DOB) approach [7], or internal model control (IMC) [6]. The problems with unstable zeros also persist in the heuristic design of PI and PID controllers using optimization methods based on the values of the integral of the absolute error (IAE) and the total variation in the control signal (TV) for setpoint and disturbance step responses. For the inverse response processes with transfer function (3), with \(a\in[0.1,1];\ b\in[0.1,3]\), controller tuning was proposed in [8]. The study was conducted at two robustness levels. It was shown that the proposed filtered PID controllers with two degrees of freedom (2 DoF)  
\[U(s)=\bigg{[}(\beta+\frac{1}{T_{i}s})W(s)-(1+\frac{1}{T_{i}s}+T_{D}s)Y(s) \bigg{]}\frac{K_{p}}{1+T_{f}s} \tag{4}\]  
provide better performance (IAE) than 2 DoF PI controllers tuned with the same approach at the same robustness level. Parameter \(\beta\) represents the setpoint weighting, \(T_{i}\) the integral time constant, \(T_{D}\) the derivative time constant, \(K_{p}\) the proportional gain and \(T_{f}\) the filter time constant. The method provided responses with low undershoot and overshoot of the controlled variable, smooth control signal and low controller effort. However, in real control applications, manipulation carried out solely with controller and process transfer functions usually does not lead to acceptable results. In practice, many other aspects of control need to be considered. Let us therefore introduce these aspects using the concept of practice-oriented controller design (POCD) as follows:  
**Definition 1** (**Practice-Oriented Controller Design**). : _Practice-oriented controller design ensures the stability of the closed loop together with the reconstruction and compensation of the acting disturbances. The aim is to achieve fast and smooth transients (characterized by the smallest number of monotonic sections of their responses to step changes in the inputs) using the simplest possible process model. At the same time, the specified attenuation of the measurement noise and the robustness of the closed loop should be achieved. The limitations of the control signal should be taken into account, while the resulting dynamics of the circuit should meet the technological requirements._  
The consideration of all the above-mentioned POCD requirements is not a matter of course even for some well-known methods for designing controllers. For example, the most commonly used linear PI controllers have no additional filters and do not allow attenuation of high-frequency noise. This problem is even more pronounced with PID controllers.