## 1 Introduction
The compensation of the reconstructed disturbance in the form of a change of the offset at the output of the stabilizing controller (for the integrating process, it is sufficient to use a simple gain) finally leads to a positive feedback of automatic reset and hyper reset controllers. Due to the fact that the disturbance observer (DOB) used in the \(PD_{n}^{m}R\) controller in Figure 1 is based on disturbance estimation from steady states (with an output derivative of zero), it was possible to reduce the number of its blocks from two (which is usually the case in the DOB [7], in the state space [19] or in the ADRC [20] based on the inverse process model [21], see also [22]) to one with \(F_{R}(s)\), which is used in HO-ARCs. Figure 1: \(PD_{n}^{m}R\) controller as a combination of ideal \(PD^{m}\) controller with a low-pass implementation filter \(Q_{n}\) giving \(PD_{n}^{m}\), saturation non-linearity, and automatic reset (AR) \(F_{R}\). **Definition 2** (HO-AR Controller). : _The \(PD_{n}^{m}R\) controller \(m\in\mathbb{Z}^{+}\); \(n\in\mathbb{Z},n\geq m\) (see Figure 1) represents a combination of the \(PD_{n}^{m}\) controller, saturation non-linearity, and automatic reset, modifying the \(PD_{n}^{m}\) offset by a positive feedback with a low-pass filter \(F_{R}\) (5); Thereby, the \(PD_{n}^{m}\) controller consists of an ideal \(PD^{m}\) controller (6) with a low-pass implementation filter \(Q_{n}\) (8)._  
To compensate for the process dead time contained in the IPDT models, the corresponding exponential term \(e^{-T_{d}s}\) can be approximated by the finite number of terms of its expansion into the Taylor series (see, e.g., [23]). The combination with the integrating process also increases the total dimension of the corresponding state vector. Its special case is the phase vector, which is formed by the output of the process and its derivatives. Thus, the automatic reset controller can be considered as a model-based solution derived for a specific IPDT model and extended to the whole family of controllers with higher-order (HO) derivatives. To simplify the controller design, only HO-ARCs with odd values of the derivative degree \(m\) are used in the proportional-derivative \(PD^{m}\) controller (Figure 1). \[PD^{m}(s)=K_{c}(1+T_{1}s+\dots+T_{m}s^{m});m=1,3,5,7 \tag{6}\]  
with a positive feedback from the limited controller output \(F_{R}\) are considered in the study. This represents a special case of HO-ARC, which was considered in [17] for \(m\in\mathbb{Z}^{+}\), \(m\in[0,5]\). The separate treatment of even and odd values of \(m\) in the constrained control case is based on the requirement to achieve conditions for absolute stability by factorizing the zeros of the controller transfer function derived using the multiple real dominant pole (MRDP) method (see, e.g., [18], which deals with the case \(m=2\), and [24], which presents a particular solution for \(m=4\)). In the proportional zone of the controller, the HO-ARC structure yields the transfer function  
\[ \begin{split}& PD^{m}R(s)=K_{c}(1+T_{1}s+\dots+T_{m}s^{m})\frac{1 +T_{i}s}{T_{i}s}=\\ &=K_{p}+\frac{K_{i}}{s}+K_{1}s+\dots+K_{m}s^{m};\;K_{i}=K_{c}/T_{ i};\;K_{p}=K_{c}(1+T_{1}/T_{i});\\ & K_{j}=K_{c}(T_{j}+T_{j+1}/T_{i};\;j=1,2,\dots,m-1;\;\;K_{m}=K_{ c}T_{m}\,.\end{split}  \tag{7}\]  
This means that this closed-loop transfer function can be considered as an alternative to the parallel HO-PID controller [25] with an explicit integrator \(K_{i}/s\) in the vicinity of steady states. The use of HO derivatives is one way of increasing the allowable gain of the system. To investigate the practical limits of this concept in the context of POCD, we will try to answer the question raised in [8], in which the PID controller with derivative term achieved better results compared to simple PI control. Is it possible to generalize the results to processes with unstable zero dynamics controlled by HO-ARCs? The implementation of the derivatives requires the inclusion of low-pass filters.