## 1 Introduction  
A whole chain of discoveries, their further modifications, improvements and patents gradually led to the development of pneumatic controllers at the beginning of the twentieth century. Due to the internal structure of the controllers, they were the heart of industrial automation [1; 2; 3]. The most important part of the structure was a positive feedback from the overall output of the controller to the output of the proportional controller, which generated a signal (automatic reset) that served to eliminate the permanent control error caused by acting disturbances. Even before the beginning of World War II, there was a generalization of the mentioned control structure with a stabilizing proportional (P) and a proportional-derivative (PD) term, also called Pre-Act (Taylor Instrument Companies), or Hyper-Reset (Foxboro). The control structures with automatic reset were later interpreted as integrating controllers [4], since automatic reset acts like an integrator in the linear domain. The name change was originally attributed to disturbance compensation control of the temperature measured by the thermocouple. By getting the potentiometer of the measuring bridge with the null balancing method, the integration of the control error was achieved by summing the increments of the angle of the motor rotating with the potentiometer [1; 2; 3]. Although both control methods with explicitly integrating controller and with automatic reset can be equivalent under certain (linear) circumstances, the equivalence is lost when working with a saturating controller (actuator). Such problems in connection with equivalent controllers with integral action became clearly apparent during the transition to digitally implemented controllers by the occurrence of so-called undesired integration, also called integrator windup [5; 6; 7; 8; 9; 10; 11; 12]. Although a number of different anti-windup controllers have been developed to address the problem, the evaluation of constrained structures based on the use of an automatic reset with a special parameterization of numerator terms of the PID controller in [13] proved to be more practical than the anti-windup structures based on integrating controllers [14]. Subsequently, the question was raised whether these useful properties could not be further improved by combining the automatic reset with a higher derivative-order controller, referred to as proportional-derivative-accelerative (PDA). However, before we start to look at the structure of the PDA-based controller, we need to clarify the terminology. The original automatic reset controller is usually referred to as a series proportional-integral (PI) controller. Similarly, an automatic reset combined with a stabilizing PD controller (originally Pre-Act) is today called a series PID controller. If an automatic reset is added to the PDA controller, it could also be called a series PIDA controller. However, all of these designations are inherently illogical because the structure under consideration does not contain an explicit integrator. The confusing terminology is due to the fact that, for a whole century since the invention of the automatic reset, it was not noticed that it can also be interpreted as a disturbance observer for the reconstruction of input disturbances. The properties of the disturbance observer become clear when analyzing the steady states of controllers designed with integrating process models [15; 16; 17; 18]. In this context, the terminology is also important from the point of view of controller tuning. Many works deal with the optimal design of PIDA controllers, also known as PIDD2, PIDD2 or PIDC controllers [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. The growing use of these controllers is the result of ever-growing demands on the performance of transients typically occurring especially in connection with speed control in high-end robotic and mechatronic applications.