###### Abstract  
Integrating systems are frequently encountered in the oil industry (oil-water-gas separators, distillation columns), power plants, paper-production plants, polymerisation processes, and in storage tanks. Due to the non-self-regulating character of the processes, any disturbance can cause a drift of the process output signal. Therefore, efficient closed-loop control of such processes is required. There are many PI and PID controller tuning methods for integrating processes. However, it is hard to find one requiring only a simple tuning procedure on the process, while the tuning method is based either on time-domain measurements or on a process transfer function of arbitrary order, which are the advantages of the magnitude optimum multiple integration (MOMI) tuning method. In this paper, we propose the extension of the MOMI tuning method to integrating processes. Besides the mentioned advantages, the extension provides efficient closed-loop control, while PI controller parameters calculation is still based on simple algebraic expressions, making it suitable for less-demanding hardware, like simpler programmable logic controllers (PLC). Additionally, the proposed method incorporates reference weighting factor \(b\) that allows users to emphasize either the disturbance-rejection or reference-following response. The proposed extension of the MOMI method (time-domain approach) was also tested on a charge-amplifier drift-compensation system, a laboratory hydraulic plant, on an industrial autoclave, and on a solid-oxide fuel-cell temperature control. All closed-loop responses were relatively stable and fast, all in accordance with the magnitude optimum criteria. m 20202  
of integrating processes (IP) is a challenging task. There are different types of integrating systems that can be classified according to the number of poles in the origin and location of other poles in the transfer function. Integrating systems are frequently encountered in the industry. For example, representatives of industrial processes that exhibit pure integrator plus time-delay (ITD) transfer function models are oil-water-gas separators in the oil industry [7], bottom-level control in a distillation column [8], composition control loop of a high-purity distillation column [9], heat-integrated distillation columns [10], isothermal continuous copolymerisation reactors [11], storage tanks with an outlet pump [12], pulp and paper plants [7], and high-pressure steam flowing to a steam-turbine generator in a power plant [13]. First-order integrating systems with/without delay (IFOTD) can be encountered in liquid-storage tanks [14], a jacketed continuous stirred tank reactor (CSTR) carrying out an exothermic reaction [15], and paper-drum-dryer cans [16]. Several tuning methods for IP have been developed so far [17, 18]. They can be classified according to models of integrating systems, i.e., tuning methods for integrating systems with time-delay [7, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], for first-order integrating systems with time-delay [18, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], for higher-order integrating systems with time-delay [52], and nonparametric tuning methods for integrating systems [53, 54, 55, 56]. Tunings methods for integrating systems with time-delay can also be applied to a first- or higher-order integrating system by using a relay feedback identification method to model the system as an integrating system with time-delay [21, 24, 26, 27]. The method of [PERSON] and [PERSON] [19] takes into account process uncertainties. The PD controller, based on the Smith predictor scheme with gain- and phase-margin specifications for an integrating-with-time-delay (ITD) process, was proposed by [PERSON] et al. [21]. [PERSON] et al. [26] developed a tuning rule derived in terms of maximal sensitivity (M\({}_{\text{S}}\)) that was based on pole-placement and frequency-response matching criteria.