## 1 Introduction  
This paper is an extension of a previous study [1] that proposed tuning of the proportional-integral (PI) controller for integrating processes. A proportional-integral-derivative (PID) controller, in single or cascade loops and with or without derivative/integral terms, is the most commonly used control algorithm in industry [2; 3; 4], although several more advanced control structures have been developed. According to the authors in Reference [5], \"Based on a survey of over eleven thousand controllers in the refining, chemicals and pulp and paper industries, 97% of regulatory controllers utilise a PID feedback control algorithm\". The disturbance-rejection and tracking performance depend on tuning the PID controller parameters. Despite the presence of auto-tuning algorithms [6; 7], PID controllers in practice are inadequately tuned due to a lack of skill and time. According to Reference [8], \"a visit to a process plant will usually show that a large number of the PID controllers are poorly tuned\". Furthermore, as stated in Reference [9], \"25% of all PID controller loops use default factory settings, implying that they have not been tuned at all\". Various tuning methods have been proposed. The first tuning rules proposed by [PERSON] [PERSON] [10] were followed by many tuning techniques, such as the Cohen-Coon, Chien-Hrones-Reswick, and refined Ziegler-Nichols rules. In addition to these typical tuning rules, more advanced methods have also been proposed. The latter usually use complex algorithms for the process identification [9; 11; 12; 13; 14; 15; 16; 17; 18; 19]. According to the literature, the number of tuning rules for stable overdamped processes is much larger than that for integrating processes. This suggests that PID tuning for integrating processes is not an extensively studied area. Integrating processes (IPs) involve at least one pole at their origin and are frequently encountered in industry. The main characteristics of IPs are their open-loop instability that leads to an unbounded output for a bounded input, i.e., in case of disturbances, the output drifts to extreme levels. Accordingly, the efficient IPs closed-loop control is a demanding task [20]. The IPs can be classified according to the process order. For example, typical representatives of the processes that employ pure integrator plus time-delay (ITD) transfer function models are the oil-water-gas separators in the oil industry [20], the bottom-level control in distillation columns [21], the composition control loops of high-purity distillation columns [22], heat-integrated distillation columns [23], isothermal continuous copolymerisation reactors [24], storage tanks with outlet pumps [25], vertical take-off airplanes [26], industrial injection moulding machines [27], pulp and paper plants [20], and the high-pressure steam flow to steam-turbine generators in power plants [28]. First-order integrating systems with/without delay (IFOTD) can be encountered in liquid-storage tanks [29], jacketed continuous stirred tank reactors (CSTRs) carrying out exothermic reactions [30], and paper-drum-dryer can [31]. The most frequently encountered example of an IP with inverse response with a time-delay (INPTD) is a boiler steam drum [32]. Additionally, the open-loop responses of various industrial processes, such as liquid-storage tanks, heating boilers, liquid level systems with pumps attached to the outflow, and batch chemical reactors, are slow, with a large dominant time constant. An IP model could be used when the required settling time is shorter than the dominant time constant of a process, even if the process dynamics are inherently of a first-order lag type [18; 33]. As observed in Reference [21], controller design based on IPs could provide better closed-loop performance than designs based on a first-order plus time delay process. Many researchers have addressed controller design methods for integrating and unstable processes.