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Eun-jin Kim1 In this contribution, fractional-order controllers of the type PD� and PI� are applied to a class of irrational transfer function models that appear in large-scale systems, such as networks of mechanical/electrical elements and distributed parameter systems. More precisely, by considering the fractional-order controller kp + k�s � in the Laplace domain with −1 ≤ � ≤ 1, a stability analysis in the parameter-space (kp, k� , �) is presented. Furthermore, as a way to measure the controller robustness, the controller’s fragility analysis using the parameter-space (kp, k� , �) is derived. Finally, several applications that demonstrate the utility of our results are included. Irrational systems (ISs) are a class of systems whose model is represented by a transfer function containing irrational orders. In [1, 2] ISs are described as implicit operators because they are solutions to an operator equation. Besides, [3] presents ISs as a type of pseudo-differential time-operators whose representation in time domain is diffusive (for further details about the diffusive representation, see ref. [4]). Practical examples of ISs can be found in various previous works across different disciplines. For instance, [2] introduces an IS model to represent the total operator describing the potential-driven fow dynamics in a large-scale self-similar tree network. In references [5–7], a version with springs and dampers of this IS is examined to propose model reductions to robotic formations or cyber-physical systems. On the other hand, infnite ladder networks can also be modelled by using an IS representation (for further details, see ref. [1]). In his famous text [8], Richard Feynman studies an infnite LC ladder circuit and proposes an expression for its total impedance in the form of an IS. Recently, this model is examined in more detail in ref. [9]. In ref. [10], an IS model for an infnite ladder of mass-springs and dampers is introduced towards the goal of modelling complex networks of mechanical systems. Furthermore, to describe the power-law behaviour in soft tissue, a hierarchical fractal ladder network model is proposed in [11]. Finally, ISs can also be found when solving partial differential equations or when modelling distributed parameter systems (for further details, see refs. [12–14]). As it can be seen from the previous discussion, ISs arise as an approximation, model reduction or exact model for certain complex or large-scale systems [5]. Regarding the design of control strategies for complex systems, many solutions have been proposed in the literature (see, for instance, refs. [15–19]), but for the case of ISs, it still remains as an open problem. Thus, by considering the benefts that ISs can bring to the modelling of complex systems, it will be of core importance to study control schemes for these mathematical models. Between the most popular control techniques, there is the well-known PID controller whose “popularity” can be attributed to its particular distinct features: simplicity and ease of implementation. In this regard, inspired by its attributes as well as the recent developments in fractional calculus, a fractional-order version of the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is