The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the actual (proportional), the accumulated (integral), and the predicted (derivative) values; the three gains depend on the magnitude of the error, the time required to eliminate the accumulated error, and the prediction horizon of the error. This paper explores the new meaning of integral and derivative actions, and gains, derived by the consideration of non-integer integration and differentiation orders, i.e., for fractional order PID controllers. The integral term responds with selective memory to the error because of its non-integer order $\lambda$ , and corresponds to the area of the projection of the error curve onto a plane (it is not the classical area under the error curve). Moreover, for a fractional proportional-integral (PI) controller scheme with automatic reset, both the velocity and the shape of reset can be modified with $\lambda$ . For its part, the derivative action refers to the predicted future values of the error, but based on different prediction horizons (actually, linear and non-linear extrapolations) depending on the value of the differentiation order, $\mu$ . Likewise, in case of a proportional-derivative (PD) structure with a noise filter, the value of $\mu$ allows different filtering effects on the error signal to be attained. Similarities and differences between classical and fractional PIDs, as well as illustrative control examples, are given for a best understanding of new possibilities of control with the latter. Examples are given for illustration purposes.

Cardiovascular diseases (CVDs) remain the leading cause of death worldwide, according to recent reports from the World Health Organization (WHO). This fact encourages research into the cardiovascular system (CVS) from multiple and different points of view than those given by the medical perspective, highlighting among them the computational and mathematical models that involve experiments much simpler and less expensive to be performed in comparison with in vivo or in vitro heart experiments. However, the CVS is a complex system that needs multidisciplinary knowledge to describe its dynamic models, which help to predict cardiovascular events in patients with heart failure, myocardial or valvular heart disease, so it remains an active area of research. Firstly, this paper presents a novel electrical model of the CVS that extends the classic Windkessel models to the left common carotid artery motivated by the need to have a more complete model from a medical point of view for validation purposes, as well as to describe other cardiovascular phenomena in this area, such as atherosclerosis, one of the main risk factors for CVDs. The model is validated by clinical indices and experimental data obtained from clinical trials performed on a pig. Secondly, as a first step, the goodness of a fractional-order behavior of this model is discussed to characterize different heart diseases through pressure–volume (PV) loops. Unlike other models, it allows us to modify not only the topology, parameters or number of model elements, but also the dynamic by tuning a single parameter, the characteristic differentiation order; consequently, it is expected to provide a valuable insight into this complex system and to support the development of clinical decision systems for CVDs.

The transition toward energy-efficient and smart buildings requires Digital Twins (DTs) that can couple real-time data with physics-based Building Energy Models (BEMs) for predictive and adaptive operation. Yet, despite rapid digitalisation, there remains a lack of practical guidance and real-world implementations demonstrating how calibrated BEMs can be effectively integrated into Building Management Systems (BMSs). This study addresses that gap by presenting a complete and reproducible end-to-end framework for embedding physics-based BEMs into operational DTs using two setups: (i) encapsulation as Functional Mock-up Units (FMUs) and (ii) containerisation via Docker. Both approaches were deployed and tested in a real educational building in Cáceres (Spain), equipped with a LoRaWAN-based sensing and actuation infrastructure. A systematic comparison highlights their respective trade-offs: FMUs offer faster execution but limited weather inputs and higher implementation effort, whereas Docker-based workflows provide full portability, scalability, and native interoperability with Internet of Things (IoT) and BMS architectures. To enable real-time operation, a surrogate modelling framework was embedded within the Docker architecture to replicate the optimisation logic of the calibrated BEM and generate predictive blind control schedules in milliseconds—bypassing simulation overhead and enabling continuous actuation. The combined Docker + surrogate setup achieved 10–15% heating energy savings during winter operation without any HVAC retrofit. Beyond the case study, this work provides a step-by-step, in-depth guideline for practitioners to integrate calibrated BEMs into real-time control loops using existing toolchains. The proposed approach demonstrates how hybrid physics- and data-driven DTs can transform building management into a scalable, energy-efficient, and operationally deployable reality.

Ionic polymer-metal composites (IPMCs) are electrically driven materials that undergo bending deformations in the presence of relatively low external voltages, exhibiting a great potential as actuators in applications in soft robotics, microrobotics, and bioengineering, among others. This paper presents an artificial eukaryotic flagellum (AEF) swimming robot made up of IPMC segments for the study of planar wave generation for robot propulsion by single and distributed actuation, i.e., considering the first flagellum link as an actuator or all of them, respectively. The robot comprises three independent and electrically isolated actuators, manufactured over the same 10 mm long IPMC sheet. For control purposes, a dynamic model of the robot is firstly obtained through its frequency response, acquired by experimentally measuring the flagellum tip deflection thanks to an optical laser meter. In particular, two structures are considered for such a model, consisting of a non-integer order integrator in series with a resonant system of both non-integer and integer order. Secondly, the identified models are analyzed and it is concluded that the tip displacement of each actuator or any IPMC point is characterized by the same dynamics, which remains unchanged through the link with mere variations of the gain for low-frequency applications. Based on these results, a controller robust to gain variations is tuned to control link deflection regardless of link length and enabling the implementation of a distributed actuation with the same controller design. Finally, the deflection of each link is analyzed to determine whether an AEF swimming robot based on IPMC is capable of generating a planar wave motion by distributed actuation.

This study focused on the relationship between sustainability reporting (SR), the circular economy (CE), and the United Nations' Sustainable Development Goals (SDGs), including an examination of the mediator role of the CE in the SR–SDGs relationship. The SR concept refers to the ways in which companies communicate with their internal and external stakeholders about their sustainability initiatives. Researchers have confirmed that the CE contributes to the achievement of SDGs based on SR. However, the literature does not offer any comprehensive frameworks that integrate these three concepts, which constitutes an important research gap. More specifically, very few studies have explored how social economy entities (SEEs) report the ways they contribute to achieving the SDGs while implementing CE practices. This research thus sought to make a significant contribution to filling this lacuna. Statistical analyses were conducted to determine how SR influences SDGs and how SR promotes these goals through the CE. Variance‐based structural equation modeling via partial least squares was applied to process data on a sample of 90 SEEs in Spain. The results show that entities that provide sufficient SR often communicate how they are implementing SDGs. CE practices have a significant effect on this SR–SDGs relationship. The findings thus make an important contribution to the literature by exploring the link between these three strategies (i.e., SR, the SDGs, and the CE) in organizational contexts and thus providing a foundation for new business models.