Fractional order calculus: basic concepts and engineering applications

The fractional order calculus (FOC) is as old as the integer one although up to recently its application was exclusively in mathematics. Many real systems are better described with FOC differential equations as it is a well-suited tool to analyze problems of fractal dimension, with long-term “memory” and chaotic behavior. Those characteristics have attracted the engineers' interest in the latter years, and now it is a tool used in almost every area of science. This paper introduces the fundamentals of the FOC and some applications in systems' identification, control, mechatronics, and robotics, where it is a promissory research field.

Condition-Based Diagnosis of Mechatronic Systems Using a Fractional Calculus Approach

While fractional calculus (FC) is as old as integer calculus, its application has been mainly restricted to mathematics. However, many real systems are better described using FC equations than with integer models. FC is a suitable tool for describing systems characterised by their fractal nature, long-term memory and chaotic behaviour. It is a promising methodology for failure analysis and modelling, since the behaviour of a failing system depends on factors that increase the model’s complexity. This paper explores the proficiency of FC in modelling complex behaviour by tuning only a few parameters. This work proposes a novel two-step strategy for diagnosis, first modelling common failure conditions and, second, by comparing these models with real machine signals and using the difference to feed a computational classifier. Our proposal is validated using an electrical motor coupled with a mechanical gear reducer.

Exposure of Lycopersicon Esculentum to Microcystin-LR: Effects in the Leaf Proteome and Toxin Translocation from Water to Leaves and Fruits

Natural toxins such as those produced by freshwater cyanobacteria have been regarded as an emergent environmental threat. However, the impact of these water contaminants in agriculture is not yet fully understood. The aim of this work was to investigate microcystin-LR (MC-LR) toxicity in Lycopersicon esculentum and the toxin accumulation in this horticultural crop. Adult plants (2 month-old) grown in a greenhouse environment were exposed for 2 weeks to either pure MC-LR (100 μg/L) or Microcystis aeruginosa crude extracts containing 100 μg/L MC-LR. Chlorophyll fluorescence was measured, leaf proteome investigated with two-dimensional gel electrophoresis and Matrix Assisted Laser Desorption Ionization Time-of-Flight (MALDI-TOF)/TOF, and toxin bioaccumulation assessed by liquid chromatography-mass spectrometry (LC-MS)/MS. Variations in several protein markers (ATP synthase subunits, Cytochrome b6-f complex iron-sulfur, oxygen-evolving enhancer proteins) highlight the decrease of the capacity of plants to synthesize ATP and to perform photosynthesis, whereas variations in other proteins (ribulose-1,5-bisphosphate carboxylase/oxygenase large subunit and ribose-5-phosphate isomerase) suggest an increase of carbon fixation and decrease of carbohydrate metabolism reactions in plants exposed to pure MC-LR and cyanobacterial extracts, respectively. MC-LR was found in roots (1635.21 μg/kg fw), green tomatoes (5.15–5.41 μg/kg fw), mature tomatoes (10.52–10.83 μg/kg fw), and leaves (12,298.18 μg/kg fw). The results raise concerns relative to food safety and point to the necessity of monitoring the bioaccumulation of water toxins in agricultural systems affected by cyanotoxin contamination.