There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Formulas For Natural Frequency And Mode Shape

The Viscosity can be imagined as the force that should be applied to a layer of fluid belonging to the plane fixed to the velocity of the layer placed at a fixed distance. For example a fluid flowing in a tube at different speeds: the minimum speed is in the edge of the section (due to friction) and the maximum speed is at the center. The viscosity in this example is that pressure which is exerted on the wall allows a constant speed on the whole section.

Non-newtonian fluids.

Explicit formula and graphs of few special functions are derived in the paper on the basis of various definitions of various fractional derivatives and various fractional integrals. Their applications are also reviewed in the paper.

/pdf/applications-of-fractional-calculus-1c2m93jo56.pdf

Applications of Fractional Calculus

Linear fractional order controllers; A survey in the frequency domain

The presence of disturbances in practical control engineering applications is unavoidable. At the same time, they drive the closed-loop system’s response away from the desired behavior. For this reason, the attenuation of disturbance effects is a primary goal of the control loop. Fractional-order controllers have now been researched intensively in terms of improving the closed-loop results and robustness of the control system, compared to the standard integer-order controllers. In this study, a novel tuning method for fractional-order controllers is developed. The tuning is based on improving the disturbance attenuation of periodic disturbances with an estimated frequency. For this, the reference–to–disturbance ratio is used as a quantitative measure of the control system’s ability to reject disturbances. Numerical examples are included to justify the approach, quantify the advantages and demonstrate the robustness. The simulation results provide for a validation of the proposed tuning method.

/pdf/fractional-order-pi-controller-design-based-on-reference-to-18eqdhi8.pdf

Fractional-Order PI Controller Design Based on Reference–to–Disturbance Ratio

Smart beams are one of the most frequently used means of studying vibrations in airplane wings. Their mathematical models have been so far solely based on classical approaches that ultimately involve integer order transfer functions. In this paper, a different approach towards modeling such smart beams is considered, an approach that is based on fractional calculus. In this way, a fractional order model of the smart beam is obtained, which is able to better capture the dynamics of the system. Based on this novel fractional order model, a fractional order PDμ controller is then tuned according to a set of three design constraints. This design leads to a closed loop system that exhibits a much smaller resonant peak compared to the uncompensated smart beam system. Experimental results are provided, considering both passive and active control responses of the smart beam, showing that a significant improvement of the closed loop behavior is obtained using the designed controller.

Fractional order modeling and control of a smart beam

This paper tackles the field of applied control engineering in the nanomedical field, both in terms of modeling and control of the dynamics of a scalable robot transiting a vascular environment. An experimental fractional order model is obtained that describes the dynamics of the robot inside a secluded part of the circulatory system in terms of velocity profile and position. The submersible has concentration sensing capabilities and when a difference is detected in the liquid's concentration, a Fractional Order Proportional Integrative (FOPI) controller is used to control the robot's velocity and position inside the vascular homology. The work is based on experimental data acquired from a dedicated platform designed and built to echo environmental traits of the circulatory system for analysis and control purposes.

Fractional Order Modeling and Control of a Carrier Prototype for Targeted Drug Delivery

The chapter is focused on presenting an experimental setup that mimics the non-Newtonian nature of blood environments. The platform was developed with didactic purposes to familiarize oneself with the interaction between submerged devices and non-Newtonian surroundings. The blood environment is a platform resembling a primary circulatory system consisting of a liquid with non-Newtonian particularities flowing inside a vein/artery. The submerged device is a small-scale robot looking like a submarine, which is equipped with a thrusting propeller and an impedance measuring sensor. The submersible travels through the blood-like environment measuring the impedance of the fluid and stops in areas where unusual impedance values are detected. The dynamics of the robot are modeled using fractional-order differentiation, and control capabilities are implemented in order to ensure proper operation.

A non-Newtonian impedance measurement experimental framework: modeling and control inside blood-like environments—fractional-order modeling and control of a targeted drug delivery prototype with impedance measurement capabilities

The paper presents a novel generalization of available integer-order event based control strategies into the fractional order field, offering a real life implementation possibility of event based fractional order control. A fractional order Proportional Integral (FOPI) controller is tuned using a set of frequency domain specifications such as gain crossover frequency, phase margin and robustness. A generalized theoretical approach is developed for FOPI event based control strategies. In addition, real-life solutions are provided in order to implement the FOPI controller on a Vertical Take-Off and Landing (VTOL) platform. The proposed methodology is validated experimentally by assessing the closed loop system performance in various scenarios such as step reference tracking, disturbance rejection and robustness to gain uncertainties.

Isabela Roxana Birs

Papers

Fractional-Order PI Controller Design Based on Reference–to–Disturbance Ratio

Fractional order modeling and control of a smart beam

Fractional Order Modeling and Control of a Carrier Prototype for Targeted Drug Delivery

A non-Newtonian impedance measurement experimental framework: modeling and control inside blood-like environments—fractional-order modeling and control of a targeted drug delivery prototype with impedance measurement capabilities

A real life implementation of fractional order event based PI control