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

Fractional order calculus has been used to generalize various types of controllers, including internal model controllers (IMC). The focus of this manuscript is towards fractional order IMCs for first order plus dead-time (FOPDT) processes, including delay and lag dominant ones. The design is novel at it is based on a new approximation approach, the non-rational transfer function method. This allows for a more accurate approximation of the process dead-time and ensures an improved closed loop response. The main problem with fractional order controllers is concerned with their implementation as higher order transfer functions. In cases where central processing unit CPU, bandwidth allocation, and energy usage are limited, resources need to be efficiently managed. This can be achieved using an event-based implementation. The novelty of this paper resides in such an event-based algorithm for fractional order IMC (FO-IMC) controllers. Numerical results are provided for lag and delay dominant FOPDT processes. For comparison purposes, an integer order PI controller, tuned according to the same performance specifications as the FO-IMC, is also implemented as an event-based control strategy. The numerical results show that the proposed event-based implementation for the FO-IMC controller is suitable and provides for a smaller computational effort, thus being more suitable in various industrial applications.

https://www.mdpi.com/2227-7390/8/8/1378/pdf

Event-Based Implementation of Fractional Order IMC Controllers for Simple FOPDT Processes

This paper proposes a framework for modelling velocity profiles and suspended objects in non-Newtonian fluid environment. A setup is proposed to allow mimicking blood properties and arterial to venous dynamic flow changes. Navier-Stokes relations are employed followed by fractional constitutive equations for velocity profiles and flow. The theoretical analysis is performed under assumptions of steady and pulsatile flow conditions, with incompressible properties. The fractional derivative model for velocity and friction drag effect upon a suspended object are determined. Experimental data from such an object is then recorded in real-time and identification of a fractional order model performed. The model is determined from step input changes during pulsatile flow for velocity in the direction of the flow. Further on, this model can be employed for controller design purposes for velocity and position in pulsatile non-Newtonian fluid flow.

/pdf/identification-for-control-of-suspended-objects-in-non-n5zp0vrcj6.pdf

Identification for control of suspended objects in non-Newtonian fluids

Classical fractional order controller tuning techniques usually consider the frequency domain specifications (phase margin, gain crossover frequency, iso-damping) and are based on knowledge of a process model, as well as solving a system of nonlinear equations to determine the controller parameters. In this paper, a novel auto-tuning method is used to tune a fractional order PI controller. The advantages of the proposed auto-tuning method are two-fold: There is no need for a process model, neither to solve the system of nonlinear equations. The tuning is based on defining a forbidden region in the Nyquist plane using the phase margin requirement and determining the parameters of the fractional order controller such that the loop frequency response remains out of the forbidden region. Additionally, the final controller parameters are those that minimize the difference between the slope of the loop frequency response and the slope of the forbidden region border, to ensure the iso-damping property. To validate the proposed method, a case study has been used consisting of a pick and place movement of an UR10 robot. The experimental results, considering two different robot configurations, demonstrate that the designed fractional order PI controller is indeed robust.

https://www.mdpi.com/1999-4893/11/7/95/pdf

Experimental Validation of a Novel Auto-Tuning Method for a Fractional Order PI Controller on an UR10 Robot

Fractional calculus has been used intensely in recent years in control engineering to extend the capabilities of the classical proportional–integral–derivative (PID) controller, but most tuning techniques are based on the model of the process. The paper presents an experimental tuning procedure for fractional-order proportional integral–proportional derivative (PI/PD) and PID-type controllers that eliminates the need of a mathematical model for the process. The tuning procedure consists in recreating the Bode magnitude plot using experimental tests and imposing the desired shape of the closed loop system magnitude. The proposed method is validated in the field of active vibration suppression by using an experimental set-up consisting of a smart beam.

An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain

https://biblio.ugent.be/publication/8624961/file/8624964.pdf

Isabela Roxana Birs

Papers

Event-Based Implementation of Fractional Order IMC Controllers for Simple FOPDT Processes

Identification for control of suspended objects in non-Newtonian fluids

Experimental Validation of a Novel Auto-Tuning Method for a Fractional Order PI Controller on an UR10 Robot

An Experimental Tuning Approach of Fractional Order Controllers in the Frequency Domain

Robust fractional order PI control for cardiac output stabilisation