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

Fractional Differential Equations

Review of fractional PID controller

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Formulas For Natural Frequency And Mode Shape

Introduction of vehicle-to-grid technology offer electric vehicles (EVs) to participate in different ancillary services under competitive electric market. EVs provide an opportunity to grow new products and services for grid management. Particularly, EVs, which is a new form of distributed energy storage, can be used to compensate the uncontracted power in the local area if the contracts between the market players are violated. This paper presents the participation of EVs for load frequency control (LFC) under deregulated environment along with other conventional sources such as hydro, thermal, and gas turbine units. An aggregate model of EV fleets and improved version of fractional order (FO) controller is provided in all the areas for robust LFC considering bilateral transactions. Flower pollination algorithm, which is one of the new proven nature inspired algorithm employed to choose the optimal parameters of the FO controllers under several scenarios. Numerous simulations are conducted to validate the superiority of the proposed control strategy.

Utilizing Electric Vehicles for LFC in Restructured Power Systems Using Fractional Order Controller

Autotuning of a Robust Fractional Order PID Controller

This paper proposes a theoretical framework for generalization of the well established first order plus dead time (FOPDT) model for linear systems. The FOPDT model has been broadly used in practice to capture essential dynamic response of real life processes for the purpose of control design systems. Recently, the model has been revisited towards a generalization of its orders, i.e., non-integer Laplace order and fractional order delay. This paper investigates the stability margins as they vary with each generalization step. The relevance of this generalization has great implications in both the identification of dynamic processes as well as in the controller parameter design of dynamic feedback closed loops. The discussion section addresses in detail each of this aspect and points the reader towards the potential unlocked by this contribution.

Generalization of the FOPDT Model for Identification and Control Purposes

This paper investigates the robustness of a fractional-order controller against the load changes of a DC motor. The gains and time constants of the DC motor are modified by means of a change in the brake. Two different setups of a DC motor, one with 25% brake and the other with 50% brake are considered in the experimental evaluation. The closed-loop performances of the fractional-order controller are compared with integer-order controller using the same performance criteria and the same tuning algorithm. Both controllers were designed based on time domain specifications. The experimental results show that the fractional-order controller outperforms the classical controller under nominal conditions as well as under gain variations situation.

/pdf/fractional-order-control-of-a-dc-motor-with-load-changes-431vu6e3bt.pdf

Fractional order control of a DC motor with load changes

Vibration suppression represents an important research topic due to the occurrence of this phenomenon in multiple domains of life. In airplane wings, vibration can cause discomfort and can even lead to system failure. One of the most frequently used means of studying vibrations in airplane wings is through the use of dedicated flexible beams, equipped with sensing and actuating mechanisms powered by suitable control algorithms. In order to optimally reject these vibrations by means of closed-loop control strategies, the availability of a model is required. So far, the modeling of these smart flexible beams has been limited to deliver integer order transfer functions models. This paper, however, describes the mathematical framework used to derive a fractional-order impedance lumped model for capturing frequency response of a flexible beam system exposed to a multisine excitation. The theoretical foundation stems from fractional calculus applied in combination with transmission line theory and wave equations. The simplified model reduces to a minimal number of parameters when converging to a limit value. It is shown that the fractional-order model outperforms an integer order model of the smart beam. Based on this novel fractional-order model, a fractional-order $$\hbox {PD}^\mu $$
 controller is then tuned. The controller design is based on shaping the frequency response of the closed-loop system such that the resonant peak is reduced in comparison to the uncompensated smart beam system and disturbances are rejected. Experimental results, considering a custom-built smart beam system, are provided, considering both passive and active control situations, showing that a significant improvement in the closed-loop behavior is obtained using the proposed controller. Comparisons with a fractional-order $$\hbox {PD}^\mu $$
 controller, tuned according to classical open-loop frequency domain design specifications, are provided. The experimental results show that the proposed tuning technique leads to similar results as the classical approach. Thus, the proposed method is a viable alternative, being based on closed-loop specifications, which is more intuitive for practitioners.

A novel fractional-order model and controller for vibration suppression in flexible smart beam

Vibrations in airplane wings have a negative impact on the quality and safety of a flight. For this reason, active vibration suppression techniques are of extreme importance. In this paper, a smart beam is used as a simulator for the airplane wings and a fractional order PD controller is designed for active vibration mitigation. To implement the ideal fractional order controller on the smart beam unit, its digital approximation is required. In this paper, a new continuous-to-discrete-time operator is used to obtain the discrete-time approximation of the ideal fractional order PD controller. The efficiency and flexibility, as well as some guidelines for using this new operator, are given. The numerical examples show that high accuracy of approximation is obtained and that the proposed method can be considered as a suitable solution for obtaining the digital approximation of fractional order controllers. The experimental results demonstrate that the designed controller can significantly improve the vibration suppression in smart beams.

https://biblio.ugent.be/publication/8584661/file/8584663.pdf

Cristina I. Muresan

Papers

Autotuning of a Robust Fractional Order PID Controller

Generalization of the FOPDT Model for Identification and Control Purposes

Fractional order control of a DC motor with load changes

A novel fractional-order model and controller for vibration suppression in flexible smart beam

Discrete-time implementation and experimental validation of a fractional order PD controller for vibration suppression in airplane wings