Performance Enhancement of Fractional $${\text{PI}}^{{{{\uplambda}}}} {\text{D}}^{{{{\upmu}}}}$$ Controllers Using Modified Grey Wolf Optimization

A new heuristic approach for minimizing possibly nonlinear and non-differentiable continuous space functions is presented. By means of an extensive testbed it is demonstrated that the new method converges faster and with more certainty than many other acclaimed global optimization methods. The new method requires few control variables, is robust, easy to use, and lends itself very well to parallel computation.

/pdf/differential-evolution-a-simple-and-efficient-heuristic-for-4vrb5o5ldc.pdf

Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces

Grey Wolf Optimizer

Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.

/pdf/an-introduction-to-the-fractional-calculus-and-fractional-2nkhvg2rdg.pdf

An Introduction to the Fractional Calculus and Fractional Differential Equations

Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI/sup /spl lambda//D/sup /spl mu//-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller for both open- and closed-loops. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI/sup /spl lambda//D/sup /spl mu//-controllers is given.

Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers

The state-of-the-art on generalized (or any order) derivatives in physics and engineering sciences, is outlined for justifying the interest of the noninteger differentiation. The problems subsequent to its use in real-time operations are then set out so as to motivate the idea of synthesizing it by a recursive distribution of zeros and poles. An analysis of the existing work is also proposed to support this idea. A comprehensive study is given of the synthesis of differentiators with integer, noninteger, real or complex orders, and whose action is limited to any given frequency bandwidth. First, a definition, in the operational and frequency domains, of a frequency-band complex noninteger order differentiator, is given in a mathematical space with four dimensions which is a Banach algebra. Then, the determination of its synthesized form, by a recursive distribution of complex zeros and poles characterized by complex recursive factors, is presented. The complex noninteger differentiation order is expressed as a function of these recursive factors. The number of zeros and poles is calculated to be as low as possible while still ensuring the stability of the synthesized differentiator to be synthesized. A time validation is presented. Finally, guidelines are proposed for the conception of the synthesized differentiator.

Performance Enhancement of Fractional $${\text{PI}}^{{{{\uplambda}}}} {\text{D}}^{{{{\upmu}}}}$$ Controllers Using Modified Grey Wolf Optimization

References

Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces

Grey Wolf Optimizer

An Introduction to the Fractional Calculus and Fractional Differential Equations

Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers

Frequency-band complex noninteger differentiator: characterization and synthesis