Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.

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Fractional order control - A tutorial

For fractional-order differentiator s/sup r/ where r is a real number, its discretization is a key step in digital implementation. Two discretization methods are presented. The first scheme is a direct recursive discretization of the Tustin operator. The second one is a direct discretization method using the Al-Alaoui operator via continued fraction expansion (CFE). The approximate discretization is minimum phase and stable. Detailed discretization procedures and short MATLAB scripts are given. Examples are included for illustration.

Discretization schemes for fractional-order differentiators and integrators

A lithographic apparatus configured to project a patterned beam of radiation onto a target portion of a substrate is disclosed. The apparatus includes a first radiation dose detector and a second radiation dose detector, each detector comprising a secondary electron emission surface configured to receive a radiation flux and to emit secondary electrons due to the receipt of the radiation flux, the first radiation dose detector located upstream with respect to the second radiation dose detector viewed with respect to a direction of radiation transmission, and a meter, connected to each detector, to detect a current or voltage resulting from the secondary electron emission from the respective electron emission surface.

Lithographic apparatus, and device manufacturing method

In recent years, it is remarkable to see the increasing number of studies related to the theory and application of fractional order controller (FOC), specially PI ? D ? controller, in many areas of science and engineering. Research activities are focused on developing new analysis and design methods for fractional order controllers as an extension of classical control theory. In this paper, a new tuning method for fractional order proportional and derivative (PD ?) or FO-PD controller is proposed for a class of typical second-order plants. The tuned FO-PD controller can ensure that the given gain crossover frequency and phase margin are fulfilled, and furthermore, the phase derivative w. r. t. the frequency is zero, i.e., the phase Bode plot is flat at the given gain crossover frequency. Consequently, the closed-loop system is robust to gain variations. The FOC design method proposed in the paper is practical and simple to apply. Simulation and experimental results show that the closed-loop system can achieve favorable dynamic performance and robustness.

A Fractional Order Proportional and Derivative (FOPD) Motion Controller: Tuning Rule and Experiments

This paper attempts to present an expository review of continued fraction expansion (CFE) based discretization schemes for fractional order differentiators defined in continuous time domain. The schemes reviewed are limited to infinite impulse response (IIR) type generating functions of first and second orders, although high-order IIR type generating functions are possible. For the first-order IIR case, the widely used Tustin operator and Al-Alaoui operator are considered. For the second order IIR case, the generating function is obtained by the stable inversion of the weighted sum of Simpson integration formula and the trapezoidal integration formula, which includes many previous discretization schemes as special cases. Numerical examples and sample codes are included for illustrations.

Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives—an Expository Review

A feedback control system compensator of the PID type is provided, wherein the proportional component of the compensator is replaced with a tilted component having a transfer function s to the power of -1/n. The resulting transfer function of the entire compensator more closely approximates an optimal transfer function, thereby achieving improved feedback controller. Further, as compared to conventional PID compensators, the TID compensator allows for simpler tuning, better disturbance rejection ratio, and smaller effects of plant parameter variations on closed loop response.

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Boris J. Lurie

Papers

Three-parameter tunable tilt-integral-derivative (TID) controller