A fractional-order integrator or just simply fractional integrator is an

integrator An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output. Integration is an importan ...

device that calculates the fractional-order integral or derivative (usually called a

differintegral In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by :\mathbb^q f is the fractional deri ...

) of an input. Differentiation or integration is a real or complex parameter. The fractional integrator is useful in fractional-order control where the history of the system under control is important to the control system output. Some industrial controllers use fractional-order PID controllers (FOPIDs), which have exceeded the performance of standard ones, to the extent that standard ones are sometimes considered as a special case of FOPIDs. Fractional-order integrators and differentiators are the main component of FOPIDs.

Overview

The

function, :

_a \mathbb^q_t \left( f(x) \right)

includes the integer order differentiation and integration functions, and allows a continuous range of functions around them. The differintegral parameters are ''a'', ''t'', and ''q''. The parameters ''a'' and ''t'' describe the range over which to compute the result. The differintegral parameter ''q'' may be any real number or

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

. If ''q'' is greater than zero, the differintegral computes a derivative. If ''q'' is less than zero, the differintegral computes an integral. The integer order integration can be computed as a Riemann–Liouville differintegral, where the weight of each element in the sum is the constant unit value 1, which is equivalent to the

Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approxima ...

. To compute an integer order derivative, the weights in the summation would be zero, with the exception of the most recent data points, where (in the case of the first unit derivative) the weight of the data point at ''t'' − 1 is −1 and the weight of the data point at ''t'' is 1. The sum of the points in the input function using these weights results in the difference of the most recent data points. These weights are computed using ratios of the

Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

incorporating the number of data points in the range 'a'',''t'' and the parameter ''q''.

Digital devices

Digital devices have the advantage of being versatile, and are not susceptible to unexpected output variation due to heat or noise. The discrete nature of a computer however, does not allow for all of history to be computed. Some finite range ,tmust exist. Therefore, the number of data points that can be stored in memory (''N''), determines the oldest data point in memory, so that the value a is never more than ''N'' samples old. The effect is that any history older than a is ''completely'' forgotten, and no longer influences the output. A solution to this problem is the Coopmans approximation, which allows old data to be forgotten more gracefully (though still with exponential decay, rather than with the power law decay of a purely

analog device Analog devices are a combination of both analog machine and analog media that can together measure, record, reproduce, receive or broadcast continuous information, for example, the almost infinite number of grades of transparency, volta ...

Analog devices

Analog devices have the ability to retain history over longer intervals. This translates into the parameter a staying constant, while ''t'' increases. There is no error due to round-off, as in the case of digital devices, but there may be error in the device due to leakages, and also unexpected variations in behavior caused by heat and noise. An example fractional-order integrator is a modification of the standard integrator circuit, where a

capacitor In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...

is used as the feedback impedance on an opamp. By replacing the capacitor with an

RC Ladder A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC cir ...

circuit, a half order integrator, that is, with :

q = -\frac,

can be constructed.

References

{{reflist Cybernetics Fractional calculus

Overview

Digital devices

Analog devices

See also

References