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 (sometime also called the derivigral) is a combined differentiation/integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by ...

) of an input. Differentiation or integration is a real or complex parameter. The fractional integrator is useful in

fractional-order control Fractional-order control (FOC) is a field of control theory that uses the fractional-order integrator as part of the control system design toolkit. The use of fractional calculus (FC) can improve and generalize well-established control methods and ...

where the history of the system under control is important to the control system output.

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. 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 approximating the area of functions or l ...

. 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 , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

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 The Coopmans approximation is a method for approximating a fractional-order integrator in a continuous process with constant space complexity. The most correct and accurate methods for calculating the fractional integral require a record of all p ...

, 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

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 A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter de ...

, where a

capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...

is used as the

feedback impedance Feedback occurs when outputs of a system are routed back as inputs as part of a Signal chain (signal processing chain), chain of Causality, cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. ...

on an

opamp An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to ...

. 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 ci ...

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

q = -\frac{2},

can be constructed.

Overview

Digital devices

Analog devices

See also