Discrete Simulation of Fractional Order Systems
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOOL OF ENGINEERING
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Fractional calculus has been shown useful for describing many real world systems, and studies are currently underway to generalize control theory to incorporate fractional states. This investigation derives a method for simulating the time response of fractional order systems using a recursive difference equation. The technique used effectively approximates a simple fractional order integrator as a summation of integer order terms. The discrete transfer function is also derived and the frequency response of the discrete algorithm is compared to the exact continuous case. Using 20 or more retained past values in the difference equation, the discrete half-order integrator demonstrates a passband of more than three decades. A slightly modified method is used to derive a recursive difference equation which simulates the response of a modal fractional order differential equation. Frequency response analysis of an example having an eigenvalue of -1 shows the characteristics of a low pass filter, effectively simulating the continuous system response for all frequencies below the Nyquist limit, even when using a small number of retained past values.
- Numerical Mathematics