Systems Identification

Part of the Advances in Industrial Control book series (AIC)


It has been experimentally observed or analytically found that both the time domain and frequency domain behaviors of some linear systems and processes do not fit the standard laws, i.e., exponential evolution in time domain or integer-order slopes in their frequency responses. In the time domain, it has been shown that these complicated dynamics can be described by, (i.e., the solutions of the constitutive equations are) generalized hyperbolic functions, \( \mathcal{F}^{k}_{\alpha , \beta} (z) \), defined as

$$ \mathcal{F}^{k}_{\alpha , \beta} (z) = C \sum^{\infty}_{n=0} \frac{k^{n}z^{\alpha n + \beta}}{(\alpha n + \beta)!} $$
. (14.1)

In particular, the Mittag–Leffler function in two parameters is defined as

$$ \mathcal{E}_{\alpha , \beta} (z) = \sum^{\infty}_{n=0} \frac{z^{n}}{\Gamma (\alpha n + \beta)} $$
, (14.2)

from which we can obtain the standard exponential, hyperbolic, or time-scaling functions as particular cases.


Fractional Calculus Constant Phase Element Transcendental Function Actuator Voltage Transfer Function Model 
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© Springer-Verlag London Limited 2010

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