Order-distributions and the Laplace-domain logarithmic operator
© Hartley and Lorenzo; licensee Springer. 2011
Received: 15 December 2010
Accepted: 30 November 2011
Published: 30 November 2011
This paper develops and exposes the strong relationships that exist between time-domain order-distributions and the Laplace-domain logarithmic operator. The paper presents the fundamental theory of the Laplace-domain logarithmic operator, and related operators. It is motivated by the appearance of logarithmic operators in a variety of fractional-order systems and order-distributions. Included is the development of a system theory for Laplace-domain logarithmic operator systems which includes time-domain representations, frequency domain representations, frequency response analysis, time response analysis, and stability theory. Approximation methods are included.
The area of mathematics known as fractional calculus has been studied for over 300 years . Fractional-order systems, or systems described using fractional derivatives and integrals, have been studied by many in the engineering area [2–9]. Additionally, very readable discussions, devoted to the mathematics of the subject, are presented by Oldham and Spanier , Miller and Ross , Oustaloup , and Podlubny . It should be noted that there are a growing number of physical systems whose behavior can be compactly described using fractional-order system theory. Specific applications are viscoelastic materials [13–16], electrochemical processes [17, 18], long lines , dielectric polarization , colored noise , soil mechanics , chaos , control systems , and optimal control . Conferences in the area are held annually, and a particularly interesting publication containing many applications and numerical approximations is Le Mehaute et al. .
The concept of an order-distribution is well documented [26–31]. Essentially, an order-distribution is a parallel connection of fractional-order integrals and derivatives taken to the infinitesimal limit in delta-order. Order-distributions can arise by design and construction, or occur naturally. In Bagley , a thermo-rheological fluid is discussed. There it is shown that the order of the rheological fluid is roughly a linear function of temperature. Thus a spatial temperature distribution inside the material leads to a related spatial distribution of system orders in the rheological fluid, that is, the position-force dynamic response will be represented by a fractional-order derivative whose order varies with position or temperature inside the material. In Hartley and Lorenzo , it is shown that various order-distributions can lead to a variety of transfer functions, many of which contain a ln(s) term, where s is the Laplace variable. Some of these results are reproduced in the tables at the end of this paper. In Adams et al. , it is shown that a conjugated-order derivative can lead directly to terms containing ln(s). In Adams et al. , it is also shown that the conjugated derivative is equivalent to the third generation CRONE control which has been applied extensively to control a variety of systems .
The next section will review the necessary results from fractional calculus and the theory of order-distributions. It will then be shown that the Laplace-domain logarithmic operator arises naturally as an order distribution, thereby providing a method for constructing a logarithmic operator either in the time or frequency domain. It is then shown that Laplace-domain logarithmic operators can be combined to form systems of logarithmic operators. Following this is the development of a system theory for Laplace-domain logarithmic operator systems which includes time-domain representations, frequency domain representations, frequency response analysis, time response analysis, and stability theory. Fractional-order approximations for logarithmic operators are then developed using finite differences. The paper concludes with some special order-distribution applications.
The Laplace-domain logarithmic operator
which is obtained from Equation 13 by letting x(t) = δ(t), a unit impulse. It is important to note that the time domain function on the right-hand side of Equation 14 is known as a Volterra function, and is defined for all positive time, not just at high frequencies (small time) .
The properties of this integral require further study, although it appears to be convergent for large time due to the gamma function going to infinity when q passes through an integer and thus driving the integrand to zero there.
Higher powers of the Laplace-domain logarithmic operator
Systems of Laplace-domain logarithmic operators
Properties of transfer functions of this type will be the subject of the remainder of the paper.
For this system, the v - plane poles are at v = -1,-2, or s = e v = e-1, e-2, which implies an unstable time response.
a real function of time. For this system, the v-plane poles are at v = +j 2, -j 2, s = e v = e j 2, e -j 2, which implies a stable and damped-oscillatory time response.
Time-response plots for this system are also shown in Figure 5 with w = 1.6, 2.5, and 3.0 in addition to w = 2.0. Note that the initial value of these functions is infinity, and that the response becomes unstable for .
Frequency responses of systems with Laplace-domain logarithmic operators
An approximation to the logarithm
This definition can be found in Spanier and Oldham .
A similar discussion can be given for a low frequency approximation.
These approximations were found to agree at high and low frequencies as predicted for 1/ln(s).
Laplace-domain logarithmic operator representation of ODE's
Equation 11 can be rewritten to demonstrate that any ODE or FODE can result from using incomplete logarithmic operators.
Notice that this equation has mixed terms, containing both an s and a ln(s), a result that is generally easy to obtain using order-distributions. A similar equation can be found for small s by reversing the limits of integration.
Thus it can be seen that in some cases, systems of order-distributions can surprisingly be represented by standard fractional-order systems.
The authors gratefully acknowledge the continued support of NASA Glenn Research Center and the Electrical and Computer Engineering Department of the University of Akron. The authors also want to express their great appreciation for the valuable comments by the reviewers.
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