Use of noninteger identification models for monitoring soil water content

Abstract

It is proposed to apply a method for identifying thermal effusivity in order to monitor the percolation of water in a soil by means of a noninteger order model. This model is expressed in the form of a linear relation connecting the fractional derivatives of the temperature at a point of the system to the fractional derivatives of the stress applied, namely a flux. These derivatives are replaced by their discrete definitions and the model coefficients are identified from experimental measurements by a method of linear least squares. Flux sensors are used to measure the flux and temperature simultaneously in the access plane to the system, enabling the thermal impedance to be calculated. The study is made up of two parts:The first part consists in establishing in the laboratory a correlation between the effusivity and water content of a sample of ground studied by traditional methods.The second part constitutes the in situ tests, where with the flux sensors and the application of the method, the effusivity of the ground can be estimated and thus the water content deduced.

Extension of the root-locus method to a certain class of fractional-order systems

“…v + · · · + a n−1 s 1 v + a n , (v > 1) (5) and k is assumed to be a positive real constant. Note that the domain of definition of P(s) is a Riemann surface with v Riemann sheets where the origin is a branch point [14].…”

“…Note that the domain of definition of P(s) is a Riemann surface with v Riemann sheets where the origin is a branch point [14]. Since the numbers are stored with a limited precision in a computer, almost all practical fractional-order transfer functions can be represented as in (5), i.e. with rational powers of s. It is a fact that when a minimal fractional-order polynomial is represented in a non-minimal form, the number of its zeros is increased but the location and the order of zeros on P remain unchanged.…”

“…Note that (8) So, according to the root-locus plot in the w-plane, any point on the positive real axis of P , the total number of real poles and zeros to the right of which is odd, lies on a root locus. But for P(s) given in (5), no line segment on R − can belong to the root locus. The reason is as follows.…”

“…Tsao et al [27] and Poinot and Trigeassou [25] clarify the identification method when the members of the model set are of fractional order. Two applications of such identifications can be found in [30,5]. Fractional-order systems are also used in the control field.…”

“…Such systems lend themselves well to some algebraic tools [17,3]. Practical examples of such systems can be found in [3,30,5]. The problem of plotting the root loci for these systems is treated in this paper.…”