An Efficient Method for Approximation of Non Rational Transfer Functions

Abstract

A method for rational approximation of linear fractional order systems (LFOS) is presented in the present paper. The method is computationally efficient, flexible and effective, as is illustrated by numerous examples. The proposed approach can also be used as an intermediate stage in designing indirect discrete rational approximations.

Stabilization of Inverted Pendulum by Fractional Order PD Controller with Experimental Validation: D-decomposition Approach

“…The physical parameters of the system used in experiment Since numerical differentiation usually introduces significant noise in velocity measurements, we estimate the corresponding velocities of the cart and pendulum from position measurements by utilizing a derivative filter given by the following fractional order transfer function TF = (s/0.02 s + 1) a , where a represents the real differentiator parameter, having in mind that a ¼ 1 for pendulums derivative filter. In order to approximate this non-rational transfer function, a computationally efficient method for rational approximation of linear fractional order systems is used, as described in [10]. The method proposed relies on the interpolation of the frequency characteristic of the system on a predefined set of target frequencies.…”

Modern methods of design, analysis, optimization and implementation of conventional control algorithms for processes with finite and infinite degrees of freedom

“…If process and controller are described with rational transfer functions, both of these approaches are equal because of effective transformations from continuous to digital domain without disturbing the quality of the control. If the controller (6), (7) or process is non-rational transfer function, the design is performed in continuous domain and then controller is approximated in continuous or digital domain [52][53][54][55][56][57]. It is important that rational approximation should include amplitude and phase frequency characteristic of complex controller with minimal deviation in order to preserve quality control.…”

“…Different proposed methods for rational approximation can be found in literature such as: interpolation of frequency characteristic (IFC) [54], ARX-based methods [57], expansions in Taylor series, use of Padé approximation [56] etc. In this way continuous transfer function becomes rational and we can apply some of the abovementioned discretization rules.…”

“…For example, it is common case for complex controllers with fractional differential and integral effect. Solution to this problem may be found in [52][53][54] where efficient techniques have been recently developed for rational approximations of complex transfer functions, which has made possible an adequate implementation of control algorithms and structures. Due to large application of PI/PID controllers, which exceeds 93% in industry according to Honeywell's surveys [55] compared to all the other regulators, many modern design methods have been developed in recent years.…”

Development of simplified mathematical model of glass melting furnace

“…Можливе послідовне застосування до однієї й тієї самої СРП різних методів апроксимації [5], що дозво-ляють, наприклад, спочатку перейти до спрощеного, що допускає точний аналітичний розв'язок, рівняння об'єкта, для якого потім знайти дробово-раціональне наближення його передатної функції [6], що визначає результуюче наближення опису вихідної моделі об'єк-та у вигляді типових моделей СЗП.…”