Applications of Fractional Calculus in Mechatronics and Control Theory: New Results

by Prof. Mihailo P. Lazarevic (Department of Mechanics, Faculty of Mechanical Engineering, University of Belgrade)

Wednesday 23 Mar 2011, 18:00 → 19:00 Europe/Belgrade

301 F (Mathematical Institute)

Abstract:
In recent years, there have been extensive research activities related to applications of fractional calculus (FC) in mechatronics and control theory. In this presentation a quite new stability test procedure is proposed for perturbed (non)linear (non) homogeneous fractional order systems with/without time delay. Results from the area of finite time and practical stability extends to (non)linear, continuous, fractional order (time-delay) systems given in state space form. Sufficient conditions of finite time stability and practical stability for this class of fractional systems are derived using generalized Gronwall inequalities. Specially, previous results can be applied for stabilization of mechatronic system where it appears a time delay in PDα fractional control system. Besides that, a PDα type of iterative learning feedback control (ILFC) is proposed for class -fractional linear time invariant system. When the structure is not known or when many parameters cannot be determined, ILC may be considered. The learning control scheme comprises two types of control laws: a PDα feedback law and a PDα feed-forward control law.A sufficient condition for convergence of a proposed ILC will be given by the theorem and proved. Using feedback loop, the PDα controller provides better stability of the system and keeps its state errors within uniform bounds. Next, it will be presented here a new algorithms of PID control based on applying FC in control of given mechatronic system for the producing of technical gases, i.e air production cryogenic. Objective is to find out optimum settings for a fractional PIα Dβ controller in order to fulfill different design specifications for the closed-loop system, taking advantage of the fractional orders and properties of liquid. Moreover, it will be shown that active control of nonlinear vibration of simply supported smart composite beam can be obtained using suitable fractional PIα Dβ controller where sensing and actuating are achieved using piezoelectric sensors and actuators. At last, the generalization of the splines (fractional B-splines), and other fractional wavelet constructions will be disscused,where we can build the wavelet bases parameterized by the continuously-varying regularity parameter α. In that way, the fractional WT technique offers very handy tool to perform the signal analysis for pattern specially welcomed in real-life non-destructing testing, which enhance already existing testing equipment with advanced signal processing - denoising, magnifying and clustering the original signal.