Modeling heat distribution with the use of a non-integer order, state space model

Abstract

A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.

Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders

“…Stabilization of non-linear systems finds applications in many areas of engineering, particularly in the fields of mechanics, robotics, electronics, and so on [1,2]. Non-linear problems arise when analyzing the dynamical behavior of systems, such as robot manipulators with flexible links, different types of oscillators, electrical and electronic circuits, buildings, bridges, and even embedded control systems [3][4][5][6][7]. In some non-linear systems, such as Chua's circuit, a phenomenon called chaos can be observed [8,9].…”

“…The meaning of the system parameters are exactly the same as described in the previous section. It is assumed that Assumptions 1-3 hold for the system (7) as well. The system model being in the form of (7) will allow, as illustrated in following sections, for creation of the terminal sliding model controller in a much more simplified form than in the case of the system (1) and (5).…”

“…Under the Assumptions 1-3, the trajectory of the system (7), when forced by the control (24), will reach the sliding surface (23), starting from the initial condition x 0 , in finite time and then remain on this surface.…”

Discrete-time feedback stabilization

“…Let us now consider the process of heating a thin rod (Oprzedkiewicz (2003(Oprzedkiewicz ( , 2016 [23,24] …”

“…Thanks to computers, nowadays, we can analyze complex mathematical models of distributed parameter systems, e.g. models of non-integer order Obrźczka (2014) [22], Sierociuk (2015) [32], Oprzêdkiewicz (2016) [24] which sometimes better describe real systems.…”