Fractional-Order Interval Parameter State Space Model of the One-Dimensional Heat Transfer Process

Abstract

In this paper, the new non-integer-order state space model of heat processes in a one-dimensional metallic rod is addressed. The fractional orders of derivatives along space and time are not exactly known, and they are described by intervals. The proposed model is the interval expanding of the state space fractional model of heat conduction and dissipation in a one-dimensional metallic rod. It is expected to better describe reality because the interval order of each real process is difficult to estimate. Using intervals enables describing the uncertainty. The presented interval model can be applied to the modeling of many real thermal processes in the industry and building. For example, it can describe the thermal conductivity of building walls. The one-dimensional approach can be applied because only the direction from inside to outside is important, and the heat distribution along the remaining directions is uniform. The paper describes the basic properties of the proposed model and supports the theory via simulations in MATLAB R2020b and experiments executed with the use of a real experimental laboratory system equipped with miniature temperature sensors and supervised by PLC and SCADA systems. The main results from the paper point out that the uncertainty of both fractional orders impacts the crucial properties of the model. The uncertainty of the derivative along the time affects only the dynamics, but the disturbance of the derivative along the length disturbs both the static and dynamic properties of the model.