Parameter Identification Of Fractional Oredr Syatem Using Operational Matriax And Its Applications

Fractional calculus is a generalization of the regular integer calculus. Compared with integer calculus, fractional calculus has overall importance, and the fractional calculus can reflects very nicely the property, that the development of system function depends on the history of the system. It has been observed that real-world physical systems are well characterised by fractional-order differential equations rather than using classical integer order models, and it can establish the complex systems more accurate using the theory of fractional calculus. Therefore, the study of fractional calculus has important theoretical and practical significance. This paper makes a study of the identification of the fractional order system. This paper fulfills the following work:Firstly, a novel method is proposed to identify the parameters of fractional-order systems. The proposed method converts the fractional differential equation to an algebraic one through a generalized operational matrix of block pulse functions. And thus, the output of the fractional system to be identified is represented by a matrix equation. The parameter identification of the fractional order system is converted to a multi-dimensional optimization problem, whose goal is to minimize the error between the output of the actual fractional order system and that of the identified system. The proposed method can simultaneously identify the parameters and the fractional differential orders of the fractional order system and avoid the drawbacks in the literature that the fractional differential orders should be known or commensurate. Furthermore, the proposed method avoids complex calculations of the fractional derivative of input and output signals. Illustrative examples covering both fractional and integer systems are given to demonstrate the validity of the proposed method.Secondly, the operational matrix of hat functions method is proposed to identify the parameters of fractional-order systems. The proposed method converts the fractional differential equation to an algebraic equation. It can simultaneously identify the parameters and the fractional differential orders of the fractional order system. Illustrative examples covering both fractional and integer time delay systems are given to demonstrate the validity of the proposed method.Again, operational matrix method is proposed to identify continuous-time fractional order system with time delay. The fractional order time delay system is converted to an algebraic equation using the operational matrix of fractional differential and integral operators, which is derived from the orthogonal expansion of a function onto a series of block pulse functions. And then, all the system parameters including the fractional differentiation orders, time delay are identified through minimizing a quadric error function, the quadric error between the output of actual system and that of identified system. The advantages of the proposed methods involve two folds. First, it can simultaneously identify the system parameters, the fractional differentiation orders and time delay since the operational matrix representation of fractional order system contains all the system parameters. It overcomes the drawbacks of existing identification methods that the differentiation orders must be known or commensurate. Second, it doesn’t involve complex calculations of fractional differentiation of input and output signal, which simplifies the identification process dramatically. Illustrative examples covering both fractional and integer time delay systems are given to demonstrate the validity of the proposed method.Finally, the operation matrix method is adopted to identify fractional real-world system. The result demonstrates the applicability of the method. The identification of real model contains the parameter and the fractional order. This paper mainly studies three models, heating furnace, viscoelastic system, and the wall problem in thermal diffusion. The result of the simulation demonstrates the validity of the proposed method.