Closed-loop time response analysis of irrational fractional-order systems with numerical Laplace transform technique
Introduction
Fractional calculus, which is the natural extension of traditional calculus, is referring to derivatives and integrals of non-integer orders. More than 300 years ago, the notion of fractional calculus was first mentioned by Leibniz [1], but it has not been widely studied until the recent decades [2], [3], [4], [5], [6]. Lately, with the development of science and technology, fractional calculus has attracted a lot of attention. Meanwhile, the extraordinary talents of this novel concept have been revealed by the increasing related studies in various fields. Several complex dynamical systems and physical phenomena can be described more accurately by non-integer models [7], [8], [9], [10], [11]. Some natural phenomena were also better modelled by distributed-order or variable-order fractional differential equations [8], [12]. Among these studies, irrational fractional-order systems are frequently used to describe physical phenomena [13], such as diffusion phenomena [14], heat flow [15] and so on [9], [16], [17]. Therefore, in this paper, an analysis and design method of systems modelled by irrational fractional-order transfer functions is presented.
Laplace transform technique is always regarded as an efficient method to solve differential equations [18], [19]. But only a few fractional differential equations which are relatively simple can be solved by Laplace transform. Moreover, the inverse Laplace transform, which is a significant and tough step in the application of Laplace transform process, usually obtains no analytical solution for fractional-order systems [19]. This is another reason that most of the existing studies in the control field are aiming at systems which can only be modelled by simple or rational fractional-order transfer functions.
To evaluate the control performance of irrational systems, several numerical algorithms are put forward [20], [21], [22]. However, few of them has been used in achieving the time response of irrational fractional-order system until now. In this paper, a numerical inverse Laplace transform technique which can be used to solve both rational and irrational fractional differential equations is discussed. The closed-loop time domain response, frequency domain analysis and stability evaluation of different kinds of fractional-order systems have also been investigated. Moreover, the MATLAB codes which are generated to solve the fractional differential equation using numerical inverse Laplace transform is provided to benefit researchers and engineers in the related fields. These codes can be used straightforward by users without a lot of experience.
The rest of this paper is organised as follows: Section 2 gives some preliminaries of fractional calculus; Section 3 presents the numerical inverse Laplace transform; the numerical solutions of irrational fractional-order systems as well as some illustrative examples are discussed in 4 Numerical solutions of irrational fractional-order systems, 5 Stability analysis of general fractional-order systems presents the stability evaluation of general fractional-order systems; Section 6 gives the time domain analysis of irrational fractional-order systems with an example of finding the roots and stability test of such systems; the frequency domain analysis of irrational fractional-order systems is also provided in Section 7; finally, the conclusion is drawn in Section 8; moreover, the MATLAB codes generated for this paper are provided in the Appendix.
Section snippets
Preliminary
The unified fractional-order integral-differential operator can be defined as:where t0, t are lower and upper limits of this operator, and v ∈ R is integral or differential order.
In detail, fractional calculus has many kinds of definitions, which have been applied in different fields of engineering and computing science. Riemann–Liouville and Caputo definitions of fractional calculus are widely used and will be introduced in the following
Numerical inverse laplace transform
Laplace transform can be considered as a useful method in solving differential equation. But the inverse Laplace transforms of a lot of functions are difficult or even unable to be obtained analytically [19]. On this occasion, numerical algorithms should be taken into consideration, especially in solving fractional differential equations. Lots of numerical inverse Laplace transform algorithms are put forward to solve this problem [24], [25], [26], [27]. In this section, the INVLAP (Inverse
Numerical solutions of irrational fractional-order systems
As it was discussed above, the analytical solutions of complex fractional-order equations are hard or even not possible to be achieved. Therefore, few study has discussed about the solutions of irrational fractional-order equations. In this section, numerical solutions of fractional-order equations are obtained according to the numerical inverse Laplace transform presented. Furthermore, the MATLAB codes of the presented algorithm are provided, which will make the simulations of fractional-order
Stability analysis of general fractional-order systems
In this section, the stability analysis of general fractional-order systems, or systems with real order transfer functions in other words, will be carried out. Theorem 1 Suppose a transfer function described byThe system is asymptotically stable, if all the roots of the characteristic equation have negative real parts. Proof Suppose that the input u(t) is a pulse function bδ(t), where δ(t) is a unit pulse function and b is a real constant. Then, the Laplace transform of u(t) is
Time domain analysis of irrational fractional-order systems
After the stability problem was solved in the last section, the most important problem now is, how all the solutions of irrational equation can be found. An algorithm is presented in below to find all the solutions of an equation. Algorithm finding all the solutions of a given set of equations Require: Anonymous function initial solution X, range of interested region A, error tolerance e Initialisation, construct initial stored solution set X, in a 3D array while true do Randomly generate an x
Frequency domain analysis of irrational fractional-order systems
Closed-loop system control performance can also be evaluated through frequency domain response. Here, the proposed method is used in frequency domain analysis of irrational fractional-order systems.
Conclusion
In this paper, the time response of irrational fractional-order system is studied based on numerical inverse Laplace transform. The numerical solutions and closed-loop responses of irrational fractional-order systems are presented. The stability evaluation of general fractional-order systems is also presented to support the results. The proposed algorithm can be used on general fractional-order system with rational or irrational transfer function. Some examples are shown to verify the accuracy
Acknowledgment
This work is partially supported by the National Natural Science Foundation of China (No. 61673094), and the Fundamental Research Funds for the Central Universities of China (Nos. G2018KY0305, G2018KY0302).
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