Description and Realization for a Class of
Irrational Transfer Functions
Abstract
This paper proposes an exact description scheme which is an extension to the wellestablished frequency distributed model method for a class of irrational transfer functions. The method relaxes the constraints on the zero initial instant by introducing the generalized Laplace transform, which provides a wide range of applicability. With the discretization of continuous frequency band, the infinite dimensional equivalent model is approximated by a finite dimensional one. Finally, a fair comparison to the wellknown Charef method is presented, demonstrating its added value with respect to the state of art.
I Introduction
Fractional calculus, as a generalization of the classical calculus of integrals and derivatives, has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering [1, 2]. It does indeed provide several potentially powerful tools for practical applications, such as fractional controller [3], constant phase element [4], fractional capacitor [5], and fractional memristor [6], etc. As for the recent relevant works, the readers can refer to the excellent papers [7, 8] and the references therein.
The numerical implementation emerged from the first beginning of applying fractional calculus in practice, since fractional order systems are essentially infinite dimensional. Though many methods were developed, none has a clear overwhelming advantage [9, 10]. This situation always existed until a recursive algorithm was proposed by Oustaloup [11]. Afterwards, a similar algorithm was introduced for fractional integrator instead of fractional differentiator [12]. Inspired from this, a series of effective and efficient method were established subsequently. For examples, fixed pole methods were constructed to reduce the degree of the approximation model [13, 14, 15]. Frequency domain identification based methods were derived to improve the approximation Accuracy [16, 17, 18]. Reference [19] considered the keeping strategy of pure integral property after approximating the fractional integrator. Similarly, some measures were taken to maintain the stability, [20, 21, 22], the monotonicity [23] and the initial value [24]. The main feature of the aforementioned methods lies in that they usually approximated the fractional operator first and then extended the related results to the target system. Nonetheless, those methods are not directly applicable to a particular yet wideclass of systems [25], i.e. .
As of today, very few research outcomes have been reported on the approximating such fractional order systems. Fortunately, there are several attempts. In view of the limitations of the wellestablished Charef’s approximation method, [26] presented a new approximation scheme based on an optimization process, which is suitable for the realization of fractional order transfer function directly. The approximation of was investigated via the distribution of the relaxation times function [27]. After deriving the numerical simulation scheme, a simple analog circuit was also designed to implement such a fractional order system. In reference [28], the state space description was deduced for by using some basic properties of Laplace transform. Also, considering the irrational continuous time transfer function, a loworder, computationally stable and efficient method was derived, resulting a discrete time rational transfer functions [29]. To be specific, some critical issues are still to be further investigated. For example, both [26] and [29] could not give the exact state space model, which might hinder the related theory analysis and further application. Additionally, the method in [28] cannot be extended to the general case. What is more, the original fractional integral/derivative is defined with nonzero initial instant. The previous discussed methods only focused on the zero initial instant situation. Therefore, a convenient and intuitive approach for implementing general irrational transfer function is urgently needed. At this point, this paper will continue to investigate the irrational transfer function. Roughly speaking, the contributions of this paper include: i) More general class of systems will be considered, such as nonzero initial instant, nonstrict proper case, discrete time case. ii) The exact state space description will be deduced rigorously. iii) The numerical realization scheme will be established accordingly.
The remainder of this paper is structured as follows. Section II provides some basic facts on this work. Section III shows the infinite dimensional nature of such fractional order system and proposes an effective approximation scheme. Section IV validates the validity of the proposed scheme by comparing to a widely used method. Section V summarizes the main outcome of this paper.
Ii Preliminaries
The concept of fractional calculus has been known because of the development of the regular calculus, with the origin probably being associated with the discussion between Leibniz and L’Hôpital in 1695. Today, there are numerous different definitions related to fractional calculus, among which RiemannLiouville and Caputo definitions are two of the most popular ones which have indeed played a striking role in engineering and science [30].
In 1847, Riemann derived a definition for fractional integral as
(1) 
which is commonly called Riemann–Liouville fractional integral, where is the integral order, is the constant initial instant and is the Euler Gamma function.
In the light of such a fractional integral in (1), two fractional derivatives were established successively, i.e., Riemann–Liouville case (in 1872)
(2) 
and Caputo case (in 1967)
(3) 
with and . It can be observed that such fractional derivatives are essentially a kind of special integral, which show the long memory characteristic of .
For convenience, the generalized Laplace transform with nonzero initial instant is defined as
(4) 
Note that when , the introduced Laplace transform degenerates into the classical one. Also, the fact can be checked directly.
Likewise, the generalized convolution product is
(5) 
Before moving on, a key lemma will be given first, which will play an essential role in developing the main results of this paper.
Lemma 1.
The objective of this paper is to investigate the realization problem of the following irrational transfer function
(7) 
In other words, finding a state space model whose transfer function is . In general, once the transfer function is realized into a state equation, it can be implemented using opamp circuits. However, due to the special long memory characteristic of fractional calculus, fractional order system is infinite dimensional in nature and the developed state space model for (7) must be infinite dimensional. As a consequence, the numerical approximation problem emerges immediately.
Iii Main Results
This section focuses on the analytical realization and numerical realization of some typical irrational transfer functions. Firstly, an underlying theorem is derived for the nonzero initial instant case.
Theorem 1.
The following statements hold

;

if is bounded on and exists;

if exists;

, .

for ;

for ;

for ;

for .
Proof.
ii) Let and . Then, the definition of leads to
(9) 
Since is bounded, there exists such that is dominated by the integrable function . Thus, by the dominated convergence theorem, it follows
(10) 
which is equivalent to statement ii).
iii) Start from (9) with an assumption . By using the the dominated convergence theorem similarly, one has
(11) 
which indicates that statement iii) hold.
iv) Defining yields
(12) 
where is adopted.
From the traditional inverse Laplace transform, one has
(13) 
and
(14) 
which is just the inverse Laplace transform with initial instant .
v) By mathematical induction, it follows
(15) 
which implies that statement v) holds for .
Suppose that statement ii) holds for , namely,
(16) 
By applying (15), one obtains
(17) 
which verifies the correctness of statement v) with . Consequently, one can conclude that for any , statement v) holds.
vi) When , it follows
(18) 
where the definition of Gamma function is adopted here.
Afterwards, the desired result in statement vi) can be deduced by applying statement i)
(19) 
vii) By adjusting the order of summation in statement v), an equivalent description follows
(20) 
viii) Define and then
(22) 
which completes the proof of statement viii). ∎
Remark 1.
With the newly introduced Laplace transform and convolution, Theorem 1 gives the Laplace transform of convolution, initial value theorem, final value theorem, inverse transform, classical derivative, Riemann–Liouville integral, Riemann–Liouville derivative and Caputo derivative with nonzero initial instant. Interestingly, the obtained results are similar to those in zero initial instant case.
Theorem 2.
Consider a system with strict proper transfer function
(23) 
where , the denominator has degree and is monic, the numerator is coprime to , , , and in nonzero.
When the fractional integrator is approximated by , redefining with , for and , then the approximation model of (23) can be expressed
(25) 
Proof.
Taking Laplace transform for (24) and defining , and , one has
(26) 
Since only the first element of vector is nonzero, only the first column of the relevant inverse matrix is given here.
(27) 
On this basis, the following formula holds
(28) 
With the help of Lemma 1, the transfer function of (24) can be deduced as
(29) 
Considering the existence and uniqueness of Laplace transform, the equivalence of systems (23) and (24) can be reached. In other words, system (24) is the realization of system (23).
In view of the infinitedimensional characteristics of system (24), it is difficult or even impossible to implement it directly. Accordingly, the following approximation scheme is applied
(30) 
Based on this point, the transfer function of (25) can be calculated as
(31) 
The proof is thus completed. ∎
Theorem 2 investigates the description and implementation problem for strict proper transfer function. For the sake of completeness, the nonstrict proper counterpart will be discussed herein.
Theorem 3.
Consider a system with proper transfer function
(32) 
where , the denominator has degree and is monic, the numerator is coprime to , and in nonzero.
When the fractional integrator is approximated by , redefining with , , , for and , then the approximation model of (33) can be expressed
(34) 
Proof.
Similarly, defining , , and taking Laplace transform for (21) and, one has
(35) 
By substituting the defined variables into (35) yields
(36) 
By applying Lemma 1 similarly, one obtains
(37) 
After investigating the rational case of and in Theorems 23, the following discussion focuses on the irrational case further.
Theorem 4.
Consider a system with proper transfer function
(39) 
where , , , , and are nonzero.
When the fractional integrator and are approximated by and , respectively, then the approximation model of (40) can be expressed
(41) 
Proof.
Likewise, defining , , and taking Laplace transform for (39), one has
(42) 
Remark 2.
Both and vary from to in (40). Even after truncating, the number of variable is still . Sometimes, to reduce the degree of approximation model without loss of precision, the principle is applied directly. Alternatively, the pole zero cancellation approach could also be performed.
Remark 3.
For the continuous time system, the stability condition is that all poles located on the principle Riemann leaf lie in the left half plane. For that reason, the approximation performance in the boundary region , is highly regarded. Since the variable in Lemma 1 is replaced by in Theorems 24, changes might be made on and to achieve a good approximation accuracy.