1. Introduction
Fractional calculus is derived from the field of mathematics and the research on this field is still ongoing [
1,
2]. In recent years, fractional calculus is being used more and more in the engineering field. Complex physical, chemical e.g., Lithium-ion batteries [
3], or biological systems e.g., blood alcohol model [
4] are increasingly being described by fractional-order models. Fractional differential equations are also used to describe parts of electrical circuits or refine the description of friction equations in mechanical systems. An overview of technical areas is provided by [
5]. Due to the non-locality and memory of fractional integration and differentiation, these models approximate partial differential equations more accurately than classic integer-order models [
6]. Over the last years, identification methods considering fractional-order models based on the modulating function method (see e.g., [
7,
8,
9,
10,
11,
12,
13,
14]) have become more prevalent. A benefit of the modulating function method is that a system of algebraic equations may be solved for the parameter identification instead of a set of differential equations and the measured signal need not be differentiated [
15]. In current approaches, a generic modulating function is chosen and parameterized heuristically. This procedure constitutes time-consuming educated guessing which does not generalize or provide clues for new applications, especially in the fractional order case. Thus, parameterizing the modulating function is a significant shortcoming of this method. For parameter identification, the algebraic equations are collected in a linear system which has to be solved. Because the modulating function is chosen and parameterized a-priori, linear dependencies between the individual algebraic equations can occur, depending on the measured signal. In this case, the parameter identification cannot be performed, which constitutes is a significant shortcoming of this method.
In this paper, we propose a systematic procedure for parameter identification using an implicit determination of a modulating function considering the current measurements. This procedure allows for direct parameter identification, without requiring the explicit calculation of modulating function. To the best of the authors’ knowledge, it is the first time that a method that automatically determines a modulating function for the parameter identification of a fractional-order model is presented. While this idea has been described for integer-order models [
16], this result does not easily generalize to the non-integer order case.
The paper is structured as follows: In
Section 2, the basics of fractional calculus are provided. In
Section 3, the modulating function method is recapped and an additional property for the modulating function method, enabling the separate identification of the parameters, are given. Afterward, the model-based auxiliary system, which is central in determining a modulating function automatically, is derived in
Section 4. The model-based auxiliary system is used to transfer the heuristic determination of a modulating function into a control problem which allows an automatic determination of the parameter-specific modulating function. A solution to the control problem is provided in
Section 5, which leads to the parameter identification and represents the main contribution of this paper. Additionally, the error occurs due to superposed noise on the input and output signal is analyzed. A numerical example in
Section 6 demonstrating the efficacy of the proposed algorithm concludes the paper.
3. Parameter-Specific Identification Using Modulating Function Method
The modulating function method is a well-known method for the parameter identification of fractional-order models (see e.g., [
7,
8,
9,
10,
11,
12]). The modulating function method was derived in [
27] for the integer-order case and has been transferred to the fractional-order case (see e.g., [
7,
8,
9,
10,
11,
12]). In
Section 3.1, we shortly recap the modulating function approach to introduce the necessary notation. Based on the recap, a parameter-specific modulating function is defined in
Section 3.2. The properties of such a parameter-specific modulating function are used to transfer the identification problem into a control problem in
Section 5.
3.1. Fractional Modulating Function Method
Suppose, a modulating function exists which fulfills Assumption A2.
Assumption A2. (Properties of Modulating Functions.)
Suppose and . The modulating function fulfills the properties: where indicates the class of continuously differentiable functions, , and .
The property
ensures that the influence of the initialization function
in (
9) is eliminated. Multiplying (
9) with a modulating function
and integrating the resulting equation by parts results in
without loss of generality we assume
. This results in the well-known lemma for the identification of the unknown parameters (see [
12]). Note, the mathematical operations changes the Caputo initialized fractional derivative used in (
9) into uninitialized Riemann-Liouville fractional derivative (
3) in (
12).
Lemma 1. (Parameter Identification Applying Modulating Function Method.)
Suppose , , the modulating-function φhas property , the system, , , , , and . Furthermore, suppose , where is given by Is regular, we get the parameters out of where and Proof. The proof can be found in [
12]. □
We note that, depending on the a-priori chosen modulating functions and their parameterization, the equations of system (
14) can be linearly dependent, which presents an obstacle for the identification of parameters.
3.2. Parameter-Specific Modulating Function
Sometimes, it is favorable to identify just one parameter at a time. Therefore, we introduce a parameter-specific modulation function approach. Suppose holds true for each parameter , where represents the s-th element of the parameter vector . Call parameter-specific modulating. In summary, in line with the assumption – we define:
Definition 4. (Parameter-Specific Modulating Function Set.)
Suppose and . The parameter-specific set is defined as follows: where the fractional derivative is given in (3), is given by In (20),
consists of the elements
and describes the j-th element of a vector, and . Theorem 1. (Parameter-Specific Identification.)
Suppose , , the fractional derivative (3) and where describes the s-th element of a the parameter vector . The s-th parameter of (12) is calculated using a parameter-specific modulating function as follows: 4. Model-Based Auxiliary System
Previous approaches using the modulating function method require such a function to be known a-priori. As motivated in the introduction and in
Section 3, finding a function that meets
–
is often a tedious task, especially for fractional systems.
In the following two sections, we, therefore, present our main result which allows parameter identification without requiring an explicit a-priori modulating function. Instead, we propose in this section the use of an auxiliary system that implicitly contains the requirements of a modulating function. In the sequel, we then demonstrate how applying an appropriate control to this dynamical system can be used to automatically retrieve a valid modulating function and achieve parameter identification at the same time.
In
Section 4.1, we introduce notation for the compact representation of an auxiliary system. The auxiliary system used for parameter identification is formally introduced in
Section 4.2.
4.1. Notations for the Model-Based Auxiliary System
Before defining the model-based auxiliary system, we here introduce notations that allow for a more compact description of the requirements needed for such a system. Afterward, we introduce additional notations regarding the dimensions of subsystems contained in the model-based auxiliary system and we define normalization parameters.
The main idea behind the model-based auxiliary system is that every expression of the modulating function can be represented as a combination of the
-th derivative
and a fractional integration with corresponding order. For example, (
19), considering
, results in
where the fractional derivative is given in (
3),
, and
. To enable a compact definition of the model-based auxiliary system, a vector collecting all possible derivative orders of (
19) is given first.
Definition 5. (Vector of Boundary Term Orders.) for the derivative orders Furthermore, suppose the two normalization parameters
and
, where
is defined by (
25). This normalization parameters ensure that all derivative orders of the model-based auxiliary system fulfill the requirement
for fractional state spaces (see [
28]). Additionally,
,
, and
are defined for a more compact representation of the subsystem dimensions.
4.2. Model-Based Auxiliary System
Equipped with the notations introduced in the previous section, we now define the model-based auxiliary system. The model-based auxiliary system is constructed by interconnecting the derivatives of the modulating function
and
(see (
12)), the identification equation of the modulating function method (
12), and the resulting boundary terms
–
. In this subsection, we start by defining the states associated with each of these parts and proceed with describing their respective state dynamics.
Remark 6. We denote the four interconnected parts of the model-based auxiliary system as its constitutive subsystems and assign each a specific identifier. Subsystem ∘ describes the equation resulting of the modulating function method (12), subsystem □ maps the connection between the derivatives of the modulating function in (12). Finally, subsystems Δ and ⋄ take the derivatives in into account. Definition 6. (Fractional State Vectors.)
Suppose , , , the fractional derivative (3), and . In Definition 6, the states ∘ represent the derivatives of the modulating function which occur in the basic equation of the modulating function method (
12). Hence, the fractional state equations for the subsystem ∘ are as given in the following Lemma 2.
Lemma 2. (Subsystem ∘.)
Suppose and or , depending on whether the output or input signal is considered. The subsystem ∘ is described using the state vector of subsystem □: Proof. Using the definition (
21), it follows directly that
The expressions related to the modulating function are equivalent to the
,
, ⋯,
elements of
which is selected by (
35). □
The connection between the derivatives of the modulating function in (
12) are mapped by subsystem □ considering that all derivatives are represented as a combination of the
-th derivative
and a fractional integration with corresponding order.
Lemma 3. (Subsystem □.)
Suppose and . Then,where is a Jordan matrix with dimensions which only has eigenvalues of 0. Proof. The connection of the derivatives of the modulating function is a chain of integrators for the input , whereby each integrator is of order . This chain can then be written as a system with a Jordan matrix where all eigenvalues are zero. □
While subsystem Δ takes the boundary terms (
19) for
into account, subsystem
◊ represents the boundary terms for
.
Lemma 4. (Subsystem Δ.)
Suppose and . Then, where is a Jordan matrix with dimensions and has only eigenvalues of 0.
Proof. The proof is analogous to the proof of Lemma 3 with the difference that the order of each integrator is 1. □
Lemma 5. (Subsystem ⋄.)
Suppose , , and . The subsystem ⋄ can be subdivided into C subsystems of dimensions :where is a Jordan matrix with dimensions which only has eigenvalues of 0. Connecting all subsystems results in Proof. The proof is analogous to the proof of Lemma 3, where the order of each integrator is . □
Finally, using the defined states and their dynamics comprising the model-based auxiliary system, we can now construct a full system description. We achieve this by simply combining all matrices and vectors of the respective subsystems into the time-variant system matrix
and the input vector of the model-based auxiliary system
with the input signal
Remark 7. The model-based auxiliary system is derived without any restrictions on the fractional order. Hence, the system must be also valid for the integer-order case. Considering , the fractional states of subsystem Δ and subsystem ⋄ are equivalent to the fractional states of subsystem □ and the model-based auxiliary system reduces to the subsystem ∘ and subsystem □. This result is in line with the results in [16]. 6. Numerical Example
The parameter identification based on implicit modulating functions is illustrated in the following. For this purpose, we consider the system
where
,
, and
are assumed to be known, and
as well as
are unknown. The simulation is started at
and the identification at
. The duration of simulation is
with a sampling time of
and the duration of identification is
. A mean-free, pseudo-random binary sequence with an amplitude of 1 is used as an input signal
[
24] (p. 165). The input, output and noisy output (
) signals are shown in
Figure 1. A green dotted line marks the starting time of the identification.
To state the model-based auxiliary system, the fractional orders of the boundary terms must first be calculated. Evaluating (
25) yields
and
. Because
, the following fractional pseudo state space yields:
Depending on the requested parameter, the final pseudo state for
is
and for
The parameter is identified by evaluating (
23) and (
56) respectively. To calculate the parameter-specific control input, (
A32) has to be evaluated. Therefore, the matrix approach of [
29] is used.
Because one measurement of the input and output signals can be used to identify all parameters, the parameter-specific control input as well as the trajectories of the parameter estimates for
and
are illustrated in
Figure 2 for the noise-free case and in
Figure 3 for the case with a noisy observation of the output signal. The final estimations
and
in the absence of noise and
and
in the case with a noisy observation of the output signal can be read at the final time of the identification
.
The results are compared to the parameter identification using a heuristically adapted modulating function. Because the system is not at rest when the identification is started, the spline-type modulating function has to be used (see [
12]). For the approach which is described in this paper, only the starting time and the duration of the identification may be chosen freely. In addition to the start and end times, the use of spline-type modulating functions requires that the number of splines and the order of the modulating function is chosen. It should be noted that the maximum order of the modulating function depends on the number of splines and the minimum order depends on the fractional order of the system (see [
12]). All the parameters of the spline-type modulating function have to be fixed a-priori. This is not necessary using the newly proposed method based on implicit modulating functions, from which the modulating function can be determined automatically.
In this example, the method described in [
12] is applied using a spline-type modulating function with 20 splines and of order 5 for the comparison. The starting time
and the identification horizon
are chosen as for the parameter-specific identification method. To derive the independent equations for the linear system (
15), the identification horizon has to be shifted. The shifting time is an additional parameter for the identification which is set to
in this example. Using the same input
and output signal
illustrated in
Figure 1, the parameters are identified to
and
. Regarding the number of shifts, each shift leads to a new independent equation. The linear system (
15) consits of
equations which means that at least
shifts are necessary to state the linear system (
15). In this example, one shift is sufficient because of
and
(see (
65)). The shifting of the identification horizon makes longer measurements necessary. Considering fractional systems with more parameters, the extension of the measurement duration can be significant. Because of the extension of the measurement duration, more data are considered for parameter identification. Nonetheless, the error of the identified parameters is with 4–5% for the method described in [
12] significant greater than the error (approx. 0.3%) made with the parameter-specific approach which is described in this paper.