How to Identify from a Step Response a Non-Commensurate Second-Species Transfer Function

Abstract

This paper proposes a time-domain identification method for second-species non-commensurate fractional-order transfer functions with exponents $\nu \in (0,2)$ and $\nu +1$. To successfully identify the system’s parameters, the method requires knowing the system’s step response, from which a limited number of points is needed. The rules developed are different depending on whether there is an overshoot or not. Performance is assessed in the presence of additive Gaussian white noise. While parameter estimation exhibits some numerical sensitivity to noisy data, the computation of the model’s natural frequency ${\omega }_n$ acts as a crucial compensator by scaling the system’s time constant, thereby arriving at useful models in most circumstances.

1. Introduction

Many real-world phenomena, such as anomalous diffusion [1], heat transfer [2,3], and viscoelasticity [4], cannot be easily modeled by integer-order equations. Fractional Calculus tackles this problem by expanding the concepts of differentiation and integration to non-integer orders that better represent these systems [5]. This extension leads to the definition of fractional-order transfer functions [6].

The method for identifying an integer second-order transfer function by analysis of its step response curve is well-known and follows well-established calculations. By extracting from said curve parameters like the maximum overshoot and the peak time, or, if there is no overshoot, the inflexion point and the settling time, it is straightforward to reach a model. The same cannot be said of fractional second-species transfer functions

$$G\left(s\right)={\displaystyle \frac{b_0}{a_0+a_{\nu }{s}^{\nu }+s^{\nu +\mu }}},\nu ,\mu \in {\mathbb{R}}^{+}.$$

(1)

Even though these are flexible models able to represent long-memory effects, oscillatory behaviour, and non-standard damping, the identification of parameters ${\omega }_n$, $\zeta$, and $\nu$ from time-domain data, particularly from a step response, is challenging. The analytical expression of the step response of (1) is not simple, and does not have a straightforward inverse. Some work has been done in this area, particularly for first-species fractional-order transfer functions ${\displaystyle \frac{b_0}{a_0+s^{\nu }}}$ [7,8,9,10], for second-species fractional-order transfer functions with two equal poles ${\displaystyle \frac{b_0}{{(a_0+s^{\nu })}^2}}$ [10], for second-species commensurate fractional-order transfer functions ${\displaystyle \frac{b_0}{a_0+a_1{s}^{\nu }+s^{2\nu }}}$ [11], and for a particular case of non-commensurate second-species transfer function ${\displaystyle \frac{b_0}{a_0+a_{\nu }{s}^{\nu }+s^2}}$ [12]. The present paper is an extension of such results.

There are, of course, other identification methods that can be applied to obtain models given by fractional first- and second-species transfer functions [13,14], including meta-heuristics [15,16,17], artificial intelligence algorithms [18], and recursive algorithms [19,20]. There are also methods to determine only the fractional order or orders of a model [21,22], allowing identification to proceed linearly from that point on. However, all such methods are necessarily non-linear, and many are iterative; as a result, they all are computationally demanding [23,24,25]. The same can be said of computing the frequency response by deconvolution and a Fourier transform in order to apply the Levy method: the intermediate steps introduce errors in the procedure, and the fractional orders must be found in some other manner anyway.

The objective of this paper is to propose a series of simple expressions, numerically established, that, from the unit step response curve, allow identification of the parameters $\nu$, $\zeta$, and ${\omega }_n$ of the second-species non-commensurate fractional-order transfer function

$$G\left(s\right)={\displaystyle \frac{1}{1+2\zeta {\left(\frac{s}{{\omega }_n}\right)}^{\nu }+{\left(\frac{s}{{\omega }_n}\right)}^{\nu +1}}},$$

(2)

which is a particular case of (1), obtained by making $\mu =1$. The behaviour of (2) has already been studied in the literature [26], and, in this way, the results in [27] are expanded. This type of identification method is important in engineering applications, where simple, straightforward methods are often required to obtain expeditious results. Although approximations are obtained, if they are accurate enough, this suffices for the purpose. Furthermore, in this paper, the sensitivity to measurement noise of the proposed identification method is verified by applying it systematically to step responses corrupted by noise.

The remainder of this paper is organised as follows: Section 2 formulates the problem; Section 3 addresses how unit step responses of (2) can be numerically found, and the conditions for stability and the existence of a resonance frequency; Section 4 presents the identification rules; Section 5 assesses their performance both without noise and in the presence of noise; and Section 6 concludes the paper.

2. Formulation of the Problem

Consider a single-input, single-output plant, to the input of which a step with amplitude L is applied at time instant $t_0$. We want to model the dynamic behaviour of the plant by transfer function

$$\begin{array}{c}\hfill G\left(s\right)={\displaystyle \frac{K{e}^{-\theta s}}{1+2\zeta {\left(\frac{s}{{\omega }_n}\right)}^{\nu }+{\left(\frac{s}{{\omega }_n}\right)}^{\nu +1}}}.\end{array}$$

(3)

The output $y\left(t\right)$ is known. $G\left(s\right)$ is linear; consequently, the following holds:

• The system’s gain K can be found as ${\displaystyle K=\frac{1}{L}\underset{t\to +\infty }{\lim }y\left(t\right)}$. This is clear from the final value theorem. Output convergence may be slow, and it may be that data was not recorded during a sufficiently long time period, but there is no theoretical difficulty involved.
• The delay time $\theta$ can be found by computing the first time instant $t_{\ne 0}$ that verifies $y\left(t\right)\ne 0$. Then, $\theta =t_{\ne 0}-t_0$. Although some issues may arise when analysing data corrupted by noise, this does not pose a challenge in an ideal environment [28,29,30].
• Consequently, the problem of identifying model (3) is reduced to the identification of the three parameters of (2), which are ${\omega }_n$, $\zeta$, and $\nu$.
• Due to linearity, we may presume without loss of generality a unit step response.
• The value of the natural frequency, ${\omega }_n$, proves to be just a scale factor in time, as will be seen below. Therefore, it is possible to assume ${\omega }_n=1$ rad/s, i.e.,

  $$G\left(s\right)={\displaystyle \frac{1}{1+2\zeta s^{\nu }+s^{\nu +1}}},$$

  (4)

  and then identify $\zeta$ and $\nu$ from the shape of the curve and finally obtain the correct value of ${\omega }_n$ from the time values involved.

3. Step Response, Stability, and Resonance Frequencies

Transfer function (2) is not commensurate. Thus, its unit step response cannot be found as the convolution of two Mittlag-Leffler functions, as in the case of a second-species commensurate transfer function [5,6]. It can be numerically found as the inverse Fourier transform of the frequency response of the desired step response:

$$y\left(t\right)={\mathcal{L}}^{-1}\left[Y\left(j\omega \right)\right]={\displaystyle \frac{1}{j\omega }}{\displaystyle \frac{1}{1+2\zeta {\left(\frac{j\omega }{{\omega }_n}\right)}^{\nu }+{\left(\frac{j\omega }{{\omega }_n}\right)}^{\nu +1}}},$$

(5)

For the purposes of all that follows, time responses were found in $[0\mathrm{s},60\mathrm{s}]$ with sampling time $T_s=0.01\mathrm{s}$. This was done by computing (5) in $[{10}^{-4}\mathrm{rad}/\mathrm{s},{10}^4\mathrm{rad}/\mathrm{s}]$ and then computing numerically (with the trapezoidal rule) its inverse Fourier transform.

The stability conditions of (2) were studied by [26,31,32]. The transfer function is stable if one of the following three conditions is verified:

• $0<\nu \le 1$ and $\zeta \ge 0$;
• $0<\nu <1$, $\zeta <0$ and ${\displaystyle \frac{-2^{\nu +1}\zeta }{\cos \left(\frac{\nu \pi }{2}\right)}}{\left(-\zeta \tan \left({\displaystyle \frac{\nu \pi }{2}}\right)\right)}^{\nu }<1$;
• $1<\nu <2$, $\zeta >0$ and ${\left(2\zeta {\omega }_n\right)}^2{\omega }_u^{2\nu }+{\omega }_u^{2\nu +2}-{\omega }_n^{2\nu +2}>0$, with ${\omega }_u=2\zeta {\omega }_n\tan \left({\displaystyle \frac{(2-\nu )\pi }{2}}\right)$.

Unstable systems have fewer applications in normal real-world problems; thus, we will only study stable cases. In particular, this restricts the fractional order to $\nu \le 2$. Stability conditions are shown in the chart of Figure 1. This figure also shows if there is a resonance frequency. There are no known analytical conditions to find out if there is; the information about this in Figure 1 was found numerically. More relevant for the identification procedure is that such resonance peaks (which only appear for $\nu <1$ if $\zeta <{\displaystyle \frac{\sqrt{2}}{2}}$, and only appear for $\zeta >0$ if $\nu >1$) may originate oscillations and overshoot in the step responses.

Actually, to develop the identification procedure, we will impose on the parameters of (2) stronger restrictions than those required for stability, namely, $0<\zeta <5$ and $0.6<\nu <2$. Fractional orders $\nu \le 0.6$ will not be considered for two reasons: First, because step responses are so slow to reach the amplitudes used in the identification rules below (and listed at the beginning of Section 4), this slow convergence conduces to significant numerical errors, even in numerically finding the step response. Second, because, in this case, the step response is likely dominated by one of the roots of the denominator, and thus it may be reasonable to use a model with only one pole instead, which can be found with the methods from [8,9,10]. Transfer functions with $\zeta \ge 5$ are mostly indistinguishable from those with lower values of $\zeta$. Consequently, this limitation does not in reality preclude finding a model for any system.

4. Identification Procedure

The step response is normalised so that its final value is 1. The following characteristics of the step response are considered in the identification procedure:

• The time $t_{0.1}$ at which the output $y\left(t\right)$ reaches $10\%$ of the steady-state value;
• The time $t_{0.2}$ at which the output $y\left(t\right)$ reaches $20\%$ of the steady-state value;
• The time $t_{0.5}$ at which the output $y\left(t\right)$ reaches $50\%$ of the steady-state value;
• The time $t_{0.7}$ at which the output $y\left(t\right)$ reaches $70\%$ of the steady-state value;
• The time $t_{0.8}$ at which the output $y\left(t\right)$ reaches $80\%$ of the steady-state value;
• The time $t_{0.9}$ at which the output $y\left(t\right)$ reaches $90\%$ of the steady-state value;
• The peak time $t_p$ at which the output $y\left(t\right)$ reaches its maximum overshoot value.

Figure 2 shows the unit step response curve of (4) when $\nu =1.3$ and $\zeta =1.8$. The points on the response indicate the characteristics from the list above which serve as inputs for the identification method described below.

Because determining ${\omega }_n$ will be left to the end, it is imperative that all variables used for system identification are dimensionless. This constraint ensures that the parameter extraction is entirely independent of the natural frequency ${\omega }_n$, preventing any time-scale coupling from biasing the estimation. Consequently, dimensionless time ratios of the temporal features above are used in what follows.

The identification rules below were found by adjusting different fits to the recorded results and choosing the best, and are divided into two cases for both $\zeta$ and $\nu$. Since the expressions were adjusted for particular ranges of the parameters, they cannot be applied successfully outside such ranges. The rules to find $\zeta$ depend on whether or not the step response has an overshoot [33]; those for $\nu$ depend on the value found for $\zeta$.

4.1. Finding the Value of $\zeta$ When There Is an Overshoot in the Step Response

If the step response has an overshoot, the parameters needed to estimate $\zeta$ are the two following time ratios:

$$\begin{array}{cc}\hfill {\tau }_1& ={\displaystyle \frac{t_{0.7}}{t_p}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\tau }_2& ={\displaystyle \frac{t_{0.8}-t_{0.5}}{t_{0.5}-t_{0.2}}}.\hfill \end{array}$$

(7)

$\zeta$ can be estimated from them by the following rule:

$$\begin{array}{cc}\hfill {\zeta }_p(\ln {\tau }_1,{\tau }_2)=& p_{00}+p_{10}\ln {\tau }_1+p_{01}{\tau }_2+p_{20}{\left(\ln {\tau }_1\right)}^2\hfill \\ & + p_{11}\ln {\tau }_1{\tau }_2+p_{02}{\tau }_2^2+p_{30}{\left(\ln {\tau }_1\right)}^3\hfill \\ & + p_{21}{\left(\ln {\tau }_1\right)}^2{\tau }_2+p_{12}\ln {\tau }_1{\tau }_2^2+p_{03}{\tau }_2^3\hfill \\ & + p_{40}{\left(\ln {\tau }_1\right)}^4+p_{31}{\left(\ln {\tau }_1\right)}^3{\tau }_2\hfill \\ & + p_{22}{\left(\ln {\tau }_1\right)}^2{\tau }_2^2+p_{13}\ln {\tau }_1{\tau }_2^3+p_{04}{\tau }_2^4\hfill \end{array}$$

(8)

The values of the coefficients are given in

Table 1. Estimated values for the polynomial regression coefficients ($p_{ij}$ and $q_{ij}$).

Coefficient	Value	Coefficient	Value
$p_{00}$	$-742$	$q_{00}$	$-11.96$
$p_{10}$	$-1661$	$q_{10}$	$-42.86$
$p_{01}$	1144	$q_{01}$	$20.31$
$p_{20}$	$1.06\times {10}^4$	$q_{20}$	$-16.48$
$p_{11}$	$2.043\times {10}^4$	$q_{11}$	$53.79$
$p_{02}$	6490	$q_{02}$	$-11.66$
$p_{30}$	$2.633\times {10}^4$	$q_{30}$	$22.84$
$p_{21}$	$4.12\times {10}^4$	$q_{21}$	$16.32$
$p_{12}$	$1.321\times {10}^4$	$q_{12}$	$-17.58$
$p_{03}$	$-637.6$	$q_{03}$	$2.585$
$p_{40}$	$1.423\times {10}^4$	$q_{40}$	$17.65$
$p_{31}$	$1.925\times {10}^4$	$q_{31}$	$-3.702$
$p_{22}$	2890	$q_{22}$	$-3.421$
$p_{13}$	$-2873$	$q_{13}$	$1.942$
$p_{04}$	$-457.1$	$q_{04}$	$-0.1983$

.

4.2. Finding the Value of $\zeta$ When There Is No Overshoot in the Step Response

If the step response does not have an overshoot, two other time ratios must be taken into account to estimate $\zeta$:

$$\begin{array}{c}\hfill {\tau }_3={\displaystyle \frac{t_{0.1}}{t_{0.5}}}\end{array}$$

(9)

$$\begin{array}{c}\hfill {\tau }_4={\displaystyle \frac{t_{0.8}}{t_{0.2}}}.\end{array}$$

(10)

$\zeta$ will then be given by the following rule:

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{n}}(\ln {\tau }_3,{\tau }_4)=& q_{00}+q_{10}\ln {\tau }_3+q_{01}{\tau }_4+q_{20}{\left(\ln {\tau }_3\right)}^2\hfill \\ & + q_{11}\ln {\tau }_3{\tau }_4+q_{02}{\tau }_4^2+q_{30}{\left(\ln {\tau }_3\right)}^3\hfill \\ & + q_{21}{\left(\ln {\tau }_3\right)}^2{\tau }_4+q_{12}\ln {\tau }_3{\tau }_4^2+q_{03}{\tau }_4^3\hfill \\ & + q_{40}{\left(\ln {\tau }_3\right)}^4+q_{31}{\left(\ln {\tau }_3\right)}^3{\tau }_4+q_{22}{\left(\ln {\tau }_3\right)}^2{\tau }_4^2\hfill \\ & + q_{13}\ln {\tau }_3{\tau }_4^3+q_{04}{\tau }_4^4\hfill \end{array}$$

(11)

The values of the coefficients are also given in Table 1.

4.3. Finding the Value of $\nu$ When $\zeta \ge 2$

If the value found for $\zeta$ is 2, or greater than 2, the value of $\nu$ can be estimated from time ratio ${\tau }_3$, previously defined in (9), and from time ratio

$${\tau }_5={\displaystyle \frac{t_{0.5}}{t_{0.9}}}$$

(12)

by the following rule:

$$\begin{array}{cc}\hfill {\nu }_1({\tau }_5,{\tau }_3)=& r_{00}+r_{10}{\tau }_5+r_{01}{\tau }_3\hfill \\ & + r_{20}{\tau }_5^2+r_{11}{\tau }_5{\tau }_3+r_{02}{\tau }_3^2\hfill \end{array}$$

(13)

The values of the coefficients are given in

Table 2. Estimated values for the polynomial regression coefficients ($r_{ij}$ and $s_{ij}$).

Coefficient	Value	Coefficient	Value
$r_{00}$	$0.9343$	$s_{00}$	$0.3404$
$r_{10}$	$1.273$	$s_{10}$	$1.623$
$r_{01}$	$-3.411$	$s_{01}$	$0.2744$
$r_{20}$	$0.3277$	$s_{20}$	$-7.181$
$r_{11}$	$1.806$	$s_{11}$	$1.577$
$r_{02}$	$4.141$	$s_{02}$	$-0.1397$
		$s_{30}$	$8.634$
		$s_{21}$	$-0.9272$
		$s_{12}$	$-0.1703$

.

4.4. Finding the Value of $\nu$ When $\zeta <2$

If the value found for $\zeta$ is less than 2, then $\nu$ can be estimated from time ${\tau }_5$, given above in (12), and from the estimated value of $\zeta$ itself, with the following rule:

$$\begin{array}{cc}\hfill {\nu }_2({\tau }_5,\zeta )=& s_{00}+s_{10}{\tau }_5+s_{01}\zeta +s_{20}{\tau }_5^2+s_{11}{\tau }_5\zeta \hfill \\ & + s_{02}{\zeta }^2+s_{30}{\tau }_5^3+s_{21}{\tau }_5^2\zeta +s_{12}{\tau }_5{\zeta }^2\hfill \end{array}$$

(14)

The values of the coefficients are given once more in Table 2.

4.5. Finding the Value of ${\omega }_n$

Once $\nu$ and $\zeta$ are identified, the natural frequency ${\omega }_n$ can be determined. The effect of changing ${\omega }_n$ is that of a scale factor in time. This is exemplified in Figure 3. Notice that the value of the overshoot and the geometric proportions of the different step responses are always the same.

There are two different possible ways of estimating ${\omega }_n$.

In one of them, the step response of (2), with ${\omega }_n=1$ rad/s and with the identified values of $\nu$ and $\zeta$, is found numerically. Choose any amplitude k, then find the time $t_{k{\omega }_n=1}$ that the step response takes to reach k and also the corresponding time $t_k$ in the step response that is being identified. Then,

$${\omega }_n={\displaystyle \frac{t_{k{\omega }_n=1}}{t_k}}$$

(15)

This is an exact relation, and thus any errors in the determination of ${\omega }_n$ are only caused by errors in the values of $\nu$ and $\zeta$ which were previously determined.

Another way is the use of the following rule:

$${\omega }_n={\displaystyle \frac{a}{t_{0.5}}},$$

(16)

where a is given by

$$\begin{array}{cc}\hfill a(\nu ,\zeta )=& \exp (u_{00}+u_{10}\nu +u_{01}\zeta \hfill \\ \hfill & + u_{20}{\nu }^2+u_{11}\nu \zeta +u_{02}{\zeta }^2)\hfill \end{array}$$

(17)

The coefficients in this rule are given in

Table 3. Estimated numerical values for the ${\omega }_n$ polynomial coefficients ($u_{ij}$).

Coefficient	Value	Coefficient	Value
$u_{00}$	$1.498$	$u_{20}$	$0.6487$
$u_{10}$	$-1.816$	$u_{11}$	$-0.165$
$u_{01}$	$0.6069$	$u_{02}$	$-0.0247$

.

5. Performance Evaluation

The identification rules in the previous section provide parameters for a model which is always only an approximation. The performance of these identification rules must be ascertained [34] first in an ideal case, when there is no measurement noise, then also when the step response used to identify the model is corrupted by noise.

5.1. Performance Without Noise

The rules were applied to the step responses of several systems given by (2) for different values of $\nu$ and $\zeta$ within the corresponding ranges of application, and with ${\omega }_n=1$ rad/s. In this way, models with identified parameters ${\nu }_i$, ${\zeta }_i$${\omega }_{ni}$ were obtained. The following results were obtained:

• Table 4. Absolute error $|\nu -{\nu }_i|$. Green: lower values of the error; yellow: higher values of the error.

  		$\mathit{\nu }$
  		0.7	0.8	0.9	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
  $\zeta$	5.0			0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.02	0.03	0.05	0.07
  	4.9			0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.02	0.03	0.05	0.07
  	4.8			0.01	0.01	0.01	0.01	0.00	0.01	0.01	0.02	0.03	0.05	0.07
  	4.7			0.01	0.01	0.01	0.01	0.00	0.00	0.01	0.02	0.03	0.05	0.07
  	4.6			0.01	0.01	0.01	0.01	0.00	0.00	0.01	0.02	0.03	0.04	0.06
  	4.5			0.01	0.01	0.01	0.01	0.00	0.00	0.01	0.01	0.03	0.04	0.06
  	4.4			0.01	0.01	0.01	0.01	0.00	0.00	0.00	0.01	0.02	0.04	0.06
  	4.3			0.01	0.01	0.01	0.01	0.00	0.00	0.00	0.01	0.02	0.04	0.06
  	4.2			0.01	0.01	0.01	0.01	0.00	0.00	0.00	0.01	0.02	0.04	0.06
  	4.1			0.00	0.01	0.01	0.01	0.00	0.00	0.00	0.01	0.02	0.03	0.05
  	4.0		0.06	0.00	0.01	0.01	0.01	0.00	0.00	0.00	0.01	0.02	0.03	0.05
  	3.9		0.06	0.00	0.02	0.01	0.01	0.00	0.00	0.00	0.00	0.01	0.03	0.05
  	3.8		0.05	0.00	0.02	0.01	0.00	0.00	0.01	0.00	0.00	0.01	0.03	0.05
  	3.7		0.05	0.00	0.02	0.01	0.00	0.00	0.01	0.01	0.00	0.01	0.03	0.04
  	3.6		0.05	0.00	0.02	0.01	0.00	0.00	0.01	0.01	0.00	0.01	0.02	0.04
  	3.5		0.05	0.01	0.02	0.01	0.00	0.01	0.01	0.01	0.01	0.00	0.02	0.04
  	3.4		0.04	0.01	0.02	0.01	0.00	0.01	0.01	0.01	0.01	0.00	0.02	0.04
  	3.3		0.04	0.01	0.02	0.02	0.00	0.01	0.02	0.02	0.01	0.00	0.01	0.03
  	3.2		0.04	0.01	0.02	0.02	0.00	0.01	0.02	0.02	0.02	0.01	0.01	0.03
  	3.1		0.03	0.02	0.03	0.02	0.00	0.01	0.02	0.02	0.02	0.01	0.01	0.02
  	3.0		0.03	0.02	0.03	0.02	0.00	0.02	0.02	0.03	0.02	0.01	0.00	0.02
  	2.9		0.03	0.02	0.03	0.02	0.00	0.02	0.03	0.03	0.03	0.02	0.00	0.02
  	2.8		0.02	0.03	0.03	0.02	0.00	0.02	0.03	0.03	0.03	0.02	0.01	0.01
  	2.7		0.02	0.03	0.03	0.02	0.01	0.02	0.04	0.04	0.04	0.03	0.01	0.00
  	2.6		0.01	0.03	0.04	0.02	0.01	0.03	0.04	0.05	0.04	0.03	0.02	0.00
  	2.5		0.00	0.04	0.04	0.01	0.01	0.03	0.05	0.05	0.05	0.04	0.03	0.01
  	2.4		0.00	0.04	0.04	0.01	0.01	0.04	0.05	0.06	0.06	0.05	0.03	0.01
  	2.3		0.01	0.05	0.04	0.01	0.02	0.04	0.06	0.06	0.06	0.06	0.04	0.02
  	2.2	0.05	0.03	0.06	0.04	0.01	0.02	0.05	0.07	0.07	0.07	0.06	0.05	0.03
  	2.1	0.07	0.01	0.06	0.05	0.01	0.03	0.06	0.07	0.08	0.08	0.07	0.06	0.04
  	2.0	0.05	0.00	0.00	0.05	0.01	0.02	0.03	0.08	0.09	0.09	0.02	0.07	0.05
  	1.9	0.04	0.01	0.02	0.05	0.01	0.02	0.02	0.02	0.10	0.10	0.09	0.08	0.04
  	1.8	0.02	0.01	0.02	0.01	0.01	0.02	0.03	0.00	0.00	0.00	0.00	0.01	0.03
  	1.7	0.01	0.02	0.03	0.02	0.02	0.02	0.01	0.00	0.01	0.01	0.00	0.00	0.04
  	1.6	0.06	0.01	0.03	0.02	0.05	0.03	0.03	0.00	0.01	0.02	0.02	0.03
  	1.5	0.04	0.01	0.03	0.01	0.08	0.03	0.05	0.05	0.01	0.03	0.02	0.01
  	1.4	0.02	0.00	0.02	0.01	0.12	0.04	0.07	0.06	0.08	0.03	0.01	0.09
  	1.3	0.00	0.01	0.01	0.03	0.16	0.05	0.08	0.08	0.10	0.02	0.04	0.07
  	1.2	0.01	0.02	0.01	0.06	0.19	0.06	0.06	0.10	0.12	0.10	0.09	0.17
  	1.1	0.01	0.03	0.02	0.09	0.22	0.04	0.08	0.13	0.14	0.10	0.13	0.17
  	1.0	0.00	0.04	0.03	0.12	0.11	0.02	0.08	0.11	0.11	0.10	0.11
  	0.9	0.00	0.02	0.03	0.13	0.07	0.00	0.11	0.14	0.12	0.10	0.15
  	0.8	0.01	0.01	0.03	0.13	0.02	0.02	0.14	0.13	0.07	0.08	0.16
  	0.7	0.00	0.05	0.03	0.10	0.01	0.11	0.15	0.06	0.01	0.06
  	0.6	0.08	0.10	0.00	0.03	0.06	0.17	0.07	0.03	0.04	0.06
  	0.5	0.22	0.14	0.22	0.02	0.17	0.10	0.07	0.13	0.08
  	0.4	0.32	0.17	0.10	0.02	0.01	0.10	0.18	0.14
  	0.3	0.31	0.23	0.14	0.13	0.19	0.25	0.21
  	0.2	0.42	0.33	0.31	0.33	0.35	0.34
  	0.1	0.61	0.54	0.69	0.63	0.67

  shows the absolute error when identifying the order $|y-y_i|$;
• Table 5. Absolute error $|\zeta -{\zeta }_i|$. Green: lower values of the error; yellow: higher values of the error.

  		$\mathit{\nu }$
  		0.7	0.8	0.9	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
  $\zeta$	5.0			0.25	0.37	0.77	0.19	0.62	0.57	0.67	0.57	0.75	0.91	0.69
  	4.9			0.21	0.30	0.70	0.08	0.27	0.30	0.31	0.45	0.85	0.62	0.52
  	4.8			0.17	0.24	0.63	0.03	0.25	0.41	0.37	0.32	0.40	0.90	0.99
  	4.7			0.13	0.18	0.56	0.17	0.22	0.12	0.42	0.67	0.48	0.59	0.82
  	4.6			0.09	0.13	0.50	0.04	0.19	0.21	0.47	0.52	0.54	0.85	0.62
  	4.5			0.06	0.07	0.44	0.10	0.14	0.30	0.06	0.36	0.59	0.52	0.41
  	4.4			0.03	0.02	0.38	0.26	0.09	0.03	0.08	0.19	0.63	0.16	0.19
  	4.3			0.00	0.03	0.32	0.08	0.03	0.04	0.10	0.49	0.13	0.39	0.61
  	4.2			0.03	0.08	0.26	0.25	0.05	0.10	0.10	0.30	0.15	0.61	0.37
  	4.1			0.05	0.12	0.21	0.09	0.14	0.14	0.09	0.09	0.16	0.22	0.10
  	4.0		0.15	0.06	0.17	0.16	0.29	0.14	0.23	0.07	0.14	0.16	0.41	0.49
  	3.9		0.14	0.08	0.20	0.11	0.15	0.03	0.21	0.04	0.14	0.14	0.01	0.20
  	3.8		0.13	0.09	0.24	0.06	0.37	0.09	0.19	0.01	0.11	0.11	0.17	0.11
  	3.7		0.11	0.10	0.27	0.02	0.25	0.22	0.19	0.06	0.13	0.07	0.31	0.24
  	3.6		0.09	0.10	0.30	0.02	0.15	0.01	0.21	0.13	0.15	0.01	0.15	0.10
  	3.5		0.07	0.10	0.33	0.05	0.41	0.15	0.23	0.22	0.07	0.06	0.02	0.22
  	3.4		0.05	0.10	0.35	0.08	0.33	0.32	0.28	0.32	0.24	0.15	0.09	0.15
  	3.3		0.02	0.09	0.37	0.11	0.26	0.13	0.33	0.44	0.05	0.26	0.18	0.13
  	3.2		0.01	0.09	0.38	0.13	0.21	0.34	0.40	0.10	0.39	0.38	0.35	0.27
  	3.1		0.04	0.08	0.40	0.14	0.18	0.17	0.49	0.24	0.23	0.04	0.28	0.02
  	3.0		0.07	0.06	0.40	0.15	0.17	0.04	0.17	0.40	0.09	0.10	0.24	0.47
  	2.9		0.09	0.05	0.40	0.16	0.17	0.28	0.29	0.12	0.48	0.27	0.21	0.24
  	2.8		0.12	0.04	0.40	0.16	0.18	0.17	0.42	0.31	0.37	0.46	0.20	0.05
  	2.7		0.14	0.03	0.40	0.15	0.22	0.09	0.17	0.07	0.28	0.10	0.20	0.11
  	2.6		0.17	0.01	0.39	0.14	0.02	0.03	0.33	0.29	0.21	0.31	0.23	0.38
  	2.5		0.18	0.00	0.37	0.12	0.05	0.02	0.12	0.09	0.16	0.02	0.27	0.24
  	2.4		0.20	0.01	0.35	0.10	0.14	0.04	0.31	0.33	0.13	0.25	0.33	0.13
  	2.3		0.21	0.02	0.33	0.07	0.01	0.04	0.14	0.17	0.11	0.00	0.41	0.03
  	2.2	0.33	0.22	0.02	0.30	0.03	0.12	0.02	0.01	0.03	0.11	0.26	0.05	0.04
  	2.1	0.12	0.22	0.03	0.27	0.01	0.18	0.03	0.24	0.31	0.13	0.04	0.05	0.09
  	2.0	0.06	0.23	0.03	0.23	0.06	0.21	0.19	0.13	0.20	0.16	0.14	0.16	0.12
  	1.9	0.20	0.24	0.04	0.19	0.12	0.21	0.11	0.06	0.11	0.20	0.14	0.29	0.14
  	1.8	0.30	0.24	0.04	0.14	0.18	0.18	0.24	0.00	0.04	0.14	0.02	0.08	0.15
  	1.7	0.35	0.25	0.05	0.08	0.24	0.14	0.13	0.04	0.01	0.08	0.15	0.06	0.20
  	1.6	0.36	0.26	0.07	0.03	0.30	0.18	0.21	0.07	0.06	0.02	0.25	0.24
  	1.5	0.33	0.27	0.08	0.03	0.35	0.13	0.25	0.29	0.10	0.27	0.01	0.11
  	1.4	0.27	0.28	0.09	0.09	0.40	0.15	0.26	0.29	0.35	0.21	0.12	0.34
  	1.3	0.19	0.29	0.10	0.14	0.43	0.16	0.26	0.29	0.36	0.16	0.21	0.25
  	1.2	0.12	0.28	0.10	0.18	0.45	0.13	0.19	0.31	0.37	0.32	0.28	0.42
  	1.1	0.06	0.25	0.09	0.20	0.44	0.07	0.20	0.32	0.37	0.28	0.32	0.37
  	1.0	0.01	0.19	0.06	0.21	0.55	0.01	0.19	0.27	0.27	0.24	0.23
  	0.9	0.03	0.09	0.02	0.21	0.26	0.02	0.22	0.29	0.24	0.20	0.26
  	0.8	0.06	0.07	0.01	0.18	0.08	0.02	0.26	0.24	0.14	0.13	0.23
  	0.7	0.03	0.21	0.02	0.13	0.04	0.19	0.25	0.11	0.01	0.06
  	0.6	0.39	0.28	0.04	0.02	0.08	0.25	0.11	0.05	0.10	0.03
  	0.5	0.59	0.28	0.30	0.03	0.25	0.14	0.12	0.23	0.18
  	0.4	0.61	0.27	0.14	0.04	0.00	0.15	0.30	0.26
  	0.3	0.42	0.32	0.17	0.16	0.27	0.39	0.35
  	0.2	0.59	0.44	0.42	0.45	0.52	0.53
  	0.1	0.93	0.78	1.22	1.06	1.32

  shows the absolute error $|\zeta -{\zeta }_i|$;
• Table 6. Absolute error $|{\omega }_n-{\omega }_{ni}|$, in rad/s, for Method (15). Green: lower values of the error; yellow: higher values of the error.

  		$\mathit{\nu }$
  		0.70	0.80	0.90	1.00	1.10	1.20	1.30	1.40	1.50	1.60	1.70	1.80	1.90
  $\zeta$	5.00			0.09	0.06	0.13	0.20	0.09	0.08	0.08	0.06	0.08	0.09	0.06
  	4.90			0.07	0.05	0.12	0.09	0.04	0.04	0.03	0.05	0.09	0.06	0.04
  	4.80			0.06	0.04	0.11	0.06	0.04	0.06	0.05	0.03	0.04	0.09	0.10
  	4.70			0.05	0.02	0.09	0.10	0.03	0.01	0.05	0.08	0.05	0.06	0.08
  	4.60			0.04	0.01	0.08	0.19	0.03	0.03	0.06	0.06	0.06	0.09	0.06
  	4.50			0.03	0.00	0.07	0.32	0.02	0.04	0.00	0.04	0.07	0.05	0.03
  	4.40			0.03	0.02	0.06	0.45	0.01	0.01	0.01	0.02	0.08	0.01	0.01
  	4.30			0.02	0.03	0.05	0.51	0.00	0.01	0.01	0.06	0.01	0.04	0.06
  	4.20			0.01	0.04	0.04	0.45	0.01	0.02	0.01	0.04	0.01	0.07	0.03
  	4.10			0.00	0.05	0.03	0.16	0.02	0.03	0.01	0.01	0.02	0.02	0.00
  	4.00		0.10	0.01	0.07	0.02	0.06	0.03	0.04	0.01	0.02	0.02	0.05	0.05
  	3.90		0.10	0.02	0.08	0.01	0.04	0.01	0.03	0.01	0.02	0.02	0.01	0.02
  	3.80		0.09	0.03	0.09	0.00	0.08	0.02	0.03	0.00	0.02	0.01	0.02	0.02
  	3.70		0.09	0.03	0.10	0.01	0.06	0.04	0.03	0.01	0.02	0.01	0.04	0.03
  	3.60		0.09	0.04	0.12	0.02	0.04	0.00	0.03	0.02	0.02	0.00	0.03	0.02
  	3.50		0.09	0.04	0.13	0.03	0.10	0.03	0.04	0.03	0.01	0.01	0.01	0.02
  	3.40		0.09	0.05	0.14	0.04	0.08	0.06	0.05	0.05	0.04	0.02	0.01	0.03
  	3.30		0.09	0.05	0.15	0.05	0.06	0.02	0.06	0.07	0.01	0.04	0.02	0.02
  	3.20		0.09	0.06	0.16	0.05	0.05	0.07	0.07	0.01	0.06	0.06	0.05	0.04
  	3.10		0.09	0.06	0.17	0.06	0.05	0.03	0.09	0.04	0.04	0.01	0.04	0.01
  	3.00		0.09	0.06	0.18	0.06	0.04	0.00	0.03	0.07	0.01	0.01	0.04	0.07
  	2.90		0.09	0.06	0.19	0.07	0.04	0.06	0.05	0.02	0.08	0.04	0.03	0.04
  	2.80		0.09	0.06	0.19	0.07	0.05	0.03	0.08	0.05	0.06	0.08	0.03	0.01
  	2.70		0.09	0.07	0.20	0.07	0.06	0.01	0.02	0.01	0.05	0.01	0.03	0.02
  	2.60		0.09	0.07	0.20	0.06	0.01	0.01	0.06	0.05	0.03	0.05	0.04	0.06
  	2.50		0.09	0.07	0.20	0.06	0.01	0.02	0.01	0.01	0.02	0.00	0.05	0.04
  	2.40		0.09	0.07	0.20	0.05	0.04	0.03	0.06	0.06	0.02	0.04	0.06	0.02
  	2.30		0.09	0.07	0.20	0.04	0.01	0.03	0.02	0.03	0.02	0.00	0.07	0.01
  	2.20	0.11	0.07	0.07	0.19	0.02	0.05	0.02	0.01	0.00	0.02	0.05	0.01	0.01
  	2.10	0.11	0.11	0.07	0.18	0.00	0.05	0.01	0.05	0.06	0.02	0.01	0.01
  	2.00	0.06	0.14	0.02	0.16	0.02	0.07	0.05	0.02	0.04	0.03	0.03	0.03	0.03
  	1.90	0.06	0.16	0.04	0.14	0.05	0.07	0.03	0.02	0.02	0.04	0.03	0.06	0.03
  	1.80	0.18	0.17	0.05	0.06	0.07	0.06	0.08	0.00	0.01	0.04	0.01	0.02	0.03
  	1.70	0.28	0.18	0.06	0.03	0.10	0.05	0.04	0.01	0.00	0.02	0.04	0.01	0.05
  	1.60	0.56	0.19	0.06	0.00	0.12	0.07	0.07	0.02	0.02	0.01	0.07	0.06
  	1.50	0.45	0.20	0.07	0.02	0.15	0.05	0.09	0.10	0.03	0.08	0.00	0.03
  	1.40	0.32	0.20	0.07	0.05	0.17	0.06	0.10	0.10	0.11	0.06	0.03	0.10
  	1.30	0.21	0.20	0.07	0.07	0.19	0.06	0.10	0.11	0.12	0.05	0.06	0.07
  	1.20	0.12	0.19	0.06	0.09	0.21	0.06	0.08	0.12	0.13	0.11	0.09	0.14
  	1.10	0.05	0.17	0.05	0.10	0.21	0.03	0.08	0.13	0.14	0.10	0.11	0.12
  	1.00	0.01	0.13	0.03	0.11	0.26	0.00	0.08	0.11	0.10	0.09	0.08
  	0.90	0.02	0.06	0.01	0.12	0.13	0.01	0.10	0.13	0.10	0.08	0.10
  	0.80	0.05	0.05	0.01	0.11	0.04	0.01	0.12	0.11	0.06	0.05	0.10
  	0.70	0.02	0.15	0.01	0.08	0.02	0.10	0.12	0.05	0.01	0.03
  	0.60	0.33	0.19	0.02	0.01	0.05	0.14	0.06	0.02	0.04	0.02
  	0.50	0.47	0.19	0.20	0.02	0.15	0.08	0.06	0.11	0.08
  	0.40	0.47	0.20	0.10	0.02	0.00	0.09	0.16	0.13
  	0.30	0.29	0.24	0.12	0.11	0.17	0.22	0.19
  	0.20	0.48	0.34	0.30	0.31	0.33	0.31
  	0.10	0.75	0.60	0.83	0.69	0.77

  and

  Table 7. Absolute error $|{\omega }_n-{\omega }_{ni}|$, in rad/s, for Methods (16) and (17). Green: lower values of the error; yellow: higher values of the error.

  		$\mathit{\nu }$
  		0.70	0.80	0.90	1.00	1.10	1.20	1.30	1.40	1.50	1.60	1.70	1.80	1.90
  $\zeta$	5.00			0.15	0.08	0.11	0.17	0.06	0.05	0.06	0.05	0.06	0.07	0.04
  	4.90			0.14	0.07	0.10	0.06	0.00	0.01	0.01	0.03	0.08	0.04	0.02
  	4.80			0.13	0.06	0.09	0.03	0.00	0.03	0.02	0.02	0.02	0.07	0.07
  	4.70			0.12	0.04	0.08	0.07	0.00	0.02	0.03	0.06	0.03	0.04	0.05
  	4.60			0.12	0.03	0.07	0.17	0.01	0.00	0.04	0.04	0.04	0.07	0.03
  	4.50			0.11	0.02	0.06	0.30	0.01	0.02	0.02	0.02	0.05	0.03	0.01
  	4.40			0.10	0.01	0.05	0.43	0.02	0.04	0.01	0.00	0.06	0.01	0.01
  	4.30			0.09	0.00	0.04	0.47	0.03	0.02	0.01	0.05	0.01	0.02	0.03
  	4.20			0.08	0.02	0.03	0.41	0.05	0.01	0.01	0.02	0.00	0.05	0.01
  	4.10			0.07	0.03	0.02	0.14	0.06	0.00	0.01	0.01	0.00	0.00	0.02
  	4.00		0.21	0.07	0.04	0.01	0.10	0.01	0.07	0.01	0.04	0.00	0.03	0.02
  	3.90		0.20	0.06	0.05	0.00	0.07	0.02	0.06	0.01	0.00	0.00	0.03	0.01
  	3.80		0.20	0.05	0.07	0.01	0.12	0.05	0.06	0.02	0.04	0.01	0.01	0.05
  	3.70		0.19	0.05	0.08	0.02	0.09	0.07	0.06	0.03	0.00	0.01	0.02	0.01
  	3.60		0.19	0.04	0.09	0.03	0.07	0.02	0.06	0.04	0.04	0.02	0.05	0.05
  	3.50		0.19	0.03	0.10	0.04	0.13	0.06	0.06	0.05	0.00	0.03	0.03	0.01
  	3.40		0.19	0.03	0.11	0.05	0.11	0.09	0.07	0.07	0.05	0.04	0.02	0.06
  	3.30		0.19	0.02	0.12	0.06	0.09	0.05	0.08	0.09	0.02	0.06	0.00	0.02
  	3.20		0.18	0.02	0.13	0.07	0.08	0.10	0.10	0.03	0.08	0.08	0.08	0.08
  	3.10		0.18	0.01	0.14	0.07	0.07	0.06	0.12	0.06	0.05	0.01	0.07	0.05
  	3.00		0.18	0.01	0.15	0.08	0.07	0.02	0.05	0.09	0.02	0.03	0.07	0.11
  	2.90		0.17	0.00	0.16	0.08	0.07	0.08	0.07	0.03	0.10	0.06	0.06	0.08
  	2.80		0.17	0.00	0.17	0.08	0.07	0.05	0.10	0.07	0.08	0.10	0.06	0.06
  	2.70		0.16	0.01	0.18	0.09	0.08	0.03	0.04	0.01	0.06	0.03	0.06	0.03
  	2.60		0.15	0.02	0.18	0.09	0.02	0.01	0.08	0.06	0.05	0.07	0.07	0.11
  	2.50		0.14	0.03	0.19	0.08	0.04	0.00	0.03	0.02	0.04	0.02	0.08	0.09
  	2.40		0.13	0.04	0.19	0.08	0.07	0.01	0.07	0.07	0.03	0.06	0.09	0.08
  	2.30		0.11	0.06	0.20	0.07	0.02	0.01	0.03	0.03	0.03	0.01	0.11	0.06
  	2.20	0.18	0.08	0.07	0.20	0.07	0.01	0.00	0.00	0.00	0.03	0.07	0.02	0.05
  	2.10	0.06	0.08	0.09	0.20	0.06	0.00	0.01	0.06	0.07	0.03	0.03	0.05	0.05
  	2.00	0.10	0.08	0.03	0.20	0.05	0.01	0.01	0.03	0.05	0.04	0.02	0.08	0.01
  	1.90	0.01	0.07	0.04	0.20	0.03	0.00	0.01	0.04	0.03	0.05	0.05	0.11	0.01
  	1.80	0.09	0.05	0.05	0.14	0.02	0.01	0.02	0.02	0.02	0.03	0.01	0.01	0.01
  	1.70	0.18	0.03	0.06	0.12	0.02	0.03	0.01	0.01	0.01	0.01	0.03	0.04	0.01
  	1.60	0.35	0.00	0.08	0.12	0.02	0.03	0.01	0.01	0.00	0.00	0.05	0.04
  	1.50	0.36	0.04	0.11	0.12	0.03	0.05	0.01	0.05	0.01	0.06	0.01	0.00
  	1.40	0.36	0.08	0.14	0.14	0.05	0.06	0.00	0.04	0.08	0.05	0.02	0.08
  	1.30	0.37	0.13	0.18	0.16	0.08	0.07	0.00	0.04	0.08	0.04	0.05	0.06
  	1.20	0.39	0.20	0.23	0.19	0.11	0.09	0.03	0.04	0.08	0.08	0.08	0.12
  	1.10	0.44	0.27	0.28	0.22	0.14	0.11	0.03	0.04	0.08	0.07	0.09	0.11
  	1.00	0.50	0.35	0.34	0.26	0.36	0.14	0.04	0.02	0.05	0.06	0.07
  	0.90	0.56	0.45	0.40	0.30	0.28	0.15	0.04	0.02	0.04	0.05	0.08
  	0.80	0.62	0.55	0.44	0.32	0.24	0.14	0.04	0.01	0.01	0.03	0.08
  	0.70	0.72	0.63	0.47	0.33	0.24	0.12	0.05	0.03	0.03	0.01
  	0.60	0.92	0.64	0.47	0.34	0.22	0.13	0.07	0.07	0.07	0.00
  	0.50	0.90	0.63	0.53	0.34	0.23	0.14	0.13	0.14	0.10
  	0.40	0.83	0.61	0.46	0.33	0.23	0.19	0.20	0.15
  	0.30	1.12	0.60	0.45	0.35	0.29	0.27	0.22
  	0.20	0.77	0.61	0.49	0.42	0.38	0.34
  	0.10	0.88	0.71	0.85	0.71	0.81

  show the absolute error $|{\omega }_n-{\omega }_{ni}|$ for Method (15) and for Methods (16) and (17), respectively;
• Table 8. RMS between the step response used for identification and the step response of the identified model. Green: lower values of the error; yellow: higher values of the error.

  		$\mathit{\nu }$
  		0.7	0.8	0.9	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
  $\zeta$	5.0			0.02	0.01	0.03	0.00	0.02	0.02	0.03	0.03	0.05	0.11	0.16
  	4.9			0.02	0.01	0.02	0.00	0.01	0.01	0.01	0.02	0.07	0.07	0.14
  	4.8			0.02	0.01	0.02	0.01	0.01	0.02	0.02	0.02	0.03	0.11	0.24
  	4.7			0.02	0.00	0.02	0.00	0.01	0.00	0.02	0.04	0.03	0.07	0.20
  	4.6			0.01	0.00	0.02	0.00	0.01	0.01	0.02	0.03	0.04	0.11	0.16
  	4.5			0.01	0.00	0.01	0.01	0.01	0.01	0.00	0.02	0.05	0.07	0.13
  	4.4			0.01	0.01	0.01	0.01	0.00	0.00	0.00	0.01	0.06	0.04	0.14
  	4.3			0.01	0.01	0.01	0.01	0.00	0.00	0.00	0.03	0.02	0.05	0.18
  	4.2			0.00	0.01	0.01	0.01	0.00	0.01	0.01	0.02	0.01	0.09	0.13
  	4.1			0.00	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.04	0.14
  	4.0		0.04	0.00	0.02	0.00	0.01	0.01	0.01	0.00	0.01	0.01	0.06	0.16
  	3.9		0.04	0.00	0.02	0.00	0.01	0.00	0.01	0.00	0.01	0.01	0.04	0.12
  	3.8		0.04	0.00	0.02	0.01	0.02	0.00	0.01	0.00	0.01	0.01	0.03	0.17
  	3.7		0.04	0.01	0.02	0.01	0.01	0.01	0.01	0.00	0.01	0.01	0.05	0.11
  	3.6		0.04	0.01	0.03	0.01	0.01	0.00	0.01	0.01	0.01	0.01	0.06	0.16
  	3.5		0.03	0.01	0.03	0.01	0.02	0.01	0.01	0.01	0.01	0.01	0.03	0.11
  	3.4		0.03	0.01	0.03	0.01	0.02	0.01	0.01	0.02	0.02	0.02	0.02	0.17
  	3.3		0.03	0.01	0.03	0.01	0.01	0.00	0.01	0.02	0.01	0.03	0.04	0.09
  	3.2		0.03	0.01	0.03	0.01	0.01	0.01	0.02	0.01	0.03	0.05	0.09	0.22
  	3.1		0.03	0.02	0.03	0.01	0.01	0.01	0.02	0.01	0.02	0.01	0.08	0.10
  	3.0		0.03	0.02	0.04	0.01	0.01	0.00	0.01	0.02	0.01	0.01	0.06	0.31
  	2.9		0.02	0.02	0.04	0.01	0.01	0.01	0.01	0.01	0.04	0.03	0.06	0.20
  	2.8		0.02	0.02	0.04	0.01	0.01	0.01	0.02	0.02	0.03	0.06	0.05	0.07
  	2.7		0.02	0.02	0.04	0.01	0.01	0.01	0.01	0.02	0.02	0.02	0.06	0.10
  	2.6		0.02	0.02	0.04	0.01	0.00	0.01	0.02	0.02	0.02	0.05	0.07	0.33
  	2.5		0.02	0.02	0.04	0.01	0.00	0.01	0.01	0.02	0.02	0.04	0.09	0.24
  	2.4		0.01	0.03	0.04	0.01	0.01	0.02	0.02	0.02	0.03	0.05	0.11	0.14
  	2.3		0.01	0.03	0.04	0.01	0.01	0.02	0.02	0.02	0.03	0.06	0.15	0.12
  	2.2	0.01	0.01	0.03	0.04	0.01	0.01	0.02	0.02	0.03	0.04	0.06	0.14	0.28
  	2.1	0.04	0.01	0.03	0.04	0.00	0.01	0.02	0.02	0.03	0.04	0.07	0.13	0.55
  	2.0	0.03	0.02	0.00	0.03	0.00	0.01	0.01	0.03	0.03	0.05	0.03	0.16	0.24
  	1.9	0.01	0.02	0.01	0.03	0.01	0.01	0.01	0.01	0.04	0.06	0.10	0.23	0.26
  	1.8	0.02	0.03	0.01	0.01	0.01	0.01	0.02	0.00	0.00	0.02	0.01	0.04	0.33
  	1.7	0.03	0.03	0.02	0.00	0.01	0.01	0.01	0.00	0.01	0.02	0.05	0.05	0.53
  	1.6	0.08	0.03	0.02	0.01	0.02	0.01	0.01	0.01	0.01	0.01	0.08	0.18
  	1.5	0.06	0.03	0.02	0.01	0.02	0.01	0.02	0.03	0.02	0.05	0.02	0.11
  	1.4	0.04	0.03	0.01	0.01	0.03	0.01	0.02	0.03	0.05	0.05	0.05	0.30
  	1.3	0.02	0.03	0.01	0.01	0.03	0.01	0.02	0.03	0.05	0.04	0.09	0.31
  	1.2	0.01	0.02	0.01	0.01	0.04	0.01	0.02	0.04	0.06	0.08	0.14	0.45
  	1.1	0.00	0.02	0.01	0.02	0.04	0.01	0.02	0.04	0.07	0.08	0.19	0.57
  	1.0	0.00	0.01	0.01	0.02	0.04	0.00	0.02	0.04	0.06	0.09	0.19
  	0.9	0.00	0.01	0.01	0.02	0.02	0.00	0.03	0.05	0.06	0.10	0.30
  	0.8	0.01	0.01	0.01	0.02	0.01	0.00	0.04	0.05	0.05	0.09	0.49
  	0.7	0.00	0.02	0.01	0.01	0.01	0.03	0.04	0.03	0.01	0.09
  	0.6	0.03	0.02	0.01	0.00	0.01	0.04	0.03	0.02	0.08	0.23
  	0.5	0.04	0.02	0.03	0.00	0.04	0.03	0.04	0.12	0.28
  	0.4	0.05	0.03	0.02	0.01	0.00	0.04	0.13	0.27
  	0.3	0.12	0.04	0.02	0.03	0.07	0.15	0.28
  	0.2	0.06	0.06	0.07	0.10	0.18	0.35
  	0.1	0.11	0.12	0.21	0.27	0.52

  shows the root mean square error (RMS) between the step response used for identification and the step response of the identified model;
• Table 9. Error evaluation metrics for the proposed identification models.

  	Without Noise		With Noise
  	Mean	Max.	Mean	Max.
  Absolute error $|\nu -{\nu }_i|$	$0.0460$	$0.6886$	$0.0657$	$0.8462$
  Absolute error $|\zeta -{\zeta }_i|$	$0.2185$	$1.3183$	$1.2004$	$32.6483$
  RMS between step responses	$0.0459$	$0.5733$	$0.1090$	$0.8265$

  sums up these results, giving the mean and maximum values.

In these tables, blue cells correspond either to unstable systems, or to systems with slow responses that did not reach in 60 s the values required to apply the identification rules.

Furthermore, to verify that the results are good not only for the combinations of values of $\nu$ and $\zeta$ used to establish the identification rules, but also for other values in the range where it is stated that they can be used, the results in Table 8 were reproduced for other pairs of $\nu ,\zeta$, as seen in

Table 10. RMS between the step response used for identification and the step response of the identified model (cf. Table 8). Green: lower values of the error; yellow: higher values of the error.

		$\mathit{\nu }$
		0.65	0.75	0.85	0.95	1.05	1.15	1.25	1.35	1.45	1.55	1.65	1.75	1.85	1.95
$\zeta$	4.95			0.04	0.01	0.01	0.04	0.13	0.02	0.02	0.02	0.03	0.06	0.12	0.28
	4.85			0.03	0.01	0.01	0.04	0.09	0.01	0.02	0.03	0.03	0.08	0.09	0.27
	4.75			0.03	0.01	0.01	0.03	0.01	0.01	0.02	0.02	0.03	0.04	0.16	0.25
	4.65			0.03	0.00	0.01	0.03	0.01	0.00	0.01	0.03	0.03	0.06	0.12	0.24
	4.55			0.03	0.00	0.00	0.03	0.01	0.01	0.00	0.02	0.03	0.08	0.08	0.24
	4.45			0.03	0.00	0.00	0.03	0.00	0.01	0.00	0.03	0.02	0.04	0.14	0.26
	4.35			0.02	0.00	0.00	0.03	0.00	0.00	0.01	0.01	0.02	0.05	0.09	0.32
	4.25			0.02	0.01	0.01	0.02	0.00	0.01	0.01	0.02	0.01	0.03	0.07	0.27
	4.15			0.02	0.01	0.01	0.02	0.00	0.00	0.00	0.00	0.01	0.02	0.09	0.23
	4.05			0.02	0.01	0.01	0.02	0.01	0.00	0.01	0.01	0.01	0.03	0.07	0.25
	3.95			0.02	0.01	0.01	0.02	0.01	0.00	0.00	0.01	0.02	0.04	0.09	0.32
	3.85			0.02	0.02	0.02	0.02	0.00	0.01	0.01	0.00	0.03	0.04	0.07	0.23
	3.75			0.01	0.02	0.02	0.01	0.00	0.00	0.00	0.01	0.01	0.03	0.08	0.23
	3.65			0.01	0.02	0.02	0.01	0.01	0.01	0.01	0.02	0.00	0.02	0.08	0.34
	3.55			0.01	0.02	0.02	0.11	0.01	0.00	0.00	0.02	0.02	0.01	0.05	0.22
	3.45			0.01	0.02	0.02	0.06	0.01	0.01	0.01	0.01	0.02	0.01	0.13	0.28
	3.35			0.01	0.03	0.02	0.02	0.01	0.01	0.01	0.01	0.00	0.01	0.07	0.47
	3.25			0.01	0.03	0.03	0.03	0.00	0.01	0.02	0.01	0.02	0.01	0.04	0.24
	3.15			0.01	0.03	0.03	0.02	0.01	0.02	0.01	0.01	0.01	0.01	0.15	0.47
	3.05			0.01	0.03	0.03	0.02	0.01	0.02	0.01	0.01	0.02	0.00	0.10	0.22
	2.95		0.06	0.01	0.03	0.03	0.02	0.00	0.01	0.03	0.01	0.05	0.01	0.05	0.54
	2.85		0.06	0.01	0.03	0.03	0.01	0.01	0.01	0.02	0.01	0.02	0.03	0.01	0.27
	2.75		0.05	0.01	0.04	0.03	0.01	0.01	0.01	0.02	0.02	0.02	0.04	0.06	0.71
	2.65		0.04	0.01	0.04	0.03	0.01	0.00	0.01	0.02	0.02	0.03	0.07	0.20
	2.55		0.03	0.01	0.04	0.03	0.01	0.01	0.01	0.02	0.03	0.03	0.05	0.19
	2.45		0.02	0.01	0.04	0.03	0.00	0.01	0.02	0.02	0.03	0.05	0.04	0.18
	2.35		0.01	0.01	0.04	0.03	0.02	0.01	0.02	0.02	0.03	0.04	0.06	0.19
	2.25		0.01	0.01	0.04	0.03	0.02	0.01	0.02	0.02	0.03	0.04	0.10	0.21
	2.15		0.01	0.02	0.04	0.02	0.01	0.01	0.02	0.03	0.04	0.06	0.09	0.25
	2.05		0.01	0.02	0.04	0.02	0.01	0.01	0.03	0.03	0.04	0.02	0.10	0.33
	1.95		0.02	0.03	0.04	0.02	0.01	0.01	0.01	0.03	0.04	0.02	0.02	0.12
	1.85		0.03	0.03	0.01	0.00	0.01	0.01	0.00	0.01	0.00	0.03	0.07	0.09
	1.75		0.03	0.03	0.01	0.00	0.01	0.01	0.01	0.01	0.03	0.04	0.04	0.05
	1.65	0.01	0.01	0.03	0.01	0.01	0.00	0.02	0.02	0.02	0.01	0.05	0.05	0.08
	1.55	0.03	0.00	0.03	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.06	0.06	0.46
	1.45	0.04	0.01	0.02	0.01	0.01	0.01	0.02	0.01	0.03	0.05	0.07	0.15	0.49
	1.35	0.04	0.02	0.02	0.01	0.02	0.01	0.02	0.02	0.03	0.03	0.08	0.09	0.55
	1.25	0.03	0.02	0.02	0.01	0.02	0.01	0.02	0.02	0.04	0.06	0.09	0.20
	1.15	0.01	0.02	0.01	0.01	0.03	0.01	0.01	0.03	0.04	0.08	0.11	0.28
	1.05	0.01	0.02	0.01	0.01	0.03	0.01	0.01	0.03	0.05	0.07	0.13	0.33
	0.95	0.03	0.02	0.00	0.01	0.03	0.01	0.01	0.04	0.05	0.06	0.15	0.50
	0.85	0.03	0.01	0.01	0.01	0.03	0.01	0.02	0.04	0.05	0.06	0.17
	0.75	0.02	0.01	0.01	0.01	0.01	0.00	0.03	0.04	0.03	0.04	0.24
	0.65	0.01	0.02	0.01	0.01	0.00	0.01	0.03	0.02	0.01	0.06	0.58
	0.55	0.04	0.03	0.01	0.02	0.01	0.03	0.01	0.03	0.09	0.15
	0.45	0.06	0.04	0.05	0.01	0.02	0.02	0.03	0.13	0.27
	0.35	0.07	0.04	0.03	0.02	0.01	0.05	0.14	0.27
	0.25	0.04	0.04	0.03	0.04	0.09	0.16	0.30
	0.15	0.07	0.07	0.08	0.14	0.20	0.39
	0.05	0.17	0.23	0.29	0.37	0.66

. The results are as good as the original ones, and the RMS values are only slightly higher for values of $\zeta$ close to 0 and values of $\nu$ close to 2.

From these tables, it can be seen that errors in $\nu$ are almost always close to zero, showing identification reliability. The identification of $\zeta$ does not have such a good performance; there are some clusters of higher absolute errors, and a maximum error of around $1.3$. Thus, discussion of the errors in ${\omega }_n$ is better left to the next subsection.

5.2. Performance in the Presence of Measurement Noise

To evaluate the identification method’s robustness when facing noisy data, the step responses were treated as follows:

• Firstly, they were corrupted with additive white Gaussian noise of variance ${\sigma }^2=0.01$. This is a typical magnitude encountered in experimental data. Since the final value of the step response was normalised to 1, the corresponding signal to noise ratio is $20{\log }_{10}{\displaystyle \frac{0.01}{1}}=40$ dB.
• Secondly, they were filtered using a centred moving average filter of order 10. This means that the filter used a window covering 11 samples in all: the last 5 samples, the current sample, and the next 5.

Table 9 compares the resulting performance indexes with those of the case in which there is no noise. It can be seen that the identification method remains quite accurate when estimating the fractional order $\nu$. There are, however, higher errors in the estimation of $\zeta$. This can be attributed to the increased difficulty in accurately extracting the required time features from the noisy signal. Since the rules rely on regression polynomials of up to order 4, slight variations are amplified, resulting in substantial deviations in the estimated value ${\zeta }_i$.

However, an interesting phenomenon occurs which minimises the importance of such errors. Although noise may lead to a poor estimate of $\zeta$, the subsequently identified natural frequency ${\omega }_n$ compensates for this. Because the natural frequency ${\omega }_n$ scales the time axis, when its value is found to adjust the overall time stretching of the guessed response, the geometric errors induced by errors in $\nu$ and $\zeta$ are effectively absorbed and neutralised, reducing the problems caused by noise in the measured step response and making the errors in ${\omega }_n$ less relevant.

An example of this is given in Figure 4 for the particular case where the plant has the true parameters $\nu =1.4$, $\zeta =2.9$, and ${\omega }_n=1.0$ rad/s. Corrupting its step response with noise, filtering it, and applying the identification rules, the estimated model has a reasonably estimated ${\nu }_i=1.42$ and a significantly mismatched ${\zeta }_i=4.31$. If the model were to use this faulty pair of parameters together with the true ${\omega }_n$, its step response, as shown in Figure 4, would be far from the one used for identification, with a slower rise time.

Nevertheless, if the natural frequency ${\omega }_n$ is computed using the second method from Section 4.5, the resulting value, together with the ${\nu }_i$ and ${\zeta }_i$ previously found, is $1.31$ rad/s. This is far from the original value, but the resulting step response is much closer to that of the original system. This can be seen from

Table 11. RMS and ITAE for the two cases in Figure 4.

Figure 4, Top		Figure 4, Bottom
RMS = 0.07	ITAE = 41.6	RMS = 0.01	ITAE = 15.8

, which compares the two cases using the RMS and the integral of the time-weighted absolute error (ITAE).

5.3. Comparison with an ARX Model

The performance of this identification method can be compared with that of an autoregressive model with an exogenous input (ARX model). This is a discrete-time model with an output that depends on past values of the output and on an input which can be manipulated. For the purpose of the present comparison, the sampling time from Section 3 was kept, and a model with two poles and one zero was chosen. This is because an integer second-order system, discretised with a zero-order hold, becomes a discrete-time transfer function with two poles and one zero; thus, a system described by a fractional-order transfer function with two poles cannot be expected to be reasonably approximated by a discrete-time transfer function with fewer parameters. On the other hand, since the final value of the transfer function must be 1 (as mentioned above in Section 2), this corresponds to three independent parameters that have to identified in

$$\begin{array}{c}\hfill G\left(z^{-1}\right)={\displaystyle \frac{b_0+b_1{z}^{-1}}{1+a_1{z}^{-1}+a_2{z}^{-2}}},\end{array}$$

(18)

which has to verify $b_0+b_1=1+a_1+a_2$. Three is the same number of parameters to be identified in (3).

The RMS error was computed for all the cases in Table 8 using the step response of a model given by (18), found using recursive least squares, over the the 60 s used in Section 3. The results are given in

Table 12. RMS between the step response used for identification and the step response of the identified ARX model given by (18). Green: lower values of the error.

		$\mathit{\nu }$
		0.7	0.8	0.9	1	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
$\zeta$	5.0	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.9	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.8	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.7	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.6	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.5	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.4	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.3	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.2	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.1	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	4.0	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	3.9	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	3.8	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	3.7	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	3.6	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	3.5	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	3.4	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	3.3	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	3.2	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	3.1	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	3.0	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.9	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.8	0.02	0.02	0.02	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.7	0.02	0.02	0.02	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.6	0.02	0.02	0.02	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.5	0.02	0.02	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.4	0.02	0.02	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.3	0.02	0.02	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.2	0.02	0.02	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.1	0.02	0.02	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.04	0.04	0.04
	2.0	0.02	0.02	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04	0.04
	1.9	0.02	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04	0.04
	1.8	0.02	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04	0.04
	1.7	0.02	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04	0.04
	1.6	0.02	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	1.5	0.02	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	1.4	0.02	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	1.3	0.01	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	1.2	0.01	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	1.1	0.01	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	1.0	0.01	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04
	0.9	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.03	0.04
	0.8	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04
	0.7	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04
	0.6	0.01	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.04
	0.5	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.03	0.03
	0.4	0.01	0.02	0.02	0.02	0.02	0.03	0.03	0.03
	0.3	0.02	0.02	0.02	0.02	0.03	0.03	0.03
	0.2	0.02	0.02	0.02	0.03	0.03	0.03
	0.1	0.02	0.02	0.03	0.03	0.03

. It can be seen that the discrete-time ARX model achieves better results close to the limits of the ranges where the rules of Section 4 can be applied, in particular for values of $\zeta$ close to 0, for values of $\nu$ close to 2, and for combinations of values of $\zeta$ and $\nu$ close to instability. On the other hand, the rules of Section 4 achieve a lower RMS for the values further from the limits of their range of application.

6. Conclusions

This paper proposed numerical time-domain identification rules for second-species non-commensurate fractional-order systems with exponents $\nu$ and $\nu +1$ based upon some points of the step response. To accommodate the higher complexity of non-commensurate systems, the resulting methodology is more complex than existing methods for first-species fractional-order transfer functions [8,9] or for second-species fractional-order transfer functions with two equal poles in [10,11]. Still, the identification method is far simpler than fitting parameters with an optimisation method.

Even when applied to noisy data, results are sufficiently accurate to result in a model with a step response close to the one being identified. The fractional-order $\nu$ is consistently correctly identified, while coefficient $\zeta$ may exhibit an estimation error due to the numerical sensitivity of the polynomials in the rules, which cannot be neglected. Despite this, the natural frequency ${\omega }_n$ acts as compensation, leading to good results for practical purposes.

Future work includes unifying identification rules for disparate fractional-order systems into a single coherent procedure [9,10,35], and combining these identification rules with established control design rules (such as integer or fractional PID controllers) to develop auto-tuned controllers.