Novel Fractional Swarming with Key Term Separation for Input Nonlinear Control Autoregressive Systems

Abstract

In recent decades, fractional order calculus has become an important mathematical tool for effectively solving complex problems through better modeling with the introduction of fractional differential/integral operators; fractional order swarming heuristics are also introduced and applied for better performance in different optimization tasks. This study investigates the nonlinear system identification problem of the input nonlinear control autoregressive (IN-CAR) model through the novel implementation of fractional order particle swarm optimization (FO-PSO) heuristics; further, the key term separation technique (KTST) is introduced in the FO-PSO to solve the over-parameterization issue involved in the parameter estimation of the IN-CAR model. The proposed KTST-based FO-PSO, i.e., KTST-FOPSO accurately estimates the parameters of an unknown IN-CAR system with robust performance in cases of different noise scenarios. The performance of the KTST-FOPSO is investigated exhaustively for different fractional orders as well as in comparison with the standard counterpart. The results of statistical indices through Monte Carlo simulations endorse the reliability and stability of the KTST-FOPSO for IN-CAR identification.

1. Introduction

1.1. Literature Review

In recent decades, fractional order calculus has become an important mathematical tool for effectively solving complex engineering problems through better modeling with the concepts of fractional calculus operators [1,2,3,4,5,6,7,8,9]. Fractional calculus operators are also exploited to design novel recursive/adaptive algorithms as well as evolutionary/swarm computation heuristics for different optimization tasks involved in engineering and science applications. For example, fractional gradient descent/fractional least mean square algorithm was proposed for various applications including recommender systems [10], channel estimation [11], automatic identification system [12], power system optimization [13], economics [14], radar signal processing [15], system identification [16,17], Hammerstein output error identification [18], wireless sensor network [19], neural network optimization [20,21,22,23,24], chaotic time-series prediction [25,26], oscillator [27], vibration rejection [28], nonlinear ARMAX identification [29] and parameter estimation of input nonlinear control autoregressive (IN-CAR) systems [29,30].

The fractional evolutionary/swarm optimization heuristics include fractional cuckoo search [31], fractional firefly [32], fractional flower pollination [33], fractional slime mould [34], fractional Archimedes optimization [35], fractional marine predator [36,37], fractional Harris hawks optimization [38,39], fractional chaotic whale optimization [40] and different variants of fractional order particle swarm optimization (FO-PSO) [41,42,43,44,45,46,47,48]. The FO-PSO algorithms are widely used and effectively exploited for solving different engineering problems, including the optimization of power amplifiers [49], filter design [50], reactive power planning [51], image thresholding [52], harmonics estimation [53], image segmentation [54] and feature selection [55]. The promising performance of the FO-PSO in terms of better modeling and accuracy over the standard counterpart motivates the authors to investigate exploiting the swarming intelligence of FO-PSO for parameter estimation in IN-CAR systems.

1.2. Contribution

The parameter estimation of the IN-CAR system has been studied through standard evolutionary optimization heuristics based on genetic algorithms and differential evolution [56,57], but fractional-order heuristics are not yet exploited in this domain. The parameter estimation of IN-CAR systems in Refs. [56,57] is conducted through an over-parameterization approach that estimates the redundant parameters due to the cross-product terms; therefore, this study first investigates exploiting the FO-PSO for parameter estimation of IN-CAR systems and then introduces the key term separation technique (KTST) [58] in the FO-PSO algorithm to avoid estimating the redundant parameters. The computational efficiency of the KTST over the standard counterpart is well established in the system identification literature [59]; further, practical application of the parameter estimation of the electrically stimulated muscle model, represented through the IN-CAR structure, is investigated using the proposed KTST-FOPSO. The key features of the current study are:

• A key term separation technique-based fractional-order particle swarm optimization, KTST-FOPSO, is presented for parameter estimation of input nonlinear control autoregressive, IN-CAR systems.
• The KTST-FOPSO is developed through the synergy of PSO swarming intelligence with concepts of fractional calculus operators and the principle of the key term separation technique.
• The KTST-FOPSO is more efficient than the conventional FO-PSO in the sense that it identifies the IN-CAR system without estimating the extra parameters involved in the over-parameterization approach.
• The accuracy, robustness, convergence, reliability, and stability of the KTST-FOPSO scheme are verified through the results of statistical indices for different fractional orders and noise levels.

1.3. Paper Outline

The remaining manuscript is prepared as follows: The mathematical model of the IN-CAR system is provided in Section 2 with the detail of the key term separation technique. The design scheme KTST-FOPSO is presented in Section 3. The simulation results with a brief discussion are given in Section 4. Finally, the concluding remarks with future works are listed in Section 5.

2. IN-CAR Identification Model

The IN-CAR system in block diagram representation is given in Figure 1; however, the mathematical representation is given as [59,60]:

$$q(t)=\frac{X(z)}{Y(z)}\overline{s}(t)+\frac{1}{Y(z)}n(t),$$

(1)

recomposing Equation (1) as

$$q(t)=\left(1-Y(z)\right)q\left(t\right)+X(z)\overline{s}(t)+n(t),$$

(2)

where q(t)/s(t) and n(t) denote the output/input and disturbance noise signal, respectively. While $X(z)$, $Y(z)$ and $\overline{s}(t)$ are defined as:

$$X\left(z\right)=x_0+x_1{z}^{-1}+x_2{z}^{-2}+,\dots ,+x_{n_x}{z}^{-n_x},$$

(3)

$$Y\left(z\right)=1+y_1{z}^{-1}+y_2{z}^{-2}+,\dots ,+y_{n_y}{z}^{-n_y}.$$

(4)

$$\overline{s}\left(t\right)=a_1{g}_1[s\left(t\right)]+a_2{g}_2[s\left(t\right)]+\dots +a_k{g}_k[s\left(t\right)],$$

(5)

Attributing Equations (3)–(5) in Equation (2), assuming $x_0=1$, and employing the principle of KTST by considering $\overline{s}(t)$ as a key term

$$\begin{array}{l}q(t)=-{\displaystyle \sum _{j=1}^{n_y}{y}_j}\left[q\left(t-j\right)\right]+{\displaystyle \sum _{j=0}^{n_x}{x}_i}\left[\overline{s}\left(t-j\right)\right]+n(t)\\ =-{\displaystyle \sum _{j=1}^{n_y}{y}_j}\left[q\left(t-j\right)\right]+x_0\left[\overline{s}\left(t\right)\right]+{\displaystyle \sum _{j=1}^{n_x}{x}_j}\left[\overline{s}\left(t-j\right)\right]+n(t)\hspace{0.17em}\\ =-{\displaystyle \sum _{j=1}^{n_y}{y}_j}\left[q\left(t-j\right)\right]+{\displaystyle \sum _{j=1}^{n_x}{x}_i}\left[\overline{s}\left(t-j\right)\right]+{\displaystyle \sum _{j=1}^k{a}_j{g}_j\left[s\left(t\right)\right]}+n(t)\end{array}.$$

(6)

Defining the information vectors in (7)–(9) and corresponding parameter vectors in (10)–(12) as

$${\beta }_y\left(t\right)={[-q\left(t-1\right),-q\left(t-2\right),\dots ,-q\left(t-n_y\right)]}^T\in {ℝ}^{n_y},$$

(7)

$${\beta }_x\left(t\right)={[\overline{s}\left(t-1\right),\overline{s}\left(t-2\right),\dots ,\overline{s}\left(t-n_x\right)]}^T\in {ℝ}^{n_x},$$

(8)

$$g\left(t\right)={[g_1[s\left(t\right)],g_2[s\left(t\right)],\dots ,g_k[s\left(t\right)]]}^T\in {ℝ}^k,$$

(9)

$$y={\left[y_1,y_2,\dots ,y_{n_y}\right]}^T\in {ℝ}^{n_y},$$

(10)

$$x={\left[x_1,x_2,\dots ,x_{n_x}\right]}^T\in {ℝ}^{n_x},$$

(11)

$$a={\left[a_1,a_2,\dots ,a_k\right]}^T\in {ℝ}^k.$$

(12)

Using (7)–(12) in (6) gives the KTST-based identification model for IN-CAR systems

$$q\left(t\right)={\beta }_y^T\left(t\right)y+{\beta }_x^T\left(t\right)x+g^T\left(t\right)a+n\left(t\right),$$

(13)

3. Proposed Methodology KTST-FOPSO

The proposed key term separation technique based on fractional order particle swarm optimization, KTST-FOPSO is presented in this section. First, the fitness/objective function for the parameter estimation of IN-CAR systems is developed and then the design procedure of KTST-FOPSO is given. The graphical abstract in terms of procedural blocks of the proposed study is provided in Figure 1.

3.1. Formulation of the Objective Function

The objective function for the parameter estimation of the KTST-based identification model of the IN-CAR systems presented in Equation (13) is developed in the mean square error sense as:

$$\mathrm{Fitness}=\frac{1}{M}{\displaystyle \sum _{i=1}^M{\left[q(t_i)-\tilde{q}(t_i)\right]}^2},$$

(14)

where M represents the number of particles, $q$ is the desired response of the IN-CAR system given by (13), and $\tilde{q}$ represents the estimated response of the model. The estimated response of the IN-CAR is given by

$$q\left(t\right)={\beta }_y^T\left(t\right)\tilde{y}+{\beta }_x^T\left(t\right)\tilde{x}+g^T\left(t\right)\tilde{a}+n\left(t\right),$$

(15)

where the estimated parameter vectors are written as:

$$\tilde{y}={\left[{\tilde{y}}_1,{\tilde{y}}_2,\dots ,{\tilde{y}}_{n_y}\right]}^T\in {ℝ}^{n_y},$$

(16)

$$\tilde{x}={\left[{\tilde{x}}_1,{\tilde{x}}_2,\dots ,{\tilde{x}}_{n_x}\right]}^T\in {ℝ}^{n_x},$$

(17)

$$\tilde{a}={\left[{\tilde{a}}_1,{\tilde{a}}_2,\dots ,{\tilde{a}}_k\right]}^T\in {ℝ}^k.$$

(18)

The objective is to minimize the fitness function (14) through the swarming intelligence of the KTST-FOPSO.

3.2. Fractional Order Swarm Optimization

The FO-PSO was developed by introducing the concept of fractional calculus in the velocity update mechanism of the standard PSO [42]. The research community appreciated the idea of FO-PSO and proposed various variants of FO-PSO for different optimization applications [43,44,45,46,47,48,49,50,51,52,53,54]. Before presenting the theoretical development of the FO-PSO, first, the velocity w and position p update relations of the conventional PSO for the nth particle with t representing flight number are given as

$$w^n(t+1)=\alpha w^n(t)+{\gamma }_1{s}_1(L^n(t)-p^n(t))+{\gamma }_2{s}_2(G^n(t)-p^n(t)),$$

(19)

$$p^n(t+1)=p^n(t)+w^n(t+1),$$

(20)

where $\alpha$ is the inertia weight, L and G represent local and best particle, respectively, ${\gamma }_1$ is cognitive acceleration coefficient, ${\gamma }_2$ is social acceleration coefficient, $s_1$ and $s_2$ are taken as pseudo-random numbers between 0 and 1.

Different fractional derivative definitions exist in the literature, including Riemann-Liouville, Caputo, Hadamard, Atangana-Baleanu, and Grünwald–Letnikov (G-L). The G-L derivative of order $\epsilon$, ${\Delta }^{\epsilon }$ for a function $r(t)$ is defined as [61]:

$${\Delta }^{\epsilon }[r(t)]=\underset{h\to 0}{\lim }\left[\frac{1}{h^{\epsilon }}{\displaystyle \sum _{l=0}^{\infty }\frac{{\left(-1\right)}^l\Gamma \left(\epsilon +1\right)r\left(t-lh\right)}{\Gamma \left(l+1\right)\Gamma \left(\epsilon -l+1\right)}}\right],$$

(21)

where Γ is the Gamma function and h is the incremental step size. Expressing (21) in discrete-time representation with sampling period T

$${\Delta }^{\epsilon }[r(t)]=\frac{1}{T^{\epsilon }}{\displaystyle \sum _{l=0}^L\frac{{\left(-1\right)}^l\Gamma \left(\epsilon +1\right)r\left(t-lT\right)}{\Gamma \left(l+1\right)\Gamma \left(\epsilon -l+1\right)}},$$

(22)

L is the truncation order representing the order of truncation and T denotes the discrete sampling period. Letting $\alpha$ as 1 in (19), T as 1 in (22), replacing $r(t)$ with $w^n(t)$ and t with t + 1 in (22), the velocity update expression of the FO-PSO is written as [61]:

$$\begin{array}{ll}{w}^n(t+1)& =-{\displaystyle \sum _{l=1}^L\frac{{\left(-1\right)}^l\Gamma \left(\epsilon +1\right)w^n(t+1-l)}{\Gamma \left(l+1\right)\Gamma \left(\epsilon -l+1\right)}}\\ & +{\gamma }_1{s}_1(L^n(t)-p^n(t))+{\gamma }_2{s}_2(G^n(t)-p^n(t))\end{array},$$

(23)

Expanding (23) for L = 1, 2, 3 and 4, respectively, as:

$$w^n(t+1)=\epsilon w^n(t)+{\gamma }_1{s}_1(L^n(t)-p^n(t))+{\gamma }_2{s}_2(G^n(t)-p^n(t)),$$

(24)

$$\begin{array}{ll}{w}^n(t+1)& =\epsilon w^n(t)+\frac{1}{2}\epsilon (1-\epsilon )w^n(t-1)\\ & +{\gamma }_1{s}_1(L^n(t)-p^n(t))+{\gamma }_2{s}_2(G^n(t)-p^n(t)),\end{array}$$

(25)

$$\begin{array}{ll}{w}^n(t+1)& =\epsilon w^n(t)+\frac{1}{2}\epsilon (1-\epsilon )w^n(t-1)+\frac{1}{6}\epsilon (1-\epsilon )(2-\epsilon )w^n(t-2)\\ & +{\gamma }_1{s}_1(L^n(t)-p^n(t))+{\gamma }_2{s}_2(G^n(t)-p^n(t)),\end{array}$$

(26)

$$\begin{array}{ll}{w}^n(t+1)& =\epsilon w^n(t)+\frac{1}{2}\epsilon (1-\epsilon )w^n(t-1)+\frac{1}{6}\epsilon (1-\epsilon )(2-\epsilon )w^n(t-2)\\ & +\frac{1}{24}\epsilon (1-\epsilon )(2-\epsilon )(3-\epsilon )w^n(t-3)\\ & +{\gamma }_1{s}_1(L^n(t)-p^n(t))+{\gamma }_2{s}_2(G^n(t)-p^n(t)),\end{array}$$

(27)

The position and velocity update expression is given in Equation (20), while Equation (23) develops the FO-PSO. A detailed description of the FO-PSO can be seen in Ref. [61]. In this study, the FO-PSO is exploited to estimate the parameters of the KTST-based IN-CAR identification model, thus being referred to as KTST-FOPSO.

The KTST-FOPSO implementation for estimation of IN-CAR parameters is given in Figure 1 while the flowchart of the proposed KTST-FOPSO scheme is presented in Figure 2 and the pseudocode is provided in Algorithm 1.

Algorithm 1: Pseudocode of the KTST-FOPSO for parameter estimation of the IN-CAR system
Inputs:	Create particle z with elements equal to the unknown parameter of the IN-CAR system
	$$z=\left[\begin{array}{ccc}y\hspace{1em}& x\hspace{1em}& a\end{array}\right]=[\begin{array}{ccc}\left(y_1,y_2,\dots ,y_{n_y}\right)& \left(x_1,x_2,\dots ,x_{n_x}\right)& \left(a_1,a_2,\dots ,a_k\right)\end{array}],$$
	and generate a swarm.
	Swarm position, $Z=\left[\begin{array}{}{z}_1\\ z_2\\ ⋮\\ z_m\end{array}\right]=\left[\begin{array}{ccc}\left(y_{1,1},y_{1,2},\dots ,y_{1,n_y}\right)& \left(x_{1,1},x_{1,2},\dots ,x_{1,n_x}\right)& \left(a_{1,1},a_{1,2},\dots ,a_{1,k}\right)\\ \left(y_{2,1},y_{2,2},\dots ,y_{2,n_y}\right)& \left(x_{2,1},x_{2,2},\dots ,x_{2,n_x}\right)& \left(a_{2,1},a_{2,2},\dots ,a_{2,k}\right)\\ ⋮& ⋮& ⋮\\ \left(y_{m,1},y_{m,2},\dots ,y_{m,n_y}\right)& \left(x_{m,1},x_{m,2},\dots ,x_{m,n_x}\right)& \left(a_{m,1},a_{m,2},\dots ,a_{m,k}\right)\end{array}\right],$
	for m number of particles z in Z. Similarly, the corresponding velocity w is generated.
Output:	The particle z of the KTST-FOPSO with the best fitness defined in (14)
Start KTST-FOPSO
Step 1:	Initialization: pseudo-real number with are Randomly generated to form initialize swarm Z with m number of particles z.
	Initialize the associated velocities w to each particle
	Set the minimum and maximum values of the velocity
	Set particle and swarm size, flights number
	Set the value of inertia weight and fractional order
	Set the values for cognitive and social acceleration coefficients
Step 2:	Fitness Calculation: Calculate the fitness of each particle using (14).
Step 3:	Termination: Terminate the KTST-FOPSO execution processing in case of the following: (a) Number of defined flights are completed; (b) Limit of tolerance, i.e., the difference between present and previous local/global best particles, achieved.
	If any of the above termination criteria are achieved, go to Step 5.
Step 4:	Update Mechanism: Update the swarm population by updating the position and velocity of the KTST-FOPSO defined in (20) and (27), respectively, and go to Step 2.
Step 5:	Fractional Order Analysis: Get the results for different fractional order values through repeating Steps 1 to 4 by considering varying fractional orders in the KTST-FOPSO.
Step 6:	Storage: Keeping the value for global best particle with corresponding fitness.
Step 7:	Robustness Analysis: Repeat Steps 1–6 by considering different noise levels in the IN-CAR systems.
Step 8:	Statistical Analysis: Obtain a dataset by repeated execution of the KTST-FOPSO scheme for parameter estimation of IN-CAR systems through multiple independent trials for reliable and accurate inferences.
End FOPSO

4. Results with Discussion

4.1. Example 1: Numerical Experimentation

The IN-CAR system considered for numerical experimentation is taken from the recent research study Ref. [58].

$$\begin{array}{ll}q(t)=\frac{X(z)}{Y(z)}\overline{s}(t)+\frac{1}{Y(z)}n(t),& \end{array}$$

$$\begin{array}{l}Y\left(z\right)=1+{1.6}_1{z}^{-1}+0.8z^{-2}\\ X\left(z\right)=0.85z^{-1}+0.65z^{-2}\\ \overline{s}\left(t\right)=a_1{g}_1\left[s\left(t\right)\right]+a_2{g}_2\left[s\left(t\right)\right]=1.0s\left(t\right)+0.5s^2\left(t\right)\end{array},$$

The desired parameter vector of the IN-CAR system is

$$\begin{array}{ll}z& ={\left[y_1,y_2,x_1,x_2{a}_1,a_2\right]}^T\\ & ={\left[1.6,0.8,0.85,0.65,1,0.5\right]}^T\end{array}$$

(28)

The simulations are conducted in MATLAB running on Windows 10 environment. The input s is a zero mean, unit variance signal and noise n is a zero mean normally distributed signal with constant variance. In simulations, the parameters of the KTST-FOSPO for the IN-CAR estimation are selected after exhaustive experimentation as: particle size = 8, flight size = 300, swarm size = 350, cognitive and global acceleration coefficients = 2, minimum velocity = −0.4, maximum velocity = 0.4, fractional order $\epsilon$ = [0.1, 0.2, …, 1] and inertia weight = 0.9$\epsilon$.

The convergence plots of the KTST-FOPSO for parameter estimation of IN-CAR system (28) are presented in Figure 3. Figure 3a,c,e presents the learning curves of odd fractional orders ($\epsilon$ = 0.1, 0.3, 0.5, 0.7, 0.9) for noise level $\eta$ = 90 dB, 60 dB and 30 dB, respectively, while the respective plots in case of even fractional orders ($\epsilon$ = 0.2, 0.4, 0.6, 0.8, 1.0) are given in Figure 3b,d,f. The learning curves indicate that the KTST-FOPSO provides fast convergence for lower fractional order values.

The fitness values along with the estimated parameters are provided in

Table 1. Parameter estimates with the level of estimation accuracy for $\epsilon$ = 0.1.

$\mathit{\eta }$	Flight	y1	y2	x1	x2	a1	a2	Fitness
90	10	1.6489	0.8404	0.8573	0.5712	1.0234	0.5045	2.345 × 10−4
	30	1.6019	0.8017	0.8519	0.6487	1.0004	0.5001	2.168 × 10−7
	70	1.5999	0.7999	0.8499	0.6500	1.0000	0.5000	2.343 × 10−9
	110	1.5999	0.8000	0.8500	0.6500	1.0000	0.5000	1.091 × 10−9
	150	1.6000	0.8000	0.8500	0.6499	1.0000	0.5000	8.772 × 10−10
60	10	1.5848	0.7862	0.8323	0.6699	0.9512	0.4723	1.362 × 10−4
	30	1.5961	0.7946	0.8581	0.6533	0.9981	0.5003	3.928 × 10−6
	70	1.5962	0.7965	0.8477	0.6497	1.0002	0.5006	1.089 × 10−6
	110	1.5972	0.7966	0.8477	0.6495	1.0002	0.5006	1.024 × 10−6
	150	1.5963	0.7962	0.8474	0.6495	1.0005	0.5004	6.665 × 10−7
30	10	1.4828	0.7867	0.7873	0.7126	0.9643	0.4703	2.121 × 10−3
	30	1.5075	0.7012	0.7570	0.7339	0.9298	0.4734	1.294 × 10−3
	70	1.5271	0.6985	0.7509	0.7387	0.9263	0.4707	9.219 × 10−4
	110	1.5271	0.6985	0.7509	0.7387	0.9263	0.4707	9.219 × 10−4
	150	1.5241	0.6990	0.7439	0.7341	0.9522	0.4710	4.919 × 10−4
Actual		1.6000	0.8000	0.8500	0.6500	1.0000	0.5000	0

,

Table 2. Parameter estimates with the level of estimation accuracy for $\epsilon$ = 0.5.

$\mathit{\eta }$	Flight	y1	y2	x1	x2	a1	a2	Fitness
90	10	1.6177	0.8238	0.9450	0.6797	0.9932	0.4992	4.443 × 10−4
	30	1.5835	0.7910	0.8387	0.6670	0.9990	0.4978	8.102 × 10−5
	70	1.6001	0.8000	0.8502	0.6520	0.9981	0.4989	3.148 × 10−7
	110	1.6002	0.8002	0.8503	0.6503	1.0000	0.5000	1.295 × 10−8
	150	1.6000	0.8001	0.8501	0.6500	1.0000	0.5000	1.701 × 10−9
60	10	1.5426	0.7515	0.6743	0.5420	0.9803	0.4701	1.685 × 10−3
	30	1.5791	0.7804	0.8256	0.6802	0.9590	0.4760	1.028 × 10−4
	70	1.5972	0.7970	0.8504	0.6556	0.9986	0.4994	2.196 × 10−6
	110	1.6001	0.7999	0.8517	0.6493	0.9994	0.4998	9.103 × 10−7
	150	1.6000	0.7998	0.8508	0.6506	0.9997	0.4999	7.752 × 10−7
30	10	1.2754	0.5619	0.8045	0.8498	0.9059	0.4525	4.647 × 10−3
	30	1.5132	0.6444	0.8622	0.5060	0.9898	0.5030	2.628 × 10−3
	70	1.5283	0.6632	0.7798	0.5217	1.0100	0.5034	1.138 × 10−3
	110	1.5283	0.6632	0.7798	0.5217	1.0100	0.5034	1.138 × 10−3
	150	1.5128	0.6866	0.7754	0.5165	1.0169	0.5051	6.623 × 10−4
Actual		1.6000	0.8000	0.8500	0.6500	1.0000	0.5000	0

and

Table 3. Parameter estimates with the level of estimation accuracy for $\epsilon$ = 1.

$\mathit{\eta }$	Flight	y1	y2	x1	x2	a1	a2	Fitness
90	10	1.5358	0.7630	0.8485	0.6824	0.9480	0.4648	1.979 × 10−3
	30	1.5358	0.7630	0.8485	0.6824	0.9480	0.4648	1.979 × 10−3
	70	1.5766	0.7786	0.9555	0.8575	0.9179	0.4705	1.317 × 10−3
	110	1.5766	0.7786	0.9555	0.8575	0.9179	0.4705	1.317 × 10−3
	150	1.5766	0.7786	0.9555	0.8575	0.9179	0.4705	1.317 × 10−3
60	10	0.9444	0.2806	0.0490	0.9061	0.7998	0.3846	1.093 × 10−2
	30	1.3685	0.6007	0.5319	0.7021	1.0508	0.5234	4.942 × 10−3
	70	1.5863	0.7789	0.8199	0.5589	1.0205	0.4909	1.604 × 10−3
	110	1.6666	0.8547	0.8232	0.4271	1.0987	0.5400	1.402 × 10−3
	150	1.6666	0.8547	0.8232	0.4271	1.0987	0.5400	1.402 × 10−3
30	10	1.1945	0.4656	0.7849	0.9205	0.7033	0.3497	6.817 × 10−3
	30	1.6540	0.8695	0.8655	0.6454	0.9308	0.4552	1.094 × 10−3
	70	1.6540	0.8695	0.8655	0.6454	0.9308	0.4552	1.094 × 10−3
	110	1.6540	0.8695	0.8655	0.6454	0.9308	0.4552	1.094 × 10−3
	150	1.6540	0.8695	0.8655	0.6454	0.9308	0.4552	1.094 × 10−3
Actual		1.6000	0.8000	0.8500	0.6500	1.0000	0.5000	0

for three fractional orders in the KTST-FOPSO ($\epsilon$ = 0.1, 0.5, 1.0). While for remaining fractional orders ($\epsilon$ = 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, 0.9), the fitness results are provided in Tables S1–S7, respectively, as Supplementary Materials. The fitness values obtained at 150 flight/iteration are 8.772 × 10⁻¹⁰, 1.701 × 10⁻⁹ and 1.317 × 10⁻³ for $\epsilon$ = 0.1, 0.5 and 1.0, respectively, in the case of $\eta$ = 90 dB. It is seen that the estimation accuracy decreases with an increase in fractional order $\epsilon$; moreover, the level of estimation accuracy decreases with an increase in the noise level $\eta$.

In order to get reliable inferences about the performance of the KTST-FOPSO scheme for IN-CAR system identification, hundred independent trials are executed and results of fitness attained against independent trials are plotted in Figure 4 for $\epsilon$ = 0.1, 0.2, 0.3, Figure 5 for $\epsilon$ = 0.4, 0.5, 0.6 and Figure 6 for $\epsilon$ = 0.7, 0.8, 0.9, 1.0. Moreover, the values of statistical indices are calculated and presented in

Table 4. Results of statistical measures for all considered $\epsilon$ and $\eta$.

$\epsilon$	$\mathit{\eta }$ = 90 dB			$\mathit{\eta }$ = 60 dB			$\mathit{\eta }$ = 30 dB
	Mini	Mean	SD	Mini	Mean	SD	Mini	Mean	SD
0.1	4.686 × 10−10	9.579 × 10−9	1.270 × 10−8	4.223 × 10−7	4.951 × 10−6	1.050 × 10−5	4.061 × 10−4	1.515 × 10−3	1.098 × 10−3
0.2	4.757 × 10−10	4.348 × 10−9	7.645 × 10−9	3.830 × 10−7	2.246 × 10−6	2.489 × 10−6	3.630 × 10−4	1.782 × 10−3	1.586 × 10−3
0.3	4.023 × 10−10	4.411 × 10−9	9.728 × 10−9	3.417 × 10−7	2.225 × 10−6	3.251 × 10−6	3.427 × 10−4	1.576 × 10−3	1.543 × 10−3
0.4	4.095 × 10−10	2.456 × 10−9	2.546 × 10−9	3.897 × 10−7	1.733 × 10−6	2.506 × 10−6	3.242 × 10−4	1.280 × 10−3	1.053 × 10−3
0.5	3.906 × 10−10	2.667 × 10−9	2.743 × 10−9	3.478 × 10−7	1.894 × 10−6	2.646 × 10−6	3.500 × 10−4	1.534 × 10−3	1.828 × 10−3
0.6	9.202 × 10−10	8.739 × 10−9	1.354 × 10−8	5.772 × 10−7	2.082 × 10−6	2.409 × 10−6	3.679 × 10−4	1.261 × 10−3	8.149 × 10−4
0.7	1.209 × 10−8	1.511 × 10−6	1.898 × 10−6	6.790 × 10−7	6.620 × 10−6	5.366 × 10−6	5.283 × 10−4	1.651 × 10−3	1.024 × 10−3
0.8	3.589 × 10−6	1.083 × 10−4	7.599 × 10−5	2.467 × 10−5	1.053 × 10−4	6.126 × 10−5	5.888 × 10−4	1.861 × 10−3	9.531 × 10−4
0.9	6.253 × 10−5	5.075 × 10−4	2.363 × 10−4	1.436 × 10−4	5.144 × 10−4	2.226 × 10−4	1.124 × 10−4	2.537 × 10−3	1.143 × 10−3
1.0	1.746 × 10−4	8.157 × 10−4	3.378 × 10−4	4.665 × 10−5	8.348 × 10−4	3.608 × 10−4	1.094 × 10−3	2.923 × 10−3	1.588 × 10−3

. The mean values for $\epsilon$ = 0.2, 0.4, 0.6, 0.8 and 1.0 are 4.348 × 10⁻⁹, 2.456 × 10⁻⁹, 8.739 × 10⁻⁹, 1.083 × 10⁻⁴ and 8.157 × 10⁻⁴, respectively, in case of $\eta$ = 90 dB; meanwhile the respective values in the case of $\eta$ = 60 dB are 2.246 × 10⁻⁶, 1.733 × 10⁻⁶, 2.082 × 10⁻⁶, 1.053 × 10⁻⁴ and 8.348 × 10⁻⁴, and values in the case of $\eta$ = 30 dB are 1.782 × 10⁻³, 1.280 × 10⁻³, 1.261 × 10⁻³, 1.861 × 10⁻³, 2.923 × 10⁻³. The results of statistical measures further endorse the observations of the single run of the KTST-FOPSO that the proposed scheme can effectively estimate the actual parameters of the IN-CAR systems without identifying the redundant parameters and the performance of the KTST-FOPSO decreases slightly for an increase in noise level $\eta$. Generally, the KTST-FOPSO provides better results for a lower value of $\epsilon$.

4.2. Example 2: Electrical Stimulated Muscle Model

In Example 2, a practical application of the IN-CAR system representing the dynamics of the electrically stimulated muscle model (SMM) is considered. The SMM is required for the rehabilitation of stroke patients whose paralyzed muscles are restored for functional use through automatic control of stimulations. The desired parameters in the case of Example 2 are taken from the real-time experiments conducted by the Signals, Images, Systems Research Group at the University of Southampton [62].

$$q(t)=\frac{X(z)}{Y(z)}\overline{s}(t)+\frac{1}{Y(z)}n(t),$$

$$\begin{array}{l}Y\left(z\right)=1-z^{-1}+0.8z^{-2}\\ X\left(z\right)=2.8z^{-1}-4.8z^{-2}& ,\\ \overline{s}\left(t\right)=1.68s\left(t\right)-2.88s^2\left(t\right)+3.42s^3\left(t\right)\end{array}$$

The desired parameter vector of the IN-CAR system in the case of Example 2 is

$$\begin{array}{l}z={\left[y_1,y_2,x_1,x_2{a}_1,a_2,a_3\right]}^T\\ ={\left[-1,0.8,2.8,-4.8,1.68,-2.88,3.42\right]}^T\end{array}$$

(29)

In Problem 2, again, the input s is generated as a zero mean, while unit variance signal and output are corrupted by 90 dB AWGN. The KTST-FOPSO parameters used in simulations are: particle size = 8, flight size = 300, swarm size = 300, cognitive and global acceleration coefficients = 1.5, minimum velocity = −2, maximum velocity = 2, fractional order $\epsilon$ = [0.1, 0.2, …, 1] and inertia weight = 0.97 $\epsilon$.

Five independent executions of the KTST-FOPSO are conducted for parameter estimation of the IN-CAR system considered in Problem 2 and results are provided based on best run out of five trials. The convergence plots are presented in Figure 7 while the parameter estimates along with the attained fitness values are provided in

Table 5. Parameter estimates with level of estimation accuracy for $\epsilon$ = 0.6.

$\mathit{\epsilon }$	y1	y2	x1	x2	a1	a2	a2	Fitness
0.1	−0.9995	0.7997	2.8060	−4.8065	1.5500	−3.1157	3.3104	2.58 × 10−3
0.2	−0.9996	0.7998	2.8353	−4.8457	1.5677	−3.0248	3.3203	5.41 × 10−3
0.3	−0.9999	0.7999	2.8004	−4.8005	1.6441	−2.9448	3.3915	1.72 × 10−4
0.4	−1.0006	0.8003	2.7683	−4.7591	1.8672	−2.5975	3.5752	6.71 × 10−3
0.5	−0.9993	0.7997	2.9522	−5.0000	1.4651	−2.9894	3.2226	8.41 × 10−2
0.6	−0.9998	0.7999	2.8065	−4.8084	1.6326	−2.9604	3.3774	4.31 × 10−4
0.7	−1.0015	0.8007	2.7166	−4.6912	2.0120	−2.3836	3.7339	3.72 × 10−2
0.8	−0.9995	0.7997	2.8167	−4.8218	1.5524	−3.0849	3.3131	2.76 × 10−3
0.9	−1.0011	0.8006	2.7766	−4.7705	1.9085	−2.4727	3.6265	9.09 × 10−3
1.0	−1.0002	0.8001	2.7968	−4.7961	1.7379	−2.7752	3.4698	4.80 × 10−4
Actual	−1.0000	0.8000	2.8000	−4.8000	1.6800	−2.8800	3.4200	0

for all considered fractional orders. The results of Figure 7 indicate that the KTST-FOPSO is convergent in the case of identification of SMM for all fractional orders; meanwhile, the parameter estimates given in Table 5 endorse the accuracy of the parameter estimation of SMM.

5. Conclusions

Following are the concluding remarks of the detailed investigation of the current study:

• A key term separation technique-based fractional order particle swarm optimization, KTST-FOPSO, is presented as an effective solution for the nonlinear system identification problem.
• The accuracy and robustness of the KTST-FOPSO are established through effective parameter estimation of input nonlinear control autoregressive, IN-CAR, systems for different fractional order and noise scenarios with generally a decreasing trend in estimation accuracy as fractional order increases from 0.1 to 1.
• The KTST-FOPSO is more efficient than the conventional FO-PSO in the sense that it avoids estimation of the redundant parameters and identifies only the actual parameters of the IN-CAR system.
• The reliability and stability of the KTST-FOPSO scheme are endorsed through results of statistical indices obtained after conducting sufficient large executions of the proposed scheme for numerical as well as practical examples of the IN-CAR system.

The concepts of the multi-innovation theory [63,64,65], auxiliary model idea [66,67], and hierarchical identification principle [68,69,70] can be integrated with the proposed methodology for better performance in system identification; moreover, the proposed scheme can be compared with other FO-PSO variants for solving complex optimization problems [71,72,73,74,75].