Nonlinear System Identification in coherence with Nonlinearity measure for Dynamic Physical systems- Case Studies


 With the recent success of using the time series to vast applications, one would expect its boundless adaptation to problems like nonlinear control and nonlinearity quantification. Though there exist many system identification methods, finding suitable method for identifying a given process is still cryptic. Moreover, to this notch, research on their usage to nonlinear system identification and classification of nonlinearity remains limited. This article hovers around the central idea of developing a ‘kSINDYc’ (key term based Sparse Identification of Nonlinear Dynamics with control) algorithm to capture the nonlinear dynamics of typical physical systems. Furthermore, existing two reliable identification methods namely NL2SQ (Nonlinear least square method) and N3ARX (Neural network based nonlinear auto regressive exogenous input scheme) are also considered for all the physical process-case studies. The primary spotlight of present research is to encapsulate the nonlinear dynamics identified for any process with its nonlinearity level through a mathematical measurement tool. The nonlinear metric Convergence Area based Nonlinear Measure (CANM) calculates the process nonlinearity in the dynamic physical systems and classifies them under mild, medium and highly nonlinear categories. Simulation studies are carried-out on five industrial systems with divergent nonlinear dynamics. The user can make a flawless choice of specific identification methods suitable for given process by finding the nonlinear metric (Δ0). Finally, parametric sensitivity on the measurement has been studied on CSTR and Bioreactor to evaluate the efficacy of kSINDYc on system identification.


Highlight of research:
1. The submission serves as a bridge to fill the leveraging gap between the computation of nonlinearity and the suitable choice of nonlinear system identification for nonlinear dynamic physical systems. 2. Given any nonlinear process, the CANM (Convergence Area based Nonlinear Measure) method categorizes it under a particular class of nonlinearity quantified within the range 0 to 1.

A notable development on the (Sparse Identification of Nonlinear Dynamics with control)
SINDYc candidate library function, by introducing the 'key nonlinear term' from the plant dynamics (kSINDYc), apart from the other higher order polynomials of the processes. 4. The research also extends to nonlinear system identification with the nonparametric N3ARX (Neural network based nonlinear auto regressive exogenous input) scheme and the most popular parametric NL2SQ (Nonlinear Least Square) method for five different dynamic processes with different levels of nonlinearity. 5. Finally, for all the case studies, a qualitative and quantitative relation is made between the nonlinear system identification methods and the nonlinearity measures.

Abstract:
With the recent success of using the time series to vast applications, one would expect its boundless adaptation to problems like nonlinear control and nonlinearity quantification. Though there exist many system identification methods, finding suitable method for identifying a given process is still cryptic. Moreover, to this notch, research on their usage to nonlinear system identification and classification of nonlinearity remains limited. This article hovers around the central idea of developing a 'kSINDYc' (key term based Sparse Identification of Nonlinear Dynamics with control) algorithm to capture the nonlinear dynamics of typical physical systems. Furthermore, existing two reliable identification methods namely NL2SQ (Nonlinear least square method) and N3ARX (Neural network based nonlinear auto regressive exogenous input scheme) are also considered for all the physical process-case studies. The primary spotlight of present research is to encapsulate the nonlinear dynamics identified for any process with its nonlinearity level through a mathematical measurement tool. The nonlinear metric Convergence Area based Nonlinear Measure (CANM) calculates the process nonlinearity in the dynamic physical systems and classifies them under mild, medium and highly nonlinear categories. Simulation studies are carried-out on five industrial systems with divergent nonlinear dynamics. The user can make a flawless choice of specific identification methods suitable for given process by finding the The main advantage of nonlinear-system-identification is the fact that, even, if an unknown system is given, just with the measured input and output data, the nonlinear dynamics of the process can be retrieved accurately. In recent years, many literatures have brought out the features, pros and cons on the usage and complexity of many notable identification algorithms for nonlinear system.
To manifest a few, Schoukens and Ljung presented a review on identification methods of linear and nonlinear systems [1]. The article also indexed an exemplary summary of many parametric identification methods. Block oriented nonlinear models can be classified under (i) Hammerstein (ii) Wieners (iii) Voltera-series. The review article conferred by [2], not only portrayed the blockoriented identification methods, but also delivered a deep thought on the most prominent nonlinear control schemes of recent time. A non-stochastic subspace algorithm was considered for multidimensional nonlinear system identification based on measured output data. However, the procedure was not tested for systems with different structural nonlinearities [3].
The autoregressive models with exogenous inputs are employed in applications where state transitions are triggered by external events [4]. Stochastic gradient parametric estimation using moving window data was presented in [5] to estimate the system's response to discrete measured data. However, the effectiveness of the method was shown only by using numerical examples and not on physical systems. The identification of LPV time-delay systems with missing output data using multiple-model approach is framed in [6]. Output-error (OE) model representing the process dynamics of CSTR and continuous fermenter, are recovered using the expectation-maximization (EM) algorithm to obtain the final global model. Reference [7] is concerned with the parametric identification of a special class of nonlinear-systems called as bilinear state space systems. Parametric identification of time-delay-systems were discussed in [8]. Multi-innovation theory is put forward in stochastic gradient algorithm based on state observer and recursive least-squared identification algorithm to improve their accuracy and convergence rate. In another work by [9], a generalized identification scheme for integral-order systems is utilized for identification of fractional-order nonlinear systems with both non-chaotic and chaotic behaviors. Being under the class of black-box modeling, Hammerstein-Wiener models can be employed for identification of complex nonlinear systems with static nonlinearity as well as dynamic linear regions [10,11].
Machine learning approaches are very powerful tools to identify a variety of highly nonlinear systems. The approaches come out with high fidelity models, that reflect the underlying physics of the nonlinear system. Many standard machine learning methods have shown spectacular performance in predicting dynamics of any interpolated system, but the resulting models usually lack generalizability and interpretability [12,13]. Recently, in one article by [14], the authors have reviewed system identification in context to powerful tools of computational intelligence methods which include genetic algorithm, particle swarm optimization and differential evolution. A variety of highly nonlinear occurrences are contemplated to assess the competence and the fast computing intelligence of genetic programming in [15].Takagi-Sugeno(TS) fuzzy modelling with unscented Kalman filter was carried over for a practical heat exchanger process in [16]. Yet, the real challenge lies on the choice of fuzzy rule numbers on the output precision.
Nonetheless there are several identification methods where the real challenge lies in developing a parsimonious model with the smallest possible number of parameters that can adequately describe the dynamics of the physical system. Also, the confrontation lies in determining the underlying dynamics of the process from the measured data.

Motivation:
The above discussion reveals that there are indigenous number of articles that discuss the concepts of system identification and measurement of nonlinear metric [21] in separate attempts. There exist enumerable identification methods and methods for quantification of nonlinearity. However, welldesigned directions or guidelines for selection of identification algorithms (based on nonlinearitymeasure) are rare and need to be established. Past literature explains that there exists a break in the continuity between these two concepts for (as mentioned above) over years. The motivation for this research comes up with a bang by readdressing the issues of system identification and the concept of nonlinear metric in a jointed venture. The research idea discussed in this article will overcome the existing disruption by relating the nonlinear metric ( 0 ∆ ) with the noteworthy system identification tools.

Contributions:
In this respect, we have established three significant system identification methods namely key term kSINDYc, N3ARX and NL2SQ methods to identify peculiar nonlinear systems from the process engineering glebe. SINDYc (Sparse Identification of Nonlinear Dynamics with control) algorithm is a symbolic sparse regression problem, which may be susceptible to over fitting problem if care is not taken to balance the model complexity [17]. This major concern is drenched here, by choosing a fewer number of relevant key terms in the candidate library of the SINDYc scheme. This paper also addresses this issue by providing the perfect choice of key terms based on the degree and type of nonlinearity of dynamic nonlinear systems. Moreover, the performance index of all the methods are correlated along with the degree of nonlinearity of each process and the outcomes are enumerated.
The paper is divided into six sections. Section 1 has introduced the literature review of many system identification methods. Section 2 examines about the CANM method to measure nonlinearity level 0 ∆ .Section 3 gives a brief review of the kSINDYc, NL2SQ and N3ARX approaches. It is followed by simulation results in section 4 which show that 0 ∆ as well as RMSE witness a major lively role in deciding the choice of nonlinear identification method for the five dynamic systems with contrasting nonlinearity. Besides, Section 4 also adds increased flavour to the current study by suggesting the suitable parsimonious model for every process. Section 5 narrates the significance of parametric sensitivity analysis of physical systems and section 6 concludes the article.
2 Proposed CANM Method 0 ∆ : The nonlinearity of the physical systems is an important issue to be addressed in controller design, bifurcation and uncertainty analysis. It varies with respect to the initial condition of state variables, excitation signals given, and input constraints associated with it. This research brings out the strength of nonlinearity of typical industrial process and its impact on the popular system identification schemes. Nevertheless, there are several nonlinear indices to mark the value of nonlinearity in the dynamic systems [18][19][20]. The concept of Convergence-area-based-nonlinearity measure (CANM) has been endorsed in the current study [21].
Without loss of generality, consider a nonlinear dynamic system of the form The CANM method conferred in this work stands distinct for its amenability in dealing with wide range of nonlinear dynamic systems. The method uses Jacobian linearization to find out linear approximation () lin yt which thoroughly depends on analysis of an operating point. The stability of the operating point decides the current dynamic behavior of the process. As many chemical, biomedical and biological processes often operate on a predesigned operating region with multiple operating points, this CANM method will be most beneficial to them. x denote the nominal input parameters and initial states of the nonlinear system respectively. The excitation signal u and the nonlinear output () act yt from the nonlinear process are treated as measured input and output data. Region I contain the proposed kSINDYC algorithm to learn the dynamics of () act yt . The most essential term in the kSINDYc library, which plays a critical role in determining the nonlinear dynamics, is weighed from the governing equations of the process. The predicted output () pred yt of Region I and measured output () act yt are used to calculate performance using RMSE criteria. On the flip side, the nonlinear system under study is linearized about its operating point j P using Jacobian linearization and is expressed as () lin yt . The 0 ∆ is computed using CANM method as given in Eq. (2). On completion of the learned dynamics using kSINDYc, Region I of Fig. 1 is replaced by N2LSQ and N3ARX identification methods. A graphical plot is made between 0 ∆ and RMSE to fill the leveraging gap between computation of nonlinearity and the suitable choice of identification for nonlinear dynamic physical systems.

Key term based SINDYc (kSINDYc):
The recent impeccable SINDYc algorithm is a celebrated parsimonious system identification technique introduced by Brunton [22]. Abundant collection of technical records is garnered with widespread curiosity on the remarkable progress made in sparse dynamics in many disciplines ranging from biology to control engineering [23][24][25]. Backdrop in this section, we provide a brief retrospect to the SINDYc algorithm, which forms the bottom line of the proposed 'kSINDYc' system identification methodology. Inspired by its application to many physical systems, regression problem using SINDYc is formulated as follows: ( ) Eq. (5) is vector that has the sparse co-efficients 12 The key terms in the library function (, ) xu Θ are given in Eq. (9) [ ] xu ⊗ denotes the vector of all product combinations in x and u .It also indicates the quadratic nonlinearities in the unknown system.
The optimization problem given in Eq. (10) can be evaluated using sparsity promoting scheme called STLS (Sequential Threshold least square method). The second part of Eq. (10) has the penalty term with the tunable weighing parameter 0 λ≥ to establish model parsimony. k ξ represents th k row of Ξ and k X  represents th k row of X  The algorithmic pseudocode for the proposed kSINDYc identification is given below:  is considered in Algorithm A1 to attain an optimal solution with minimum error and maximum convergence rate. The specific convergence criteria for STLS algorithm in SINDYc framework is provided in [26]. kSINDYc can handle large candidate library with the regularizing tuner λ . Not limitingly, the choice of the sparsity knob λ is made in such a way that there is a tradeoff between accuracy and complexity of the kSINDYc algorithm.

NL2SQ method:
Many works on parametric system identification have used least square method to estimate the numerical values of parameters [27][28][29]. Other techniques to estimate the parameters of the physical systems include Subspace identification methods and Hammerstein-wiener models, [4,12]. NL2SQ with Levenberg-Marquardt algorithm is another established method used for optimizing the process parameters in the field nonlinear system identification [16,28,30]. The The model parameters θ are estimated with the basic requirement of minimizing the objective function in Eq. (11).
The Jacobian matrix () J θ of () H θ has to be found out to optimize θ for m number of samples. Using LM algorithm, the objective function for NL2SQ method is modified as LM h [29].
The damping factor is always 0 µ≥ ,for which the following effects are observed. When 0 µ> , the co-efficient () T JJ I +µ is positive definite and so LM h is in descent direction. If µ is very and goes into the steepest descent direction. On the other hand, if µ is very small LM GN hh = , the LM algorithm converges with the Gauss Newton method. In the gradient descent method, the LM h is minimized by refreshing the parameters in the steepest-descent direction. The gradients of the process are calculated using automatic differentiation. On the other hand, in the Gauss-Newton method LM h is reduced by considering the least square module to be locally quadratic to its parameters and sorting out the minimum value from this quadratic term. The LM algorithm operates similar to gradient-descent method when the parameters are away from their optimal value, and behaves more like a Gauss-Newton scheme when the parameters are very near to the optimal point. It can be concluded that the LM algorithm involves the cross combination of gradient descent and Gauss-Newton methods.

N3ARX Method:
Neural Networks is another computational intelligence approach for identifying nonlinear systems in real world scenario with accurate estimations [31,33,34]. Neural Networks can be employed without knowing any prior knowledge about the dynamics of the system The NARX method is a standard identification technique and is found in enormous literatures [32,35]. A novel optimal identification algorithm is presented for NMPC based on the Neural network model for different operating regions of highly nonlinear dynamic processes in [36]. Hybrid combination of Neural network algorithm with NARX method is investigated in this research to make a strong comparison with the kSINDYc method of identification. A simple Neural network structure is taken with 1 hidden layer, and linear activation function in Matlab. The number of neurons required to identify each process will differ depending upon the nonlinearity and operating region. The number of hidden layer nodes in N3ARX method can be chosen iteratively.

Simulation study:
The simulation study is carried out here for typical Industrial processes from chemical to biological field. The learned models are developed using kSINDYc, N3ARX and NL2SQ identification

Example 1: Three Tank Process
A three-tank hydraulic process with the configuration of first pump supplying a liquid to first tank is considered in the present work. The objective is to control liquid levels in each tank by measuring the level of 3 rd tank. The dynamic equations and the associated process parameters 12 23 3 1 , ,, c c cA 12 3 , h h and h 31 1 qm s − . The nonlinear differential equations of the three-tank system are given by Eq. (20) to (22). The initial level of the all the tanks is assumed to be zero. 1 q denotes the inflow rate of the liquid in the first tank with the constraint 31 1 u qm s − = is presented in Fig.2. The three-tank process has a large settling time of around 5000 sec with a weak nonlinear behavior.

here]
A 10% ± variation in feed flow rate from nom u also termed as prbs u is adopted (through PRBS mode) to check the open loop response of tank level 3 h in Fig. 3. It has been observed that NL2SQ identification approach outperforms kSINDYc in the case of a sluggish nonlinear system like three-tank process.

Example 2: CSTR
An exothermal, continuous stirred tank reactor (CSTR) is widely used to convert reactants to products ( is considered as manipulated input and temperature of the reactor T is the output variable. The states are concentration of reactants a C and temperature of reactor T . The nominal operating data for the reaction is available in [38]. The initial states and steady state points of the Concentration gradient a C of the species A and the effluent temperature of the reactor T are assumed to be the same where By carefully observing Eqs. (24) and (25), we can clearly understand that the activation energy level E has an effect on rate-constant of reaction which further influences the outputs of the CSTR, and depends upon the operating conditions and mechanism of species A undergoing the reaction dynamics (outputs from sparse space) from kSINDYc identification method outperforms the other two methods for a nonlinear CSTR process.

Example 3: Heat Exchanger
A Heat Exchanger (HE) is a device where a cold fluid is heated by another hot stream mostly by convection principle. Recently, first principle modelling of a heat exchanger for a high temperature milk pasteurization unit was enumerated using log mean temperature difference approach [39]. In our research, a nonlinear physical model of a fluid-fluid HE process is detailed in this section, adopted from [40].

Example 4: Bio reactor
A bioreactor otherwise called a fermenter, a special type of heterogeneous reactor, is an essential automated system used in food processing and pharmaceutical industries. A fed-batch reactor with the manipulated input of dilution rate D and the process output, biomass concentration X is adopted from [41]. The mass balance equations representing the kinetic model of the bioreactor are given in Eqs (31)(32)(33). At high substrate concentration, S, rate of product formation is independent of S due to limited amount of enzyme; at low substrate concentration, the rate of product formation becomes proportional to S and follows first-order kinetics. Fermenters generally produce heat respiration and maintenance of bio-chemical pathways by microbes. Control becomes essential in large scale installations. However lack in proper knowledge behind kinetic pathways, calculation of cooling, aeration, pH, and agitations need attention. Here growth The density of microbial cells also called biomass concentration X of any microorganism grows by consuming the substrate S fed to it.
where a Haldane type of specific growth rate is given by The nonlinearity of the bioreactor varies w.r.t the specific growth rate () S µ , the type of excitation given, initial states of , SXand the operating region of dilution rate u .Therefore a bioreactor can be contemplated as a very sensitive nonlinear system, subjected to the above factors.
[Insert Fig. 8 here] [Insert Fig. 9 here] The response of (/ ) X g litre due to PRBS input in the dilution rate of the feed flow 10% prbs nom uu = ± is portrayed in Fig. 9. It can be noticed that the three methods of identification showed excellent tracking of the biomass-concentration (nonlinear dynamics) with very sharp variations.

Example 5: Distillation Column
A 9 stage ( 9 s n = ) binary Distillation Column (DC), to separate methanol-water mixture, operated in the LV (liquid-vapour) configuration with the manipulated variable as reflux rate to the column -1 nom u L kmol min = is taken for the study from [41]. The distillate composition D x () kmol which is the top most product is the output variable y . The feed mixture containing 50% Methanol has to be rectified continuously to 98% purity. The common problems are vapor cross-flow channeling, foaming and unaccounted interactions. The presence of many state-variables and process parameters make the simulation of DC model more complex. Accordingly, certain process assumptions are made as follows for an easier analysis: A perfect binary mixture with constant pressure, no vapor holdup, and constant relative volatility, on all stages are considered. The ordinary differential equations governing the DC are Eqs. (35)(36)(37)(38)(39) [Insert Fig. 10 here] The DC model when operated in a wider operating region instead of a fixed input at nom u imparts a massive nonlinear phenomenon and does not suit the normal operation. Therefore, the initial conditions of distillate D [Insert Table 1 here] Table 1, summarizes the major key terms selected in the candidate library function of each process.
The nonlinear function of each process is decided by the key term nl k . The candidate library Θ has relatively a fewer functional terms. nl k term is chosen carefully by checking the influencing terms from the dynamic equations of every process. Moreover, introducing the key nonlinear terms in the candidate library function of kSINDYc is intended to build models of dynamic physical systems with diverse nonlinear behavior.
[Insert Fig. 11 here] Fig. 11 is an important graphical representation that relates the nonlinearity of each process with the three nonlinear system identification methods in terms of the objective function. The criteria to be accounted for is the trade-off between Goodness of fit and the complexity of the learned model () pred yt .
The RMSE from Eq. (41) is found for all the processes described in the previous sections for N samples. Table 2 will provide the inference made on the choice of the appropriate system identification method, on the basis of nonlinearity level and the RMSE of every process. As seen from Fig.11, the crucial factor in determining the choice of system identification rests on the minimum RMSE between the actual and learned dynamics of the identified model among all the three identified models.
[Insert Table 2 (42) For all the case studies discussed in this section, a qualitative comparison is made between the RMSE of all the nonlinear system identification methods and 0 ∆ in Fig.11. The identification method which gives least RMSE under each class of 0 ∆ imply a better estimate on all basis and is exclusively picked up for the perfect choice of system identification. The physical quantities of each process addressed in Table 3, have different orders of magnitude. To sustain a uniform scale in measuring the nonlinearity, the time period t , input u and output variable () act yt of all nonlinear physical process are normalized between 0 and 1, and thereafter the CANM method 0 ∆ is intended from Eq. (2) [Insert Table 3 here] Table 3, affirms the better choice for system identification of medium and high nonlinear process to be kSINDYc approach whereas the mild nonlinear systems can follow NL2SQ method to learn the process dynamics. The effectiveness of N3ARX approach trails behind kSINDYc and NL2SQ in terms of RMSE and execution time. Table 3 also presents assertive conclusive remarks on the selection of identification method based on strength of nonlinearity 0 ∆ .

Remark 1:
N3ARX method requires large number of training data set to provide accurate solution.
The learned dynamics using this approach did not meet the acceptable limit at nom u .Consequently, it can be used for systems with broader range of excitation signals prbs u and chirp u (CHIRP excitation). The major setback of N3ARX method compared to other methods is its long computation time in MATLAB.

Remark 2:
The learned dynamics using NL2SQ is satisfactory for mild and medium nonlinear systems. As it is a parametric identification scheme it requires a healthy knowledge of the process parameters and the nominal operating regions. This method fails to identify the process model whose measured data is within the low data limit.
Remark 3: kSINDYc is computationally attractive, requires less data, assumes a few number of candidate terms in Θ to make an interpretable efficient model at nom u and prbs u . The method outstrips NL2SQ and N3ARX by accurately following the plant dynamics of medium and highly nonlinear systems.
Parametric variations occurring in a unit in an industrial process is a decisive topic to be separately discussed. The uncertainities and the parametric fluctuations will definitely affect the dynamic stability of the process which in turn will be a matter of concern for nonlinear identification of dynamic systems. Local and global sensitivity analysis were carried out in semi-batch reactors to find out the intensity of interactions between the input-output parameters and to understand the variations in output variables for infinitesimal disturbances in input parameters [42]. This section gives a precise view of global parametric sensitivity analysis and the parametric influence on output variables. The model variance i V is used to compute the sensitivity index Si . The effect of each individual parameter on the output can be measured from Si [42]. The parametric sensitivity of each process is evaluated considering uniform sampling for 500 P N = samples. The simulation study in Section 4 confirms that the CSTR and Bioreactor processes are immensely sensitive to nonlinearity and excitation signal u . Therefore the metric 0 ∆ changes with respect to operating point P , inputs nom u , prbs u and Sensitivity index Si .
[Insert Fig. 12 here] Fig.12 shadows the sensitivity of the bioreactor with respect to variations in each parameter for around 10% variations. The yield coefficient XS Y gives a maximum sensitivity index Si as seen from Fig.12 (a) for a particular time instant 10 t hours = .A scatter plot was obtained between the most influencing parameter XS Y and the output variable X in Fig. 12 and maximum Si at 10 t hrs = .
[Insert Fig. 13 here] Referring from Eqs. (24,25) in the exothermal CSTR, the variations in input parameter / ER has a significant effect on the output as viewed from Fig.13 (a). For a parametric variation 8000 10000 / ER <≤ , the reactor temperature () TK , swirls around 460 K in Fig. 13(b). However, for 10000 / ER ≥ , the temperature makes a sudden fall to 350 K .Also in Fig.13(c), we can see a nonlinear relationship between the sensitivity index / Si ER and the output variable () TK. It is evident from Figs. 12 and Fig.13, that the PSA has a major implication on identifying the dynamics of nonlinear systems, which in turn will affect the identification of the nonlinear systems.

Conclusion:
In the present work, we have devised kSINDYc, N3ARX and NL2SQ system identification methods which are popular for identification of many nonlinear dynamic physical systems. Input-() pred yt from the three identification methods kSINDYc, N3ARX and NL2SQ are validated with the true response () act yt through RMSE for all the case studies. In particular, the kSINDYc identification method is formulated by tuning the sparcification knob λ set in the ratio between 0 and 10. Additionally, the CANM 0 ∆ targets to find out the degree of nonlinearity of those nonlinear examples. A fresh quantitative analysis is exemplified to cohere the nonlinear metric 0 ∆ with the above said identification methods. Finally, a global parametric sensitivity analysis was investigated on CSTR and Bioreactor to perceive the changes in the nonlinear dynamics caused by the varying the sensitive process parameters. The presented adaptation of these advisable system identification methods is therefore considered important for all users who are interested in finding an interpretable, identification method for complex and diverse nonlinear systems.

Scope:
This article exemplifies a fresh quantitative analysis that correlates the nonlinear metric 0 ∆ with the most admired non-parametric and parametric system identification schemes. Initially, the nonlinear dynamic processes are assorted in a single framework, as mild, medium and highly nonlinear using CNLM method. However, if the process is unstable, the index 0 ∆ fails to measure the nonlinearity, due to the inadequacy in finding steady state operating point. Consequently, the measure may be deficient, when the process is operated in a region far beyond the operating point. Such deprivation issues have to be addressed in the sequel while measuring nonlinearity. Conclusively, this research study will be definitely instrumental for the researchers and academicians of nonlinear dynamics community, but needs to be further tested in real-world physical systems.