1 Introduction

UAVs have been a growing resource for remote sensing of precise agricultural, military, civic, and industrial applications [1]. The UAVs are equipped with cognitive equipment that allows them to find their position, recognise obstacles, and govern their motions using proper navigational procedures. An effective UAV strategy can minimise search time and autonomy capital costs. Gul et al. [2] established the importance of proper selection of navigational strategies for course planning, optimization and obstacle avoidance. The challenges posed by the diversity of the UAVs design and models were worked upon by technological advancements in aviation sector and ground control vehicles [3,4,5,6,7,8,9,10,11,12,13,14,15] and development of hi-fidelity systems [16,17,18,19,20,21,22,23,24,25,26,27,28,29].

The main advantage of UAVs is their ability to collect diverse information in limited time and even limited resources. Most important factor for a successful UAV mission is a robust control systems. for which extensive research has been made in linear and non-linear domain [3, 4, 8, 9]. Work that is comparable to that which has been done in the field of ground robotics has also been done by Gul et al. [16, 19, 20, 30,31,32].

System identification overcomes the cost and time limitation of wind tunnel testing, which is the commonly used method for determining the parameters of these dynamic systems. Extensive research has been made available on system identification techniques applied in different walks of life and UAVs as well. These research works show that the autopilot control system of the UAV can be designed and verified using the developed dynamic model [32, 33].

1.1 Related Work

The proprietary nature of UAVs can be attributed to the sparseness of the open literature available on design and development [34]. the research work presented by Mir et al. in [3,4,5,6] provides a valuable addition in the design and optimization of small-scale quad-copters. However, the literature shows that research carried out based on system identification techniques is mostly limited to one or two techniques. Other UAV design optimization and controllability approaches, like a variable-span morphing wing UAV, have been extensively discussed by Mestrinho et al. [35]. Similarly, Mir et al. [3,4,5, 7,8,9,10,11,12] have made a significant impact on the development of a bio-inspired UAV’s soaring energetics.

It (grey box modeling approach) has been used by Hopping and Garrett [34] to simulate the Taurus UAV’s longitudinal dynamics using prediction error method (PEM). Same approach was further used by Ahsan et al. [36] for modeling the lateral dynamics of the same UAV. Stillfurther, Rasheed [37] used the same approach and developed the longitudinal and lateral dynamics of SmartOne UAV.

Belge et al. [38] obtains model parameters of a UAV with different SI modle structures including ARX, ARMAX and OE models under external disturbances. Similarly, ALtan et al. ( [39, 40]), used ARX model structure to design Load Transporting System (LTS) model analyzing helicopter flight dynamics using transfer functions. Dube and Pedro [41] model a quadrotor-based aerial manipulator’s system identification. ARX and ARMAX models are based on linear and yaw accelerations [41].

Cavanini et al. [42] offers an innovative online estimating method based on the LPV-ARX model that is low-cost and light on storage space. A key benefit of his methodology is that it eliminates the need for a time-consuming training step, which allows for the suggested approach to be used online without degrading its knowledge base. Aerosonde is a UAV designed to collect meteorological data and uses a Model Predictive Control (MPC)-based autopilot which is presented by [43]. His design is based on data driven LPV models derived using two different MPC-LPV based algorithms. Belge et al. [44] used a hybrid meta-heuristic optimization algorithm called Harris hawk optimization (HHO)-grey wolf optimization (GWO) to find the best way to plan and track a path. His novel approach enables the UAV to actualize the payload hold-release mission avoiding obstacles with an additional benefit of generating robust optimal path without getting held in local minima. SISO mathematical model based on frequency response has been obtained by Saengphet et al. [45] using the post-flight input and output data of a Tailess UAV.

Using a Box-Jenkins model structure, Bnhamdoon et al. [46] identification provides a unique approach for identifying a quad-copter autopilot system in noisy conditions. Real-time detection of quadrotor UAV dynamics through deep learning approaches has also been investigated [47]. Munguia et al. [48] provided an additional online estimating approach for UAV utilising Extended Kalman Filter (EKF) technology.

Puttigue et al. [49] makes use of nonlinear and complicated UAV models created by a system identification technique based on artificial neural networks. These models may be accessed both online and offline (ANN). For the graph partitioning problem, an approximate solution is provided by applying the neural network technique [50]. To get a high-quality solution, the algorithm follows the route of a barrier problem’s minimum points as the barrier parameter decreases from a big positive integer to zero. Hoffer et al. [51] and Mir et al. [12, 15] conducted a survey of several system identification approaches and their applicability for tiny low-cost aerial vehicles. Sierra et al. [52] identify alternative control-oriented models of a quad-rotor UAV. Similarly, Khalil et al. [53] compares ARX linear estimation with Hammerstein-Wiener nonlinear estimation for ARF-60 UAV models.

The cited literature is only a small sample of the work that has been done by other scholars. Aerial drones with fixed wings or multiple rotors receive the majority of the attention in the research literature. There is still a lot of room for advancement in study in the field of comparing and analysing various linear and non-linear system identification approaches side-by-side for UAVs [5, 6].

1.2 Motivating Behind this Research

The costs and collateral effects of a collision make model prediction and performance analysis using trial flights for UAVs or other aerial vehicles impractical. An alternative to wind tunnels and CFD research for aircraft mathematical modelling is system identification. Although the literature shows the use of system identification in model estimation of UAVs, the number of model structures used and compared for these research works is very limited. Similarly, linear and nonlinear predicted models have not been compared. Researchers wanted a platform to evaluate linear and non-linear model estimates and validation. This lack of research on multiple model structures encouraged the authors for this paper.

1.3 Contributions by this Study

As shown in the prologue of linked study, relatively little free literature compares system identification strategies for UAVs. This study attempts to provide a unified platform for estimating any UAV model combining linear and nonlinear regression approaches, individual training, and viewpoint analysis of prediction models. The authors of this paper provides the readers with a platform giving a comprehensive comparison of linear and nonlinear model structures. the proposed scheme can be applied for different platforms.

1.4 Sequence of this Paper

Following is the paper’s structure. In Sect. 2, a quick overview of system identification is presented. The findings and analysis section offers linear and nonlinear parametric model responses and residue analysis. Final model selection involves comprehensive investigation and comparison. The author verified the final model by simulating based UAV mission data. Finally, the conclusion and limits are given.

2 Problem Formulation and Methodology

Ultimately, the purpose of this work is to provide a precise model for UAV. MATLAB was used to identify the system. To begin with, any simulation programme can be used to gather data for the proposed study method. Research is based on the pre-processing and filtering of simulation data that covers the whole regime from takeoff to cruise, integrating varied heights and environmental variables of a UAV. Next, several linear and non-linear identification approaches are applied to the simulation data to identify the model. The next step is to train a model to do each method on its own.The best-fitting model is then selected based on model quality parameters such as final prediction error, percentage of fit to simulation data, and analysis of the residuals. Validation of the chosen model on varied simulated data and analysis of the findings are the last steps.

2.1 Model Structures

In this research, a number of model structures are used to mimic the UAV’s MIMO dynamics. Inputs consisted of aileron deflection(deltaa) and Vtail deflection (deltaEe), while outputs included yaw rate (P), pitch rate (Q), and roll rate (R). Firstly a basic idea of the system under test is taken using the Finite Impulse response (FIR) Model, which is a non-parametric estimation technique. FIR model provides the reaction of the system to an impulse or shock in one or more variable using the expression given in Eq. (1), where output y(t) is given by the convolution sum of the input u(t) [54].

$$\begin{aligned} \begin{aligned} y(t)= \int _0^t \! h(t-z)\! u(z) \, \mathrm {d}z \end{aligned} \end{aligned}$$
(1)

The gathered system information, such as Time delay and model order, was then used to determine parametric models.

2.1.1 Linear Parametric Models

In the class of linear model structures, ARX, ARMAX, Box Jenkins, Output Error and state space model structures have been used for analysis and comparison. Equations (2), (3), (4), (6) and (7) represent the mathematical expressions used for ARX, ARMAX, Box Jenkins, Output Error and state space model structures respectively as given by Fatima et al. [54].

$$\begin{aligned} \begin{aligned} \! A(q) \! y(t) = \sum _{i=0}^{nu} \! B_i(q) \!u_i(t-nk_i) + \!e(t) \end{aligned} \end{aligned}$$
(2)

ARX model

$$\begin{aligned} \! A(q) \! y(t)= & {} \sum _{i=0}^{nu} \! B_i(q) \!u_i(t-nk_i) + \!C(q) \!e(t) \end{aligned}$$
(3)
$$\begin{aligned} \! y(t)= & {} \sum _{i=0}^{nu} \frac{\! B_i(q)}{\! F_i(q)} \!u_i(t-nk_i) + \frac{\!C(q)}{\!D(q)} \!e(t) \end{aligned}$$
(4)
$$\begin{aligned} \! y(t)= & {} \sum _{i=0}^{nu} \frac{\! B_i(q)}{\! F_i(q)} \!u_i(t-nk_i) + \!e(t) \end{aligned}$$
(5)
$$\begin{aligned} \! y(t)= & {} \sum _{i=0}^{nu} \frac{\! B_i(q)}{\! F_i(q)} \!u_i(t-nk_i) + \!e(t) \end{aligned}$$
(6)
$$\begin{aligned} \! x(t+1)\,=\, & {} A \!x(t) + B \! u(t) + K \!e(t)\nonumber \\ \! y(t)\,=\, & {} C \! x(t) + D \! u(t) + \!e(t) \end{aligned}$$
(7)

Where A, B, C, D, and K are system matrices represented in \(q-1\) time shift operator.

2.1.2 Nonlinear Parametric Models

This model uses a weighted sum of linear predictors, which is a combination of both new data and data from the past to forecast future outcomes. Using F as a model regressor in non-linear ARX provides more structural flexibility. Multilayered neural network regressors were employed in conjunction with wavelet network non-linearity, tree partition non-linearity, and non-linearity in wavelet networks. The output of the first two estimators is computed using offset, linear weights, and non-linearity functions, the latter of which uses radial input combinations. In the multilayered neural network estimator, there are input, output, and perhaps hidden layers. This approach can detect complex correlations between inputs and outcomes. After being trained and selected, the network will produce correct answers for the regime it was taught, but its results may be misleading for input and output values outside the regime.

3 Results and Analysis

The authors present the findings of several models to describe the MIMO dynamics of a UAV employing aileron deflections and the Vtail deflections as inputs and yaw rate (P), pitch rate (Q) and roll rate (R) as the outputs. As per the literature available, system identification methodology splits the available data into two parts: the first portion is used to estimate and validate the model, which is then re-used for the second part of the simulation data. For each class of model structures, we went through several stages of development and training. FPE was used to choose the best model, and the quality of that model is further examined using FPE, residual analysis, percentages of fit between model predicted output and outputs from simulations and MSE.

Further consideration is given to the results by presenting the reader with a comparative study of the best model of each approach, and the final model is picked based on an examination of the model’s quality using the previously mentioned criteria.

3.1 FIR Model Response

A zero \(T_s\) time delay is assumed in the following procedures based on an analysis of the FIR model response shown in Fig. 1. There is a 53.59 percent match between the modelled and real pitch, roll, and yaw rates for the FIR model’s FPE and MSE, as well as a 34.31 percent fit for the model’s MSE.

Fig. 1
figure 1

Finite Impulse Response

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3.2 Linear Parametric Models Response

Several combinations of the order of polynomials A(q), B(q), C(q), D(q) and F(q) (represented by \(N_a\), \(N_b\), \(N_c\), \(N_d\) and \(N_f\)) were tried out iteratively to arrive at the final model order for the each of the linear model structures referred in Sect. 2.1.1. In the end, the best model with a minimal FPE and MSE was chosen for each model class after approximately 1000 models were generated for each structure. Residue corelation plots for each model structure is shown in Figs. 2, 3, 4, 5 and 6. these plots indicate that the part of the simulated response which was not being predicted by models is well within the confidence region which in this case has been taken to be 95 %.

Fig. 2
figure 2

Residue Correlation: ARX Model

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Fig. 3
figure 3

Residue Correlation: ARMAX Model

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Fig. 4
figure 4

Residue Correlation: Box Jenkin’s Model

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Fig. 5
figure 5

Residue Correlation: Output Error Model

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Fig. 6
figure 6

Residue Correlation: State Space Model

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Similarly, Figs. 7, 8 and 9 shows the fit percentages between 1-step ahead predicted response of each of the selected linear parametric model structure and simulated data.

Fig. 7
figure 7

Linear Parametric Model Response: Roll Rate

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Fig. 8
figure 8

Linear Parametric Model Response: Pitch Rate

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Fig. 9
figure 9

Linear Parametric Model Response: Yaw Rate

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3.3 Non-linear Parametric Model Responses

Further extending our proposed methodology, the nonlinear parametric model structures were used for analysis and comparison of the UAV model. Figure  10 illustrate the comparison between the simulation data outputs and the predicted outputs of our nonlinear models. Figures 11, 12, 13 and 14 show residue correlation graphs of non-linear ARX models employing tree partition, wavelet network, and neural network estimators. Estimators created the models. For comparisons, the order of the model was set to the best-fit ARX model.

Fig. 10
figure 10

NLARX Model Responses

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Fig. 11
figure 11

Residue Correlation:NLARX Model (Tree Partition)

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Fig. 12
figure 12

Residue Correlation: NLARX Model (Wavelet Network)

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Fig. 13
figure 13

Residue Correlation:NLARX Model(Neural Network)

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4 Comparative Analysis

For this research, all linear and non-linear parametric model estimating methodologies used are compared in Table1, which lists the best model based on various criteria, including model order, FPE, Mean Squared Error (MSE), percentage of fit of roll, pitch, and yaw rates, number of free parameters, viewpoint analysis of residue correlation, and the number of free parameters.

Table 1 Comparison Analysis of Linear & Nonlinear Parametric Model Responses
Full size table

ARAMX and Nonlinear ARX models with wavelet network estimator deliver superior FPE, MSE, fit percentages, etc. Residue analysis chose the ARMAX model. Next, the selected model was validated using similar simulation data with different height and velocity profile. Figures 15 and 16 compare projected ARMAX model output with validation data. The ARMAX model’s fit percentages and residue correlation graphs are found to be adequate.

Fig. 14
figure 14

Residue Correlation:NLARX Model(Neural Network)

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Fig. 15
figure 15

Validation of ARMAX Model with Different Simulation Data

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Fig. 16
figure 16

Residue Correlation: ARMAX Model (Validation Sortie)

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4.1 Observations and Opinions

As shown in Table 1, linear and non-linear parametric models produce acceptable results for future design revisions and simulations of the test UAV. High discrepancies between expected and observed responses make the output error model unusable. The output error model predicts model response using only prior inputs, not inputs and outputs.

Model order is higher in linear ARX than in linear ARMAX. To provide a more accurate forecast, ARMAX models are often used with smaller model orders. There are fewer coefficients in Box Jenkins and state space models since they have a smaller model order. In terms of fit percentages, mean squared error, and ultimate prediction error, the ARMAX and ARX models are more valuable than other alternatives.

Complexity and processing time restrict the usage of nonlinear ARX models with tree partition, wavelet network, and neural network. Residual analysis is the final component in choosing the appropriate model structure. ARMAX model likewise performs well on this parameter. ARMAX model was chosen to verify simulation data.

4.2 Research Limitations

The results for the current study are obtained using simulations data. The validity of the research can be further enhanced using post flight data for any aerial vehicle.

4.3 Future Work

This study may be expanded to examine the impact of modifying the model identification inputs. It is possible to add inputs like as height, speed, and push to observe the reaction of each model (input parametric sweep). Similarly, data from different simulations with broader parametric ranges may be used to investigate the influence on each model’s predictive capacity.

5 Conclusion

Authors developed a UAV fLight dynamics model employing system identification. Various parametric linear and nonlinear models with diverse architectures were created. The dataset comprised impulse response, ARX, ARMAX, Box Jenkin’s, Output Error, State Space, Nonlinear ARX models with tree partition, wavelet network, and neural network models. In order to select the best model for a given structure, several models were constructed and trained. A full perspective analysis was performed on all models, and after comparison and analysis, the ARMAX model with FPE 0.0022 was chosen as the best match. The model was then used to forecast the output of a new sortie with the same flying regime. The roll rate (P), pitch rate (Q), and yaw rate (R) were all within a tolerable range, standing at 88.91%, 71.42% and 79.69%.