# Nonlinear Aeroelastic System Identification Based on Neural Network
## 2 Neural Network System Identification  
Traditional identification algorithms, such as the Volterra kernel, Hammerstein-Winner, and harmonic balance methods, need to establish approximate models corresponding to actual systems. However, the actual aeroelastic system is complex and influenced by many nonlinear factors. Thus, establishing corresponding mathematical models of the system is difficult. ANN has good nonlinear mapping ability, self-learning adaptability, and parallel information processing ability. This method can identify essential nonlinear systems and is an ideal tool for identifying unknown and uncertain nonlinear systems. The neural network is also a physical realization of the system and can be used for online control.  
The TDRNN consists of input, hidden, and output layers. The hidden layer contains the input and feedback nodes and nonlinear and linear neurons. A feedback connection is present in the hidden layer. The time-delay memory elements (\(Z^{-1}\)) are used to store previous values of the input signal and output (feedback) signal. \(W\) denotes the weight of the neuron. The weights (\(W\)) can be adjusted during the training process until accuracy of the neural network meets requirements. b represents the bias of the neuron. \(f_{1}\) and \(f_{2}\) are transfer functions. The structure of the TDRNN is shown in Figure 1.  
The input-output relationship of the TDRNN is written as follows:  
\[y(t)=f(y(t-1),y(t-2),\ldots,y(t-p),x(t-1),\ldots,x(t-q)) \tag{1}\]  
Figure 1: Structure of the time-delay recurrent neural network (TDRNN).  
where \(x(t)\) is the input value at \(t\) moment, \(y(t)\) is the output value at \(t\) moment. The output signal \(y(t)\) is regressed on previous values of the output signal \((y(t-1),y(t-2),\ldots,y(t-p))\) and previous values of an independent (exogenous) input signal \((x(t-1),\ldots,x(t-q))\).  
The mean squared error (MSE) of training data is used to evaluate performance of the proposed TDRNN, which is defined as:  
\[MSE=\sum_{t=1}^{N}(\hat{y}(t)-y(t))^{2}/N \tag{2}\]  
where \(\hat{y}(t)\) is the estimated value of the TDRNN, \(y(t)\) is the real value of the nonlinear aeroelastic system, which is obtained from the numerical simulation or wind-tunnel experiment. \(N\) is the number of the calculated data values.  
Taking control surface flap deflection (\(\beta\)) as the system input signal and pitch angle (\(\alpha\)) as the only considered output signal, the aeroelastic system is a single-input-single-output (SISO) system. The flowchart for neural network identification is shown in Figure 2. Measurement noise with an intensity of signal-to-noise ratio (SNR) = 20 dB is added to the output signal to simulate the actual environment. A TDRNN is used to approach the nonlinear aeroelastic system. The input and output data obtained from the simulation is divided into three parts. The first part is the training set (70%), which is used to compute the gradient and update network weights and biases. The second part is the validation set (15%), which is used to evaluate the network during the training process. The third part is the test set (15%), which is used to compare different models. After the network has been trained, it can be used to estimate the response of the nonlinear aeroelastic system.