## 3 Fault-Tolerant Aircraft Controller Design
### Design of the Gradient Descent-Based Spatio-Temporal RBF Neural Network  
Radial basis function neural networks have many advantages in function approximation due to their simple structure, low computational complexity, and small required amount of data, and have arbitrary nonlinear mapping ability, which can be used to estimate uncertain parameters in nonlinear systems. Therefore, considering the nonlinear characteristics of the input and output data of the actuator fault model and wind disturbance model, radial basis neural networks were used in this study for parameter estimation. In contrast, conventional signal processing is mostly conducted with a single temporal or spatial signal. However, combining time-domain information with spatial information can provide more advantages for tracking target information. Referring to the ST-RBFNN used for chaotic time series prediction in [32], we propose a new RBFNN to estimate the actuator fault data and external disturbance data using its advantages in processing time dynamic characteristics and spatial nonlinear (complexity) signals, as shown in Figure 3.  
A Spatio-Temporal RBF Neural Network (ST-RBFNN)  
Initially, the input of the input layer was defined as \(\left[ \begin{array}{cc}x(t)&x(t-1)\end{array} \right]^{T}\), where \(x(t)\in\mathbb{R}^{P}\), \(x(t-1)\in\mathbb{R}^{P}\), \(x(t-1)\) is the sampled value of the previous moment of \(x(t)\). \(x(t)=\left[ \begin{array}{cc}\phi\;\psi\;\phi\;\phi\;\phi\;\psi\;u_{1}\;u_{2} \;u_{3}\;u_{4}\end{array} \right]\) includes the attitude angle, angular velocity, and control law signal of the aircraft. Subsequently, when designing the nonlinear hidden layer, the temporal expansion was considered in the kernel space of the neural network in the signal processing. As a result, two parallel time layers corresponding to the input were designed for mapping the dynamic and nonlinear characteristics of the signal in time.The output layer adopted a linear combination output. Consider ST-RBFNN for parameter estimation \(\varsigma(k)\). The expected value of the \(k\)th target signal can then be defined as \(d(k)\). The error between the network estimate and the network expected value is \(\delta(k)=d(k)-\varsigma(k)\). The corresponding cost function is  
\[\epsilon(k)=\frac{1}{2}(d(k)-\varsigma(k))^{2}=\frac{1}{2}\delta^{2}(k). \tag{21}\]  
The overall mapping of the spatio-temporal radial basis function neural network adopted in this study was derived as follows:  
\[\varsigma(k)=\sum_{i=1}^{p}\sum_{t=1}^{T}w_{(i,t)}(k)\psi\Big{(}\Big{\|}x-c_{( i,t)}\Big{\|}\Big{)}+b(k), \tag{22}\]  
Figure 3: Structure of ST-RBFNN. where \(\varsigma:\mathbb{R}^{P}\rightarrow\mathbb{R}^{1}\), \(w_{(i,t)}(k)\) is the current weight value of the hidden layer to the output layer, \(p\) is the number of neurons in the hidden layer of the neural network, \(T\) is the cutoff time, \(c_{(i,t)}\in R^{m_{0}}\) is the central value of the neural network, and \(b(k)\) is the deviation term for the output neuron. \(w_{(i,t)}(k)\) and \(b(k)\) is updated with each iteration, and \(\psi_{(i,t)}\) is the basis function of the hidden layer of the neural network. First, the common Gaussian kernel function was adopted by [36]  
\[\psi_{i1}\Big{(}\Big{\|}x-c_{(i,t)}\Big{\|}\Big{)}=\exp\Bigg{(}\frac{-\Big{\|} x-c_{(i,t)}\Big{\|}^{2}}{\sigma^{2}}\Bigg{)}. \tag{23}\]  
The gradient descent learning algorithm of RBFNN based on spatio-temporal extension was designed as follows:  
\[w_{(i,t)}(k+1)=w_{(i,t)}(k)-\kappa\delta_{w_{(i,t)}}\varepsilon(k), \tag{24}\]  
where \(w_{(i,t)}(k+1)\) is the updated weight value, \(\kappa\) is the learning step length, \(\delta_{w_{(i,t)}}\) is the regular derivative, and \(-\delta_{w_{(i,t)}}\varepsilon(k)\) can be estimated by the differential chain rule.