## Appendix C  
This section describes the stability and convergence analysis of the proposed RBFNNPID algorithm. The basic knowledge of the stability and convergence analysis is Lyapunov theorem, also known as Lyapunov stability. \(V(x_{1},x_{2},K,x_{N})\) is an arbitrary function defined in the neighborhood of the origin \(\Omega\), where \(\Omega\) is a state of equilibrium and \(x_{1},x_{2},K,x_{N}\) are variables, then  
\[|x_{i}|\leq H,i=1,2,K,N \tag{1}\]  
where, \(H\) is a positive constant. Assuming that \(V\) is a continuous differentiable function in \(\Omega\) and \(V(0,0,K,0)=0\). Such that  
1. \(V(x)>0\) is positive definite or \(V(x)<0\) is negative definite, \(x\in\Omega\) and \(x\
eq 0\);
2. \(V(x)>0\) is positive semi-definite or \(V(x)<0\) is negative semidefinite, \(x\in\Omega\);  
Consider an autonomous nonlinear dynamical system  
\[\dot{x}=f(x) \tag{2}\]  
where, \(f(0)=0\). Assuming \(x_{i}=x_{i}(t)\), (\(i=1,2\), \(K\), \(N\)) is the solution of the system (2). We can obtain the derivation  
\[\frac{dV}{dt}=\frac{\partial V}{\partial x_{1}}\frac{\partial x_{1}}{\partial t }+\frac{\partial V}{\partial x_{2}}\frac{\partial x_{2}}{\partial t}+K+\frac{ \partial V}{\partial x_{N}}\frac{\partial x_{N}}{\partial t} \tag{3}\]  
Introducing the gradient vector (Equation (A4)) into Equation (3)  
\[\
abla V(x)=\left[\frac{\partial V}{\partial x_{1}},\frac{\partial V}{\partial x _{2}},L,\frac{\partial V}{\partial x_{N}}\right]^{T} \tag{4}\]  
We will arrive at the final equation as below:  
\[V=\left[\
abla V(x)\right]^{T}\!f(x)=\omega(x) \tag{5}\]  
The following conclusions can be made from the above analysis:  
1. If \(V(x)\) is positive (or negative) definite, and if derivation \(V=\omega(x)\) is negative (or positive) semi-definite, the system is said to be Lyapunov stable at the equilibrium of the origin;
2. If \(V(x)\) is positive (or negative) definite, and if derivation \(V=\omega(x)\) is negative (or positive) definite, the system is said to be exponentially stable at the equilibrium of the origin;
3. If \(V(x)\) is positive (or negative) definite, and if derivation \(V=\omega(x)\) is also positive (or negative) definite, the system is said to be unstable at the equilibrium of the origin;  
For the adjustment of the weights of the neural network, we need a parameter called learning rate \(\eta\). If \(\eta\) is too large, NN will be unstable; but if \(\eta\) is too small, the convergence rate will be too slow. Therefore, the selection of the value of learning rate is crucial to the stability and convergence of the system. Assuming the indicator function of the RBFNNPID controller  
\[J(k)=\frac{1}{2}[y(k)-y_{m}(k)]^{2}=\frac{1}{2}e^{2}(k) \tag{6}\]  
where, \(e(k)\) is the learning error of the network. In order to ensure the adjustment of the weight coefficient is carried out in the direction of the negative gradient relative to \(\omega(k)\), there must be  
\[\omega(k+1)=\omega(k)-\eta\frac{\partial J(k)}{\partial\omega(k)}\] (A7)  
From Equations (A6) and (A7), we can get  
\[\Delta\omega(k)=\frac{\partial J(k)}{\partial\omega(k)}=e(k)\frac{\partial e( k)}{\partial\omega(k)}=e(k)\frac{\partial e(k)}{\partial\Delta u(k)}\frac{ \partial\Delta u(k)}{\partial\omega(k)}\] (A8)  
Defining a Lyapunov function of a discrete-time systems as  
\[v(k)=\frac{1}{2}e^{2}(k)\] (A9)  
As we introduced in the paper, the gradient descent method is used as the change of the network-learning algorithm. \[\Delta v(k)=v(k+1)-v(k)=\frac{1}{2}e^{2}(k+1)-\frac{1}{2}e^{2}(k)=\frac{1}{2} \Delta e(k)[2e(k)+\Delta e(k)]\] (A10)  
where, \(e(0)=0\), and,  
\[\dot{x}=f(x) \tag{2}\]0 (A11)  
It can be obtained from Equation (A8), that  
\[\dot{x}=f(x) \tag{2}\]1 (A12)  
and  
\[\dot{x}=f(x) \tag{2}\]2 (A13)  
By substituting Equation (A13) into Equation (A12)  
\[\dot{x}=f(x) \tag{2}\]3 (A14)  
Then, substituting Equation (A14) into Equation (A10)  
\[\dot{x}=f(x) \tag{2}\]4 (A15)  
Knowing from the Lyapunov stability theory, the system is stable when \(\Delta v<0\).