## 2 Materials and Methods
### A Neural Network Based Adaptive PID Algorithm  
#### 2.2.1 Radial Basis Function (RBF) Neural Network  
Artificial neural network (ANN) is an artificial intelligence system to imitate biological neural networks (BNN). It uses nonlinear processing unit to simulate biological neurons for simulating the behavior of biological synapses among neurons by adjusting the variable weights between connected units. The specific topological structure of the network is organized from each processing unit in a certain connected form. Parallel processing ability and distributed storage are the main features of ANN. Furthermore, it has strong fault tolerance and nonlinear mapping ability with self-organization, self-learning and adaptive reasoning ability [33]. BP (backpropagation) network and RBF network are the most widely used forms of ANN. It is easily to be seen in the widely uses of pattern recognition, prediction, automatic control, etc. [34]. BP algorithm, a supervised learning algorithm, is based on gradient descent algorithm. The drawbacks of BP include an easy fall into local optimum, slow convergence speed, and disunity network structure. RBF network is a feedforward network based on the function approximation theory. It has strong global approximation ability, which can guarantee the network to approximation any kind of nonlinear function with arbitrary accuracy. It can fundamentally overcome the problem of local optimum occurs in BP network. The RBF network has the advantages of simple structure, fast convergence speed and strong generalization ability [35]. Radial basis function (RBF) neural network used in this paper is a three-layer forward network, which is a local approximation method of neural networks. The RBF neural network is composed of three layers, the input layer, the hidden layer and the output layer as shown in Figure 3. The mapping of the input layer to the output layer is nonlinear and the mapping of the space from the hidden layer to the output layer is linear. This kind of mapping configuration itself can speed up the learning rate and avoid the problem of local minima [18]. In Figure 3, the input vector of the input layer of the neural network is represented as:  
\[X=[x_{1},x_{2},\cdots,x_{s},\cdots,x_{n}]^{T} \tag{4}\]  
where, \(x_{s}\) = [\(u_{s}\)(\(k\)), \(y_{s}\)(\(k\)), \(y_{s}\)(\(k-1\))], s = 1, 2, \(\ldots\), \(n\); \(u\)(\(k\)) is the output of the controller; \(y\)(\(k\)) is the present (measured) output of the system (or process), that is, the measured value of DO concentration; \(y\)(\(k-1\)) is the last measured value of DO concentration output from the process. Figure 3: Topology of a radial basis function (RBF) neural network. The middle layer is the hidden layer. The activation function of the hidden layer is composed of radial basis functions. Each array of computing units of hidden layers is called node. The radial basis vector of the nodes in the RBF neural network is shown in Equation (5). \[T=[h_{1},h_{2},\cdots,h_{j},\cdots,h_{m}]^{T} \tag{5}\]  
where, \(h_{j}\) is Gaussian function,  
\[h_{j}=\exp(\frac{\|X-C_{j}\|}{2b_{j}^{2}}) \tag{6}\]  
where, \(j=1\), \(2\),..., \(m\). \(C_{j}\) is the central vector of the first \(j\) node of the hidden layer of the RBF neural network,  
\[C_{j}=[c_{j1},c_{j2},\cdots,c_{ji},\cdots,c_{jn}]^{T} \tag{7}\]  
where, \(i=1\), \(2\),..., \(n\). The basic width vector of the hidden layer node of the RBF neural network is  
\[B=[b_{1},b_{2},\cdots,b_{j},\cdots,b_{m}]^{T} \tag{8}\]  
where, \(b_{j}\) is the parameter of the first \(j\) node and \(j=1\), \(2\),..., \(m\). The weight vector of RBF neural network \(W\) is given by:  
\[W=[w_{1},w_{2},\cdots,w_{j},\cdots,w_{m}]^{T} \tag{9}\]  
Then, the estimated output of the RBF network is defined as:  
\[y_{m}=w_{1}h_{1}+w_{2}h_{2}+\cdots+w_{m}h_{m} \tag{10}\]  
The performance index function of the RBF neural network is set as follows:  
\[E_{1}=\frac{1}{2}(y(k)-y_{m}(k))^{2} \tag{11}\]  
where, \(y(k)\) is the system output and \(y_{m}(k)\) is the estimated output of the RBF network.