# An Online Energy Management Control for Hybrid Electric Vehicles Based on Neuro-Dynamic Programming
## 2 Preparation of Knowledge about the NDP EMS
### Common Wavelet Neural Network  
The wavelet neural network is based on a wavelet transformation theory and neural network. Different from broadly used back propagation neural network (BPNN) and RBFNN, a wavelet neural network, which we call a common wavelet neural network (CWNN), utilizes a wavelet function as the activation function in the hidden layer. Compared with a RBFNN, a CWNN does not need a clustering preprocessing on the input data, and has same approximation ability. A commonly used topology of a CWNN is shown in Figure 2.  
In the CWNN, the input of hidden layer is as follows:  
\[h^{\prime}_{h}=\sum_{i=1}^{I}\omega_{ijk}x_{i},\quad h=1,2,\cdots,H, \tag{10}\]  
Figure 2: Topology of CWNN.  
1. Causality and consistency. \[\cdots\subset V_{m+1}\subset V_{m}\subset V_{m-1}\subset\cdots,\,m\in Z\] (15)
2. Scaling regularity. When we approximate a function \(f(x)\in L^{2}(R)\) with \(f_{m}(x)=A_{m}f(x)\) at the resolution \(2^{m}\), where \(A_{m}\) is a projection of \(f(x)\) in subspace \(V_{m}\). Then the following equation will be satisfied: \[f_{m}(x)=A_{m}f(x)\in V_{m}\Leftrightarrow f_{m-1}(x)=A_{m-1}f(2x)\in V_{m-1}, \,\,m\in Z.\] (16)
3. Gradual completeness. \[\underset{m=-\infty}{\cap}V_{m}=\{0\},\underset{m=-\infty}{\cup}V_{m}=L^{2}(R)\] (17)
4. Orthogonal basis existence. \[V_{m}=\overline{linear\,span\{\phi_{mk},k\in Z\}},\] (18) where \(\phi(t)\in V_{0}\) is a unique function, called a scaling function. Meanwhile, \(\phi_{mk}(x)=\sqrt{2^{-m}}\phi(2^{-m}x-k)\), \(m\in Z\) and \(k\in Z\).  
When we approximate a function \(f(x)\) with \(f_{m}(x)=A_{m}f(x)\) at the resolution \(2^{m}\), we can conclude the following equation based on Feature (4):  
\[f_{m}(x)=A_{m}f(x)=\underset{k=-\infty}{\overset{+\infty}{\sum}}a_{mk}\phi_{ mk}(x). \tag{19}\]  
We define \(W_{m}\) be the orthogonal space of \(V_{m}\) in \(V_{m-1}\). In other words, \(V_{m-1}=V_{m}\oplus W_{m}\) and \(V_{m}\bot W_{m}\). Then,  
\[f_{m-1}(x)=A_{m-1}f(x)=[A_{m}\oplus D_{m}]f(x)=A_{m}f(x)\oplus D_{m}f(x), \tag{20}\]  
where \(D_{m}\) is a projection of \(f(x)\) on the space \(W_{m}\). We can say \(D_{m}f(x)\) is a detail of function \(f(x)\) at resolution \(2^{m}\). [PERSON] has pointed out that a unique wavelet \(\psi(x)\) exists, whose dilations and translations constitute a family of functions \(\{\psi_{mk}\},m\in Z,k\in Z\). These \(\{\psi_{mk}\}\) are the orthonormal basis of space \(W_{m}\). As a result, we get the following equation:  
\[D_{m}f(x)=\underset{k=-\infty}{\overset{+\infty}{\sum}}d_{mk}\psi_{mk}(x), \tag{21}\]  
where \(d_{mk}\) is the projection of \(f(x)\) on the orthonormal basis \(\psi_{mk}(x)\). We obtain  
where \(L\) is the lowest resolution and \(n_{L}\) is the total number of scaling functions. In a real application, we can obtain \(L\) according to the following equation:  
\[L=\text{int}(\text{lg}(x_{\text{max}}-x_{\text{min}})/\text{lg2})\text{ or }L=\text{int}(\text{lg}(x_{\text{max}}-x_{\text{min}})/\text{lg2})\pm 1, \tag{24}\]  
where \(x_{\text{max}}\), and \(x_{\text{min}}\) are the maximum and minimum value of \(x\). Using Equation (19), we will get the following equation at resolution \(2^{0}\):  
\[f(x)\approx f_{0}(x)=f_{1}(x)+\sum_{k=1}^{2^{L-1}n_{L}}d_{1k}\psi_{1k}(x)=\sum _{k=1}^{n_{L}}a_{Lk}\phi_{Lk}(x)+\sum_{m=1}^{L}\sum_{k=1}^{2^{L-m}n_{L}}d_{mk} \psi_{mk}(x). \tag{25}\]  
Equations (19) and (21) are the multi-resolution analysis of functions proposed by [PERSON] in Moody (1977).