# A Novel Reconstruction Method for Irregularly Sampled Observation Sequences for Digital Twin
## 4 Simulation Experiments and Analysis  
This section selects the furnace of the boiler system as the test object to evaluate the reconstruction performance of the developed model. We use the nominal data of a 600 MW coal-fired power plant under 100% rated conditions to train the sequence reconstruction model and validate its effectiveness. Under the rated conditions, it is required that the coal flow rate is about 74.5 kg/s and the excess air coefficient is 1.2; thus, the gas oxygen content in the boiler system can be maintained around 3.2% and the boiler system can operate stably. The reference observation sequence is generated by the digital twin model of the furnace, which is defined by the following equations [31, 32, 33]:  
\[\left\{ \begin{array}{l}\mathbf{x}(k+1)=\mathbf{Ax}(k)+\mathbf{Bu}(k)\\ \mathbf{y}(k+1)=\mathbf{Cx}(k+1)\end{array} \right. \tag{19}\]  
where \(\mathbf{x}\), \(\mathbf{u}\), and \(\mathbf{y}\) represent the system's state, input, and observation vectors, respectively, and they are defined as follows:  
\[ \begin{split}\mathbf{x}&=\left[\begin{array}{ccc}V_{gs}&\rho_{b }&T_{gs}&\mathcal{O}_{cp}\end{array}\right]^{\mathrm{T}}\\ \mathbf{u}&=\left[\begin{array}{ccc}W_{cf}&V_{sa}\end{array}\right]^{\mathrm{T} }\\ \mathbf{y}&=\left[\begin{array}{ccc}V_{gs}&T_{gs}&P_{b}&\mathcal{O}_{cp}\end{array} \right]^{\mathrm{T}}\end{split} \]The matrices \(\mathbf{A},\mathbf{B},\) and \(\mathbf{C}\) are given by:  
\[\mathbf{A}= \begin{bmatrix}a_{11}&0&0&0\\ 0&a_{22}&0&0\\ 0&0&a_{33}&0\\ a_{41}&0&0&a_{44}\end{bmatrix} \quad\mathbf{B}= \begin{bmatrix}0&b_{12}\\ b_{21}&b_{22}\\ b_{31}&b_{32}\\ 0&0\end{bmatrix} \quad\mathbf{C}= \begin{bmatrix}1&0&0&0\\ 0&0&1&0\\ 0&c_{32}&0&0\\ 0&0&0&1\end{bmatrix} \]  
where  
\[\left\{ \begin{array}{l}a_{11}=1-T_{a}\\ a_{22}=1-V_{gs}/V_{b}\\ a_{33}=1-\frac{V_{gs}}{V_{b}}-K_{sl}\big{(}\frac{T_{gs}^{4}-T_{gs}^{4}}{(S_{ gtp}V_{b}T_{gs})}\big{)}\\ a_{41}=21\Big{(}V_{sa}-V_{0}\cdot W_{cf}\Big{)}/(V_{sa}\cdot V_{b})\\ a_{44}=1-V_{gs}/V_{b}\end{array} \right.\quad\left\{ \begin{array}{l}b_{12}=T _{a}\\ b_{21}=1/V_{b}\\ b_{22}=p_{a}/V_{b}\\ b_{31}=Q_{net,ar}\big{/}\big{(}C_{gs}\rho_{b}V_{b}\big{)}\\ b_{32}=H_{a}\rho_{a}/\big{(}C_{gs}\rho_{b}V_{b}\big{)}\end{array} \right.\]  
The physical meanings of the parameters and the value ranges of the input and output quantities are shown in Table 1. To validate the model's performance, we compare the proposed model with the RNN-NODE method from [22] and the improved RNN-NCDE method. The network architecture parameters for each model are listed in Table 2.  
The model training process uses the following settings: number of iterations--200; initial learning rate--\(2\times 10^{-3}\); minimum learning rate--\(2\times 10^{-4}\); decay rate--0.999; hyperparameter--\(\lambda=0.005\). The thresholds for the reference latent feature weights are set as \(r_{1}=5\times 10^{-4}\), \(r_{2}=5\times 10^{-3}\), \(q_{1}=5\times 10^{-6}\), and \(q_{2}=0.9\). The desired regular sequence length is set to \(N^{\prime}=500\), with a sampling interval of \(1s\). The control input \(\mathbf{u}\) is normalized to the range \([0,1]\), and the observation \(\mathbf{y}\) is normalized to the range \([-1,1]\). The experimental analysis focuses on two aspects: (1) performance under different sampling numbers; (2) performance under high noise levels. The relevant information about the parameters of the experimental environment, including CPU, GPU, memory, Matlab and deep learning toolbox version, are detailed in Table 3.