## 4 Simulation Example  
Consider the steam water heat exchanger system in Figure 1, where the steam flow \(u_{1}\) and the water flow \(u_{2}\) are two manipulated variables to control the water temperature \(y\). The discrete input-output representation of the process is expressed as the following equation. \[y(t) = z^{-d_{1}}B_{1}(z)u_{1}(t)+z^{-d_{2}}B_{2}(z)u_{2}(t)+v(t)\] \[B_{1}(z) = 0.95z^{-1}+0.72z^{-2}+0.40z^{-3},\] \[B_{2}(z) = -0.63z^{-1}-0.47z^{-2}-0.25z^{-3},\] \[d_{1} = 42,d_{2}=20\]  
Let \(l=50\); then the parameter vector to be identified is:  
\[\boldsymbol{\theta}= \left[\mathbf{0}_{42},0.95,0.72,0.40,\mathbf{0}_{25},-0.63,-0.47, -0.25,\mathbf{0}_{27}\right]^{\intercal}\in\mathbb{R}^{N}\]  
It can be seen from the above equation that the dimension of the parameter vector \(\boldsymbol{\theta}\) is \(N=100\), but only six of the parameters are non-zeros. In the identification, the inputs \(\{u_{i}(t)\}\), \(i=1,2\) are taken as independent persistent excitation signal sequences with zero mean and unit variances and \(\{v(t)\}\) as white noise sequences with zero mean and variance \(\sigma^{2}=0.10^{2}\). Let \(\epsilon=0.05\), applying the proposed TH-OMP algorithm and the least squares (LS) algorithm to estimate the system with \(m\) measurements. The parameter estimation errors \(\delta:=(\|\boldsymbol{\theta}-\hat{\boldsymbol{\theta}}\|)/\|\boldsymbol{ \theta}\|\)_versus_\(m\) are shown in Figure 2.  
From Figure 2, we can see that under a limited dataset \((m<100)\), the parameter estimation error of the TH-OMP algorithm approaches zero, and the estimation accuracy is much higher than that of the LS algorithm. Let \(m=80\); using the TH-OMP algorithm to estimate \(\boldsymbol{\theta}\), the parameter estimation errors \(\delta\)_versus_ iteration \(k\) are shown in Figure 3; the non-zero parameter estimates _versus_ iteration \(k\) are shown in Table 1 and Figure 4, where the dash dot lines are true parameters and the solid lines are parameter estimates. Figure 1: The steam water heat exchanger. Figure 4: The parameters estimates _versus_ iteration \(k\). Figure 2: The estimation error \(\delta\)_versus_\(m\) measurements. After 10 iterations, the estimation error is \(\delta=0.8841\%\), and the estimated parameter vector is:  
\[\hat{\mathbf{\theta}}_{\epsilon=0.05}=[\mathbf{0}_{42},0.9472,0.7105,0.3921,\mathbf{0}_{25}, 0.6246,0.4751,0.2524,\mathbf{0}_{27}]^{\mathrm{ T}}\in\mathbb{R}^{N} \tag{16}\]  
If we choose \(\epsilon=0.01\), then we can get the estimated parameter vector as:  
\[B_{1}(z) = 0.95z^{-1}+0.72z^{-2}+0.40z^{-3},\]0 \[B_{1}(z) = 0.95z^{-1}+0.72z^{-2}+0.40z^{-3},\]1  
It is obvious that the parameter vector \(\mathbf{\theta}\) has only three zero blocks in this example. However, we can see from the above equation that there exist three undesirable parameter estimates \(\hat{\theta}_{5}\), \(\hat{\theta}_{31}\) and \(\hat{\theta}_{87}\). Moreover, the three parameter estimates are much smaller than other parameters. Therefore, according to the structure of \(\mathbf{\theta}\), the parameters \(\hat{\theta}_{5}\), \(\hat{\theta}_{31}\) and \(\hat{\theta}_{87}\) can be set to zeros. Then, the estimation result is the same as the case when \(\epsilon=0.05\). This implies that even if a too small \(\epsilon\) is chosen, we can still obtain the effective parameter estimates based on the model structures. From Equations (15) and (16), we can obtain the orders \(n_{b_{1}}=n_{b_{2}}=3\) and get the time delay estimates as:  
\[B_{1}(z) = 0.95z^{-1}+0.72z^{-2}+0.40z^{-3},\]2  
From Figures 2-4, Table 1 and Equations (16) and (17), we can conclude that the TH-OMP algorithm converges faster than the LS algorithm, can achieve a high estimation accuracy by \(K\) iterations with finite measurements (\(m<N\)) and can effectively estimate the time delay of each channel. measuring cost, as well as improve the identification efficiency. The simulation results are given to demonstrate the performance of the proposed method.