## 4 Performance of KF Used for Parameter Estimation in RTHS
### Tracking Performance with Prescribed Displacement Command  
For convenience, the control plant is modeled by a second-order transfer function, which is given by  
\[T(s)=\frac{\omega^{2}}{s^{2}+2\zeta\omega s+\omega^{2}}, \tag{22}\]  
where \(\omega\) and \(\zeta\) are the equivalent circular frequency and damping ratio, respectively, and the two parameters are set to be 50 rad/s and 0.8, respectively. By using the forward-difference method in Equation (22), a three-parameter discrete control plant model can be obtained, namely  
\[d_{k+1}^{\mathrm{c}}=x_{1}d_{k+1}^{\mathrm{m}}+x_{2}d_{k}^{\mathrm{m}}+x_{3}d _{k-1}^{\mathrm{m}}, \tag{23}\]  
and the corresponding feedforward compensator is given by  
\[d_{k+1}^{\mathrm{c}}=\hat{x}_{1}d_{k+1}+\hat{x}_{2}d_{k}+\hat{x}_{3}d_{k-1}. \tag{24}\]  
In this section, the sampling frequency is 4096 Hz, resulting in the true parameters for the three-parameter discrete model of the second-order control plant model in Equation (22) being \(\mathbf{x}_{\mathrm{true}}=[0.6777\)\(-1.3422\)\(0.6646]^{\mathrm{T}}\times 10^{4}\). The initial value of \(\mathbf{x}\), which is denoted by \(\mathbf{x}_{0}\), is determined by employing the least-square method, where the displacement command and measurement originated from the control plant without using any compensation method. Thus, the initial value \(\mathbf{x}_{0}\) is \([0.5817\)\(-1.152\)\(0.5704]^{\mathrm{T}}\times 10^{4}\). The initial covariance matrix \(\mathrm{P}_{0}\) is assumed to be diagonal, which means that the initial values of the parameters are statistically uncorrelated. The diagonal entries of \(\mathrm{P}_{0}\) are computed as \((q_{\mathrm{I}}\)\(\times\)\(x_{i})^{2}\)[41], where \(i=1\), 2, 3, and \(q_{\mathrm{f}}=0.1\). The measurement noise \(R\) is set to be 0.001. It should be noted that the measurement noise was modeled by the random number, and a KF was used to suppress the effect of noise.  
The tracking performance of the second-order transfer system is shown in Figure 4, where the displacement command is a swept signal with frequency ranging from 0.1 Hz to 20 Hz, and has an amplitude of 100 mm. For comparison, the response without any compensation method is also given in the figure. It is seen that as the input frequency increases, the tracking performance without compensation decreases rapidly, which is detrimental to RTHS. On the contrary, it can be found in Figure 4a that the measurement displacement provided by the proposed KF-ADC method agrees well with the displacement command. The enlarged view is shown in Figure 4b, and it can be seen that the measurement displacement is almost identical to the displacement command. This reveals that by compensating using the proposed method, the control plant exhibits a very good tracking performance.  
Figure 4: Tracking performance of the second-order transfer function. (**a**) Overview; (**b**) Enlarged view.  
The evolutions of the estimated system model parameter values are shown in Figure 5. It is seen in the figure that the estimated system model parameters vary slightly compared with the initial value after the first 0.1 s. Then, the estimated parameters move rapidly toward the true values between 0.1 s and 0.5 s. Subsequently, the estimated parameters remain relatively stable, and are almost identical to the true ones. This result shows that once the dynamics of the control plant are excited, the discrete-time model of the control plant can be quickly estimated using KFs with high-accuracy, and the time-delay can be compensated effectively. This indicates that the KF has a fast convergence speed and high estimation accuracy.