## 3 Identification Model Analysis
### Numerical Simulation  
MATLAB simulation software was used as the experimental tool. Referring to the nonlinear state space model established in Section 2.3:  
\[\begin{array}{rcl}\dot{x}(t)&=&f(t,x(t),u(t)),\\ y(t)&=&h(t,x(t)).\end{array} \tag{55}\]In the simulation, (55) was discretized by the fourth-order Runge-Kutta method(R-K), the sampling step \(h\) was taken as 0.01 s, and its implementation was as in (56).  
\[x_{k+1} = x_{k}+\frac{h}{\xi}(\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4})\] \[\alpha_{1} = f(t_{k},x_{k})\] \[\alpha_{2} = f(t_{k}+\frac{h}{2},x_{k}+\frac{h}{2}\alpha_{1})\] \[\alpha_{3} = f(t_{k}+\frac{h}{2},x_{k}+\frac{h}{2}\alpha_{2})\] \[\alpha_{4} = f(t_{k}+h,x_{k}+h\alpha_{3}) \tag{56}\]  
The input of the system is \(u_{g}=2+\sin(2\pi t)\) in Figure 3, the simulation data of system parameters are shown in Table 1, and total times \(T=10s\), \(x_{1}(t)\), \(x_{3}(t)\), \(x_{4}(t)\) and \(x_{5}(t)\) were obtained from (56). The data length was \(N=T/h=1000\).  
T-D parameters were \(r=100\) and \(h_{0}=0.05\). In order to obtain the first-order differentiation, \(x_{1}(t),x_{3}(t),x_{4}(t)\) and \(x_{5}(t)\) were used as the input signals of T-D to obtain the piston rod velocity data \(\hat{x}_{1}(t)=x_{2}(t)\), the hydraulic cylinder left and right chamber pressure derivative data \(\hat{x}_{3}(t)\) and \(\hat{x}_{4}(t)\), and the servo valve velocity data \(\hat{x}_{5}(t)=x_{6}(t)\). To obtain the second-order differentiation, \(x_{2}(t)\) and \(x_{6}(t)\) obtained by T-D serve as the input signal of T-D, and the corresponding differentiation signals \(\hat{x}_{2}(t)\) and \(\hat{x}_{6}(t)\) were obtained, respectively.  
Afterwards, the parameter estimates were obtained on the basis of the identification algorithm in Section 3.2. In this experiment, the Coulomb friction factor \(f_{c}\) was taken as 5 and the viscous friction factor \(f_{\rm{p}}\) was taken as 20; neglecting the Stribeck friction parameter \(f_{s}\), (26) and (27) became:  
\[{\rm{Y}}_{1} = [\hat{x}_{2},x_{2},{\rm{sign}}(x_{2}),x_{1}], \tag{57}\] \[{\rm{\Psi}}_{1} = [m,B+f_{\rm{p}},f_{\rm{c}},K_{t}]^{T}. \tag{58}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Physical Implication** & **Value** \\ \hline \(K_{\rm{sp}}\) & Servo valve amplification factor & 0.00025 \\ \(\omega_{\rm{sp}}\) & Servo valve natural frequency & 100 \\ \(\zeta_{\rm{sp}}\) & Servo valve damping ratio & 0.5 \\ \(K_{\rm{p}}\) & Proportional controller factor & 1 \\ \(K_{f}\) & Displacement sensor gain & 40 \\ \(A_{1}\) & Effective area of the piston on the left side & 0.0085 \\ \(A_{2}\) & Effective area of the piston on the right side & 0.004 \\ \(C_{\rm{ip}}\) & Internal leakage factor & \(3\times 10^{-11}\) \\ \(C_{\rm{ep}}\) & External leakage factor & 0 \\ \(\beta_{c}\) & Effective bulk modulus of elasticity & 700,000,000 \\ \(K_{t}\) & Spring stiffness & 1,000,000 \\ \(m\) & Total mass & 120 \\ \(B\) & Viscous damping factor & 6000 \\ \(F_{f}\) & Stribeck friction & 0 \\ \(F_{L}\) & External force & 4000 \\ \(V_{01}\) & Given by (12) & 0.11 \\ \(V_{02}\) & Given by (13) & 0.09 \\ \(V_{L1}\) & Given by (12) & 0 \\ \(V_{L2}\) & Given by (13) & 0 \\ \(C_{d}\) & Orifice flow factor & 0.6 \\ \(W\) & Orifice area gradient & 0.03 \\ \(\rho\) & Oil density & 880 \\ \(p_{s}\) & System oil supply pressure & 12 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary table of system parameters.  
Figure 5: \(x_{2}\) and its differential signal \(\dot{x}_{2}\).  
Figure 6: \(x_{3}\) and its differential signal \(x_{3}\).  
Figure 7: \(x_{4}\) and its differential signal \(\dot{x}_{4}\).  
Figure 8: \(x_{5}\) and its differential signal \(\hat{x}_{6}\).  
Figure 10: Results of the identification algorithm for \(\Psi_{1}\).  
Figure 9: \(x_{6}\) and its differential signal \(\hat{x}_{6}\).  
**Remark 2**.: _The identification algorithm in this paper is based on a mathematical model of a hydraulic system. It can be applied to other hydraulic systems by making appropriate adjustments as long as the mathematical model is established._