## 3 Identification Model Analysis
### Identification Model  
Referring to [39], the Stribeck friction is modeled as an easily identifiable parametric linearization as follows:  
\[F_{f}=f_{c}\text{sign}(\dot{x}_{1})+f_{v}\dot{x}_{1}+f_{s}\dot{x}_{1}^{\frac{ 1}{3}}, \tag{23}\]  
where \(f_{c}\) is the Coulomb friction factor; \(f_{v}\) is the Viscous friction factor; \(f_{s}\) is the Stribeck friction parameter.  
From (23) and (17), we have:  
\[Q_{1}=\text{Y}_{1}\cdot\Psi_{1}, \tag{24}\]  
where:  
\[Q_{1} = A_{1}x_{3}-A_{2}x_{4}-F_{L}; \tag{25}\] \[\text{Y}_{1} = [\dot{x}_{2},x_{2},\text{sign}(x_{2}),(x_{2})^{\frac{1}{3}},x_{1}];\] (26) \[\Psi_{1} = [m,B+f_{v},f_{c},f_{s},K_{t}]^{T}. \tag{27}\]  
By (7), (12) and (18), we obtain:  
\[Q_{2}=\text{Y}_{2}\cdot\Psi_{2}, \tag{28}\]  
where \(\left[\begin{array}{c}d_{1}(k)\\ d_{2}(k)\end{array}\right]\) is the zero-mean random noise vector. Using the least squares recursive algorithm, it follows that the identification algorithm of (43) is:  
\[\widehat{\Psi}_{1}(k) = \widehat{\Psi}_{1}(k-1)+K_{1}(k)[Q_{1}(k)-Y_{1}(k)\Psi_{1}(k-1)], \tag{46}\] \[K_{1}(k) = P_{1}(k-1)\Upsilon_{1}^{T}(k)[Y_{1}(k)P_{1}(k-1)\Upsilon_{1}^{T} (k)+\frac{1}{\Lambda_{1}(k)}]^{-1},\] (47) \[P_{1}(k) = [I_{1}-K_{1}(k)Y_{1}(k)]P_{1}(k-1); \tag{48}\]  
the identification algorithm of (44) is:  
\[\widehat{\Psi}_{4}(k) = \widehat{\Psi}_{4}(k-1)+K_{2}(k)[Q_{4}(k)-Y_{4}(k)\Psi_{4}(k-1)], \tag{49}\] \[K_{2}(k) = P_{2}(k-1)\Upsilon_{4}^{T}(k)[Y_{4}(k)P_{2}(k-1)\Upsilon_{4}^{T }(k)+\frac{1}{\Lambda_{2}(k)}]^{-1},\] (50) \[P_{2}(k) = [I_{2}-K_{2}(k)Y_{4}(k)]P_{2}(k-1); \tag{51}\]  
and the identification algorithm of (45) is:  
\[\widehat{\Psi}_{2}(k) = \hat{\Psi}_{2}(k-1)+K_{3}(k)[\left[\begin{array}{c}Q_{2}(k)\\ Q_{3}(k)\end{array}\right]-\left[\begin{array}{c}Y_{2}(k)\\ Y_{3}(k)\end{array}\right]\Psi_{2}(k-1)], \tag{52}\] \[K_{3}(k) = P_{3}(k-1)\left[\begin{array}{c}Y_{2}(k)\\ Y_{3}(k)\end{array}\right]^{T}[\left[\begin{array}{c}Y_{2}(k)\\ Y_{3}(k)\end{array}\right]P_{3}(k-1)\left[\begin{array}{c}Y_{2}(k)\\ Y_{3}(k)\end{array}\right]^{T}+\left[\begin{array}{c}\frac{1}{\Lambda_{3}(k )}\end{array}\right]]^{-1},\] (53) \[P_{3}(k) = [I_{3}-K_{3}(k)\left[\begin{array}{c}Y_{2}(k)\\ Y_{3}(k)\end{array}\right]|P_{3}(k-1). \tag{54}\]  
From the identification algorithm, it can be found that the differential signal is necessary in the input or output data. Here, we choose the T-D, replace the differentiation with the difference, and take the step length as \(h_{s}\); at this time there will be moment inequality, so the first-order differential time is pushed back one data, the second-order differential time is pushed back two data, the principle of T-D is as follows:  
\[\left\{\begin{array}{c}fh=\mbox{fhan}((\mu_{1}(k)-x(k)),\mu_{2}(k),r,h_{0}) \\ \mu_{1}(k+1)=\mu_{1}(k)+h_{s}\cdot\mu_{2}(k)\\ \mu_{2}(k+1)=\mu_{2}(k)+h_{s}\cdot fm\end{array}\right.\]  
where \(x(k)\) is the input signal; \(\mu_{1}(k)\) is the tracking signal; \(\mu_{2}(k)\) is the first-order differential signal for the input signal \(x(t)\); \(h_{0}\) is a variable independent of \(h_{s}\) and takes a value appropriately larger than \(h_{s}\). Function \(\mbox{fhan}(\gamma_{1},\gamma_{2},\tau,h_{0})\) is:  
\[\left\{\begin{array}{c}\sigma=\left\{\begin{array}{c}\gamma_{2}+\frac{ \alpha-i}{2}\mbox{sign}(\zeta),|\zeta|>d\\ \gamma_{2}+\frac{\zeta}{h_{0}},|\xi|\leq d\end{array}\right.\\ \mbox{fhan}=-\left\{\begin{array}{c}\tau\mbox{sign}(\sigma),|\sigma|>\iota\\ \tau\frac{\zeta}{\iota},|\sigma|\leq\iota\end{array}\right.\end{array}\right.\]  
where \(\iota=\tau h_{0}\); \(d=lh_{0}\); \(\xi=\gamma_{1}+h_{0}\gamma_{2}\); \(a=(\iota^{2}+8\tau|\xi|)^{1/2}\).