## 2 Theoretical Aspects
### A Nonlinear Hybrid Control Strategy Combining Dry Friction and Active Control Force  
#### 2.2.1 Governing Motion Equation in the First State Space  
Since the friction force is a function of relative velocity \(\dot{x_{i}}-\dot{x_{j}}\) and relative displacement \(x_{i}-x_{j}\), in order to facilitate the design of control law, the above physical space is transformed into a state space where the relative displacements are set as new variables in the new governing equation. Supposing \(y_{1}=x_{1}-x_{2}\) and \(y_{2}=x_{1}+x_{2}\), then the friction force will be decoupled in the new state space, only engaged in the equation of \(y_{1}=x_{1}-x_{2}\), while disappearing in the equation of \(y_{2}=x_{1}+x_{2}\). The governing equation in the new state space is expressed as  
\[\left\{ \begin{array}{c}m\ddot{y_{1}}=-\frac{1}{2}(k_{1}+4k_{2}+k_{3})y_{1}+ \frac{1}{2}(k_{3}-k_{1})-k_{3}y_{3}+2f+f_{exc}+f_{c_{1}}-f_{c_{2}}\\ my_{2}=\frac{1}{2}(k_{3}-k_{1})y_{1}-\frac{1}{2}(k_{2}+k_{2})y_{2}+k_{3}y_{3}+ f_{exc}+f_{c_{1}}+f_{c_{2}}\\ m\ddot{y_{3}}=-\frac{k_{2}}{2}y_{1}+\frac{k_{3}}{2}y_{2}-(k_{3}+k_{4})y_{3} \end{array} \right.. \tag{4}\]  
The corresponding matrix form is written as  
\[\left[ \begin{array}{cc}m_{1}&0&0\\ 0&m_{2}&0\\ 0&0&m_{3}\end{array} \right]\left[ \begin{array}{c}\ddot{y_{1}}\\ \ddot{y_{2}}\\ \ddot{y_{3}}\end{array} \right]=-\frac{1}{2}\left[ \begin{array}{cc}k_{1}+4k_ {2}+k_{3}&k_{1}-k_{3}&2k_{3}\\ k_{1}-k_{3}&k_{1}+k_{2}&-2k_{3}\\ k_{3}&-k_{3}&2(k_{3}+k_{4})\end{array} \right]\left[ \begin{array}{c}y_{1}\\ y_{2}\\ y_{3}\end{array} \right]+\left[ \begin{array}{c}2f\\ 0\\ 0\end{array} \right]+\left[ \begin{array}{c}f_{exc}\\ f_{exc}\\ 0\end{array} \right]+\left[ \begin{array}{c}f_{c_{1}}-f_{c_{2}}\\ f_{c_{1}}+f_{c_{2}}\\ 0\end{array} \right]. \tag{5}\]  
\[\text{where }\mathbf{Y}=\left[ \begin{array}{c}y_{1}\\ y_{2}\\ y_{3}\end{array} \right]=\left[ \begin{array}{c}x_{1}-x_{2}\\ x_{1}+x_{2}\\ x_{3}\end{array} \right]\text{ is the displacement vector in the newly defined state space;}\]  
\[\mathbf{M_{m}}=\left[ \begin{array}{cc}m&0&0\\ 0&m&0\\ 0&0&m\end{array} \right]\text{ is the mass matrix in the newly defined state space;}\]  
\[\mathbf{K_{m}}=\frac{1}{2}\left[ \begin{array}{cc}k_{1}+4k_{2}+k_{3}&k_{1}-k_{3 }&2k_{3}\\ k_{1}-k_{3}&k_{1}+k_{2}&-2k_{3}\\ k_{3}&-k_{3}&2(k_{3}+k_{4})\end{array} \right]\text{ is the stiffness matrix in the newly defined state space;}\]  
\[\mathbf{F_{nl_{m}}}=\left[ \begin{array}{c}2f\\ 0\\ 0\end{array} \right]=-\left[ \begin{array}{c}2\mu\cdot F_{N}\cdot\text{sgn}( \dot{x_{1}}-\dot{x_{2}})\\ 0\\ 0\end{array} \right]\text{ is the friction vector in the newly defined state space;}\]  
\[\mathbf{F_{exc_{m}}}=\left[ \begin{array}{c}f_{exc}\\ f_{exc}\\ 0\end{array} \right]\text{ is the exterior excitation vector in the newly defined state space;}\]  
\[\mathbf{F_{c_{m}}}=\left[ \begin{array}{c}f_{c_{1}}-f_{c_{2}}\\ f_{c_{1}}+f_{c_{2}}\\ 0\end{array} \right]\text{ is the active control force vector in the newly defined state space, and the corresponding matrix form is written as  
\[\mathbf{M_{m}}\ddot{\mathbf{Y}}=-\mathbf{K_{m}}\mathbf{Y}+\mathbf{F_{nl_{m}}}+\mathbf{F_{exc_{m}}}+\bm {F_{c_{m}}}, \tag{6}\]  
The transformation matrix \(\mathbf{P}\) bring the physical coordinates into the state space is defined as  
\[\mathbf{Y}=\left[ \begin{array}{c}y_{1}\\ y_{2}\\ y_{3}\end{array} \right]=\mathbf{P}\cdot\mathbf{X}, \tag{7}\]where \(\mathbf{P}=\left[ \begin{array}{ccc}1&-1&0\\ 1&1&0\\ 0&0&1\end{array} \right]\), which is reversible, and vice versa, \(\mathbf{X}=P^{-1}\mathbf{Y}\). Employing this transformation matrix, the physical space can be easily transformed into state space.