## 2 Establishment of Disturbance Model
### Disturbance Model of Single Support Cylinder  
#### 2.2.1 Support Cylinder Structure  
The support cylinder is composed of two chambers: an upper chamber and a lower chamber. The structures of the support cylinder are shown in Figure 2.  
The movement of the support cylinder can be expressed as  
\[\Delta L=\frac{\Delta V_{u}-\Delta V_{d}}{2\cdot S} \tag{2}\]  
where \(\Delta L\) is the displacement of support cylinder, \(\Delta V_{u}\) is the volume change of the upper chamber, \(\Delta V_{d}\) is the volume change of the lower chamber, \(S\) is the cross-sectional area of support piston. Figure 1: Model of the flexible support system (FSS). (**a**) 3D model of the FSS. (**b**) Arrangement of support cylinder. When the support shaft of support cylinder moves \(\Delta L\), the volume of the upper and lower chambers can be expressed as:  
\[V_{u}=f\big{(}\mathbf{W},\mathbf{L},\Delta L,\Delta L_{1},\Delta L_{3},\Delta L_{4}, \Delta L_{6},R_{p}\big{)} \tag{3}\]  
\[V_{d}=f\big{(}\mathbf{W},\mathbf{L},\Delta L,\Delta L_{2},\Delta L_{3},\Delta L_{5}, \Delta L_{6},R_{p}\big{)} \tag{4}\]  
where \(\mathbf{W}=\left[ \begin{array}{cc}W_{1}&\\ &\ddots&\\ &&W_{m}\end{array} \right]\) and \(\mathbf{L}=\left[ \begin{array}{cc}L_{1}&\\ &\ddots&\\ &&L_{n}\end{array} \right]\) are the structural parameters of support cylinder, \(m\) and \(n\), respectively, indicate the number of radial and axial structural parameters of support cylinder, \(\Delta L_{1}\) is the extension quantity of the small rolling diaphragm in upper chamber, \(\Delta L_{2}\) is the extension quantity of the small rolling diaphragm in lower chamber, \(\Delta L_{3}\) is the extension quantity of the large rolling diaphragm, \(\Delta L_{4}\) is the compression quantity of the small rolling diaphragm in upper chamber, \(\Delta L_{5}\) is the compression quantity of the small rolling diaphragm in lower chamber, \(\Delta L_{6}\) is the compression quantity of the large rolling diaphragm, \(R_{p}\) is the chamfer radius of the chamber. During the work of the support cylinder, the hydraulic oil in the upper chamber exerts a force on the small rolling diaphragm in the upper chamber and the large rolling diaphragm, and the hydraulic oil in the lower chamber will exert a force on the small rolling diaphragm in the lower chamber and the large rolling diaphragm. Because the pressures in the upper and lower chambers of the support cylinder, as well as the deformation of the small rolling diaphragm in the upper and lower chambers, are different, the deformations of the rolling diaphragms in the upper and lower chambers need to be calculated separately. (1) Deformations of the small rolling diaphragms in the upper and lower chambers are:  
\[\Delta L_{1}=\frac{W_{2}L_{1}}{2 EW_{h1}}P_{u} \tag{5}\]  
\[\Delta L_{2}=\frac{W_{2}L_{1}}{2 EW_{h1}}P_{d} \tag{6}\]  
Figure 2: Structure of support cylinder. where \(P_{u}\) is the pressure of upper chamber, \(P_{d}\) is the pressure of lower chamber, \(E\) is the modulus of elasticity, \(W_{h1}\) is the thickness of small rolling diaphragm, \(W_{2}\) is the gap between outer piston and cylinder wall, \(L_{1}\) is the mounting height of small rolling diaphragm. (2) Deformation of the large rolling diaphragm can be expressed as:  
\[\Delta L_{3}=\frac{W_{3}L_{3}}{2 EW_{h2}}(P_{u}-P_{d}) \tag{7}\]  
where \(W_{h2}\) is the thickness of large rolling diaphragm, \(W_{3}\) is the gap between internal piston and cylinder wall, \(L_{3}\) is the installation height of large rolling diaphragm. (3) Compression of the rolling diaphragm  
The compression process of the rolling diaphragm, owing to the relatively small amount of deformation, can be simplified to a simple compression deformation. \[P_{1}\cdot S_{p}=E\cdot S_{p}\cdot\varepsilon \tag{8}\]  
where \(\sigma\) is the stress on rolling diaphragm, \(P_{1}\) is the pressure on rolling diaphragm, \(S_{p}\) is the area under compression, \(\varepsilon\) is the unit deformation.