## 2 Establishment of Disturbance Model
### Disturbance Model of Single Support Cylinder
#### 2.2.2 Working Medium  
The working medium used in the FSS is hydraulic oil, which has fixed physical properties at an ambient temperature and pressure, mainly the compressibility and expansion. The state equation of hydraulic oil can be expressed as  
\[f(\rho,P,T)=0 \tag{10}\]  
where \(\rho\) is the density of hydraulic oil, \(P\) is the pressure of hydraulic oil, \(T\) is the temperature of hydraulic oil. The state equation can be Taylor expanded at \(\rho_{0}\). \[ \begin{array}{l}\rho=\rho_{0}(P_{0},T_{0})+\left.\frac{\partial\rho(P,T)}{ \partial P}\right|_{T_{0}}(P-P_{0})+\left.\frac{\partial\rho(P,T)}{\partial T} \right|_{P_{0}}(T-T_{0})+\left.\frac{1}{2!}\frac{\partial^{2}\rho(P,T)}{ \partial^{2}P}\right|_{T_{0}}(P-P_{0})^{2}+\\ \left.\frac{1}{2!}\frac{\partial^{2}\rho(P,T)}{\partial P\partial T}\right|_{ T_{0},P_{0}}(P-T_{0})+\left.\frac{1}{2!}\frac{\partial^{2}\rho(P,T)}{ \partial T\partial P}\right|_{P_{0},T_{0}}(P-P_{0})(T-T_{0})+\left.\frac{1}{2! }\frac{\partial^{2}\rho(P,T)}{\partial^{2}T}\right|_{P_{0}}(T-T_{0})^{2}+ \cdots+\sigma^{n}\end{array}  \tag{11}\]  
where \(\sigma^{n}\) is the Lagrange remainder. \[ \begin{array}{l}\sigma^{n}=\frac{1}{(n+1)!}\left(\frac{\partial\rho(P,T)}{ \partial P}+\frac{\partial\rho(P,T)}{\partial T}\right)^{n+1}\rho(P_{0}+ \theta P,T_{0}+\theta T)\end{array} \]  
where \(0<\theta<1\). Written in incremental form:  
\[\begin{array}{l}\rho-\rho_{0}(P_{0},T_{0})=d\rho=\left.\frac{\partial\rho(P,T )}{\partial P}\right|_{T_{0}}dP+\left.\frac{\partial\rho(P,T)}{\partial P} \right|_{P_{0}}dT+\left.\frac{1}{2!}\frac{\partial^{2}\rho(P,T)}{\partial P} \right|_{T_{0}}d^{2}P\\ +\left.\frac{1}{2!}\frac{\partial^{2}\rho(P,T)}{\partial P\partial T}\right| _{T_{0},P_{0}}dPdT+\left.\frac{1}{2!}\frac{\Owing to the compression of density relative to pressure and the difficulty in measuring temperature, a relative quantity is used for analysis and calculation:  
\[\frac{d\rho}{\rho_{0}}=kdP+\alpha_{oil}dT+\frac{1}{2!}k^{2}d^{2}P+\frac{1}{2!}k \alpha_{oil}PdT+\frac{1}{2!}k\alpha_{oil}dTdP+\frac{1}{2!}\alpha_{oil}{}^{2}d^ {2}T+\cdots+\frac{1}{(n+1)!}(\alpha_{oil}+k)^{n+1} \tag{13}\]  
where \(k\) is the volume compression coefficient of hydraulic oil, and \(\alpha_{oil}\) is the expansion of hydraulic oil. (1) Expansion of hydraulic oil  
Assuming that the volume of hydraulic oil is \(V_{0}\) at 0 \({}^{\circ}\)C and the volume is \(V_{T}\) at \(T\)\({}^{\circ}\)C, then the volume expansion coefficient is:  
\[V_{T}=V_{0}(1+\alpha T) \tag{14}\]  
Because the volume expansion coefficient of hydraulic oil is very small, for the convenience of calculation when the temperature is not high, the following equation can be directly used for calculation: not taking \(V_{0}\) into consideration at 0 \({}^{\circ}\)C, \(\alpha\) is the volume expansion coefficient. \[V_{2}=V_{1}[1+\alpha(T_{2}-T_{1})] \tag{15}\]  
When the hydraulic oil contains other substances:  
\[V_{t}=V_{1}[1+\alpha_{1}(T_{2}-T_{1})]+V_{2}[1+\alpha_{2}(T_{2}-T_{1})]+\cdots +V_{n}[1+\alpha_{n}(T_{2}-T_{1})] \tag{16}\]  
where \(V_{1}\), \(V_{2}\),..., \(V_{n}\) is the volume fraction of the components of mixed liquid at a temperature of \(T_{1}\), \(\alpha_{1}\), \(\alpha_{2}\),..., \(\alpha_{n}\alpha_{n}\) is the average value of volume expansion coefficients of the mixed liquid components when the temperature rises from \(T_{1}\) to \(T_{2}\). (2) Compressibility of hydraulic oil  
When hydraulic oil is mixed with air, the influence of the dissolved air on the elastic modulus can be ignored, and the air in the form of bubbles will affect the effective volume elastic modulus of hydraulic oil. Because the support cylinder is in a low-pressure working state, in order to facilitate the calculation, the following assumptions are made for the derivation of the theoretical model: (i) The variation of the hydraulic line with changes in temperature and pressure change may be ignored. (ii) The quality of air contained in hydraulic oil does not change with time.