# Practical Control of a Cold Milling Machine using an Adaptive PID Controller
## 2 Control Model
### System Model
#### 2.2.2 Model of Drive System  
The flux function of the variable hydraulic displacement pump can be described as,  
\[Q_{b}=K_{b}i_{e}\omega_{e}\phi_{b}-C_{ib}(P-P_{r})-C_{eb}P \tag{3}\]  
where \(Q_{b}\) is the flux of the hydraulic pump, \(K_{b}\) is the displacement gradient of the hydraulic pump, \(i_{e}\) is the transmission ratio of the gearbox between the engine and the pump, \(C_{ib}\) is the internal leakage coefficient of the pump, \(P\) is the work pressure of the pump, \(P_{r}\) is the backpressure of the hydraulic system, \(C_{eb}\) is the external leakage coefficient of the pump, and \(\phi_{b}\) is the angle of the sliding shoes in the pump. Here, \(\phi_{b}\) is proportional to the control voltage \(u\) of the hydraulic pump [27]:  
\[\phi_{b}=k_{\phi}u \tag{4}\]  
where \(k_{\phi}\) is a constant. Substituting Equation (4) into Equation (3), and taking the result linearization and Laplace transform, it yields  
\[Q_{b}(s)=K_{b}i_{e}k_{\phi}\omega_{e0}u(s)+K_{b}i_{e}k_{\phi}u_{0}\omega_{e}(s )-C_{b}P(s) \tag{5}\]  
where \(C_{b}=C_{ib}+C_{eb}\) is the total leakage coefficient of the hydraulic pump, \(\omega_{e0}\) and \(u_{0}\) are the initial values. \(P_{r}\) is the back pressure of the hydraulic system which is a constant in Equation (3), when we take Laplace transform of Equation (3), \(P_{r}\) is zero. The flux continuity function of the motor is given by [27]:  
\[Q_{b}-C_{im}(P-P_{r})-C_{em}P=D_{m}\overset{\bullet}{\partial_{m}}+\frac{V_{0 }}{\beta_{e}}P \tag{6}\]  
Taking Laplace transform, Equation (6) is rewritten as  
\[Q_{b}(s)-C_{m}P(s)=D_{m}s\theta_{m}(s)+\frac{V_{0}}{\beta_{e}}sP(s) \tag{7}\]  
where \(C_{im}\) is the internal leakage coefficient of the hydraulic motor; \(C_{em}\) is the external leakage coefficient of the motor, \(D_{m}\) is the displacement of the hydraulic motor, \(\theta_{m}\) is the rotation angle of the motor axle, \(V_{0}\) is the total volume of the high pressure chamber of the pump, motor and hydraulic pipe, \(\beta_{e}\) is the modulus elastic of the hydraulic oil volume, and \(C_{m}\) is the total leakage coefficient of the hydraulic motor. The torque balance function of the hydraulic motor can be derived as  
\[D_{m}(P-P_{r})=J_{m}\overset{\bullet\bullet}{\partial_{m}}+B_{m}\overset{ \bullet}{\partial_{m}}+M_{L} \tag{8}\]  
where \(M_{L}\) is the load of the hydraulic motor. The transform function of Equation (9) is given as  
\[D_{m}P(s)=(J_{m}s^{2}+B_{m}s)\theta_{m}(s)+M_{L}(s) \tag{9}\]  
The relationship between the work speed and the angular velocity of the hydraulic motor can be expressed as  
\[v_{T}(s)=\frac{2\pi R_{S}s\theta_{m}(s)}{i_{g}} \tag{10}\]  
Where \(i_{g}\) is the transmission ratio from the hydraulic motor to the drive wheel. Combining Equations (5) (7) (9) and (10), the model of the pump control motor system can be established as  
\[K_{\texttt{\tiny{T}}T}v_{T}(s)+K_{T_{L}}M_{L}(s)=K_{b}k_{\phi}a_{\phi 0}u(s)+K_{b}e_{k}k_{\phi}u_{0}a_{\phi}(s) \tag{11}\]  
where  
\[K_{T_{L}}=\frac{C_{t}}{D_{m}}\bigg{(}1+\frac{V_{0}}{C_{t}\beta_{e}s}\bigg{)},C _{t}=C_{b}+C_{m},K_{\texttt{\tiny{T}}T}=\frac{D_{m}}{2\pi R_{g}}\bigg{[}\frac{ J_{m}V_{0}}{D_{m}^{2}\beta_{e}}s^{2}+\bigg{(}\frac{B_{m}V_{0}}{D_{m}^{2}\beta_{e}}+ \frac{C_{t}J_{m}}{D_{m}^{2}}+J_{m}\bigg{)}s+B_{m}\bigg{(}1+\frac{C_{t}}{D_{m}^ {2}}\bigg{)}\bigg{]}\]  
The torque balance function of the milling drum is expressed as  
\[\phi_{b}=k_{\phi}u \tag{4}\]0  
where \(M_{de}\) is the part of engine output torque transmitted from Gearbox II in Figure 3, \(J_{d}\) the moment of inertia of the milling rotor, \(C_{d}\) the rotational damp of the milling rotor. \(\omega_{d}\) the rotary speed of the cutting drum, \(M_{d}\) the resistance torque of the milling rotor, and \(M_{de}\) the input torque of the milling rotor.