Dynamics of Hydrostatic Transmission

The hydrostatic transmission dynamics is represented by a nonlinear fourth order state-space model

$$\begin{align*} \dot{q}_{1}(t) &= -a_{11}q_{1}(t)+b_{11}u_{1}(t) \\ \dot{q}_{2}(t) &= -a_{22}q_{2}(t)+b_{22}u_{2}(t) \\ \dot{q}_{3}(t) &= a_{31}q_{1}(t)p(t)-a_{33}q_{3}(t)-a_{34}q_{2}(t)q_{4}(t) \\ \dot{q}_{4}(t) &=a_{43}q_{2}(t)q_{3}(t)-a_{44}q_{4}(t), \end{align*}$$

where $q_1(t)$ is the normalized hydraulic pump angle, $q_2(t)$ is the normalized hydraulic motor angle, $q_3(t)$ is the pressure difference [bar], $q_4(t)$ is the hydraulic motor speed [rad/s], $p(t)$ is the speed of hydraulic pump [rad/s], $u_1(t)$ is the normalized control signal of the hydraulic pump, and $u_2(t)$ is the normalized control signal of the hydraulic motor. It is supposed that the external variable $p(t)$ , as well as the second state variable $q_2(t)$ are measurable. In given working point the model parameters are

$\eqalignno{& a_{11}=7.6923 \qquad a_{22}=4.5455 \quad a_{33}=7.6054.10^{-4} \cr &a_{31}=0.7877 \qquad a_{34}=0.9235\quad\ b_{11}=1.8590.10^{3} \cr &a_{43}=12.1967 \quad\ \ a_{44}=0.4143\quad b_{22}= 1.2879.10{}^{{3}}}$