Electro-hydraulically controlled wheel loader

The overall model consists of hydraulics (cylinders, valves, pump, etc.) and two degree of freedom linkage. The equations can be combined to form a MIMO state space system. The state vector, $x$, is defined in the table below. The input is the current in two valve solenoids as follows: $u = [i_1, i_2]^T$ . The output is given as, $y = [θ_1, θ_{21}]$. The states of the model are summarized in the table below. The dynamic equations for the linkage and electrohydraulic system in terms of state variables can be written as follows:

$$\begin{align*} \dot{x}_1 &= x_3\\ \dot{x}_2 &= x_4\\ \begin{bmatrix} \dot{x}_3 \\ \dot{x}_4 \end{bmatrix} &= M^{-1}(\tau(x_9, x_{10}, x_{11}, x_{12}, x_{13}) - h(x_1, x_2, x_3, x_4))\\ \dot{x}_5 &= \dfrac{\beta}{V_p}(\omega_px_6/2\pi - x_5K_{Lp}-(Q_{P A,1}(x_5, x_8, x_9) + Q_{P B, 1}(x_8, x_{10} + Q_{P A,2}(x_5, x_{11}, x_{12}) + Q_{P B,2}(x_5, x_{11}, x_{13})))\\ \dot{x}_6 &= [x_7 - x_5 + P_{margin}] G_p\\ \dot{x}_7 &= (\max(x_9,x_{10},x_{12},x_{13})-x_7)1/\tau_p\\ \dot{x}_8 &= (-x_8 + G_vu_1)1/\tau_v\\ \dot{x}_9 &= \dfrac{\beta}{V_{A,1}(x_1)}(Q_{PA,1(x_5,x_8,x_9}) + Q_{TA,1}(x_8,x_9-\dot{V}_{A,1}(x_3))\\ \dot{x}_{10} &= \dfrac{\beta}{V_{B,1}(x_1)}(Q_{PB,1(x_8,x_{10}}) + Q_{TB,1}(x_8,x_{10}-\dot{V}_{B,1}(x_3))\\ \dot{x}_{11} &= (-x_{11} + G_vu_2)1/\tau_v \dot{x}_{12} &= \dfrac{\beta}{V_{A,12}(x_2)}(Q_{PA,2(x_5,x_{11},x_{12}}) + Q_{TA,2}(x_{11},x_{12}-\dot{V}_{A,2}(x_4))\\ \dot{x}_{13} &= \dfrac{\beta}{V_{B,2}(x_2)}(Q_{PB,2(x_5,x_{11},x_{13}}) + Q_{TB,2}(x_{11},x_{13}-\dot{V}_{B,2}(x_4))\\ \end{align*}$$