Relative Degree Two MIMO Nonlinear System:

Consider the following relative degree two MIMO nonlinear systems:

$$\begin{align*} \dot x_1 &= x_2 + \vartheta _1 x_1 \sin \left(t \right) + \Delta _1 \left({x_1 } \right) \\ \dot x_2 &= u + \vartheta _2 \left[{\matrix{{\left({x_1 + x_{2,1} } \right)\sin ^3 \left(t \right)} \cr {x_{2,1} + 2x_{2,2} } \cr}} \right] + \left[{\matrix{1 \cr {x_1 + x_{2,2} } \cr}} \right]\Delta _2 \left({x_{2,1} } \right) \\ y &= \left[{x_1,x_{2,2} } \right]^{\mathrm T}, \end{align*}$$

where $\Delta_1(x_1)=d_1\sin{(r_1x_1)}$ and $\Delta_2(x_{2,1})=d_2\tan{(r_2,x_{2,1})}.$ $\vartheta_1, \vartheta_2,\Delta_1,\Delta_2$ satisfy

$\displaylines{2 \le \vartheta _1 \le 4, - 4 \le \vartheta _2 \le - 1, \left\vert {\Delta _1 \left({x_1 } \right)} \right\vert \le \delta _1 = 40\cr \left\vert {\Delta _2 \left({x_{2,1} } \right)} \right\vert \le \delta _2 = 20. }$

The initial conditions are assumed to be $x_1(0)=0.5,x_{2,1}(0)=0$ and $x_{2,2}(0)=0.2$.

The actual plant parameters are

$\theta _1 = 3$, $\theta _2 = - 3$, $d_1 = - 30$, $d_2 = - 15$, $r_1 = 2$, $r_2 = 0.05.$