(Solved): Then, we can obtain the linearized state equations of this system in standard form \[ \begin{array} ...

Then, we can obtain the linearized state equations of this system in standard form \[ \begin{array}{l} \dot{\mathbf{x}}=\mathbf{A} \mathbf{x}+\mathbf{B} u \\ \mathbf{y}=\mathbf{C x}+\mathbf{D} u \end{array} \] where \[ \begin{array}{l} \mathbf{A}=\left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & -\frac{1}{4} m_{p}^{2} L_{p}^{2} L_{r} g / J & -\left(J_{p}+\frac{1}{4} m_{p} L_{p}^{2}\right) B_{r} / J & \frac{1}{2} m_{p} L_{p} L_{r} B_{p} / J \\ 0 & \frac{1}{2} m_{p} L_{p} g\left(J_{r}+m_{p} L_{r}^{2}\right) / J & \frac{1}{2} m_{p} L_{p} L_{r} B_{r} / J & -\left(J_{r}+m_{p} L_{r}^{2}\right) B_{p} / J \end{array}\right] \\ \mathbf{B}=\frac{1}{J}\left[\begin{array}{c} 0 \\ 0 \\ J_{p}+\frac{1}{4} m_{p} L_{p}^{2} \\ -\frac{1}{2} m_{p} L_{p} L_{r} \end{array}\right] \\ \mathbf{C}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right] \quad \mathbf{D}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] \\ \end{array} \] where \[ J=J_{p} m_{p} L_{r}^{2}+J_{r} J_{p}+\frac{1}{4} J_{r} m_{p} L_{p}^{2} . \] Note that these state equations are valid only if the system is close to the equilibrium state. The outputs \( y_{1} \) and \( y_{2} \) are defined as the angles \( \theta \) and \( \alpha \) respectively. This means that we have two sensors in the system measuring these values. Using MATLAB, do the following: (a) Substitute the numerical values and determine the matrices. (b) Determine the eigenvalues of the system matrix. Show that the system is unstable. (c) Determine the transfer function of the system. Determine the poles. Verify that they are the same as the eigenvalues in part (b). (d) Let the initial state be given as \( x(0)=\left[\begin{array}{llll}0 & 0.1 & 0 & 0\end{array}\right]^{T} \). Explain in words what this initial state corresponds to, by referring to the figure above if necessary. Plot the response of the system for this initial condition and zero input. (e) Is the system controllable? Is it observable? Make the necessary calculations and explain.