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1) Please obtain the state-space representation of the plant below using MATLAB. Then obtain the eigen values of the matrix A, again using MATLAB (Hint: These could be useful tf2ss(), and eig()). Can you place all of the poles of this system to \( -5 \) using the states in your model? If so, how? \[ G_{(s)}=\frac{(s+10)\left(s^{2}+2 s+25\right)}{s^{2}\left(s^{2}+2 s+2\right)} \] 2) Suppose \( A, B \), and \( C \) below represents the parameters of a state-space model of a plant. Please manually obtain the poles and zeros of the system. Show all the steps. No MATLAB use is allowed for this question. \[ \begin{aligned} A & =\left[\begin{array}{ll} -2 & 1 \\ -2 & 0 \end{array}\right] \quad B=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] \\ C & =\left[\begin{array}{ll} 1 & 0 \end{array}\right] \end{aligned} \] \[ \begin{array}{ll} A & =\left[\begin{array}{ccc} 2 / 3 & 5 / 3 & -1 / 3 \\ 2 & 1 & -1 \\ 4 / 3 & 4 / 3 & -2 / 3 \end{array}\right] \\ C & =\left[\begin{array}{lll} 0 & 1 & 1 \end{array}\right] \end{array} \] 3) Suppose \( A, B \), and \( C \) above represents the parameters of a state-space model of a plant. Assuming that we can measure all of the states of the system, can we place the poles of the system to \( -1,-2 \), and \( -3 \) using state feedback? 4) Consider the state-space model given above. This time assume that we can only measure the output of the system. Can we still place the poles of the system to \( -1,-2 \), and \( -3 \) using the state-space model above? 5) Consider the state-space model given above. Suppose the output of interest is changed and the corresponding new \( \mathrm{C} \) matrix is as given below. Can we place the poles of the system to \( -1,-2 \), and \( -3 \) using the new state-space model? \[ C=\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right] \]

Answer: % define variable s s = tf('s'); % Define G(s) G = (s+10)*(s^2+2*s+25) / (s^2*(s^2+2*s+2)); % Ext