(Solved): This homework will use MATLAB functions using the problems in the state-space representation of HW6 ...

This homework will use MATLAB functions using the problems in the state-space representation of HW6 of Problem 2a-3. For all a-d, a) \( \ddot{x}+2 \dot{x}+x=3+t \),output \( y=x \), I.C. \( x(0)=4, \dot{x}(0)=2 \) b) \( \ddot{x}+7 \dot{x}+3 x=-4 \), output \( y_{1}=x, y_{2}=\dot{x} \), I.C. \( x(0)=0, \dot{x}(0)=-4 \) c) \( \ddot{x}+\dot{x}+3 x=2-3 t \), output \( y=x+7 \dot{x} \), I.C. \( x(0)=2, \dot{x}(0)=3 \) d) \( \dddot{x}+2 \ddot{x}+7 \dot{x}+10 x=5 \), output \( y_{1}=x-2, y_{2}=\dddot{x} \), I.C. \( x(0)=4, \dot{x}(0)=2, \ddot{x}(0)=0 \) 1. Use function eig on your A matrix to find the roots of the characteristic equation 2. Use function poly on the solution from the result of eig(A) to re-create the characteristic equation. 3. Find the system settling time, and write the general form of the homogenous solution with constants. (e.g, \( x_{h}=A_{1} e^{-3 t}+A_{2} e^{-t} \), you don't need to compute \( A_{1} \) and \( A_{2} \) ) 4. Use function initial to find the homogenous response plot when \( u(t)=0 \) 5. Use function step to find the unit step response plot when all initial conditions are 0 6. Use function Isim to find the complete response plot for the given initial conditions. Before you proceed, set the \( \vec{u} \) as following and compute matrices B and D to match where \( \vec{u}=[1, t] \) for problems a) \& c) and \( \vec{u}=[1] \) for problems b) \& d)