Consider the mechanical system given in Figure $2.d_1(t)$ and $d_2(t)$ are the displacements of the Consider the mechanical system given in Figure $2.d_1(t)$ and $d_2(t)$ are the displacements of the

two blocks with respect to their nominal positions $($when no any force exist$).v_1(t)$ and $v_2(t)$ are

the velocities of the two blocks, respectively.

Figure $2.$ Mechanical System

two blocks with respect to their nominal positions $($when no any force exist$).v_1(t)$ and $v_2(t)$ are

the velocities of the two blocks, respectively.

Figure $2.$ Mechanical SystemAn external force $f(t)$ is applied to the system as the input. Assume that the movement of blocks

$1$ and $2$ is opposed by viscous friction forces given by

$f_{D_1}(t)=D_1{v}_1(t),f_{D_2}(t)=D_2{v}_2(t)$

Choose the state vector to be

$x(t)={\left[\begin{array}{cccc}{d}_1(t)& d_2(t)& v_1(t)& v_2(t)\end{array}\right]}^T$

$(1)$ Derive the state$-$space model of the system. Assume the output of the system, $y(t),$ is the

velocity of the second block.

$(2)$ Analyse the stability, controllability and observability of the system. Assume the

parameters are given by:

$M_1=10[kg],M_2=5[kg],k_1=20[N{m}^{-1}],k_2=35[N{m}^{-1}]$

$b=5[Ns{m}^{-1}],D_1=6[Ns{m}^{-1}],D_2=10[Ns{m}^{-1}]$

$(3)$ Design state feedback control law $u=-Kx$ such that the poles of the closed$-$loop system

are $-3,-11,-12,-13.$ Show the state trajectory and output trajectory in MATLAB for a given

initial state.

$(4)$ Next, choose the state vector as

$\stackrel{}{x}(t)={\left[\begin{array}{cccc}{d}_1(t)& d_2(t)-d_1(t)& v_1(t)& v_2(t)-v_1(t)\end{array}\right]}^T$

Apply similarity transformation to obtain the new state space model using the new state vector.

Analyse the stability, controllability and observability of the new system.