Design, model, and simulate a DC motor system using MATLAB and Simulink. The project involves deriving mathematical equations, solving them using multiple methods, analyzing system performance. The tasks are as follows: $1.$ DC Motor Overview $$ Explain the principle of operation of a DC motor, emphasizing torque generation and angular motion. $$ Identify the basic components of the DC motor, such as: Stator, Rotor, Armature winding and Commutator $($include a schematic $($Reference$)).2.$ DC Motor Parameters Use the following parameters for analysis: $$ Armature resistance, Ra$=1\mathrm{.}5$ Armature inductance, La$=0\mathrm{.}3$H $$ Motor torque constant, Kt$=0\mathrm{.}08$Nm$/$A $$ Back EMF constant, Ke$=0\mathrm{.}08$V$/$rad$/$s $$ Rotor inertia, J$=0\mathrm{.}02$kg$/$m$2$ Damping coefficient, B$=0\mathrm{.}03$Ns$/$m $3.$ Mathematical Modeling $$ Derive the electrical equation for the armature circuit $$ Derive the mechanical equation $$ Combine the equations to derive the complete mathematical model relating input voltage to angular position $4.$ Solution of the Mathematical Model $$ Solve the equations using the following methods: $1.$ State$-$Space Variables: Represent the system using state variables and derive the state$-$space representation. $2.$ Transfer Function: Derive the transfer function relating input voltage to angular position in the Laplace domain.