Brushed DC motor

Introduction

If you have never heard about Brushed DC motor, then the Wikipedia is a really good start.

A short description about the direct current motor: when a rectangular coil carrying current is placed in a magnetic field, a torque acts on the coil which rotates it continuously. When the coil rotates, the shaft attached to it also rotates and thus it is able to do mechanical work.

When the coil is powered, a magnetic field is generated around the armature. The left side of the armature is pushed away from the left magnet and drawn towards the right, causing rotation.

When the coil turns through 90°, the brushes lose contact with the commutator and the current stops flowing through the coil. However the coil keeps turning because of its own momentum. Now when the coil turns through 180°, the sides get interchanged 

 Motor Model

The following model is a equivalent-circuit representation of the brushed DC motor.

 In the equivalent-circuit above the electrical system equation is the following - using Kirchhoff's voltage and current law:
$$V_{a}=iR_{a}+L_{a}\frac{di}{dt}+e$$ $$e=K_{e}\omega_{m}$$  Mechanical system equation:
$$T_{l} = -J\frac{d\omega_{m}}{dt}-b\omega_{m}+K_{m}i_{a}$$ From the equation above we can write the following equations:
$$\frac{di}{dt} = - \frac{R_{a}}{L_{a}}i_{a} - \frac{K_{e}}{L_{a}}\omega_{m} + \frac{1}{L_{a}}V_{a}$$ $$\frac{d\omega_{m}}{dt} = \frac{K_{m}}{J}i_{a} - \frac{b}{J}\omega_{m} - \frac{1}{J}T_{l}$$  
Where:
$J$: Motor and load Inertia
$b$ : Viscous damping coefficient
$K_{e}$: Speed constant
$K_{m}$: Torque constant
$L_{a}$: Motor armature coil inductance
$R_{a}$: Motor armature coil resistance

State Space Method

The state space method takes from the
$$\dot{x(t)} = Ax(t) + B u(t)$$ $$y(t) = Cx(t) + Du(t)$$
Where the state-space matrices are the following:
$$\dot{x(t)} =\left[ {\begin{array}{cc} \frac{di}{dt} \\ \frac{d\omega}{dt} \\ \frac{d\theta}{dt} \end{array} } \right] A =\left[ {\begin{array}{cc} -\frac{R_{a}}{L_{a}} & -\frac{K_{e}}{L_{a}} &  0 \\ \frac{K_{m}}{J} & -\frac{b}{J} & 0 \\ 0 & 1 & 0 \end{array} } \right] B=\left[ {\begin{array}{cc} \frac{1}{L_{a}} \\ 0 \\ 0 \end{array} } \right] C=\left[ {\begin{array}{cc} 0 \\ 1 \\ 0 \end{array} } \right] D=\left[ {\begin{array}{cc} 0 \end{array} } \right]$$