How To model a DC motor using state-space representation in MATLAB

To model a DC motor using state-space representation in MATLAB, follow these steps:

Step 1: Define the DC Motor Dynamics

A DC motor can be modeled using two state variables:

Angular velocity ( $ω$ ) of the rotor.
Armature current ( $i_{a}$ ).

Parameters:

$R$ = Armature resistance (Ω)
$L$ = Armature inductance (H)
$K_{t}$ = Torque constant (N·m/A)
$K_{e}$ = Back-emf constant (V·s/rad)
$J$ = Rotor inertia (kg·m²)
$B$ = Damping coefficient (N·m·s/rad)

Equations:

Electrical equation: $L d t d i ^{a} = V - R i_{a} - K_{e} ω$
Mechanical equation: $J d t d ω = K_{t} i_{a} - B ω$

Step 2: State-Space Representation

Define the state vector $x = [i^{a} ω]$ and input $u = V$ .

State-Space Matrices:

$x ˙ = A x + B u y = C x + D u$

State matrix: $A = [- L R J K ^{t} - L K ^{e} - J B]$
Input matrix: $B = [L 1 0]$
Output matrix (measuring angular velocity $ω$ ): $C = [01], D = 0$

Step 3: Implement in MATLAB

Example Code:

% Define DC motor parameters
R = 2.0;    % Ohms
L = 0.5;    % Henries
Kt = 0.1;   % N·m/A
Ke = 0.1;   % V·s/rad (Ke ≈ Kt for SI units)
J = 0.02;   % kg·m²
B = 0.01;   % N·m·s/rad

% State-space matrices
A = [-R/L  -Ke/L;
      Kt/J  -B/J];
B = [1/L; 0];
C = [0 1];   % Output is angular velocity (ω)
D = 0;

% Create state-space model
motor_ss = ss(A, B, C, D);

% Step response
figure;
step(motor_ss);
title('Step Response of DC Motor (Angular Velocity)');
xlabel('Time (s)');
ylabel('ω (rad/s)');

% Simulate with arbitrary input (e.g., PWM voltage)
t = 0:0.01:2;
u = 12 * ones(size(t));  % 12V input
figure;
lsim(motor_ss, u, t);
title('Voltage Input Response');

Step 4: Extend for Angular Position

If the output is angular position ( $θ$ ), add a third state:

$d t d θ = ω$

Update the state vector to $x = i^{a} ω θ$ , and adjust the matrices:

A_extended = [-R/L  -Ke/L  0;
              Kt/J  -B/J   0;
              0      1     0];
B_extended = [1/L; 0; 0];
C_extended = [0 0 1];  % Output is angular position (θ)
D_extended = 0;

motor_ss_pos = ss(A_extended, B_extended, C_extended, D_extended);

Step 5: Validate with Simulink

Build a Simulink model using:
- State-Space Block: Load your A, B, C, D matrices.
- Voltage Source: Use a step input or PWM signal.
- Scope: Monitor angular velocity/position.
Compare results with the MATLAB script.