How to Model a DC Motor in State-Space with Load Torque Disturbance Rejection?

I’m working on a DC motor control system and need to implement a state-space model that accounts for load torque disturbances. The motor is armature-controlled with parameters:

Inertia (J), damping (B), torque constant (Kt), back-EMF (Ke).

States: Angular velocity (ω) and armature current (iₐ).

Problem:

State-Space Setup: How to properly structure the state-space matrices to include both control input (Va) and disturbance input (Tl)?

Disturbance Rejection:

If using a PID controller, how to ensure the integral term compensates for Tl?

Is a disturbance observer (DOB) better for this case?

Simulation:

In MATLAB, how to simulate the system with lsim when there are two inputs (Va and Tl)?

Should I manually inject Tl into the derivative equations?

Request:

Theoretical guidance on state-space disturbance modeling.

MATLAB/Simulink implementation tips.

Answer 1

[Sabin]

The state-space can be constructed as xdot=A.x+B.u+w, where w is the disturbance. If the disturbance is known a coordinate transformation can be derived to cast the state-space system as zdot=F.z+G.s. The integral term will ensure zero steady-state error. An alternative is to use a disturbance observer. Perhaps you can try the Active Disturbance Rejection Control instead of PID. There are also many relevant examples on mathworks.com:

https://www.mathworks.com/help/sps/ug/dc-motor-control-state-feedback-and-observer.html

https://www.mathworks.com/help/slcontrol/ug/design-adrc-for-bldc-motor.html