Adding integral action to MPC for offset-free tension tracking.
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MPC with model mismatch may have steady-state error. Integral action eliminates offset by accumulating error over time.
u_I = gamma * u_I - c_I * dt * (T - T_ref)
u_total = u_mpc + u_I
Where:
Integral gain c_I:
Decay factor gamma:
Limit integral term to prevent windup during saturation:
u_I = np.clip(u_I, -max_integral, max_integral)
Apply integral action per tension section:
for i in range(num_sections):
u_I[i] = gamma * u_I[i] - c_I * dt * (T[i] - T_ref[i])
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