Integral Action for Offset-Free Control

Why Integral Action?

MPC with model mismatch may have steady-state error. Integral action eliminates offset by accumulating error over time.

Implementation

u_I = gamma * u_I - c_I * dt * (T - T_ref)
u_total = u_mpc + u_I

Where: - u_I: Integral control term - gamma: Decay factor (0.9-0.99) - c_I: Integral gain (0.1-0.5 for web systems) - dt: Timestep

Tuning Guidelines

Integral gain c_I: - Too low: Slow offset correction - Too high: Oscillations, instability - Start with 0.1-0.3

Decay factor gamma: - gamma = 1.0: Pure integral (may wind up) - gamma < 1.0: Leaky integrator (safer) - Typical: 0.9-0.99

Anti-Windup

Limit integral term to prevent windup during saturation:

u_I = np.clip(u_I, -max_integral, max_integral)

For R2R Systems

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])