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
python
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:
python
u_I = np.clip(u_I, -max_integral, max_integral)
For R2R Systems
Apply integral action per tension section:
python
for i in range(num_sections):
u_I[i] = gamma * u_I[i] - c_I * dt * (T[i] - T_ref[i])