Introduction & Context

This reference sheet quantifies the relative merits of feedback (FB) versus feed-forward (FF) control for a distillation column separating ethanol–water. The comparison is based on closed-loop time constants, disturbance frequency response, effective delay mismatch, and implementation complexity. The worksheet is used during the front-end engineering design (FEED) phase to decide whether the extra instrumentation and modelling effort required for feed-forward compensation is justified by the expected reduction in product variability.

Methodology & Formulas

Disturbance frequency
Convert the observed disturbance period \(T_{\rm d}\) to angular frequency: \[ \omega_{\rm d} = 2\pi/T_{\rm d}. \]
Feedback closed-loop time constant
With a PI controller tuned for tight control, the closed-loop time constant is: \[ \tau_{\rm FB} = \frac{\tau_{\rm proc}}{1 + K_{\rm c}K_{\rm p}} \] where \(K_{\rm c}\) is the controller gain and \(K_{\rm p}\) the process steady-state gain.
Frequency–time constant product
The dimensionless group: \[ \omega_{\rm d}\tau_{\rm FB} \] determines whether feedback alone is fast enough to attenuate the disturbance. Berk’s rule states that feed-forward becomes economically attractive when this product is ≤ 1.5.
Feed-forward effective delay mismatch
The normalised delay error is: \[ \varepsilon = \frac{|\theta_{\rm FF} - \theta_{\rm proc}|}{\tau_{\rm proc}} \] where \(\theta_{\rm FF}\) is the sensor–model delay in the feed-forward path and \(\theta_{\rm proc}\) the true process dead-time.
Relative capital cost increment
An empirical cost penalty for the extra hardware and modelling effort is: \[ C_{\rm rel} = 0.4(N_{\rm sens} - 1) + 0.2(N_{\rm model} - 1) \] with \(N_{\rm sens}\) the number of sensors (FB = 1, FF adds one extra) and \(N_{\rm model}\) the modelling effort (empirical tuning assumed 2).

Validity regimes and warning thresholds
Parameter	Range / Criterion	Interpretation
\(\tau_{\rm FB}/\tau_{\rm proc}\)	0.33 – 1.0	Acceptable tuning aggressiveness
\(\varepsilon\)	< 0.3	Feed-forward delay mismatch acceptable
\(\omega_{\rm d}\tau_{\rm FB}\)	> 1.5	FB alone sufficient; FF not justified
Reynolds number Re	< 10,000	Laminar flow assumption valid

Feed-forward is preferred when the disturbance can be measured before it upsets the controlled variable and when the process model is well-known. Use feedback when disturbances are unmeasurable, the model is uncertain, or tight steady-state accuracy is required. Many plants combine both: feed-forward rejects the measured upsets and feedback trims the residual error.

Step 1: Identify the process, disturbance, and valve gains plus dead-times.
Step 2: Tune the feedback PID first with the feed-forward off; aim for a conservative setting to avoid oscillations.
Step 3: Add the feed-forward compensator (static gain and dynamic lag/lead) and verify steady-state compensation with a step disturbance.
Step 4: Fine-tune the feed-forward dynamic block so the disturbance effect is cancelled within one process dead-time.
Step 5: Re-check feedback tuning; reduce proportional band slightly if the combined response shows overshoot.

An inaccurate model causes incomplete disturbance rejection and can even amplify the upset. The feedback controller will eventually correct the offset, but performance degrades. Monitor the feedback controller output: if it continuously drifts away from its normal bias, the feed-forward gain or dynamics need adjustment. Periodic model re-identification and adaptive gain scheduling help maintain effectiveness.

Yes, provided you can measure the disturbance and add a summer block in the DCS. Insert the feed-forward signal just after the PID output summer so the PID still sees the true error. Make sure the feed-forward path includes a dynamic compensation block (lead-lag) to match process timing. Commission with the feedback in manual first, then switch to auto and retune the PID if necessary.

Worked Example: Feed-Forward vs. Feedback Control of a Distillation Column

A binary distillation column is used to separate ethanol–water. The operator wants to keep the overhead ethanol mole fraction at 80.0 mol % despite disturbances in the feed composition. We will compare the closed-loop performance of a feedback PI controller with a combined feed-forward/feedback scheme.

Feed flow rate: 100.0 kmol h⁻¹
Nominal feed composition: 10.0 mol % ethanol
Measured feed-composition variability: ±2.0 mol %
Overhead specification: 80.0 mol % ethanol
Overhead tolerance: ±0.5 mol %
Process gain: 0.04 (mol % ethanol)/(mol % feed)
Process time constant: 6.0 min
Process dead time: 1.0 min
Disturbance dead time for feed-forward: 0.5 min
PI controller gain: 20.0 %/%
Integral time: 8.0 min

Compute the feedback-only closed-loop time constant: \[ \tau_{\text{FB}} = \frac{\tau_{\text{proc}}}{1 + K_c K_p} = \frac{6.0}{1 + 20.0 \times 0.04} = 3.333\ \text{min} \]
Determine the dimensionless disturbance frequency: \[ \omega_d = \frac{2\pi}{T_d} = \frac{2\pi}{8.0} = 0.785\ \text{rad min}^{-1} \]
Evaluate the dimensionless frequency–time-constant product: \[ \omega_d \tau_{\text{FB}} = 0.785 \times 3.333 = 2.618 \]
Compute the maximum expected feedback error (first-order approximation): \[ \epsilon_{\text{FB}} = \frac{K_p \Delta z}{1 + K_c K_p} = \frac{0.04 \times 2.0}{1 + 20.0 \times 0.04} = 0.083\ \text{mol %} \]
Estimate the feed-forward model accuracy factor: \[ C_{\text{rel}} = \frac{N_{\text{sens}}}{N_{\text{model}}} \cdot \frac{\theta_{\text{FF}}}{\theta_{\text{proc}}} = \frac{2}{2} \cdot \frac{0.5}{1.0} = 0.600 \]
Predict the residual error with combined feed-forward + feedback: \[ \epsilon_{\text{FF+FB}} = C_{\text{rel}} \cdot \epsilon_{\text{FB}} = 0.600 \times 0.083 = 0.050\ \text{mol %} \]

Final Answer: The feedback-only strategy yields a worst-case steady-state offset of 0.083 mol % ethanol, exceeding the 0.5 mol % tolerance band. Adding feed-forward reduces the residual error to 0.050 mol %, keeping the overhead composition within specification.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle