Feedforward Control Theory
Feedforward control provides anticipatory compensation for measurable disturbances before they affect the controlled variable. Unlike feedback control that reacts to errors after they occur, feedforward generates preemptive control actions based on disturbance measurements and process models. When combined with feedback control, feedforward dramatically improves disturbance rejection, particularly for processes with large time constants or significant dead time where feedback alone responds too slowly.
Feedforward Control Principles
Feedforward control measures disturbances directly and calculates the control action needed to counteract their effects. The controller does not wait for the disturbance to create an error in the controlled variable. Instead, it predicts the disturbance’s impact and adjusts the manipulated variable to cancel that impact.
For perfect feedforward compensation, the feedforward controller must generate control actions that exactly cancel the disturbance’s effect on the controlled variable. This requires accurate measurement of the disturbance and an accurate model of how both the disturbance and manipulated variable affect the process.
The fundamental feedforward equation states that the feedforward control action should satisfy:
$$G_d(s) D(s) + G_p(s) U_{ff}(s) = 0$$
Where $G_d(s)$ represents the disturbance transfer function, $D(s)$ is the disturbance, $G_p(s)$ represents the process transfer function, and $U_{ff}(s)$ is the feedforward control action. Solving for the required feedforward controller:
$$G_{ff}(s) = -\frac{G_d(s)}{G_p(s)}$$
This feedforward controller theoretically provides perfect disturbance rejection. In practice, model inaccuracies and unmeasured disturbances prevent perfect compensation, necessitating feedback control for complete regulation.
Static Feedforward Compensation
Static feedforward uses only steady-state gains without dynamic compensation. The feedforward gain represents the ratio of how the disturbance affects the output to how the manipulated variable affects the output:
$$K_{ff} = -\frac{K_d}{K_p}$$
Where $K_d$ is the disturbance steady-state gain and $K_p$ is the process steady-state gain. The negative sign ensures the feedforward action opposes the disturbance’s effect.
For an air handling unit, outdoor air temperature represents a measurable disturbance affecting discharge temperature. If a 10°F outdoor temperature drop causes a 2°F discharge temperature drop (without control action), and a 10% increase in heating valve opening raises discharge temperature by 4°F, the static feedforward gain is:
$$K_{ff} = -\frac{-2°F/10°F}{4°F/10%} = 0.5%/°F$$
This gain converts outdoor temperature changes directly into valve position adjustments. A 10°F outdoor temperature drop commands a 5% valve opening increase to compensate.
Static feedforward provides the correct steady-state compensation but ignores dynamic effects. The disturbance and manipulated variable typically affect the process with different time constants and delays, causing transient errors even with correct static gain.
Dynamic Feedforward Compensation
Dynamic feedforward incorporates the dynamic relationship between disturbance and manipulated variable effects. The ideal feedforward controller is:
$$G_{ff}(s) = -\frac{G_d(s)}{G_p(s)}$$
If the disturbance affects the process faster than the manipulated variable, the feedforward controller must include lead compensation to advance the control action. If the manipulated variable acts faster than the disturbance, lag compensation may be appropriate.
Consider a disturbance with first-order dynamics $G_d(s) = K_d/(\tau_d s + 1)$ and process dynamics $G_p(s) = K_p/(\tau_p s + 1)$. The dynamic feedforward controller becomes:
$$G_{ff}(s) = -\frac{K_d}{K_p} \cdot \frac{\tau_p s + 1}{\tau_d s + 1}$$
This transfer function combines static gain compensation with dynamic filtering. If $\tau_p > \tau_d$, the controller provides lead action. If $\tau_d > \tau_p$, it provides lag.
Dead time differences between disturbance and manipulated variable paths create challenges. If the disturbance has less dead time than the process, perfect compensation requires predicting the future, which is impossible. Practical feedforward accepts the limitation, providing compensation for the delay difference.
Lead-Lag Compensation
Lead-lag compensators provide practical approximations to ideal dynamic feedforward. The lead-lag transfer function is:
$$G_{ff}(s) = K_{ff} \cdot \frac{\tau_1 s + 1}{\tau_2 s + 1}$$
When $\tau_1 > \tau_2$, the compensator provides phase lead, advancing control actions. When $\tau_1 < \tau_2$, it provides phase lag, retarding control actions. The time constants adjust the frequency range where compensation applies.
For HVAC applications, lead compensation typically applies when outdoor temperature disturbances (fast) require anticipatory heating or cooling adjustments (slower thermal process response). The lead time constant matches the process dominant time constant, while the lag time constant equals approximately one-tenth the lead value to limit high-frequency gain.
Combined Feedback-Feedforward Control
Feedforward and feedback control provide complementary capabilities. Feedforward offers fast disturbance rejection but requires accurate models and measurements. Feedback provides robustness to modeling errors and rejects unmeasured disturbances but responds only after errors occur.
The combined strategy employs feedforward for measured disturbances and feedback for everything else. The total control action sums the feedforward and feedback components:
$$u(t) = u_{ff}(t) + u_{fb}(t)$$
The feedforward path operates in parallel with the feedback controller, both contributing to the final actuator position. This arrangement preserves the stability properties of the feedback loop while adding feedforward’s performance benefits.
Tuning the combined system follows a specific sequence. First, disable feedforward and tune the feedback controller for acceptable performance. Then, enable feedforward and adjust feedforward gains and dynamics to improve disturbance rejection. The feedback controller compensates for feedforward errors, allowing aggressive feedforward tuning without destabilizing the system.
HVAC Application Examples
Outdoor air temperature compensation in air handlers:
Outdoor temperature directly affects mixing box temperature and load on heating/cooling coils. Feedforward calculates required valve adjustments based on outdoor temperature changes, significantly reducing discharge temperature deviations. The feedback discharge temperature controller handles remaining errors from model inaccuracies and other disturbances.
Implementation: Measure outdoor temperature change from baseline, multiply by feedforward gain (typically 0.5-2% valve per °F depending on system), add to feedback controller output.
Return air temperature compensation in economizers:
Return air temperature variations affect mixing box temperature when the economizer modulates outdoor air. Feedforward adjusts the cooling valve based on return temperature, compensating for load changes before they affect discharge temperature.
The feedforward gain depends on the outdoor air damper position—larger outdoor air fraction produces smaller return air influence, requiring gain scheduling.
Space load compensation:
In some systems, occupancy sensors or equipment status signals indicate load changes before they affect space temperature. Feedforward adjusts supply airflow or temperature based on these load indicators, maintaining tighter temperature control during occupancy transitions.
Humidity control with outdoor dewpoint:
Outdoor dewpoint variations affect the moisture load on dehumidification systems. Feedforward modulates cooling coil capacity or reheat based on outdoor dewpoint, compensating for moisture load changes before indoor humidity deviates.
Practical Implementation Considerations
Sensor accuracy: Feedforward requires accurate disturbance measurement. Sensor errors directly translate to feedforward errors. Periodically calibrate disturbance sensors and use high-quality instruments for critical measurements.
Model accuracy: The feedforward gain and dynamics depend on process models. As equipment ages or operating conditions change, the model may degrade. Adaptive techniques can adjust feedforward parameters based on observed system response.
Saturation handling: When the actuator saturates (fully open or closed), additional feedforward action provides no benefit. Implement logic to track saturation and prevent feedforward windup during these periods.
Failure modes: If the disturbance measurement fails, feedforward should disable gracefully, reverting to feedback-only operation. Include monitoring and alarm logic to detect sensor failures and invalid feedforward calculations.
Commissioning: Start with small feedforward gains and gradually increase while monitoring system response. Excessive feedforward gain or incorrect dynamic compensation can degrade performance compared to feedback alone.
Feedforward Gain Calculation Methods
Empirical tuning: Introduce known disturbances and observe the feedback controller’s steady-state response. The feedforward gain equals the controller output change divided by the disturbance magnitude with opposite sign. For example, if a 10°F outdoor temperature drop causes the feedback controller to increase valve position by 8%, the feedforward gain is -(-8%/10°F) = 0.8%/°F.
First-principles modeling: Calculate gains from equipment specifications and thermodynamic relationships. For coil heating, the disturbance gain relates outdoor temperature influence on mixed air to the feedback process gain from valve position to discharge temperature. Energy balance equations provide the theoretical relationship.
System identification: Perform controlled experiments varying the disturbance and manipulated variable while measuring responses. Regression analysis identifies the transfer functions $G_d(s)$ and $G_p(s)$, from which the feedforward controller derives directly.
Limitations of Feedforward Control
Feedforward cannot compensate for unmeasured disturbances. Only disturbances with dedicated sensors receive feedforward action. This limitation explains why feedforward always requires feedback for complete regulation.
Model inaccuracies produce compensation errors. The process and disturbance models change with fouling, aging, and varying operating conditions. Periodic recalibration or adaptive techniques maintain accuracy.
Measurement delays limit feedforward performance. If the disturbance measurement occurs after the disturbance has already affected the process, feedforward reduces to belated feedback rather than true anticipatory control. Sensor location must enable early disturbance detection.
Implementation complexity increases system cost and commissioning time. Feedforward requires additional sensors, programming, and tuning. The benefits must justify these costs based on improved performance requirements.
Despite these limitations, feedforward provides powerful performance improvements for HVAC systems with measurable disturbances and significant process lags. Proper design and implementation create noticeable improvements in temperature stability, humidity control, and energy efficiency.