Model-based control of zone dampers in an HVAC system

1. A heating, ventilation, and air conditioning (HVAC) system comprising:

HVAC equipment with a fan configured to provide conditioned air to a conditioned space that is divided into zones, and that is set up in an installation with ductwork that defines an air circulation path, and zone dampers controllable to regulate airflow of the conditioned air to respective ones of the zones; and

control circuitry operably coupled to the HVAC equipment and configured to at least:

receive an indication of zone airflow target values for the zones of the conditioned space;

access a duct model including a first term that describes a flow factor for a zone damper as a nonlinear function of damper position, a second term that describes zone airflow as a function of the flow factor and a branch pressure across zone-specific branches of the air circulation path, and at least one or more unknown parameters that are specific to the installation and are determined using a backpropagation-based regression analysis of training data and an estimate of the unknown parameters;

apply the zone airflow target values to the duct model to determine a damper position set; and

actuate the zone dampers to respective damper positions of the damper position set.

2. The HVAC system of claim 1 , wherein the control circuitry is configured to apply the zone airflow target values to the duct model to further determine a value of total airflow to achieve the zone airflow target values with a total pressure target, and

wherein the control circuitry is further configured to cause the fan to provide the conditioned air with the value of the total airflow.

3. The HVAC system of claim 1 , wherein the control circuitry is further configured to at least:

determine values of the unknown parameters that are specific to the installation; and

fit the duct model with the values of the unknown parameters.

4. The HVAC system of claim 3 , wherein the control circuitry configured to determine the values of the unknown parameters includes the control is configured to at least:

receive an indication of zone sizes, each of the zone sizes expressed as a maximum zone airflow for the zone with the zone damper in a fully-open position; and

determine the values of the unknown parameters using the duct model and the zone sizes.

5. The HVAC system of claim 3 , wherein the duct model further includes a third term that describes a shared pressure across a shared portion of the air circulation path as a function of total airflow, and the control circuitry configured to determine the values of the unknown parameters includes the control circuitry configured to at least:

obtain training data that includes observations of the total airflow, observations of total pressure, and damper position sets, wherein the total pressure is a sum total of the branch pressure and the shared pressure;

perform a regression analysis of the duct model using the training data and an estimate of the unknown parameters to determine updated estimates of the unknown parameters; and

determine the values of the unknown parameters from the updated estimates of the unknown parameters.

6. The HVAC system of claim 5 , wherein the control circuitry configured to obtain the training data includes the control circuitry configured to at least:

obtain state vectors that represent states of the installation, and that include respective ones of the observations of the total airflow, the observations of the total pressure, and the damper position sets;

select or reject the state vectors based on acceptability of the states and variation across the states of the installation, the acceptability of a state determined based on at least a metric that characterizes the state as steady-state or transient based on a respective state vector, the state being acceptable when the state is at least characterized as steady-state; and

add the observations of the total airflow, the observations of the total pressure, and the damper position sets for those of the state vectors that are selected to the training data.

7. The HVAC system of claim 5 , wherein the control circuitry configured to perform the regression analysis includes the control circuitry configured to at least:

access, from the training data, a respective observation of the total airflow, a respective observation of the total pressure, and a respective damper position set;

apply the respective observation of the total pressure, the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine a predicted total airflow;

determine an error between the predicted total airflow and the respective observation of the total airflow; and

perform the regression analysis to update the estimates of the unknown parameters to the updated estimates of the unknown parameters that cause the predicted total airflow to be closer to the respective observation of the total airflow, and thereby reduce the error.

8. The HVAC system of claim 7 , wherein the control circuitry configured to apply the respective observation of the total pressure, the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine the predicted total airflow includes the control circuitry configured to at least:

determine estimates of the flow factor for the zone dampers using the respective damper position set, and the first term of the duct model that describes the flow factor;

perform a recursion to determine predicted zone airflows using the respective observation of the total pressure, the estimates of the flow factor, the second term of the duct model that describes the zone airflow, and the third term of the duct model that describes the shared pressure; and

determine the predicted total airflow from the predicted zone airflows.

9. The HVAC system of claim 5 , wherein the control circuitry configured to perform the regression analysis includes the control circuitry configured to at least:

access, from the training data, a respective observation of the total airflow, a respective observation of the total pressure, and a respective damper position set;

apply the respective observation of the total airflow and the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine a predicted total pressure;

determine an error between the predicted total pressure and the respective observation of the total pressure; and

perform the regression analysis to update the estimates of the unknown parameters to the updated estimates of the unknown parameters that cause the predicted total pressure to be closer to the respective observation of the total pressure, and thereby reduce the error.

10. The HVAC system of claim 9 , wherein the control circuitry configured to apply the respective observation of the total airflow and the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine the predicted total pressure includes the control circuitry configured to at least:

determine estimates of the flow factor for the zone dampers using the respective damper position set and the first term of the duct model that describes the flow factor; and

determine the predicted total pressure using the respective observation of the total airflow, the estimates of the flow factor, the second term of the duct model restated to describe the branch pressure as a function of the total airflow and the flow factor for the zone dampers, and the third term of the duct model that describes the shared pressure.

11. The HVAC system of claim 1 , wherein the control circuitry configured to apply the zone airflow target values to the duct model to determine the damper position set includes the control circuitry configured to at least:

determine estimated maximums of the flow factor for the zone dampers in a fully-open position from the first term of the duct model that describes the flow factor;

determine a maximum value of the branch pressure using the zone airflow target values, the estimated maximums of the flow factor, and the second term of the duct model restated to describe the branch pressure as a function of the zone airflow and the flow factor;

determine estimates of the flow factor for the zone dampers using the maximum value of the branch pressure, the zone airflow target values, and the second term of the duct model restated to describe the flow factor as a function of the zone airflow and the branch pressure; and

determine the damper position set using the estimates of the flow factor and the first term of the duct model restated to describe the damper position as a nonlinear function of the flow factor.

12. The HVAC system of claim 11 , wherein the duct model further includes a third term that describes a shared pressure across a shared portion of the air circulation path as a function of total airflow, the zone airflow target values are applied to the duct model to further determine a value of the total airflow to achieve the zone airflow target values with a branch pressure target, and the control circuitry configured to apply the zone airflow target values to the duct model to determine the value of the total airflow includes the control circuitry configured to at least:

determine an estimate of the total pressure using the maximum value of the branch pressure, an estimate of the total airflow as a sum total of the zone airflow target values, and the third term that describes the shared pressure;

perform a recursion to determine predicted zone airflows using the estimate of the total pressure, the estimates of the flow factor, the second term of the duct model that describes the zone airflow, and the third term of the duct model that describes the shared pressure; and

determine the value of the total airflow from the predicted zone airflows, and

wherein the control circuitry is further configured to cause the fan to provide the conditioned air with the value of the total airflow.

13. A method of controlling a heating, ventilation, and air conditioning (HVAC) system that includes HVAC equipment with a fan configured to provide conditioned air to a conditioned space that is divided into zones, and that is set up in an installation with ductwork that defines an air circulation path, and zone dampers controllable to regulate airflow of the conditioned air to respective ones of the zones, the method comprising:

receiving an indication of zone airflow target values for the zones of the conditioned space;

accessing a duct model including a first term that describes a flow factor for a zone damper as a nonlinear function of damper position, and a second term that describes zone airflow as a function of the flow factor, a branch pressure across zone-specific branches of the air circulation path, and at least one or more unknown parameters that are specific to the installation and are determined using a backpropagation-based regression analysis of training data and an estimate of the unknown parameters;

applying the zone airflow target values to the duct model to determine a damper position set; and

actuating the zone dampers to respective damper positions of the damper position set.

14. The method of claim 13 , wherein the zone airflow target values are applied to the duct model to further determine a value of total airflow to achieve the zone airflow target values with a total pressure target, and

wherein the method further comprises causing the fan to provide the conditioned air with the value of the total airflow.

15. The method of claim 13 , wherein the method further comprises:

determining values of the unknown parameters that are specific to the installation; and

fitting the duct model with the values of the unknown parameters.

16. The method of claim 15 , wherein determining the values of the unknown parameters comprises:

receiving an indication of zone sizes, each of the zone sizes expressed as a maximum zone airflow for the zone with the zone damper in a fully-open position; and

determining the values of the unknown parameters using the duct model and the zone sizes.

17. The method of claim 15 , wherein the duct model further includes a third term that describes a shared pressure across a shared portion of the air circulation path as a function of total airflow, and determining the values of the unknown parameters comprises:

obtaining training data that includes observations of the total airflow, observations of total pressure, and damper position sets, wherein the total pressure is a sum total of the branch pressure and the shared pressure;

performing a regression analysis of the duct model using the training data and an estimate of the unknown parameters to determine updated estimates of the unknown parameters; and

determining the values of the unknown parameters from the updated estimates of the unknown parameters.

18. The method of claim 17 , wherein obtaining the training data comprises:

obtaining state vectors that represent states of the installation, and that include respective ones of the observations of the total airflow, the observations of the total pressure, and the damper position sets;

selecting or rejecting the state vectors based on acceptability of the states and variation across the states of the installation, the acceptability of a state determined based on at least a metric that characterizes the state as steady-state or transient based on a respective state vector, the state being acceptable when the state is at least characterized as steady-state; and

adding the observations of the total airflow, the observations of the total pressure, and the damper position sets for those of the state vectors that are selected to the training data.

19. The method of claim 17 , wherein performing the regression analysis comprises:

accessing, from the training data, a respective observation of the total airflow, a respective observation of the total pressure, and a respective damper position set;

applying the respective observation of the total pressure, the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine a predicted total airflow;

determining an error between the predicted total airflow and the respective observation of the total airflow; and

performing the regression analysis to update the estimates of the unknown parameters to the updated estimates of the unknown parameters that cause the predicted total airflow to be closer to the respective observation of the total airflow, and thereby reduce the error.

20. The method of claim 19 , wherein applying the respective observation of the total pressure, the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine the predicted total airflow comprises:

determining estimates of the flow factor for the zone dampers using the respective damper position set, and the first term of the duct model that describes the flow factor;

performing a recursion to determine predicted zone airflows using the respective observation of the total pressure, the estimates of the flow factor, the second term of the duct model that describes the zone airflow, and the third term of the duct model that describes the shared pressure; and

determining the predicted total airflow from the predicted zone airflows.

21. The method of claim 17 , wherein performing the regression analysis comprises:

accessing, from the training data, a respective observation of the total airflow, a respective observation of the total pressure, and a respective damper position set;

applying the respective observation of the total airflow and the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine a predicted total pressure;

determining an error between the predicted total pressure and the respective observation of the total pressure; and

performing the regression analysis to update the estimates of the unknown parameters to the updated estimates of the unknown parameters that cause the predicted total pressure to be closer to the respective observation of the total pressure, and thereby reduce the error.

22. The method of claim 21 , wherein applying the respective observation of the total airflow and the respective damper position set, and the estimate of the unknown parameters, to the duct model to determine the predicted total pressure comprises:

determining estimates of the flow factor for the zone dampers using the respective damper position set and the first term of the duct model that describes the flow factor; and

determining the predicted total pressure using the respective observation of the total airflow, the estimates of the flow factor, the second term of the duct model restated to describe the branch pressure as a function of the total airflow and the flow factor for the zone dampers, and the third term of the duct model that describes the shared pressure.

23. The method of claim 13 , wherein applying the zone airflow target values to the duct model to determine the damper position set comprises:

determining estimated maximums of the flow factor for the zone dampers in a fully-open position from the first term of the duct model that describes the flow factor;

determining a maximum value of the branch pressure using the zone airflow target values, the estimated maximums of the flow factor, and the second term of the duct model restated to describe the branch pressure as a function of the zone airflow and the flow factor;

determining estimates of the flow factor for the zone dampers using the maximum value of the branch pressure, the zone airflow target values, and the second term of the duct model restated to describe the flow factor as a function of the zone airflow and the branch pressure; and

determining the damper position set using the estimates of the flow factor and the first term of the duct model restated to describe the damper position as a nonlinear function of the flow factor.

24. The method of claim 23 , wherein the duct model further includes a third term that describes a shared pressure across a shared portion of the air circulation path as a function of total airflow, the zone airflow target values are applied to the duct model to further determine a value of the total airflow to achieve the zone airflow target values with a branch pressure target, and applying the zone airflow target values to the duct model to determine the value of the total airflow comprises:

determining an estimate of the total pressure using the maximum value of the branch pressure, an estimate of the total airflow as a sum total of the zone airflow target values, and the third term that describes the shared pressure;

performing a recursion to determine predicted zone airflows using the estimate of the total pressure, the estimates of the flow factor, the second term of the duct model that describes the zone airflow, and the third term of the duct model that describes the shared pressure; and

determining the value of the total airflow from the predicted zone airflows, and wherein the method further comprises causing the fan to provide the conditioned air with the value of the total airflow.