Patent Publication Number: US-11644207-B2

Title: Smart thermostat with model predictive control

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 16/185,274 filed Nov. 9, 2018, which is a continuation of U.S. patent application Ser. No. 15/625,830 filed Jun. 16, 2017, and granted Dec. 4, 2018 as U.S. Pat. No. 10,146,237, which claims the benefit of and priority to U.S. Provisional Patent Application No. 62/491,545 filed Apr. 28, 2017. All of these patent applications and patents are incorporated by reference herein in their entireties. 
    
    
     BACKGROUND 
     The present disclosure relates generally to a smart thermostat and more particularly to a smart thermostat with model predictive control. Thermostats are often configured to monitor and control the temperature of a building zone or other space. For example, a thermostat can be mounted on a wall within the building zone and configured to measure the temperature of the building zone. Thermostats typically send commands to HVAC equipment (e.g., on/off commands, heating/cooling commands, etc.) to cause the HVAC equipment to affect the temperature of the building zone. 
     Conventional thermostats operate according to a fixed temperature setpoint schedule which defines the temperature setpoints for the thermostat at various times. The temperature setpoint schedule is typically set by a user via a local user interface on the thermostat. In many implementations, a fixed temperature setpoint schedule leads to suboptimal control of the HVAC equipment, which can increase the cost of heating/cooling the building zone. It would be desirable to automatically determine optimal temperature setpoints for a thermostat in order to take advantage of time-varying energy prices, zone heat transfer characteristics, and/or other factors that can affect the cost of heating/cooling the building zone. 
     SUMMARY 
     One implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes an equipment controller and a model predictive controller. The equipment controller is configured to drive the temperature of the building zone to an optimal temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the optimal temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps. 
     In some embodiments, the model predictive controller is configured to determine the cost of operating the HVAC equipment during each of the plurality of time steps using a set of time-varying utility rates comprising a utility rate value for each time step. The time-varying utility rates may be received from a utility provider or predicted by the model predictive controller. 
     In some embodiments, the model predictive controller is configured to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory comprising a temperature setpoint for each of the plurality of time steps. 
     In some embodiments, the model predictive controller is configured to optimize the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range. 
     In some embodiments, the model predictive controller is configured to generate the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the model predictive controller is configured to predict a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. 
     In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     In some embodiments, the model predictive controller is configured to predict the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period. 
     In some embodiments, the constraint on the predicted temperature of the building zone requires the model predictive controller to maintain the predicted temperature of the building zone within a first zone temperature range during a first time step of the optimization period and within a second zone temperature range, different from the first zone temperature range, during another time step of the optimization period subsequent to the first time step. 
     Another implementation of the present disclosure is a method performed by a thermostat for a building zone for monitoring and controlling temperature of the building zone. The method includes generating a cost function that accounts for a cost operating HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps, and operating HVAC equipment to provide heating or cooling to the building zone to drive the temperature of the building zone to the optimal temperature setpoints. 
     In some embodiments, the method includes receiving a set of time-varying utility rates from a utility provider or predicting the time-varying utility rates. The set of time-varying utility rates may include a utility rate value for each time step. The method may include determining the cost of operating the HVAC equipment during each of the plurality of time steps using the set of time-varying utility rates. 
     In some embodiments, using the predictive model to predict the temperature of the building zone includes predicting the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory that includes a temperature setpoint for each of the plurality of time steps. 
     In some embodiments, optimizing the cost function includes optimizing the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range. 
     In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the method includes predicting a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. 
     In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     In some embodiments, the method includes predicting the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period. 
     Another implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes an equipment controller and a model predictive controller. The equipment controller is configured to drive the temperature of the building zone to a zone temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the zone temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and performing an optimization of the cost function subject to a constraint on the predicted temperature of the building zone to determine a temperature setpoint trajectory including a temperature setpoint value for each of the plurality of time steps. 
     Another implementation of the present disclosure is a model predictive control system for monitoring and controlling temperature of a building zone. The model predictive control system includes a thermostat and a model predictive controller. The thermostat is configured to drive the temperature of the building zone to an optimal temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the optimal temperature setpoint and provide the optimal temperature setpoint to the thermostat via a communications network. The model predictive controller determines the optimal temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps. 
     In some embodiments, the model predictive controller is configured to determine the cost of operating the HVAC equipment during each of the plurality of time steps using a set of time-varying utility rates that includes a utility rate value for each time step. The time-varying utility rates may be received from a utility provider or predicted by the model predictive controller. 
     In some embodiments, the model predictive controller is configured to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory comprising a temperature setpoint for each of the plurality of time steps. 
     In some embodiments, the model predictive controller is configured to optimize the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range. 
     In some embodiments, the model predictive controller is configured to generate the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the model predictive controller is configured to predict a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. 
     In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     In some embodiments, the model predictive controller is configured to predict the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period. 
     In some embodiments, the constraint on the predicted temperature of the building zone requires the model predictive controller to maintain the predicted temperature of the building zone within a first zone temperature range during a first time step of the optimization period and within a second zone temperature range, different from the first zone temperature range, during another time step of the optimization period subsequent to the first time step. 
     Another implementation of the present disclosure is a method for monitoring and controlling temperature of a building zone. The method includes generating a cost function that accounts for a cost operating HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, optimizing the cost function at a model predictive controller subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps, and providing an optimal temperature setpoint from the model predictive controller to a thermostat for the building zone via a communications network. The method further includes, at the thermostat, driving the temperature of the building zone to the optimal temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. 
     In some embodiments, the method includes receiving a set of time-varying utility rates from a utility provider or predicting the time-varying utility rates. The set of time-varying utility rates may include a utility rate value for each time step. The method may include determining the cost of operating the HVAC equipment during each of the plurality of time steps using the set of time-varying utility rates. 
     In some embodiments, using the predictive model to predict the temperature of the building zone includes predicting the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory that includes a temperature setpoint for each of the plurality of time steps. 
     In some embodiments, optimizing the cost function includes optimizing the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range. 
     In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the method includes predicting a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. 
     In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     In some embodiments, the method includes predicting the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period. 
     Another implementation of the present disclosure is a model predictive control system for monitoring and controlling temperature of a building zone. The model predictive control system includes a thermostat and a model predictive controller. The thermostat is configured to drive the temperature of the building zone to a zone temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the zone temperature setpoint and provide the zone temperature setpoint to the thermostat via a communications network. The model predictive controller determines the zone temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and performing an optimization of cost function subject to a constraint on the predicted temperature of the building zone to determine the temperature setpoint trajectory. 
     Another implementation of the present disclosure is a model predictive controller for monitoring and controlling temperature of a building zone. The model predictive controller includes a system identifier and a predictive optimizer. The system identifier is configured to operate the model predictive controller in a system identification mode. Operating in the system identification mode includes performing a system identification process to automatically generate a predictive model based on heat transfer characteristics of the building zone. The predictive optimizer is configured to operate the model predictive controller in an operational mode. Operating in the operational mode includes using the predictive model to predict the temperature of the building zone. The model predictive controller is configured to automatically transition from the operational mode to the system identification mode in response to a determination that a prediction error of the predictive model exceeds a threshold value. 
     In some embodiments, operating in the operational mode includes optimizing a cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of a plurality of time steps in an optimization period. In some embodiments, the cost function accounts for a cost of operating HVAC equipment to provide heating or cooling to the building zone during each of the plurality of time steps in the optimization period. 
     In some embodiments, performing the system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of a temperature setpoint for the building zone and the temperature of the building zone during each of a plurality of time steps during a learning period. 
     In some embodiments, the set of input-output data includes a discrete HVAC staging trajectory for staged HVAC equipment. The HVAC staging trajectory may include a discrete HVAC equipment load at each of the plurality of time steps during the learning period. In some embodiments, performing the system identification process includes filtering the input-output data to generate a continuous HVAC equipment load signal from the discrete HVAC staging trajectory. 
     In some embodiments, collecting the set of input-output data includes modulating a temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint. In some embodiments, performing the system identification process comprises filtering the input-output data to remove oscillations in the temperature of the building zone around the temperature setpoint resulting from activating and deactivating staged HVAC equipment. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. 
     In some embodiments, the model predictive controller includes a load/rate predictor configured to predict a value of an unmeasured heat load disturbance experienced by the building zone at each of a plurality of time steps in an optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance. 
     In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by HVAC equipment controlled by the model predictive controller as a function of the temperature of the building zone and a temperature setpoint for the building zone. 
     Another implementation of the present disclosure is a method for monitoring and controlling temperature of a building zone. The method includes operating a model predictive controller in a system identification mode. Operating in the system identification mode includes performing a system identification process to automatically generate a predictive model based on heat transfer characteristics of the building zone. The method further includes operating the model predictive controller in an operational mode. Operating in the operational mode includes using the predictive model to predict the temperature of the building zone. The method further includes automatically transitioning from the operational mode to the system identification mode in response to a determination that a prediction error of the predictive model exceeds a threshold value. 
     In some embodiments, operating in the operational mode includes optimizing a cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of a plurality of time steps in an optimization period. In some embodiments, the cost function accounts for a cost of operating HVAC equipment to provide heating or cooling to the building zone during each of the plurality of time steps in the optimization period. 
     In some embodiments, performing the system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of a temperature setpoint for the building zone and the temperature of the building zone during each of a plurality of time steps during a learning period. 
     In some embodiments, the set of input-output data includes a discrete HVAC staging trajectory for staged HVAC equipment. The HVAC staging trajectory may include a discrete HVAC equipment load at each of the plurality of time steps during the learning period. In some embodiments, performing the system identification process includes filtering the input-output data to generate a continuous HVAC equipment load signal from the discrete HVAC staging trajectory. 
     In some embodiments, collecting the set of input-output data comprises modulating a temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint. In some embodiments, performing the system identification process includes filtering the input-output data to remove oscillations in the temperature of the building zone around the temperature setpoint resulting from activating and deactivating staged HVAC equipment. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. 
     In some embodiments, the method includes predicting a value of an unmeasured heat load disturbance experienced by the building zone at each of a plurality of time steps in an optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance. 
     In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by HVAC equipment controlled by the model predictive controller as a function of the temperature of the building zone and a temperature setpoint for the building zone. 
     Another implementation of the present disclosure is a model predictive controller for monitoring and controlling temperature of a building zone. The model predictive controller includes a state/disturbance estimator and a predictive optimizer. The state/disturbance estimator is configured to estimate an initial state of the building zone at a beginning of an optimization period. The state of the building zone includes a temperature of air within the building zone and a temperature of solid mass within the building zone. The predictive optimizer is configured to use a predictive model of the building zone to predict the state of the building zone at each of a plurality of time steps of the optimization period based on the estimated initial state of the building zone and a temperature setpoint trajectory comprising a temperature setpoint for each of the plurality of time steps. The predictive optimizer is configured to generate optimal temperature setpoints for each of the plurality of time steps by optimizing a cost function that accounts for a cost operating HVAC equipment during each of the plurality of time steps. 
     In some embodiments, the predictive optimizer is configured to generate the cost function and determine the cost operating the HVAC equipment during each time step using a set of time-varying utility rates that includes a utility rate value for each time step. The set of time-varying utility rates can be received from a utility provider or predicted by the model predictive controller. 
     In some embodiments, the predicted state of the building zone includes a predicted temperature of the building zone. The predictive optimizer may be configured to optimize the cost function subject to a constraint on the predicted temperature of the building zone. 
     In some embodiments, the model predictive controller includes a load/rate predictor configured to predict a value of an unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance experienced by the building zone at each time step. 
     In some embodiments, the predictive model includes a mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     In some embodiments, the predictive optimizer is configured to predict the cost of operating the HVAC equipment as a function of an amount of heating or cooling provided by the HVAC equipment at each time step of the optimization period. 
     In some embodiments, the model predictive controller includes a system identifier configured to generate the predictive model by performing a system identification process. The system identification process may include collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period. 
     In some embodiments, performing system identification process includes modulating the temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint. 
     Another implementation of the present disclosure is a method for monitoring and controlling temperature of a building zone. The method includes estimating an initial state of the building zone at a beginning of an optimization period. The state of the building zone includes a temperature of air within the building zone and a temperature of solid mass within the building zone. The method includes using a predictive model of the building zone to predict the state of the building zone at each of a plurality of time steps of the optimization period based on the estimated initial state of the building zone and a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps. The method includes generating optimal temperature setpoints for each of the plurality of time steps by optimizing a cost function that accounts for a cost operating HVAC equipment during each of the plurality of time steps. 
     In some embodiments, the method includes generating the cost function and determining the cost operating the HVAC equipment during each time step using a set of time-varying utility rates comprising a utility rate value for each time step. The set of time-varying utility rates can be received from a utility provider or predicted as part of the method. 
     In some embodiments, the predicted state of the building zone includes a predicted temperature of the building zone. The method may include optimizing the cost function subject to a constraint on the predicted temperature of the building zone. 
     In some embodiments, the method includes predicting a value of an unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance experienced by the building zone at each time step. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     In some embodiments, the method includes predicting the cost of operating the HVAC equipment as a function of an amount of heating or cooling provided by the HVAC equipment at each time step of the optimization period. 
     In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period. 
     Another implementation of the present disclosure is a model predictive controller for monitoring and controlling temperature of a building zone. The model predictive controller includes a state/disturbance estimator and a predictive optimizer. The state/disturbance estimator is configured to estimate an initial state of the building zone at a beginning of an optimization period. The state of the building zone includes a temperature of air within the building zone and a temperature of solid mass within the building zone. The predictive optimizer is configured to use a predictive model of the building zone to predict the state of the building zone at each of a plurality of time steps of the optimization period based on the estimated initial state of the building zone and a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps. The predictive optimizer is configured to generate temperature setpoints for each of the plurality of time steps by optimizing a cost function that accounts for a cost operating HVAC equipment during each of the plurality of time steps. 
     Another implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes a load/rate predictor and a predictive optimizer. The load/rate predictor is configured to predict a price of one or more resources consumed by HVAC equipment to generate heating or cooling for the building zone at each of a plurality of time steps of an optimization period. The predictive optimizer is configured to generate a cost function that accounts for a cost of operating the HVAC equipment during each time step as a function of the predicted prices at each time step, use a predictive model to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps, and optimize the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps. 
     In some embodiments, the thermostat includes an equipment controller configured to drive the temperature of the building zone to an optimal temperature setpoint by operating the HVAC equipment to provide heating or cooling to the building zone. 
     In some embodiments, the predictive model defines the temperature of the building zone as a function of an unmeasured heat load disturbance. In some embodiments, the load/rate predictor is configured to predict a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period as a function of at least one of day type, time of day, building occupancy, outdoor air temperature, and weather forecasts. 
     In some embodiments, the thermostat includes a system identifier configured to generate the predictive model by performing a system identification process. The system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period. 
     In some embodiments, the system identification process includes modulating the temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     In some embodiments, the predictive optimizer is configured to predict the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period. 
     Another implementation of the present disclosure is a method performed by a thermostat for a building zone for monitoring and controlling temperature of the building zone. The method includes predicting a price of one or more resources consumed by HVAC equipment to generate heating or cooling for the building zone at each of a plurality of time steps of an optimization period, generating a cost function that accounts for a cost of operating the HVAC equipment during each time step as a function of the predicted prices at each time step, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps, and optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps. 
     In some embodiments, the method includes driving the temperature of the building zone to an optimal temperature setpoint by operating the HVAC equipment to provide heating or cooling to the building zone. 
     In some embodiments, the predictive model defines the temperature of the building zone as a function of an unmeasured heat load disturbance. In some embodiments, the method includes predicting a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period as a function of at least one of day type, time of day, building occupancy, outdoor air temperature, and weather forecasts. 
     In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process may include collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period. 
     In some embodiments, performing the system identification process includes modulating the temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint. 
     In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint. 
     Another implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes a load/rate predictor and a predictive optimizer. The load/rate predictor is configured to predict a price of one or more resources consumed by HVAC equipment to generate heating or cooling for the building zone at each of a plurality of time steps of an optimization period. The predictive optimizer is configured to generate a cost function that accounts for a cost of operating the HVAC equipment during each time step as a function of the predicted prices at each time step, use a predictive model to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps, and perform an optimization of the cost function subject to a constraint on the predicted temperature of the building zone to determine temperature setpoints for each of the plurality of time steps. 
     Those skilled in the art will appreciate that the summary is illustrative only and is not intended to be in any way limiting. Other aspects, inventive features, and advantages of the devices and/or processes described herein, as defined solely by the claims, will become apparent in the detailed description set forth herein and taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a drawing of a smart thermostat having a landscape aspect ratio, according to some embodiments. 
         FIG.  2    is a drawing of a smart thermostat having a portrait aspect ratio, according to some embodiments. 
         FIG.  3    is a block diagram of a model predictive control system including a model predictive controller and a smart thermostat which communicates with the model predictive controller via a network, according to some embodiments. 
         FIG.  4    is a block diagram of a model predictive control system including a smart thermostat that includes a model predictive controller and an equipment controller, according to some embodiments. 
         FIG.  5    is a block diagram illustrating the model predictive controller of  FIGS.  3 - 4    in greater detail, according to some embodiments. 
         FIG.  6    is a block diagram illustrating various types of predictive models, according to some embodiments. 
         FIG.  7    is a graph of a temperature excitation signal over a one-day period, according to some embodiments. 
         FIG.  8    is a thermal circuit diagram that models the heat transfer characteristics of a building zone, according to some embodiments. 
         FIG.  9    is a flowchart of a process which can be performed by the model predictive controller of  FIG.  5    to identify parameters of a predictive model, according to some embodiments. 
         FIG.  10    is a graph of an unfiltered temperature signal and an unfiltered sensible load signal which can be received as feedback from a controlled HVAC system, according to some embodiments. 
         FIG.  11    is a graph of a filtered temperature signal and a filtered sensible load signal which can be generated by filtering the signals shown in  FIG.  10   , according to some embodiments. 
         FIG.  12    is a flowchart of a process for operating the model predictive controller of  FIG.  5    in a system identification mode and in an operational mode, according to some embodiments. 
         FIG.  13    is a flowchart of a process which can be performed by the model predictive controller of  FIG.  5    to determine optimal temperature setpoints when operating in the operational mode, according to some embodiments. 
         FIG.  14    is a flowchart of a process which can be performed by the model predictive controller of  FIG.  5    to generate a predictive model and use the predictive model to determine optimal temperature setpoints for a building zone, according to some embodiments. 
         FIG.  15    is a graph comparing the energy consumption of the model predictive control systems of  FIGS.  3 - 4    with a baseline system that does not use model predictive control, according to some embodiments. 
         FIG.  16    is a graph of time-varying electricity prices illustrating different electricity prices at different times, according to some embodiments. 
         FIG.  17    is a graph comparing the building mass temperature in the model predictive control systems of  FIGS.  3 - 4    with a baseline system that does not use model predictive control, according to some embodiments. 
         FIG.  18    is a graph comparing the zone air temperature in the model predictive control systems of  FIGS.  3 - 4    with a baseline system that does not use model predictive control, according to some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Overview 
     Referring generally to the FIGURES, a smart thermostat with model predictive control and components thereof are shown according, to some embodiments. The smart thermostat can be configured to monitor and control one or more environmental conditions of a building zone (e.g., temperature, humidity, air quality, etc.). In some embodiments, the smart thermostat is mounted on a wall within the building zone and configured to measure the temperature, humidity, and/or other environmental conditions of the building zone. The smart thermostat can be configured to communicate with HVAC equipment that operates to affect the measured environmental conditions. 
     In some embodiments, the smart thermostat includes a model predictive controller and an equipment controller. In other embodiments, the model predictive controller is separate from the smart thermostat and communicates with the smart thermostat via a communications network (e.g., the Internet, a building network, etc.). The model predictive controller can be configured to determine optimal temperature setpoints T sp  for the building zone for each of a plurality of time steps in an optimization period. The equipment controller can be configured to receive the optimal temperature setpoints T sp  from the model predictive controller and can operate HVAC equipment to drive the temperature of the building zone to the optimal temperature setpoints. 
     To determine the optimal temperature setpoints T sp , the model predictive controller can optimize an objective function (i.e., a cost function) that accounts for the cost of operating the HVAC equipment over the duration of the optimization period. The costs of operating the HVAC equipment can include, for example, the costs of resources consumed by the HVAC equipment during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of the HVAC equipment. The optimization performed by the model predictive controller is described in greater detail below. 
     The model predictive controller can optimize the objective function subject to a set of constraints. The constraints may include temperature constraints (e.g., a maximum temperature limit and a minimum temperature limit for the building zone), equipment capacity constraints, load change constraints, thermal mass storage constraints, HVAC load constraints, and/or other constraints that limit the operation of the HVAC equipment and/or describe the temperature evolution of the building zone. The model predictive controller can automatically generate the optimization constraints by performing a system identification process and generating predictive models that describe the controlled system (i.e., the HVAC equipment and the building zone). 
     In some embodiments, the model predictive controller generates a thermal mass storage model that describes the temperature of the air within the building zone T ia  as a function of several physical parameters and variables. The physical parameters in the thermal mass storage model may include, for example, the thermal capacitance C ia  of the air in the building zone, the thermal capacitance C m  of the solid mass within the building zone, the thermal resistance R mi  between the air within the building zone and the solid mass, the thermal resistance R oi  between the air within the building zone and the ambient environment, etc. The variables in the thermal mass storage model may include the temperature of air within the building zone T ia , the temperature of the solid mass within the building zone T m , the outside air temperature T oa , the amount of heating or cooling provided by the HVAC equipment {dot over (Q)} HVAC , and/or a thermal energy load disturbance (i.e., a heat load disturbance) {dot over (Q)} other . 
     In some embodiments, the model predictive controller generates an HVAC load model. The HVAC load model may describe the amount of heating or cooling {dot over (Q)} HVAC  provided by the HVAC equipment as a function of the temperature setpoint T sp  and the temperature of the air T ia  within the building zone. The model predictive controller can use the thermal mass storage model and the HVAC load model to predict system states (i.e., building mass temperature T m , building air temperature T ia , etc.), predict the load disturbance {dot over (Q)} other , and establish constraints on the optimization performed by the model predictive controller. 
     In some embodiments, the model predictive controller uses the thermal mass storage model and the HVAC load model to take advantage of the building mass as a thermal energy storage medium. For example, thermal energy can be stored in the solid building mass (i.e., precooling or preheating the building zone) to allow the model predictive controller to take advantage of time-varying utility rates and demand charges when determining the optimal temperature setpoints T sp . To precool the building zone, thermal energy may be removed from the thermal mass during periods when energy prices are lowest. During periods when energy prices are largest, thermal energy may be moved back into the solid mass from the zone air, which reduces the sensible HVAC load {dot over (Q)} HVAC  needed to maintain the zone air within a comfortable temperature range. Similarly, the model predictive controller can preheat the solid mass within the building zone when energy prices are low and use the stored thermal energy to warm the zone air when energy prices are higher. 
     In some embodiments, the model predictive controller is configured to operate in two distinct modes: (1) parameter identification mode and (2) operational mode. In the parameter identification mode, the model predictive controller can manipulate the temperature setpoint T sp  provided to the smart thermostat and/or the equipment controller to induce a dynamic response in the zone temperature T ia . This procedure of inducing a dynamic response is referred to as the parameter identification experiment or system identification process and can be used to identify the values of the parameters in the thermal mass storage model and the HVAC load model. In the operational mode, the model predictive controller can solve an optimal control problem to determine the optimal temperature setpoint trajectory T sp  (i.e., a time series of temperature setpoints) that minimizes the cost of energy consumed by the HVAC equipment over the duration of the optimization period. The optimal temperature setpoints T sp  can be provided to the smart thermostat and/or the equipment controller for use in generating operating commands for the HVAC equipment. 
     The setpoint T sp  computed for the first time step of the optimization period can be sent to the smart thermostat and/or the equipment controller to be implemented during the first time step. The smart thermostat and/or the equipment controller can turn stages of HVAC equipment on and/or off such that the temperature T ia  of the zone is forced to and then maintained at the temperature setpoint T sp . At the next time step, the model predictive controller receives updated information (i.e., feedback on the zone conditions), resolves the optimal control problem, and sends the temperature setpoint T sp  for the next time step to the smart thermostat and/or the equipment controller. 
     The model predictive controller can collect feedback on the zone conditions (e.g., zone temperature T ia , outdoor air temperature T oa , heating/cooling load {dot over (Q)} HVAC ) when operating in the parameter identification mode. The zone conditions can be monitored and recorded over the identification experiment, along with the commanded zone temperature setpoint T sp , as input-output data (i.e., training data). The model predictive controller can use the input-output data to compute the zone thermal model parameters by performing a system identification process. The feedback received from the smart thermostat may include the current minimum and maximum allowable setpoint to limit the adjustments to the zone temperature setpoint T sp  made during the identification experiment. These and other features of the smart thermostat and/or the model predictive controller are described in greater detail below. 
     Throughout this disclosure, the terms “optimal,” “optimized,” “optimum,” and the like are used to refer to values (e.g., temperature setpoint values, HVAC load values, etc.) that are determined by performing an optimization process. Similarly, the term “optimizing” is used to refer to the process of performing an optimization. In some instances, the values resulting from the optimization process are true optimal values (i.e., values that achieve the minimum possible value or maximum possible value for a performance variable given the constraints on the optimization process). In other instances, the values resulting from the optimization process are not true optimal values. This can result from imperfect information used to perform the optimization, inaccuracies in the predictive model used to constrain the optimization, and/or various other factors that can prevent the optimization from converging on the true optimal values. The terms “optimal,” “optimized,” “optimum,” and the like should be interpreted to include any values that result from performing an optimization, regardless of whether the values are true optimal values. Similarly, the term “optimizing” should be interpreted to include the process of performing an optimization, regardless of whether that optimization converges on the true optimal values. 
     Smart Thermostat 
     Referring now to  FIGS.  1 - 2   , a smart thermostat  100  is shown, according to some embodiments. Smart thermostat  100  can be configured to monitor and control one or more environmental conditions of a building zone (e.g., temperature, humidity, air quality, etc.). In some embodiments, smart thermostat  100  is mounted on a wall within the building zone and configured to measure the temperature, humidity, and/or other environmental conditions of the building zone. Smart thermostat  100  can be configured to communicate with HVAC equipment that operates to affect the measured environmental conditions. 
     Smart thermostat  100  can be installed in a home, office building, school, hospital, or any other environment-controlled space. In residential implementations, the HVAC equipment controlled by smart thermostat  100  may include a home furnace, air conditioning unit, and/or other types of residential HVAC equipment. In commercial implementations, the HVAC equipment may include one or more chillers, boilers, air handling units, rooftop units, dampers, or other types of HVAC equipment configured to affect the environment of a building zone. It is contemplated that smart thermostat  100  can be configured to control any type of HVAC equipment. 
     In some embodiments, smart thermostat  100  includes a communications interface that enables smart thermostat  100  to connect to a communications network (e.g., a local area network, the Internet, a cellular network, etc.) and communicate with various external systems and devices (e.g., user devices, remote servers, remote controllers, etc.). For example, smart thermostat  100  can be configured to receive weather forecasts from a remote weather service via the Internet. In some embodiments, smart thermostat  100  receives temperature setpoints from a model predictive controller via the communications network. In other embodiments, the model predictive controller is a component of smart thermostat  100 . 
     Smart thermostat  100  is shown to include a display screen  102  and a base  104 . Base  104  can attach to a wall or other surface upon which smart thermostat  100  is mounted. Display screen  102  may be transparent or semi-transparent (e.g., an organic LED display) and can be configured to display text, graphics, and other information for presentation to a user. In some embodiments, display screen  102  is touch-sensitive (e.g., a capacitive or resistive touch screen) and configured to receive user input. Display screen  102  may have a landscape aspect ratio as shown in  FIG.  1    (i.e., the width of display screen  102  may exceed the height of display screen  102 ) or a portrait aspect ratio as shown in  FIG.  2    (i.e., the height of display screen  102  may exceed the width of display screen  102 ). Display screen  102  may be cantilevered from base  104  (i.e., attached to base  104  along a single edge of display screen  102 ) and may extend from base  104  in a direction substantially parallel to the mounting surface. 
     Smart thermostat  100  can include some of all of the features of the thermostats described in U.S. patent application Ser. No. 15/143,373 filed Apr. 29, 2016, U.S. patent application Ser. No. 15/146,763 filed May 4, 2016, U.S. patent application Ser. No. 15/146,749 filed May 4, 2016, U.S. patent application Ser. No. 15/146,202 filed May 4, 2016, U.S. patent application Ser. No. 15/146,134 filed May 4, 2016, U.S. patent application Ser. No. 15/146,649 filed May 4, 2016, U.S. Provisional Patent Application No. 62/331,863 filed May 4, 2016, U.S. Provisional Patent Application No. 62/352,955 filed Jun. 21, 2016, U.S. patent application Ser. No. 15/298,191 filed Oct. 19, 2016, U.S. patent application Ser. No. 15/336,793 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/336,792 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/336,789 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/338,221 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/338,215 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/336,791 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/397,722 filed Jan. 3, 2017, and/or U.S. Provisional Patent Application No. 62/446,296 filed Jan. 13, 2017. The entire disclosure of each of these patent applications is incorporated by reference herein. 
     Model Predictive Control Systems 
     Referring now to  FIGS.  3 - 4   , a pair of model predictive control (MPC) systems  300  and  400  are shown, according to some embodiments. MPC system  300  is shown to include a model predictive controller  302 , a communications network  304 , smart thermostat  100 , HVAC equipment  308 , and a building zone  310 . In MPC system  300 , model predictive controller  302  is separate from smart thermostat  100  and configured to provide a temperature setpoint T sp  to smart thermostat  100  via communications network  304 . In MPC system  400 , model predictive controller  302  is a component of smart thermostat  100  along with an equipment controller  406 . 
     Building zone  310  can include one or more rooms or zones within a home, office building, school, hospital, or any other environment-controlled space. HVAC equipment  308  can include any type of equipment operable to affect the temperature, humidity, and/or other environmental conditions of building zone  310 . For example, HVAC equipment  308  can include a home furnace, air conditioning unit, one or more chillers, boilers, air handling units, rooftop units, dampers, or other types of HVAC equipment configured to affect the environment of building zone  310 . 
     Model predictive controller  302  can be configured to determine an optimal temperature setpoint T sp  for smart thermostat  100  for each of a plurality of time steps during an optimization period. Smart thermostat  100  can use the temperature setpoints T sp  provided by model predictive controller  302  to generate equipment commands for HVAC equipment  308 . The equipment commands can be generated by smart thermostat  100  and/or equipment controller  406  (e.g., an on/off controller, an equipment staging controller, a proportional-integral (PI) controller, a proportional-integral-derivative (PID) controller, etc.). HVAC equipment  308  operate according to the equipment commands to provide variable amount of heating or cooling {dot over (Q)} HVAC  to building zone  310 . By controlling the temperature setpoint T sp , model predictive controller  302  can modulate the amount of heating or cooling {dot over (Q)} HVAC  provided by HVAC equipment  308 , thereby affecting the temperature of building zone  310 . 
     To determine the optimal temperature setpoints T sp , model predictive controller  302  can optimize an objective function (i.e., a cost function) that accounts for the cost of operating HVAC equipment  308  over the duration of the optimization period. The costs of operating HVAC equipment  308  can include, for example, the costs of resources consumed by HVAC equipment  308  during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of HVAC equipment  308 . The optimization performed by model predictive controller  302  is described in greater detail with reference to  FIG.  5   . 
     Model predictive controller  302  can optimize the objective function subject to a set of constraints. The constraints may include temperature constraints (e.g., a maximum temperature limit and a minimum temperature limit for building zone  310 ), equipment capacity constraints, load change constraints, thermal mass storage constraints, HVAC load constraints, and/or other constraints that limit the operation of HVAC equipment  308  and/or describe the temperature evolution of building zone  310 . Model predictive controller  302  can automatically generate the optimization constraints by performing a system identification process and generating predictive models that describe the controlled system (i.e., HVAC equipment  308  and building zone  310 ). 
     In some embodiments, model predictive controller  302  generates a thermal mass storage model that describes the temperature of the air within building zone  310  T ia  as a function of several physical parameters and variables. The physical parameters in the thermal mass storage model may include, for example, the thermal capacitance C ia  of the air in building zone  310 , the thermal capacitance C m  of the solid mass within building zone  310 , the thermal resistance R mi  between the air within building zone  310  and the solid mass, the thermal resistance R oi  between the air within building zone  310  and the ambient environment, etc. The variables in the thermal mass storage model may include the temperature of air within building zone  310  T ia , the temperature of the solid mass within building zone  310  T m , the outside air temperature T oa , the amount of heating or cooling provided by HVAC equipment  308  {dot over (Q)} HVAC , and/or a thermal energy load disturbance {dot over (Q)} other . 
     In some embodiments, model predictive controller  302  generates a HVAC load model. The HVAC load model may describe the amount of heating or cooling {dot over (Q)} HVAC  provided by HVAC equipment  308  as a function of the temperature setpoint T sp  and the temperature of the air T ia  within building zone  310 . Model predictive controller  302  can use the thermal mass storage model and the HVAC load model to predict system states (i.e., building mass temperature T m , building air temperature T ia , etc.), predict the load disturbance {dot over (Q)} other , and establish constraints on the optimization performed by model predictive controller  302 . 
     In some embodiments, model predictive controller  302  uses the thermal mass storage model and the HVAC load model to take advantage of the building mass as a thermal energy storage medium. For example, thermal energy can be stored in the solid building mass (i.e., precooling or preheating building zone  310 ) to allow model predictive controller  302  to take advantage of time-varying utility rates and demand charges when determining the optimal temperature setpoints T sp . To precool building zone  310 , thermal energy may be removed from the thermal mass during periods when energy prices are lowest. During periods when energy prices are largest, thermal energy may be moved back into the solid mass from the zone air, which reduces the sensible HVAC load {dot over (Q)} HVAC  needed to maintain the zone air within a comfortable temperature range. Similarly, model predictive controller  302  can preheat the solid mass within building zone  310  when energy prices are low and use the stored thermal energy to warm the zone air when energy prices are higher. 
     In some embodiments, model predictive controller  302  is configured to operate in two distinct modes: (1) parameter identification mode and (2) operational mode. In the parameter identification mode, model predictive controller  302  can manipulate the temperature setpoint T sp  provided to smart thermostat  100  and/or equipment controller  406  to induce a dynamic response in the zone temperature T ia . This procedure of inducing a dynamic response is referred to as the parameter identification experiment or system identification process and can be used to identify the values of the parameters in the thermal mass storage model and the HVAC load model. In the operational mode, model predictive controller  302  can solve an optimal control problem to determine the optimal temperature setpoint trajectory T sp  (i.e., a time series of temperature setpoints) that minimizes the cost of energy consumed by HVAC equipment  308  over the duration of the optimization period. The optimal temperature setpoints T sp  can be provided to smart thermostat  100  and/or equipment controller  406  for use in generating operating commands for HVAC equipment  308 . 
     The setpoint T sp  computed for the first time step of the optimization period can be sent to smart thermostat  100  and/or equipment controller  406  to be implemented during the first time step. Smart thermostat  100  and/or equipment controller  406  can turn stages of HVAC equipment  308  on and/or off such that the temperature T ia  of the zone is forced to and then maintained at the temperature setpoint T sp . At the next time step, model predictive controller  302  receives updated information (i.e., feedback on the zone conditions), resolves the optimal control problem, and sends the temperature setpoint T sp  for the next time step to smart thermostat  100  and/or equipment controller  406 . 
     Model predictive controller  302  can collect feedback on the zone conditions (e.g., zone temperature T ia , outdoor air temperature T oa , heating/cooling load {dot over (Q)} HVAC ) when operating in the parameter identification mode. The zone conditions can be monitored and recorded over the identification experiment, along with the commanded zone temperature setpoint T sp , as input-output data (i.e., training data). Model predictive controller  302  can use the input-output data to compute the zone thermal model parameters by performing a system identification process. The feedback received from smart thermostat  100  may include the current minimum and maximum allowable setpoint to limit the adjustments to the zone temperature setpoint T sp  made during the identification experiment. 
     In some embodiments, a pseudo-random binary signal (PRBS) is generated that is subsequently used to produce a signal that takes values in the set {T sp,1 , T sp,2 }, where T sp,1  and T sp,2  are maximum and minimum allowable temperature setpoints depending on mode of operation of smart thermostat  100  (e.g., heating, cooling, home, sleep, or away mode). The minimum and maximum temperature setpoints may be time-varying to account for the different operational modes of smart thermostat  100  (e.g., home, sleep, and away). In some embodiments, smart thermostat  100  uses occupancy detection and/or user feedback to change the mode. 
     To determine the optimal temperature setpoints T sp  in the operational mode, model predictive controller  302  can use the thermal model of building zone  310  to predict the zone temperature T ia  and HVAC fuel consumption over the optimization period, given a forecast of the weather and heat disturbance load on the zone (e.g., heat generated from people and electrical equipment and gained through solar radiation). The zone temperature T ia  may be subject to comfort constraints. The predicted trajectories of the zone temperature T ia , temperature setpoint T sp , and HVAC fuel consumption can be sent to smart thermostat  100  to be displayed to the user allowing the user to better understand the setpoint decision made by model predictive controller  302 . 
     In some embodiments, smart thermostat  100  allows a user to override the optimal temperature setpoints computed by model predictive controller  302 . Smart thermostat  100  may include a savings estimator configured to determine a potential cost savings resulting from the optimal temperature setpoints relative to the user-specified temperature setpoints. Smart thermostat  100  can be configured to display the potential cost savings via display screen  102  to inform the user of the economic cost predicted to result from overriding the optimal temperature setpoints. These and other features of model predictive controller  302  are described in greater detail below. 
     Model Predictive Controller 
     Referring now to  FIG.  5   , a block diagram illustrating model predictive controller  302  in greater detail is shown, according to come embodiments. Model predictive controller  302  is shown to include a communications interface  502  and a processing circuit  504 . Communications interface  502  may facilitate communications between model predictive controller  302  and external systems or devices. For example, communications interface  502  may receive measurements of zone conditions from smart thermostat  100 , HVAC equipment  308 , and/or building zone  310 . Zone conditions may include the temperature T ia  of the air within building zone  310 , the outside air temperature T oa , and/or the amount of heating/cooling {dot over (Q)} HVAC  provided to building zone  310  at each time step of the optimization period. Communications interface  502  may also receive temperature constraints for building zone  310  and/or equipment constraints for HVAC equipment  308 . In some embodiments, communications interface  502  receives utility rates from utilities  524  (e.g., time-varying energy prices, demand charges, etc.), weather forecasts from a weather service  526 , and/or historical load and rate data from historical data  528 . In some embodiments, model predictive controller  302  uses communications interface  502  to provide temperature setpoints T sp  to smart thermostat  100  and/or equipment controller  406 . 
     Communications interface  502  may include wired or wireless communications interfaces (e.g., jacks, antennas, transmitters, receivers, transceivers, wire terminals, etc.) for conducting data communications external systems or devices. In various embodiments, the communications may be direct (e.g., local wired or wireless communications) or via a communications network (e.g., a WAN, the Internet, a cellular network, etc.). For example, communications interface  502  can include an Ethernet card and port for sending and receiving data via an Ethernet-based communications link or network. In another example, communications interface  502  can include a Wi-Fi transceiver for communicating via a wireless communications network or cellular or mobile phone communications transceivers. 
     Processing circuit  504  is shown to include a processor  506  and memory  508 . Processor  506  may be a general purpose or specific purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable processing components. Processor  506  is configured to execute computer code or instructions stored in memory  508  or received from other computer readable media (e.g., CDROM, network storage, a remote server, etc.). 
     Memory  508  may include one or more devices (e.g., memory units, memory devices, storage devices, etc.) for storing data and/or computer code for completing and/or facilitating the various processes described in the present disclosure. Memory  508  may include random access memory (RAM), read-only memory (ROM), hard drive storage, temporary storage, non-volatile memory, flash memory, optical memory, or any other suitable memory for storing software objects and/or computer instructions. Memory  508  may include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present disclosure. Memory  508  may be communicably connected to processor  506  via processing circuit  504  and may include computer code for executing (e.g., by processor  506 ) one or more processes described herein. When processor  506  executes instructions stored in memory  508  for completing the various activities described herein, processor  506  generally configures controller  302  (and more particularly processing circuit  504 ) to complete such activities. 
     Still referring to  FIG.  5   , model predictive controller  302  is shown to include a system identifier  510 , a load/rate predictor  518 , a state/disturbance estimator  520 , and a predictive optimizer  522 . In brief overview, system identifier  510  can be configured to perform a system identification process to identify the values of the parameters in the building thermal mass model and the HVAC load model. The parameters identified by system identifier  510  may define the values of the system matrices A, B, C, and D and the estimator gain K. State/disturbance estimator  520  can be configured to compute an estimate of the current state {circumflex over (x)}(k) and unmeasured disturbance {circumflex over (d)}(k). The estimation performed by state/disturbance estimator  520  can be performed at each sample time and may use the system matrices A, B, C, D and estimator gain K identified by system identifier  510 . 
     Load/rate predictor  518  can generate predictions or forecasts of the internal heat load {dot over (Q)} other , the non-HVAC building electricity consumption (used to calculate electrical demand charges), and the utility rates at each time step of the optimization period as a function of historical data  528  and several key predictor variables (e.g., time-of-day, outdoor air conditions, etc.). The models used by load/rate predictor  518  may be different than that identified by system identifier  510 . In some embodiments, load/rate predictor  518  uses the prediction techniques described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, the entire disclosure of which is incorporated by reference herein. 
     Predictive optimizer  522  can manipulate the temperature setpoint T sp  of building zone  310  to minimize the economic cost of operating HVAC equipment  308  over the duration of the optimization period using the building thermal mass storage model, the HVAC load model, and state and disturbance estimates and forecasts. In some embodiments, predictive optimizer  522  uses open-loop predictions to optimize the temperature setpoint T sp  trajectory over the optimization period. At each time step of the optimization period, the predictions can be updated based on feedback from the controlled system (e.g., measured temperatures, measured HVAC loads, etc.). 
     Before discussing system identifier  510 , load/rate predictor  518 , state/disturbance estimator  520 , and predictive optimizer  522  in detail, the notation used throughout the remainder of the present disclosure and the class of system model used by model predictive controller  302  are explained. The set of integers is denoted by   and the set of positive integers is denoted by    ≥0 , and the set of integers contained in the interval [a, b] is denoted by    a.b . For a time-dependent vector x(k)∈   n , {circumflex over (x)}(i|k)∈   n  denotes the estimated value of x at time step i∈   ≥0  given the measurement at time step k∈   ≥0  where i≥k and {tilde over (x)}(i|k)∈   n  denotes the (open-loop) predicted value of x at time step i∈   ≥0  with the prediction initialized at time k∈   ≥0  (i≥k). For notational simplicity, the notation of the predicted value of x at time i starting from time k is abbreviated to {tilde over (x)}(i). Boldface letters are used to represent a sequence with cardinality N∈   ≥0  (i.e., x:={x(0), . . . , x(N−1)}). The notation ⋅* (e.g., x*) denotes an optimal quantity with respect to some optimization problem. 
     In some embodiments, the model generated and used by model predictive controller  302  is a discrete-time linear time-invariant model, as shown in the following equation:
 
 x ( k+ 1)= Ax ( k )+ Bu ( k )
 
 y ( k )= Cx ( k )+ Du ( k )  (Equation 1)
 
where k∈   ≥0  is the time index, x(k)∈   n  is the state vector, y(k)∈   p  is the measured output, and u(k)∈   m  is the input vector. The matrix A can be assumed to be stable; that is the real parts of the eigenvalues of A lie within the unit circle. The pair (A, B) can be assumed to be controllable and the pair (A, C) can be assumed to be observable. Some HVAC equipment  308  and devices are only operated at finite operating stages. In the building models considered, this gives rise to discrete outputs whereby the measured output takes values within a finite set of numbers: y i (k)∈{y i,stage,1 y i,stage,2 , . . . , y i,stage,m } for some i. In this case, the output does not evolve according to Equation 1.
 
     In some embodiments, the model generated and used by model predictive controller  302  is a grey box continuous-time models derived from thermal resistance-capacitance (RC) modeling principals. Model predictive controller  302  can be configured to parameterize the resulting continuous-time thermal models and identify the continuous-time model parameters. The continuous-time version of the model of Equation 1 can be represented as shown in the following equation
 
 {dot over (x)} ( t )= A   c   x ( t )+ B   c   u ( t )
 
 y ( t )= C   c   x ( t )+ D   c   u ( t )  (Equation 2)
 
where t≥0 is continuous time (the initial time is taken to be zero) and A c , B c , C c , D c  are the continuous-time versions of the discrete-time system matrices A, B, C, and D. The values of the parameters in the matrices A, B, C, and D as well as an estimator gain K can be determined by system identifier  510  by performing a system identification process (described in greater detail below)
 
State/Disturbance Estimator
 
     State/disturbance estimator  520  can be configured to compute an estimate of the current state {circumflex over (x)}(k) and unmeasured disturbance {circumflex over (d)}(k). The estimation performed by state/disturbance estimator  520  can be performed at each sample time (i.e., when operating in the operational mode) and may use the system matrices A, B, C, D and estimator gain K identified by system identifier  510 . In some embodiments, state/disturbance estimator  520  modifies the system model to account for measurement noise and process noise, as shown in the following equation:
 
 x ( k+ 1)= Ax ( k )+ Bu ( k )+ w ( k )
 
 y ( k )= Cx ( k )+ Du ( k )+ v ( k )  (Equation 3)
 
where w(k) is the process noise and v(k) is the measurement noise.
 
     State/disturbance estimator  520  can use the modified system model of Equation 3 to estimate the system state (k) as shown in the following equation:
 
 {tilde over (x)} ( k+ 1| k )= A{tilde over (x)} ( k|k− 1)+ Bu ( k )+ K ( y ( k )− ŷ ( k|k− 1))
 
 ŷ ( k|k− 1)= C{tilde over (x)} ( k|k− 1)+ Du ( k )  (Equation 4)
 
where {circumflex over (x)}(k+1|k) is the estimated/predicted state at time step k+1 given the measurement at time step k, ŷ(k|k−1) is the predicted output at time step k given the measurement at time step k−1, and K is the estimator gain. Equation 4 describes the prediction step of a Kalman filter. The Kalman filter may be the optimal linear filter under some ideal assumptions (e.g., the system is linear, no plant-model mismatch, the process and measurement noise are follow a white noise distribution with known covariance matrices).
 
     In some embodiments, the estimated state vector {circumflex over (x)}(k+1|k), the output vector ŷ(k|k−1), and the input vector u(k) are defined as follows: 
                   x   ^     ⁡     (       k   +   1     ❘   k     )       =     [               T   ^     ia     ⁡     (       k   +   1     ❘   k     )                     T   ^     m     ⁡     (       k   +   1     ❘   k     )                   I   ^     ⁡     (       k   +   1     ❘   k     )             ]       ,     
     ⁢         y   ^     ⁡     (     k   ❘     k   -   1       )       =             [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                       Q   ^     .     HVAC     ⁡     (     k   ❘     k   -   1       )             ]     ,       u   ⁡     (   k   )       =     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]                   
where {circumflex over (T)} ia  is an estimate of the zone air temperature T ia , {circumflex over (T)} m  is an estimate of the zone mass temperature T m , Î is an estimate of the integrating disturbance I, {circumflex over ({dot over (Q)})} HVAC  is an estimate of the heating or cooling load provided by HVAC equipment  308 , T sp  is the temperature setpoint, and T oa  is the outdoor air temperature.
 
     The state estimation described by Equation 4 is effective in addressing stationary noise. However, unmeasured correlated disturbances are also common in practice. Correlated disturbances can lead to biased estimates/predictions. In some embodiments, state/disturbance estimator  520  accounts for disturbances by adding a disturbance state to the system model, as shown in the following equation: 
                       [             x   ^     (       k   +   1     ❘   k     )                 d   ^     (       k   +   1     ❘   k     )           ]     =         [         A         B   d             0       I         ]     [             x   ^     (       k   +   1     ❘   k     )                 d   ^     (       k   +   1     ❘   k     )           ]     +       [   ⁠         B           0         ]     ⁢     u   ⁡   (   k   )       +       [           K   x               K   d           ]     ⁢     (       y   ⁡   (   k   )     -       y   ^     (     k   ❘     k   -   1       )       )           ⁢   
   	         y   ^     (     k   ❘     k   -   1       )     =         [         C         C   d           ]     [             x   ^     (     k   ❘     k   -   1       )                 d   ^     (     k   ❘     k   -   1       )           ]     +     Du   ⁡   (   k   )                 (     Equation   ⁢         5     )               
where {circumflex over (d)}(k) is the unmeasured disturbance at time step k. The effect of adding the disturbance state is similar to that of the integral action of a proportional-integral controller. State/disturbance estimator  520  may select values of B d  and C d  and modify the estimation problem as shown in Equation 5. In some embodiments, state/disturbance estimator  520  selects values of B d  and C d  such that the system model is an integrating input and/or output disturbance model.
 
     In some embodiments, the choice of disturbance models may have a significant impact on the ability of model predictive controller  302  to adequately estimate the lumped thermal building mass temperature T m . Accurate estimation of the thermal mass temperature T m  enables model predictive controller  302  to effectively use the solid building mass to store thermal energy when preheating or precooling building zone  310  in order to time-shift the thermal energy load and minimize the cost of operating HVAC equipment  308 . 
     System Identifier 
     System identifier  510  can be configured to perform a system identification process to identify the values of the parameters in the building thermal mass model and the HVAC load model. The functions performed by system identifier  510  may be performed when operating in the parameter identification mode. The parameters identified by system identifier  510  may define the values of the system matrices A, B, C, and D and the estimator gain K. System identifier  510  can perform the system identification process during the commissioning phase of the control system and/or after the detection of a substantial change in the model parameters that has resulted in significant closed-loop performance deterioration. In some embodiments, the system identification process performed by system identifier  510  is the same or similar to the system identification process described in U.S. Pat. No. 9,235,657 titled “System Identification and Model Development” and granted Jan. 12, 2016. The entire disclosure of U.S. Pat. No. 9,235,657 is incorporated by reference herein. 
     System identification (SI) is the art and science of building mathematical models of dynamic systems from observed input-output data. In general, a model for a system may be developed using three paradigms: white box, grey box or black box models. These three system identification modeling paradigms are summarized in  FIG.  6   . 
     For applications that require high fidelity models, white box models may be derived via detailed first-principles modeling approaches. This approach often requires a high degree of knowledge of the physical system and may result in a high-order nonlinear dynamic model. While the resulting model may capture most of the relevant physical behavior a system, there is usually a very high engineering investment required to develop such a model. Given the complexities of the resulting models, often white box models cannot be used in real-time predictive controllers owing to real-time computational restrictions. Nevertheless, white box models may help to evaluate control approaches in simulation to identify and mitigate potential pitfalls of the approach. 
     In a black box modeling paradigm, little a priori knowledge about the model structure is assumed except for standard assumptions like linearity and stability of the model. Black box methods fit the best model to the input-output data. While there is little to no assumed a priori knowledge about the model, which offers a deal of flexibility, there are a few drawbacks to the approach. Given that little about the model structure is assumed, black box models often tend to have a large amount of parameters, which requires a large amount of data to properly select the model order and appropriate input-output representation. To capture the necessary amount of input-output data needed for black box methods, an undesirable length for the system identification experiment may be required. 
     Finally, grey box modeling assumes that there is a known model structure often derived through simplified first-principals or semi-physical modeling. The model structure is parameterized and parameter estimation techniques are used to estimate the parameters. Owing to the incorporation of more a priori knowledge into grey box models, the resulting number of parameters for these types of models are often less than black box models. As a result of fewer parameters as well as the fact that the model structure (given it is valid) already encodes important relationships, less input-output data is typically needed to fit a good model with respect to that needed to fit a good model for black box methods. 
     In some embodiments, system identifier  510  uses a grey box model to represent HVAC equipment  308  and building zone  310  (i.e., the controlled system). In a grey box model system identification approach, a model structure   is selected, which is subsequently parameterized with a parameter θ∈ ⊂   d  where   is the set of admissible model parameter values. The resulting possible models are given by the set:
 
 M ={ (θ)|θ∈ }  (Equation 6)
 
     To fit a proper parameter vector to the controlled system, system identifier  510  can collect input-output data. The input-output data may include samples of the input vector u(k) and corresponding samples of the output vector y(k), as shown in the following equation:
 
 Z   N =[ y (1), u (1), y (2), u (2), . . . , y ( N ), u ( N )]  (Equation 7)
 
where N&gt;0 is the number of samples collected. Parameter estimation is the problem of selecting the proper value {circumflex over (θ)} N  of the parameter vector given the input-output data Z N  (i.e., it is the mapping: Z N →{circumflex over (θ)} N  ∈ ).
 
     In some embodiments, system identifier  510  uses a prediction error method to select the values of the parameter vector {circumflex over (θ)} N . A prediction error method (PEM) refers to a particular family of parameter estimation methods that is of interest in the context of the present disclosure. The prediction error method used by system identifier  510  may include fitting the parameter vector {circumflex over (θ)} N  by minimizing some function of the difference between the predicted output and observed output. For example, let ŷ(k, θ) be the predicted output at time step k given the past input-output sequence Z k-1  and the model  (θ) for θ∈ . The prediction error at time step k is given by:
 
ε( k, 0):= y ( k )− ŷ ( k ,θ)  (Equation 8)
 
     In some embodiments, system identifier  510  filters the prediction error sequence through a stable linear filter and defines the following prediction performance metric: 
                       V   N     (     θ   ,     Z   N       )     =       1   N     ⁢       ∑     k   =   1     N           ℓ   ⁡   (       ε   F     (     k   ,   θ     )     )                 (     Equation   ⁢         9     )               
where l(⋅) is the cost function (e.g., a positive definite function) and ε F  (k, θ) is the filtered prediction error. The parameter estimate {circumflex over (θ)} N  of a prediction error method is then given by:
 
{circumflex over (θ)} N ={circumflex over (θ)} N ( Z   N )= arg     V   N (θ, Z   N )  (Equation 10)
 
     In various embodiments, system identifier  510  can use any parameter estimation method that corresponds to Equation 10. For example, system identifier  510  can define a quadratic cost function that is the square of the prediction errors, as shown in the following equation:
 
 l (ε( k ,θ))=− l ( y ( k )− ŷ ( k|k− 1,θ))=∥ y ( k )− ŷ ( k|k− 1,θ)∥ 2   2   (Equation 11)
 
where ŷ(k|k−1, θ) denotes the one-step ahead prediction of the output using the model  (θ). When the prediction errors are independently and identically distributed random variables from a normal distribution (i.e., the process and measurement noise is Gaussian white noise) and the model being identified is linear, the cost function of Equation 11 is optimal from a statistical point-of-view.
 
     Still referring to  FIG.  5   , system identifier  510  is shown to include a thermal mass storage model generator  512 , a HVAC load model generator  514 , and an estimator gain identifier  516 . Thermal mass storage model generator  512  can be configured to estimate or identify the parameters of a thermal mass storage model that describes the relationship between indoor air temperature T ia , building mass temperature T m , outdoor air temperature T oa , internal load {dot over (Q)} other , and sensible HVAC load {dot over (Q)} HVAC . The indoor air temperature T ia  represents the temperature of the air within building zone  310 . The building mass temperature T m  represents the temperature of the solid objects within building zone  310 . The internal load {dot over (Q)} other  represents heat generated internally within building zone  310  by building occupants, electronics, resistive loads, and other sources of heat. The sensible HVAC load {dot over (Q)} HVAC  represents the amount of heating or cooling applied to building zone  310  by HVAC equipment  308 . 
     HVAC load model generator  514  can be configured to generate a HVAC load model that describes the relationship between indoor air temperature T ia , the temperature setpoint T sp , and the sensible HVAC load {dot over (Q)} HVAC . In some embodiments, the HVAC load model describes the dynamics of HVAC equipment  308  when providing heating or cooling to building zone  310 . For example, the HVAC load model may represent HVAC equipment  308  as a proportional-integral (PI) control system that provides the sensible HVAC load {dot over (Q)} HVAC  as a function of the indoor air temperature T ia  and the temperature setpoint T sp . 
     Estimator gain identifier  516  can be configured to calculate the state/disturbance estimation gain (i.e., the estimator gain K). The estimator gain K may be referred to in other contexts as the disturbance matrix or Kalman filter gain. The operation of thermal mass storage model generator  512 , HVAC load model generator  514 , and estimator gain identifier  516  are described in greater detail below. 
     Thermal Mass Storage Model Generator 
     Thermal mass storage model generator  512  can be configured to estimate or identify the parameters of a thermal mass storage model that describes the relationship between indoor air temperature T ia , building mass temperature T m , outdoor air temperature T oa , internal load {dot over (Q)} other , and sensible HVAC load {dot over (Q)} HVAC . In some embodiments, thermal mass storage model generator  512  performs a multi-stage system identification process to generate the parameters of the thermal mass storage model. The stages of the system identification process may include input-output data collection, generating a parameterized building thermal model, and estimating (i.e., fitting) the parameters of the parameterized building thermal model using a parameter identification algorithm. 
     The first stage of the system identification process is input-output data collection. The type of input-output data collected to fit the model parameters may have a significant impact in the usefulness and accuracy of the resulting model. In some embodiments, thermal mass storage model generator  512  manipulates the temperature setpoint T sp  in an open-loop fashion as to sufficiently and persistently excite the system. For example, thermal mass storage model generator  512  can persistently excite (PE) a signal {s(k): k=0, 1, 2, . . . } with spectrum ϕ s (ω) if ϕ s (ω)&gt;0 for almost all co where: 
     
       
         
           
             
               
                 
                   
                     
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     Input-output data collected during closed-loop operation may not provide an information-rich dataset with respect to system identification because the inputs tend to be correlated to themselves as well as to the unmeasured disturbances as a result of closing the loop with a controller. As a result input-output data obtained via closed-loop operation may not meet the PE condition without adding perturbations to the inputs similar to that used in extremum seeking control. 
     Instead of closed-loop identification, thermal mass storage model generator  512  can perform a system identification experiment is to meet the PE condition. In some embodiments, thermal mass storage model generator  512  performs the system identification experiment while building zone  310  is occupied. Accordingly, it may be desirable to design the system identification experiment to be as short as possible in order to prevent or minimize disruptions with building occupants. In some embodiments, thermal mass storage model generator  512  makes small changes to the temperature setpoints T sp  (e.g., changes of 3-5° F.) in order to have minimal impact on the comfort of the building occupants. 
     In some embodiments, thermal mass storage model generator  512  generates the excitation signals using a pseudorandom binary sequence (PRBS).  FIG.  7    shows a graph  700  of an example excitation temperature setpoint signal  702  over a one day period. Thermal mass storage model generator  512  can be configured to generate and provide temperature setpoint signal  702  to smart thermostat  100  during the system identification experiment. Temperature setpoint signal  702  can be generated a priori and tailored to meet the comfort constraints. The minimum and maximum temperature values of temperature setpoint signal  702  may be different depending on whether HVAC equipment  308  are operating in a heating dominated or cooling dominated mode. The values of temperature setpoint signal  702  may define the values of temperature setpoint variable T sp  in the input data set used to fit the model parameters. 
     For variable speed systems, excitation signals generated from a PRBS may not be suitable in practice owing to concerns over saturation effects leading to nonlinearities in the experimental dataset. This issue can be monitored and analyzed via high-fidelity building simulations and setpoint experiments on buildings. Modifications to the excitation signal generation methodology can be evaluated first in simulation to understand the technical trade-offs and limitations. Experiments on buildings can also be performed to help assess the use of other types of excitation signals. 
     Thermal mass storage model generator  512  can generate a set of input data by modulating the temperature setpoint T sp  and recording the values of the temperature setpoint T sp  and the outside air temperature T oa , at each time step during the system identification period. The set of input data may include a plurality of values of the temperature setpoint T sp  and the outside air temperature T oa . Thermal mass storage model generator  512  can generate a set of output data by monitoring and recording zone conditions at each time step of the system identification period. The zone conditions may include the zone air temperature T ia  and the heating/cooling load {dot over (Q)} HVAC . The set of output data may include a plurality of values of the zone air temperature T ia , and the heating/cooling load {dot over (Q)} HVAC . 
     The second stage of the system identification process is generating a parameterized building thermal model. The parameterized building thermal model may describe the relationship between zone temperature T ia , and sensible HVAC load {dot over (Q)} HVAC . In some embodiments, the parameterized building thermal model is based on a thermal circuit representation of building zone  310 . An example a thermal circuit representing the heat transfer characteristics of building zone  310  is shown in  FIG.  8   . 
     Referring now to  FIG.  8   , a thermal circuit  800  representing the heat transfer characteristics of building zone  310  is shown, according to some embodiments. Thermal circuit  800  is a two thermal resistance, two thermal capacitance (2R2C) control-oriented thermal mass model. The thermal energy balance within thermal circuit  800  is given by the following linear differential equations: 
                         C   ia     ⁢       T   .     ia       =         1     R   mi       ⁢     (       T   m     -     T   ia       )       +       1     R   oi       ⁢     (       T   oa     -     T   ia       )       +       Q   .     HVAC     +       Q   .     Other         ⁢   
   	         C   m     ⁢       T   .     m       =       1     R   mi       ⁢     (       T   ia     -     T   m       )                 (     Equation   ⁢         13     )               
where T ia , represents the indoor air temperature, T oa  represents the outdoor air temperature, T m  represents the thermal mass temperature (i.e., the average temperature of solid objects in building zone  310 ), C m  represents thermal capacitance of the thermal mass, C ia  represents the indoor air thermal capacitance, R mi  represents the thermal resistance between the indoor air and the thermal mass, R oi  represents the thermal resistance between the indoor air and the outdoor air, {dot over (Q)} HVAC  represents the sensible heat added to building zone  310  by HVAC equipment  308  (or removed from building zone  310  if cooling is provided), and {dot over (Q)} other  represents the heat load disturbance (e.g., internal heat generation within building zone  310  via solar radiation, occupancy, electrical equipment, etc.).
 
     In some embodiments, heat transfer between adjacent zones is modeled as part of the heat load disturbance {dot over (Q)} other  and thus, not explicitly accounted for in the model. In other embodiments, thermal interactions between adjacent zones can be accounted for by modeling the temperature of adjacent zones as additional temperature nodes and adding a thermal resistor between each adjacent zone and building zone  310 . For buildings that have significant thermal coupling between zones (e.g., large stores, open floorplan buildings, etc.), it may be desirable to model the thermal interactions between adjacent zones. 
     The thermal dynamic model of Equation 13 describes the heat transfer between the indoor air and the thermal mass, the heat transfer through the building walls, the sensible heat/cooling duty of HVAC equipment  308  {dot over (Q)} HVAC , and the internal heat load {dot over (Q)} other  on the space generated by the occupants, electrical plug and lighting, and solar irradiation. Throughout this disclosure, the model of Equation 13 is referred to as the thermal model. The thermal model has been demonstrated to yield acceptable prediction accuracy while managing the trade-off between model complexity (e.g., number of model parameters) and model prediction accuracy. In some embodiments, the internal heat load {dot over (Q)} other  cannot be measured and is therefore treated as an unmeasured, time-varying disturbance. 
     HVAC Load Model Generator 
     HVAC load model generator  514  can be configured to generate a HVAC load model that describes the relationship between indoor air temperature T ia , the temperature setpoint T sp , and the sensible HVAC load {dot over (Q)} HVAC . In some embodiments, the HVAC load model describes the dynamics of HVAC equipment  308  when providing heating or cooling to building zone  310 . For example, the HVAC load model may represent HVAC equipment  308  as a proportional-integral (PI) control system that provides the sensible HVAC load {dot over (Q)} HVAC  as a function of the indoor air temperature T ia  and the temperature setpoint T sp . 
     In some embodiments, the sensible load {dot over (Q)} HVAC  provided by HVAC equipment  308  is a function of the temperature setpoint T sp  as well as other factors including the heat gained/lost through the building walls, the internal heat load {dot over (Q)} other , and the heat transferred to/removed from the building thermal mass. To provide or remove heat to a particular zone, various actions may be performed by HVAC equipment  308 . For example, for a chilled water cooled building with a central air handling unit (AHU) serving several variable air volume (VAV) terminal boxes, a chiller may generate chilled water. The chilled water can be pumped to the AHU cooling coil and air can be forced over the cooling coils of the AHU to cool the air to a supply air temperature setpoint (e.g., 55° F.). The cooled air can be delivered through the supply air duct to terminal VAV boxes at the location of building zone  310 . The flow rate of cooled air can be adjusted via a damper such that building zone  310  reaches and/or stays at the temperature setpoint T sp . 
     In some embodiments, the dynamics of the entire process of providing cooling or heating to building zone  310  is non-negligible on the time-scale of interest for utilizing the energy storage of a building mass. To account for the dynamics of the heating/cooling process, HVAC load model generator  514  can model the dynamics between the temperature setpoint T sp  and the HVAC sensible load {dot over (Q)} HVAC . An example of a model that can be model that can be generated by HVAC load model generator  514  is the following proportional-integral (PI) controller model:
 
 {dot over (Q)}   HVAC,j   =K   p,j ε sp   +K   I,j ∫ 0   t ε sp ( s ) ds  
 
ε sp   =T   sp,j   −T   ia   (Equation 14)
 
where j∈{clg, hlg} is the index that is used to denote either heating or cooling mode. In some embodiments, different sets of parameters are generated for the heating and cooling modes. In other words, HVAC load model generator  514  can generate a first set of HVAC load model parameters for the cooling mode and a second set of HVAC load model parameters for the heating mode. In some embodiments, the heating and cooling load is constrained to the following set: {dot over (Q)} HVAC,clg  ∈[−{dot over (Q)} clg,cap , 0] and {dot over (Q)} HVAC,htg  ∈[0, {dot over (Q)} htg,cap ] where {dot over (Q)} HVAC &lt;0 indicates that HVAC equipment  308  are providing cooling and {dot over (Q)} HVAC  &gt;0 indicates that HVAC equipment  308  are providing heating
 
     The model of Equation 14 includes a proportional gain term (i.e., K p,j ε sp ) that accounts for the error ε sp  between the indoor air temperature T ia  and the setpoint T sp,j . The model of Equation 14 also includes an integrator term (i.e., K I,j ∫ 0   t ε sp (s) ds) to account for the time-varying nature of the HVAC load {dot over (Q)} HVAC,j  required to force the indoor temperature T ia  to its setpoint T sp,j . The integrator value may be correlated with the internal heat load disturbance {dot over (Q)} other  and thus, may play the role of an integrating disturbance model. In some embodiments, the model of Equation 14 does not represent a particular PI controller, but rather is a lumped dynamic model describing the relationship between the HVAC load {dot over (Q)} HVAC,j  and the temperature setpoint T sp,j . In some embodiments, the HVAC load model defines the HVAC load {dot over (Q)} HVAC,j  as a function of the zone air temperature T ia , the temperature setpoint T sp,j , the zone air humidity H ia , a zone air humidity setpoint H sp,j , zone heat load {dot over (Q)} other , and/or any of a variety of factors that affect the amount of heating or cooling provided by HVAC equipment. It is contemplated that the HVAC load model can include any combination of these or other variables in various embodiments. 
     In some embodiments, HVAC load model generator  514  accounts for staged HVAC equipment in the HVAC load model. For staged equipment, the sensible heating and cooling load {dot over (Q)} HVAC,j  can be modeled to take values in the countable set {dot over (Q)} HVAC,j ={{dot over (Q)} s1 , {dot over (Q)} s2 , . . . , {dot over (Q)} sn }, where {dot over (Q)} si  i∈{1, . . . , n} is the HVAC load with the first through ith stage on. The staging down and staging up dynamics can be assumed to be negligible. Alternatively, the staging down and staging up dynamics can be captured using a low-order linear model. 
     The model of Equation 14 allows for continuous values of {dot over (Q)} HVAC,j  and may not be directly applicable to staged HVAC equipment. However, HVAC load model generator  514  can compute the time-averaged HVAC load {dot over (Q)} HVAC,j  over a time period, which gives a continuous version of the HVAC load {dot over (Q)} HVAC,j  needed to meet the temperature setpoint T sp,j . For example, HVAC load model generator  514  can apply a filtering technique to the equipment stage on/off signal to compute a time-averaged version of the HVAC load {dot over (Q)} HVAC,j . The filtered version of the HVAC load {dot over (Q)} HVAC,j  can then be used as an input to the model of Equation 14 to describe the relationship between the HVAC load {dot over (Q)} HVAC,j  and the temperature setpoint T sp,j . Throughout this disclosure, the variable {dot over (Q)} HVAC,j  is used to represent both the continuous HVAC load when HVAC equipment  308  have a variable speed and to represent the filtered HVAC load when HVAC equipment  308  are staged. 
     Incorporating the thermal and the HVAC load models together and writing the system of equations as a linear system of differential equations gives the following state space representation: 
               [             T   .     ia                 T   .     m               I   .           ]     =         [             1     C   ia       ⁢     (       K     p   ,   j       -     1     R   mi       -     1     R   oi         )             1       C   ia     ⁢     R   mi               -         K     p   ,   j       ⁢     K     I   ,   j           C   ia                   1       C   m     ⁢     R   mi               -     1       C   m     ⁢     R   mi               0             -   1         0       0         ]     [   ⁠           T   ia               T   m             I         ]     +                [   ⁠           -       K     p   ,   j         C   ia               1       C   ia     ⁢     R   oi                 0       0           1       0         ⁠   ]     [           T     sp   ,   j                 T   oa           ]     +       [           1     C   ia               0           0         ]     ⁢         Q   .     other     
   	   [           T   ia                 Q   .       HVAC   ,   j             ]         =         [         1       0       0             -     K     p   .   j             0         K     I   ,   j             ]     [           T   ia               T   m             I         ]     +       [         0       0             K     p   ,   j           0         ]     [           T     sp   ,   j                 T   oa           ]                   
or more compactly:
 
 {dot over (x)}=A   c (θ) x+B   c (θ) u+B   d   d  
 
 y=C   c (θ) x+D   c (θ) u   (Equation 15)
 
where x T =[T ia , T m , I], u T =[T sp,j , T oa ], d={dot over (Q)} other /C ia , y T =[T ia , {dot over (Q)} HVAC,j ], θ is a parameter vector containing all non-zero entries of the system matrices, and
 
     
       
         
           
             
               
                 
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     The third stage of the system identification process includes fitting the parameters θ of the parameterized building thermal model (i.e., the model of Equation 15) to the input-output data using a parameter identification algorithm.  FIG.  9    is a flowchart of a process  900  which can be performed by system identifier  510  to fit the parameters θ of the parameterized building thermal model. If HVAC equipment  308  is staged, system identifier  510  can filter input-output data to remove the high frequency dither in the indoor air temperature T ia  and can compute a time-averaged version of the HVAC load {dot over (Q)} HVAC  from the discrete HVAC staging trajectory (step  902 ). System identifier  510  can then fit the thermal model parameters and HVAC load model parameters θ to the input-output data (or filtered input-output data) obtained during the first stage of the system identification process (step  904 ). System identifier  510  can augment the resulting state-space model with another integrating disturbance model and can estimate the Kalman filter gain for the resulting model (step  906 ). Using data obtained in a secondary experiment or under normal operation, system identifier  510  can validate the model through the use of statistics that capture the multi-step prediction accuracy of the resulting model (step  908 ). Each of these steps is described in detail below. 
     Step  902  may include filtering the input-output data to remove a high frequency dither in the indoor air temperature T ia  and computing a time-averaged version of the HVAC load {dot over (Q)} HVAC  from the discrete HVAC staging trajectory. When HVAC equipment  308  are staged, the indoor air temperature trajectory may have a high frequency dither caused by HVAC equipment  308  staging up and down. For example, when operating in a cooling mode, HVAC equipment  308  can be staged up (e.g., by activating discrete chillers) to drive the indoor air temperature T ia  to the temperature setpoint T sp  minus a deadband. When the indoor air temperature T ia  reaches the temperature setpoint T sp  minus the deadband, HVAC stages can be deactivated, which results in an increase in the temperature T ia . The indoor air temperature T ia  may increase until it reaches the temperature setpoint T sp  plus the deadband, at which time the HVAC stages can be activated again. The observed effect may be a high frequency dither or oscillation of the temperature T ia  around the setpoint T sp  (shown in  FIG.  10   ). Also, the HVAC load trajectory may take discrete values as stages are turned on and off. The duty cycle of the on/off trajectory of the stages may depend on the heat transfer to/from building zone  310 , the error between the setpoint and indoor air temperature, and the setpoint. Thus, a good indication of the HVAC load {dot over (Q)} HVAC  on a continuous scale is some time-average of the HVAC load trajectory or stage on/off trajectories. 
     To remove the high frequency dither and to perform the time-averaging on the HVAC load trajectory, system identifier  510  can filter the input-output data obtained from the SI experiment. In some embodiments, system identifier  510  uses a first-order SavitzkyGolay filter (SGF) to filter the input-output data. The SGF involves fitting a polynomial to a set of data samples and evaluating the resulting polynomial at a single point within the approximation interval. The SGF is equivalent to performing discrete convolution with a fixed impulse response. However, the SGF is a non-casual filter. To address this issue, system identifier  510  can use the nearest past filtered measurements in the system identification process. For example, let Δt filter  be the filtering window of the SGF. For the filter to compute its smoothed estimate of the data at a given time step t, it can use data from t−Δt filter /2 to t+Δt filter /2. To provide the filtered measurements (indoor air temperature T ia  and HVAC load {dot over (Q)} HVAC ) at a sample time t k , the filtered measurement from t k −Δt filter /2 can be used. 
       FIG.  10    is a pair of graphs  1000  and  1050  illustrating an example of the indoor air temperature T ia  trajectory and the HVAC load {dot over (Q)} HVAC  trajectory before filtering with the SGF. Line  1002  represents the outdoor air temperature T oa  and line  1004  represents the unfiltered indoor air temperature T ia . In steady-state, the indoor air temperature T ia , may vary around the temperature setpoint within a region that is plus and minus the control deadband  1106 . Line  1052  represents the unfiltered sensible load {dot over (Q)} HVAC  for staged HVAC equipment. 
       FIG.  11    is a pair of graphs  1100  and  1150  illustrating an example of the indoor air temperature T ia , trajectory and the HVAC load {dot over (Q)} HVAC  trajectory after filtering with the SGF. Line  1102  represents the outdoor air temperature T oa  and line  1004  represents the filtered indoor air temperature T ia . In some embodiments, the filtering removes the variation of temperature about the setpoint to smooth the temperature trajectory. Line  1152  represents the filtered sensible load {dot over (Q)} HVAC  for staged HVAC equipment and may be a time-averaged version of the unfiltered sensible load  1052  {dot over (Q)} HVAC  shown in  FIG.  10   . 
     Referring again to  FIG.  9   , step  904  may include fitting the thermal model parameters and HVAC load model parameters to the input-output data (or filtered input-output data) obtained during the first stage of the system identification process. In step  904 , the time-varying heat load disturbance d may be omitted. System identifier  510  can identify the parameters for the model given by: 
                         x   .     (   t   )     =         [           θ   1           θ   2           θ   3               θ   4           -     θ   4           0             -   1         0       0         ]     ⁢     x   ⁡   (   t   )       +       [             θ   5     ⁢     θ   7               θ   5     ⁢     θ   6               0       0           1       0         ]     ⁢     u   ⁡   (   t   )           ⁢   
       y   ⁡   (   t   )     =         [         1       0       0             -     θ   7           0         θ   8           ]     ⁢     x   ⁡   (   t   )       +       [         0       0             θ   7         0         ]     ⁢     u   ⁡   (   t   )                   (     Equation   ⁢         16     )               
where the effect of the time-varying disturbance d is accounted for via the integrator model.
 
     Using the prediction error method (PEM) described previously, system identifier  510  can obtain an estimate of the model parameters θ. In some embodiments, system identifier  510  estimates the model parameters θ using the functions provided by the Matlab System Identification toolbox. For example, the identification can be performed using the function greyest with the initial state option set to estimate, the disturbance model option set to none, and the estimation focus option is set to simulation. Other options may not be specifically set and therefore default values can be used. 
     The initial condition of the model of Equation 16 may be unknown, but can be from the data. The disturbance component (i.e., the Kalman gain estimate K) may not be included in the model of Equation 16. The estimation focus can be set to prediction so that the algorithm automatically computes the weighting function of the prediction errors. The stability estimation focus option may perform the same weighting as the prediction option, but also enforces model stability. Under the parameterization of the model of Equation 16, the values of the resistance, capacitance, and PI parameters may be computed from the parameter vector θ if necessary. Given that the resulting PEM optimization problem is a nonlinear, non-convex problem, the initial guess on the parameter values may have a significant impact on the parameter values obtained, and multiple executions of the PEM solver provided a different initial guess each time may converge to different local minima. 
     To perform step  904  autonomously, model verification can be performed before process  900  continues to  906 . If any of the verification steps are not satisfied, the PEM solver can be reinitialized with a different random initial guess. The model verification steps that can be performed in step  904  may include ensuring stability by checking the eigenvalues of the obtained A. In some embodiments, the model verification steps include ensuring that the identified parameters are greater than zero. This is a condition arising from the physical meaningfulness of the model. The model verification steps may include ensuring that the thermal capacitance of the air C ia  is less than the thermal capacitance C m  of the building mass. In some embodiments, the model verification steps include verifying that the matrix A is well conditioned in order to avoid prediction problems and model sensitivity to noise. If the condition number of A is less than 10500, the conditioning of A may be deemed to be acceptable. 
     Once the obtained model satisfies all the checks, process  900  may proceed to step  906 . It is possible that the conditions described above are not ever satisfied. However, this could be due to a number of reasons that may be difficult to autonomously diagnose and potentially, be associated with a problem beyond the limitations of the SI procedure. To address this problem, a limit can be set for the number of times through the steps above. If the parameter acceptability criteria is not satisfied after the maximum number of iterations through step  904 , system identifier  510  can report an error for a human operator to investigate and address. 
     Estimator Gain Identifier 
     As discussed above, the model of Equation 16 may only account for the main thermal dynamics of the system and may not identify an appropriate estimator gain. In step  906 , the model of Equation 16 can be augmented with another integrating disturbance model and the estimator gain can be identified. In some embodiments, step  906  is performed by estimator gain identifier  516 . Estimator gain identifier  516  can be configured to augment the model of Equation 16 to produce the augmented model: 
                 [               x   ^     .     (   t   )                   d   ^     .     (   t   )           ]     =         [           A   c           B   d             0       0         ]     [             x   ^     (   t   )                 d   ^     (   t   )           ]     +       [           B   c             0         ]     ⁢     u   ⁡   (   t   )       +         [             K   x     (   ϕ   )                 K   d     (   ϕ   )           ]       ︸     =     :     K   ⁡   (   ϕ   )             ⁢     (       y   ⁡   (   t   )     -       y   ^     (   t   )       )           ⁢   
         y   ^     (   t   )     =         [           C   c           C   d           ]     [             x   ^     (   t   )                 d   ^     (   t   )           ]     +       D   c     ⁢     u   ⁡   (   t   )                 
where the parameters of A c , B c , C c  and D c  are the same as the parameters θ determined in step  904 . The disturbance model can be selected such that B d =1 and C d =0, which is an input integrating disturbance model. K(ϕ) is the identified estimator gain parameterized with the parameter vector ϕ as shown in the following equation:
 
                         K   x     (   ϕ   )     =     [           ϕ   1           ϕ   2             0       0             ϕ   3           ϕ   4           ]       ,         K   d     (   ϕ   )     =     [           ϕ   5           ϕ   6           ]               (     Equation   ⁢         17     )               
where the elements corresponding to the mass temperature T m  (i.e., the middle row of the matrix K x (ϕ)) are set to zero. If these elements are allowed to take non-zero values, poor estimation performance may result. Accordingly, estimator gain identifier  516  may set the elements corresponding to the mass temperature T m  to zero. The values for the parameters ϕ can be obtained using the function greyest on the augmented model, using the same input-output data as that used in step  904 .
 
     Estimator gain identifier  516  can be configured to generate the following augmented matrices: 
                 A   aug     =     [           A   c           B   d             0       0         ]       ,       B   aug     =     [           B   c             0         ]       ,       C   aug     =     [           C   c           C   d           ]       ,       D   aug     =     D   c             
Similar to the verifications performed in step  904 , estimator gain identifier  516  can check for the stability of the observer system A aug −KC aug , and its conditioning number. In this case, the conditioning number of A aug −KC aug  may be deemed acceptable if it is less than 70,000.
 
     Step  908  may include validating the augmented model generated in step  906 . As discussed above, system identifier  510  may select the optimal model parameters that fit the input-output data with respect to an objective function that depends on the one-step error prediction. In step  904 , the PEM is the one-step ahead prediction without estimation (i.e., open-loop estimation), whereas step  906  represents the one-step ahead closed-loop estimation problem. However, within the context of MPC, multi-step open-loop predictions can be used to select the optimal input trajectory over the prediction horizon. Thus, the model validation performed in step  908  may ensure that the estimator is tuned appropriately and that the multi-step open-loop prediction performed with the identified model provides sufficient accuracy. 
     In some embodiments, system identifier  510  collects another input-output data set for model validation. This data may be collected during normal operation or from another SI experiment. Using this validation data set, system identifier  510  can generate one or more metrics that quantify the multi-step prediction accuracy. One example of a metric which can be generated in step  908  is the Coefficient of Variation Weighted Mean Absolute Prediction Error (CVWMAPE). CVWMAPE is an exponentially weighted average of the absolute prediction error at each time step that quantities the prediction error over time. System identifier  510  can calculate the CVWMAPE as follows: 
                       CVWMAPE   ⁡   (   k   )     =               ∑     i   =   k       k   +     N   h     -   1           e       -   i       N   h         ⁢       ❘   &#34;\[LeftBracketingBar]&#34;         y   ⁡   (   i   )     -       y   ^     (     i   |   k     )         ❘   &#34;\[RightBracketingBar]&#34;                 ∑     i   =   k       k   +     N   h     -   1           e       -   i       N   h         ⁢       ❘   &#34;\[LeftBracketingBar]&#34;       y   ⁡   (   i   )       ❘   &#34;\[RightBracketingBar]&#34;               ,     k   =   0     ,   1   ,   2   ,   …           (     Equation   ⁢         18     )               
where N h  ∈   &gt;0  is the prediction horizon, y(i) is the measured output at time step i, and ŷ(i|k) is the predicted output with the identified model given a measurement at time step k and the input sequence u(k), u(k+1), . . . , u(i−1). For notational simplicity the variable y is used to denote one of the two outputs (i.e., in this subsection, y is a scalar). The CVWMAPE can be computed for both outputs.
 
     Another prediction error to consider is prediction error with respect to the q-step ahead prediction (q∈   ≥0 ). System identifier  510  can calculate the Coefficient of Variation Root Mean Squared Prediction Error (CVRMSPE) is to evaluate the q-step ahead prediction error. For example, system identifier  510  can calculate the CVRMSPE for a range of values of q from zero-step ahead prediction up to N a -step ahead prediction. Given a set of measured output values {y(0), . . . , y(M)} for M∈   ≥0 , the CVRMSPE is given by: 
                     CVRMSPE   ⁡   (   q   )     =       RMSPE   ⁡   (   q   )         y   _     (   q   )               (     Equation   ⁢         19     )                         where                                     RMSPE   ⁡   (   q   )     =           ∑     i   =     q   +   1       M           (       y   ⁡   (   i   )     -       y   ^     (     i   ❘     i   -   q       )       )         M   -   q           ,           (     Equation   ⁢         20     )                                 y   _     (   q   )     =         ∑     i   =   q     M       y   ⁡   (   i   )         M   -   q         ,           (     Equation   ⁢         21     )               
for all q∈{0, N h −1}. The CVRMSPE helps identify the prediction error over the duration of the optimization period.
 
     Once the system identification process is complete, system identifier  510  can provide the system matrices A, B, C, and D with identified parameters θ and the estimator gain K with identified parameters ϕ to state disturbance estimator  520  and predictive optimizer  522 . State disturbance estimator  520  can use the system matrices A, B, C, and D and the estimator gain K to compute an estimate of the current state {circumflex over (x)}(k) and unmeasured disturbance {circumflex over (d)}(k) at each sample time. 
     Load/Rate Predictor 
     Referring again to  FIG.  5   , model predictive controller  302  is shown to include a load/rate predictor  518 . Load/rate predictor  518  can be configured to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k) and the utility rates {circumflex over (r)} elec (k) for each time step k (e.g., k=1 . . . n) of an optimization period. Load/rate predictor  518  can provide the load and rate predictions to predictive optimizer  522  for use in determining the optimal temperature setpoints T sp . 
     In some embodiments, load/rate predictor  518  predicts the value of {circumflex over ({dot over (Q)})} other (k) at each time step k using a history of disturbance estimates {circumflex over (d)} hist  generated by state/disturbance estimator  520 . As described above, the state/disturbance estimation problem is given by: 
                       [               T   ^     ia     (       k   +   1     ❘   k     )                   T   ^     m     (       k   +   1     ❘   k     )                 I   ^     (       k   +   1     ❘   k     )                 d   ^     (       k   +   1     ❘   k     )           ]     =         A   aug     [               T   ^     ia     (     k   ❘     k   -   1       )                   T   ^     m     (     k   ❘     k   -   1       )                 I   ^     (     k   ❘     k   -   1       )                 d   ^     (     k   ❘     k   -   1       )           ]     +       B   aug     [             T   sp     (   k   )                 T   oa     (   k   )           ]     +       [           K   x               K   d           ]     ⁢     (       [             T   ia     (   k   )                   Q   .     HVAC     (   k   )           ]     -     [               T   ^     ia     (     k   ❘     k   -   1       )                     Q   ^     .     HVAC     (     k   ❘     k   -   1       )           ]       )           ⁢   
       [               T   ^     ia     (     k   ❘     k   -   1       )                     Q   ^     .     HVAC     (     k   ❘     k   -   1       )           ]     =         C   aug     [               T   ^     ia     (     k   ❘     k   -   1       )                   T   ^     m     (     k   ❘     k   -   1       )                 I   ^     (     k   ❘     k   -   1       )                 d   ^     (     k   ❘     k   -   1       )           ]     +     D   [             T   sp     (   k   )                 T   oa     (   k   )           ]         ⁢   
         where   ⁢           A   aug       =     [         A         B   d             0       I         ]       ,       B   aug     =     [         B           0         ]       ,       C   aug     =       [         C       0         ]     .                 (     Equation   ⁢         22     )               
State/disturbance estimator  520  can perform the state/disturbance estimation for each time step k to generate a disturbance estimate {circumflex over (d)}(k) for each time step k. State/disturbance estimator  520  can record a history of disturbance estimates {circumflex over (d)} hist :={{circumflex over (d)}(k|k−1)} k=1   N     hist    and provide the history of disturbance estimates {circumflex over (d)} hist  to k=1 load/rate predictor  518 .
 
     Load/rate predictor  518  can use the history of disturbance estimates {circumflex over (d)} hist  to predict the value of {circumflex over ({dot over (Q)})} other (k) at each time step k. In some embodiments, the heat load disturbance {circumflex over ({dot over (Q)})} other (k) is a function of the time of day, the day type (e.g., weekend, weekday, holiday), the weather forecasts, building occupancy, and/or other factors which can be used by load/rate predictor  518  to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k). For example, load/rate predictor  518  can predict the heat load disturbance using the equation:
 
{circumflex over ( {dot over (Q)} )} other ( k )= f   pred ( kΔ+t   O   {circumflex over (d)}   hist )  (Equation 23)
 
where Δ&gt;0 is the sample period and t 0  is an initial time. In some embodiments, load/rate predictor  518  uses weather forecasts from a weather service  526  and/or load history data from historical data  528  to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k) at each time step.
 
     In some embodiments, load/rate predictor  518  uses a deterministic plus stochastic model trained from historical load data to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k). Load/rate predictor  518  may use any of a variety of prediction methods to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k) (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). In some embodiments, load/rate predictor  518  makes load/rate predictions using the techniques described in U.S. patent application Ser. No. 14/717,593. 
     Load/rate predictor  518  is shown receiving utility rates from utilities  524 . Utility rates may indicate a cost or price per unit of a resource (e.g., electricity, natural gas, water, etc.) provided by utilities  524  at each time step k in the prediction window. In some embodiments, the utility rates are time-variable rates. For example, the price of electricity may be higher at certain times of day or days of the week (e.g., during high demand periods) and lower at other times of day or days of the week (e.g., during low demand periods). The utility rates may define various time periods and a cost per unit of a resource during each time period. Utility rates may be actual rates received from utilities  524  or predicted utility rates estimated by load/rate predictor  518 . 
     In some embodiments, the utility rates include demand charges for one or more resources provided by utilities  524 . A demand charge may define a separate cost imposed by utilities  524  based on the maximum usage of a particular resource (e.g., maximum energy consumption) during a demand charge period. The utility rates may define various demand charge periods and one or more demand charges associated with each demand charge period. In some instances, demand charge periods may overlap partially or completely with each other and/or with the prediction window. Predictive optimizer  522  may be configured to account for demand charges in the high level optimization process performed by predictive optimizer  522 . Utilities  524  may be defined by time-variable (e.g., hourly) prices, a maximum service level (e.g., a maximum rate of consumption allowed by the physical infrastructure or by contract) and, in the case of electricity, a demand charge or a charge for the peak rate of consumption within a certain period. Load/rate predictor  518  may store the predicted heat load disturbance {circumflex over ({dot over (Q)})} other (k) and the utility rates in memory  508  and/or provide the predicted heat load disturbance {circumflex over ({dot over (Q)})} other (k) and the utility rates to predictive optimizer  522 . 
     Predictive Optimizer 
     Still referring to  FIG.  5   , model predictive controller  302  is shown to include a predictive optimizer  522 . Predictive optimizer  522  can manipulate the temperature setpoint T sp  when operating in the operational mode to minimize the economic cost of operating HVAC equipment  308  over the duration of the optimization period. To determine the temperature setpoints T sp , predictive optimizer  522  can optimize an objective function (i.e., a cost function) that accounts for the cost of operating HVAC equipment  308  over the duration of the optimization period. The costs of operating HVAC equipment  308  can include, for example, the costs of resources consumed by HVAC equipment  308  during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of HVAC equipment  308 . 
     An example of an objective function which can be optimized by predictive optimizer  522  is shown in the following equation: 
                     min       T   sp     ,     ϵ   T     ,     δ   ⁢     T   sp                   ∑     k   =   0       N   -   1             (           r   elec     (   k   )     ⁢         ℓ   ^     elec     (   k   )       +       p     ϵ   T       ⁢       ϵ   T     (   k   )       +       p     δ   ⁢   T       ⁢   δ   ⁢       T   sp     (   k   )         )               (     Equation   ⁢         24     )               
where r elec (k) is the price of electricity at time step k, {circumflex over (l)} elec (k) is the predicted electricity consumption of HVAC equipment  308  at time step k, ϵ T (k) is a number of degrees by which the zone temperature constraints are violated at time step k, p ϵT  is a zone air temperature penalty coefficient applied to ϵ T (k), δT sp (k) is a number of degrees by which the zone temperature setpoint T sp  changes between time step k−1 and time step k, and p δT     sp    is a temperature setpoint change penalty coefficient applied to δT sp (k). Equation 24 accounts for the cost of electricity consumption (r elec (k){circumflex over (l)} elec (k)), penalizes violations of the indoor air temperature bounds (p ϵ     T   ϵ T (k)), and penalizes changes in the temperature setpoint (p δT δT sp (k)).
 
     If additional resources (other than electricity) are consumed by HVAC equipment  308 , additional terms can be added to Equation 24 to represent the cost of each resource consumed by HVAC equipment  308 . For example, if HVAC equipment  308  consume natural gas and water in addition to electricity, Equation 24 can be updated to include the terms r gas (k){circumflex over (l)} gas (k) and r water (k){circumflex over (l)} water (k) in the summation, where r gas (k) is the cost per unit of natural gas at time step k, {circumflex over (l)} gas (k) is the predicted natural gas consumption of HVAC equipment  308  at time step k, r water (k) is the cost per unit of water at time step k, and {circumflex over (l)} water (k) is the predicted water consumption of HVAC equipment  308  at time step k. 
     Predictive optimizer  522  can be configured to estimate the amount of each resource consumed by HVAC equipment  308  (e.g., electricity, natural gas, water, etc.) as a function of the sensible heating or cooling load {circumflex over ({dot over (Q)})} HVAC . In some embodiments, a constant efficiency model is used to compute the resource consumption of HVAC equipment  308  as a fixed multiple heating or cooling load {circumflex over ({dot over (Q)})} HVAC  (e.g., {circumflex over (l)} elec (k)=η{circumflex over ({dot over (Q)})} HVAC (k)) In other embodiments, equipment models describing a relationship between resource consumption and heating/cooling production can be used to approximate the resource consumption of the HVAC equipment  308 . The equipment models may account for variations in equipment efficiency as a function of load and/or other variables such as outdoor weather conditions (e.g., {circumflex over (l)} elec (k)=f({circumflex over ({dot over (Q)})} HVAC (k),T oa (k),η base )). Predictive optimizer  522  can convert the estimated heating/cooling load {circumflex over ({dot over (Q)})} HVAC (k) at each time step to one or more resource consumption values (e.g., {circumflex over (l)} elec (k), {circumflex over (l)} gas (k), {circumflex over (l)} water (k), etc.) for inclusion in the objective function. 
     In some embodiments, predictive optimizer  522  is configured to modify the objective function to account for various other costs associated with operating HVAC equipment  308 . For example, predictive optimizer  522  can modify the objective function to account for one or more demand charges, peak load contribution charges, equipment degradation costs, equipment purchase costs, revenue generated from participating in incentive-based demand response programs, economic load demand response, and/or any other factors that can contribute to the cost incurred by operating HVAC equipment  308  or revenue generated by operating HVAC equipment  308 . Predictive optimizer  522  can add one or more additional terms to the objective function to account for these or other factors. Several examples of such functionality are described in U.S. patent application Ser. No. 15/405,236 filed Jan. 12, 2017, U.S. patent application Ser. No. 15/405,234 filed Jan. 12, 2017, U.S. patent application Ser. No. 15/426,962 filed Feb. 7, 2017, U.S. patent application Ser. No. 15/473,496 filed Mar. 29, 2017, and U.S. patent application Ser. No. 15/616,616 filed Jun. 7, 2017. The entire disclosure of each of these patent applications is incorporated by reference herein. In some embodiments, model predictive controller  302  includes some or all of the functionality of the controllers and/or control systems described in these patent applications. 
     Predictive optimizer  522  can be configured to automatically generate and impose constraints on the optimization of the objective function. The constraints may be based on the system model provided by system identifier  510 , the state estimates provided by state/disturbance estimator  520 , the heat load disturbance predictions provided by load/rate predictor  518 , and constraints on the zone air temperature T ia . For example, predictive optimizer  522  can generate and impose the following constraints for each time step k∈{0, . . . , N−1}: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               T 
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                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             
                               T 
                               m 
                             
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             I 
                             ⁡ 
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       A 
                       [ 
                       
                         
                           
                             
                               
                                 T 
                                 ia 
                               
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 m 
                               
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               I 
                               ⁡ 
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     + 
                     
                       B 
                       [ 
                       
                         
                           
                             
                               
                                 T 
                                 sp 
                               
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 oa 
                               
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     + 
                     
                       
                         B 
                         d 
                       
                       ⁢ 
                       
                         
                           
                             Q 
                             . 
                           
                           other 
                         
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     25 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       [ 
                       
                         
                           
                             Q 
                             . 
                           
                           
                             HVAC 
                             . 
                           
                         
                         ( 
                         k 
                         ) 
                       
                       ] 
                     
                     = 
                     
                       
                         C 
                         [ 
                         
                           
                             
                               
                                 
                                   T 
                                   ia 
                                 
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   m 
                                 
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 I 
                                 ⁡ 
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         D 
                         [ 
                         
                           
                             
                               
                                 
                                   T 
                                   sp 
                                 
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   T 
                                   oa 
                                 
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     26 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         
                           ℓ 
                           ^ 
                         
                         elec 
                       
                       ( 
                       k 
                       ) 
                     
                     = 
                     
                       η 
                       ⁢ 
                       
                         
                           
                             
                               Q 
                               ^ 
                             
                             . 
                           
                           HVAC 
                         
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     27 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         T 
                         ia 
                       
                       ( 
                       k 
                       ) 
                     
                     ≤ 
                     
                       
                         
                           T 
                           max 
                         
                         ( 
                         k 
                         ) 
                       
                       + 
                       
                         T 
                         deadband 
                       
                       + 
                       
                         
                           ϵ 
                           T 
                         
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     28 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         T 
                         ia 
                       
                       ( 
                       k 
                       ) 
                     
                     ≥ 
                     
                       
                         
                           T 
                           min 
                         
                         ( 
                         k 
                         ) 
                       
                       - 
                       
                         T 
                         deadband 
                       
                       - 
                       
                         
                           ϵ 
                           T 
                         
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     29 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         T 
                         sp 
                       
                       ( 
                       k 
                       ) 
                     
                     ∈ 
                     
                       [ 
                       
                         
                           
                             T 
                             
                               sp 
                               , 
                               min 
                             
                           
                           ( 
                           k 
                           ) 
                         
                         , 
                         
                           
                             T 
                             
                               sp 
                               , 
                               max 
                             
                           
                           ( 
                           k 
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     30 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         
                           T 
                           sp 
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       - 
                       
                         
                           T 
                           sp 
                         
                         ( 
                         k 
                         ) 
                       
                     
                     ≤ 
                     
                       δ 
                       ⁢ 
                       
                         
                           T 
                           sp 
                         
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     31 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         - 
                         
                           
                             T 
                             sp 
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           T 
                           sp 
                         
                         ( 
                         k 
                         ) 
                       
                     
                     ≤ 
                     
                       δ 
                       ⁢ 
                       
                         
                           T 
                           sp 
                         
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     32 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         ϵ 
                         T 
                       
                       ( 
                       k 
                       ) 
                     
                     ≥ 
                     0 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     33 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                  
                   
                     
                       
                         
                           T 
                           ia 
                         
                         ( 
                         0 
                         ) 
                       
                       = 
                       
                         
                           T 
                           ^ 
                         
                         
                           ia 
                           , 
                           0 
                         
                       
                     
                     , 
                     
                       
                         
                           T 
                           m 
                         
                         ( 
                         0 
                         ) 
                       
                       = 
                       
                         
                           T 
                           ^ 
                         
                         
                           m 
                           , 
                           0 
                         
                       
                     
                     , 
                     
                       
                         I 
                         ⁡ 
                         ( 
                         0 
                         ) 
                       
                       = 
                       
                         
                           I 
                           ^ 
                         
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     34 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation 25 is based on the thermal mass storage model provided by thermal mass storage model generator  512 . The thermal mass storage model predicts the system states at each time step k+1 as a function of the system states x(k), inputs u(k), and heat load disturbance at the previous time step k. Specifically, Equation 25 defines the relationship between the system states T ia (k+1), T m (k+1), and I(k+1) at time step k+1, the system states T ia (k), T m (k), and I(k) at time step k, the controlled or measured inputs T sp (k) and T oa (k) at time step k, and the estimated heat load disturbance {dot over (Q)} other (k) at time step k. The parameters in the matrices A and B can be determined by system identifier  510 , as previously described. The value of {dot over (Q)} other (k) at each time step k can be determined by load/rate predictor  518 . 
     Equation 26 is based on the HVAC load model provided by HVAC load model generator  514 . The HVAC load model defines the heating/cooling load {dot over (Q)} HVAC (k) at each time step k as a function of the system states T ia (k), T m (k), and I(k) at time step k and the controlled or measured inputs T sp (k) and T oa (k) at time step k. The parameters in the matrices C and D can be determined by system identifier  510 , as previously described. 
     Equation 27 defines the relationship between the predicted electric consumption {circumflex over (l)} elec (k) at time step k and the heating/cooling load {dot over (Q)} HVAC (k) at time step k. If additional resources (other than electricity) are consumed by HVAC equipment  308 , additional constraints can be added to define the relationship between {dot over (Q)} HVAC (k) and the amount of each resource consumed by HVAC equipment  308 . 
     Equations 28 and 29 constrain the zone air temperature T ia (k) at each time step k between the minimum temperature threshold T min (k) and the maximum temperature threshold T max (k) at that time step k, plus or minus the temperature deadband T deadband and a temperature error ϵ T (k). In some embodiments, the values of the minimum temperature threshold T min (k) and the maximum temperature threshold T max (k) can vary over the duration of the optimization period. For example, the minimum temperature threshold T min (k) and the maximum temperature threshold T max (k) may define a first zone temperature range (i.e., T min (k) to T max (k)) during time step k and a second zone temperature range (i.e., +1) to T max (k+1)) during time step k+1. The values of T min  and T max  at each time step can vary as a function of the time of day, day of the week, building occupancy, or other factors. Predictive optimizer  522  may be allowed to violate the temperature constraints if necessary, but any deviation from the defined temperature range (i.e., between T min (k)−T deadband  and T max (k)+T deadband ) may be penalized in the objective function by imposing a penalty p ϵ     T    on the temperature error ϵ T (k). Equation 33 requires the temperature error ϵ T (k) to be non-negative at each time step k. 
     Equation 30 limits the temperature setpoint T sp (k) at each time step between a minimum temperature setpoint T sp,min (k) and a maximum temperature setpoint T sp,max (k), whereas Equations 31-32 define the changes δT sp (k) in the temperature setpoint T sp (k) between consecutive time steps. The minimum temperature setpoint T sp,min (k) and the maximum temperature setpoint T sp,max (k) may be different from the minimum temperature threshold T min (k) and the maximum temperature threshold T max (k) used to constrain the zone air temperature T ia (k). In some embodiments, the values of the minimum temperature setpoint T sp,min (k) and the maximum temperature setpoint T sp,max (k) can vary over the duration of the optimization period. For example, the minimum temperature setpoint T sp,min (k) and the maximum temperature setpoint T sp,max (k) may define a first temperature setpoint range (i.e., T sp,min (k) to T sp,max (k)) during time step k and a second temperature setpoint range (i.e., T sp,min (k+1) to T sp,max (k+1)) during time step k+1. The values of T sp,min  and T sp,max  at each time step can vary as a function of the time of day, day of the week, building occupancy, or other factors. Changes in the temperature setpoint T sp  may be penalized in the objective function by imposing a penalty p δT     sp    on the temperature change δT sp (k). 
     Equation 34 sets the initial system states for the zone air temperature T ia (0), the zone mass temperature T m (0), and the integrated disturbance I(0) at the first time step k=0 of the optimization period. The initial system states T ia (0), T m (0), and I(0) may be set to the initial state estimates {circumflex over (T)} ia,0 , {circumflex over (T)} m,0 , and Î 0  provided by state/disturbance estimator  520 . In other words, the predictive model shown in Equations 25-26 may be initialized using the state estimates provided by state/disturbance estimator  520 . 
     Predictive optimizer  522  can be configured to optimize the objective function (Equation 24) subject to the optimization constraints (Equations 25-34) to determine optimal values of the temperature setpoint T sp (k) at each time step k of the optimization period. Predictive optimizer  522  can use any of a variety of optimization techniques to perform the optimization. For example, the optimization problem can be formulated as a linear program (Equations 24-34) and solved using a linear optimization technique (e.g., a basis exchange algorithm, an interior point algorithm, a cutting-plane algorithm, a branch and bound algorithm, etc.) using any of a variety solvers and/or programming languages. In some embodiments, predictive optimizer  522  performs the optimization at the beginning of each time step k to determine optimal temperature setpoints T sp (k) for the next N time steps. 
     Once the optimal temperature setpoints T sp  have been determined, predictive optimizer  522  can provide the optimal temperature setpoint T sp (k) for the current time step k to smart thermostat  100  and/or equipment controller  406 . State/disturbance estimator  520  can use the temperature setpoint T sp (k) for the current time step k along with feedback received from HVAC equipment  308  and building zone  310  during time step k (e.g., measurements of T ia (k), T oa (k), {dot over (Q)} HVAC (k), etc.) to update the state/disturbance estimates the next time the optimization is performed. 
     Model Predictive Control Processes 
     Referring now to  FIGS.  12 - 14   , several flowcharts of model predictive control processes  1200 - 1400  are shown, according to some embodiments. Processes  1200 - 1400  can be performed by one or more components of model predictive control systems  300 - 400 , as described with reference to  FIGS.  3 - 11   . For example, processes  1200 - 1400  can be performed by model predictive controller  302 , smart thermostat  100 , equipment controller  406 , and/or HVAC equipment  308  to control the temperature of building zone  310 . 
     Referring specifically to  FIG.  12   , process  1200  is shown to include operating a model predictive controller in system identification mode to identify parameters of a thermal mass storage model and a HVAC load model for a building zone (step  1202 ). In some embodiments, step  1202  is performed by system identifier  510 . Step  1202  can include generating a thermal mass storage model by modeling the heat transfer characteristics of the building zone using a thermal circuit  800 , as described with reference to  FIG.  8   . Step  1202  may also include generating a HVAC load model by modeling the amount of heating/cooling {dot over (Q)} HVAC  provided by HVAC equipment  308  as a function of the temperature setpoint T sp  and the zone air temperature T ia . The thermal mass storage model and the HVAC load model can be used to generate a system of differential equations that define the relationship between zone air temperature T ia , zone mass temperature T m , outdoor air temperature T oa , the amount of heating or cooling {dot over (Q)} HVAC  provided by HVAC equipment  308 , the heat load disturbance {dot over (Q)} other , and the temperature setpoint T sp . 
     The unknown parameters in the thermal mass storage model and the HVAC load model (e.g., R oi , R mi , C m , C ia ) can be organized into parameter matrices A, B, C, and D, as shown in the following equations: 
               [             T   .     ia                 T   .     m               I   .           ]     =       [             1     C   ia       ⁢     (       K     p   ,   j       -     1     R   mi       -     1     R   oi         )             1       C   ia     ⁢     R   mi               -         K     p   ,   j       ⁢     K     I   ,   j           C   ia                   1       C   m     ⁢     R   mi               -     1       C   m     ⁢     R   mi               0             -   1         0       0         ]     ⁢               [           T   ia               T   m             I         ]     +       [           -       K     p   ,   j         C   ia               1       C   ia     ⁢     R   oi                 0       0           1       0         ]     ⁡     [           T     sp   ,   j                 T   oa           ]       +       [           1     C   ia               0           0         ]     ⁢         Q   .     other     ⁢     
     ⁢           [           T   ia                 Q   .       HVAC   ,   j             ]         =         [         1       0       0             -     K     p   ,   j             0         K     I   ,   j             ]     ⁡     [           T   ia               T   m             I         ]       +       [         0       0             K     p   ,   j           0         ]     ⁡     [           T     sp   ,   j                 T   oa           ]                     
or more compactly:
 
 {dot over (x)}=A   c (θ) x+B   c (θ) u+B   d   d  
 
 y=C   c (θ) x+D   c (θ) u  
 
where x T =[T ia ,T m ,I], u T =[T sp,j , T oa ], d={dot over (Q)} other /C ia , y T =[T ia ,{dot over (Q)} HVAC,j ], θ is a parameter vector containing all non-zero entries of the system matrices, and
 
     
       
         
           
             
               
                 
                   A 
                   c 
                 
                 ⁡ 
                 
                   ( 
                   θ 
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         - 
                         
                           
                             θ 
                             5 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 θ 
                                 1 
                               
                               + 
                               
                                 θ 
                                 2 
                               
                               + 
                               
                                 θ 
                                 4 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           θ 
                           5 
                         
                         ⁢ 
                         
                           θ 
                           2 
                         
                       
                     
                     
                       
                         
                           θ 
                           3 
                         
                         ⁢ 
                         
                           θ 
                           4 
                         
                         ⁢ 
                         
                           θ 
                           5 
                         
                       
                     
                   
                   
                     
                       
                         
                           θ 
                           6 
                         
                         ⁢ 
                         
                           θ 
                           2 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             θ 
                             6 
                           
                         
                         ⁢ 
                         
                           θ 
                           2 
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       
                         - 
                         1 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   B 
                   c 
                 
                 ⁡ 
                 
                   ( 
                   θ 
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           θ 
                           1 
                         
                         ⁢ 
                         
                           θ 
                           5 
                         
                       
                     
                     
                       
                         
                           θ 
                           4 
                         
                         ⁢ 
                         
                           θ 
                           5 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 B 
                 d 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         θ 
                         5 
                       
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   C 
                   c 
                 
                 ⁡ 
                 
                   ( 
                   θ 
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         - 
                         
                           θ 
                           4 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         
                           θ 
                           3 
                         
                         ⁢ 
                         
                           θ 
                           4 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   C 
                   d 
                 
                 ⁡ 
                 
                   ( 
                   θ 
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   D 
                   c 
                 
                 ⁡ 
                 
                   ( 
                   θ 
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         θ 
                         4 
                       
                     
                     
                       0 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     The unknown parameters θ in the thermal mass storage model and the HVAC load model (i.e., the parameters in matrices A, B, C, and D) can be identified by fitting the parameters θ to a set of input-output data (e.g., sets of values T ia , T oa , T sp , and {dot over (Q)} HVAC ) collected from the HVAC system when operating in the system identification mode. In some embodiments, the input-output data are collected by modulating the temperature setpoint T sp  and observing the corresponding values of T ia , T oa , and {dot over (Q)} HVAC  for a plurality of time steps during the system identification mode. The process of fitting the model parameters θ to the set of input-output data can be accomplished by performing process  900 , as described with reference to  FIG.  9   . 
     If HVAC equipment  308  are staged, step  1202  can include filtering input-output data to remove the high frequency dither in the indoor air temperature T ia  and can compute a time-averaged version of the HVAC load {dot over (Q)} HVAC  from the discrete HVAC staging trajectory. The thermal model parameters and HVAC load model parameters θ can then be fit to the input-output data (or filtered input-output data). In some embodiments, step  1202  includes augmenting the resulting state-space model with another integrating disturbance model and estimating the Kalman filter gain for the resulting model. Using data obtained in a secondary experiment or under normal operation, step  1202  can include validating the model through the use of statistics that capture the multi-step prediction accuracy of the resulting model. 
     Still referring to  FIG.  12   , process  1200  is shown to include operating the model predictive controller in operational mode to determine temperature setpoints T sp  for the building zone (step  1204 ). In some embodiments, step  1204  is performed by predictive optimizer  522 , as described with reference to  FIG.  5   . To determine the temperature setpoints T sp , step  1204  can include optimizing an objective function (i.e., a cost function) that accounts for the cost of operating HVAC equipment  308  over the duration of the optimization period. The costs of operating HVAC equipment  308  can include, for example, the costs of resources consumed by HVAC equipment  308  during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of HVAC equipment  308 . The objective function can be optimized subject to constraints provided by the thermal mass storage model and the HVAC load model, as well as constraints on the zone temperature T ia . Step  1204  can be accomplished by performing process  1300 , described in greater detail with reference to  FIG.  13   . 
     In some embodiments, process  1200  includes automatically switching between the system identification mode and the operational mode based on a prediction error of the thermal mass storage model and/or the HVAC load model. For example, process  1200  can include comparing the value of the zone temperature {circumflex over (T)} ia (k|k−1) predicted by the thermal mass storage model to the actual measured value of the zone temperature T ia (k). If the difference between the predicted value T ia (k|k−1) and the actual value T ia (k) exceeds an error threshold, process  1200  may automatically return to step  1202  and repeat the system identification process. Similarly, process  1200  can include comparing the value of the HVAC equipment load {circumflex over ({dot over (Q)})} HVAC (k|k−1) predicted by the HVAC load model to the actual value of the HVAC load {dot over (Q)} HVAC (k). If the difference between the predicted value {circumflex over ({dot over (Q)})} HVAC  (k|k−1) and the actual value {dot over (Q)} HVAC (k) exceeds an error threshold, process  1200  may automatically return to step  1202  and repeat the system identification process. 
     Process  1200  is shown to include providing the optimal temperature setpoints T sp  to an equipment controller (step  1206 ) and using the equipment controller to operate HVAC equipment to drive the temperature of the building zone T ia  to the optimal temperature setpoints T sp  (step  1208 ). Step  1206  can include sending the optimal temperature setpoints T sp  from model predictive controller  302  to an equipment controller  406 . In some embodiments, both model predictive controller  302  and equipment controller  406  are components of a smart thermostat  100  (as shown in  FIG.  4   ). In other embodiments, model predictive controller  302  is separate from smart thermostat  100  and configured to send the optimal temperature setpoints to smart thermostat  100  via a communications network  304  (as shown in  FIG.  3   ). Smart thermostat  100  and/or equipment controller  406  can use the optimal temperature setpoints T sp  to generate control signals for HVAC equipment  308  which operate to drive the temperature of the building zone T ia  to the optimal temperature setpoints T sp . 
     Referring now to  FIG.  13   , a flowchart of a process  1300  for generating optimal temperature setpoints for a building zone is shown, according to some embodiments. In some embodiments, process  1300  is performed by model predictive controller  302 , as described with reference to  FIGS.  5 - 11   . Process  1300  can be performed to accomplish step  1204  of process  1200 . 
     Process  1300  is shown to include predicting a heat load disturbance {dot over (Q)} other  experienced by a building zone at each time step (k=0 . . . N−1) of an optimization period (step  1302 ). In some embodiments, step  1302  is performed by load/rate predictor  518  and state/disturbance predictor  520 . Step  1302  can include generating a disturbance estimate {circumflex over (d)}(k) for each time step k in the optimization period. In some embodiments, step  1302  includes estimating the disturbance state {circumflex over (d)}(k) at each time step k in the optimization period using the following state/disturbance model: 
               [               T   ^     ia     ⁡     (       k   +   1     ❘   k     )                     T   ^     m     ⁡     (       k   +   1     ❘   k     )                   I   ^     ⁡     (       k   +   1     ❘   k     )                   d   ^     ⁡     (       k   +   1     ❘   k     )             ]     =           A   avg     ⁡     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     T   ^     m     ⁡     (     k   ❘     k   -   1       )                   I   ^     ⁡     (     k   ❘     k   -   1       )                   d   ^     ⁡     (     k   ❘     k   -   1       )             ]       +       B   avg     ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]       +       [           K   x               K   d           ]     ⁢       (       [             T   ia     ⁡     (   k   )                     Q   .     HVAC     ⁡     (   k   )             ]     -     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     Q     ^   .       HVAC     ⁡     (     k   ❘     k   -   1       )             ]       )     ⁢     
     ⁢           [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     Q     ^   .       HVAC     ⁡     (     k   ❘     k   -   1       )             ]         =         C   avg     ⁡     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     T   ^     m     ⁡     (     k   ❘     k   -   1       )                   I   ^     ⁡     (     k   ❘     k   -   1       )                   d   ^     ⁡     (     k   ❘     k   -   1       )             ]       +     D   ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]                               ⁢         where   ⁢           ⁢     A   avg       =     [         A         B   d             0       I         ]       ,       B   avg     =     [         B           0         ]       ,       C   avg     =       [     C   ⁢           ⁢   0     ]     .               
Step  1302  can include recording a history of disturbance estimates {circumflex over (d)} hist :={{circumflex over (d)}(k|k−1)} k=1   N     hist    predicted by the state/disturbance model.
 
     Step  1302  can include using the history of disturbance estimates {circumflex over (d)} hist  to predict the value of {circumflex over ({dot over (Q)})} other (k) at each time step k. In some embodiments, the heat load disturbance {circumflex over ({dot over (Q)})} other (k) is a function of the time of day, the day type (e.g., weekend, weekday, holiday), the weather forecasts, building occupancy, and/or other factors which can be used by load/rate predictor  518  to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k). For example, step  1302  can include predicting the heat load disturbance using the equation:
 
 {circumflex over ({dot over (Q)})}   other ( k )= f   pred ( kΔ+t   0   ,{circumflex over (d)}   hist )
 
where Δ&gt;0 is the sample period and t 0  is an initial time. In some embodiments, step  1302  includes using weather forecasts from a weather service  526  and/or load history data from historical data  528  to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k) at each time step.
 
     In some embodiments, step  1302  includes using a deterministic plus stochastic model trained from historical load data to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k). Step  1302  can include using any of a variety of prediction methods to predict the heat load disturbance {circumflex over ({dot over (Q)})} other (k) (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). In some embodiments, step  1302  includes predicting the heat load disturbance {circumflex over ({dot over (Q)})} other (k) using the techniques described in U.S. patent application Ser. No. 14/717,593. 
     Still referring to  FIG.  13   , process  1300  is shown to include estimating initial system states {circumflex over (x)}(0) for a first time step (k=0) of the optimization period (step  1304 ). The initial system states {circumflex over (x)}(0) may include the estimated zone air temperature {circumflex over (T)} ia (0), the estimated zone mass temperature {circumflex over (T)} m  (0), and the estimated integrating disturbance Î(0) at the beginning of the optimization period. In some embodiments, step  1304  is performed by state/disturbance estimator  520 . The initial system states {circumflex over (x)}(0) can be estimated using same the state/disturbance estimation model used in step  1302 . For example, step  1304  can include estimating the system states {circumflex over (x)}(k) at each time step k using the following equation:
 
 {circumflex over (x)} (++1| k )= A{circumflex over (x)} ( k|k− 1)+ Bu ( k )+ K ( y ( k )− ŷ ( k|k− 1))
 
 ŷ ( k|k− 1)= C{circumflex over (x)} ( k|k− 1)+ Du ( k )
 
where {circumflex over (x)}(k+1|k) is the estimated/predicted state at time step k+1 given the measurement at time step k, ŷ(k|k−1) is the predicted output at time step k given the measurement at time step k−1, and K is the estimator gain.
 
     In some embodiments, the estimated state vector {circumflex over (x)}(k+1|k), the output vector ŷ(k|k−1), and the input vector u(k) are defined as follows: 
                   x   ^     ⁡     (       k   +   1     ❘   k     )       =     [               T   ^     ia     ⁡     (       k   +   1     ❘   k     )                     T   ^     m     ⁡     (       k   +   1     ❘   k     )                   I   ^     ⁡     (       k   +   1     ❘   k     )             ]       ,     
     ⁢         y   ^     ⁡     (     k   ❘     k   -   1       )       =     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     Q     ^   .       HVAC     ⁡     (     k   ❘     k   -   1       )             ]       ,     
     ⁢       u   ⁡     (   k   )       =     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]             
where {circumflex over (T)} ia  is an estimate of the zone air temperature T ia , {circumflex over (T)} m  is an estimate of the zone mass temperature T m , Î is an estimate of the integrating disturbance I, {circumflex over ({dot over (Q)})} HVAC  is an estimate of the heating or cooling load provided by HVAC equipment  308 , T sp  is the temperature setpoint, and T oa  is the outdoor air temperature. The estimated system states {circumflex over (x)}(0) at the first time step k=0 can be used as the initial system states.
 
     Still referring to  FIG.  13   , process  1300  is shown to include using a thermal mass storage model and a heat load model to predict the temperature of the building zone {circumflex over (T)} ia  and the HVAC equipment load {circumflex over ({dot over (Q)})} HVAC  at each time step as a function of the estimated system states {circumflex over (x)}, temperature setpoint trajectory T sp , and estimated heat load disturbance {circumflex over ({dot over (Q)})} other  (step  1306 ). The thermal mass storage model may be defined as follows: 
               [               T   ^     ia     ⁡     (     k   +   1     )                     T   ^     m     ⁡     (     k   +   1     )                   I   ^     ⁡     (     k   +   1     )             ]     =       A   ⁡     [               T   ^     ia     ⁡     (   k   )                     T   ^     m     ⁡     (   k   )                   I   ^     ⁡     (   k   )             ]       +     B   ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]       +       B   d     ⁢           Q   .     ^     other     ⁡     (   k   )                 
where {circumflex over (T)} ia (k+1) is the predicted temperature of the building zone at time step k+1, {circumflex over (T)} m (k+1) is the predicted temperature of the building mass at time step k+1, Î(k+1) is the predicted value of the integrating disturbance at time step k+1, {circumflex over (T)} ia (k) is the predicted temperature of the building zone at time step k, {circumflex over (T)} m (k) is the predicted temperature of the building mass at time step k, Î(k) is the predicted value of the integrating disturbance at time step k, T sp (k) is the temperature setpoint at time step k, T oa (k) is the outdoor air temperature (measured) at time step k, and {circumflex over ({dot over (Q)})} other (k) is the estimated heat load disturbance at time step k.
 
     The HVAC load model may be defined as follows: 
               [           Q   .     ^     HVAC     ⁡     (   k   )       ]     =       C   ⁡     [               T   ^     ia     ⁡     (   k   )                     T   ^     m     ⁡     (   k   )                   I   ^     ⁡     (   k   )             ]       +     D   ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]               
where {circumflex over ({dot over (Q)})} HVAC (k) is the predicted HVAC equipment load at time step k and {circumflex over (T)} ia (k), {circumflex over (T)} m (k), Î(k), T sp (k), and T oa (k) are the same as the corresponding variables in the thermal mass storage model.
 
     Process  1300  is shown to include optimizing an economic cost function subject to constraints on the predicted temperature of the building zone {circumflex over (T)} ia  to determine optimal temperature setpoints T sp  for the building zone at each time step of the optimization period (step  1308 ). An example of an objective function which can be optimized in step  1308  is shown in the following equation: 
               min       T   sp     ,       ϵ   ⁢             T     ,     δ   ⁢           ⁢     T   sp           ⁢       ∑     k   =   0       N   -   1       ⁢     (           r   elec     ⁡     (   k   )       ⁢         ℓ   ^     elec     ⁡     (   k   )         +       p     ϵ   T       ⁢       ϵ   T     ⁡     (   k   )         +       p     δ   ⁢           ⁢   T       ⁢   δ   ⁢           ⁢       T   sp     ⁡     (   k   )           )             
where r elec (k) is the price of electricity at time step k, {circumflex over (l)} elec (k) is the predicted electricity consumption of HVAC equipment  308  at time step k, ϵ T (k) is a number of degrees by which the zone temperature constraints are violated at time step k, p ϵ     T    is a zone air temperature penalty coefficient applied to ϵ T (k), δD sp (k) is a number of degrees by which the zone temperature setpoint T sp  changes between time step k−1 and time step k, and p δT     sp    is a temperature setpoint change penalty coefficient applied to δD sp (k). Equation 24 accounts for the cost of electricity consumption (r elec (k){circumflex over (l)} elec (k)), penalizes violations of the indoor air temperature bounds (p ϵ     T   ϵ T (k)), and penalizes changes in the temperature setpoint (p δT δD sp (k)).
 
     If additional resources (other than electricity) are consumed by HVAC equipment  308 , additional terms can be added to the objective function to represent the cost of each resource consumed by HVAC equipment  308 . For example, if HVAC equipment  308  consume natural gas and water in addition to electricity, the objective function can be updated to include the terms r gas (k){circumflex over (l)} gas (k) and r water (k){circumflex over (l)} water (k) in the summation, where r gas (k) is the cost per unit of natural gas at time step k, {circumflex over (l)} gas (k) is the predicted natural gas consumption of HVAC equipment  308  at time step k, r water (k) is the cost per unit of water at time step k, and {circumflex over (l)} water (k) is the predicted water consumption of HVAC equipment  308  at time step k. 
     In some embodiments, step  1308  includes estimating the amount of each resource consumed by HVAC equipment  308  (e.g., electricity, natural gas, water, etc.) as a function of the sensible heating or cooling load {circumflex over ({dot over (Q)})} HVAC . In some embodiments, a constant efficiency model is used to compute the resource consumption of HVAC equipment  308  as a fixed multiple heating or cooling load {circumflex over ({dot over (Q)})} HVAC  (e.g., {circumflex over (l)} elec (k)=η{circumflex over ({dot over (Q)})} HVAC (k)). In other embodiments, equipment models describing a relationship between resource consumption and heating/cooling production can be used to approximate the resource consumption of the HVAC equipment  308 . The equipment models may account for variations in equipment efficiency as a function of load and/or other variables such as outdoor weather conditions (e.g., {circumflex over (l)} elec (k)=f({circumflex over ({dot over (Q)})} HVAC (k),T oa (k), η base )). Step  1308  can include converting the estimated heating/cooling load {circumflex over ({dot over (Q)})} HVAC (k) at each time step to one or more resource consumption values (e.g., {circumflex over (l)} elec (k), {circumflex over (l)} gas (k), {circumflex over (l)} water (k), etc.) for inclusion in the objective function. 
     Step  1308  can include optimizing the objective function to determine optimal values of the temperature setpoint T sp (k) for each time step in the optimization period. The objective function can be optimized subject to a set of constraints (e.g., Equations 25-34). The constraints may include the thermal mass storage model and the HVAC load model, which define the relationship between {circumflex over ({dot over (Q)})} HVAC , the temperature setpoints T sp , and the zone air temperature T ia  at each time step. The constraints may also include constraints on the zone air temperature T ia (k) and constraints that define the penalty terms in the objective function. 
     In some embodiments, step  1308  includes performing the optimization at the beginning of each time step k to determine optimal temperature setpoints T sp (k) for the next N time steps. Once the optimal temperature setpoints T sp  have been determined, the optimal temperature setpoint T sp (k) for the current time step k can be provided to smart thermostat  100  and/or equipment controller  406  and the optimization period can be shifted forward in time by one time step. The temperature setpoint T sp (k) for the current time step k along with feedback received from HVAC equipment  308  and building zone  310  during time step k (e.g., measurements of T ia (k), T oa (k), {dot over (Q)} HVAC (k), etc.) can be used to update the state/disturbance estimates the next time the optimization is performed. Steps  1304 - 1308  can be repeated at the beginning of each time step to determine the optimal setpoint trajectory T sp  for the new (i.e., time shifted) optimization period. 
     Referring now to  FIG.  14   , a flowchart of a process  1400  for controlling the temperature of a building zone using model predictive control is shown, according to some embodiments. Process  1400  can be performed by one or more components of model predictive control systems  300 - 400 , as described with reference to  FIGS.  3 - 11   . For example, process  1400  can be performed by model predictive controller  302 , smart thermostat  100 , equipment controller  406 , and/or HVAC equipment  308  to control the temperature of building zone  310 . 
     Process  1400  is shown to include identifying parameters of a thermal mass storage model and a HVAC load model for a building zone (step  1402 ). In some embodiments, step  1202  is performed by system identifier  510 . Step  1402  can include generating a thermal mass storage model by modeling the heat transfer characteristics of the building zone using a thermal circuit  800 , as described with reference to  FIG.  8   . Step  1402  may also include generating a HVAC load model by modeling the amount of heating/cooling {dot over (Q)} HVAC  provided by HVAC equipment  308  as a function of the temperature setpoint T sp  and the zone air temperature T ia . The thermal mass storage model and the HVAC load model can be used to generate a system of differential equations that define the relationship between zone air temperature T ia , zone mass temperature T m , outdoor air temperature T oa , the amount of heating or cooling {dot over (Q)} HVAC  provided by HVAC equipment  308 , the heat load disturbance {dot over (Q)} other , and the temperature setpoint T sp . 
     The unknown parameters in the thermal mass storage model and the HVAC load model (e.g., R oi , R mi , C m , C ia ) can be organized into parameter matrices A, B, C, and D, as shown in the following equations: 
               [             T   .     ia                 T   .     m               I   .           ]     =       [             1     C   ia       ⁢     (       K     p   ,   j       -     1     R   mi       -     1     R   oi         )             1       C   ia     ⁢     R   mi               -         K     p   ,   j       ⁢     K     I   ,   j           C   ia                   1       C   m     ⁢     R   mi               -     1       C   m     ⁢     R   mi               0             -   1         0       0         ]     ⁢               [           T   ia               T   m             I         ]     +       [           -       K     p   ,   j         C   ia               1       C   ia     ⁢     R   oi                 0       0           1       0         ]     ⁡     [           T     sp   ,   j                 T   oa           ]       +       [           1     C   ia               0           0         ]     ⁢         Q   .     other     ⁢     
     ⁢           [           T   ia                 Q   .       HVAC   ,   j             ]         =         [         1       0       0             -     K     p   ,   j             0         K     I   ,   j             ]     ⁡     [           T   ia               T   m             I         ]       +       [         0       0             K     p   ,   j           0         ]     ⁡     [           T     sp   ,   j                 T   oa           ]                     
or more compactly:
 
 {dot over (x)}=A   c (θ) x+B   c (θ) u+B   d   d  
 
 y=C   c (θ) x+D   c (θ) u  
 
where x T =[T ia ,T m ,I], u T =[T sp,j ,T oa ], d={dot over (Q)} other /C ia , y T =[T ia ,{dot over (Q)} HVAC,j ], θ is a parameter vector containing all non-zero entries of the system matrices, and
 
     
       
         
           
             
               
                 
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     The unknown parameters θ in the thermal mass storage model and the HVAC load model (i.e., the parameters in matrices A, B, C, and D) can be identified by fitting the parameters θ to a set of input-output data (e.g., sets of values T ia , T oa ,T sp , and {dot over (Q)} HVAC ) collected from the HVAC system when operating in the system identification mode. In some embodiments, the input-output data are collected by modulating the temperature setpoint T sp  and observing the corresponding values of T ia , T oa , and {dot over (Q)} HVAC  for a plurality of time steps during the system identification mode. The process of fitting the model parameters θ to the set of input-output data can be accomplished by performing process  900 , as described with reference to  FIG.  9   . 
     If HVAC equipment  308  are staged, step  1402  can include filtering input-output data to remove the high frequency dither in the indoor air temperature T ia  and can compute a time-averaged version of the HVAC load {dot over (Q)} HVAC  from the discrete HVAC staging trajectory. The thermal model parameters and HVAC load model parameters θ can then be fit to the input-output data (or filtered input-output data). In some embodiments, step  1402  includes augmenting the resulting state-space model with another integrating disturbance model and estimating the Kalman filter gain for the resulting model. Using data obtained in a secondary experiment or under normal operation, step  1402  can include validating the model through the use of statistics that capture the multi-step prediction accuracy of the resulting model. 
     Still referring to  FIG.  14   , process  1400  is shown to include predicting a heat load disturbance {dot over (Q)} other  experienced by a building zone at each time step (k=0 . . . N−1) of an optimization period (step  1404 ). In some embodiments, step  1404  is performed by load/rate predictor  518  and state/disturbance predictor  520 . Step  1404  can include generating a disturbance estimate {circumflex over (d)}(k) for each time step k in the optimization period. In some embodiments, step  1404  includes estimating the disturbance state {circumflex over (d)}(k) at each time step k in the optimization period using the following state/disturbance model: 
               [               T   ^     ia     ⁡     (       k   +   1     ❘   k     )                     T   ^     m     ⁡     (       k   +   1     ❘   k     )                   I   ^     ⁡     (       k   +   1     ❘   k     )                   d   ^     ⁡     (       k   +   1     ❘   k     )             ]     =           A   avg     ⁡     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     T   ^     m     ⁡     (     k   ❘     k   -   1       )                   I   ^     ⁡     (     k   ❘     k   -   1       )                   d   ^     ⁡     (     k   ❘     k   -   1       )             ]       +       B   avg     ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]       +       [           K   x               K   d           ]     ⁢       (       [             T   ia     ⁡     (   k   )                     Q   .     HVAC     ⁡     (   k   )             ]     -     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     Q     ^   .       HVAC     ⁡     (     k   ❘     k   -   1       )             ]       )     ⁢     
     ⁢           [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     Q     ^   .       HVAC     ⁡     (     k   ❘     k   -   1       )             ]         =         C   avg     ⁡     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     T   ^     m     ⁡     (     k   ❘     k   -   1       )                   I   ^     ⁡     (     k   ❘     k   -   1       )                   d   ^     ⁡     (     k   ❘     k   -   1       )             ]       +     D   ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]                               ⁢         where   ⁢           ⁢     A   avg       =     [         A         B   d             0       I         ]       ,       B   avg     =     [         B           0         ]       ,       C   avg     =       [     C   ⁢           ⁢   0     ]     .               
Step  1404  can include recording a history of disturbance estimates {circumflex over (d)} hist :={{circumflex over (d)}(k|k−1)} k=1   N     hist    predicted by the state/disturbance model.
 
     Step  1404  can include using the history of disturbance estimates {circumflex over (d)} hist  to predict the value of {dot over ({circumflex over (Q)})}other (k) at each time step k. In some embodiments, the heat load disturbance {dot over ({circumflex over (Q)})} other (k) is a function of the time of day, the day type (e.g., weekend, weekday, holiday), the weather forecasts, building occupancy, and/or other factors which can be used by load/rate predictor  518  to predict the heat load disturbance {dot over ({circumflex over (Q)})} other (k). For example, step  1404  can include predicting the heat load disturbance using the equation:
 
{dot over ( {circumflex over (Q)} )} other ( k )= f   pred ( kΔ+t   0   ,{circumflex over (d)}   hist )
 
where Δ&gt;0 is the sample period and t 0  is an initial time. In some embodiments, step  1404  includes using weather forecasts from a weather service  526  and/or load history data from historical data  528  to predict the heat load disturbance {dot over ({circumflex over (Q)})} other  (k) at each time step.
 
     In some embodiments, step  1404  includes using a deterministic plus stochastic model trained from historical load data to predict the heat load disturbance {dot over ({circumflex over (Q)})} other  (k). Step  1404  can include using any of a variety of prediction methods to predict the heat load disturbance {dot over ({circumflex over (Q)})} other (k) (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). In some embodiments, step  1404  includes predicting the heat load disturbance {dot over ({circumflex over (Q)})} other (k) using the techniques described in U.S. patent application Ser. No. 14/717,593. 
     Still referring to  FIG.  14   , process  1400  is shown to include generating an economic cost function that accounts for a cost of resources consumed by HVAC equipment at each time step (k=0 . . . N−1) of an optimization period (step  1406 ). An example of an objective function which can be generated in step  1406  is shown in the following equation: 
               min       T   sp     ,       ϵ   ⁢             T     ,     δ   ⁢           ⁢     T   sp           ⁢       ∑     k   =   0       N   -   1       ⁢     (           r   elec     ⁡     (   k   )       ⁢         ℓ   ^     elec     ⁡     (   k   )         +       p     ϵ   T       ⁢       ϵ   T     ⁡     (   k   )         +       p     δ   ⁢           ⁢   T       ⁢   δ   ⁢           ⁢       T   sp     ⁡     (   k   )           )             
where r elec (k) is the price of electricity at time step k, {circumflex over (l)} elec (k) is the predicted electricity consumption of HVAC equipment  308  at time step k, ϵ T (k) is a number of degrees by which the zone temperature constraints are violated at time step k, p ϵ     T    is a zone air temperature penalty coefficient applied to ϵ T (k), δT sp (k) is a number of degrees by which the zone temperature setpoint T sp  changes between time step k−1 and time step k, and p δT     sp    is a temperature setpoint change penalty coefficient applied to δT sp (k). Equation 24 accounts for the cost of electricity consumption (r elec (k){circumflex over (l)} elec (k)), penalizes violations of the indoor air temperature bounds (p ϵ     T   ϵ T (k)), and penalizes changes in the temperature setpoint (p δT δT sp (k)).
 
     If additional resources (other than electricity) are consumed by HVAC equipment  308 , additional terms can be added to the objective function to represent the cost of each resource consumed by HVAC equipment  308 . For example, if HVAC equipment  308  consume natural gas and water in addition to electricity, the objective function can be updated to include the terms r gas (k){circumflex over (l)} gas (k) and r water (k){circumflex over (l)} water (k) in the summation, where r gas (k) is the cost per unit of natural gas at time step k, {circumflex over (l)} gas (k) is the predicted natural gas consumption of HVAC equipment  308  at time step k, r water (k) is the cost per unit of water at time step k, and {circumflex over (l)} water (k) is the predicted water consumption of HVAC equipment  308  at time step k. 
     In some embodiments, step  1406  includes estimating the amount of each resource consumed by HVAC equipment  308  (e.g., electricity, natural gas, water, etc.) as a function of the sensible heating or cooling load {dot over ({circumflex over (Q)})} HVAC . In some embodiments, a constant efficiency model is used to compute the resource consumption of HVAC equipment  308  as a fixed multiple heating or cooling load {dot over ({circumflex over (Q)})} HVAC  (e.g., {circumflex over (l)} elec (k)=η{dot over ({circumflex over (Q)})} HVAC (k)) In other embodiments, equipment models describing a relationship between resource consumption and heating/cooling production can be used to approximate the resource consumption of the HVAC equipment  308 . The equipment models may account for variations in equipment efficiency as a function of load and/or other variables such as outdoor weather conditions (e.g., {circumflex over (l)} elec (k)=f({dot over ({circumflex over (Q)})} HVAC (k),T oa (k),η base )). Step  1406  can include converting the estimated heating/cooling load {dot over ({circumflex over (Q)})} HVAC (k) at each time step to one or more resource consumption values (e.g., {circumflex over (l)} elec (k), {circumflex over (l)} gas (k), {circumflex over (l)} water (k), etc.) for inclusion in the objective function. 
     Still referring to  FIG.  14   , process  1400  is shown to include estimating initial system states {circumflex over (x)}(0) for a first time step (k=0) of the optimization period (step  1408 ). The initial system states {circumflex over (x)}(0) may include the estimated zone air temperature {circumflex over (T)} ia (0), the estimated zone mass temperature {circumflex over (T)} m  (0), and the estimated integrating disturbance Î(0) at the beginning of the optimization period. In some embodiments, step  1408  is performed by state/disturbance estimator  520 . The initial system states {circumflex over (x)}(0) can be estimated using same the state/disturbance estimation model used in step  1404 . For example, step  1408  can include estimating the system states {circumflex over (x)}(k) at each time step k using the following equation:
 
 {circumflex over (x)} ( k+ 1| k )= A{circumflex over (x)} ( k|k− 1)+ Bu ( k )+ K ( y ( k )− ŷ ( k|k− 1))
 
 ŷ ( k|k− 1)= C{circumflex over (x)} ( k|k− 1)+ Du ( k )
 
where {circumflex over (x)}(k+1|k) is the estimated/predicted state at time step k+1 given the measurement at time step k, ŷ(k|k−1) is the predicted output at time step k given the measurement at time step k−1, and K is the estimator gain.
 
     In some embodiments, the estimated state vector {circumflex over (x)}(k+1|k), the output vector ŷ(k|k−1), and the input vector u(k) are defined as follows: 
                   x   ^     ⁡     (       k   +   1     ❘   k     )       =     [               T   ^     ia     ⁡     (       k   +   1     ❘   k     )                     T   ^     m     ⁡     (       k   +   1     ❘   k     )                   I   ^     ⁡     (       k   +   1     ❘   k     )             ]       ,     
     ⁢         y   ^     ⁡     (     k   ❘     k   -   1       )       =     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     Q     ^   .       HVAC     ⁡     (     k   ❘     k   -   1       )             ]       ,     
     ⁢       u   ⁡     (   k   )       =     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]             
where {circumflex over (T)} ia  is an estimate of the zone air temperature T ia , {circumflex over (T)} m  is an estimate of the zone mass temperature T m , Î is an estimate of the integrating disturbance I, {dot over ({circumflex over (Q)})} HVAC  is an estimate of the heating or cooling load provided by HVAC equipment  308 , T sp  is the temperature setpoint, and T oa  is the outdoor air temperature. The estimated system states {circumflex over (x)}(0) at the first time step k=0 can be used as the initial system states.
 
     Still referring to  FIG.  14   , process  1400  is shown to include using a thermal mass storage predictive model to constrain system states {circumflex over (x)}(k+1|k) at time step k+1 to the system states {circumflex over (x)}(k) and the temperature setpoint T sp (k) at time step k (step  1410 ). An example of a constraint which can be generated based on the thermal mass storage predictive model is: 
               [               T   ^     ia     ⁡     (     k   +   1     )                     T   ^     m     ⁡     (     k   +   1     )                   I   ^     ⁡     (     k   +   1     )             ]     =       A   ⁡     [               T   ^     ia     ⁡     (   k   )                     T   ^     m     ⁡     (   k   )                   I   ^     ⁡     (   k   )             ]       +     B   ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]       +       B   d     ⁢           Q   .     ^     other     ⁡     (   k   )                 
where {circumflex over (T)} ia (k+1) is the predicted temperature of the building zone at time step k+1, {circumflex over (T)} m (k+1) is the predicted temperature of the building mass at time step k+1, Î(k+1) is the predicted value of the integrating disturbance at time step k+1, {circumflex over (T)} ia (k) is the predicted temperature of the building zone at time step k, T oa (k) is the predicted temperature of the building mass at time step k, Î(k) is the predicted value of the integrating disturbance at time step k, T sp (k) is the temperature setpoint at time step k, T oa (k) is the outdoor air temperature (measured) at time step k, and {dot over ({circumflex over (Q)})} other (k) is the estimated heat load disturbance at time step k.
 
     Process  1400  is shown to include using a HVAC load predictive model to constrain the HVAC equipment load {dot over ({circumflex over (Q)})} HVAC (k) at time step k to systems states {circumflex over (x)}(k) and the temperature setpoint T sp (k) at time step k (step  1412 ). An example of a constraint which can be generated based on the HVAC load model is: 
               [           Q   .     ^     HVAC     ⁡     (   k   )       ]     =       C   ⁡     [               T   ^     ia     ⁡     (   k   )                     T   ^     m     ⁡     (   k   )                   I   ^     ⁡     (   k   )             ]       +     D   ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]               
where {dot over ({circumflex over (Q)})} HVAC (k) is the predicted HVAC equipment load at time step k and {circumflex over (T)} ia (k), {circumflex over (T)} m (k), Î(k), T sp (k), and T oa (k) are the same as the corresponding variables in the thermal mass storage model.
 
     Still referring to  FIG.  14   , process  1400  is shown to include optimizing the economic cost function subject to constraints defined by the predictive models and constraints on the predicted zone temperature {circumflex over (T)} ia (k) to determine optimal temperature setpoints T sp  for the building zone for each time step (k=0 . . . N−1) of the optimization period (step  1414 ). The objective function can be optimized subject to a set of constraints (e.g., Equations 25-34). The constraints may include the thermal mass storage model and the HVAC load model, which define the relationship between {dot over ({circumflex over (Q)})} HVAC , the temperature setpoints T sp , and the zone air temperature T ia  at each time step. The constraints may also include constraints on the zone air temperature T ia (k) and constraints that define the penalty terms in the objective function. The optimization performed in step  1414  may generate a temperature setpoint T sp (k) for each time step in the optimization period. 
     Process  1400  is shown to include providing the temperature setpoint T sp  for the first time step (k=0) of the optimization period to an equipment controller (step  1416 ). Step  1416  can include sending the optimal temperature setpoints T sp  from model predictive controller  302  to an equipment controller  406 . In some embodiments, both model predictive controller  302  and equipment controller  406  are components of a smart thermostat  100  (as shown in  FIG.  4   ). In other embodiments, model predictive controller  302  is separate from smart thermostat  100  and configured to send the optimal temperature setpoints to smart thermostat  100  via a communications network  304  (as shown in  FIG.  3   ). In some embodiments, only the temperature setpoint for the first time step (i.e., T sp  (0)) is provided to the equipment controller in step  1416 . The remaining temperature setpoints are for future time steps and may be updated the next time the optimization is performed (e.g., at the beginning of the next time step). 
     In some embodiments, step  1416  includes providing the entire set of temperature setpoints T sp  (i.e., a temperature setpoint T sp (k) for each time step k) to the equipment controller. In the event that communication between model predictive controller  302  and equipment controller  406  is lost (e.g., network connectivity is disrupted), equipment controller  406  can use the set of temperature setpoints provided in step  1416  until communications between model predictive controller  302  and equipment controller  406  are restored. For example, equipment controller  406  can use each temperature setpoint T sp (k) received in step  1416  to control HVAC equipment  308  during the corresponding time step k until communications between model predictive controller  302  and equipment controller  406  are restored. 
     Still referring to  FIG.  14   , process  1400  is shown to include using feedback collected from the HVAC system during the first time step (k=0) to update the thermal mass storage predictive model (step  1418 ). The feedback collected from the HVAC system may include measurements of the zone air temperature T ia  and/or measurements of the HVAC equipment load {dot over (Q)} HVAC . Step  1418  may include calculating the difference between the value of the zone air temperature {circumflex over (T)} ia (k|k−1) predicted in step  1408  and the actual measured value of the zone air temperature T ia (k) collected in step  1418 . Similarly, step  1418  may include calculating the difference between the value of the HVAC equipment load {dot over ({circumflex over (Q)})} HVAC (k|k−1) predicted in step  1408  and the actual measured value of the HVAC equipment load {dot over (Q)} HVAC (k) collected in step  1418 . 
     The updating performed in step  1418  may include multiplying the Kalman gain matrix K generated in step  1402  by the vector of differences, as shown in the following equation: 
               [               T   ^     ia     ⁡     (       k   +   1     ❘   k     )                     T   ^     m     ⁡     (       k   +   1     ❘   k     )                   I   ^     ⁡     (       k   +   1     ❘   k     )                   d   ^     ⁡     (       k   +   1     ❘   k     )             ]     =         A   avg     ⁡     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     T   ^     m     ⁡     (     k   ❘     k   -   1       )                   I   ^     ⁡     (     k   ❘     k   -   1       )                   d   ^     ⁡     (     k   ❘     k   -   1       )             ]       +       B   avg     ⁡     [             T   sp     ⁡     (   k   )                   T   oa     ⁡     (   k   )             ]       +       [           K   x               K   d           ]     ⁢     (       [             T   ia     ⁡     (   k   )                     Q   .     HVAC     ⁡     (   k   )             ]     -     [               T   ^     ia     ⁡     (     k   ❘     k   -   1       )                     Q     ^   .       HVAC     ⁡     (     k   ❘     k   -   1       )             ]       )               
In some embodiments, the updating performed in step  1418  is equivalent to the updating step of a Kalman filter in which the predicted states {circumflex over (x)}(k+1|k) are calculated based on the error between the predicted system output ŷ(k|k−1) at time step k and the actual/measured values of the predicted output variables y(k).
 
     Process  1400  is shown to include shifting the optimization forward in time by one time step (step  1420 ) and returning to step  1408 . Steps  1408 - 1420  can be repeated at the beginning of each time step to generate a set of temperature setpoints T sp  for each time step in the shifted optimization period. The first temperature setpoint T sp (0) generated for the shifted optimization period can be provided to the equipment controller and used to control the HVAC equipment. 
     Performance Graphs 
     Referring now to  FIGS.  15 - 18   , several graphs  1500 - 1800  illustrating the performance of model predictive control (MPC) systems  300 - 400  are shown, according to some embodiments. Graph  1500  compares the energy consumption of MPC systems  300 - 400  to the energy consumption of a baseline temperature control system without MPC. Graph  1600  indicates the cost of electricity (i.e., $/kWh) which varies over time. Line  1502  represents the energy consumption of the baseline system, whereas line  1504  represents the energy consumption of MPC systems  300 - 400 . MPC systems  300 - 400  use more energy than the baseline system during an off-peak period  1506  when energy prices are low. However, MPC systems  300 - 400  use less energy than the baseline system during a peak period  1508  when energy prices are high. Advantageously, this results in a cost savings relative to the baseline system. 
     Graph  1700  compares the building mass temperature trajectories of the baseline system and MPC systems  300 - 400 . Line  1702  represents the temperature T m  of the solid mass within building zone  310  when the baseline system is used to provide temperature control. Since the baseline system does not store energy in the building mass, the temperature T m  remains relatively constant over the duration of the optimization period. Line  1704  represents the temperature T m  of the solid mass within building zone  310  when MPC systems  300 - 400  are used to provide temperature control. During the off-peak period  1506 , building zone  310  is precooled, which results in a decrease in the temperature T m  of the building mass. During the peak period  1508 , thermal energy from the air within building zone  310  is moved into the building mass, which increases the temperature T m  of the building mass and provides cooling for the air within building zone  310 . 
     Graph  1800  compares the zone air temperature trajectories of the baseline system and MPC systems  300 - 400 . Line  1802  represents the temperature T ia  of the air within building zone  310  when the baseline system is used to provide temperature control. The temperature T ia  remains relatively constant over the duration of the optimization period. Line  1704  represents the zone air temperature T ia  within building zone  310  when MPC systems  300 - 400  are used to provide temperature control. During the off-peak period  1506 , building zone  310  is precooled to the minimum comfortable zone temperature T min  (e.g., 70° F.). The zone air temperature T ia  remains relatively constant during the off-peak period  1506 , but the temperature of the building mass T m  decreases as heat is removed from building zone  310 . During the peak period  1508 , the building zone temperature T ia  is allowed to increase to the maximum comfortable zone temperature T max  (e.g., 76° F.). Heat from the air within building zone  310  flows into the building mass, which provides cooling for the air within building zone  310  and increases the temperature of the building mass T m . 
     Configuration of Exemplary Embodiments 
     The construction and arrangement of the systems and methods as shown in the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.). For example, the position of elements can be reversed or otherwise varied and the nature or number of discrete elements or positions can be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps can be varied or re-sequenced according to alternative embodiments. Other substitutions, modifications, changes, and omissions can be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present disclosure. 
     The present disclosure contemplates methods, systems and program products on any machine-readable media for accomplishing various operations. The embodiments of the present disclosure can be implemented using existing computer processors, or by a special purpose computer processor for an appropriate system, incorporated for this or another purpose, or by a hardwired system. Embodiments within the scope of the present disclosure include program products comprising machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine-readable media can be any available media that can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a general purpose or special purpose computer or other machine with a processor. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions include, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing machines to perform a certain function or group of functions. 
     Although the figures show a specific order of method steps, the order of the steps may differ from what is depicted. Also two or more steps can be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various connection steps, processing steps, comparison steps and decision steps.