Newton-based extremum-seeking control system

1. A system for operating and assessing performance of building equipment, the system comprising:

a dither signal generator configured to:

perturb a setpoint for a controller by applying a dither signal to the setpoint, resulting in a perturbed setpoint, and

provide the perturbed setpoint to the controller, wherein the controller uses the perturbed setpoint to generate one or more control inputs for the building equipment, and the building equipment operate in accordance with the control inputs to affect a performance variable indicating a performance of the building equipment; and

a performance assessment system configured to:

receive, from the building equipment, an output signal and obtain values of the performance variable based on the output signal, the values of the performance variable resulting from operating the building equipment based on the perturbed setpoint;

determine, from the values of the performance variable, a gradient of the performance variable with respect to the perturbed setpoint and a Hessian of the performance variable affected by the building equipment with respect to the setpoint;

determine an adjustment to the setpoint predicted to drive the performance variable to an extremum based on the gradient of the performance variable and the Hessian of the performance variable.

2. The system of claim 1 , wherein the system is configured to determine the adjustment based at least on multiplying the gradient and an inverse of the Hessian of the performance variable.

3. The system of claim 1 , wherein the system determines the gradient and the Hessian of the performance variable by passively observing an effect of the perturbed setpoint on the output signal without additional perturbations to the control input after the dither signal is applied.

4. The system of claim 1 , wherein the performance assessment system is configured to apply the adjustment to the setpoint, the adjustment causing the controller to adjust an operation of the equipment to drive the performance variable to the extremum.

5. The system of claim 1 , wherein the system is configured to determine the gradient of the performance variable and the Hessian of the performance variable by applying a plurality of demodulation signals to the output signal,

wherein one demodulation signal of the demodulation signals is associated with a perturbation matrix that generates an estimate of the Hessian, and

wherein a second demodulation signal of the demodulation signals is a sinusoidal signal that generates the gradient.

6. The system of claim 1 , wherein the system is configured to determine the gradient of the performance variable and the Hessian of the performance variable by:

converting the values of the output signal to a vector to create a data sample vector;

generating a multivariable form of the data sample vector using a least squares technique; and

obtaining a gradient vector and the Hessian based on the multivariable form of the data sample vector.

7. The system of claim 6 , wherein the system is configured to determine the adjustment based on recursively implementing the converting, generating, and obtaining operations until the performance variable reaches the extremum.

8. The system of claim 1 , wherein the performance variable indicates a performance characteristic of the building equipment affected by the setpoint to the controller.

9. The system of claim 8 , wherein the performance characteristic is a total power consumption of the building equipment affected by the setpoint to the controller.

10. A method for operating and assessing performance of building equipment, the method comprising:

perturbing a setpoint for a controller by applying a dither signal to the setpoint, resulting in a perturbed setpoint, and

providing the perturbed setpoint to the controller, wherein the controller uses the perturbed setpoint to generate one or more control inputs for the building equipment, and the building equipment operate in accordance with the control inputs to affect a performance variable indicating a performance of the building equipment;

receiving, from the building equipment, an output signal and obtaining values of the performance variable based on the output signal, the values of the performance variable resulting from operating the building equipment based on the perturbed setpoint;

determining, from the values of the performance variable, a gradient of the performance variable with respect to the perturbed setpoint and a Hessian of the performance variable affected by the building equipment with respect to the setpoint;

determining an adjustment to the setpoint predicted to drive the performance variable to an extremum based on the gradient of the performance variable and the Hessian of the performance variable.

11. The method of claim 10 , wherein determining the adjustment is based at least on multiplying the gradient and an inverse of the Hessian of the performance variable.

12. The method of claim 10 , wherein determining the gradient and the Hessian of the performance variable comprises passively observing an effect of the perturbed setpoint on the output signal without additional perturbations to the control input after the dither signal is applied.

13. The method of claim 10 , comprising applying the adjustment to the setpoint, the adjustment causing the controller to adjust an operation of the equipment to drive the performance variable to the extremum.

14. The method of claim 10 , wherein determining the gradient of the performance variable and the Hessian of the performance variable comprises applying a plurality of demodulation signals to the output signal,

wherein one demodulation signal of the demodulation signals is associated with a perturbation matrix that generates an estimate of the Hessian, and

wherein a second demodulation signal of the demodulation signals is a sinusoidal signal that generates the gradient.

15. The method of claim 10 , wherein determining the gradient of the performance variable and the Hessian of the performance variable comprises:

converting the values of the output signal to a vector to create a data sample vector;

generating a multivariable form of the data sample vector using a least squares technique; and

obtaining a gradient vector and the Hessian based on the multivariable form of the data sample vector.

16. The method of claim 15 , wherein determining the adjustment is based on recursively implementing the converting, generating, and obtaining operations until the performance variable reaches the extremum.

17. A non-transitory computer-readable storage medium having instructions stored thereon that, upon execution by a processor, cause the processor to perform operations to operate and assess performance of building equipment, the operations comprising:

perturbing a setpoint for a controller by applying a dither signal to the setpoint, resulting in a perturbed setpoint;

providing the perturbed setpoint to the controller, wherein the controller uses the perturbed setpoint to generate one or more control inputs for the building equipment, and the building equipment operate in accordance with the control inputs to affect a performance variable indicating a performance of the building equipment;

receiving, from the building equipment, an output signal and obtaining values of the performance variable based on the output signal, the values of the performance variable resulting from operating the building equipment based on the perturbed setpoint;

determining, from the values of the performance variable, a gradient of the performance variable with respect to the perturbed setpoint and a Hessian of the performance variable affected by the building equipment with respect to the setpoint; and

determining an adjustment to the setpoint predicted to drive the performance variable to an extremum based on the gradient of the performance variable and the Hessian of the performance variable.

18. The non-transitory computer-readable storage medium of claim 17 , wherein determining the gradient of the performance variable and the Hessian of the performance variable comprises applying one or more demodulation signals to the output signal,

wherein one demodulation signal of the one or more demodulation signals is associated with a perturbation matrix that generates an estimate of a Hessian, and

wherein a second demodulation signal of the one or more demodulation signals is a sinusoidal signal that generates the gradient.

19. The non-transitory computer-readable storage medium of claim 17 , wherein determining the gradient of the performance variable and the Hessian of the performance variable comprises:

converting the values of the output signal to a vector to create a data sample vector;

generating a multivariable form of the data sample vector using a least squares technique; and

obtaining a gradient vector and a Hessian based on the multivariable form of the data sample vector.

20. The non-transitory computer-readable storage medium of claim 17 , wherein the operations comprise applying the adjustment to a setpoint input, the adjustment causing the controller to adjust an operation of the equipment to drive the performance variable to the extremum.