Extremum-seeking control system with saturation constraints
An extremum seeking controller includes a processing circuit configured to modulate a manipulated variable provided as an input to a plant using an extremum-seeking control technique to drive a gradient of an objective function with respect to the manipulated variable toward an extremum. The objective function includes a performance variable characterizing a performance of the plant responsive to the manipulated variable. The objective function also includes a saturation adjustment term that becomes active as the plant approaches a saturation point and remains active as the plant operates within a saturated region past the saturation point. The saturation adjustment term causes the processing circuit to adjust the manipulated variable toward a value of the manipulated variable that returns the plant from the saturated region to a non-saturated region.
1. An extremum-seeking controller comprising a processing circuit configured to:
modulate a manipulated variable provided as an input to a plant using an extremum- seeking control technique to drive a gradient of an objective function with respect to the manipulated variable toward an extremum;
wherein the objective function comprises:
a performance variable characterizing a performance of the plant responsive to the manipulated variable; and
a saturation adjustment term that becomes active as the plant approaches a saturation point and remains active as the plant operates within a saturated region past the saturation point, the saturation adjustment term causing the processing circuit to adjust the manipulated variable toward a value of the manipulated variable that returns the plant from the saturated region to a non-saturated region, wherein the performance variable remains substantially constant responsive to changes in the manipulated variable when the plant operates within the saturated region.
2. The controller of claim 1 , wherein the saturation adjustment term is based on both:
a value of a saturation variable affected by the manipulated variable and capable of becoming saturated at a subset of values of the manipulated variable; and
a value of the manipulated variable at which the gradient is calculated relative to the value of the manipulated variable that returns the plant from the saturated region to the non-saturated region.
3. The controller of claim 1 , wherein the saturation adjustment term comprises a barrier function that has a value of substantially zero when the plant operates within the non-saturated region and increases as the plant approaches the saturated region.
4. The controller of claim 1 , wherein the gradient of the objective function with respect to the manipulated variable comprises:
a first gradient of the performance variable with respect to the manipulated variable; and
a second gradient of a saturation variable with respect to the manipulated variable, the second gradient being substantially zero when the saturation variable is saturated.
5. The controller of claim 1 , wherein the gradient of the objective function with respect to the manipulated variable comprises a difference between (1) a value of the manipulated variable at which the gradient is calculated and (2) the value of the manipulated variable that returns the plant to the non-saturated region.
6. The controller of claim 1 , wherein the manipulated variable comprises a fan speed and the performance variable comprises a system power.
7. The controller of claim 1 , wherein the manipulated variable comprises a compressor suction or discharge pressure setpoint.
8. An extremum-seeking controller comprising a processing circuit configured to:
modulate a manipulated variable provided as an input to a plant using an extremum- seeking control technique to drive a gradient of an objective function with respect to the manipulated variable toward an extremum;
wherein the objective function comprises a performance variable characterizing a performance of the plant responsive to the manipulated variable and a saturation adjustment term based on both:
a value of a saturation variable capable of becoming saturated at a subset of values of the manipulated variable, wherein the performance variable remains substantially constant responsive to changes in the manipulated variable when the saturation variable becomes saturated; and
a value of the manipulated variable at which the gradient is calculated relative to a value of the manipulated variable at which the saturation variable is non-saturated.
9. The controller of claim 8 , wherein the saturation adjustment term comprises a barrier function that has a value of substantially zero when the saturation variable is non-saturated and increases as the saturation variable becomes saturated.
10. The controller of claim 8 , wherein the saturation adjustment term causes the processing circuit to adjust the manipulated variable toward the value of the manipulated variable at which the saturation variable is non-saturated.
11. The controller of claim 8 , wherein the gradient of the objective function with respect to the manipulated variable comprises:
a first gradient of the performance variable with respect to the manipulated variable; and
a second gradient of the saturation variable with respect to the manipulated variable, the second gradient being substantially zero when the saturation variable is saturated.
12. The controller of claim 8 , wherein the gradient of the objective function with respect to the manipulated variable comprises a difference between the value of the manipulated variable at which the gradient is calculated and the value of the manipulated variable at which the saturation variable is non-saturated.
13. The controller of claim 8 , wherein the manipulated variable comprises a fan speed and the performance variable comprises a system power.
14. A real-time optimization controller comprising a processing circuit configured to:
provide a value of manipulated variable as an input to a plant;
obtain (1) a value of a performance variable characterizing a performance of the plant responsive to the manipulated variable and (2) a value of a saturation variable affected by the manipulated variable and capable of becoming saturated at a subset of values of the manipulated variable, wherein the performance variable remains substantially constant responsive to changes in the manipulated variable when the saturation variable becomes saturated;
evaluate an objective function comprising a performance variable term and a saturation adjustment term, the performance variable term comprising the performance variable, and the saturation adjustment term comprising (1) a function of the saturation variable and (2) a function of the value of manipulated variable provided as the input to the plant relative to a value of the manipulated variable at which the saturation variable is non-saturated; and
adjust the value of the manipulated variable provided as the input to the plant using a real-time optimization technique to drive a gradient of the objective function with respect to the manipulated variable toward an extremum.
15. The controller of claim 14 , wherein the real-time optimization technique is an extremum-seeking control technique.
16. The controller of claim 14 , wherein the function of the saturation variable comprises a barrier function that has a value of substantially zero when the saturation variable is within a predefined range and increases based on an amount the saturation variable deviates from the predefined range.
17. The controller of claim 14 , wherein the function of the manipulated variable causes the processing circuit to adjust the manipulated variable toward the value of the manipulated variable at which the saturation variable is non-saturated.
18. The controller of claim 14 , wherein the gradient of the objective function with respect to the manipulated variable comprises:
a first gradient of the performance variable with respect to the manipulated variable; and
a second gradient of the saturation variable with respect to the manipulated variable, the second gradient being substantially zero when the saturation variable is saturated.
19. The controller of claim 14 , wherein the gradient of the objective function with respect to the manipulated variable comprises a difference between the value of the manipulated variable provided as the input to the plant and the value of the manipulated variable at which the saturation variable is non-saturated.
20. The controller of claim 14 , wherein the manipulated variable comprises a fan speed and the performance variable comprises a system power.