Control of processing parameters during substrate polishing using constrained cost function

1. A computer program product for controlling a polishing system, the computer program product residing on a non-transitory computer readable medium, the computer program product comprising instructions for causing one or more computers to:

receive from an in-situ monitoring system, for each region of a plurality of regions on a substrate being processed by the polishing system, a sequence of characterizing values for the region;

for each region, determine a polishing rate for the region; and

calculate an adjustment for at least one processing parameter, wherein calculation of the adjustment includes minimizing a cost function that includes, for each region, a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and wherein optimization of the cost function is subject to at least one constraint.

2. The computer program product of claim 1 , wherein optimization of the cost function is subject to at least one inter-zone constraint.

3. The computer program product of claim 2 , wherein the inter-zone constraint comprises a maximum difference in a processing parameter between zones.

4. The computer program product of claim 3 , wherein the inter-zone constraint comprises a maximum difference in a processing parameter between adjacent zones.

5. The computer program product of claim 4 , wherein the inter-zone constraints comprise a maximum difference in pressure between adjacent zones.

6. The computer program product of claim 2 , wherein the inter-zone constraint comprises a maximum difference of a pressure in a zone from an average pressure of multiple zones.

7. The computer program product of claim 1 , wherein optimization of the cost function is subject to a parameter constraint.

8. The computer program product of claim 7 , wherein the parameter constraint comprises a maximum or minimum parameter value.

9. The computer program product of claim 8 , wherein the parameter constraint comprises a maximum pressure.

10. The computer program product of claim 1 , wherein the at least one constraint comprises a linear inequality constraint.

11. The computer program product of claim 1 , wherein each characterizing value of the sequence of characterizing values is a thickness.

12. The computer program product of claim 1 , wherein the parameter is pressure of a chamber in a carrier head of the polishing system.

13. A polishing system, comprising:

a platen to support a polishing pad;

a carrier head to hold a substrate in contact with the polishing pad;

a motor to generate relative motion between the carrier head and the polishing pad;

an in-situ monitoring system to, for each region of a plurality of regions on the substrate being polished, generate a sequence of characterizing values for the region; and

a controller configured to

receive from the in-situ monitoring system, for each region of a plurality of regions on the substrate, the sequence of characterizing values for the region,

for each region, determine a polishing rate for the region, and

calculate an adjustment for at least one processing parameter, wherein calculation of the adjustment includes minimizing a cost function that includes, for each region, a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and wherein optimization of the cost function is subject to at least one constraint.

14. The system of claim 13 , wherein optimization of the cost function is subject to at least one inter-zone constraint.

15. The system of claim 14 , wherein the inter-zone constraint comprises a maximum difference in a processing parameter between zones.

16. The system of claim 14 , wherein the inter-zone constraint comprises a maximum difference of a pressure in a zone from an average pressure of multiple zones.

17. The system of claim 13 , wherein optimization of the cost function is subject to a parameter constraint.

18. The system of claim 17 , wherein the parameter constraint comprises a maximum or minimum parameter value.

19. A method for controlling a polishing system, the method comprising:

receiving from an in-situ monitoring system, for each region of a plurality of regions on a substrate being processed by the polishing system, a sequence of characterizing values for the region;

for each region, determining a polishing rate for the region; and

calculating an adjustment for at least one processing parameter, wherein calculating the adjustment includes minimizing a cost function that includes, for each region, a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and wherein optimization of the cost function is subject to at least one constraint.

20. A computer program product for controlling a semiconductor processing system, the computer program product residing on a non-transitory computer readable medium, the computer program product comprising instructions for causing one or more computers to:

receive from an in-situ monitoring system, for each region of a plurality of regions on a substrate being processed by a processing system, a sequence of characterizing values for the region;

for each region, determine a rate of change of the characterizing value for the region; and

calculate an adjustment for at least one processing parameter, wherein calculation of the adjustment includes minimizing a cost function that includes, for each region, a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and wherein optimization of the cost function is subject to at least one constraint.