Methods and systems for analyzing system operator coupling susceptibility

• US 7,272,473 B2
• Filed: 09/17/2004
• Issued: 09/18/2007
• Est. Priority Date: 09/17/2004
• Status: Expired due to Fees

• Current Assignee: The Boeing Co.
• Original Assignee: The Boeing Co.
• Inventors: Lee, Brian P., Hiltner, Dale W.
• Primary Examiner(s): Keith; Jack
• Assistant Examiner(s): Diacou; Ari M.
• Application Number:
  US10/944,128
• Publication Number:
  US 20060064207A1
• Time in Patent Office: 1,096 Days
• Field of Search: 701/3, 701/8, 701/14
• US Class Current: 701/9
• CPC Class Codes: G05D 1/0066 for limitation of acceleration or stress

Abstract

Methods for analyzing system operator coupling susceptibility are described herein. In one embodiment, an open loop frequency response can be compared to a related over-steer point. An over-steer point can be the condition where, as operator gain is increased for an open loop system, the corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain. In other embodiments, an open loop frequency response can be compared to a critical gain rate. The critical gain rate can be the rate of change of open loop gain per open loop phase angle of the corresponding open loop system response at an over-steer point that is associated with a zero decibel closed loop gain.

Claims

1. A method for evaluating the susceptibility of a system to operator coupling, the method comprising:

providing data relationships for multiple over-steer points, each over-steer point being the condition where, as operator gain is increased for a corresponding open loop system, the corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain, and wherein each of the data relationships include a corresponding open loop phase angle, open loop gain, and gain rate, and further wherein the gain rate is the rate of change of open loop gain per open loop phase angle of the corresponding open loop system response at the over-steer point;

providing a selected open loop frequency response; and

determining if the selected open loop frequency response has at least one open loop phase angle where a gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships at the corresponding open loop phase angle; and

if so

finding the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships;

at least one of storing the at least one open loop phase angle, displaying the at least one open loop phase angle, printing the at least one open loop phase angle, and using the at least open loop phase angle to evaluate the susceptibility of system to operator coupling.

2. The method of claim 1 wherein increasing operator gain includes increasing a magnitude of an operators input.

3. The method of claim 1 wherein:

providing data relationships for multiple over-steer points includes providing at least a portion of the data relationships for multiple over-steer points stored on at least one of a computing system and a computer readable medium;

determining if the selected open loop frequency response has at least one open loop phase angle where the gain rate of the selected opeii loop frequency response is at least approximately equal to the gain rate of one of the data relationships includes determining if the selected open loop frequency response has at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships using a computing system; and

finding the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships includes finding the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships.

4. The method of claim 1 wherein:

providing data relationships for multiple over-steer points includes providing at least a portion of the data relationships for multiple over-steer points in graphical form;

determining if the selected open loop frequency response has at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships includes plotting the selected open loop frequency response in graphical form and comparing the selected open loop frequency response to the data relationships; and

finding the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships includes comparing the selected open loop frequency response to the data relationships.

5. The method of claim 1 wherein the method further comprises comparing an open loop gain of the selected open loop frequency at the at least one open loop phase angle to the open loop gain of the data relationship at the at least one open loop phase angle.

6. The method of claim 1 wherein providing data relationships for multiple over-steer points includes providing data relationships that include a corresponding closed loop gain, and wherein the method further comprises:

determining if the closed loop gain of the data set corresponding to the at least one open loop phase angle is equal to zero decibels or less than zero decibels; and

if so

comparing an open loop gain of the selected open loop frequency at the at least one open loop phase angle to an open loop gain associated with the combination of the at least one open loop phase angle and a closed loop gain of zero decibels.

7. The method of claim 1 wherein the method further comprises computing an over-steer margin for the at least one of the open loop phase angles, the over-steer margin being the open loop gain of the data relationship at the at least one open loop phase angle minus the open loop gain of the selected open loop frequency at the at least one open loop phase angle.

8. The method of claim 1 wherein providing data reiationsnips for multiple over-steer points includes providing data relationships that include a corresponding closed loop gain, and wherein the method further comprises:

determining if the closed loop gain of the data set corresponding to the at least one open loop phase angle is equal to zero decibels or less than zero decibels; and

if so

computing an over-steer margin for the at least one of the open loop phase angles, the over-steer margin being an open loop gain associated with the combination of the at least one open loop phase angle and a closed loop gain of zero decibels minus the open loop gain of the selected open loop frequency at the at least one open loop phase angle.

9. The method of claim 1 wherein providing data relationships for multiple over-steer points includes providing data relationships that include a corresponding closed loop gain, and wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining if the gain rate of the open loop frequency response at the at least one open loop phase angle is equal to the critical gain rate or less than the critical gain rate; and

if so

comparing an open loop gain of the selected open loop frequency at the at least one open loop phase angle to an open loop gain associated with the combination of the at least one open loop phase angle and a closed loop gain of zero decibels.

10. The method of claim 1 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining if the selected open loop frequency response has at least one phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate; and

if so

finding the at least one open loop phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate; and

comparing the at least one open loop phase angle where the gain rate of the selected open loop frequency response is less than the crtical gain rate to the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships, if one exists.

11. The method of claim 1 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the selected open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

finding the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees;

comparing the open loop gain of the imaginary open loop frequency response and the selected open loop frequency response at −

180 degrees.

12. The method of claim 1 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the selected open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

finding the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

computing a PIO margin, the PIO margin being the open loop gain of the imaginary open loop frequency response minus the open loop gain of ihe selected open loop frequency response at −

180 degrees.

13. The method of claim 1 wherein the method further comprises:

computing the over-steer margin for at least one of the open loop phase angles where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships, the over-steer margin being the open loop gain of the data relationship at the at least one open loop phase angle minus the open loop gain of the selected open loop frequency at the at least one open loop phase angle;

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the selected open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

finding the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees;

computing a PIO margin, the PIO margin being the open loop gair of the imaginary open loop frequency response minus the open loop gain of the selected open loop frequency response at −

180 degrees; and

comparing the over-steer margin to the PIO margin.

14. The method of claim 1 wherein the method further comprises:

finding the open loop gain of the selected open loop frequency at the at least one open loop phase angle;

determining the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

comparing the open loop gain of the selected open loop frequency at the at least one open loop phase angle and the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees.

15. The method of claim 1 wherein the method further comprises:

finding the open loop gain of the selected open loop frequency at the at least one open loop phase angle;

determining the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

computing an over-steer to PIO margin, the over-steer to PIO imagine being the open loop gain of the selected open loop frequency at the at least one open loop phase angle minus the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees.

16. The method of claim 1 wherein the selected open loop frequency response includes an open loop frequency response of an aircraft.

17. The method of claim 1 wherein providing a selected open loop frequency response includes providing a first open loop frequency response, and wherein the method further includes:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the first open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

providing a second open loop frequency response; and

comparing the open loop gain of the imaginary open loop frequency response at a phase angle of −

180 degrees to the open loop gain of the second open loop frequency response at a phase angle of −

180 degrees.

18. A method for developing data to be used to evaluate the susceptibility of an operator coupling with an operator controlled system, the method comprising:

determining an over-steer point for each of multiple open loop systems having corresponding open loop frequency responses, each over-steer point being the condition where, as the operator gain is increased for each corresponding open loop system, the corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain; and

determining data relationships for each over-steer point, each data relationship including a corresponding open loop phase angle, an open loop gain, and a gain rate, the gain rate being the rate of change of open loop gain per open loop phase angle of the corresponding open loop system response;

storing the data relationships.

19. The method of claim 18 wherein:

determining an over-steer point for each of multiple of open loop frequency responses includes determining an over-steer point for each of multiple open loop frequency responses using a computing system; and

determining data relationships for each over-steer point includes determining data relationships for each over-steer point using a computing system.

20. The method of claim 18 wherein storing the data relationships includes storing at least a portion of the data relationships in at least one of a computing system and a computer readable medium.

21. The method of claim 18 wherein storing the data relationships includes storing at least a portion of the data relationships on paper in at least one of a tabular form and a graphical form.

22. A computer readable medium having conterts configured to carry out a method for evaluating the susceptibility of a system to operator coupling, the method comprising:

at least one of storing and computing data relationships for multiple over-steer points, each over-steer point being the condition where, as operator gain is increased for a corresponding open loop system, the corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain, and each of the data relationships including a corresponding open loop phase angle, an open loop gain, and a gain rate, the gain rate being the rate of change of open loop gain per open loop phase angle of the corresponding open loop system response; and

determining if a selected open loop frequency response has at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships at the corresponding open loop phase angle; and

if so

determining the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships.

23. The computer readable medium of claim 22 wherein the method further comprises comparing the open loop gain of the selected open loop frequency at the at least one open loop phase angle to the open loop gain of the data relationship at the at least one open loop phase angle.

24. The computer readable medium of claim 22 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining if the selected open loop frequency response has at least one phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate; and

if so

determining the at least one open loop phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate; and

comparing the at least one open loop phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate to the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships, if one exists.

25. The computer readable medium of claim 22 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the selected open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

determining the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

comparing the open loop gain of the imaginary open loop frequency response and the selected open loop frequency response at −

180 degrees.

26. The computer readable medium of claim 22 wherein the method further comprises:

finding the open loop gain of the selected open loop frequency at the at least one open loop phase angle;

determining the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

comparing the open loop gain of the selected open loop frequency at the at least one open loop phase angle and the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees.

27. The computer readable medium of claim 22 wherein determining if a selected open loop frequency response has at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships at the corresponding open loop phase angle includes determining if a first open loop frequency response has at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships at the corresponding open loop phase angle, and wherein the method further includes:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the first open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

providing a second open loop frequency response; and

comparing the open loop gain of the imaginary open loop frequency response at a phase angle of −

180 degrees to the open loop gain of the second open loop frequency response at a phase angle of −

180 degrees.

28. A system for evaluating the susceptibility of a system to operator coupling, the system comprising:

means for providing data relationships for multiple over-steer points, each over-steer point being the condition where, as operator gain is increased for a corresponding open loop system, the corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain, and each of the data relationships including a corresponding open loop phase angle, an open loop gain, and a gain rate, the gain rate being the rate of change of open loop gain per open loop phase angle of the corresponding open loop system response;

means for receiving a selected open loop frequency response; and

means for determining if the selected open loop frequency response has at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships at the corresponding open loop phase angle; and

if so

means for finding the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships.

29. The system of claim 28 wherein the system further comprises means for comparing the open loop gain of the selected open loop frequency at the at least one open loop phase angle to the open loop gain of the data relationship at the at least one open loop phase angle.

30. The system of claim 28 wherein the means for providing data relationships for multiple over-steer points includes a means for providing data relationships that include a corresponding closed loop gain, and wherein the system further comprises:

means for determining if the closed loop gain of the data set corresponding to the at least one open loop phase angle is equal to zero decibels or less than zero decibels; and

if so

means for comparing an open loop gain of the selected open loop frequency at the at least one open loop phase angle to an open loop gain associated with the combination of the at least one open loop phase angle and a closed loop gain of zero decibels.

31. The system of claim 28 wherein the system further comprises:

means for providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

means for determining if the selected open loop frequency response has at least one phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate; and

if so

means for finding the at least one open loop phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate; and

means for comparing the at least one open loop phase angle where the gain rate of the selected open loop frequency response is less than the critical gain rate to the at least one open loop phase angle where the gain rate of the selected open loop frequency response is at least approximately equal to the gain rate of one of the data relationships, if one exists.

32. The system of claim 28 wherein the system further comprises:

means for providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

means for determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the selected open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

means for finding the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

means for comparing the open loop gain of the imaginary open loop frequency response and the selected open loop frequency response at −

180 degrees.

33. The system of claim 28 wherein the system further comprises:

means for finding the open loop gain of the selected open loop frequency at the at least one open loop phase angle;

means for determining the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

means for comparing the open loop gain of the selected open loop frequency at the at least one open loop phase angle and the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees.

34. The system of claim 28 wherein means for receiving a selected open loop frequency response includes means for receiving a first open loop frequency response, and wherein the system further includes:

means for providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

means for determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate between an open loop gain and open loop phase angle of the first open loop frequency at the at least one open loop phase angle and a phase angle of −

180 degrees, the constant gain rate being at least approximately equal to the critical gain rate;

means for receiving a second open loop frequency response; and

means for comparing the open loop gain of the imaginary open loop frequency response at a phase angle of −

180 degrees to the open loop gain of the second open loop frequency response at a phase angle of −

180 degrees.

35. A system for developing data to be used to evaluate the susceptibility of a system to operator coupling, the system comprising:

means for determining an over-steer point for each of multiple open loop systems having open loop frequency responses, each over-steer point being the condition where, as operator gain is increased for each corresponding open loop system, the corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain; and

means for determining data relationships for each over-steer point, each data relationship including a corresponding open loop phase angle, an open loop gain, and a gain rate, the gain rate being the rate of change of open loop gain per open loop phase angle of the corresponding open loop system response.

36. A system of claim 35 wherein the system further comprises means for storing the data relationships.

37. A method for evaluating the susceptibility of an operator coupling with an operator controlled system, the method comprising:

providing an open loop frequency response for an open loop system having a selected operator gain;

determining if there is at least one operator gain for the open loop system that provides at least one over-steer point, the at least one over-steer point being the condition where, as the operator gain is increased for the open loop system, the corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain; and

if there is at least one over-steer point;

determining an open loop phase angle associated with the at least one over-steer point;

at least one of storing the open loop phase angle, displaying the open loop phase angle, printing the open loop phase angle, and using the open loop phase angle to evaluate the susceptibility of the operator coupling with the operator controlled system.

38. The method of claim 37 wherein the method further comprises:

determining an open loop gain at the at least one over-steer point; and

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point.

39. The method of claim 37 wherein:

determining if there is at least one operator gain for the open loop system that provides at least one over-steer point includes determining if there is at least one operator gains for the open loop system that provides at least one over-steer point using a computing system; and

determining an open loop phase angle associated with the at least one over-steer point includes determining an open loop phase angle associated with the at least one over-steer point using a computing system.

40. The method of claim 37 wherein the method further comprises:

determining an open loop gain at the at least one over-steer point using a computing system; and

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point using a computing system.

41. The method of claim 37 wherein the method further comprises:

determining an open loop gain at the at least one over-steer point;

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point; and

comparing the open loop gain at the at least one over-steer point to the open loop gain of the open loop frequency response at the open bop phase angle associated with the at least one over-steer point.

42. The method of claim 37 wherein the method further comprises:

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

finding a closed loop gain at the at least one over-steer point;

determining if the closed loop gain at the at least one over-steer point is equal to zero decibels or less than zero decibels; and

if so

comparing the open loop gain of the open loop frequency response at the at least one open loop phase angle to an open loop gain associated with the combination of the at least one open loop phase angle and a closed loop gain of zero decibels.

43. The method of claim 37 wherein the method further comprises:

determining an open loop gain at the at least one over-steer point;

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point; and

computing an over-steer margin for the open loop frequency response at the open loop phase angle associated with the at least one over-steer point by subtracting the open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point from the open loop gain at the at least one over-steer point.

44. The method of claim 37 wherein the method further comprises:

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

finding a closed loop gain at the at least one over-steer point;

determining if the closed loop gain at the at least one over-steer point is equal to zero decibels or less than zero decibels; and

if so

computing an over-steer margin for the open loop frequency response at the open loop phase angle associated with the at least one over-steer point by subtracting the open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point from an open loop gain associated with the combination of the at least one open loop phase angle and a closed loop gain of zero decibels.

45. The method of claim 37 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining if the open loop frequency response has at least one phase angle where the gain rate of the open loop frequency response is less than the critical gain rate; and

if so

finding the at least one open loop phase angle where the gain rate of the open loop frequency response is less than the critical gain rate.

46. The method of claim 37 wherein the method further comprises:

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining if the gain rate of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point is equal to the critical gain rate or less than the critical gain rate; and

if so

comparing the open loop gain of the open loop frequency at the at least one open loop phase angle to an open loop gain associated with the combination of the at least one open loop phase angle and a closed loop gain of zero decibels.

47. The method of claim 37 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate at least approximately equal to the critical rate between an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point and a phase angle of −

180 degrees;

finding the open loop gain of the open loop frequency response at a phase angle of −

180 degrees; and

comparing the open loop gain of the imaginary open loop frequency response at −

180 degrees to the open loop gain of the open loop frequency response at a phase angle of −

180 degrees.

48. The method of claim 37 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate at least approximately equal to the critical rate between an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point and a phase angle of −

180 degrees;

finding the open loop gain of the open loop frequency response at a phase angle of −

180 degrees; and

computing a PIO margin, the PIO margin being the open loop gain of the imaginary open loop frequency response at −

180 degrees minus the open loop gain of the open loop frequency response at a phase angle of −

180 degrees.

49. The method of claim 37 wherein the method further comprises:

finding an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

determining an open loop gain of the open loop frequency response at a phase angle of −

180 degrees; and

comparing the open loop gain of the open loop frequency at the at least one open loop phase angle and the open loop gain of the open loop frequency response at a phase angle of −

180 degrees.

50. The method of claim 37 wherein the method further comprises:

finding an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

determining an open loop gain of the open loop frequency response at a phase angle of −

180 degrees; and

computing an over-steer to PIO margin by subtracting the open loop gain of the open loop frequency response at a phase angle of −

180 degrees from the open loop gain of the open loop frequency response at the at least one open loop phase angle.

51. The method of claim 37 wherein the open loop frequency response includes an open loop frequency response of an aircraft.

52. The method of claim 37 wherein providing an open loop frequency response includes providing a first open loop frequency response, and wherein the method further includes:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate at least approximately equal to the critical rate between an open loop gain of the first open loop frequency response at the open loop phase angle associated with the at least one over-steer point and a phase angle of −

180 degrees;

providing a second open loop frequency response; and

comparing the open loop gain of the imaginary open loop frequency response at a phase angle of −

180 degrees to the open loop gain of the second open loop frequency response at a phase angle of −

180 degrees.

53. A computer readable medium having contents configured to carry out a method for evaluating the susceptibility of a system to operator coupling, the method comprising:

receiving an open loop frequency response for an open loop system having a selected operator gain;

determining if there is at least one operator gain for the open loop system that provides at least one over-steer point, the over-steer point being the condition where, as the operator gain is increased for the open loop system, a corresponding closed loop gain at a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain; and

if there is at least one over-steer point; and

determining an open loop phase angle associated with the at least one over-steer point.

54. The computer readable medium of claim 53 wherein the method further comprises:

determining an open loop gain at the at least one over-steer point; and

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point.

55. The computer readable medium of claim 53 wherein the method further comprises:

determining an open loop gain at the at least one over-steer point;

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point; and

comparing the open loop gain at the at least one over-steer point to open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point.

56. The computer readable medium of claim 53 wherein the method further comprises:

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining if the open loop frequency response has at least one phase angle where the gain rate of the open loop frequency response is less than the critical gain rate; and

if so

finding the at least one open loop phase angle where the gain rate of the open loop frequency response is less than the critical gain rate.

57. The computer readable medium of claim 53 wherein the method further comprises:

determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate at least approximately equal to the critical gain rate between an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point and a phase angle of −

180 degrees;

finding the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

comparing the open loop gain of the imaginary open loop frequency response at −

180 degrees to the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees.

58. The computer readable medium of claim 53 wherein the method further comprises:

finding an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

determining an open loop gain of the open loop frequency response at a phase angle of −

180 degrees; and

comparing the open loop gain of the open loop frequency at the at least one open loop phase angle and the open loop gain of the open loop frequency response at a phase angle of −

180 degrees.

59. The computer readable medium of claim 53 wherein receiving an open loop frequency response includes receiving a first open loop frequency response, and wherein the method further includes:

determining an open loop gain of the first open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate at least approximately equal to the critical gain rate between an open loop gain of the first open loop frequency response at the open loop phase angle associated with the at least one over-steer point and a phase angle of −

180 degrees;

providing a second open loop frequency response; and

comparing the open loop gain of the imaginary open loop frequency response at a phase angle of −

180 degrees to the open loop gain of the second open loop frequency response at a phase angle of −

180 degrees.

60. A system for evaluating the susceptibility of a system to operator coupling, the system comprising:

means for receiving an open loop frequency response for an open loop system having a selected operator gain;

means for determining if there is at least one operator gain for the open loop system that provides at least one over-steer point, the over-steer point being the condition where, as the operator gain is increased for the open loop frequency response, the corresponding closed loop gain al a corresponding resonance peak increases by at least approximately the same amount as the increase in operator gain or at least approximately a selected larger amount than the increase in operator gain; and

if there is at least one over-steer point; and

means for determining an open loop phase angle associated with the at least one over-steer point.

61. The system of claim 60 wherein the system further comprises:

means for determining an open loop gain at the at least one over-steer point; and

means for determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point.

62. The system of claim 60 wherein the system further comprises:

means for determining an open loop gain at the at least one over-steer point;

means for determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point; and

means for comparing the open loop gain at the at least one over-steer point to open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point.

63. The system of claim 60 wherein the system further comprises:

means for providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

means for determining if the open loop frequency response has at least one phase angle where the gain rate of the open loop frequency response is less than the critical gain rate; and

if so

means for finding the at least one open loop phase angle where the gain rate of the open loop frequency response is less than the critical gain rate.

64. The system of claim 60 wherein the system further comprises:

means for determining an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

means for providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

means for determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate at least approximately equal to the critical rate between an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point and a phase angle of −

180 degrees;

means for finding the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees; and

means for comparing the open loop gain of the imaginary open loop frequency response at −

180 degrees to the open loop gain of the selected open loop frequency response at a phase angle of −

180 degrees.

65. The system of claim 60 wherein the system further comprises:

means for finding an open loop gain of the open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

means for determining an open loop gain of the open loop frequency response at a phase angle of −

180 degrees; and

means for comparing the open loop gain of the open loop frequency response at the at least one open loop phase angle and the open loop gain of the open loop frequency response at a phase angle of −

180 degrees.

66. The system of claim 60 wherein means for receiving an open loop frequency response includes means for receiving a first open loop frequency response, and wherein the system further includes:

means for determining an open loop gain of the first open loop frequency response at the open loop phase angle associated with the at least one over-steer point;

means for providing a critical gain rate, the critical gain rate being the rate of change of open loop gain per open loop phase angle of an open loop frequency response for an open loop system having an over-steer point with a closed loop gain at least approximately equal to zero decibels;

means for determining the open loop gain of an imaginary open loop frequency response at a phase angle of −

180 degrees, the imaginary open loop frequency response having a constant gain rate at least approximately equal to the critical rate between an open loop gain of the first open loop frequency response at the open loop phase angle associated with the at least one over-steer point and a phase angle of −

180 degrees;

providing a second open loop frequency response; and

comparing the open loop gain of the imaginary open loop frequency response at a phase angle of −

180 degrees to the open loop gain of the second open loop frequency response at a phase angle of −

180 degrees.