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Multivariable Control Systems ECSE 6460

Multivariable Control Systems ECSE 6460. Fall 2009 Lecture 26: 4 December 2009. Uncertain MIMO Plant. G p (s). Frequency Response. Feedback Control Loop. K(s). G p (s). +. -. practically zero steady state error for step reference rise time less than 1 second overshoot less than 5%

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Multivariable Control Systems ECSE 6460

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  1. Multivariable Control SystemsECSE 6460 Fall 2009 Lecture 26: 4 December 2009

  2. Uncertain MIMO Plant Gp(s)

  3. Frequency Response

  4. Feedback Control Loop K(s) Gp(s) + - • practically zero steady state error for step reference • rise time less than 1 second • overshoot less than 5% • robustly for any value of c

  5. Menu for the day • Representing the uncertainty • a la unstructured uncertainty • a la structured uncertainty • Assessment of robust stability • a la unstructured uncertainty • a la structured uncertainty • Assessment of robust stability • a la unstructured uncertainty • a la structured uncertainty

  6. Representing Uncertainty

  7. Unstructured Uncertainty Wo(s) D(s) Output uncertainty + G(s) + Wi(s) D(s) Input uncertainty + + G(s)

  8. Output Uncertainty Wo(s) D(s) Output uncertainty + G(s) + We need to select a weight such that:

  9. Input Uncertainty Wi(s) D(s) Input uncertainty + + G(s) We need to select a weight such that:

  10. Weight Wo(s) Refer to uncert1.m

  11. Structured Uncertainty

  12. Structured Uncertainty Ws(s) D relax to + G(s) +

  13. Relaxation from real to complex uncertainty

  14. Robust Stability

  15. Assessment of Robustness • In this topic, we first design a controller for the nominal system, then we assess the robustness properties of the system. • Two control designs are explored, decentralized inverse-based, and H-infinity synthesis. K(s) Gp(s) + -

  16. Decentralized Inverse Based Based on nominal plant model Refer to decent.m

  17. Nominal Performance

  18. H synthesis for nominal plant • Use S/T synthesis with: Refer to hinf_ex.m

  19. Nominal Performance

  20. Robust Stability Assessment Unstructured Uncertainty Wo(s) D(s) + K(s) G(s) + + -

  21. Robust Stability Assessment Wo(s) + K(s) G(s) + -

  22. Robust Stability Assessment Refer to decent.m

  23. Robust Stability Assessment Refer to hinf_ex.m

  24. Robust Stability Assessment Structured Uncertainty Ws(s) D + K(s) + G(s) + -

  25. Robust Stability Assessment Structured Uncertainty Ws(s) + K(s) + G(s) + -

  26. Structured Singular Value Peak at 0.130 For decentralized controller

  27. Perturbed system responses

  28. Structured Singular Value Peak at 0.150

  29. Perturbed system responses

  30. Robust Performance

  31. Nominal Performance Refer to hinf_ex.m

  32. Robust Performance Assessment Unstructured Uncertainty Wo(s) + + K(s) G(s) WP(s) + + -

  33. Robust Performance Assessment Unstructured Uncertainty Wo(s) + + K(s) G(s) WP(s) + + -

  34. Robust Performance Assessment DP D N

  35. Robust Performance Assessment

  36. Robust Performance Assessment Structured Uncertainty Ws(s) + + K(s) G(s) WP(s) + + -

  37. Robust Performance Assessment Structured Uncertainty Ws(s) + + K(s) G(s) WP(s) + + -

  38. Robust Performance Assessment DP D N

  39. Robust Performance Assessment

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