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Chapter 10 – Feedback Linearization

Chapter 10 – Feedback Linearization. Big Picture:. Linear System. Nonlinear System. Control Input Transformation. Linear Controller. When does such a transformation exist? How do we find it? (not really control design at this point).

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Chapter 10 – Feedback Linearization

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  1. Chapter 10 – Feedback Linearization Big Picture: Linear System Nonlinear System Control Input Transformation Linear Controller When does such a transformation exist? How do we find it? (not really control design at this point)

  2. Other than here, square brackets still indicate a matrix

  3. Matrix Vectors Lie Brackets

  4. = Lie Bracket

  5. Only [f1,f2] here

  6. Control Law Reminder from Chapter 2 – Linear approximation of a system System Taylor series at origin System This is not what we will do in this section!

  7. Transformation Complete: Transformed the nonlinear system into system that is linear from the input perspective. Control Design: Use linear control design techniques to design v.

  8. A A B is 2x1, “directs u to a specific row” Scalar 1 Transformation Complete: Transformed the nonlinear system into a linear system. Control Design: Use linear control design techniques to design v: 1

  9. Control Law Example 7: Implementing the Result Looks like a linear system Linearizing Control System

  10. A Transformation Complete: Transformed the nonlinear system into a linear system. Control Design: Use linear control design techniques to design v.

  11. Affine in u We are restricted to this type of system

  12. Feedback linearization and transformation process: T(x) u(x) Using standard linear control design techniques v(z) Stable closed-loop system

  13. Not matched to u Matched to u Note that n=2 in the above procedure.

  14. (continued) 1

  15. (continued) Will only depend on T2 Moved the nonlinearities to the bottom equation where they are matched with u

  16. (continued)

  17. Control Law Example 9: Implementing the Result System Linearizing Control

  18. Always good to try this approach but may not be able to find a suitable transformation. Conditions on the determinant u “lives” behind g, so g must possess certain properties so that u has “enough access” to the system • Questions: • What is adf g(x)? Review Lie Bracket • What is a span? set of vectors that is the set of all linear combinations of the elements of that set. • What is a distribution? Review Distributions sections • What is an involutive distribution? Lie Brackets Matrix

  19. = n=2 (size of x)

  20. System

  21. System u y ?

  22. =

  23. Summary Main Idea Nonlinear System Linear System Control Input Transformation Linear Controller Input to state linearization Input to output linearization

  24. Homework 10.1

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