6_cont_obs - Control of Nonlinear Dynamic Systems: Theory...

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 62 6 ` Controllability and Observability of Nonlinear Systems Controllability for Nonlinear Systems The Use of Lie Brackets: Definition Consider two vector fields f(x) and g(x) in n . Then the Lie bracket operation generates a new vector field: [] g x f f x g g f , Also, higher order Lie brackets can be defined: Key points Nonlinear observability is intimately tied to the Lie derivative. The Lie derivative is the derivative of a scalar function along a vector field. Nonlinear controllability is intimately tied to the Lie bracket. The Lie bracket can be thought of as the derivative of a vector field with respect to another. References o Slotine and Li, section 6.2 (easiest) o Sastry, chapter 11 pages 510-516, section 3.9 and chapter 8 o Isidori, chapter 1 and appendix A (hard)
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 63 [] g f g ad f , ) , ( 1 g f f g ad f , , ) , ( 2 [ ] ) , ( , ) , ( 1 g ad f g ad k f k f Note: the “ad” is read “adjoint”. Re-writing controllability conditions for linear systems using this notation: m m u B u B u B Ax Bu Ax x + + + + = + = ... 2 2 1 1 & Ax x f = ) (, i i B x g = ) ( i m i i m i i i u B f x A u AB x A x A x = = = + = = 1 2 1 2 , & & & How this came about… A x f = , 0 = = x B x g So for example: [ ] 1 1 1 1 1 ) ( , , AB B x Ax Ax x B B Ax B f = = = If we keep going: ∑∑ == + = + = m i m i i i f i i u B ad x A u B A x A x 11 2 3 2 3 , & & & Notice how this time the minus signs cancel out. + = + = = m i m i i i n f n n i i n n n n n u B ad x A u B A x A dt x d x 1 1 1 ) ( , ) 1 ( Re-writing the controllability condition: [ ] [ ] [ ] [ ] [ ] m n f n f m f f m B ad B ad B ad B ad B B C , ,... , ,... , ,... , , ,..., 1 1 1 1 1 = The condition has not changed – just the notation.
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 64 The terms B 1 through B m correspond to the B term in the original matrix, the terms with ad f correspond to the AB terms, the terms with ad f n-1 correspond to the A n-1 B terms. Recap – Controllability for Linear Systems Bu Ax x + = & [ ] B A AB B C n 1 | ... | | = Local conditions (linear systems) Bu Ax x + = & Let u = constant (otherwise no pb, but you get u u & & & , etc…) ABu x A Bu Ax A x + = + = 2 ) ( & & Bu A x A x n n n 1 ) ( + = For linear systems, you get nothing new after the nth derivative because of the Cayley- Hamilton theorem. Nonlinear Systems Assume we have an affine system: = + = m i i i u x g x f x 1 ) ( ) ( & The general case is much more involved and is given in Hermann and Krener. If we don’t have an affine system, we can sometimes ruse: ) , ( u x f x = & Let v u u = + & τ Select a new state: = u x z and v is my control the system is affine in (z,v), and pick τ to be OK.
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6_cont_obs - Control of Nonlinear Dynamic Systems: Theory...

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