As I understand, stability of such a closed-loop system is discussed in two steps.
1- observer error dynamics must go to zero exponentially fast.
2- state variables must always remain in the attraction region of the controller.
Khalil ("Nonlinear Systems" 3rd edition, page 617) has explained this in a very complicated way.
Can anyone give a better and formal explanation?
Thank you Morteza, but I did not see any section about high-gain observers in the Slotine book.
You are only asking about nonlinear high-gain observer, yet you enter the nonlinear systems stability issue, which may not be simple at all.
All above mentioned books are excellent and good to read (and I myself also see Slotine and Li as most readable). Still, all stability proofs for nonlinear nonautonomous systems (i.e., where the time appears explicitly, either via input commands or as time varying parameters) are based on Barbalat’s Lemma and require strong conditions of uniform continuity.
This is based on the fact that, when these books and people in general mention LaSalle’s (or Krasovsky-LaSalle) Invariance Principle, they all mean 1950-1960 works, which only cover autonomous systems of the form xdot=f(x) (i.e., where time does not appear).
However, at least for LaSalle, those works are only an early advance to his REAL contribution and REAL Invariance Principle of 1976-1980, where he extended it to nonautonomous systems, yet , for some amazing reason, have remained unknown by the large nonlinear systems community, including those otherwise excellent books.
Besides, new works continue along the lines of those LaSalle’s latest works and simplify nonlinear systems analysis even more.
A discussion and the corresponding references can be found on Research Gate at
https://www.researchgate.net/post/Is_anyone_here_familiar_with_the_Barbalat_lemma_and_its_application_in_control_engineering#view=56e81ba83d7f4bc87d4a9801.
Hope it helps.
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