Nonlinear control: Difference between revisions

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== Properties of nonlinear systems ==

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Some properties of nonlinear dynamic systems are

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== Analysis and control of nonlinear systems ==

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There are several well-developed techniques for analyzing nonlinear feedback systems:

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* The [[Kalman's conjecture]].

Graphically, these conjectures can be interpreted in terms of graphical restrictions on the graph of Φ(''y'') ''x'' ''y'' or also on the graph of ''d''Φ/''dy'' ''x'' Φ/''y''.<ref>{{Cite journal|lastlast1=Naderi|firstfirst1=T.|last2=Materassi|first2=D.|last3=Innocenti|first3=G.|last4=Genesio|first4=R.|date=2019|title=Revisiting Kalman and Aizerman Conjectures via a Graphical Interpretation|journal=IEEE Transactions on Automatic Control|volume=64|issue=2|pages=670–682|doi=10.1109/TAC.2018.2849597|s2cid=59553748 |issn=0018-9286}}</ref> There are counterexamples to Aizerman's and Kalman's conjectures such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists with a stable periodic solution—[[hidden oscillation]].

There are two main theorems concerning the Lur'e problem which give sufficient conditions for absolute stability:

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== Theoretical results in nonlinear control ==

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=== Frobenius theorem ===

The [[Frobenius theorem (differential topology)|Frobenius theorem]] is a [[deep result]] in differential geometry. When applied to nonlinear control, it says the following: Given a system of the form

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== Further reading ==

{{refbegin}}

*{{cite journal |firstfirst1=A. I. |lastlast1=Lur'e |first2=V. N. |last2=Postnikov |title=К теории устойчивости регулируемых систем |trans-title=On the Theory of Stability of Control Systems |journal=Prikladnaya Matematika I Mekhanika |volume=8 |issue=3 |year=1944 |pages=246–248 |language=Russian }}

*{{cite book |first=M. |last=Vidyasagar |title=Nonlinear Systems Analysis |edition=2nd |publisher=Prentice Hall |___location=Englewood Cliffs |year=1993 |isbn=978-0-13-623463-0 }}

*{{cite book |first=A. |last=Isidori |title=Nonlinear Control Systems |edition=3rd |publisher=Springer |___location=Berlin |year=1995 |isbn=978-3-540-19916-8 }}

*{{cite book |first=J. |last=Adamy |title=Nonlinear Systems and Controls |edition=1st |publisher=Springer |___location=Berlin |year=2022 |isbn=978-3-662-65633-4 }}

*{{cite book |first=H. K. |last=Khalil |title=Nonlinear Systems |edition=3rd |publisher=Prentice Hall |___location=Upper Saddle River |year=2002 |isbn=978-0-13-067389-3 }}

*{{cite book |firstfirst1=B. |lastlast1=Brogliato |first2=R. |last2=Lozano |first3=B. |last3=Maschke |first4=O. |last4=Egeland |title=Dissipative Systems Analysis and Control |publisher=Springer |___location=London |edition=3rd |year=2020 }}

*{{cite journal

|author1=Leonov G.A. |author2=Kuznetsov N.V. | year = 2011

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| url = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf

| doi = 10.1134/S1064562411040120

| issue = 1|s2cid=120692391 }}

*{{cite journal

|author1=Bragin V.O. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |author4=Leonov G.A. | year = 2011

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| url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf

| doi = 10.1134/S106423071104006X

| issue = 5|s2cid=21657305 }}

*{{cite journal

| author = Leonov G.A., Kuznetsov N.V.