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[[File:Block scheme of control system.jpg|thumb|
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Fig. 1. Block scheme of control system. ''G''(''s'') – linear transfer function, ''f''(''e'') – single-valued, continuous, differentiable function]]


'''Kalman's conjecture''' or '''Kalman problem''' is a [[conjecture]] on absolute stability of
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[[nonlinear control]] system  with one scalar nonlinearity, which belongs to the sector of linear stability. Kalman's conjecture is a strengthening of [[Aizerman's conjecture]]
and is a special case of [[Markus–Yamabe conjecture]].  This conjecture was proven false but led to the (valid) [[Nonlinear_control#Absolute_stability_problem|sufficient criteria on absolute stability]].
 
== Mathematical statement of Kalman's conjecture (Kalman problem)==
In 1957 [[R. E. Kalman]] in his paper
<ref name=1957-Kalman>{{cite journal
| author = Kalman R.E.  
| year = 1957
| title = Physical and Mathematical mechanisms of instability in nonlinear automatic control systems
| journal = Transactions of ASME
| volume = 79
| number = 3
| pages = 553&ndash;566
}}
</ref> stated the following:
<blockquote>
If ''f''(''e'') in Fig. 1 is replaced by constants ''K'' corresponding to all possible values of ''f''<nowiki>'</nowiki>(''e''), and it is found that the closed-loop system is stable for all such ''K'', then it intuitively clear that the system must be monostable; i.e., all transient solutions will converge to a unique, stable critical point.
</blockquote>
 
Kalman's statement can be reformulated in the following conjecture:<ref name=2011-DAN-Kalman>{{cite journal
| author = Leonov G.A., Kuznetsov N.V.
| year = 2011
| title = Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems
| journal = Doklady Mathematics
| volume = 84
| number = 1
| pages = 475&ndash;481
| url = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
| doi = 10.1134/S1064562411040120}}
</ref>
<blockquote>
Consider a system with one scalar nonlinearity''
:<math>
  \frac{dx}{dt}=Px+qf(e),\quad e=r^*x \quad x\in R^n,
</math>
where ''P'' is a constant ''n''×''n'' matrix, ''q'', ''r'' are constant ''n''-dimensional vectors, ∗ is an operation of transposition, ''f''(''e'') is scalar function, and ''f''(0)&nbsp;=&nbsp;0. Suppose, ''f''(''e'') is a differentiable function and the following condition''
:<math>
k_1 < f'(e)< k_2. \,
</math>
is valid. Then Kalman's conjecture is that the system is stable in large (i.e. unique stationary point is global [[attractor]]) if all linear systems with ''f''(''e'')&nbsp;=&nbsp;''ke'', ''k''&nbsp;∈&nbsp;(''k''<sub>1</sub>,&nbsp;''k''<sub>2</sub>) are asymptotically stable.
</blockquote>
 
In [[Aizerman's conjecture]] in place of condition on the derivative of nonlinearity it is required that the nonlinearity itself belongs to linear sector.
 
Kalman's conjecture is true for ''n''&nbsp;≤&nbsp;3 and for ''n''&nbsp;>&nbsp;3 there are effective methods for construction of counterexamples:<ref>{{cite journal
| author = Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A.
| year = 2011
| title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
| journal = Journal of Computer and Systems Sciences International
| volume = 50
| number = 5
| pages = 511&ndash;543
| url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
| doi = 10.1134/S106423071104006X}}
</ref><ref name=2011-IJBC-Hidden-attractors>{{cite journal |
author = Leonov G.A., Kuznetsov N.V. |
year = 2013 |
title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
journal = International Journal of Bifurcation and Chaos |
volume = 23 |
issue = 1 |
pages = art. no. 1330002|
url = http://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024|
doi = 10.1142/S0218127413300024}}
</ref> nonlinearity derivative belongs to the sector of linear stability, and unique stable equilibrium coexists with a stable periodic solution ([[hidden oscillation]]).
 
== References ==
{{Reflist}}
 
== Further reading ==
*{{cite journal
| author = Leonov G.A., Kuznetsov N.V.
| year = 2011
| title = Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems
| journal = IFAC Proceedings Volumes (IFAC-PapersOnline)
| volume = 18
| number = 1
| pages = 2494&ndash;2505
| url = http://www.math.spbu.ru/user/nk/PDF/2011-IFAC-Hidden-oscillations-control-systems-Aizerman-problem-Kalman.pdf
| doi = 10.3182/20110828-6-IT-1002.03315}}
 
== External links ==
* [http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf Analytical-numerical localization of hidden oscillation in counterexamples to Aizerman's and Kalman's conjectures]
 
[[Category:Conjectures]]
[[Category:Nonlinear control]]

Revision as of 22:07, 3 March 2014

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