Aizerman's conjecture explained

In nonlinear control, Aizerman's conjecture or Aizerman problem states that a linear system in feedback with a sector nonlinearity would be stable if the linear system is stable for any linear gain of the sector. This conjecture, proposed by Mark Aronovich Aizerman in 1949,^[1] was proven false but led to the (valid) sufficient criteria on absolute stability.^[2]

Mathematical statement of Aizerman's conjecture (Aizerman problem)

Consider a system with one scalar nonlinearity

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)=0. Suppose that the nonlinearity f is sector bounded, meaning that for some real and with , the function satisfies Then Aizerman'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)=ke, k ∈(k1,k2) are asymptotically stable.

There are counterexamples to Aizerman's conjecture such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists with a stable periodic solution, i.e. a hidden oscillation.^{[3] [4] [5] [6]} However, under stronger assumptions on the system, such as positivity, Aizerman's conjecture is known to hold true.^[7]

Variants

• Strengthening of Aizerman's conjecture is Kalman's conjecture (or Kalman problem) where in place of condition on the nonlinearity it is required that the derivative of nonlinearity belongs to linear stability sector.
• A multivariate version of Aizerman's conjecture holds true over the complex field,^[8] and it can be used to derive the circle criterion for the stability of nonlinear time-varying systems.^[9]

Further reading

• Atherton, D.P. . Siouris, G.M. . 1977 . Nonlinear Control Engineering . IEEE Transactions on Systems, Man, and Cybernetics . 7 . 7 . 567–568 . 10.1109/TSMC.1977.4309773.

External links

• Web site: Counterexamples to Aizerman's and Kalman's conjectures and describing function method.

Notes and References

1. Book: https://www.worldscientific.com/doi/10.1142/9789812799852_0006 . 10.1142/9789812799852_0006 . The Aizerman conjecture: The Popov method . Mathematical Problems of Control Theory . Series on Stability, Vibration and Control of Systems, Series A . 2001 . 6 . 155–166 . World Scientific . 978-981-02-4694-5 .
2. Liberzon . M. R. . 2006 . Essays on the absolute stability theory . Automation and Remote Control . en . 67 . 10 . 1610–1644 . 10.1134/S0005117906100043 . 120434924 . 0005-1179.
3. Leonov G.A. . Kuznetsov N.V. . 2011 . Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems . Doklady Mathematics . 84 . 1 . 475–481 . 10.1134/S1064562411040120. 120692391 .
4. Bragin V.O. . Vagaitsev V.I. . Kuznetsov N.V. . Leonov G.A. . 2011 . Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits . Journal of Computer and Systems Sciences International . 50 . 5 . 511–543 . 10.1134/S106423071104006X. 21657305 .
5. Kuznetsov N.V. . 2020. Theory of hidden oscillations and stability of control systems. Journal of Computer and Systems Sciences International. 59. 5. 647 - 668. 10.1134/S1064230720050093. 225304463.
6. Leonov G.A. . Kuznetsov N.V. . 2013 . Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits . International Journal of Bifurcation and Chaos . 23 . 1 . 1330002–219. 10.1142/S0218127413300024. 2013IJBC...2330002L .
7. Drummond . Ross . Guiver . Chris . Turner . Matthew C. . 2023 . Aizerman Conjectures for a Class of Multivariate Positive Systems . IEEE Transactions on Automatic Control . 68 . 8 . 5073–5080 . 10.1109/TAC.2022.3217740 . 260255282 . 0018-9286.
8. Hinrichsen . D. . Pritchard . A. J. . 1992-01-01 . Destabilization by output feedback . Differential and Integral Equations . 5 . 2 . 10.57262/die/1371043976 . 118359120 . 0893-4983. free .
9. Jayawardhana . Bayu . Logemann . Hartmut . Ryan . Eugene . 2011 . The Circle Criterion and Input-to-State Stability . IEEE Control Systems Magazine . 31 . 4 . 32–67 . 10.1109/MCS.2011.941143 . 84839719 . 1066-033X.