Aizerman's conjecture

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

${\displaystyle {\frac {dx}{dt}}=Px+qf(e),\quad e=r^{*}x\quad x\in \mathbb {R} ^{n},}$

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${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$ with ${\displaystyle k_{1}<k_{2}}$, the function ${\displaystyle f}$ satisfies

${\displaystyle k_{1}<{\frac {f(e)}{e}}<k_{2},\quad \forall \;e\neq 0.}$

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]

References

1. "The Aizerman conjecture: The Popov method". Mathematical Problems of Control Theory. Series on Stability, Vibration and Control of Systems, Series A. Vol. 6. World Scientific. 2001. pp. 155–166. doi:10.1142/9789812799852_0006. ISBN   978-981-02-4694-5.
2. Liberzon, M. R. (2006). "Essays on the absolute stability theory" . Automation and Remote Control. 67 (10): 1610–1644. doi:10.1134/S0005117906100043. ISSN   0005-1179. S2CID   120434924.
3. Leonov G.A.; Kuznetsov N.V. (2011). "Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems" (PDF). Doklady Mathematics. 84 (1): 475–481. doi:10.1134/S1064562411040120. S2CID   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" (PDF). Journal of Computer and Systems Sciences International. 50 (5): 511–543. doi:10.1134/S106423071104006X. S2CID   21657305.
5. Kuznetsov N.V. (2020). "Theory of hidden oscillations and stability of control systems" (PDF). Journal of Computer and Systems Sciences International. 59 (5): 647–668. doi:10.1134/S1064230720050093. S2CID   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. Bibcode:2013IJBC...2330002L. doi:10.1142/S0218127413300024.
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. doi:10.1109/TAC.2022.3217740. ISSN   0018-9286. S2CID   260255282.
8. Hinrichsen, D.; Pritchard, A. J. (1992-01-01). "Destabilization by output feedback". Differential and Integral Equations. 5 (2). doi: 10.57262/die/1371043976 . ISSN   0893-4983. S2CID   118359120.
9. Jayawardhana, Bayu; Logemann, Hartmut; Ryan, Eugene (2011). "The Circle Criterion and Input-to-State Stability" . IEEE Control Systems Magazine. 31 (4): 32–67. doi:10.1109/MCS.2011.941143. ISSN   1066-033X. S2CID   84839719.

Further reading

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

External links

• "Counterexamples to Aizerman's and Kalman's conjectures and describing function method" (PDF).