Kalman's conjecture

Kalman's conjecture or Kalman problem is a disproved conjecture on absolute stability of nonlinear control system with one scalar nonlinearity, which belongs to the sector of linear stability.

This conjecture was proven false but led to the (valid) sufficient criteria on absolute stability.

1 is replaced by constants K corresponding to all possible values of f'(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.

Kalman's statement can be reformulated in the following conjecture:[2] 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.

Kalman's conjecture is true for n ≤ 3 and for n > 3 there are effective methods for construction of counterexamples:[3][4] the nonlinearity derivative belongs to the sector of linear stability, and a unique stable equilibrium coexists with a stable periodic solution (hidden oscillation).