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. 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 sufficient criteria on absolute stability.

Mathematical statement of Kalman's conjecture (Kalman problem)

In 1957 R. E. Kalman in his paper
stated the following:

If f in Fig. 1 is replaced by constants K corresponding to all possible values of f', 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:

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 is scalar function, and f = 0. Suppose, f is a differentiable function and the following condition
is valid. Then Kalman's conjecture is that the system is stable in the large if all linear systems with f = ke, k ∈ are asymptotically stable.

In Aizerman's conjecture in place of the condition on the derivative of nonlinearity it is required that the nonlinearity itself belongs to the linear sector.
Kalman's conjecture is true for n ≤ 3 and for n > 3 there are effective methods for construction of counterexamples: the nonlinearity derivative belongs to the sector of linear stability, and a unique stable equilibrium coexists with a stable periodic solution.
In discrete-time, the Kalman conjecture is only true for n=1, counterexamples for n ≥ 2 can be constructed.