Kalman's conjecture

Fig. 1. Block scheme of control system. G(s) - liner transfer function, f(e) - single-valued, continuous, differentiable function

In 1957 year R.E. Kalman in his paper stated the following: “If f(e) in Fig. 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 fall 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

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

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, f(e) is a piecewise-differentiable function and in the points of differentiability the following condition

$k1<f'(e)<k2.$

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)=ke, k ∈(k1,k2) are asymptotically stable.

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

External links

• Analytical-numerical localization of hidden oscillation in counterexamples to Aizerman's and Kalman's conjectures