Kalman's conjecture or Kalman problem is a disproved

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

on absolute stability of

nonlinear control Nonlinear control theory is the area of control theory which deals with systems that are nonlinear system, nonlinear, time-variant system, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that ...

system with one scalar nonlinearity, which belongs to the sector of linear stability. Kalman's conjecture is a strengthening of

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 Ar ...

and is a special case of Markus–Yamabe conjecture. This conjecture was proven false but led to the (valid) 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''(''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 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 : $\frac=Px+qf(e),\quad e=r^*x \quad x\in R^n,$ 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 differentiable function and the following condition : $k_1 < f'(e)< k_2. \,$ is valid. Then Kalman's conjecture is that the system is stable in the large (i.e. a unique stationary point is a global
attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...
) if all linear systems with ''f''(''e'') = ''ke'', ''k'' ∈ (''k''₁, ''k''₂) are asymptotically stable.

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 (

hidden oscillation In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounde ...

). In discrete-time, the Kalman conjecture is only true for n=1, counterexamples for ''n'' ≥ 2 can be constructed. The development of Kalman's ideas on global stability based on the stability of linear approximation for a cylindrical phase space gave rise to the Viterbi problem on the coincidence of phase-locked loop ranges.

References

External links

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

Discrete-time counterexample in Maplecloud
Disproved conjectures Nonlinear control Hidden oscillation

Mathematical statement of Kalman's conjecture (Kalman problem)

References

Further reading

External links