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 (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

on absolute stability of

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

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 was proven false but ...

and is a special case of

Markus–Yamabe conjecture In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks ...

. 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 ...
) 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.

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