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  "body": "\n## Counterexample\n\nKalman's conjecture concerns systems of the form\n$$\\dot x = P x + q\\, f(e), \\qquad e = r^{T} x.$$\nHere $x \\in \\R^n$, $P$ is a constant real $n \\times n$ matrix, and $q, r \\in \\R^n$ are constant vectors.\nThe function $f \\colon \\R \\to \\R$ is differentiable with $f(0) = 0$.\nAssume that $k_1 \u003c f'(e) \u003c k_2$ for all $e$, and that for every $k \\in (k_1, k_2)$ the [[linear system|037ZNY7NAG]] $\\dot x = (P + k\\, q r^{T}) x$ is [[asymptotically stable|0NB689782W]].\nThe conjecture claims that then the [[equilibrium|0N8NGKDZDZ]] $x = 0$ attracts every solution.\n\nThe claim is false in general.\nFor $n \\le 3$ it is true.\nBut for some $n \u003e 3$ there are $P$, $q$, $r$ and $f$ that satisfy all the hypotheses, while the system has a non-constant [[periodic orbit|0AA97XA70K]].\nA solution on that orbit does not tend to $0$, so $0$ is not a global attractor.\n\n## Remarks\n\nKalman stated the problem in 1957.\nIn the counterexamples constructed by Leonov, Kuznetsov and coauthors, the periodic orbit is stable and coexists with the stable equilibrium at $0$.\nIts basin of attraction does not touch a small neighbourhood of the equilibrium, so it is a hidden oscillation: simulation from initial points near $0$ does not reveal it.\nOnce a counterexample exists in dimension $n$, adding decoupled coordinates with $\\dot w = -w$ gives one in every larger dimension.\nFor the discrete-time analogue the conjecture holds only for $n = 1$.\n\n## Explanations\n\n### L1 human\n\nReplace a nonlinear part of a feedback loop by straight lines whose slopes lie in the range of the slopes of the nonlinearity.\nIf every such linear system is stable, it is tempting to think that the nonlinear system is stable as well.\nKalman's conjecture says so, and it is true in low dimension.\nIn higher dimensions it can fail: the nonlinear system can keep oscillating forever, although every linear comparison system is stable.\n",
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