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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be open, and let $f, g \\colon \\Omega \\to \\R^n$ be smooth [[vector fields|0299VCW1TH]].\nConsider the single-input control-affine [[control system|ZPDTVZQN5K]] $x' = f(x) + g(x) u$ with $u \\in \\R$, and let $x_0 \\in \\Omega$.\nThe system is *locally feedback linearizable* at $x_0$ if there are the following data.\nAn open neighbourhood $W \\subseteq \\Omega$ of $x_0$, and a smooth [[diffeomorphism|0KZZ70ZHXP]] $\\Phi$ from $W$ onto an open subset of $\\R^n$.\nSmooth functions $\\alpha, \\beta \\colon W \\to \\R$ with $\\beta(x) \\ne 0$ for all $x \\in W$.\nThese data must satisfy two conditions:\n\n1. under the feedback $u = \\alpha(x) + \\beta(x) v$, the new state $z = \\Phi(x)$ satisfies $z' = A z + B v$ for a real $n \\times n$ matrix $A$ and a vector $B \\in \\R^n$;\n2. the linear system $z' = A z + B v$ is [[controllable|A9WFM77Z4P]].\n\n*Feedback linearization* is the construction of such $\\Phi$, $\\alpha$ and $\\beta$.\n\n## Remarks\n\nExample: the pendulum $x_1' = x_2$, $x_2' = -\\sin x_1 + u$ has $f(x) = (x_2, -\\sin x_1)$ and $g(x) = (0, 1)$; with $\\Phi = \\id$, $\\alpha(x) = \\sin x_1$ and $\\beta = 1$ one gets $z_1' = z_2$, $z_2' = v$, which is controllable.\nNon-example: with $g \\equiv 0$ the input has no effect, so clause 2 fails for every choice of $\\Phi$, $\\alpha$ and $\\beta$.\nWrite $\\mathrm{ad}_f g = [f, g]$ for the [[Lie bracket|68FGQEBD6J]].\nThe system is locally feedback linearizable at $x_0$ if and only if two conditions hold.\nThe vectors $g, \\mathrm{ad}_f g, \\ldots, \\mathrm{ad}_f^{n-1} g$ are linearly independent at $x_0$.\nThe distribution spanned by $g, \\mathrm{ad}_f g, \\ldots, \\mathrm{ad}_f^{n-2} g$ is involutive near $x_0$.\nThe related input–output linearization only makes the map from $v$ to a chosen output linear, and may leave internal dynamics.\n",
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