For every simple Lie algebra $mathfrak{g}$ we consider the associated Takiff algebra $mathfrak{g}^{}_{ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $mathfrak{g}$. We use a matrix presentation of $mathfrak{g}^{}_{ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${rm U}(mathfrak{g}^{}_{ell})$. A similar matrix presentation for the affine Kac--Moody algebra $widehat{mathfrak{g}}^{}_{ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $mathfrak{g}^{}_{ell}$.
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