  Controlled frames have been recently introduced in Hilbert spaces to improve
the numerical efficiency of interactive algorithms for inverting the frame
operator. In this paper, unlike the cross-Gram matrix of two different
sequences which is not always a diagnostic tool, we define the controlled-Gram
matrix of a sequence as a practical implement to diagnose that a given sequence
is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that
the operator associated to controlled Gram matrix will be bounded, invertible,
Hilbert-Schmidt or a trace-class operator. Similar to standard frames, we
present an explicit structure for controlled Riesz bases and show that every
$(U, C)$-controlled Riesz basis $\{f_{k}\}_{k=1}^{\infty}$ is in the form
$\{U^{-1}CMe_{k}\}_{k=1}^{\infty}$, where $M$ is a bijective operator on $H$.
Furthermore, we propose an equivalent accessible condition to the sequence
$\{f_{k}\}_{k=1}^{\infty}$ being a $(U, C)$-controlled Riesz basis.
