An analytic approximation of the maximal invariant ellipsoid for a discrete-time linear system with bounded controls is derived. The approximation is expressed explicitly in terms of the coefficient matrices of the system and the positive definite matrix that represents the shape of the invariant ellipsoid. It is shown that this approximation is very close to the exact maximal invariant ellipsoid obtained by solving either an LMI-based optimization problem or a nonlinear algebraic equation. Furthermore, the necessary and sufficient condition for such an approximation to be equal to the exact maximal invariant ellipsoid is established. On the other hand, the monotonicity of the maximal invariant ellipsoid resulting from the "minimal energy control with guaranteed convergence rate" problem is established that shows a trade-off between increasing the size of the invariant ellipsoid and increasing the convergence rate of the closed-loop system under a bounded control. Two illustrative examples demonstrate of the effectiveness of the results.