This work studies the ${H}_{\infty }$ -based minimal energy control with a preset convergence rate (PCR) problem for a class of disturbed linear time-invariant continuous-time systems with matched external disturbance. This problem aims to design an optimal controller so that the energy of the control input satisfies a predetermined requirement. Moreover, the closed-loop system asymptotic stability with PCR is ensured simultaneously. To deal with this problem, a modified game algebraic Riccati equation (MGARE) is proposed, which is different from the game algebraic Riccati equation in the traditional ${H}_{\infty }$ control problem due to the state cost being lost. Therefore, a unique positive-definite solution of the MGARE is theoretically analyzed with its existing conditions. In addition, based on this formulation, a novel approach is proposed to solve the actuator magnitude saturation problem with the system dynamics being exactly known. To relax the requirement of the knowledge of system dynamics, a model-free policy iteration approach is proposed to compute the solution of this problem. Finally, the effectiveness of the proposed approaches is verified through two simulation examples.