Modular multilevel converter (MMC) gained popularity in modern power applications due to its attributes like scalability and modularity. But, the number of submodules of an MMC and its corresponding voltage sensing devices increase with the increase in voltage levels, which makes the design complex, bulky, and expensive. Hence, an observer for estimating the voltages across the capacitors makes the complete design inexpensive, simple, and compact. This article presents an exponential power rate discrete-time sliding mode observer (EPR-DTSMO) for the observation of voltage across the capacitors of the half-bridge submodule of an MMC. In this article, the hybrid nature of an MMC is investigated, and the MMC system is demonstrated to be observable using the $Z(T_{N})$ observability criteria for hybrid systems. The proposed EPR-DTSMO is then designed for the MMC, and its convergence for the observation errors is proved. The proposed observer is validated using an experimental setup after verifying with numerical simulations. Moreover, its performance is compared to other observer techniques present in the literature. Both simulation and experimental results show that the proposed EPR-DTSMO is well suited for the sensor-less operation of MMC without using any voltage sensors for submodule capacitors.