We developed a theory that describes fast mode-hop-free tuning of an external cavity diode laser in Littrow configuration with two acousto-optic modulators (AOMs) inside the laser cavity. The theory is based on synchronous shifting of the external cavity modes and the Littrow grating selectivity. It allows calculating the driving signals of both AOMs in order to reach a desired temporal variation of the laser frequency, including particularly fast tuning as well as an arbitrary shape of the tuning function. Furthermore, we present a laser setup for which the needed signals for both AOMs are generated by two direct digital synthesizer circuits. Thereby we were able to verify the theoretical predictions experimentally, achieving, e.g., sinusoidal single-mode tuning of the laser frequency over $40\mathrm{GHz}$ at a repetition rate of $10\mathrm{kHz}$ and over $12\mathrm{GHz}$ at $25\mathrm{kHz}$ . Finally, the limitations of the theory are discussed.