Stability Analysis Using Describing Function For Non-Linear Attitude Control Loop with Slosh Dynamics of Chandrayaan-3 Landing Mission

Chandrayaan-3 became the first Indian Lunar Lander mission which landed on Moon's south pole in 2023. The lunar landing mission consists of multiple phases where primary deceleration was done using liquid engines. The navigation, guidance, and control system dictates the trajectory and descent operations. Guidance generates the attitude reference profile to be tracked by the inner control loop. Attitude control is performed using non-linear reaction control system which operates in on-off mode. A high-fidelity mechanical analogous slosh (pendulum) model is used to analyze the sloshing phenomena and control stability during the powered descent phase. Propellant sloshing induces disturbance which can interact with the inner attitude control loop and impact the overall stability margins. The pulse width pulse frequency modulator (PWPFM) converts the linear control commands to non-linear on-off pulses for the reaction control system (RCS). PWPFM is the non-linear element present in the attitude control loop which will also impact the stability margins and introduce limit cycles. In this paper, the linear stability analysis of the attitude control loop with propellant slosh dynamics is extended to incorporate the PWPFM non-linearity using the describing function analysis. An algorithm to numerically compute describing function is proposed which resolves the initialization issue with integrators. The stability margins and stability of limit cycle as a function of time has been computed. Finally, a correlation between frequency domain analyses and high-fidelity time domain simulation is presented.