<p>The time-dependent, dynamical nature of Hall effect thrusters (HETs) has been well-documented throughout decades of research. Recent advances in high-speed data acquisition have provided readily available time-dependent observables from Hall thruster experiments that can be used to add valuable insight into thruster operation that otherwise would not be gleaned using time-averaged quantities alone. This work explores the use of phase space embeddings of the time-dependent discharge current of a Hall thruster as a means to track thruster stability. These embeddings provide higher-dimensional representations of time-dependent signals that, according to Takens’ Theorem, contain information into the state of the entire system at a given time. Two such embeddings can be compared to determine the error between them at a low computational cost, enabling such analysis to run in real time with experimental data collection. As such, these embeddings may be useful in determining the length of startup transients in a Hall thruster, as the settling of the discharge current dynamics should be evident in the error between subsequently measured traces. A distance-based cost metric is used to evaluate changes in the discharge current phase portraits during thruster startup to study the rate of change in the system dynamics. Convoluted results, where the rate of change in error between phase portrait traces can capture the large initial transients of thruster startup but not the longer-lasting gradual settling, are discussed in relation to discharge power and thruster temperature. The impact of performing active control on the discharge power is discussed in an effort to provide more insight into whether the phase portraits of the discharge current are a sufficient tool for stability analysis.</p>