Identification of Unstable Linear Systems using Data-driven Koopman Analysis

System identification plays a crucial role in modern control techniques, enabling the data-driven learning of input-output maps or mathematical models. However, practical applications face challenges as the actual number of states is often unknown, and observed variables may be limited. Additionally, unstable systems present further difficulties, as their outputs rapidly diverge or saturate, hindering long-term measurement. This paper addresses these challenges by proposing a novel input-aware modeling method for unstable linear systems using data-driven Koopman analysis. Unlike traditional Koopman analysis which focuses solely on state dynamics, our method explicitly incorporates the influence of the input function
$u(t)$
. This enables us to accurately capture the complete behavior of the system, even under the influence of external control signals. By leveraging Koopman operator theory on augmented state-input data, we capture both the intrinsic dynamics and the sensitivity to external control, crucial for accurate prediction and control of unstable systems. This input-aware approach extends the capabilities of data-driven Koopman analysis to improve modeling and control of complex unstable systems in various applications.