Is Stabilizing an Unknown Linear System Easier than Model Identification

Abstract

Recently, directly learning an optimal LQR policy of an unknown linear system via policy optimization methods has attracted significant interest. These methods typically require a stabilizing gain to initiate iterative updates, leading to the impression that obtaining a stabilizing gain is relatively easy. In this work, however, we delve into the problem whether stabilizing an unknown linear system is easier than model identification, and our findings yield a negative answer. To support this conclusion, we introduce a novel concept named sufficient richness of input sectional data for direct data-driven property analysis. We then establish that an input sectional data is sufficiently rich if and only if it spans a property-dependent minimum linear subspace. Furthermore, by proving that the certain subspace for stabilizability is Rn+m, we essentially show that obtaining a stabilizing gain is just as challenging as identifying the unknown linear system in terms of sample complexity.