Error bounds in observer-based LTI system identification

In this paper, we study the estimation accuracy of deadbeat observers to identify multi-input/single-output (MISO) state-space (SS) models from input–output data. It is demonstrated that the ordinary least-squares method (LSM) to estimate the observer parameters has an asymptotic error proportional to the noise-to-signal ratio (NSR) when the output measurements are corrupted by zero-mean white-noise uncorrelated with the inputs. This result warrants a formal study of guaranteed performance for the interpolatory estimators that explain the input–output data. To this end, an asymptotic upper bound on the feasible parameter set diameter of all interpolatory estimators, including the nonlinear least-squares, the lasso, the elastic net, the analytic center, the Chebyshev, and the minimax estimators, is derived assuming that the root-mean-square (RMS) value of noise is bounded. This bound holds with probability one (w.p.1) and is proportional to the √ N S R . An asymptotic lower bound on the feasible parameter diameter that is proportional to the N S R and holds w.p.1 is also derived using the LSM estimate. A key assumption in the paper is that the inputs are quasi-stationary and persistently exciting (PE) of a sufficient order. An experimental study confirms the derived results.