On Optimality and Robustness in Linear Dynamic System Identification

Strong consistency and asymptotic error distribution for a new class of nonlinear recursive parameter estimation algorithms of an approximate Newton–Raphson type are established. The system model is given in the discrete-time domain by a linear difference equation with constant parameters. The parameter estimator design is based on martingale theory and the Cramér–Rao (CR) theorem, providing a maximum likelihood (ML)-type optimal recursive parameter identification algorithm, whose minimum asymptotic estimation error covariance matrix achieves the CR bound under the worst-case pdf within a specified class; this worst-case pdf simultaneously yields the maximum asymptotic error covariance matrix within the class. However, the worst-case pdf does not generally exist, making the min–max optimal design indeterminable. Therefore, an approximate ML (AML)-type optimal on a class design, based on a suboptimal worst-case pdf, minimizing the scalar Fisher information within the specified class, has also been developed. The proposed design minimizes the conditional estimation error covariance under the specified suboptimal worst-case pdf. Such an approach results in Huber’s M-robustified version of the AML-based optimal design on the class of contaminated Gaussian pdfs. The practical performance of the proposed approach is analyzed through statistical validation, based on the relative asymptotic estimation efficiency measure and Monte Carlo simulations.