The Cramer-Rao lower bound for bilinear systems

Estimation of the unknown parameters that characterize a bilinear system is of primary importance in many applications. The Cramer-Rao lower bound (CRLB) provides a lower bound on the covariance matrix of any unbiased estimator of unknown parameters. It is widely applied to investigate the limit of the accuracy with which parameters can be estimated from noisy data. Here it is shown that the CRLB for a data set generated by a bilinear system with additive Gaussian measurement noise can be expressed explicitly in terms of the outputs of its derivative system which is also bilinear. A connection between the nonsingularity of the Fisher information matrix and the local identifiability of the unknown parameters is exploited to derive local identifiability conditions of bilinear systems using the concept of the derivative system. It is shown that for bilinear systems with piecewise constant inputs, the CRLB for uniformly sampled data can be efficiently computed through solving a Lyapunov equation. In addition, a novel method is proposed to derive the asymptotic CRLB when the number of acquired data samples approaches infinity. These theoretical results are illustrated through the simulation of surface plasmon resonance experiments for the determination of the kinetic parameters of protein-protein interactions.