Abstract
The problem of reduced-order adaptive estimation for singularly perturbed systems, which are two-time scale continuous-time stochastic systems, is considered in the framework of nonlinear filtering. It is shown that in some case the parameter estimate by a reduced-order adaptive estimator designed for a reduced-order model has a bias even in the limit when fast variables are infinitely fast. This is in contrast with the linear estimation for a singularly perturbed system where the difference between state estimates by a reduced-order Kalman filter and a full-order Kalman filter vanishes in the limit when fast variables are infinitely fast. It is then shown that the bias of the parameter estimate is attributable to the problem of stochastic integrals, i.e., a part of the Itô integrals appeared in the nonlinear filter equations is actually calculated as Stratonovich-like integrals when the reduced-order adaptive estimator is used. Numerical examples are given to illustrate the bias of the parameter estimate. Finally, conditions are given under which the bias of parameter estimate vanishes.
