The simultaneous on-line estimation of parameters and states in linear systems

The practical implementation of adaptive controllers using minicomputers requires algorithms which are both numerically economical and robust. The problem of combined state and parameter estimation for adaptive controllers was originally posed as a nonlinear filtering problem. The only known nonlinear filter which can be practically implemented on a small computer is the extended Kalman filter. The extended Kalman filter, however, often diverges, thus, there is a need for economical, robust parameter-state estimators. A simple suboptimal parameter and state estimator is presented which fills this need. The filter is based on a particular canonical form for the state-space equations of a linear system which allows the parameters and states to be estimated separately using two linear estimators. If an innovations model is used, the steady-state Kalman filter gains can be estimated and thus, during steady-state operation, the estimates of the states can be easily obtained. Numerical exampies are presented which demonstrate the increased robustness and speed of the proposed linear estimator over the extended Kalman filter.