New Results on Optimal Joint Parameter and State Estimation of Linear Stochastic Systems

In this paper a complete presentation of a new canonical representation of multiinput, multioutput linear stochastic systems is given. Its equivalence with operator form directly linked with ARMA processes as well as with classical state space representation is given, and a transfer matrix interpretation is developed in an example. The importance of the new representation is mainly in the fact that in the joint state and parameters estimation problem, all unknown parameters appear linearly when an input-output record is available. Moreover, if noises are Gaussian and their statistics are known, a conditionally time varying Kalman-Bucy type filter gives the recursive optimal estimation of parameters and state. Historical comments and remarks about the adaptive version of this algorithm are given. Finally an illustrative low order example is described.