Selective Forgetting in Adaptive Procedures

Abstract The efficient signal processing necessary in adaptive procedures, requires a steady flow of data to be treated, conr:entrated and used without loss of information in a short period of time. Traditionally, in an abundance of statistical techniques, the parameters are regarded as constants, and therefore possible to estimate from a long enough sequence of measurements. However, It is often, e.g. in connection with adaptive control or prediction or in process supervision, unrealistic to assume fixed parameters. One then instead has to construct recursive algorithms for the estimation in time-varying dynamic systems able to handle the slow as well as the abrupt timevariations of the system parameters. When the parameters vary regularly, i.e. without abrupt changes, the rate of variation might differ considerably between individual parameters. Alternatively, the now of information about the evolution of the parameters might be limited to certain directions in parameter space. In this paper an algorithm is presented, which can be used in an adaptive context in order to handle Such different rates of variation or limited flow of information. In adaptive procedure:s a model for the parameter evolution has to be explicitely or implicitcly postulated. Such a model with common exponential growt.h in the parameter covariance update, leads to conventional forgetting methods. These suffer from the drawback of having the same rate of forgetting for all parameters and postulating a homogeneous flow of information. This might e.g. result. in covariance windup if the forgetting is turned to the parameters with the fastest variation. In the presented algorithm however, the parameter evolution model is changed in such a manner that the it is possible to tailor the forgetting properties of the algorithm to the rate of information flow in parameter space or to the rate of variation of the individual parameters or groups of parameters. Hrnce covariance windup can be avoided. The essential steps in the algorithm are the separation of time and measurement updates of the parameter covariance matrix and the recursive analysis of the information pattern. This also forms the basis for the analysis of the algorithm. The paper thus contains presentation, analysis, illustration and application of a proposed algorithm with selective forgetting.