On Covariance Modification and Re - Gularization in Recursive Least Squa

In this paper the relationships between covariance modiication and regularization in recursive least squares identiication are investigated. An update equation for the information matrix is derived and it is shown how regularization of the information matrix can be expressed as a particular type of covariance matrix modiication. The paper also presents an analysis of the eeects of a covariance modiication for obtaining regularization that was proposed in Salgado et al. (1988). 1. INTRODUCTION The recursive least squares (RLS) algorithm is a useful tool in automatic control and signal processing for identifying systems and signals with time varying character. See e.g. Ljung and SS oderstrr om (1983) for an introduction. In order for the algorithm to maintain its tracking ability it is however necessary to have a mechanism that prevents the algorithm gain from tending to zero. Such mechanisms involve the use of, for example, forgetting factor or covari-ance modiication, see e.g. Goodwin and Sin (1984). These diierent methods however introduce a new problem since the algorithm becomes sensitive for poor excitation in the input signal. Running the RLS algorithm with a forgetting factor implies that periods with poor excitation can result in an exponential growth of the co-variance matrix in the algorithm. This phenomenon , which sometimes is denoted estimator windup, is discussed in e.g. Astrr om and Witten-mark (1989). This problem makes is necessary to introduce some safety action that handles this situation. One method that has been proposed, see Astrr om and Wittenmark (1989), is to monitor the trace of the covariance matrix in the algorithm, and if necessary normalize this matrix. Another alternative is to concentrate on the information matrix, i.e. the inverse of the covariance matrix, and prevent it from becoming singular. This is achieved using so called regularization and this is the focus of this paper. Of particular interest will be a method proposed in Salgado et al. (1988).