Projection-based Bayesian recursive estimation of ARX model with uniform innovations

Autoregressive model with exogenous inputs (ARX) is a widely-used black-box type model underlying adaptive predictors and controllers. Its innovations, stochastic unobserved stimulus of the model, are white, zero mean with time-invariant variance. Mostly, the innovations are assumed to be normal. It induces least squares as the adequate estimation procedure. The light tails of the normal distribution allow one to accept the unbounded support as a reasonable approximate description of bounded physical quantities. In some cases, however, this approximation is too crude or does not fit subsequent processing, for instance, robust control design. Then, techniques that deal with unknown-but-bounded equation errors are used. More often than not these techniques give up a stochastic interpretation of innovations and develop estimation algorithms of a min-max type.The paper assumes bounded innovations but stays within the standard Bayesian estimation framework by assuming uniformly distributed innovations. The posterior probability density function (pdf) is first described and approximated by a pdf with a fixed-dimensional statistic. Consequently, the estimation can run in real time. Moreover, its limited memory allows for tracking time-varying parameters. In this manner, an alternative to popular forgetting techniques is also obtained.The paper provides a complete algorithmic solution and illustrates its behavior.