Sequential deconvolution input reconstruction

Abstract

The reconstruction of inputs from measured outputs is examined. It is shown that the rank deficiency that arises in de-convolving non-collocated arrangements is associated with a kernel that is non-zero only over the part of the time axis where delay from wave propagation prevents uniqueness. Input deconvolution, therefore, follows in the same manner for collocated and non-collocated scenarios, collocation being the special case where the prediction lag can be zero. This paper illustrates that deconvolution carried out on a sliding window is a conditionally stable process and the condition for stability is derived. Examination of the Cramer-Rao Lower Bound of the inputs in frequency shows that the inference model should be formulated such that the spectra of the inputs to be reconstructed, and of the realized measurement noise, are within the model bandwidth. An expression for the error in the reconstructed input as a function of the noise sequence is developed and is used to control the regularization, when regularization is needed. The paper brings attention to the fact that finite dimensional models cannot display true dead time and that failure to recognize this matter has led to algorithms that, in general, propose to violate the physical constraints.