Translation of bounds on time-domain behaviour of dynamical systems into parameter bounds for discrete-time rational transfer-function models

Computation of bounds on the parameters of a Linear model of a dynamical sy stem, given observations of the system input and output and bounds on the m odel-output error has developed into an interesting alternative to paramete r estimation by least-squares, maximum-likelihood or recursive prediction-e rror methods. It has potential, so far unexploited, for using prior knowled ge of bounds on plant behaviour to augment the information in the observati ons. The paper examines the forms of bounds on the parameters of a discrete -time rational transfer-function model implied by bounds on physically mean ingful parameters such as time constants, modal amplitudes, steady-state ga ins and ringing frequency. Bounds on a single time constant are found to yi eld parameter bounds which are mainly linear but have a non-linear section, of degree rising rapidly with model order. Simultaneous bounds on two or m ore time constants give overall parameter bounds ranging from polytopes, ea sy to handle, to intractably high-degree surfaces, depending on model order and how the original bounds overlap. Bounds on amplitude and steady-state gain of a real mode prove to be linear. Oscillatory modes yield quadratic b ounds, ellipsoidal in the numerator- and denominator-parameter subspaces bu t not overall. Bounds on the initial phase of the ringing bound a bilinear form in the numerator and denominator parameters, at any given value of amp litude. Simultaneous bounds on amplitude and phase look intractable. The ri nging frequency of an oscillatory mode is shown to impose parameter bounds of a degree which doubles for each additional pole, but bounds on damping g ive lower degrees. The practical implications of these results are discusse d. (C) 1999 IMACS/Elsevier Science B.V. All rights reserved.