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 system, given observations of the system input and output and bounds on the model-output error, has developed into an interesting alternative to parameter estimation by least-squares, maximum-likelihood or recursive prediction-error methods. It has potential, so far unexploited, for using prior knowledge of bounds on plant behaviour to augment the information in the observations. The paper examines the forms of bounds on the parameters of a discrete-time rational transfer-function model implied by bounds on physically meaningful parameters such as time constants, modal amplitudes, steady-state gains and ringing frequency. Bounds on a single time constant are found to yield parameter bounds which are mainly linear but have a non-linear section, of degree rising rapidly with model order. Simultaneous bounds on two or more time constants give overall parameter bounds ranging from polytopes, easy 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 bounds, ellipsoidal in the numerator- and denominator-parameter subspaces but 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 amplitude. Simultaneous bounds on amplitude and phase look intractable. The ringing frequency of an oscillatory mode is shown to impose parameter bounds of a degree which doubles for each additional pole, but bounds on damping give lower degrees. The practical implications of these results are discussed.