Optimality properties in finite sample l_iidentification with bounded noise

Abstract

In this paper we investigate finite sample optimality properties for worst-case /I identification of the impulse response of discrete time, linear, time-invariant systems. The experimental conditions we consider consist of rn experiments of length N. The measured outputs are corrupted by component-wise bounded additive disturbances with known bounds. The quantification of the identification error is given by the maximum /I-norm of the difference between the true impulse response samples and the estimated ones, where the maximum is taken with respect to all admissible plants and all admissible disturbances.First we show that for any given experimental condition, almost-optimal (within a factor of two) estimates can be obtained by solving suitable linear programmes. Then we study how experimental conditions affect the identification error. Optimality of the experimental data is measured by the diameter of information, a quantity which is at most twice as large as the minimal worst-case error.We show that the minimum number of experiments allowing us to minimize the diameter of information is m* = 2N. The values of the diameter of information and the corresponding optimal inputs are derived for the two extreme experimental conditions rn = 1 and rn = 2N.

H/sub ∞/ identification and model quality evaluation

“…To make a minimal use of a priori information, we consider information of the residual type, i.e., giving constraints on the for only. In particular we assume to know and such that (31) Letting and using the well-known ordering relation between and norms, the result is (32) and consequently (33) Remark 1: The measurements give information only on the first samples of , while the prior knowledge is residual, giving information on samples from and continuing. As a consequence, there is no problem of consistency between the two types of information.…”

“…It can be observed that what we really need is only the value of in (32). Then a more general residual a priori information could be considered .…”

“…The advantages of such an approach, over the nonparametric one, have been investigated in statistical and other SM identification settings. In particular, it is known that the "informational complexity" of the nonparametric approach in identification is "high," i.e., the number of measurements needed to assure a given level of identification error grows exponentially with the number of impulse response samples to be estimated [32]- [35]. The informational complexity may be largely reduced using mixed parametric and nonparametric models, as shown in [36].…”

Convergence analysis of central and minimax algorithms in scalar regressor models

“…Finite sample properties of system identification methods have been studied before in different settings. In the worst-case deterministic setting, finite sample properties have been studied in [3,14,[16][17][18]20,22,24,25]. In these studies, disturbances are allowed to be correlated with regressors, which cannot occur in the setting of this paper.…”