Optimality properties in finite sample li identification with bounded noise

In this paper we investigate finite sample optimality properties for worst-case l2 identification of the impulse response of discrete time, linear, time-invariant systems. the experimental conditions we consider consist of m 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 l1-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− = 2−N. the values of the diameter of information and the corresponding optimal inputs are derived for the two extreme experimental conditions m = 1 and m = 2−N.

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