Learning HModel Sets from Data: The Set Membership Approach

  • Mario Milanese
  • Michele Taragna
Part of the Trends in Mathematics book series (TM)


The paper investigates the problem of identifying time-invariant, discrete-time, exponentially stable, possibly infinite dimensional, linear systems from noisy experimental data. The aim is to deliver not a single model, but a set of models whose size inH∞norm measures the uncertainty in the identification. The noise assumptions can account for information on its maximal magnitude and deterministic uncorrelation properties.

The paper overviews recent results of the authors, focusing on the optimality properties for finite data and on the tradeoff between optimality and complexity of the identified model set.A method is given for solving the consistency/prior validation problem, requiring the solution of one linear programming problem. An algorithm is presented for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency, i.e., the set of responses at that frequency of all systems not falsified by data. Such approximations provide a method for computing the identification error of any identified model set within any desired accuracy, while in the literature only bounds are available, which are often quite conservative. A method for evaluating an optimal model set at any given number of frequencies is presented. This optimal model set may prove to be too complex and simpler model sets may be looked for, at the expense of identification accuracy degradation. This is measured by the optimality level of the identified model set, i.e., the ratio between the achieved identification error and the minimal one. By suitably approximating the optimal model set, a ’-optimal algorithm is derived, called “nearly optimal”, thus improving over the 2-optimality of “almost optimal” algorithms available in the literature. Using these results, reduced-order model sets with nominal models inRH O are derived which tightly include the optimal model set, and their optimality levels are evaluated. Since the optimality level of reduced-order model sets can be made near to\by increasing their order, the model order selection can be performed by suitably trading off between model set complexity and optimality level degradation.


Set Membership identification Identification for control Model sets Model quality Finite samples Complex systems 


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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Mario Milanese
    • 1
  • Michele Taragna
    • 1
  1. 1.Dipartimento di Automatica e InformaticaPolitecnico di TorinoTorinoItaly

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