Necessary and sufficient conditions for uniqueness of the minimum in Prediction Error Identification

  • Authors:

  • A. S. Bazanella;X. Bombois;M. Gevers

  • Affiliations:

  • Electrical Engineering Department, Universidade Federal do Rio Grande do Sul, Porto Alegre-RS, Brazil;Delft Center for Systems and Control, Delft University of Technology, The Netherlands;ICTEAM, University of Louvain, B1348 Louvain-la-Neuve, Belgium and Dept ELEC, Vrije Universiteit Brussel, Pleinlaan 2, B1050 Brussels, Belgium

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 2012

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Abstract

The contribution of this paper is to establish computable necessary and sufficient conditions on the model structure and on the experiment under which the Prediction Error Identification (PEI) criterion has a unique global minimum. We consider a broad class of rational model structures whose numerator and denominator are affine in the unknown parameter vector; this class encompasses all classical model structures used in system identification. The main results in this paper rely on the standard assumption that the system is in the model set, while some intermediate results are valid even when this assumption does not hold (in particular Theorem 4.2 and Lemma 6.1). This is achieved by first establishing necessary and sufficient conditions on the model structure and on the experiment under which a global minimum is isolated; these conditions must hold locally, at the global minimum. A second contribution is to show that these conditions are equivalent to the nonsingularity of the information matrix at that minimum. For open loop identification and, with some additional constraints also for closed loop identification, the nonsingularity of the information matrix is then also equivalent to the uniqueness of the global minimum.