By Eric Walter, Luc Pronzato, J. Norton

The presentation of a coherent method for the estimation of the parameters of mathematical versions from experimental info is tested during this quantity. Many themes are coated together with the alternative of the constitution of the mathematical version, the alternative of a functionality criterion to match types, the optimization of this functionality criterion, the review of the uncertainty within the expected parameters, the layout of experiments for you to get the main correct information and the serious research of effects. There also are numerous beneficial properties targeted to the paintings akin to an up to date presentation of the technique for checking out versions for identifiability and distinguishability and a accomplished remedy of parametric optimization including larger give some thought to ation of numerical points and which examines recursive and non-recursive tools for linear and nonlinear versions.

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9 As for identifiability, there are cases where no structural conclusion can he drawn. Consider for instance two model structures, with transfer functions 1\ I 1\ H(s, p) = 2 S 1\ 1\ + PIS + P2 and H(s, p) = (s 1 + Pl)(S + P2) , with p and p belong)ng to ]R2. ]f l1(s, p) has two real poles. H(s, p) is indistinguishable from 'R(s, p); otherwise, H(s, p) is distinguishable from if(s, p), because of the restriction of p to real values. None of these situations can a priori be 0 considered atypical.

P p Al(p) u(t) -----I Ym(t, p. 5. VI(p) ::: M(p). Note the asymmetry of the previous definition. The fact that t! d. from M does not imply that the converse is true. One class of models may include the other (without this being obvious at first sight). d. d. d. • 1985). Note, however, that one now hopes to prove the non-existence of a solution for p, whereas in identifiability studies one hoped to prove the uniqueness of this solution. \"!. ~2(0) = 0, 1\ Ym Xl· The assoc'iated transfer functions, when put in the same canonical form, can respectively be written as H(s, p) = -----~---- and 1\ 1\ H(s, p) 1\ 1\ 1\ 1\ - + PI + P3 =S 2 +s(PlS +P2+P3)+PIP2 1\ 1\ 1\ • The identity of input-output behaviour therefore translates into p For any p, it is possible to find such that these equations are satisfied, and vice versa.

Once 12,h lib, lie and llr have been chosen, the unknown parameters arc Such a structure is called AR1I1AX (AutoRegressive-Movillg Average with eXogel/ous variable) or CARMA (Colltrolled AutoRegressive Moving Average). It extends the ARMA structure to the case where controlled inputs are present (Box and Jenkins, 1976). Removing the autoregressive part of an ARMAX structure, one gels a Fillite Impulse Response (FIR) structure. Sometimes, replacing y(t) by ily(t) = y(t) - y(t-I) and 11(1) by L1u(t) = u(t) - u(l-l) allows one to get rid of very slowly varying perturbations, such as offsets, by working on increments.