Separate discussions should be submitted for the individual papers in this symposium.
The following lemma is a discrete multiple-input version of an earlier result (Beck 1978). It is assumed here that the sampling interval I1.t is sufficiently small so that all the modal frequencies of the model lie below the Nyquist frequency. It can be concluded that if predictions are to be made at the roof, then any of these optimal models will give the same results, while if the re- sponse at a lower degree of freedom is to be predicted, the predictions of all optimal models must be included, with their probabilistic predictions appropriately weighted through the coefficients Wk as shown in Beck and Katafygiotis (1998).
This leads to the concept of model identifiability, which is addressed in this paper (Bellman and Astrom 1970). are the "free" parameters that need to be assigned values from a bounded region S(a) in order to choose a particular model M(a) in . M.) E RNd is the model output vector at Nd degrees of freedom (DOF) at time tn =mit where dt is a prescribed sampling interval; and Z7 = is the input up to this time. Also, the theory can be easily extended to the cases where the initial conditions and/or the mass matrix are uncertain.
Let Q~(a; 2~) = denote the model output history at a set of No observed DOFs for a given input history 27 and for a structural model M(a) E . Here, So E RNo XNd is a selection matrix with elements equal to one or zero depending on which DOFs are observed. M-Identifiability of Structural Model Parameters The issue here is the M-identifiability of the parameter vec- tor a at some optimal value a, o~ equ~valently,~the TM-identi- fiability of the parameters 6 at 6 = [ell ... , 'Nf· ~ It can be easily shown that the model response Q~(a; Z~) calculated following a modal analysis approach is uniquely determined by the following vector of uncertain parameters: W(a) == [wr C9), ,,, I3j;)(9), bjl)(9): r == I, 2, ... , NI , i E ~, I E ,;eaf (9) where w, = rth modal frequency; ~, = damping ratio of the rth mode' 13(~) = effective participation factor of the rth mode, , (/) 0 at the ith DOF for the kth input (Beck 1978); b, == ith D F component of the pseudostatic influence coefficient vector cor- responding to the lth input (acceleration input), and ,;eo C and ,;ea C denote the sets of integers corresponding to the observed DOFs and to the com- ponents of the input vector z E RN, representing externally applied accelerations, respectively. The vector W(a) by its definition uniquely determmes the model output history Q~(a; Z~). , NI , i E ~, I E ,;ely (10) denote the vector W(a). Effective Participation Factors p I" of All Output-Equiv- alent Optimal Solutions Shown in Table 1 for First Three Modes r = 1, 2, 3 and for All OOF 1= 1, ... As before, the values of these coefficients are found to be invariant with respect to the assumed input history and damping coefficients.
A valuable observation in solving the system identifiability problem is that all models that are output-equivalent to a given optimal model are also optimal. denote a class of structural models that prescribes, or implies, a functional relationship q(n; a) for the input-out- put behavior of a structure, where a E S(a) e RN. Under the above assumptions, the vector of uncertain-val- ued parameters for the class of models MNd is a = [91, 92, ... , 'NJT (8) Note that although (7) is chosen here to describe the relation- 464/ JOURNAL OF ENGINEERING MECHANICS / APRIL 1998 ship K = K(6), the discussion and algorithm presented in the following are not restricted to a specific choice for K(6).
Utilizing this lemma, it follows that out of the elements of a, the vector t is globally M-identifiable and, ther~fore, only the M-identifiability of the stiffness parameters at 9 needs to be examined. (%) (1 ) (2) (3) (4) (5) (6) (7) (8) 1 1.0000 ooסס.1 ooסס.1 OOסס.1 ooסס.1 OOסס.1 21.35 2 1.5848 0.6963 1.2875 0.7574 1.1766 0.7898 13.49 3 1.9970 0.7980 0.7095 1.3848 0.7113 0.8980 4.91 4 2.0000 ooסס.1 ooסס.1 0.5000 1.0000 ooסס.1 21.35 5 2.0932 1.0476 0.7240 0.7374 0.6705 1.2738 17.07 6 2.2911 0.6304 0.9321 1.1774 0.9515 0.6631 6.46 7 2.4913 0.8777 0.6514 1.1106 0.6672 0.9475 7.40 8 2.8252 0.6753 0.8826 0.9021 0.8753 0.7520 7.97 lowing this curve, the value of w I(O) is monitored to check whether there exist other points 8(/), I is followed and the value of W2(0) is_ monitored. It is also clear that if a different OOF instead of, or in addition to, the one corresponding to the roof was measured, then the number of output-equivalent optimal models would be dras- tically reduced.
Udwadia and Shah (1978) showed that the prob- lem of identifying the stiffness distribution of an Nrstory shear building from its base input and its roof response is a nonunique problem with an upper bound of Nd ! It is found that the previously identified point 0(2) = [2.0, O. D ow nl oa de d fr om a sc el ib ra ry .o rg b y U ni ve rs ity o f Sa sk at ch ew an o n 06 /0 9/ 14 . F or p er so na l u se o nl y; a ll ri gh ts r es er ve d. For example, it can be shown that if the OOF corresponding to the third floor is measured, then only the fourth solution in Table I is output-equivalent to the initial uniform model.
A new algorithm is presented to solve this problem for a class of multi-degree-of-freedom linear structural models, and the approach is illustrated with examples. The interest here lies in investigating the model identifia- bility of a at some optimal value ii of the model parameters in the sense of Beck and Katafygiotis (1998). "Classical normal modes in damped linear dynamic systems." J.
The goal is to find all the structural models within a specified class that produce the same output at a set of observed degrees of freedom as a given model in the class when they are all subjected to the same input. This technical note is part of the ]ou17Ul1 of Engi- neering Mechanics, Vol. ©ASCE, ISSN 0733- 9399/98/0004-0463-0467/.00 $.50 per page. Note that if a J is globally M-identifiable at ii, then it is also M-identi- fiable at ii. A parameter aj of a is locally M-identifiable at ii for class . and input 27 if it is M-identifiable but not globally M-identifiable. For example, the model parameter vector a is (globally) M-identifiable at ii if all of its elements are (globally) M-identifiable at ii, whereas a is locally M-identifiable at ii if it is M-identifiable but at least one of its elements is only locally M-identifiable at ii.
Algorithm for Resolving M-Identifiability of Stiffness Parameters Assume that there is a finite number of structural models M(a(k» =: M(9(k, t): k == 1, 2, ... D ow nl oa de d fr om a sc el ib ra ry .o rg b y U ni ve rs ity o f Sa sk at ch ew an o n 06 /0 9/ 14 . F or p er so na l u se o nl y; a ll ri gh ts r es er ve d. However, in this case no more curves need to be followed because CI(O; 6(2)) == CI(O; 6(1») and C2(0; 6(2)) == ci O; 6(1»). It is ~oncluded that in this example two optimal solutions exist: 0(1) = [1.0, 1.0f and 0(2) = [2.0, The proposed methodology first searches the domain S(O) = (0, 3) X (0, 3) for all solutions 6(/) of the problem '! CONCLUSION An algorithm to investigate model identifiability in struc- tural model updating is presented.