7 System Identification of an MDOF Experimental Structure with a View Towards Validation and Verification 59 Fig. 7.2 ‘FRFs’ between the base acceleration of the rig and the relative accelerations of the upper floors 7.3 System Identification Using Self-Adaptive Differential Evolution (SADE) For the sake of completeness, a brief overview of the basic Differential Evolution (DE) and SADE algorithms will be given here; for more detail, the reader is referred to the original papers [11, 12]. As in all evolutionary optimisation procedures, a population of possible solutions (here, the vector of parameter estimates), is iterated in such a way that succeeding generations of the population contain better solutions to the problem in accordance with the Darwinian principle of survival of the fittest. The problem is framed here as a minimisation problem with the cost function defined as a normalised mean-square error between the measured data and that predicted using a given parameter estimate. Having established by FRF analysis that the base-excited system appears to correspond well to a three-DOF system, the model equations considered were, m1Rz1 Cc1Pz1 Cc2.Pz1 Pz2/ Ck1z1 Ck2.z1 z2/ D m Ry0 m2Rz2 Cc2.Pz2 Pz1/ Cc3.Pz2 Pz3/ Ck2.z2 z1/ Ck3.z2 z3/ D m Ry0 m3Rz3 Cc3.Pz3 Pz3/ Ck3.z3 z2/ D m Ry0 (7.1) where the fzi D yi y0 W i D 1; : : : ;3g are displacement coordinates relative to the base displacement. As discussed above, the rig was operated in its linear condition in order to acquire data for the identification. As it is not clear what the actual masses are prior to the identification, an estimate m is used for the RHS of the equations. The estimate is based on the physical masses of the shelves and associated fixings, Including m1, m2 and m3 in the parameter vector D .m1;m2;m3;c1;c2;c3;k1;k2;k3/ allows the identification algorithm to correct for the contribution of the vertical beams etc. Based on the design geometry and materials, m was taken here as 5.475 kg. The cost function referred to above was defined here in terms of the prediction errors associated with each DOF. A set of Normalised Mean-Square-Errors (NMSEs) Ji were defined by, Ji . / D 100 N 2 Rzi N Xi D1 .Rzi ORzi . // 2 (7.2) where 2 Rzi is the variance of the measured sequence of relative accelerations and the caret denotes a predicted quantity; N is the number of ‘training’ points used for identification. The total cost function J was then taken as the average of the Ji . Previous experience has shown that a cost value of less than 5.0 represents a good set of model predictions (or parameter estimates). In order to generate the predictions ORzi , the coupled Eqs. (7.1) were integrated forward in time in Matlab [8] using a fixedstep fourth-order Runge–Kutta scheme for initial value problems [10]. A better solution could potentially be found by using
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