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29 SYSTEM SYSTEM SYSTEM + + INPUT OUTPUT ERROR + ERROR INVERSE M1 M2-1 MODEL: M MODEL: M-1 GENERALIZED ERROR (a) Forward Model (b) Inverse Model (c) Generalized Model Figure 25. Methods for system identification process (29). should be adjusted until the error is reduced sufficiently (29). error criterion. Figure 26 depicts use of the forward model in The system identification process considers three different error the system identification process including the parameter minimization models depending on the choice of residuals adjustment and adaptation algorithm for the reflection crack- combined with the model: forward model, inverse model, and ing model calibration. generalized model shown in Figure 25. The forward approach minimizes the output error between the model and the system Parameter Adjustment while using the same input. In the inverse approach, the input and Adaption Algorithm error is minimized based on the same output. The generalized model is a combination of the forward and inverse approach A parameter adjustment and adaptation algorithm was when the model is invertible (29). developed based on the Taylor series expansion as follows (30) When the system output is fixed because it is observed or obtained from an actual system, the output from the model must be refined to calibrate the mathematical model includ- [ Fki ]{ i } = {rk } (9) ing its parameters. In this project, the reflection cracking amount and severity model (mathematical model) was cali- where m n f pi brated based on observed reflection crack data (actual system [Fki] = sensitivity matrix = pk (m n matrix ) ; output) to produce the predicted crack data (model output) k =1 i =1 i fk which is close to the observed crack data. m, n = number of output data and model parameters, An optimal model for the physical system is obtained when respectively; the output error between the system and the model is small fk = mathematical model; enough to meet an error criterion. However, if the error does pi = model parameters; not meet the criterion, the parameters in the mathematical {i} = change vector (relative change of parameters) = model should be corrected by a parameter adjustment and [1, 2, . . . , n]T; and adaptation algorithm. The correction process is performed {rk} = residual vector (error between system and model iteratively until the error becomes small enough to meet the outputs) = [r1, r2, . . . , rm]T. System Output from System Fields (Observed Cracks) Output error Parameters of Model Model Good (, ) Reflection Cracking Output from Model Model (Predicted Cracks) No Good Parameter Adjustment and Adaption Algorithm Figure 26. Scheme of system identification process.