Optimal weight problem for response sensitivity-based damage identification

The response sensitivity approach has been shown to be effective for structural damage identification. The goal function involved is of the form g ( α )   =   ‖ R ( α )   −   R ^ ‖ W 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315189390/3a3a1415-aecc-43fa-9f80-ac1a274361fa/content/eq1260.tif"/> with R ( α ) denoting the response function derived via governing equations with given parameters α and R ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315189390/3a3a1415-aecc-43fa-9f80-ac1a274361fa/content/eq1261.tif"/> the measured data. In many cases of response sensitivity analysis, the weight matrix W is simply set to be de identity matrix and few studies have been presented alone on how to properly choose the weight matrix. This paper tries to present a simple theoretical and numerical investigation on the optimal weight problem for response sensitivity-based damage identification with measurement and model errors. By minimizing the expectation of the squares error in identification results, the optimal weight matrix is found to be W o p t   =   c Q R − 1 ,   c   >   0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315189390/3a3a1415-aecc-43fa-9f80-ac1a274361fa/content/eq1262.tif"/> when only measurement errors is involved with covariance Q _R . Futher analysis shows that the optimal weight matrix can also make the ‘relative error’ - square root of expectation of squares relative error in every individual damage parameter minimized. What’s more, if model errors, such as errors in the damping and load parameters are additionally considered, the optimal matrix is W o p t   =   c ( Q R   +   S β Q β   S β T ) − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315189390/3a3a1415-aecc-43fa-9f80-ac1a274361fa/content/eq1263.tif"/> , c > 0, where Q β   =   E [ ε β ε β T ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315189390/3a3a1415-aecc-43fa-9f80-ac1a274361fa/content/eq1264.tif"/> and S β   =   ∇ β R ( α e x ,   β ^ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315189390/3a3a1415-aecc-43fa-9f80-ac1a274361fa/content/eq1265.tif"/> with ε_β being model errors of the model parameters β ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315189390/3a3a1415-aecc-43fa-9f80-ac1a274361fa/content/eq1266.tif"/> . Numerical tests on a plane frame are conducted to see the effect of the proposed optimal weight matrix.