## 5 Damage Characterization, Objective Functions, and Optimization Algorithm
### Objective Features and Functions  
Since damage detection becomes an optimization problem eventually, the capability of a damage identification method strongly depends on the selected features and constructed objective functions that have a profound impact on the behavior of the optimization algorithm. Usually, we do not directly use the raw data in the time domain to compare two system states. Instead, we extract so-called features, for example, mean values, variances, maximum values, crest factors or other statistical characteristics of signals, correlation functions, spectral densities, frequency response functions, impedance spectra, eigenfrequencies, modal shapes, or any other features [13]. Meanwhile, the objective function is normally built up using the residuals between the measured responses and numerical predictions, such as a sum of squared difference [148] as,  
\[f(\theta)=\frac{1}{2}\sum_{j=1}^{m}\bigl{|}z_{j}(\theta)-\widetilde{z}_{j} \bigr{|}^{2}=\frac{1}{2}\sum_{j=1}^{m}r_{j}(\theta)^{2}=\frac{1}{2}\|r(\theta )\|^{2} \tag{1}\]  
where each \(z_{j}(\theta)\) represents an analytical quantity that is a nonlinear function of the variables \(\theta\), \(\widetilde{z}\) refers to the measured value of the quantity \(z\), and \(\|\bullet\|\) denotes the Euclidean norm. In order to obtain a unique solution, the number \(m\) of residuals \(r_{j}=z_{j}-\widetilde{z}_{j}\) should be greater than the number \(n\) of unknowns \(\theta\). There are various expressions that can be potentially used for this purpose, such as the frequency residual, mode shape related function, modal flexibility residual, and a combination of them [149]. The best features for damage detection are typically application specific, which can come from the time, frequency, or modal domains depending on the employed level of data consideration. Research into methods based on all the three domains is unlikely to stop because no single method has yet been able to identify every type of damage in every type of structures [3]. Time domain features may be represented by the peak accelerations, the root-mean squares of responses, the temporal moments, etc. The direct use of time response in a model-based identification was proposed by [150; 151] that used a least-squares sensitivity-based iterative parameter estimation routine. [PERSON] et al. [152] proposed a neural network based detection scheme that uses time responses both in the training and detection phases. [PERSON] et al. [153] and [PERSON] and [PERSON] [154] studied the direct use of time histories of vehicle or bridge acceleration in sensitivity-based model updating method. Frequency domain features are commonly the shift in the resonance and anti-resonances or changes affecting the amplitudes. If the excitation is known, the FRF can provide abundant information on damage in a desired frequency range. With the objective of eliminating the modal extraction error, damage detection using measured FRF has been studied by a number of researchers [155; 156; 157; 158] and is very promising because the FRF data can be obtained directly without any further extraction and processing. Meanwhile, the transmissibility functions employ only the measurement of the structural responses and do not require knowledge of the excitation input. This operational advantage is coupled with remarkable damage sensitivity as proven in the work by [71; 72; 78; 159]. [PERSON] et al. [160]published a comparative study of using the FRF curvature, transmissibility, and the strain energy obtained directly from FRFs as damage indicators. Modal domain features are physically meaningful and easier to be interpreted than the time or frequency domain features.