# Tracking of Stiffness Variation in Structural Members Using Input Error Function Observers
## 4 Optimization of Input Error Function Three DOF Mass-Spring-Damper System  
Instead of using a bank of input error observers for estimating change in stiffness, input error minimization of a damaged structural member is proposed for tracking the stiffness variation of a structural system in real-time. Figure 4 below shows the workflow of the entire process. Using the input error function formulation, the error signal for each structural member is established. The interaction matrix \(M_{i}\) formulation and introduction of vector \(N_{i}\) has completely decoupled the influence of stiffness variation in individual structural members on the output measurements. This enables tracking the stiffness variation in each structural member by using the error function formulation in Equation (14). As indicated in Figure 3, a zero-error signal is obtained when the correct stiffness estimation is used. The stiffness estimation process to obtain a zero-error signal is automated by implementing an optimization technique. The optimization parameter (stiffness value) corresponding to the optimized zero-error signal will be indicative of the stiffness of the structural member under consideration. Otherwise, for the intertwined influence of stiffness variations on output measurements, using an interaction matrix formulation, a one-to-one correlation is established between an error function and an individual structural member. This unique relationship between the structural members, as represented by blocks \"Member N\" and an error signal block \"EN\" in Figure 4, makes it possible to optimize the error signal, which is a function of the stiffness of a structural member. Thus, the structural stiffness of each individual member can be obtained to establish an accurate analytical model of the system. Input-output measurements are generally corrupted with unwanted, random signals, referred to as noise. The presence of noise in the input-output measurements affect the system identification results. Even a small percentage of noise affects the estimation of the stiffness matrix. To study the effect of noise, output measurements are corrupted with 2%, 5% and 10% rms white noise for the three DOF spring-mass-damper system. It is assumed that noise at output measurements is a random phenomenon and there is no correlation between different output measurements. Unbiased randomness of the noise in output measurements can be addressed by applying an averaging scheme, which should cancel out the effects of noise in estimating the stiffness of individual springs. Monte-Carlo  
Figure 4: Block diagram showing input error functions for tracking of member stiffness. simulations for the input-output measurements are carried out, and using 100 different random input-output measurements, the stiffness of each spring is estimated. Figure 5 plots the variation in estimation of stiffness in springs K1, K2 and K3 for different % rms white noise in output measurements. The data bars in \"red\" indicate the correct stiffness of the corresponding spring used in numerical simulations, and the data bars in \"blue\" indicate the estimated stiffness with optimization of an input error function. It should be noted that data presented in \"blue\" are the estimated stiffness values in the range as indicated along the \(x\)-axis and the number of estimations normalized with the total number of Monte-Carlo simulations as indicated along the \(y\)-axis. As the % rms noise is increased from 2% to 10%, more scatter in the estimation of stiffness can be observed. Table 1 summarizes the data for 100 iterations. It can be observed that even in the presence of 10% rms noise, stiffness estimation for individual springs is remarkably close to the actual stiffness. Although stiffness estimation is close to the actual stiffness, an increase in % rms noise results in an increase in scatter, as evident from plots in Figure 5 and the standard deviation listed in Table 1. A greater standard deviation indicates less confidence in the estimation of stiffness and vice versa.