# Tracking of Stiffness Variation in Structural Members Using Input Error Function Observers
## 2 Theoretical Background  
In this section, the mathematical formulation for the equivalence of stiffness change and actuator failure is derived. The second-order dynamic system in physical coordinate \(q(t)\), with the external input \(u(t)\) is denoted as  
\[M\hat{q}(t)+\varnothing\hat{q}(t)+Kq(t)=B_{f}u(t) \tag{1}\]  
\(M\), \(\varnothing\) and \(K\) represent the mass, damping and stiffness matrices, respectively. The input influence matrix \(B_{f}\) associates each input to the corresponding degree of freedom. The structural changes to an individual structural member updates the corresponding entries in the global stiffness matrix. When compared to global undamaged stiffness matrix \(K\), changes due to structural changes in stiffness of the members can be expressed as \((\Delta K)\)  
\[M\tilde{q}(t)+\varnothing\hat{q}(t)+(K\pm\Delta K)q(t)=B_{f}u(t) \tag{2}\]  
Any structural updates resulting in changes in stiffness of the members are captured by the term \((\Delta K)\). Structural damage results in decreasing stiffness, while structural strengthening results in increasing the stiffness of the member. The term \(\Delta Kq(t)\) can be viewed as input error \(\Delta u(t)\), and Equation (2) can be rewritten as  
\[M\tilde{q}(t)+\varnothing\hat{q}(t)+(K)q(t)=B_{f}u(t)\pm\Delta u(t) \tag{3}\]  
Since each output is influenced by all other inputs, it is not obvious to identify the location of structural damage. An actuator failure detection algorithm [13; 14] presents a formulation that can decouple the influence of one actuator on the output measurements. Previously, actuator failure formulation was implemented by [PERSON] et al. [29] using an analogy between stiffness change and input error for structural damage detection and localization. Stiffness variation in individual spring elements for a three DOF spring-mass-damper system is identified in real-time; however, no attempt is made to estimate the severity of damage. For the sake of completeness and ease of reference, only critical steps for the input error function formulation [44] are included here. The development of the input error function is in a state-space framework, and hence, Equation (1) is rewritten in first-order form. Assuming an \(n\)-th order, \(r\)-input, \(m\)-output discrete-time model of a system in state-space format  
\[ \begin{array}{l}x(k+1)=Ax(k)+Bu(k)\\ y(k)=Cx(k)+Du(k)\end{array}  \tag{4}\]  
The special interaction matrix formulation is introduced below, which will be a key in establishing a connection between state space models and input-output models [17]. Substituting repeatedly for some \(p\geq 0\), Equation (4) takes the following form  
\[ \begin{array}{l}x(k+p)=A^{p}x(k)+\mathbb{O}_{\text{on}}nt_{p}(k)\\ y_{p}(k)=O_{bs}x(k)+T_{op}u_{p}(k)\end{array}  \tag{5}\]  
starting with \(y(k)\) and \(u(k)\) and going \(p\) steps into the future, column vectors of output data \(y_{p}(k)\) and input data \(u_{p}(k)\) are obtained  
\[ \begin{array}{l}y_{p}(k)=\left[\begin{array}{c}y(k)\\ y(k+1)\\ \vdots\\ y(k+p-1)\end{array}\right],\,u_{p}(k)=\left[\begin{array}{c}u(k)\\ u(k+1)\\ \vdots\\ u(k+p-1)\end{array}\right]\end{array}  \tag{6}\]  
Controllability matrix \(\mathbb{O}_{ent}\) of the order of \(n\times pr\), an observability matrix, \(O_{bs}\) of the order of \(pm\times n\) and the \"Toeplitz\" matrix of the system of Markov parameters \(T_{op}\) of the order of \(pm\times pr\) from Equation (5) can be written in terms of state space formulation, as shown in Equation (7)  
\[ \begin{array}{l}\mathbb{O}_{\text{ont}}\;=\left[A^{p-1}B,\cdots,AB,B\right],O_{bs}=\left[C,CA,\cdots,CA^{p-1}\right],\\ T_{op}=\left[\begin{array}{ccccc}D&0&0&\cdots&0\\ CB&D&\cdots&\cdots&0\\ CAB&CB&D&\ldots&\vdots\\ \vdots&\vdots&\vdots&\vdots&0\\ CA^{p-2}B&\cdots&CAB&CB&D\end{array}\right]\end{array}  \tag{7}\]Considering the contribution of each individual input \(i=1,2,\ldots,r\), Equation (5) can be rewritten as  
\[