# Tracking of Stiffness Variation in Structural Members Using Input Error Function Observers
## 2 Theoretical Background
\begin{array}{l}x(k+p)=&A^{p}x(k)+\mathbb{\hat{U}}_{onti}u_{1}^{(p)}(k)+\cdots +\mathbb{\hat{U}}_{ontr}u_{r}^{(p)}(k)+B_{1}u_{1}(k+p-1)\\ &+\cdots+B_{r}u_{r}(k+p-1)\\ y_{p}(k)=&O_{bs}x(k)+T_{op1}u_{1}^{(p)}(k)+\cdots+T_{opr}u_{r}^{(p)}(k)+D_{1}u_ {1}(k+p-1)+\cdots\\ &+D_{r}u_{r}(k+p-1)\end{array}  \tag{8}\]  
where  
\[ \begin{array}{l}B=[B_{1},B_{2},\ldots,B_{r}],\\ u_{i}^{p}(k)=\left[\begin{array}{c}u_{i}(k)\\ u_{i}(k+1)\\ \vdots\\ u_{i}(k+p-1)\end{array}\right],\quad T_{opi}=\left[\begin{array}{cccc}O_{ onti}=\left[A^{p-1}B_{i},\ldots,A^{2}B_{i}\ AB_{i}\right],\\ D&0&0&\cdots&0\\ CB_{i}&D&\cdots&\cdots&0\\ CAB_{i}&CB_{i}&D&\cdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&0\\ CA^{p-2}B_{i}&\cdots&CAB_{i}&CB_{i}&D\end{array}\right]\end{array}  \tag{9}\]  
The \(D\) matrix becomes a null matrix if the measurements are displacements. Acceleration measurements can be easily accommodated with a simple modification [17]. With the introduction of an interaction matrix \(M_{i}\), Equation (8) can be written as  
\[ \begin{array}{l}x(k+p)\quad=\left(A^{p}+M_{i}O\right)x(k)+\left(\mathbb{ \hat{U}}_{ont1}+CM_{i}T_{op1}\right)u_{1}^{(p)}(k)+\cdots\\ \quad+\left(\mathbb{\hat{U}}_{ontr}+CM_{i}T_{opr}\right)u_{r}^{(p)}(k)+B_{1}u_ {1}(k+p-1)+\cdots\\ \quad+B_{r}u_{r}(k+p-1)-M_{i}y_{p}(k)\end{array}  \tag{10}\]  
Interaction matrix \(M_{i}\) enables decoupling the influence of changes to the stiffness of the members in output measurements except, for the term \(CB_{i}\). Pre-multiplying the Equation (10) by \(\mathbb{\hat{U}}\) results in  
\[ \begin{array}{l}\mathrm{y}(k+p)\quad=\left(CA^{p}+CM_{i}O\right)x(k)+\left( C\mathbb{\hat{U}}_{ont1}+CM_{i}T_{op1}\right)u_{1}^{(p)}(k)+\cdots\\ \quad+\left(C\mathbb{\hat{U}}_{ontr}+CM_{i}T_{opr}\right)u_{r}^{(p)}(k)+CB_{1} u_{1}(k+p-1)+\cdots\\ \quad+CB_{r}u_{r}(k+p-1)-CM_{i}y_{p}(k)\end{array}  \tag{11}\]  
The number of independent sensors needs to be at least equal to the number of structural members (\(m\geq r\)) to ensure the existence of \(M_{i}\) such that failure among each structural member can be distinguished. This also ensures an orthogonal row vector \(N_{i}\) to all remaining column vectors \(CB_{j},\ j\
eq i\)  
\[ \begin{array}{l}CA^{p}+CM_{i}O_{bs}=0\\ \mathbb{\hat{C}}\mathbb{\hat{U}}_{ontrj}+CM_{i}T_{opj}=0,\ \forall j\
eq i\\ N_{i}^{T}(CB_{i})\quad=0,\ \forall j\
eq i\end{array}  \tag{12}\]  
Introduction of an orthogonal row vector \(N_{i}\) has eliminated the influence of the term \(CB_{i}\) in Equation (11), provided conditions in Equation (12) are satisfied. Equation (11) thus can be written in terms of each input \(i\), by pre multiplying by a vector \(N_{i}\)  
\[ \begin{array}{l}N_{i}^{T}y(k+p)=\quad N_{i}^{T}\big{(}C\mathbb{\hat{U}}_{onti }+CM_{i}T_{opi}\big{)}u_{i}^{(p)}(k)+N_{i}^{T}CB_{i}u_{i}(k+p-1)\\ \quad-N_{i}^{T}CM_{i}y_{p}(k)\end{array}  \tag{13}\]  
Using Equation (13), input error function can be written in general form by accounting for actuator error \(z_{i}(k)\) such that \(u_{i}(k)=\overline{u_{i}}(k)+z_{i}(k)\) using coefficients \(\alpha_{\mathrm{i}}\) and \(\beta_{\mathrm{i}}\). Introduction of coefficients \(\alpha_{\mathrm{i}}\) and \(\beta_{\mathrm{i}}\) are critical in transforming Equation (14) to a scalar equation. A unique relationship is established between an error function and a particular structural member. This is a key to uncoupling the influence of structural members on all output measurements except one and enables tracking the stiffness variation in structural members using the input error function in Equation (13). \[ \begin{array}{ll}e_{i}(k)=\alpha_{0}y(k)&+\alpha_{1}y(k-1)+\cdots+\alpha_{p}y(k -p)+\beta_{1}\overline{u_{i}}(k-1)+\beta_{2}\overline{u_{i}}(k-2)\\ &+\cdots+\beta_{p}\overline{u_{i}}(k-p)\end{array}  \tag{14}\]  
Coefficients \(\alpha_{0},\alpha_{1},\ldots,\alpha_{p}\) (are \(1\times m\) row vectors) and \(\beta_{1},\beta_{2},\ldots,\beta_{p}\) (scalars) in Equation [14] above are related to the terms in state space formulation as indicated in Equation (15) below  
\[