# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 2 Component Modal Synthesis
### State-Space Representation of Reduced-Order Model
\begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{2R-1}\\ \dot{x}_{2R}\end{array} \right\}=\left[ \begin{array}{ccccc}0&1&\ldots&0&0\\ -\omega_{*1}^{2}&0&\ldots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\ldots&0&1\\ 0&0&\ldots&-\omega_{*R}^{2}&0\end{array} \right]\left\{ \begin{array}{c}x_{1} \\ x_{2}\\ \vdots\\ x_{2R-1}\\ x_{2R}\end{array} \right\}\] \[+[\phi]^{\intercal}\left\{ \begin{array}{c}0\\ p_{1}\\ \vdots\\ 0\\ p_{R}\end{array} \right\} \tag{33}\]  
The modal displacements and velocities are obtained by solving Equation (33). For example, \(x_{1}\), \(x_{2}\) represent the modal displacement and velocity of the first retained degree of freedom. Let \(D\) be the number of degrees of freedom where the structure's displacement and velocity are to be monitored. Then, suppose \((S_{i}^{D}|i\in\{1,\ldots,D\})\) and \((S_{j}|j\in\{1,\ldots,R\})\) are the sequence of the set of degrees of freedom where the kinematics to be observed and the sequence of the set of retained degrees of freedom in the reduced structure, respectively. Let us now define a matrix \([\bar{C}]_{2D\times 2R}\) in the following way:  
\[\bar{C}_{2i-1,2j+1} =1\colon S_{i}^{D}=S_{j},\forall i\in\{1,\ldots,D\},\forall j\in \{1,\ldots,R\}\] \[\bar{C}_{2i,2j+2} =1\colon S_{i}^{D}=S_{j},\forall i\in\{1,\ldots,D\},\forall j\in \{1,\ldots,R\} \tag{34}\] \[\bar{C}_{i,j} =0\colon\text{otherwise}\]  
Let us define \(\Lambda^{ss}\) and \(B^{ss}\) as follows:  
and  
\[B^{ss}=\{\beta^{ss}\in\mathbb{N}^{+}|\beta^{ss}\text{ is even and }\beta^{ss}\leq 2R\}\]  
The original modal matrix \([\phi]\) of the reduced structure can then be modified as follows:  
\[\phi^{ss}_{i,j}= \begin{cases}\phi_{\frac{i}{2},\frac{j}{2}}&\colon\forall i= \lambda^{ss}\in\Lambda^{ss},\forall j=\beta^{ss}\in B^{ss}\\ 0&\colon\text{otherwise}\end{cases}  \tag{35}\]  
where \([\phi^{ss}]\) is the reduced structure's modified modal matrix. The necessity for the aforementioned modification of the modal matrix is to accommodate the additional states related to the system velocity. The kinematic vector at selected degrees of freedom in physical coordinates can be derived using Equations (27), (34), and (35), yielding  
\[\{y\}=[C]\{x\} \tag{36}\]  
where \([C]=[\bar{C}][\phi^{ss}]\), and \(\{\text{x}\}\) is the system kinematics vector in modal coordinates. Furthermore, only a subset of the retained degrees of freedom in a reduced finite element model of a structure is subjected to external excitation. Suppose \(r_{in}\) is the number of excitations applied at distinct degrees of freedom of the structure, where \(r_{in}\leq R\). Let us further assume \((S_{i}^{r_{in}}|i\in\{1,\ldots,r_{in}\})\) is the sequence of the set of degrees of freedom where the external excitations are applied. As a result, Equation (33) can be rewritten as  
\[\left\{ \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\\ \vdots\\ \dot{x}_{2R-1}\\ \dot{x}_{2R}\end{array} \right\} =\left[ \begin{array}{ccccc}0&1&\ldots&0&0\\ -\omega_{*1}^{2}&0&\ldots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\ldots&0&1\\ 0&0&\ldots&-\omega_{*R}^{2}&0\end{array} \right]\left\{ \begin{array}{c}x_{1} \\ x_{2}\\ x_{2R-1}\\ x_{2R}\end{array} \right\}\] \[\quad+[\phi^{ss}]_{2R\times 2R}[\beta^{p}]_{2R\times r_{in}}\left\{ \begin{array}{c}|p_{1}|\\ \vdots\\ |p_{r_{in}}|\end{array} \right\}_{r_{in}\times 1} \tag{37}\]  
where \(|p_{j}|\)\(\forall j\in\{1,\ldots,r_{in}\}\) is the magnitude of \(j\)th externally applied excitation in physical coordinates, and \([\beta^{p}]\) is a Boolean matrix defined as follows  
\[\beta^{p}_{2i,j} =1 \quad:S_{j}^{r_{in}}=S_{i},\forall i\in\{1,\ldots,R\},\forall j\in \{1,\ldots,r_{in}\} \tag{38}\] \[\beta^{p}_{i,j} =0 \quad:\text{otherwise}\]Finally, the state-space representation of a reduced finite element model can be expressed using Equation (37) as  
\[\{\dot{x}\} =[A]\{x\}+[B]\{z\} \tag{39}\] \[\{y\} =[C]\{x\}+[D]\{z\}\]  
where the matrices are represented as  
\[[A] =