## 2 Acceleration Based Moving Force Identification Theory
### MFI State Space Equations and Their General Solution  
The state-space form is used to replace the 2 nd order differential equation with two 1 st order equations. Thus, acceleration is represented as the first derivative of unknown velocity (as opposed to the 2 nd derivative of unknown displacement). Hence, Equation (1) is rewritten as:  
\[\{\dot{v}\}=\left[M_{g}\right]^{-1}\{\left[-K_{g}\right]\{u\}-\left[C_{g} \right]\{\dot{u}\}\}+\left[M_{g}\right]^{-1}\!\left[L\right]\{g(t)\} \tag{2}\]  
The vector, \(\{X\}\) is used to represent displacements and velocities of nodes as follows:  
\[\{X\}=\left\{\begin{array}{c}u\\ \dot{u}\end{array}\right\} \tag{3}\]  
The first derivative of \(\{X\}\) can be written as:  
\[\left\{\frac{dX}{dt}\right\}=\{\dot{X}\}=\left\{\begin{array}{c}\dot{u}\\ \ddot{u}\end{array}\right\}=\left\{\begin{array}{c}\dot{u}\\ M_{g}{}^{-1}\{\left[-K_{g}\right]\{u\}-\left[C_{g}\right]\{\dot{u}\}+\left[M_{ g}\right]^{-1}\!\left[L\right]\{g(t)\}\end{array}\right\} \tag{4}\]  
Rearranging Equation (4) gives:  
\[\left\{\frac{dX}{dt}\right\}=\left[A\right]\{X\}+\left[D\right]\{g(t)\} \tag{5}\]where,  
\[[A]=\left[\begin{array}{cc}[0]&[I]\\ -\big{[}M_{g}\big{]}^{-1}\big{[}K_{g}\big{]}&-\big{[}M_{g}\big{]}^{-1}\big{[}C_{g }\big{]}\end{array}\right] \tag{6}\]  
\[[D]=\left[\begin{array}{cc}[0]\\ \big{[}M_{g}\big{]}^{-1}\big{[}L\big{]}\end{array}\right] \tag{7}\]  
In Equation (6), \([I]\) denotes the identity matrix and \([0]\) denotes the null matrix. Many identities and null matrices are used in this paper and their size changes according to the context. For example, the identity and null matrices in Equation (6) are the same size as \([M_{g}]^{-1}\big{[}K_{g}\big{]}\) and \([M_{g}]^{-1}\big{[}C_{g}\big{]}\). Hence, the size of the identity matrix and null matrix in Equation (6) is \([n_{dof}\times n_{dof}]\). Equation (5) is the state-space form of the equation of motion. To solve these differential equations, [PERSON] and [PERSON] [34] use Laplace transforms and Pade Approximations to give the following solution:  
\[\{X\}_{j+1}=\Big{[}e^{[A]h}\big{]}\{X\}_{j}+\left[\Big{[}e^{[A]h}-[I]\Big{]} \big{[}A]^{-1}\Bigg{(}\begin{array}{cc}[0]\\ \big{[}M_{g}\big{]}^{-1}\big{[}L\big{]}\end{array}\Bigg{)}\right]\{g\}_{j} \tag{8}\]  
where \(j\) is the time step number and \(h\) is the time difference between the \(j\)th and \(j\)+1 th time steps. In Equation (8), \(e^{[A]h}\) is referred to as the exponential of a matrix, which arises in the solution to a system of differential equations. The size of \(e^{[A]h}\) is \([2n_{dof}\times 2n_{dof}]\). From Equation (8), this time step variable \(\{X_{j+1}\}\) can be solved given the previous time step variable \(\{X_{j}\}\). To simplify the format of Equation (8), the following is defined:  
\[[M]=\Big{[}e^{[A]h}\Big{]} \tag{9}\]  
\[[P]=\left[\Big{[}e^{[A]h}-[I]\Big{]}\big{[}A]^{-1}\Bigg{(}\begin{array}{cc} [0]\\ \big{[}M_{g}\big{]}^{-1}\big{[}L\big{]}\end{array}\Bigg{)}\right] \tag{10}\]  
Then, Equation (8) can be written as:  
\[\{X\}_{j+1}=[M]\{X\}_{j}+[P]\{g\}_{j} \tag{11}\]  
The solution of Equation (11) has been validated and its accuracy deemed acceptable by others [30]. The size of \([P]\) is \((2n_{dof}\times n_{dof})\) and the size of \([M]\) is \((2n_{dof}\times 2n_{dof})\). However, to improve the accuracy of the force vector solution, [PERSON] et al. [18] applied first-order regularization to the force vector to achieve the following solution:  
\[\left\{\begin{array}{cc}X\\ g\end{array}\right\}_{j+1}=\left[\begin{array}{cc}[M]&\big{[}P_{j}\big{]} \\ \big{[}0\big{]}&[I]\end{array}\right]\left\{\begin{array}{cc}X\\ g\end{array}\right\}_{j}+\left[\begin{array}{cc}[0]\\ [I]\end{array}\right]\{r\}_{j} \tag{12}\]  
where, \(r_{j}\) is the first derivative of force, \(g_{j}\). [PERSON] et al. [17] show that the first-order regularization approach significantly improves the accuracy of the solution. The regularization technique is applied to the ill-conditioned problem in this paper to smooth the force vector results.