## 2 Nonlinear Subspace Identification of Mechanical Systems
### Estimation of Underlying Linear System and Nonlinear Coefficients  
As mentioned previously, the estimated matrices \(\hat{\mathbf{A}}_{\mathrm{d}},\hat{\mathbf{B}}_{\mathrm{d}}^{\mathrm{e}},\hat {\mathbf{C}}_{\mathrm{d}}\), and \(\hat{\mathbf{D}}_{\mathrm{d}}^{\mathrm{e}}\) are within a similarity transformation. Therefore, the structural parameters can not be extracted directly from these matrices. Fortunately, it is not necessary to compute the similarity transformation matrix that the system parameters can be obtained based on an invariant matrix called extended FRF matrix defined as  
\[\mathbf{H}_{\mathrm{e}}(\omega)=\mathbf{D}_{\mathrm{c}}^{\mathrm{e}}+\mathbf{ C}_{\mathrm{c}}(j\omega\mathbf{I}-\mathbf{A}_{\mathrm{c}})^{-1}\mathbf{B}_{ \mathrm{c}}^{\mathrm{e}} \tag{10}\]  
where the subscript e stands for _extended_. Here, only in Equation (10), \(j=\sqrt{-1}\) is the imaginary unit and \(\omega\) is the angular frequency. The invariance property of the extended FRF matrix up to the similarity transformation has been proofed [27]. However, the identified matrices are discrete-time state-space model and need to transform to the corresponding continuous-time system. In general, the zero-order hold assumption is considered and the transformations are \[\begin{split}\mathbf{A}_{\mathbf{d}}&=e^{\mathbf{A}_{\mathrm{c}} \Delta t}\\ \mathbf{B}_{\mathbf{d}}^{\mathbf{e}}&=(e^{\mathbf{A}_{ \mathrm{c}}\Delta t}-\mathbf{I})\mathbf{A}_{\mathrm{c}}^{-1}\mathbf{B}_{ \mathrm{c}}^{\mathbf{e}}\\ \mathbf{C}_{\mathbf{d}}&=\mathbf{C}_{\mathrm{c}}\\ \mathbf{D}_{\mathbf{d}}^{\mathbf{e}}&=\mathbf{D}_{ \mathrm{c}}^{\mathbf{e}}\end{split} \tag{11}\]  
One should point out that, it is difficult to maintain zero-order-hold assumption if the sample frequency is too low. Then, the bilinear transformation should be performed while mapping the discrete-time matrices to continuous-time domain [28].  
Assuming the FRF matrix of underlying linear system is \(\mathbf{H}\) defined likely as Equation (10), the relationship between \(\mathbf{H}_{\mathrm{e}}\) and \(\mathbf{H}\) is  
\[\begin{split}\mathbf{H}_{\mathrm{e}}&=\mathbf{0}+ \left[\begin{array}{cccc}\mathbf{I}&\mathbf{0}\end{array}\right]\left[ \begin{array}{cccc}\mathbf{p}_{11}&\mathbf{P}_{12}\\ \mathbf{P}_{21}&\mathbf{P}_{22}\end{array}\right]\left[\begin{array}{cccc} \mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{M}^{-1}&\mathbf{M}^{-1}\mu_{1}\mathbf{L}_{1}&\cdots&\mathbf{M}^{-1}\mu _{n}\mathbf{L}_{n}\end{array}\right]\\ &=\mathbf{P}_{12}\mathbf{M}^{-1}\left[\begin{array}{cccc}\mathbf{I}&\mu_{1} \mathbf{L}_{1}&\cdots&\mu_{n}\mathbf{L}_{n}\end{array}\right]\\ &=\frac{1}{j\omega}\left(\mathbf{M}^{-1}\mathbf{C}+j\omega\mathbf{I}+ \mathbf{M}^{-1}\mathbf{K}\frac{1}{j\omega}\right)^{-1}\mathbf{M}^{-1}\left[ \begin{array}{cccc}\mathbf{I}&\mu_{1}\mathbf{L}_{1}&\cdots&\mu_{n}\mathbf{L }_{n}\end{array}\right]\\ &=\left(\mathbf{K}+j\omega\mathbf{C}-\omega^{2}\mathbf{M}\right)^{-1}\mathbf{M }\mathbf{M}^{-1}\left[\begin{array}{cccc}\mathbf{I}&\mu_{1}\mathbf{L}_{1}& \cdots&\mu_{n}\mathbf{L}_{n}\end{array}\right]\\ &=\left[\begin{array}{cccc}\mathbf{H}&\mathbf{H}\mu_{1}\mathbf{L}_{1}&\mathbf{ H}\mu_{2}\mathbf{L}_{2}&\cdots&\mathbf{H}\mu_{n}\mathbf{L}_{n}\end{array}\right] \end{split} \tag{12}\]  
where  
\[\begin{split}\mathbf{N}&=j\omega\mathbf{I}-\mathbf{A}_{ \mathrm{c}}=\left[\begin{array}{cc}\mathbf{N}_{11}&\mathbf{N}_{12}\\ \mathbf{N}_{21}&\mathbf{N}_{22}\end{array}\right]\\ \mathbf{P}&=(j\omega\mathbf{I}-\mathbf{A}_{\mathrm{c}})^{-1}=\left[ \begin{array}{cc}\mathbf{P}_{11}&\mathbf{P}_{12}\\ \mathbf{P}_{21}&\mathbf{P}_{22}\end{array}\right]\\ \mathbf{P}_{12}&=-\mathbf{N}_{11}^{-1}\mathbf{N}_{12}\mathbf{S}_{ \mathbf{N}_{11}}^{-1}\\ \mathbf{S}_{\mathbf{N}_{11}}^{-1}&=(\mathbf{N}_{22}-\mathbf{N}_{21}\mathbf{N}_ {11}^{-1}\mathbf{N}_{12})\end{split} \tag{13}\]  
The underlying linear system and the nonlinear coefficients can be identified from Equation (12).