## 2 Nonlinear Subspace Identification of Mechanical Systems
### State-Space Nonlinear Feedback Description  
The governing equation of vibration of the nonlinear mechanical system under generalized coordinates is  
\[\mathbf{M}\ddot{\boldsymbol{q}}(t)+\mathbf{C}\dot{\boldsymbol{q}}(t)+\mathbf{K }\boldsymbol{q}(t)+\sum_{j=1}^{n}\mu_{j}\mathbf{L}_{j}\delta_{j}(\boldsymbol{q }(t),\dot{\boldsymbol{q}}(t))=\boldsymbol{p}(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\in\mathbb{R}^{m\times m}\) are the linear mass, damping, and stiffness matrices, respectively. The superscript \(m\) means the nonlinear mechanical system is modeled by an \(m\)-DOF finite-element (FE) model. \(\boldsymbol{p}(t)\) and \(\boldsymbol{q}(t)\) are the generalized external force and displacement vectors, respectively. The nonlinear components are modeled by a lumped form that contains \(n\) nonlinearities in the system. \(g_{j}\) is the mathematical description of the \(i\)-th nonlinearity and needs to be known previously. The location of the \(j\)-th nonlinearity is specified by a vector \(\mathbf{L}_{j}\), the element of which is 0, 1, or \(-1\). The unknown nonlinear coefficient is \(\mu_{j}\). As subspace methods can obtain an identified state-space model, the second-order vibrating equation in Equation (1) is transformed into a first-order state-space description for better understanding. Defining the state vector \(\boldsymbol{x}=( \begin{array}{cc}\boldsymbol{q}^{\mathrm{T}}&\dot{ \boldsymbol{q}}^{\mathrm{T}}\end{array} )^{\mathrm{T}}\in\mathbb{R}^{N}\) and moving the nonlinear term to the right-hand side by considering the feedback interpretation [26] as shown in Figure 1, the dynamic equations are recast into  
\[ \begin{cases}\dot{\boldsymbol{x}}&=\mathbf{A}_{\mathrm{c}}\boldsymbol{x}+ \mathbf{B}_{\mathrm{c}}^{\mathrm{e}}\boldsymbol{u}\\ \boldsymbol{q}&=\mathbf{C}_{\mathrm{c}}\boldsymbol{x}+\mathbf{D}_{\mathrm{c}}^ {\mathrm{e}}\boldsymbol{u}\end{cases}  \tag{2}\]  
where the subscript c indicates the _continuous-time system_; superscript e stands for _extended_; \(\mathbf{A}_{\mathrm{c}}\in\mathbb{R}^{N\times N}\), \(\mathbf{B}_{\mathrm{c}}^{\mathrm{e}}\in\mathbb{R}^{N\times(m+n)}\), \(\mathbf{C}_{\mathrm{c}}\in\mathbb{R}^{m\times N}\), and \(\mathbf{D}_{\mathrm{c}}^{\mathrm{e}}\in\mathbb{R}^{m\times(m+n)}\) are the state, extended input, output, and extended direct feed-through matrices, respectively; and \(N\) = 2 m. The extended input vector is \(\boldsymbol{u}=( \begin{array}{cc}\boldsymbol{p}(t)^{\mathrm{T}}&g_{1}(t)^{ \mathrm{T}}&g_{2}(t)^{\mathrm{T}}&\cdots&g_{n}(t)^{\mathrm{T}}\end{array} )^{ \mathrm{T}}\in\mathbb{R}^{m+n}\). The transform between state-space and physical-space model is  
Figure 1: The feedback interpretation of nonlinear vibrating system. \[ \begin{split}\mathbf{A}_{\mathrm{c}}&=\left[\begin{array}{ c c c c}\mathbf{0}&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array}\right]\\ \mathbf{B}_{\mathrm{c}}^{\mathrm{e}}&=\left[\begin{array}{ c c c c}\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{M}^{-1}&\mathbf{M}^{-1}\mu_{1}\mathbf{L}_{1}&\mathbf{M}^{-1}\mu_{2} \mathbf{L}_{2}&\cdots&\mathbf{M}^{-1}\mu_{n}\mathbf{L}_{n}\end{array}\right] \\ \mathbf{C}_{\mathrm{c}}&=\left[\begin{array}{ c c c c}\mathbf{1}&\mathbf{0}\end{array}\right]\\ \mathbf{D}_{\mathrm{c}}^{\mathrm{e}}&=\left[\begin{array}{ c c c c}\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}\end{array}\right] \end{split}  \tag{3}\]  
Equation (2) describe a continuous-time state-space system. However, in practice, the excited and observed signals of a vibrating system are measured by discrete-time sampling. A discrete-time translation of Equation (2) is introduced, and the discrete-time state-space system is  
\[ \begin{cases}\dot{\boldsymbol{x}}(k+1)&=\mathbf{A}_{\mathrm{d}}\boldsymbol{x} (k)+\mathbf{B}_{\mathrm{d}}^{\mathrm{e}}\boldsymbol{u}(k)\\ \boldsymbol{q}(k)&=\mathbf{C}_{\mathrm{d}}\boldsymbol{x}(k)+\mathbf{D}_{ \mathrm{d}}^{\mathrm{e}}\boldsymbol{u}(k)\end{cases}