# Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network
## 5 Linear System Analysis  
First, we consider the case where the inertial magnetic mass and the tip magnet are far apart so that the magnetic repulsive force can be neglected and the cantilever beam behaves as a linear inverted pendulum. The elastic potential of the beam is assumed of the form \(U(x)=kx^{2}/2\). Substituting the potential into the SDEs system (5), a linear system of SDEs for the dimensionless variables can be obtained using the diagonal matrix  
\[\mathbf{P}=\text{diag}[l_{0}^{-1},Tl_{0}^{-1},C_{\text{P}2}Q_{0}^{-1},TQ_{0}^{ -1},C_{\text{P}2}Q_{0}^{-1}] \tag{17}\]  
where \(l_{0}\), \(Q_{0}\) are normalizing constants equal to one, with dimension of a length and of a charge, respectively, and \(T=1/\omega=\sqrt{m/k}\) is a normalization time. The following SDEs system is obtained:  
\[d\mathbf{X}_{t}=\mathbf{A}\,\mathbf{X}_{t}\,dt+\varepsilon\,\mathbf{B}\,dW_{t} \tag{18}\]  
where  
\[\mathbf{A}=\left[ \begin{array}{cccc}0&1&0&0&0\\ -1&-\rho&-\beta&0&0\\ 0&\alpha&0&-1&0\\ 0&0&\mu&0&-\mu\\ 0&0&0&\eta&-\delta\end{array} \right],\qquad\mathbf{B}=[0,\sigma,0,0,0]^{T} \tag{19}\]  
and with the following values of the parameters  
\[\rho=\frac{\gamma}{\sqrt{mk}},\quad\beta=\frac{\alpha}{C_{p2}k},\quad\mu= \frac{m}{k\,C_{\text{P}2}\,L_{S}}\]  
\[\eta=\frac{C_{p2}}{C_{p}},\quad\delta=\frac{G_{L}}{C_{p}}\sqrt{\frac{m}{k}}, \quad\sigma=\frac{1}{m}\Big{(}\frac{m}{k}\Big{)}^{\frac{3}{4}} \tag{20}\]  
Note that, by definition, all parameters are positive. First of all, we observe that for the set of parameters that we shall use, all eigenvalues of the matrix \(\mathbf{A}\) will have negative real parts. Taking the stochastic expectation in Equation (18), we obtain the ordinary differential equations (ODEs) system  
\[\frac{d}{dt}\text{E}[\mathbf{X}_{t}]=\mathbf{A}\,\text{E}[\mathbf{X}_{t}] \tag{21}\]  
Because matrix \(\mathbf{A}\) is stable, it follows that \(\text{E}[\mathbf{X}_{t}]\to 0\) for \(t\to+\infty\). For the second-order moments, we evaluate  
\[d\left(\mathbf{X}_{t}\mathbf{X}_{t}^{T}\right)= d\mathbf{X}_{t}\,\mathbf{X}_{t}^{T}+\mathbf{X}_{t}\,d\mathbf{X}_{t}^{T}+d \mathbf{X}_{t}\,d\mathbf{X}_{t}^{T}\] \[= \Big{(}\mathbf{A}\,\mathbf{X}_{t}\,\mathbf{X}_{t}^{T}+\mathbf{X}_{ t}\mathbf{X}_{t}^{T}\,\mathbf{A}^{T}+\varepsilon^{2}\mathbf{B}\,\mathbf{B}^{T} \Big{)}dt+\varepsilon\Big{(}\mathbf{B}\mathbf{X}_{t}^{T}+\mathbf{X}_{t}\mathbf{ B}^{T}\Big{)}dW_{t} \tag{22}\]  
where we use [PERSON]'s lemma, e.g., \(dt^{2}=dt\,dW_{t}=0\), and \(dW_{t}^{2}=dt\). Taking expectations on both sides, using the martingale property of [PERSON]'s integral and the fact that asymptotically \(\mathrm{E}[\mathbf{X}_{t}]=0\), we obtain the Lyapunov equation  
\[\frac{d\boldsymbol{\sigma}}{dt}=\mathbf{A}\,\boldsymbol{\sigma}+\boldsymbol{ \sigma}\,\mathbf{A}^{T}+\varepsilon^{2}\mathbf{B}\,\mathbf{B}^{T} \tag{23}\]  
where \(\boldsymbol{\sigma}=\mathrm{E}[\mathbf{X}_{t}\mathbf{X}_{t}^{T}]\) is the covariance matrix, and we used the fact that \(\boldsymbol{\sigma}\) is symmetric. Because matrix \(\mathbf{A}\) is stable and \(\mathbf{B}\mathbf{B}^{T}\) is symmetric, the solution of (23) is unique. The asymptotic solution is obtained solving the stationary Lyapunov equation  
\[\mathbf{A}\,\boldsymbol{\sigma}+\boldsymbol{\sigma}\,\mathbf{A}^{T}+ \varepsilon^{2}\mathbf{B}\,\mathbf{B}^{T}=0 \tag{24}\]  
To optimize the matching network, we solved the stationary Lyapunov Equation (24) for many values of the parameters \(L_{S}\) and \(C_{P}\) within a given interval. Figure 3 shows the root mean square value of the output voltage \(v_{o(\mathrm{rms})}=\sqrt{\mathrm{E}[v_{o}^{2}(t)]}\) versus the values of \(L_{S}\) and \(C_{P}\). The values of the other parameters used in our analysis are summarized in Table 1, and they are well comparable to those used in several other recent works, e.g., [33; 38; 45].