# Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network
## 6 Nonlinear System Analysis
* We solve the moment ODEs system in the weak noise limit, for all values of the matching network parameters \(L_{S}\) and \(C_{P}\) in a given interval, finding the optimal values that maximize the output average power and power efficiency. * In the strong noise limit, we solve the full SDEs system numerically, using the values of \(L_{S}\) and \(C_{P}\) determined in the previous step, to verify whether the matched load offers better performance in terms of output average power and power efficiency with respect to the resistive load setup. To begin with, we introduce the centered variable \(\mathbf{\overline{X}}_{t}=\mathbf{X}_{t}-\mathbf{n}\). Then, \(\mathbf{\overline{X}}_{t}\sim\mathcal{N}(0,\overline{\mathbf{\sigma}})\), where \(\overline{\mathbf{\sigma}}=\mathrm{E}[\mathbf{\overline{X}}_{t}\mathbf{\overline {X}}_{t}^{T}]\) is the covariance matrix, and  
\[d\mathbf{\overline{X}}_{t}=\big{(}\mathbf{A}_{0}(\mathbf{\overline{X}}_{t}+ \mathbf{\mu})+\mathbf{n}(\mathbf{\overline{X}}_{t}+\mathbf{\mu})\big{)}dt+\epsilon \,\mathbf{B}\,dW_{t} \tag{27}\]  
By definition, asymptotically \(\mathbf{\overline{X}}_{t}\to 0\) for \(t\to+\infty\), and therefore, taking expectations, \(\mathbf{\mu}\) solves  
\[\mathbf{A}_{0}\mathbf{\mu}+\mathrm{E}[\mathbf{n}(\mathbf{\overline{X}}_{t}+\mathbf{ \mu})]=0 \tag{28}\]  
For a multivariate centered normal distribution, \(k\)-th order central moments are null if \(k\) is odd, while even orders can be expressed in terms of the elements of the covariance matrix. In particular,  
\[\mathrm{E}[(\overline{X}_{1}+\mu_{1})^{3}]=3\mu_{1}\overline{\mathbf{\sigma}}_{1 1}+\mu_{1}^{3} \tag{29}\]  
and using the definition of \(\mathbf{n}(\mathbf{\overline{X}}_{t}+\mathbf{\mu})\), we have  
\[\mathrm{E}[\mathbf{n}(\mathbf{\overline{X}}_{t}+\mathbf{\mu})]=-\kappa\mu_{1} \left[ \begin{array}{c}0\\ \mu_{1}^{2}+3\overline{\mathbf{\sigma}}_{11}\\ 0\\ \vdots\\ 0\end{array} \right] \tag{30}\]  
For the second-order central moment, repeating the steps (22) and (23), we obtain the nonlinear Lyapunov equation  
\[\frac{d\overline{\mathbf{\sigma}}}{dt}=\mathbf{A}_{0}\overline{\mathbf{\sigma}}+ \overline{\mathbf{\sigma}}\mathbf{A}_{0}^{T}+\mathrm{E}[\mathbf{n}(\mathbf{ \overline{X}}_{t}+\mathbf{\mu})\mathbf{\overline{X}}_{t}^{T}]+\mathrm{E}[\mathbf{ X}_{t}\mathbf{n}^{T}(\mathbf{\overline{X}}_{t}+\mathbf{\mu})]+\varepsilon^{2} \mathbf{B}\mathbf{B}^{T} \tag{31}\]A straightforward calculation yields  
\[\mathbf{n}(\overline{\mathbf{X}}_{t}+\boldsymbol{\mu})\mathbf{X}_{t}^{T}=-\kappa \left[ \begin{array}{cccc}0&0&\ldots&0\\ (\overline{X}_{1}+\mu_{1})^{3}\overline{X}_{1}&(\overline{X}_{1}+\mu_{1})^{3} \overline{X}_{2}&\ldots&(\overline{X}_{1}+\mu_{1})^{3}\overline{X}_{n}\\ 0&0&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&0\end{array} \right] \tag{32}\]  
For a multivariate centered normal distribution,  
\[\mathrm{E}[(\overline{X}_{1}+\mu_{1})^{3}\overline{X}_{1}] =3\,\overline{\boldsymbol{\sigma}}_{11}\Big{(}\overline{ \boldsymbol{\sigma}}_{11}+\mu_{1}^{2}\Big{)} \tag{33a}\] \[\mathrm{E}[(\overline{X}_{1}+\mu_{1})^{3}\overline{X}_{j}] =3\overline{\boldsymbol{\sigma}}_{1j}\Big{(}\overline{ \boldsymbol{\sigma}}_{11}+\mu_{1}^{2}\Big{)} \tag{33b}\]  
and therefore,  
\[\mathrm{E}[\mathbf{n}(\overline{\mathbf{X}}_{t}+\boldsymbol{\mu}) \mathbf{X}_{t}^{T}]+\mathrm{E}[\mathbf{X}_{t}\,\mathbf{n}^{T}(\overline{ \mathbf{X}}_{t}+\boldsymbol{\mu})]=\\ -3\kappa\Big{(}\overline{\boldsymbol{\sigma}}_{11}+\mu_{1}^{2} \Big{)}\left[ \begin{array}{ccccc}0&\overline{\boldsymbol{\sigma}}_{11}&0& \ldots&0\\ \overline{\boldsymbol{\sigma}}_{11}&2\overline{\boldsymbol{\sigma}}_{12}& \overline{\boldsymbol{\sigma}}_{13}&\cdots&\overline{\boldsymbol{\sigma}}_{1n} \\ 0&\overline{\boldsymbol{\sigma}}_{13}&0&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&\overline{\boldsymbol{\sigma}}_{1n}&0&\cdots&0\end{array}