多个WET Stations情况下非线性EHmodel的凸性

Convexity Analysis of Nonlinear Wireless Power Transfer with Multiple RF Sources. Xiaopeng Yuan et.al. IEEE Transactions on Vehicular Technology, 2022 (pdf) (Citations 0)

Quick Overview

• 用Bruno给的能量收割模型，进行分析多个WET Stations联合供能的情况下，结合非线性，分析最终接收能量和入射能量的关系（最终接收能量对于入射能量的倒数是凸的）
• 多个WET stations 联合优化，就意味着可以通过在多个独立信号之间进行功率分配，这就是一个新的优化方向。

System model

单个能量接收： \[ \begin{aligned} e^{\frac{R_{\mathrm{L}} I_{\mathrm{out}}}{n v} t}\left(I_{\mathrm{out}}+I_{s}\right) & \approx I_{s}+\sum_{i=1}^{n_{0}} \bar{k}_{i} R_{\mathrm{ant}}^{\frac{i}{2}} \mathbb{E}\left\{y_{\mathrm{in}}^{i}\right\} \\ & \approx I_{s}+\sum_{i=1}^{n_{0}} \beta_{i} \mathbb{E}\left\{y_{\mathrm{in}}^{i}\right\} \end{aligned} \] 然后得到：

最终得到单个的表达式： \[ P_{\mathrm{dc}}=I_{\mathrm{out}}^{2} R_{\mathrm{L}} \approx\left(\frac{1}{a} W_{0}\left(a e^{a I_{s}}\left(I_{s}+\sum_{i=1}^{n_{0}} \beta_{i} \mathbb{E}\left\{y_{\mathrm{in}}^{i}\right\}\right)\right)-I_{s}\right)^{2} R_{L} \]

Main Idea

由于 \[ I_{\mathrm{out}} \approx \frac{1}{a} W_{0}\left(a e^{a I_{s}} \varphi(\mathbf{Q})\right)-I_{s} \] 且其Hessian矩阵为： \[ \nabla_{\mathbf{u}}^{2} I_{\text {out }}(\mathbf{Q})=\frac{1}{\left(a \varphi(\mathbf{Q})+e^{a I_{\text {out }}(\mathbf{Q})}\right) \varphi(\mathbf{Q})}\left(\nabla_{\mathbf{u}}^{2} \varphi(\mathbf{Q}) \cdot \varphi(\mathbf{Q})-\nabla_{\mathbf{u}} \varphi(\mathbf{Q}) \cdot \nabla_{\mathbf{u}}^{T} \varphi(\mathbf{Q})+\frac{e^{2 a I_{\text {out }}(\mathbf{Q})} \nabla_{\mathbf{u}} \varphi(\mathbf{Q}) \cdot \nabla_{\mathbf{u}}^{T} \varphi(\mathbf{Q})}{\left(a \varphi(\mathbf{Q})+e^{a I_{\text {out }}(\mathbf{Q})}\right)^{2}}\right) \] 所以有\(\varphi(\mathbf{Q})>0\)，且当满足 \[ \nabla_{\mathbf{u}}^{2} \varphi(\mathbf{Q}) \cdot \varphi(\mathbf{Q})-\nabla_{\mathbf{u}} \varphi(\mathbf{Q}) \cdot \nabla_{\mathbf{u}}^{T} \varphi(\mathbf{Q}) \] 为半正定的时候，\(I_{out}(\mathbf{Q})\)对于\(\textbf{u}\)是凸的。其实等价于\(f(\mathbf{Q}(\mathbf{u}))\)的Hessian矩阵是半正定的。

换句话说，保证\(f(\mathbf{Q}(\mathbf{u}))=\ln \varphi(\mathbf{Q}(\mathbf{u}))\)对于\(\mathbf{u}\)是凸的，\(I_{out}\)就是对于\(\mathbf{u}\)是凸的，而\(P_{dc}=I^2_{out}R_L\)也就是对于\(\mathbf{u}\)是凸的。

取\(u_{m}=\ln Q_{m} \text {, i.e., } Q_{m}\left(u_{m}\right)=e^{u_{m}}, \forall m \in \mathcal{M}\)

然后证明了\(g(\mathbf{z})\)对于\(\mathbf{z}\)而言是半正定的，利用了

具有非负对角元的Hermit对角占优（主对角大于该列所有元素）矩阵是半正定的

性质

由于 \[ \nabla_{\mathbf{z}}^{2} \hat{g}(\mathbf{z})=\frac{1}{\left(\sum_{n=1}^{N} e^{z_{n}}\right)^{2}}\left\{\begin{array}{cclc} e^{z_{1}} \sum_{n \neq 1} e^{z_{n}} & -e^{z_{1}} e^{z_{2}} & \ldots & -e^{z_{1}} e^{z_{N}} \\ -e^{z_{1}} e^{z_{2}} & e^{z_{2}} \sum_{n \neq 2} e^{z_{n}} & \ldots & -e^{z_{2}} e^{z_{N}} \\ \ldots & \ldots & \ldots & \ldots \\ -e^{z_{1}} e^{z_{N}} & -e^{z_{2}} e^{z_{N}} & \ldots & e^{z_{N}} \sum_{n \neq N} e^{z_{n}} \end{array}\right\} \] 所以\(P_{dc}\)对于\(\text{ln}Q_m\)而言是凸的。

然后由保凸原则得到

进一步得到：

The convexity properties proved in Theorem 1-4 hold for any given truncation order n0, namely they also hold if the truncation order n approaches infifinity