# Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network
## 2 Stochastic Differential Equations and Energy Harvesting System Modeling  
Mechanical vibrations are random in nature, and thus they are best described as a stochastic process. Let \((\Omega,\mathcal{F},P)\) be a probability space, where \(\Omega\) is the sample space, \(\mathcal{F}=(\mathcal{F}_{t})_{t\geq 0}\) is a filtration, i.e., the \(\sigma\)-algebra of all the events, and \(P\) a probability measure. A vector-valued stochastic process \(\mathbf{X}_{t}\) is a vector of random variables parameterized by \(t\in T\). We adopt the standard notation used in probability: capital letters denote random variables, while lower case letters denote their possible values. The parameter space \(T\) is usually the half-line \([0,+\infty[\). Alternatively, the stochastic process can be thought of as the function \(\mathbf{X}_{t}:\Omega\times T\mapsto\mathbb{R}^{d}\). The energy of mechanical vibrations is typically concentrated at low frequencies, but if the frequency spectrum is wide enough, and the noise correlation time is negligible, a white Gaussian noise process may represent a reasonable approximation. A truly white noise process cannot exist in the real world, as the flat spectrum characteristic of a white process implies an infinite power. Even if the white source overestimates the available power, a finite bandwidth is usually recovered since the deterministic dynamics normally embeds  
Figure 1: Schematic representation of a piezoelectric cantilever beam energy harvester. a low-pass behavior. Therefore, the white process approximation is quite common in the literature, and a well-developed theory is available, making the white noise approximation very convenient from the mathematical point of view [22; 23; 25; 30; 31; 39; 40; 41; 42; 43]. White Gaussian noise is the \"formal\" derivative of a Wiener process. A one-dimensional Wiener process \(W_{t}=W(t)\) is characterized by \(\mathrm{E}[W_{t}]=0\) (symbol \(\mathrm{E}[X_{t}]\) denotes expectation of the stochastic process \(X_{t}\) with respect to the measure \(P\)), covariance \(\mathrm{cov}(W_{t},W_{s})=\mathrm{E}[W_{t},W_{s}]=\mathrm{min}(t,s)\) and \(W_{t}\sim\mathcal{N}(0,t)\), where symbol \(\sim\) means \"distributed as\", and \(\mathcal{N}(0,t)\) denotes the normal distribution, centered at zero. A \(d\)-dimensional system of stochastic differential equations (SDEs) driven by the one-dimensional Wiener process \(W_{t}\) reads  
\[d\mathbf{Z}_{t}=\mathbf{a}(\mathbf{Z}_{t})dt+\mathbf{B}(\mathbf{Z}_{t})dW_{t} \tag{1}\]  
where \(\mathbf{Z}_{t}:\Omega\times T\mapsto\mathbb{R}^{d}\) is a vector valued stochastic process, the vector valued function \(\mathbf{a}:\mathbb{R}^{d}\mapsto\mathbb{R}^{d}\) is called the drift function, and \(\mathbf{B}:\mathbb{R}^{d}\mapsto\mathbb{R}^{d}\) is termed diffusion. Drift and diffusion are measurable functions satisfying a global Lipschitz condition to ensure the existence and uniqueness solution theorem [31]. If function \(\mathbf{B}(\mathbf{Z}_{t})\) is constant, noise is called unmodulated or additive; otherwise, it is modulated or multiplicative. The SDEs for the energy harvesting system can be derived from classical mechanics, from the characterization of piezoelectric materials, and from the circuit description of the electrical load [18; 19; 28]. For the mechanical part and the piezoelectric transducer, we have  
\[m\dot{x}+\gamma\dot{x}+U^{\prime}(x)+\ae= f_{\mathrm{ext}}(t) \tag{2a}\] \[C_{\mathrm{p}\dot{x}}+\alpha\dot{x}+I_{\mathrm{L}}=0 \tag{2b}\]  
Equation (2a) describes the mechanical domain.