# FOG De-Noising Algorithm Based on Augmented Nonlinear Differentiator and Singular Spectrum Analysis
## 2 Algorithm  
### Augmented Nonlinear Differentiator  
In this section, some fundamental theorems on designing augmented nonlinear differentiator (AND) is introduced.  
**Lemma 1**[23]: _Let \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}\) be a locally Lipschitz continuous function with \(f(0,0)=0\), the following system is globally asymptotically stable with respect to the equilibrium point \((0,0)\):_  
\[\left\{\begin{array}{l}\dot{x}_{1}(t)=x_{2}(t),x_{1}(0)=x_{10}\\ \dot{x}_{2}(t)=f(x_{1}(t),x_{2}(t)),x_{2}(0)=x_{20}\end{array}\right. \tag{1}\]  
_provided that the solution of differential system (1) satisfies \((x_{1}(t),x_{2}(t))\to 0\) as \(t\rightarrow\infty\), where \((x_{10},x_{20})\) is any given initial value. Then for any differential and bounded input signal \(v(t)\) with the condition of \(A=\sup_{t\in[0,\infty)}\bigl{|}\dot{v}(t)\bigr{|}<\infty\), the solution of the following tracking differentiator:_  
\[\left\{\begin{array}{l}\dot{z}_{1R}(t)=z_{2R}(t),z_{1R}(0)=z_{10}\\ \dot{z}_{2R}(t)=R^{2}f(z_{1R}(t)-v(t),z_{2R}(t)/R),z_{2R}(0)=z_{20}\end{array}\right. \tag{2}\]  
_is convergent in the sense that: For every \(a>0\), \(z_{1R},z_{2R}\) are uniformly convergent to the input signal \(v(t)\) and its derivative on \([a,\infty)\) as \(R\rightarrow\infty\), respectively. Additionally, \(R>0\) is referred as the acceleration factor._  
**Lemma 2**[23]: _Suppose that there exist some positive constants \(\delta_{1},\delta_{2}>0\) such that \(sig(x_{1})\geq\delta_{1}x_{1}\), \(sig(x_{2})\geq\delta_{2}x_{2}\), \(0<sig^{\prime}(x_{0})<\delta_{1}\delta_{2}\) for any \(x_{0},x_{1},x_{2}\in\mathbb{R}\), then the following system is globally asymptotically stable with respect to the equilibrium point \((0,0,0)\) on the condition that \(a_{0}>0,a_{1}>0,a_{2}>0,b>0\):_  
\[\left\{\begin{array}{l}\dot{x}_{0}(t)=x_{1}(t),x_{0}(0)=x_{00}\\ \dot{x}_{1}(t)=x_{2}(t),x_{1}(0)=x_{10}\\ \dot{x}_{2}(t)=-sig(x_{0}(t);a_{0},b)-sig(x_{1}(t);a_{1},b)-sig(x_{2}(t);a_{2 },b),x_{2}(0)=x_{20}\end{array}\right. \tag{3}\]  
_where \(sig()\) denotes the sigmoid function. Then for any bounded input signal \(v(t)\), satisfying \(M=\sup_{t\in[0,\infty)}\bigl{|}v(t)\bigr{|}<\infty\), the solution of the following augmented nonlinear differentiator:_  
\[\left\{\begin{array}{l}\dot{z}_{0R}(t)=z_{1R}(t),z_{0R}(0)=z_{00}\\ \dot{z}_{1R}(t)=z_{2R}(t),z_{1R}(0)=z_{10}\\ \dot{z}_{2R}(t)=-R^{3}\bigl{[}sig(z_{0R}(t)-\int v(\tau)d\tau;a_{0},b)+sig(z_{ 1R}(t)/R;a_{1},b)+sig(z_{2R}(t)/R^{2};a_{2},b)\bigr{]},z_{2R}(0)=z_{20}\end{array}\right. \tag{4}\]  
_is convergent in the sense that: for every \(a>0\), \(z_{1R},z_{2R}\) are uniformly convergent to the input signal \(v(t)\) and its derivative on \([a,\infty)\) as \(R\rightarrow\infty\), respectively. Additionally \(R>0\) is referred to as the acceleration factor._  
It should be noted that in the absence of measurement noise, the estimation quality of the described tracking differentiator, such as the dynamic performance, is mainly dependent on the characteristic of acceleration function designed. Furthermore, Lemmas 1 and 2 also indicate that the approximation error can be made sufficiently small if acceleration factor is chosen as a high-gain than necessarily.