Algebraic differentiators: A detailed introduction¶
This notebook includes a detailed introduction into the theoretical background of algebraic differentiators and shows how to use the proposed implementation.
Content of this notebook¶
Theoretical background: Time-domain and frequency-domain analysis
Numerical differentiation of sample signals: The numerical estimation of derivatives of simulation and experimental data.
Theoretical background¶
Time-domain analysis¶
The algebraic derivative estimation methods were initially derived using differential algebraic manipulations of truncated Taylor series \cite{mboup2007b,mboup2009a}. Later works \cite{liu2011a,kiltz2017} derived these filters using an approximation theoretical approach that yields a straightforward analysis of the filter characteristics, especially the estimation delay. Using this approach, the estimate of the $n$-th order derivative of a signal ${t\mapsto y(t)}$ denoted $\hat{y}^{(n)}$ can be approximated as
\begin{equation}
\hat{{y}}^{(n)}(t)=\int_{0}^{T}g^{(n)}_T(\tau)y(t-\tau)\textrm{d}\tau,
\end{equation}
with the filter kernel
\begin{equation}
g_T(t)=\frac{2}{T}\sum_{i=0}^{N}\frac{P_i^{(\alpha,\beta)}(\vartheta)}{\big\lVert{P_i^{(\alpha,\beta)}\big\rVert}^2}w^{(\alpha,\beta)}\left(\nu(t)\right)P_i^{(\alpha,\beta)}\left(\nu(t)\right).
\end{equation}
In the latter equation $\nu(t)=1-\frac{2}{T}t$, $\big\lVert{x}\big\rVert=\sqrt{\langle x,x\rangle}$ is the norm induced by the inner product
\begin{equation}
\langle x,y\rangle=\int_{-1}^{1}w^{(\alpha,\beta)}(\tau)x(\tau){y(\tau)}\textrm{d}\tau,
\end{equation}
with the weight function
\begin{equation}
w^{(\alpha,\beta)}(\tau)=\begin{cases}
(1-\tau)^\alpha(1+\tau)^\beta,&\quad\tau\in[-1,1],\\
0,&\quad\text{otherwise},
\end{cases}
\end{equation}
$N$ is the degree of the polynomial approximating the signal $y^{(n)}$ in the time window $[t-T,t]$, $T$ is the filter window length, and $\vartheta$ parameterizes the estimation delay as described below.
This approach yields a straightforward analysis of the estimation delay $\delta_t$ and the degree of exactness $\gamma$. The degree of exactness was introduced in \cite{kiltz2017} as the polynomial degree up to which the derivative estimation is exact. If $\gamma=2$ for example, the first and second time derivatives of a polynomial signal of degree two are exact up to an estimation delay. The delay and the degree of exactness are given as
\begin{align}
\gamma&=\left\{\begin{matrix}
n+N+1,\quad &\text{if } N=0\vee\vartheta=p_{N+1,k}^{(\alpha,\beta)}\\[4pt]
n+N,\quad&\text{otherwise,}
\end{matrix}\right.\\
\delta_t&=\left\{\begin{matrix}
\frac{\alpha+1}{\alpha+\beta+2}T,\quad&\text{if } N=0\\[4pt]
\frac{1-\vartheta}{2}T,\quad&\text{otherwise,}
\end{matrix}\right.
\end{align}
with $p_{N+1,k}^{(\alpha,\beta)}$ the $k$-th zero of the Jacobi polynomial of degree $N+1$.
