	\begin{algorithmic}[1]
		\State Select an initial estimate $\hat{\mathbf{c}}^{(1)}$, a maximum number of iterations $M_{\textnormal{iter}}$, and a tolerance factor $\epsilon$
          \State $j\gets 1$, $\textnormal{flag}\gets 1$
		\While{$j\leq M_{\textnormal{iter}}$ and $\textnormal{flag}=1$}
		\State \textbf{E-step}: Compute the expectation
		\begin{equation}
			\label{qfunction}
			Q(\mathbf{c},\hspace{-0.02cm}\hat{\mathbf{c}}^{(j)}\hspace{-0.01cm}) \hspace{-0.04cm}= \hspace{-0.04cm}\mathbb{E}\hspace{-0.03cm}\left\{ \log \textnormal{p}(\mathbf{z}_{1:N}\hspace{-0.02cm},\hspace{-0.02cm}\mathcal{Y}_{1:N}|\mathbf{c}\hspace{-0.02cm})|\mathcal{Y}_{1:N}\hspace{-0.02cm},\hspace{-0.02cm}\hat{\mathbf{c}}^{(j)}\hspace{-0.04cm}\right\}\hspace{-0.03cm}.\hspace{-0.1cm}
		\end{equation} 
		\State \textbf{M-step}: Solve the optimization problem
		\begin{equation}
			\label{mstep}
			\hat{\mathbf{c}}^{(j+1)} = \underset{\mathbf{c}\in \mathbb{R}^N}{\arg \max} \left( Q(\mathbf{c},\hat{\mathbf{c}}^{(j)}) -\frac{\gamma\mathbf{c}^{\top}\mathbf{Kc}}{2} \right).
		\end{equation}
		\If{$\dfrac{\|\hat{\mathbf{c}}^{(j+1)}-\hat{\mathbf{c}}^{(j)}\|_2}{\|\hat{\mathbf{c}}^{(j)}\|_2}<\epsilon$} 		\State $\textnormal{flag} \gets 0$
		\EndIf
		\State $j \gets j+1$
		\EndWhile
	\end{algorithmic}

\begin{algorithmic}
[1]
\State Select an initial estimate $\hat{\mathbf{c}}^{(1)}$, a maximum number of iterations $M_{\textnormal{iter}}$, and a tolerance factor $\epsilon$
\State $j\gets 1$, $\textnormal{flag}\gets 1$
\While{$j\leq M_{\textnormal{iter}}$ and $\textnormal{flag}=1$}
\State \textbf{E-step}: Compute the expectation
\begin{equation*}
Q(\mathbf{c},\hspace{-0.02cm}\hat{\mathbf{c}}^{(j)}\hspace{-0.01cm}) \hspace{-0.04cm}= \hspace{-0.04cm}\mathbb{E}\hspace{-0.03cm}\left\{ \log \textnormal{p}(\mathbf{z}_{1:N}\hspace{-0.02cm},\hspace{-0.02cm}\mathcal{Y}_{1:N}|\mathbf{c}\hspace{-0.02cm})|\mathcal{Y}_{1:N}\hspace{-0.02cm},\hspace{-0.02cm}\hat{\mathbf{c}}^{(j)}\hspace{-0.04cm}\right\}\hspace{-0.03cm}.\hspace{-0.1cm}
\end{equation*}
\State \textbf{M-step}: Solve the optimization problem
\begin{equation*}
\hat{\mathbf{c}}^{(j+1)} = \underset{\mathbf{c}\in \mathbb{R}^N}{\arg \max} \left( Q(\mathbf{c},\hat{\mathbf{c}}^{(j)}) -\frac{\gamma\mathbf{c}^{\top}\mathbf{Kc}}{2} \right).
\end{equation*}
\If{$\dfrac{\|\hat{\mathbf{c}}^{(j+1)}-\hat{\mathbf{c}}^{(j)}\|_2}{\|\hat{\mathbf{c}}^{(j)}\|_2}<\epsilon$}   \State $\textnormal{flag} \gets 0$
\EndIf
\State $j \gets j+1$
\EndWhile
\end{algorithmic}
{
    "title": "kernel-based identification using lebesgue-sampled data",
    "id": "2303.06045",
    "abstract": "sampling in control applications is increasingly done non-equidistantly in time. this includes applications in motion control, networked control, resource-aware control, and event-based control. some of these applications, like the ones where displacement is tracked using incremental encoders, are driven by signals that are only measured when their values cross fixed thresholds in the amplitude domain. this paper introduces a non-parametric estimator of the impulse response and transfer function of continuous-time systems based on such amplitude-equidistant sampling strategy, known as lebesgue sampling. to this end, kernel methods are developed to formulate an algorithm that adequately takes into account the bounded output uncertainty between the event timestamps, which ultimately leads to more accurate models and more efficient output sampling compared to the equidistantly-sampled kernel-based approach. the efficacy of our proposed method is demonstrated through a mass-spring damper example with encoder measurements and extensive monte carlo simulation studies on system benchmarks.",
    "categories": "eess.sy cs.sy",
    "doi": "",
    "created": "2023-03-10",
    "updated": "2024-02-23",
    "authors": [
        "rodrigo a. gonz\u00e1lez",
        "koen tiels",
        "tom oomen"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2303.06045"
}
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