\begin{algorithm}
\caption{Terminal set construction via symbolic model}\label{safetygame}
\textbf{Input:} $\mathcal{X}$ (specification set), $\mathcal{D}_{N}$ (training dataset), $\eta_x, \eta_u, \varepsilon, \widetilde{\gamma}$ (some parameters for symbolic model), $\gamma_i$ for $i \in \mathbb{N}_{1: n_x}$ (a set of scalars);

\textbf{Output:} $\mathcal{X}_f$ (terminal set), $\mathcal{X}_S$ ($\tilde{\gamma}$-contractive set), $C$ (refined controller);
\begin{algorithmic}[1]

\State $\mathsf{q} \leftarrow (\eta_x, \eta_u, \varepsilon)$;\label{tripleline}

\State ${\Sigma}_{\mathsf{q}\tilde{\gamma}} \leftarrow (\mathcal{X}_{\mathsf{q}\tilde{\gamma}}, \mathcal{U}_{\mathsf{q}\tilde{\gamma}}, G_{\mathsf{q}\tilde{\gamma}})$;\label{symbolicline}

\State $\ell \leftarrow 0$;

\State $\mathcal{Q}_{\ell} \leftarrow \left[\bigcap_{i=1}^{n_x}\mathsf{Int}_{d_i}(\mathcal{X}; \epsilon_i)\right]_{{\eta}_x}$\label{finitestatesetline};

\Repeat\label{solvesafetygamestart}
    \State $\mathcal{Q}_{\ell + 1} \leftarrow {\rm Pre}_{{\Sigma}_{\mathsf{q}\tilde{\gamma}}}(\mathcal{Q}_{\ell})$; \label{predecessoralg}
    \State $\ell \leftarrow \ell + 1$;
\Until{$\mathcal{Q}_{\ell - 1} = \mathcal{Q}_{\ell}$;}\label{solvesafetygameend}

\State $\mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}} \leftarrow \mathcal{Q}_{\ell};$\label{compinvariantdisline}

\State $\mathcal{X}_S \leftarrow \{{x} \in \mathcal{X}: \ ({x}_{\mathsf{q}\tilde{\gamma}}, {x}) \in R(\varepsilon), \exists{x}_{\mathsf{q}\tilde{\gamma}} \in \mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}}\};$ \label{compinvariantline}

\State $C_{\mathsf{q}\tilde{\gamma}} \leftarrow \{{u}_{\mathsf{q}\tilde{\gamma}} \in \mathcal{U}_{\mathsf{q}\tilde{\gamma}}: \widetilde{G}({x}_{\mathsf{q}\tilde{\gamma}}, {u}_{\mathsf{q}\tilde{\gamma}}) \subseteq \mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}}, \ \forall {x}_{\mathsf{q}\tilde{\gamma}} \in \mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}}\};$\label{descontroller}

\State $C \leftarrow \{C_{\mathsf{q}\tilde{\gamma}}({x}_{\mathsf{q}\tilde{\gamma}}): ({x}_{\mathsf{q}\tilde{\gamma}}, {x}) \in R(\varepsilon), \ \forall {x} \in \mathcal{X}_S \}$; \label{refinecontroller}

\State $\mathcal{X}_f \leftarrow [\bigcap_{i=1}^{n_x}\mathsf{Int}_{\tilde{\gamma}_i}(\mathcal{X}_S), \bigcap_{i=1}^{n_x}\mathsf{Int}_{\gamma_i}(\mathcal{X}_S)]$;\label{terminalsetline}

\State \textbf{return} $\mathcal{X}_f, \mathcal{X}_S, C$;
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Terminal set construction via symbolic model}\textbf{Input:} $\mathcal{X}$ (specification set), $\mathcal{D}_{N}$ (training dataset), $\eta_x, \eta_u, \varepsilon, \widetilde{\gamma}$ (some parameters for symbolic model), $\gamma_i$ for $i \in \mathbb{N}_{1: n_x}$ (a set of scalars);
\textbf{Output:} $\mathcal{X}_f$ (terminal set), $\mathcal{X}_S$ ($\tilde{\gamma}$-contractive set), $C$ (refined controller);
\begin{algorithmic}
[1]
\State $\mathsf{q} \leftarrow (\eta_x, \eta_u, \varepsilon)$;
\State ${\Sigma}_{\mathsf{q}\tilde{\gamma}} \leftarrow (\mathcal{X}_{\mathsf{q}\tilde{\gamma}}, \mathcal{U}_{\mathsf{q}\tilde{\gamma}}, G_{\mathsf{q}\tilde{\gamma}})$;
\State $\ell \leftarrow 0$;
\State $\mathcal{Q}_{\ell} \leftarrow \left[\bigcap_{i=1}^{n_x}\mathsf{Int}_{d_i}(\mathcal{X}; \epsilon_i)\right]_{{\eta}_x}$;
\Repeat \State $\mathcal{Q}_{\ell + 1} \leftarrow {\rm Pre}_{{\Sigma}_{\mathsf{q}\tilde{\gamma}}}(\mathcal{Q}_{\ell})$; \State $\ell \leftarrow \ell + 1$;
\Until{$\mathcal{Q}_{\ell - 1} = \mathcal{Q}_{\ell}$;}
\State $\mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}} \leftarrow \mathcal{Q}_{\ell};$
\State $\mathcal{X}_S \leftarrow \{{x} \in \mathcal{X}: \ ({x}_{\mathsf{q}\tilde{\gamma}}, {x}) \in R(\varepsilon), \exists{x}_{\mathsf{q}\tilde{\gamma}} \in \mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}}\};$
\State $C_{\mathsf{q}\tilde{\gamma}} \leftarrow \{{u}_{\mathsf{q}\tilde{\gamma}} \in \mathcal{U}_{\mathsf{q}\tilde{\gamma}}: \widetilde{G}({x}_{\mathsf{q}\tilde{\gamma}}, {u}_{\mathsf{q}\tilde{\gamma}}) \subseteq \mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}}, \ \forall {x}_{\mathsf{q}\tilde{\gamma}} \in \mathcal{X}_{S, {\mathsf{q}\tilde{\gamma}}}\};$
\State $C \leftarrow \{C_{\mathsf{q}\tilde{\gamma}}({x}_{\mathsf{q}\tilde{\gamma}}): ({x}_{\mathsf{q}\tilde{\gamma}}, {x}) \in R(\varepsilon), \ \forall {x} \in \mathcal{X}_S \}$;
\State $\mathcal{X}_f \leftarrow [\bigcap_{i=1}^{n_x}\mathsf{Int}_{\tilde{\gamma}_i}(\mathcal{X}_S), \bigcap_{i=1}^{n_x}\mathsf{Int}_{\gamma_i}(\mathcal{X}_S)]$;
\State \textbf{return} $\mathcal{X}_f, \mathcal{X}_S, C$;
\end{algorithmic}
\end{algorithm}
{
    "title": "learning-based event-triggered mpc with gaussian processes under   terminal constraints",
    "id": "2110.12214",
    "abstract": "event-triggered control strategy is capable of significantly reducing the number of control task executions without sacrificing control performance. in this paper, we propose a novel learning-based approach towards an event-triggered model predictive control (mpc) for nonlinear control systems whose dynamics are unknown apriori. in particular, the optimal control problems (ocps) are formulated based on predictive states learned by gaussian process (gp) regression under a terminal constraint constructed by a symbolic abstraction. the event-triggered condition proposed in this paper is derived from the recursive feasibility so that the ocps are solved only when an error between the predictive and the actual states exceeds a certain threshold. based on the event-triggered condition, we analyze the stability of the closed-loop system and show that the finite-time convergence to the terminal set is achieved as the uncertainty of the gp model becomes smaller. moreover, in order to reduce the uncertainty of the gp model and increase efficiency to find the optimal solution, we provide an overall learning-based event-triggered mpc algorithm based on an iterative task. finally, we demonstrate the proposed approach through a tracking control problem.",
    "categories": "math.oc",
    "doi": "",
    "created": "2021-10-23",
    "updated": "2024-01-01",
    "authors": [
        "yuga onoue",
        "kazumune hashimoto",
        "akifumi wachi"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2110.12214"
}
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