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zswAXJ;&-m3O5%58g_g?Q=DSv8DRM+nv8ATG=c5u!t-97nrIxy($wz%HrN2tl&wo;T z=lH0<rS`A$QJJMK?Df$A%k6*AM*}UlX1k9DS?;CBeN=9_ch~r+!gAl->Z8GyyYE&X z4YAyFk&lL2Uhj8(G|ciE-uBUO%iFfcN0pX$c(#v5s2acHqbf`HSni{dmM(kCN24q~ z?EifkZRv||@zEGd-?qa?tl$&Bdqnr!b(ZXjmh71-E$OGP5xajZo|zT=57A(r+6rZE pWOH<KWnpa!Wo~3|VrmL=bYXIIcyeWC3OG49F)}v_B_%~qMhdn5aZ>;Q diff --git a/AMPC_Summary.tex b/AMPC_Summary.tex index 7c1896d..64fbd63 100644 --- a/AMPC_Summary.tex +++ b/AMPC_Summary.tex @@ -1205,6 +1205,8 @@ \end{whitebox} \begin{whitebox}{\textbf{RECOVERY-INITIALIZATION}} + \blue{Noise} Unimodal + \blue{PRS -- Probabalistic Reachable Sets} \begin{highlightbox}{} $\F_i^p$ is i-step PRS for process $\{d(k)\}$ initialized at $d(0)$ if @@ -1241,20 +1243,130 @@ \end{whitebox} \begin{whitebox}{\textbf{INDIRECT FEEDBACK}} + \blue{Noise} any + \blue{MPC Formulation} \begin{highlightbox}{} \footnotesize \vspace{-1em} - \begin{align*} - \min_{\bar{u}_i} \ &\textstyle\sum_{i=0}^{N-1} \|\bar{x}_i\|_Q^2 + \|\bar{u}_i\|_R^2 + \|\bar{x}_N\|_P^2 & \text{uncertain for cost}\\ - \mathrm{s.t.} \ &\bar{x}_{i+1} = A\bar{x}_i + B\bar{u}_i \\ - &{\color{color2} z_{i+1} = A z_i + B v_i} \\ + \begin{alignat*}{2} + \min_{\bar{u}_i} \ &\textstyle\sum_{i=0}^{N-1} \|\bar{x}_i\|_Q^2 + \|\bar{u}_i\|_R^2 + \|\bar{x}_N\|_P^2 \\ + \mathrm{s.t.} \ &\bar{x}_{i+1} = A\bar{x}_i + B\bar{u}_i \quad && \text{uncertain for performance}\\ + &{\color{color2} z_{i+1} = A z_i + B v_i} && {\color{color2}\text{nominal for safety}} \\ &\bar{e}_i = \bar{x}_i - z_i \\ - &{\color{color2} z_i \in \X \ominus \ \F_{i+k}^p} & \text{nominal for constraints}\\ + &{\color{color2} z_i \in \X \ominus \ \F_{i+k}^p} && \\ &{\color{color2} z_N \in \mathcal{Z}_f \subset \X \ominus \F_\infty^p} \\ &\bar{x}_0 = x(k), \quad {\color{color2} z_0 = z(k) = z_{1|k-1}} + \end{alignat*} + \end{highlightbox} + \end{whitebox} + + \section{RECAP -- REGRESSION} + + \section{LBMPC -- LEARNING-BASED MPC} + + \begin{whitebox}{\textbf{RP-LBMPC -- Robust Performance LBMPC}} + + \blue{System} + \begin{whitebox}{} + \footnotesize + \vspace{-1em} + \begin{align*} + x(k+1) = + \underbrace{Ax(k) + Bu(k)}_{\text{nominal dynamics}} + + + \underbrace{g(x(k),u(k)) + w(k)}_{:= \bar{w}(k) \text{ model error}} \end{align*} + \end{whitebox} + + \blue{Models} + \begin{whitebox}{} + \scriptsize + \centering + \begin{tabular}{l l l} + Nominal Lin Model & $z_{i+1} = A z_i + Bu_i$ & constraints $\leadsto$ safety \\ + Learned N-Lin model & $x_{i+1} = Ax_i + Bu_i + \mathcal{O}_k(x_i,u_i)$ & cost $\leadsto$ performance + \end{tabular} + \end{whitebox} + + \blue{Robust Performance LBMPC} + \begin{highlightbox}{} + \footnotesize + \vspace{-1em} + \begin{alignat*}{2} + \min_{\bar{u}_i} \ &\textstyle\sum_{i=0}^{N-1} l(x_i, u_i) + l_f(x_N) \\ + \mathrm{s.t.} \ &x_{i+1} = Ax_i + Bu_i + \mathcal{O}_k(x_i, u_i) \qquad && \text{for performance} \\ + &{\color{color2}z_{i+1} = Az_i + Bu_i} && {\color{color2}\text{for safety}} \\ + &{\color{color2}z_i \in \X \ominus \F_i}, \quad u_i \in \U \ominus K\F_i \\ + &{\color{color2}z_N \in \X_f \ominus \F_N} \\ + &\bar{x}_0 = x(k), \quad {\color{color2}z_0 = x(k)} + \end{alignat*} \end{highlightbox} + + \begin{minipage}[t]{0.48\linewidth} + \blue{Terminal Invariant Set} + \begin{whitebox}{} + \scriptsize + Assuming stable $K$, find RPI set $\X_f$ + \begin{align*} + &\X_f \subseteq \{x \mid x\in\X, Kx \in \U \} \\ + &(A+BK)\X_f \oplus \W \subseteq \X_f + \end{align*} + \end{whitebox} + \end{minipage} + \begin{minipage}[t]{0.48\linewidth} + \blue{Constraint Tightening} + \begin{whitebox}{} + \scriptsize + Assuming $\bar{w}(k)\in\W$ + \begin{align*} + \F_0 &= 0, \quad + \F_i &= \bigoplus_{j=0}^{i-1}(A+BK)^j\W + \end{align*} + \end{whitebox} + \end{minipage} + + \blue{Performance \& Safety} + \begin{whitebox}{} + \scriptsize + \begin{itemize}[leftmargin=1.5em] + \item Cost functino depends on states of learned model, which uses $\mathcal{O}_k$ + \item Recursive feasibility / robust constraint satisfaction not depend on $\mathcal{O}$ + \item $\mathcal{O}_k$ can be time-varying as we want to update learned model + \item \textbf{While performance model can be updated online, uncertainty set $\W$ cannot as cannot ensure $\W_{k+1} \subseteq \W_k$} + \end{itemize} + \end{whitebox} + + \blue{Robust Feasibility \& Constraint Satisfaction} + \begin{whitebox}{} + \scriptsize + If $\X_f$ is robust terminal invar set and $U$ is feasible for RP-LBMPC scheme, then applying $u(k)=u_0^\star$ gives + \begin{enumerate}[leftmargin=3em, label=\textbf{(\arabic*)}] + \item Recursive feasibility + \item Robust constraint satisfaction $x\in\X, u\in\U$ + \end{enumerate} + \end{whitebox} + + \blue{Lemma -- Continuity of Value Function} + \begin{whitebox}{} + \scriptsize + Let $\X_N$ be feasible set of RP-LBMPC. IF functions $l, l_f, \mathcal{O}_k$ continuous, then $J_k^\star(x(k))$ continuous on $\mathrm{int}(\X_N)$ + \end{whitebox} + + \blue{ISS -- Input To State Stability} + \begin{whitebox}{} + \scriptsize + Assume + \begin{enumerate}[leftmargin=3em, label=\textbf{(\arabic*)}] + \item Terminal set $\X_f$ RPI + \item LBMPC opt problem feasible for $x(0)$ + \item Stage cost $l$, terminal cost $l_f$ \textbf{strictly convex, continuous, time-invariant} + \item System controlled with RP-LBMPC and $\mathcal{O}_k(x,u)=0$ is ISS + \end{enumerate} + Then system in CL with RP-LBMPC is ISS wrt model mismatch $g(x,u)+w$ and oracle $\mathcal{O}_k(x,u)$ + \end{whitebox} + + \end{whitebox} \end{multicols*} -- GitLab