Auto sync: 2025-09-14 17:33:58 (3 files changed)

M  .task/backlog.data

M  .task/pending.data

M  .task/undo.data

.task/backlog.data

@@ -31,3 +31,6 @@
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.task/completed.data

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.task/pending.data

@@ -2,7 +2,6 @@
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 [description:"Look over obsidian tasks and see if anything is worth moving over" entry:"1757443262" modified:"1757443262" status:"pending" tags:"taskwarrior" tags_taskwarrior:"x" uuid:"c1a5390d-5b84-4f9c-8acb-ffb970682660"]
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.task/undo.data

@@ -109,3 +109,15 @@ time 1757621896
 old [description:"Read Branicky Lyapunov paper" entry:"1757615669" modified:"1757620907" project:"Thesis" start:"1757620907" status:"pending" tags:"reading" tags_reading:"x" uuid:"f4d32147-ec74-4f68-b8d5-146c0b6f35df"]
 new [description:"Read Branicky Lyapunov paper" entry:"1757615669" modified:"1757621896" project:"Thesis" status:"pending" tags:"reading" tags_reading:"x" uuid:"f4d32147-ec74-4f68-b8d5-146c0b6f35df"]
 ---
+time 1757864178
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+---
+time 1757885608
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+---

Writing/ERLM/goals-and-outcomes/v4.tex

@@ -1,4 +1,4 @@
-\section{Goals and Outcomes - REVISED}
+\section{Goals and Outcomes}
 
 The goal of this research is to develop a unified framework combining temporal
 logic synthesis with continuous-time verification methods to create autonomous

Writing/ERLM/main.aux

@@ -1,7 +1,11 @@
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 \bibstyle{unsrt}
 \providecommand \oddpage@label [2]{}
-\@writefile{toc}{\contentsline {section}{\numberline {1}Goals and Outcomes - ORIGINAL}{1}{}\protected@file@percent }
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-\@writefile{toc}{\contentsline {section}{\numberline {2}Goals and Outcomes - REVISED}{2}{}\protected@file@percent }
 \bibdata{references}
+\@writefile{toc}{\contentsline {section}{\numberline {2}State of the art and limits of current practice}{3}{}\protected@file@percent }
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Writing/ERLM/main.bbl

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Writing/ERLM/main.blg

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Writing/ERLM/main.fdb_latexmk

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 []\OT1/phv/b/n/10 Prove that hy-brid sys-tem im-ple-men-ta-tions achieve safety and per-for-mance spec-i-fi-ca-tions across
  []
 
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+                                                  )\) can be much more
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+                                                  )\) is that it must
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+and I'll forget about whatever was undefined.
+
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+                  )\) can depend a control input \(u\), or can be
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+
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Writing/ERLM/main.tex

@@ -4,10 +4,9 @@
 
 \maketitle
 
-\input{goals-and-outcomes/v3}
-
-\newpage
 \input{goals-and-outcomes/v4}
+\newpage
+\input{state-of-the-art/v1.tex}
 
 \bibliography{references}
 

Writing/ERLM/state-of-the-art/v1.tex

@@ -1,13 +1,75 @@
 \section{State of the art and limits of current practice}
 
-%Start with some discussion about what a hybrid system is. Lay it out with some
+This research will improve our ability to build high assurance autonomous hybrid
-%knowledge about jumps and flow and what not. Lay down a mathematical
+control systems by connecting tools from different areas of the scientific
-%preliminary here.
+literature. We will combine control and dynamics theory of hybrid systems with
+the discrete systems analysis found in the computer science space. These two
+fields are disparate, but both approach the problem of high assurance hybrid
+systems. First, we will discuss the control theory side. Hybrid systems from a
+differential equation perspective will be considered, and comparisons to the
+current state of the art of nuclear power control will be discussed. Then, we
+will discuss approaches for discrete and hybrid systems from the logical and
+computer science perspective. These gap between these two fields is the crux of
+the intellectual merit of this research.
 
-\subsection{Discrete Systems and Reactive Synthesis}
+\subsection{Control Theory and Hybrid Systems}
 
-% LIMITATION: Discrete synthesis is predicated upon 
+Hybrid systems have two components to their behavior. They have continuous
+dynamics called 'flow', and have discrete dynamics called 'jumps'. Hybrid
+systems can often be described as a set of differential and difference
+equations. An hybrid system can be defined as follows:
 
-\subsection{Continuous System Verification}
+\begin{align}
+  \dot{x}(t) &= f(x(t), q(t), u(t)) \\
+  q(k+1) &= \nu(x(k), q(k), u(k))
+  \label{eq:hybrid-sys}
+\end{align}
 
-\subsection{Hybrid System Verification Approaches}
+In this description there are two functions: \(f(\cdot)\) and
+\(\nu(\cdot)\). \(f(\cdot)\) defines the continuous dynamics, while
+\(\nu(\cdot)\) defines the discrete dynamics. These functions take three
+inputs: the continuous states \(x\), the discrete state \(q\), and an optional
+control input \(u\). \(f_i(\cdot)\) here is a nonlinear function, where \(q\)
+changes the mode of the continuous dynamics, but is otherwise dependent on the
+current continuous states and control input. \(\nu_i(\codt)\) can be much more
+generally created. The only restriction on \(\nu_i(\codt)\) is that it must
+not create an infinite number of jumps in a finite amount of time.
+\(\nu_i(\codt)\) can depend a control input \(u\), or can be
+\textit{autonomous}, where the discrete mode is defined by the system states
+\(x\) and \(q\). While an autonomous system may not have a control input, that
+does not mean the system is not controlled, as the continuous dynamics
+\(f_i(\cdot)\) can be constructed such that the system is controlled by the
+continuous and discrete states \(x\) and \(q\).
+ 
+The hybrid systems we are most interested in here are \textit{continuous} autonomous
+hybrid systems. These systems are hybrid systems whose continuous states \(x\)
+do not change during a jump, and do not rely on an external control input. This
+research is primarily aimed at control of physical systems whose continuous
+states \(x\) are physically required to be continuous across jumps. For example,
+a nuclear reactor control rod system may jump from a warm-up ramp control mode to a load
+following controller, but the continuous states of the reactor such as
+temperature and rod position can not instantaneously change.
+
+Continuous hybrid systems are often constructed by piecing together control
+systems for different regions in the state space. This way of building a hybrid
+control system is intuitive--controllers are built for local regions with local
+objectives. The problem arises when analysis of these local controllers is
+generalized to the entire, global, hybrid system. Even in the most simple case
+of linear time invariant controller modes, global guarantees of stability can
+not be made using linear control theory. Instead, approaches using stitched
+networks of Lyapunov functions have been introduced. %CITE CITE CITE%
+These Lyapunov functions are difficult to find, but can for some linear switched
+hybrid systems provide a way to verification.
+
+Another way of verifying stability of hybrid control systems includes performing
+reachability analysis. Reachability is a formal methods tool that uses
+abstractifications of a system to determine output ranges of dynamic systems for
+a given input. Reachability is not limited to linear systems, and can be
+performed on nonlinear systems as well. Reachability for hybrid systems suffers
+from two main deficiencies. First, the analysis can take a very long time to
+compute. Second, reachability uses approximations. Hybrid systems that have
+trajectories of significantly long times can cause problems when using
+reachability because either the computation time makes the analysis infeasible,
+or the compounding approximation errors makes the analysis meaningless.
+
+\subsection{Formal Methods and Reactive Synthesis}