Auto sync: 2025-09-14 17:33:23 (16 files changed)
M .task/backlog.data M .task/completed.data M .task/pending.data M .task/undo.data M Writing/ERLM/goals-and-outcomes/v4.tex M Writing/ERLM/main.aux A Writing/ERLM/main.bbl A Writing/ERLM/main.blg
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---
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\section{Goals and Outcomes - REVISED}
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\section{Goals and Outcomes}
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The goal of this research is to develop a unified framework combining temporal
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logic synthesis with continuous-time verification methods to create autonomous
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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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\input{goals-and-outcomes/v3}
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\input{goals-and-outcomes/v4}
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@ -1,13 +1,75 @@
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\section{State of the art and limits of current practice}
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||||
%Start with some discussion about what a hybrid system is. Lay it out with some
|
||||
%knowledge about jumps and flow and what not. Lay down a mathematical
|
||||
%preliminary here.
|
||||
This research will improve our ability to build high assurance autonomous hybrid
|
||||
control systems by connecting tools from different areas of the scientific
|
||||
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}
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user