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addressed all split notes. Citations file still needs updated and pdflatex is throwing fits about citations

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57
1-goals-and-outcomes/goals.tex
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@ -3,12 +3,12 @@
% GOAL PARAGRAPH
The goal of this research is to develop a methodology for creating autonomous
hybrid control systems with mathematical guarantees of safe and correct
behavior.\splitnote{Clear thesis statement. Gets right to it.}
behavior.
% INTRODUCTORY PARAGRAPH Hook
Nuclear power plants require the highest levels of control system reliability,
where failures can result in significant economic losses, service interruptions,
or radiological release.\splitnote{Stakes established immediately — good hook.}
or radiological release.
% Known information
Currently, nuclear plant operations rely on extensively trained human operators
who follow detailed written procedures and strict regulatory requirements to
@ -16,22 +16,16 @@ manage reactor control. These operators make critical decisions about when to
switch between different control modes based on their interpretation of plant
conditions and procedural guidance.
% Gap
This reliance on human operators prevents autonomous control capabilities and
creates a fundamental economic challenge for next-generation reactor
designs.\splitsuggest{The ``and'' here joins two distinct issues (autonomy
barrier + economics). Consider making the causal link explicit: ``This reliance
on human operators not only prevents autonomous control capabilities but also
creates...'' or split into two sentences.}
Small modular reactors, in particular, face per-megawatt staffing costs far
exceeding those of conventional plants and threaten their economic viability.
This reliance on human operators not only prevents autonomous control
capabilities but creates a fundamental economic challenge for next-generation
reactor designs. Small modular reactors, in particular, face per-megawatt
staffing costs far exceeding those of conventional plants and threaten their
economic viability.
% Critical Need
What is needed is a method to create autonomous control systems that safely
manage complex operational sequences with the same assurance as human-operated
systems, but without constant human supervision.\splitpolish{``What is needed
is'' — Gopen would call this a weak topic position. The sentence buries the
subject. Try: ``Autonomous control systems must safely manage complex
operational sequences...'' Puts the actor in the topic position.}
Autonomous control systems must safely manage complex operational sequences with
the same assurance as human-operated systems, but without constant human
supervision.
% APPROACH PARAGRAPH Solution
To address this need, we will combine formal methods with control theory to
build hybrid control systems that are correct by construction.
@ -41,33 +35,22 @@ mirroring how operators change control strategies. Existing formal methods can
generate provably correct switching logic from written requirements, but they
cannot handle the continuous dynamics that occur during transitions between
modes. Meanwhile, traditional control theory can verify continuous behavior but
lacks tools for proving correctness of discrete switching
decisions.\splitnote{Excellent setup of the gap — shows why neither approach
alone is sufficient.}
lacks tools for proving correctness of discrete switching
decisions. This work is conducted within the University of Pittsburgh Cyber Energy Center,
which provides access to industry collaboration and Emerson control hardware,
ensuring that developed solutions align with practical implementation
requirements.
% Hypothesis
By synthesizing discrete mode transitions directly from written operating
procedures and verifying continuous behavior between transitions, we can create
hybrid control systems with end-to-end correctness guarantees. If existing
procedures can be formalized into logical specifications and continuous dynamics
verified against transition requirements, then autonomous controllers can be
built that are provably free from design
defects.\splitnote{Hypothesis is clear and testable.}
built that are provably free from design defects.
% Pay-off
This approach will enable autonomous control in nuclear power plants while
maintaining the high safety standards required by the industry.
% Qualifications
This work is conducted within the University of Pittsburgh Cyber Energy Center,
which provides access to industry collaboration and Emerson control hardware,
ensuring that developed solutions align with practical implementation
requirements.\splitsuggest{This qualifications paragraph feels orphaned here.
It's important context but reads as an afterthought. Consider integrating it
into the approach paragraph (``...demonstrated on Emerson hardware through our
partnership with the Cyber Energy Center'') or moving to a ``Why This Will
Succeed'' framing later.}
% OUTCOMES PARAGRAPHS
If this research is successful, we will be able to do the following:
@ -107,8 +90,7 @@ If this research is successful, we will be able to do the following:
reactor simulation using industry-standard control hardware. This
demonstration will prove correctness across multiple coordinated control
modes from cold shutdown through criticality to power
operation.\splitnote{``cold shutdown through criticality to power
operation'' — concrete and impressive scope.}
operation.
% Outcome
We will demonstrate that autonomous hybrid control can be realized in the
nuclear industry with current equipment, establishing a path toward reduced
@ -119,7 +101,7 @@ If this research is successful, we will be able to do the following:
% IMPACT PARAGRAPH Innovation
The innovation in this work is unifying discrete synthesis with continuous
verification to enable end-to-end correctness guarantees for hybrid
systems.\splitnote{Clear ``what's new'' statement.}
systems.
% Outcome Impact
If successful, control engineers will create autonomous controllers from
existing procedures with mathematical proof of correct behavior. High-assurance
@ -130,5 +112,4 @@ nuclear power. Small modular reactors offer a promising solution to growing
energy demands, but their success depends on reducing per-megawatt operating
costs through increased autonomy. This research will provide the tools to
achieve that autonomy while maintaining the exceptional safety record the
nuclear industry requires.\splitnote{Strong closing — ties technical work to
real-world impact and economic necessity.}
nuclear industry requires.

59
2-state-of-the-art/state-of-art.tex
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@ -6,7 +6,7 @@ first understand how nuclear reactors are operated today. This section examines
reactor operators and the operating procedures we aim to leverage, then
investigates limitations of human-based operation, and concludes with current
formal methods approaches to reactor control
systems.\splitnote{Good roadmap — tells reader exactly what's coming.}
systems.
\subsection{Current Reactor Procedures and Operation}
@ -20,15 +20,7 @@ developed using guidance from NUREG-0899~\cite{NUREG-0899, 10CFR50.34}, but thei
development relies fundamentally on expert judgment and simulator validation
rather than formal verification. Procedures undergo technical evaluation,
simulator validation testing, and biennial review as part of operator
requalification under 10 CFR 55.59~\cite{10CFR55.59}. Despite this rigor,
procedures fundamentally lack formal verification of key safety properties. No
mathematical proof exists that procedures cover all possible plant states, that
required actions can be completed within available timeframes, or that
transitions between procedure sets maintain safety
invariants.\splitsuggest{This paragraph is doing a lot. Consider splitting:
first paragraph on the hierarchy and compliance, second on the lack of formal
verification. The ``No mathematical proof exists...'' sentence is powerful and
deserves emphasis.}
requalification under 10 CFR 55.59~\cite{10CFR55.59}.
\textbf{LIMITATION:} \textit{Procedures lack formal verification of correctness
and completeness.} Current procedure development relies on expert judgment and
@ -37,9 +29,7 @@ possible plant states, that required actions can be completed within available
timeframes, or that transitions between procedure sets maintain safety
invariants. Paper-based procedures cannot ensure correct application, and even
computer-based procedure systems lack the formal guarantees that automated
reasoning could provide.\splitpolish{This repeats the ``No mathematical
proof exists...'' sentence almost verbatim from the paragraph above. Either
cut it from the paragraph or from the LIMITATION box.}
reasoning could provide.
Nuclear plants operate with multiple control modes: automatic control, where the
reactor control system maintains target parameters through continuous reactivity
@ -59,8 +49,7 @@ actuation, containment isolation, and basic process
control~\cite{WRPS.Description, gentillon_westinghouse_1999}. Human operators,
however, retain control of strategic decision-making: power level changes,
startup/shutdown sequences, mode transitions, and procedure
implementation.\splitnote{This is the key insight — the hybrid nature is
already there, just not formally verified.}
implementation.
\subsection{Human Factors in Nuclear Accidents}
@ -75,8 +64,7 @@ operator requires several years of training.
The persistent role of human error in nuclear safety incidents---despite decades
of improvements in training and procedures---provides the most compelling
motivation for formal automated control with mathematical safety
guarantees.\splitnote{Strong thesis for this subsection.}
Operators hold legal authority under 10 CFR Part 55 to make critical decisions,
guarantees. Operators hold legal authority under 10 CFR Part 55 to make critical decisions,
including departing from normal regulations during emergencies. The Three Mile
Island (TMI) accident demonstrated how a combination of personnel error, design
deficiencies, and component failures led to partial meltdown when operators
@ -87,8 +75,7 @@ the licensee without formal verification that operators can fulfill this
responsibility does not guarantee safety. This tension between operational
flexibility and safety assurance remains unresolved: the person responsible for
reactor safety is often the root cause of
failures.\splitnote{``the person responsible for reactor safety is often the
root cause of failures'' — devastating summary. Very effective.}
failures.
Multiple independent analyses converge on a striking statistic: 70--80\% of
nuclear power plant events are attributed to human error, versus approximately
@ -100,15 +87,14 @@ of 190 events at Chinese nuclear power plants from
2007--2020~\cite{zhang_analysis_2025} found that 53\% of events involved active
errors, while 92\% were associated with latent errors---organizational and
systemic weaknesses that create conditions for
failure.\splitnote{Strong empirical grounding. The Chinese plant data is a
nice addition — shows this isn't just a Western regulatory perspective.}
failure.
\textbf{LIMITATION:} \textit{Human factors impose fundamental reliability limits
that cannot be overcome through training alone.} The persistent human
error contribution despite four decades of improvements demonstrates that these
limitations are fundamental rather than a remediable part of human-driven
control.\splitnote{Well-stated. The ``four decades'' point drives it home.}
control.
\subsection{Formal Methods}
\subsubsection{HARDENS}
@ -138,7 +124,7 @@ models of sensors, actuators, and compute infrastructure. Automatic code
synthesis generated verifiable C implementations and SystemVerilog hardware
implementations directly from Cryptol models---eliminating the traditional gap
between specification and implementation where errors commonly
arise.\splitnote{Good technical depth on HARDENS toolchain.}
arise.
Despite its accomplishments, HARDENS has a fundamental limitation directly
relevant to hybrid control synthesis: the project addressed only discrete
@ -151,7 +137,7 @@ reactor safety depends on the interaction between continuous
processes---temperature, pressure, neutron flux---evolving in response to
discrete control decisions. HARDENS verified the discrete controller in
isolation but not the closed-loop hybrid system
behavior.\splitnote{Clear articulation of the gap your work fills.}
behavior.
\textbf{LIMITATION:} \textit{HARDENS addressed discrete control logic without
continuous dynamics or hybrid system verification.} Verifying discrete control
@ -189,13 +175,11 @@ dynamic logic (dL). dL introduces two additional operators
into temporal logic: the box operator and the diamond operator. The box operator
\([\alpha]\phi\) states that for some region \(\phi\), the hybrid system
\(\alpha\) always remains within that region. In this way, it is a safety
ivariant being enforced for the system.\splitfix{Typo: ``ivariant'' should be
``invariant''} The second operator, the diamond
invariant being enforced for the system. The second operator, the diamond
operator \(<\alpha>\phi\) says that for the region \(\phi\), there is at least
one trajectory of \(\alpha\) that enters that region. This is a declaration of a
liveness property.
liveness property\cite{platzer2018}.
%source: https://symbolaris.com/logic/dL.html
While dL allows for the specification of these liveness and safety properties,
actually proving them for a given hybrid system is quite difficult. Automated
@ -210,12 +194,8 @@ readers a sense of the scaling problem.}
%gyroscopes overloding and needing to dump speed all the time
Approaches have been made to alleviate
these issues for nuclear power contexts using contract and decomposition based
methods, but are far from a complete methodology to design systems
with.\splitpolish{``but are far from a complete methodology to design systems
with'' — awkward ending preposition. Try: ``but remain far from a complete
design methodology'' or ``but do not yet constitute a complete design
methodology.''}
%source: Manyu's thesis.
methods, but do not yet constitute a complete design
methodology\cite{kapuria2025}.%source: Manyu's thesis.
Instead, these approaches have been used on systems that have been designed a
priori, and require expert knowledge to create the system proofs.
@ -223,8 +203,9 @@ priori, and require expert knowledge to create the system proofs.
%very much, so the limitation is that logic based hybrid system approaches have
%not been used in the DESIGN of autonomous controllers, only in the analysis of
%systems that already exist.
\splitinline{Your comment here is spot-on. You should add a LIMITATION box:
\textit{Differential dynamic logic has been used for post-hoc analysis of
existing systems, not for the constructive design of autonomous controllers.}
This is exactly the gap you're filling — you're doing synthesis, not just
verification.}
\textbf{LIMITATION:} \textit{Differential dynamic logic has been used for
post-hoc analysis of existing systems, not for the constructive design of
autonomous controllers.} Current formal methods based approaches can in theory
completely describe the behavior of a hybrid autonomous control system, but in
practice remain difficult to implement, and have no straightforward application
to the design of a hybrid autonomous control system.

165
2-state-of-the-art/v1.tex
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@ -1,165 +0,0 @@
\section{State of the Art and Limits of Current Practice}
The principal aim of this research is to create autonomous reactor control
systems that are tractably safe. To understand what is being automated, we must
first understand how nuclear reactors are operated today. This section examines
reactor operators and the operating procedures we aim to leverage, then
investigates limitations of human-based operation, and concludes with current
formal methods approaches to reactor control systems.
\subsection{Current Reactor Procedures and Operation}
Nuclear plant procedures exist in a hierarchy: normal operating procedures for
routine operations, abnormal operating procedures for off-normal conditions,
Emergency Operating Procedures (EOPs) for design-basis accidents, Severe
Accident Management Guidelines (SAMGs) for beyond-design-basis events, and
Extensive Damage Mitigation Guidelines (EDMGs) for catastrophic damage
scenarios. These procedures must comply with 10 CFR 50.34(b)(6)(ii) and are
developed using guidance from NUREG-0900~\cite{NUREG-0899, 10CFR50.34}, but their
development relies fundamentally on expert judgment and simulator validation
rather than formal verification. Procedures undergo technical evaluation,
simulator validation testing, and biennial review as part of operator
requalification under 10 CFR 55.59~\cite{10CFR55.59}. Despite this rigor,
procedures fundamentally lack formal verification of key safety properties. No
mathematical proof exists that procedures cover all possible plant states, that
required actions can be completed within available timeframes, or that
transitions between procedure sets maintain safety invariants.
\textbf{LIMITATION:} \textit{Procedures lack formal verification of correctness
and completeness.} Current procedure development relies on expert judgment and
simulator validation. No mathematical proof exists that procedures cover all
possible plant states, that required actions can be completed within available
timeframes, or that transitions between procedure sets maintain safety
invariants. Paper-based procedures cannot ensure correct application, and even
computer-based procedure systems lack the formal guarantees that automated
reasoning could provide.
Nuclear plants operate with multiple control modes: automatic control, where the
reactor control system maintains target parameters through continuous reactivity
adjustment; manual control, where operators directly manipulate the reactor; and
various intermediate modes. In typical pressurized water reactor operation, the
reactor control system automatically maintains a floating average temperature
and compensates for power demand changes through reactivity feedback loops
alone. Safety systems, by contrast, operate with implemented automation. Reactor
Protection Systems trip automatically on safety signals with millisecond
response times, and engineered safety features actuate automatically on accident
signals without operator action required.
The division between automated and human-controlled functions reveals the
fundamental challenge of hybrid control. Highly automated systems handle reactor
protection---automatic trips on safety parameters, emergency core cooling
actuation, containment isolation, and basic process
control~\cite{WRPS.Description, gentillon_westinghouse_1999}. Human operators,
however, retain control of strategic decision-making: power level changes,
startup/shutdown sequences, mode transitions, and procedure implementation.
\subsection{Human Factors in Nuclear Accidents}
Current-generation nuclear power plants employ over 3,600 active NRC-licensed
reactor operators in the United States~\cite{operator_statistics}. These
operators divide into Reactor Operators (ROs), who manipulate reactor controls,
and Senior Reactor Operators (SROs), who direct plant operations and serve as
shift supervisors~\cite{10CFR55}. Staffing typically requires at least two ROs
and one SRO for current-generation units~\cite{10CFR50.54}. Becoming a reactor
operator requires several years of training.
The persistent role of human error in nuclear safety incidents---despite decades
of improvements in training and procedures---provides the most compelling
motivation for formal automated control with mathematical safety guarantees.
Operators hold legal authority under 10 CFR Part 55 to make critical decisions,
including departing from normal regulations during emergencies. The Three Mile
Island (TMI) accident demonstrated how a combination of personnel error, design
deficiencies, and component failures led to partial meltdown when operators
misread confusing and contradictory readings and shut off the emergency water
system~\cite{Kemeny1979}. The President's Commission on TMI identified a
fundamental ambiguity: placing responsibility for safe power plant operations on
the licensee without formal verification that operators can fulfill this
responsibility does not guarantee safety. This tension between operational
flexibility and safety assurance remains unresolved: the person responsible for
reactor safety is often the root cause of failures.
Multiple independent analyses converge on a striking statistic: 70--80\% of
nuclear power plant events are attributed to human error, versus approximately
20\% to equipment failures~\cite{WNA2020}. More significantly, the root cause of
all severe accidents at nuclear power plants---Three Mile Island, Chernobyl, and
Fukushima Daiichi---has been identified as poor safety management and safety
culture: primarily human factors~\cite{hogberg_root_2013}. A detailed analysis
of 190 events at Chinese nuclear power plants from
2007--2020~\cite{zhang_analysis_2025} found that 53\% of events involved active
errors, while 92\% were associated with latent errors---organizational and
systemic weaknesses that create conditions for failure.
\textbf{LIMITATION:} \textit{Human factors impose fundamental reliability limits
that cannot be overcome through training alone.} The persistent human
error contribution despite four decades of improvements demonstrates that these
limitations are fundamental rather than a remediable part of human-driven control.
\subsection{HARDENS and Formal Methods}
The High Assurance Rigorous Digital Engineering for Nuclear Safety (HARDENS)
project represents the most advanced application of formal methods to nuclear
reactor control systems to date~\cite{Kiniry2024}.
HARDENS aimed to address a fundamental dilemma: existing U.S. nuclear control
rooms rely on analog technologies from the 1950s--60s. This technology is
obsolete compared to modern control systems and incurs significant risk and
cost. The NRC contracted Galois, a formal methods firm, to demonstrate that
Model-Based Systems Engineering and formal methods could design, verify, and
implement a complex protection system meeting regulatory criteria at a fraction
of typical cost. The project delivered a Reactor Trip System (RTS)
implementation with full traceability from NRC Request for Proposals and IEEE
standards through formal architecture specifications to verified software.
HARDENS employed formal methods tools and techniques across the verification
hierarchy. High-level specifications used Lando, SysMLv2, and FRET (NASA Formal
Requirements Elicitation Tool) to capture stakeholder requirements, domain
engineering, certification requirements, and safety requirements. Requirements
were analyzed for consistency, completeness, and realizability using SAT and SMT
solvers. Executable formal models used Cryptol to create a behavioral model of
the entire RTS, including all subsystems, components, and limited digital twin
models of sensors, actuators, and compute infrastructure. Automatic code
synthesis generated verifiable C implementations and SystemVerilog hardware
implementations directly from Cryptol models---eliminating the traditional gap
between specification and implementation where errors commonly arise.
Despite its accomplishments, HARDENS has a fundamental limitation directly
relevant to hybrid control synthesis: the project addressed only discrete
digital control logic without modeling or verifying continuous reactor dynamics.
The Reactor Trip System specification and verification covered discrete state
transitions (trip/no-trip decisions), digital sensor input processing through
discrete logic, and discrete actuation outputs (reactor trip commands). The
project did not address continuous dynamics of nuclear reactor physics. Real
reactor safety depends on the interaction between continuous
processes---temperature, pressure, neutron flux---evolving in response to
discrete control decisions. HARDENS verified the discrete controller in
isolation but not the closed-loop hybrid system behavior.
\textbf{LIMITATION:} \textit{HARDENS addressed discrete control logic without
continuous dynamics or hybrid system verification.} Verifying discrete control
logic alone provides no guarantee that the closed-loop system exhibits desired
continuous behavior such as stability, convergence to setpoints, or maintained
safety margins.
HARDENS produced a demonstrator system at Technology Readiness Level 2--3
(analytical proof of concept with laboratory breadboard validation) rather than
a deployment-ready system validated through extended operational testing. The
NRC Final Report explicitly notes~\cite{Kiniry2024} that all material is
considered in development, not a finalized product, and that ``The demonstration
of its technical soundness was to be at a level consistent with satisfaction of
the current regulatory criteria, although with no explicit demonstration of how
regulatory requirements are met.'' The project did not include deployment in
actual nuclear facilities, testing with real reactor systems under operational
conditions, side-by-side validation with operational analog RTS systems,
systematic failure mode testing (radiation effects, electromagnetic
interference, temperature extremes), NRC licensing review, or human factors
validation with licensed operators in realistic control room scenarios.
\textbf{LIMITATION:} \textit{HARDENS achieved TRL 2--3 without experimental
validation.} While formal verification provides mathematical correctness
guarantees for the implemented discrete logic, the gap between formal
verification and actual system deployment involves myriad practical
considerations: integration with legacy systems, long-term reliability
under harsh environments, human-system interaction in realistic
operational contexts, and regulatory acceptance of formal methods as
primary assurance evidence.

97
3-research-approach/approach.tex
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@ -30,7 +30,7 @@ The challenge of hybrid system verification lies in the interaction between
discrete and continuous dynamics. Discrete transitions change the governing
vector field, creating discontinuities in the system's behavior. Traditional
verification techniques designed for purely discrete or purely continuous
systems cannot handle this interaction directly.\splitpolish{Missing space before ``Our} Our methodology addresses this
systems cannot handle this interaction directly. Our methodology addresses this
challenge through decomposition. We verify discrete switching logic and
continuous mode behavior separately, then compose these guarantees to reason
about the complete hybrid system. This two-layer approach mirrors the structure
@ -47,7 +47,7 @@ This means that the system does not have external input and that continuous
states do not change instantaneously when discrete states change. For our
systems of interest, the continuous states are physical quantities that are
always Lipschitz continuous. This nomenclature is borrowed from the Handbook on
Hybrid Systems Control \cite{HANDBOOK ON HYBRID SYSTEMS}, but is redefined here
Hybrid Systems Control \cite{lunze2009}, but is redefined here
for convenience:
\begin{equation}
@ -75,12 +75,13 @@ The creation of a HAHACS amounts to the construction of such a tuple together
with proof artifacts demonstrating that the intended behavior of the control
system is satisfied by its actual implementation. This approach is tractable now
because the infrastructure for each component has matured. The novelty is not in
the individual pieces, but in the architecture that connects them.\splitnote{This is your key insight — the novelty is compositional, not component-level.} By defining
entry, exit, and safety conditions at the discrete level first, we transform the
intractable problem of global hybrid verification into a collection of local
verification problems with clear interfaces. Verification is performed per mode
rather than on the full hybrid system, keeping the analysis tractable even for
complex reactor operations.
the individual pieces, but in the architecture that connects them.
By defining entry, exit, and safety conditions at the discrete level first, we
transform the intractable problem of global hybrid verification into a
collection of local verification problems with clear interfaces. Verification is
performed per mode rather than on the full hybrid system, keeping the analysis
tractable even for complex reactor operations.
\begin{figure}
\centering
@ -293,16 +294,8 @@ implementation. Second, it clearly demonstrates where natural language documents
are insufficient. These procedures may still be used by human operators, so any
room for interpretation is a weakness that must be addressed.
(Some examples of where FRET has been used and why it will be successful here)
%%% NOTES (Section 2):
% - Add concrete FRET example showing requirement → FRETish → LTL
% - Discuss hysteresis and how to prevent mode chattering near boundaries
% - Address sensor noise and measurement uncertainty in threshold definitions
% - Consider numerical precision issues when creating discrete automata
% ----------------------------------------------------------------------------
% 3. DISCRETE CONTROLLER SYNTHESIS
% ----------------------------------------------------------------------------
\splitinline{Some examples of where FRET has been used and why it will be successful
here}
Once system requirements are defined as temporal logic specifications, we use
them to build the discrete control system. To do this, reactive synthesis tools
@ -339,19 +332,10 @@ system using temporal logics and synthesizing the controller using deterministic
algorithms, we are assured that strategic decisions will always be made
according to operating procedures.
(Talk about how one would go from a discrete automaton to actual code)
\splitinline{Talk about how one would go from a discrete automaton to actual
code}
(Examples of reactive synthesis in the wild)
%%% NOTES (Section 3):
% - Mention computational complexity of synthesis (doubly exponential worst case)
% - Discuss how specification structure affects synthesis tractability
% - Reference GR(1) fragment as a tractable subset commonly used in practice
% - May want to include an example automaton figure
% ----------------------------------------------------------------------------
% 4. CONTINUOUS CONTROLLERS
% ----------------------------------------------------------------------------
\splitinline{Examples of reactive synthesis in the wild}
\subsection{Continuous Control Modes}
@ -380,7 +364,7 @@ continuous state space among several discrete vector fields has traditionally
been a difficult problem for validation and verification. The discontinuity of
the vector fields at discrete state interfaces makes reachability analysis
computationally expensive, and analytic solutions often become intractable
\cite{MANYUS THESIS}.
\cite{kapuria2025, lang2021}.
We circumvent these issues by designing our hybrid system from the bottom up
with verification in mind. Each continuous control mode has an input set and
@ -404,18 +388,9 @@ $q_i$ must drive the system from any state in $\mathcal{X}_{entry,i}$ to some
state in $\mathcal{X}_{exit,i}$ while remaining within $\mathcal{X}_{safe,i}$.
We classify continuous controllers into three types based on their objectives:
transitory, stabilizing, and expulsory.\splitnote{This three-mode taxonomy is elegant — maps verification tools to control objectives cleanly.} Each type has distinct verification
transitory, stabilizing, and expulsory. Each type has distinct verification
requirements that determine which formal methods tools are appropriate.
%%% NOTES (Section 4):
% - Add figure showing the relationship between entry/exit/safety sets
% - Discuss how standard control techniques (LQR, MPC, PID) fit into this framework
% - Mention assume-guarantee reasoning for compositional verification
% ----------------------------------------------------------------------------
% 4.1 TRANSITORY MODES
% ----------------------------------------------------------------------------
\subsubsection{Transitory Modes}
Transitory modes are continuous controllers designed to move
@ -466,16 +441,6 @@ systems require more conservative over-approximations using techniques such as
Taylor models or polynomial zonotopes. For this work, we will select tools
appropriate to the fidelity of the reactor models available.
%%% NOTES (Section 4.1):
% - Add timing constraints discussion: what if the transition takes too long?
% - Consider timed reachability for systems with deadline requirements
% - Mention that the Mealy machine perspective unifies this: continuous system
% IS the transition, entry/exit conditions are the discrete states
% ----------------------------------------------------------------------------
% 4.2 STABILIZING MODES
% ----------------------------------------------------------------------------
\subsubsection{Stabilizing Modes}
Stabilizing modes are continuous controllers with an objective of maintaining a
@ -523,16 +488,6 @@ For example, a lower fidelity model can be used for controller design, but a
higher fidelity model can be used for the actual validation of that stabilizing
controller.
%%% NOTES (Section 4.2):
% - Clarify relationship between barrier certificates and Lyapunov stability
% - Discuss what happens at mode boundaries: barrier for this mode vs guard
% for transition
% - Mention tools: SOSTOOLS, dReal, barrier function synthesis methods
% ----------------------------------------------------------------------------
% 4.3 EXPULSORY MODES
% ----------------------------------------------------------------------------
\subsubsection{Expulsory Modes}
Expulsory modes are continuous controllers responsible for
@ -582,16 +537,6 @@ plant dynamics. The expulsory mode must handle the worst-case dynamics within
this envelope. This is where conservative controller design is appropriate as
safety margins will matter more than performance during emergency shutdown.
%%% NOTES (Section 4.3):
% - Discuss sensor failures vs actual plant failures
% - Address unmodeled disturbances that aren't failures
% - How much parametric uncertainty is enough? Need methodology for bounds
% - Mention graceful degradation: graded responses vs immediate SCRAM
% ----------------------------------------------------------------------------
% 5. INDUSTRIAL IMPLEMENTATION
% ----------------------------------------------------------------------------
\subsection{Industrial Implementation}
The methodology described above must be validated on realistic
@ -619,17 +564,9 @@ the success and impact of this work. We will directly address the gap of
verification and validation methods for these systems and industry adoption by
forming a two-way exchange of knowledge between the laboratory and commercial
environments. This work stands to be successful with Emerson implementation
because we will have access to system experts\splitfix{Typo: ``excess should be ``access} at Emerson to help with the fine
because we will have access to system experts at Emerson to help with the fine
details of using the Ovation system. At the same time, we will have the benefit
of transferring technology directly to industry with a direct collaboration in
this research, while getting an excellent perspective of how our research
outcomes can align best with customer needs.
%%% NOTES (Section 5):
% - Get specific details on ARCADE interface from Emerson collaboration
% - Mention what startup sequence will be demonstrated (cold shutdown →
% criticality → low power?)
% - Discuss how off-nominal scenarios will be tested (sensor failures,
% simulated component degradation)
% - Reference Westinghouse relationship if relevant

285
3-research-approach/v1.tex
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@ -1,285 +0,0 @@
\section{Research Approach}
This research will overcome the limitations of current practice to build
high-assurance hybrid control systems for critical infrastructure. Building
these systems with formal correctness guarantees requires three main thrusts:
\begin{enumerate}
\item Translate operating procedures and requirements into temporal logic
formulae
\item Create the discrete half of a hybrid controller using reactive synthesis
\item Develop continuous controllers to operate between modes, and verify
their correctness
\end{enumerate}
Commercial nuclear power operations remain manually controlled by human
operators, yet the procedures they follow are highly prescriptive and
well-documented. This suggests that human operators may not be entirely
necessary given current technology. Written procedures and requirements are
sufficiently detailed that they may be translatable into logical formulae with
minimal effort. If successful, this approach enables automation of existing
procedures without system reengineering. To formalize these procedures, we will
use temporal logic, which captures system behaviors through temporal relations.
The most efficient path for this translation is NASA's Formal Requirements
Elicitation Tool (FRET). FRET employs a specialized requirements language called
FRETish that restricts requirements to easily understood components while
eliminating ambiguity~\cite{katis_capture_2022}. FRETish bridges natural language
and mathematical specifications through a structured English-like syntax
automatically translatable to temporal logic.
FRET enforces this structure by requiring all requirements to contain six
components: %CITE FRET MANUAL
\begin{enumerate}
\item Scope: \textit{What modes does this requirement apply to?}
\item Condition: \textit{Scope plus additional specificity}
\item Component: \textit{What system element does this requirement affect?}
\item Shall
\item Timing: \textit{When does the response occur?}
\item Response: \textit{What action should be taken?}
\end{enumerate}
FRET provides functionality to check system \textit{realizability}. Realizability
analysis determines whether written requirements are complete by examining the
six structural components. Complete requirements neither conflict with one
another nor leave any behavior undefined. Systems that are not realizable from
their procedure definitions and design requirements present problems beyond
autonomous control implementation. Such systems contain behavioral
inconsistencies---the physical equivalent of software bugs. Using FRET during
autonomous controller development allows systematic identification and
resolution of these errors.
The second category of realizability issues involves undefined behaviors
typically left to human judgment during operations. This ambiguity is
undesirable for high-assurance systems, since even well-trained humans remain
prone to errors. Addressing these specification gaps in FRET during development
yields controllers free from these vulnerabilities.
FRET exports requirements in temporal logic format compatible with reactive
synthesis tools. Linear Temporal Logic (LTL) builds upon modal logic's
foundational operators for necessity ($\Box$, ``box'') and possibility
($\Diamond$, ``diamond''), extending them to reason about temporal
behavior~\cite{baier_principles_2008}. The box operator $\Box$ expresses that a
property holds at all future times (necessarily always), while the diamond
operator $\Diamond$ expresses that a property holds at some future time
(possibly eventually). These are complemented by the next operator ($X$) for the
immediate successor state and the until operator ($U$) for expressing
persistence conditions.
Consider a nuclear reactor SCRAM requirement expressed in natural language:
\textit{``If a high temperature alarm triggers, control rods must immediately
insert and remain inserted until operator reset.''} This plain language
requirement can be translated into a rigorous logical specification:
\begin{equation}
\Box(HighTemp \rightarrow X(RodsInserted \wedge (\neg
RodsWithdrawn\ U\ OperatorReset)))
\end{equation}
This specification precisely captures the temporal relationship between the
alarm condition, the required response, and the persistence requirement. The
necessity operator $\Box$ ensures this safety property holds throughout all
possible future system executions, while the next operator $X$ enforces
immediate response. The until operator $U$ maintains the state constraint until
the reset condition occurs. No ambiguity exists in this scenario because all
decisions are represented by discrete variables. Formulating operating rules in
this logic enforces finite, correct operation.
Reactive synthesis is an active research field focused on generating discrete
controllers from temporal logic specifications. The term ``reactive'' indicates
that the system responds to environmental inputs to produce control outputs.
These synthesized systems are finite, with each node representing a unique
discrete state. The connections between nodes, called \textit{state
transitions}, specify the conditions under which the discrete controller moves
from state to state. This complete mapping of possible states and transitions
constitutes a \textit{discrete automaton}. Discrete automata can be represented
graphically as nodes (discrete states) with edges indicating transitions between
them. From the automaton graph, one can fully describe discrete system dynamics
and develop intuitive understanding of system behavior. Hybrid systems naturally
exhibit discrete behavior amenable to formal analysis through these finite state
representations.
We will employ state-of-the-art reactive synthesis tools, particularly Strix,
which has demonstrated superior performance in the Reactive Synthesis
Competition (SYNTCOMP) through efficient parity game solving
algorithms~\cite{meyer_strix_2018,jacobs_reactive_2024}. Strix translates linear
temporal logic specifications into deterministic automata automatically while
maximizing generated automata quality. Once constructed, the automaton can be
implemented using standard programming control flow constructs. The graphical
representation enables inspection and facilitates communication with controls
programmers who lack formal methods expertise.
We will use discrete automata to represent the switching behavior of our hybrid
system. This approach yields an important theoretical guarantee: because the
discrete automaton is synthesized entirely through automated tools from design
requirements and operating procedures, the automaton---and therefore our hybrid
switching behavior---is \textit{correct by construction}. Correctness of the
switching controller is paramount. Mode switching represents the primary
responsibility of human operators in control rooms today. Human operators
possess the advantage of real-time judgment: when mistakes occur, they can
correct them dynamically with capabilities extending beyond written procedures.
Autonomous control lacks this adaptive advantage. Instead, autonomous
controllers replacing human operators must not make switching errors between
continuous modes. Synthesizing controllers from logical specifications with
guaranteed correctness eliminates the possibility of switching errors.
While discrete system components will be synthesized with correctness
guarantees, they represent only half of the complete system. Autonomous
controllers like those we are developing exhibit continuous dynamics within
discrete states. These systems, called hybrid systems, combine continuous
dynamics (flows) with discrete transitions (jumps). These dynamics can be
formally expressed as~\cite{branicky_multiple_1998}:
\begin{equation}
\dot{x}(t) = f(x(t),q(t),u(t))
\end{equation}
\begin{equation}
q(k+1) = \nu(x(k),q(k),u(k))
\end{equation}
Here, $f(\cdot)$ defines the continuous dynamics while $\nu(\cdot)$ governs
discrete transitions. The continuous states $x$, discrete state $q$, and
control input $u$ interact to produce hybrid behavior. The discrete state $q$
defines which continuous dynamics mode is currently active. Our focus centers
on continuous autonomous hybrid systems, where continuous states remain
unchanged during jumps---a property naturally exhibited by physical systems. For
example, a nuclear reactor switching from warm-up to load-following control
cannot instantaneously change its temperature or control rod position, but can
instantaneously change control laws.
The approach described for producing discrete automata yields physics-agnostic
specifications representing only half of a complete hybrid autonomous
controller. These automata alone cannot define the full behavior of the control
systems we aim to construct. The continuous modes will be developed after
discrete automaton construction, leveraging the automaton structure and
transitions to design multiple smaller, specialized continuous controllers.
Notably, translation into linear temporal logic creates barriers between
different control modes. Switching from one mode to another becomes a discrete
boolean variable. \(RodsInserted\) or \(HighTemp\) in the temporal
specifications are booleans, but in the real system they represent physical
features in the state space. These features mark where continuous control modes
end and begin; their definition is critical for determining which control mode
is active at any given time. Information about where in the state space these
conditions exist will be preserved from the original requirements and included
in continuous control mode development, but will not appear as numeric values in
discrete mode switching synthesis.
The discrete automaton transitions are key to the supervisory behavior of the
autonomous controller. These transitions mark decision points for switching
between continuous control modes and define their strategic objectives. We
will classify three types of high-level continuous controller objectives based
on discrete mode transitions:
\begin{enumerate}
\item \textbf{Stabilizing:} A stabilizing control mode has one primary
objective: maintaining the hybrid system within its current discrete mode.
This corresponds to steady-state normal operating modes, such as a
full-power load-following controller in a nuclear power plant. Stabilizing
modes can be identified from discrete automata as nodes with only incoming
transitions.
\item \textbf{Transitory:} A transitory control mode has the primary goal of
transitioning the hybrid system from one discrete state to another. In
nuclear applications, this might represent a controlled warm-up procedure.
Transitory modes ultimately drive the system toward a stabilizing
steady-state mode. These modes may have secondary objectives within a
discrete state, such as maintaining specific temperature ramp rates before
reaching full-power operation.
\item \textbf{Expulsory:} An expulsory mode is a specialized transitory mode
with additional safety constraints. Expulsory modes ensure the system is
directed to a safe stabilizing mode during failure conditions. For example,
if a transitory mode fails to achieve its intended transition, the
expulsory mode activates to immediately and irreversibly guide the system
toward a globally safe state. A reactor SCRAM exemplifies an expulsory
continuous mode: when initiated, it must reliably terminate the nuclear
reaction and direct the reactor toward stabilizing decay heat removal.
\end{enumerate}
Building continuous modes after constructing discrete automata enables local
controller design focused on satisfying discrete transitions. The primary
challenge in hybrid system verification is ensuring global stability across
transitions~\cite{branicky_multiple_1998}. Current techniques struggle with this
problem because dynamic discontinuities complicate
verification~\cite{bansal_hamilton-jacobi_2017,guernic_reachability_2009}. This
work alleviates these problems by designing continuous controllers specifically
with transitions in mind. Decomposing continuous modes according to their
required behavior at transition points avoids solving trajectories through the
entire hybrid system. Instead, local behavior information at transition
boundaries suffices. To ensure continuous modes satisfy their requirements, we
employ three main techniques: reachability analysis, assume-guarantee contracts,
and barrier certificates.
Reachability analysis computes the reachable set of states for a given input
set. While trivial for linear continuous systems, recent advances have extended
reachability to complex nonlinear
systems~\cite{frehse_spaceex_2011,mitchell_time-dependent_2005}. We use
reachability to define continuous state ranges at discrete transition boundaries
and verify that requirements are satisfied within continuous modes.
Assume-guarantee contracts apply when continuous state boundaries are not
explicitly defined. For any given mode, the input range for reachability
analysis is defined by the output ranges of discrete modes that transition to
it. This compositional approach ensures each continuous controller is prepared
for its possible input range, enabling reachability analysis without global
system analysis. Finally, barrier certificates prove that mode transitions are
satisfied. Barrier certificates ensure that continuous modes on either side of a
transition behave appropriately by preventing system trajectories from crossing
a given barrier. Control barrier functions certify safety by establishing
differential inequality conditions that guarantee forward invariance of safe
sets~\cite{prajna_safety_2004}. For example, a barrier certificate can guarantee
that a transitory mode transferring control to a stabilizing mode will always
move away from the transition boundary, rather than destabilizing the target
stabilizing mode.
This compositional approach has several advantages. First, this approach breaks
down autonomous controller design into smaller pieces. For designers of future
autonomous control systems, the barrier to entry is low, and design milestones
are clear due to the procedural nature of this research plan. Second, measurable
design progress also enables measurement of regulatory adherence. Each step in
this development procedure generates an artifact that can be independently
evaluated as proof of safety and performance. Finally, the compositional nature
of this development plan enables incremental refinement between control system
layers. For example, difficulty developing a continuous mode may reflect a
discrete automaton that is too restrictive, prompting refinement of system
design requirements. This synthesis between levels promotes broader
understanding of the autonomous controller.
To demonstrate this methodology, we will develop an autonomous startup
controller for a Small Modular Advanced High Temperature Reactor (SmAHTR). We
have already developed a high-fidelity SmAHTR model in Simulink that captures
the thermal-hydraulic and neutron kinetics behavior essential for verifying
continuous controller performance under realistic plant dynamics. The
synthesized hybrid controller will be implemented on an Emerson Ovation control
system platform, representative of industry-standard control hardware deployed
in modern nuclear facilities. The Advanced Reactor Cyber Analysis and
Development Environment (ARCADE) suite will serve as the integration layer,
managing real-time communication between the Simulink simulation and the Ovation
controller. This hardware-in-the-loop configuration enables validation of the
controller implementation on actual industrial control equipment interfacing
with a realistic reactor simulation, assessing computational performance,
real-time execution constraints, and communication latency effects.
Demonstrating autonomous startup control on this representative platform will
establish both the theoretical validity and practical feasibility of the
synthesis methodology for deployment in actual small modular reactor systems.
This unified approach addresses a fundamental gap in hybrid system design by
bridging formal methods and control theory through a systematic, tool-supported
methodology. Translating existing nuclear procedures into temporal logic,
synthesizing provably correct discrete switching logic, and developing verified
continuous controllers creates a complete framework for autonomous hybrid
control with mathematical guarantees. The result is an autonomous controller
that not only replicates human operator decision-making but does so with formal
assurance that switching logic is correct by construction and continuous
behavior satisfies safety requirements. This methodology transforms nuclear
reactor control from a manually intensive operation requiring constant human
oversight into a fully autonomous system with higher reliability than
human-operated alternatives. More broadly, this approach establishes a
replicable framework for developing high-assurance autonomous controllers in any
domain where operating procedures are well-documented and safety is paramount.

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\section{Research Approach}
\iffalse
HACS: hybrid autonomous control system
HAHACS: High-Assurance Hybrid AUtonomous Control System
The research approach here needs to clearly outline the solution the the problem
and identify the actions taken that will advance knowledge and solve the
problem.
First, what is the problem?
\textit{
Inhibition to adopt hybrid autonomous control in critical infrastructure is
rooted in safety concerns of system stability. Without a human in the loop
with general intelligence, HACS have not been trusted where failure modes can
be unique and novel.
}
So, what's the solution?
\textit{
This research approach develops a methodology to build HACS that are provably
safe. This methodology builds on existing technologies, and unifies different
research thrusts to build a complete hybrid control system. To do this, the
problem of a HAHCS is broken into three distinct pieces:
\begin{enumerate}
\item System specification: properties of the HAHaCS such as transition
between control modes and system invariants are specified using a formal
methods tool.
- This provides exact behavior
- allows realizabillity checking of controller specs. Can a controller
actually be built from these specs?
- ?
- ?
\item Discrete Behavior Synthesis: The discrete component of the controller
is synthesized directly from system specifications using reactive
synthesis.
- This ELIMINATES wholesale the possibility of introducing logical bugs
in the creation of the strategic part of the HAHCS. Critical decisions
that are normally made by a human are automated directly from the
formal specifications.
- This does two critical things:
- It makes the creation of the controller tractable. The reasons the
controller changes between modes acn be traced back to the
specification (and thus any requirements), which is a trace for
liability and justification of system behavior
- Discrete control decisions made by humans are reliant on the human
operator operating correctly. Humans are intrinsically probabalistic
creatures who cannot eliminate human error. By defining the behavior
of this system using temporal logics and synthesizing the controller
using deterministic algorithims, we are assured that strategic
decisions will always be made as according to operating procedures.
\item Continuous Behavior Synthesis and Verification: The continuous
components of the controller are built using existing dynamics and control
theory but then verified using reachability and barrier certificats.
- It's very challenging (nigh impossible) to say for certain how to
build any continuous control mode. That is honestly going to be have to
left to the specific control system and its objectives. It's not really
the point of this PhD to say how to do that. For that reason, I'm going
to assume that controllers between modes are generally possible to
build. That is to say that there exists a controller that can transition
between modes, but it is a human hunt to find it.
- To check if a candidate controller does transition between discrete
modes, we do two things:
- Check invariants using reachability. Specifications will require
that control modes transiiton from one mode to the next, where
appropriate. When this is the case, these invariants are extracted to
be checked using reachability. The control mode is given the possible
entry conditions of the 'entry' mode, and the possible 'exit' states
are analyzed. A cont. controller passes this reachability test if
there is no reachable state that is not at the exit condition of the
state transition.
--- This needs flushed out more. I think this can really be clarified
using entry and exit conditions of Mealy machines. The continuous
system IS the transition, and the reachabililty test is saying whether
or not the physical system actually satisfies the entry and exit
conditions.
- Then, for systems that need to STAY within one mode, we will use
barrier certificates. These can let us define a continuous state
boundary, and define for a discrete controller state, the total
controller will NOT leave the continuous boundary.
- One thing that must be considered is the idea that this analysis is
predicated on the physical system being correct to the model. If this
isn't true, we must define continuous modes that catch failure states.
If transition invariants are violated, we must shut down the system, and
build safety oriented control modes that we can be sure with a much
broader set of entry conditions will safely shut down the plant.
-- Q for dan: is it critical to really have software to namedrop or is it
better to stay amorphous on the technology? Iirc Manyu did a little bit of
both.
\end{enumerate}
What's the intellectual merit?
\textit{
There is no outstanding way to build HAHACS. This methodology provides a
basis for systems engineers to think about the components of a HAHACS as
interlocking pieces whos verification interlinks into a broader system.
This will also motivate the adoption of temporal logic to define autonomous
control systems, by allowing a close connection and tracability between
requirements from regulations to system specifications.
}
}
Some thoughts on invariants, and how they fit here: There are several types of
safety invariants that HAHACS might have.
1. Conditions that initiate a switch between control modes (reactiive synthesis
relevant)
2. Invariants about the stability of discrete states (barrier certificates)
3. Invariants ensuring the transition between discrete states (reachability)
4. Invariants about the timeliness of discrete transitions (??? Reachability?)
How do we reason about all of these invariants. Well, fundamentally they can
all be reasoned about with temporal logic statements. Using next and eventually
operators, we can get to the fundamental behavior of all of these modes. What's
challenging is the fact that we ensure that all of these specifications are
validated differs between the type of invariant. This is really the beauty of
this approach, and the intellectual merit. This proposal provides a way for
hybrid control systems to be verified for autonomous control systems by
diversifying the way that the invariants are checked.
Reactive synthesis helps us build discrete controllers using specifications
that have conditons that don't depend on time. These invariants generally are
strategic decisions, such as changing between operating modes, initiating power
level changes, or perhaps doing a refueling or shutdown routine. These
specifications are able to be nearly directly drawn from operating procedures,
and should be closely tied to instructions that would be used for human
operators. They have checkpoints for the continuous system in between different
control implements. An example is, raise power at a certain rate while ensure
temperature remains between certain bounds. These conditions are physical
states, but they are a binary result. The condition is really binary, desipite
perhaps having units of celsius or %power. When we build discrete controllers
from these specifications, we get the validation of the controller of these
specs for free by nature of reactive synthesis tools. We get direct
traceability from the operating procedure to the discrete controller
implementation with minimal human effort.
That being said, there are no free lunches here. Ultimately, we're controlling
physical systems, and while we can automate the controller building between
stratgic objectives, it is not trivial to do so for the controller of the
physical process. These controllers are going to have to be built manually,
with the continuous dynamics of the system in mind. Helpfully, if
specifications are complete first, one can obtain discrete controller before
building physical controllers. The result of this is a simplification of
controller design, becuase the operational goals of each continuous controller
is clearly outlined by the invariants that define the goal of each discrete
mode. While for reactive synthesis purposes conditions such as a certain
temperature being reached or power level attained are binary variables, the
continuous physical meaning becomes important in the design and analysis of the
physical controllers. The continuous value of these conditions becomes the goal
of the continuous controller design, while also providing a basis to check
controller performance.
To check continuous controllers are valid, we can split continuous controller
objectives into two types. First, we have continuous controllers that are
designed to move the plant between two different discrete modes. These will be
called 'transitory' controllers, because their entire purpose is to transition
the plant betweeen between discrete control modes. Because of the specification
of the hybrid control system a priori, we will have defined what the invariants
of these transitions are in continuous state space. Then, once a continuosu
controller design is developed, it can be validated using reachability
analysis. The input set for the analysis is the possible states that enter this
transitory mode, while the reachable states must be entirely contained within
the exit invariant for the controller to pass. At the time of writing this
proposal, it is not clear what the most efficient way to obetain this
continuous controller is, but is generally beyond the scope of this work. It is
assumed that they generally won't be so difficult to find for most systems, as
the refinement of the discrete controller should simplify the control
objectives of the physical controllers significantly.
The second type of continuous controller that may be utilized in a HAHAHCS is a
controller that tries to maintaine a continuous steady state, such that no
discrete transitions are triggered. Reachability on these systems may not prove
a prudent approach to validating this behavior for a candidate continuous
controller, and instead, barrier certificates must be used. Barrier
certificates analyze the dynamics of the system to say whether or not flux
across a given boudnary exists. That is to say that they evaluate whether or
not there is a trajectory or not that leaves a given boundary. This definition
is exactly what defines the validity of a stabilizing continuous control mode.
Once again, because the design of the discrete controller defines careful
boundaries in continuous state space, the barrier is known a priori of which we
must satisfy this condition. This will eliminate the search for such a barrier,
and minimze complicatoin in validating stabilizing continuous control modes.
Finally, consideration must be paid for when errors occur. The validation of
these continuous control modes hinges upon having an assumption ofcorrect
model, which in the case of a mechanical failure will almsot certainly be
invalidated. Special continuous controllers for these conditions must be
created, called 'explusory' control modes. These controllers will be
responsible for ensuring safety in case of failure, and will be designed with
reachability, but in this case, additional allocation for the allowing of
physical parameters will be allowed in the analysis. Traditional safety
analysis will also be used to identify potential failure modes, and the
modelling of their worst case dynamics. The HAHCS will be able to idenfity why
such a fault occors because an discrte boundary condition will be violated by
the continuous physical controller. That is to say, since we will have
validated the continuous control modes using reachability and barrier
certificates a priori, we will know with certainty that the only room for
dynamics to change is a shift in the plant dynamics, not that of the proven
controller.
\fi
%%%%%%%%%% TABLESETTING
% what is a hybrid system really for this proposal
% Define: A hybrid system with continuous state space X ⊆ ℝⁿ and discrete modes Q = {q₁, q₂, ..., qₘ}
% Each discrete mode qᵢ has an associated continuous state region Xᵢ ⊆ X
% The discrete controller manages transitions between modes based on continuous state thresholds
% what are requirements, anyways?
% why do we care about defining the whole hybrid system into requirements?
% How do different requirements line up into different parts of the system?
% (operational vs strategic requirements and their relevance to different parts
% of our system)
Autonomous control systems are fundamentally different from automatic control
systems. The difference between these systems is the level at which
they operate. Automatic control systems are purely operational systems,
To build a high-assurance hybrid autonomous control system (HAHACS), a
mathematical description of the system must be established. This work will make
use of automata theory while including logical statements and control theory.
The nomenclature and lexicon between these fields is far from homogenous, and
the reviewer of this proposal is not expected to be an expert in all fields
simultaneously. To present the research ideas as clearly as possible in this
section, the following syntax is explained.
A hybrid system is a dynamical system that has both continuous and discrete
states. The specific type of system discussed in this proposal are continuous
autonomous hybrid systems. This means that these systems a) do not have
external input \footnote{This is not strictly true in our case because we allow
strategic inputs. For example, a remote powerplant may receive a start-up or
shutdown command from a different location, but only this binary high level
input is a strategic input.} and b) continuous states do not change
instantaneously when discrete states change. For our systems of interest, the
continuous states are physical, and are always Lipschitz continuous. This
nomenclature is heavily borrowed from \cite{HANDBOOK ON HYBRID SYSTEMS CONTROL},
but is redefined here for convenience:
\begin{equation}
H = (\mathcal{Q}, \mathcal{X}, \mathbf{f}, Init, \mathcal{G}, \mathcal{R}, Inv)
\end{equation}
where:
\begin{itemize}
\item \( \mathcal{Q}\): is the discrete states of the system
\item \( \mathcal{X}\): is the continuous states of the system
\item \(\mathbf{f}: \mathcal{Q} \times \mathbb{R} \rightarrow \mathbb{R} \), where
\(\mathbf{f}_i\) is a
vector field that defines the continuous dynamics for each \(q_i\)
\item \(Init\): the initial states of \(q\) and \(x\)
\item \( G\): guard
conditions that define when discrete state transitions occur
\item \(\delta: \mathcal{Q} \times G \rightarrow \mathcal{Q}\), are the
discrete state transition functions
\item \mathcal{R}: Reset maps that define state 'jumps'
\item \(Inv\): Safety invariants on the continuous dynamics
\end{itemize}
The creation of a HAHACS essentially boils down to the creation of such a tuple
where there are proof artifacts that the intended behavior of the control system
are satisfied by the actual implementation of the control systems. But to create
such a HAHACS, we must first completely describe its behavior.
%% Brief discussion on what each part of this tuple means for us
\subsection{System Requirement and Specifications}
Temporal logic is a powerful set of semantics to build systems that can have
complex but deterministic behavior.
%%%%%%%%%%% Building discrete controllers
% Buildout of requirements from written procedures (this is easy for critical
% systems - we already have the requirements)
% What happens to the invariants that specify a continuous space? Save em for
% later. Here they become binary for our purposes
% KEY POINT: We don't IMPOSE discrete abstraction - we FORMALIZE existing practice
% Operating procedures (esp. nuclear) already define go/no-go conditions as discrete predicates
% e.g., "WHEN coolant temp >315°C AND pressurizer level 30-60% THEN MAY initiate load following"
% These thresholds come from design-basis safety analysis, validated over decades
% Our methodology assumes this domain knowledge exists and provides formalization framework
% The discrete predicates p₁, p₂, ... are Boolean functions over continuous state: pᵢ: X → {true, false}
% Q: How do we rigorously set thresholds for continuous→discrete abstraction?
% Q: How do we handle hysteresis to prevent mode chattering near boundaries?
% Q: How do we account for sensor noise and measurement uncertainty?
% Q: How do we handle numerical precision issues when creating discrete automata? (relates to task 36)
% Discrete controller implementation can be realized with reactive synthesis.
% LTL specs to automata
% talk a bit about tools here like FRET. Talk about previous attempts.
\begin{figure}[htbp]
\centering
\framebox[0.8\textwidth]{\rule{0pt}{3cm}\textit{Strategic, operational,
tactical placeholder}}
\caption{Breakdown of control scope}
\label{fig:strat_op_tact}
\end{figure}
Human control of nuclear power can be divided into three different scopes:
strategic, operational, and tactical. Strategic control is the high-level and
long term decision making for the plant. This level has objectives that are
complex and economic in scale, such as managing labor needs and supply chains to
optimize sheduled maintenence and downtime. The time scale on this level of
control is long, often over months or years. The lowest level of control is the
tactical level. This is the individual control of pumps, turbines, and
chemistry of the plant. This level of control has already been somewhat
automated today in nuclear power, and is generally considered 'automatic
control' when autonomous. These controls are almost always continuous systems,
and have a direct impact on the physical state of the plant. Tactical control
objectives are things like maintaining a pressurizer level, maintaining a
certain core temperature, or adjusting reactivity with a chemical shim. The level of
control linking these two levels, then, is the operational control scope.
Operational control is the primary responsibility of human operators today.
Operational control takes the current strategic objective, and implements
tactical control objectives to drive the plant towards strategic goals. In this
way, it is the bridge between high and low level goals. A strategic goal may be
to perform refueling at a certain time, while the tactical level of the plant
currently is focused on mainting a certain core temperature. The operational
level is what issues the shutdown procedure of the plant, using several smaller
tactical goals along the way to achieve this objective. Thus, the combination of
the operational and tactical level of the plant fundamentally forms a hybrid
controller. The tactical level is the continuous evolution of the plant
according to the control input and control law, while the operational level is a
discrete state evolution which determines the tactical control law to reach
different operational states.
This operational control level is the main reason for the requirement of human
opeartors in nuclear control today. The hybrid nature of this control system
makes it difficult to prove that a controller will perform according to the
strategic requirements, as the infrastructure to build hybrid systems today
dooes not exist. Humans have been used for this layer because the general
intelleigence of humans has be relied upon as a safe way to manage the hybrid
nature of this system. But, these operators are using prescriptive operating
manuals to perform their control with strict procedures on what control to
implement at a given time. These procedures are the key to the operational
control scope.
The method of constructing a HAHACS in this proposal leverages two key points of
the way this control scope is done today: first, the operational scope control
is effectively discrete control. Second, the rules of implementing this control
are described a priori to their implementation in operating procedures. We can
make great use of these facts by formalizing the rules for transitioning between
discrete states. To do this, we will use temporal logic to formalize discrete
switching behavior.
Temporal logic is a rich syntax that allows for the definition of logical
calculations including time related bounds. For this reason, we can make
statements relating discrete control modes to one another. Using temporal logic,
we can effectively describe all of the requirements of a HAHACS. The guard
conditions \(G\) are easily defined by determining boundary conditions between
discrete states and defining their behavior, while continuous mode invariants
can be defined using temporal logic statements as well. These form the basis of
any proofs about a HAHACS, and are the fundamental 'truth' statements about what
the behavior of the system is designed to be.
To build these temporal logic statements, an intermediary tool called FRET is
planned to be used. FRET stands for Formal Requirements Elicitation Tool, and
was designed by NASA to build high assurance timed systems. FRET is an
intermediarly language between temporal logic and natural language that allows
for rigid definitions of temporal behvarior while using a logic-novice friendly
syntax. This benefit is crucial for the feasibility of this methodology for
industry, as minimizing the barrier to formal methods is a critical component of
their scucess. By reducing the expert knowledge required to use these tools,
their adoption with current workforce becomes easier.
A key feature of FRET is the ability to start with logically imprecise
statements and consecutively refine them into a well-posited specification. We
can use this to our advantage by directly dumping in operating procedures and
design requirements into FRET in natural language, and iteratively refining them
into the specifications for a HAHACS. This has two distinct but important
benefits. First, it allows us to draw a direct link from the design
documentation to the digital system implementation. Second, it clearly
demonstrates where the natural language documents are insufficient. These
procedures may still be used by human operators, so any wiggle room for
interpretation is a weakness that must be addressed.
%Talk about how we go from temp logic to reactive synth. Metnion fret can
%export, or naturlly support reactive synth solver ltlsynt, a sota react synth
%solver
Once system requirements we defined as temporal logic specifications we will use
the specifications to build the discrete control system. To do this, reactive
synthesis tools will be utilized. Reactive synthesis is a field in computer
science that deals with the automated creation of reactive programs from
temporal logic specifications. A reactive program is one that for a given state
takes an input, and produces an output. Our systems, such as the discrete
portion of the controller, fit exactly this this mold. The current discrete
state, and status of guard conditions are the input to the system, while the
output is the next discrete state. The output of a reactive synthesis algorithim
is a discrete automata.
Reactive synthesis' main advantage is the fact that at no point in the
production of a discrete automata of the program is human engineering required.
The resultant automata is correct by construction. This method of construction
eliminates the possibility of human error outright at the implementation state.
Instead, the effort on the human designer is directed at the specification of
the system behavior itself.
% talk about what the benefits of reactive synth are. Proof chain, machine
% checkable, blah blah blah
%%%%% I NEED TO WRITE ABOUT HOW REQUIREMENTS ARE EXTRACTED AND WHAT BECOME
CONTINUOUS CONTROLLER TRANSITIONS VS DISCRETE GUARD CONDITIONS
%%%%%%%%%%%% Building continuous controllers
\subsection{Continuous Controllers}
% The whole point of a hybrid system is that there are continuous components
% underneath the digital system. We built the discrete like the physical doesn't
% exist, but it really does. So how do we capture the physical system too?
% SCOPE FRAMING: This methodology VERIFIES continuous controllers, not SYNTHESIZES them
% Compare to model checking: doesn't tell you HOW to design software, verifies if it satisfies specs
% We assume controllers can be designed using standard control theory techniques
% Our contribution: verification that candidate controllers compose correctly with discrete layer
% What are the main different kinds of continuous modes we may see?
% Mathematical structure: Each discrete mode qᵢ provides three key pieces of information:
% 1. Entry conditions: X_entry,i ⊆ X (initial state set)
% 2. Exit conditions: X_exit,i ⊆ X (target state set)
% 3. Invariants: X_safe,i ⊆ X (safety envelope during operation)
% These come from the discrete controller synthesis and define objectives for continuous control
% Q: Who designs the continuous controllers and how? This methodology verifies
% them, but doesn't synthesize them. Is this a scope problem?
The synthesis of the discrete operational controller is only half of an
autonomous controller. These control systems are hybrid, with both discrete and
continuous components. In this section, we will talk about the continuous
control modes that are the transitions between discrete modes, how they may be
synthesized, and how we plan to verify them.
The operational control scope defines go/no-go decisions that themselves are
deciding what kind of continuous control to implement. To this end, the entry or
exit of a discrete state triggers are themselves the guard conditions \(G\) that
define the barriers of the continuous controller. These continuous controllers
all share a large state space, but each individual continuous control mode
operates within it's own partition defined by the discrete state \(q_i\) and
guard conditions \(G\). This partitioning of the continuous state space amongst
several discrete vector fields controlled by the given \(q_i\) has traditionally
been a difficult problem for validation and verification of systems properties.
Typically, the discontinuity of the vector fields at discrete state interfaces
make things like reachability analysis computationally expensive, and analytic
solutions become intractable.
We circumnavigate these issues by designing our hybrid system from the bottom up
with this verification in mind. Each continuous control mode has an input and
output set clearly defined by our discrete transitions \textit{a priori}.
Consider that we define the continuous state space as \(X\). Whenever we create
guard functions from our design requirements for a given system, we are
effectively creating subsets \(X_{entry,i}\) and \(X_{exit,i}\) for each
discrete mode \(q_i\). These subsets define when the state transitions occur
between discrete modes, but more importantly when building continuous control
modes, they become control objectives.
% Start talking about what it means to build controlelrs to the objectives
% rahter than the other why around. ALso why it makes things much easier to
% verify and validate
%%%%%% Transitory modes
% entry and exit conditions
% the goal is getting from one physical state to another
% MATHEMATICAL FORMULATION:
% Control objective: reach(X_entry,i) → reach(X_exit,i) while maintaining x(t) ∈ X_safe,i
% Standard control techniques (LQR, MPC, trajectory optimization) applied with these constraints
%
% VERIFICATION: Reachability analysis confirms ALL trajectories starting in X_entry,i
% reach X_exit,i without violating X_safe,i
% Formally: Reach(X_entry,i, f(x,u), T) ⊆ X_exit,i ∪ X_safe,i
% where f(x,u) is the closed-loop continuous dynamics
%
% we have the physical requirements from earlier specifications. Here we use
% them in a reachability analysis. This time, we use the actual physical values
% instead of the binary yes/no we used for discrete
% Q: How do we verify timing constraints? If a transitory controller eventually
% reaches the exit condition but takes too long, that violates safety. Timed
% automata? Timed reachability?
% Q: Should formalize the Mealy machine perspective - continuous system IS the
% transition, and entry/exit conditions are the discrete states. This could be
% a unifying conceptual framework.
%%%%%% stabilizing modes
% these are control modes with an objective of KEEPING a certain discrete state
% stable
%
% MATHEMATICAL FORMULATION:
% Control objective: remain(X_target,i) where X_target,i ⊂ X_safe,i
% Standard feedback control (PID, state feedback, LQG) applied to maintain equilibrium
%
% VERIFICATION: Barrier certificates prove closed-loop dynamics cannot escape X_safe,i
% Formally: Find B(x) s.t. ∇B(x)·f(x,u) ≤ 0 for all x ∈ ∂X_safe,i
% This proves no trajectory can cross the boundary (no flux out of safety region)
%
% we also have the physical requirements for this. These can be used for barrier
% certificates. We can prove that our model won't leave a given area without
% some disturbance.
%%%%%% expulsory modes
% I've made an implicit assumption when talking about transitory and stabilizing
% modes. That our model is correct. This might not be true
% In the case of a failure, our model will almost certainly be incorrect. For
% this, we have to build safe shutdown modes too, since a human won't be in the
% loop to shut things down.
%
% MATHEMATICAL FORMULATION:
% Control objective: reach(X_current) → reach(X_safe_shutdown) under parameter uncertainty
% where X_current may be anywhere in X (worst-case entry conditions)
% Dynamics have parametric uncertainty: f(x,u,θ) where θ ∈ Θ_failure
%
% VERIFICATION: Parametric reachability analysis with robustness margins
% Reach(X_current, f(x,u,θ), T) ⊆ X_safe_shutdown for all θ ∈ Θ_failure
% Conservative bounds on Θ_failure come from FMEA/traditional safety analysis
% WE can detect physical failures exist because our physical controllers have
% previously been proven as correct by reachability and barrier certificates. We
% KNOW our controller cannot be incorrect for the plant, so if an invariant is
% violated, we KNOW it's the plant that has changed.
% Q: What about sensor failures (wrong readings vs actual plant failure)?
% Q: What about unmodeled disturbances that aren't failures?
% Q: What if model uncertainty was too optimistic to begin with?
% Need to be more precise about what "model failure" means and detect-ability.
% We do this using continuous modes that shutdown the system, and using
% reachability analysis with parametric uncertainty, we can prove for a range of
% error conditions we can maintain safe shutdown.
% Q: How much parametric uncertainty is enough? How do we determine bounds for
% worst-case failure dynamics? Need methodology for this.
%%%%%%%%%%%% Implementation with industrial partnerships
%%%%%%% Emerson
%talk about this
% ovation system
% scenic? Is that what they call it?
% ripe partnership with Westinghouse
% Likely build a model with a ccng plant. They already have sophisticated models
% of them
% build controller with simplified model, then test with high fidelity digital
% twin
%
%%%%%%%%%%

18
4-metrics-of-success/metrics.tex
View File

@ -4,9 +4,7 @@ This research will be measured by advancement through Technology Readiness
Levels, progressing from fundamental concepts to validated prototype
demonstration. This work begins at TRL 2--3 and aims to reach TRL 5, where
system components operate successfully in a relevant laboratory
environment.\splitnote{TRL as primary metric is smart — speaks industry
language.}
This section explains why TRL advancement provides the most appropriate success
environment. This section explains why TRL advancement provides the most appropriate success
metric and defines the specific criteria required to achieve TRL 5.
Technology Readiness Levels provide the ideal success metric because they
@ -15,9 +13,7 @@ deployment---precisely what this work aims to bridge. Academic metrics like
papers published or theorems proved cannot capture practical feasibility.
Empirical metrics like simulation accuracy or computational speed cannot
demonstrate theoretical rigor. TRLs measure both dimensions
simultaneously.\splitnote{Good framing — explains why other metrics are
insufficient.}
Advancing from TRL 3 to TRL 5 requires maintaining theoretical rigor while
simultaneously. Advancing from TRL 3 to TRL 5 requires maintaining theoretical rigor while
progressively demonstrating practical feasibility. Formal verification must
remain valid as the system moves from individual components to integrated
hardware testing.
@ -72,9 +68,10 @@ across the full operational envelope. The controller must handle off-nominal
scenarios to validate that expulsory modes function correctly. For example,
simulated sensor failures must trigger appropriate fault detection and mode
transitions, and loss-of-cooling scenarios must activate SCRAM procedures as
specified. Graded responses to minor disturbances are outside this work's
scope.\splitsuggest{Consider noting why graded responses are out of scope —
is it time, complexity, or scope creep? Brief justification helps.}
specified. Graded responses to minor disturbances are outside this work's scope.
If this work is successful, graded responses are an extension that requires
additional labor, but not new capability as the mechanics of a graded response
are an addition of special discrete modes and continuous controllers to match.
Formal verification results must remain valid, with discrete behavior matching
specifications and continuous trajectories remaining within verified bounds.
This proves that the methodology produces verified controllers implementable on
@ -92,5 +89,4 @@ operating on industrial control hardware through hardware-in-the-loop testing in
a relevant laboratory environment. This establishes both theoretical validity
and practical feasibility, proving that the methodology produces verified
controllers and that implementation is achievable with current
technology.\splitnote{Clear success criteria. Committee will know exactly
what ``done'' looks like.}
technology.

33
5-risks-and-contingencies/risks.tex
View File

@ -1,7 +1,7 @@
\section{Risks and Contingencies}
This research relies on several critical assumptions that, if invalidated, would
require scope adjustment or methodological revision.\splitnote{Honest acknowledgment of risks with clear contingencies — committee will appreciate this.} The primary risks to
require scope adjustment or methodological revision. The primary risks to
successful completion fall into four categories: computational tractability of
synthesis and verification, complexity of the discrete-continuous interface,
completeness of procedure formalization, and hardware-in-the-loop integration
@ -49,10 +49,8 @@ rather than a failure.
The second critical assumption concerns the mapping between boolean guard
conditions in temporal logic and continuous state boundaries required for mode
transitions. This interface represents the fundamental challenge of hybrid
systems: relating discrete switching logic to continuous dynamics. Temporal
logic operates on boolean predicates, while continuous control requires
reasoning about differential equations and reachable sets. Guard conditions
requiring complex nonlinear predicates may resist boolean abstraction, making
systems: relating discrete switching logic to continuous dynamics. Guard conditions
requiring complex predicates may resist boolean abstraction, making
synthesis intractable. Continuous safety regions that cannot be expressed as
conjunctions of verifiable constraints would similarly create insurmountable
verification challenges. The risk extends beyond static interface definition to
@ -88,18 +86,6 @@ structures the problem to minimize complex transitions, with critical safety
properties concentrated in expulsory modes that can receive additional design
attention.
Mitigation strategies focus on designing continuous controllers with discrete
transitions as primary objectives from the outset. Rather than designing
continuous control laws independently and verifying transitions post-hoc, the
approach uses transition requirements as design constraints. Control barrier
functions provide a systematic method to synthesize controllers that guarantee
forward invariance of safe sets and convergence to transition boundaries. This
design-for-verification approach reduces the likelihood that interface
complexity becomes insurmountable. Focusing verification effort on expulsory
modes---where safety is most critical---allows more complex analysis to be
applied selectively rather than uniformly across all modes, concentrating
computational resources where they matter most for safety assurance.
\subsection{Procedure Formalization Completeness}
The third assumption is that existing startup procedures contain sufficient
@ -143,16 +129,3 @@ valuable to both the nuclear industry and formal methods community, establishing
clear requirements for next-generation procedure development and autonomous
control specification languages.
Early-stage procedure analysis with domain experts provides the primary
mitigation strategy. Collaboration through the University of Pittsburgh Cyber
Energy Center enables identification and resolution of ambiguities before
synthesis attempts, rather than discovering them during failed synthesis runs.
Iterative refinement with reactor operators and control engineers can clarify
procedural intent before formalization begins, reducing the risk of discovering
insurmountable specification gaps late in the project. Comparison with
procedures from multiple reactor designs---pressurized water reactors, boiling
water reactors, and advanced designs---may reveal common patterns and standard
ambiguities amenable to systematic resolution. This cross-design analysis would
strengthen the generalizability of any proposed specification language
extensions, ensuring they address industry-wide practices rather than specific
quirks.

2
6-broader-impacts/impacts.tex
View File

@ -66,6 +66,6 @@ applies to any hybrid system with documented operational requirements. Potential
applications include chemical process control, aerospace systems, and autonomous
transportation, where similar economic and safety considerations favor increased
autonomy with provable correctness guarantees. Demonstrating this approach in
nuclear power---one of the most regulated and safety-critical domains\splitnote{``If it works here, it works anywhere — strong closing argument.}---will
nuclear power---one of the most regulated and safety-critical domains---will
establish both the technical feasibility and regulatory pathway for broader
adoption across critical infrastructure.

2
8-schedule/schedule.tex
View File

@ -93,4 +93,4 @@ methodology. M5 (Month 20) achieves TRL 5 by demonstrating practical
implementability on industrial hardware. This milestone delivers a conference
paper submission to NPIC\&HMIT or CDC documenting hardware implementation and
experimental validation. M6 (Month 24) completes the dissertation documenting
the entire methodology, experimental results, and research contributions.\splitnote{Clear timeline with publication targets — shows you have a plan.}
the entire methodology, experimental results, and research contributions.

231
main.aux
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@ -1,103 +1,24 @@
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@ -107,106 +28,106 @@
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\begin{document}
\pagenumbering{roman}
% \pagenumbering{roman}
\maketitle
\input{1-goals-and-outcomes/research-statement.tex}
% \input{1-goals-and-outcomes/research-statement.tex}
\newpage
\tableofcontents
\newpage
\pagenumbering{arabic}
\input{1-goals-and-outcomes/goals}
\input{1-goals-and-outcomes/goals.tex}
\newpage
\input{2-state-of-the-art/state-of-art}

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\contentsline {section}{Contents}{iii}{}%
\contentsline {section}{Contents}{1}{}%
\contentsline {section}{\numberline {1}Goals and Outcomes}{1}{}%
\contentsline {section}{\numberline {2}State of the Art and Limits of Current Practice}{3}{}%
\contentsline {subsection}{\numberline {2.1}Current Reactor Procedures and Operation}{3}{}%