\section{Research Approach}

% ============================================================================
% STRUCTURE (maps to Thesis.RA tasks):
%   1. Introduction + Hybrid Systems Definition (Task 34)
%   2. System Requirements and Specifications (Task 35)
%   3. Discrete Controller Synthesis (Task 36)
%   4. Continuous Controllers Overview (Task 37)
%      4.1 Transitory Modes (Task 38)
%      4.2 Stabilizing Modes (Task 39)
%      4.3 Expulsory Modes (Task 40)
%   5. Industrial Implementation (Task 41)
% ============================================================================

% ----------------------------------------------------------------------------
% 1. INTRODUCTION AND HYBRID SYSTEMS DEFINITION
% ----------------------------------------------------------------------------
Previous approaches to autonomous control verified discrete switching logic or continuous control behavior—never both simultaneously. Extensive simulation trials validate continuous controllers. Simulated control room testing and human factors research evaluate discrete switching logic. Despite consuming enormous resources, neither method provides rigorous guarantees of control system behavior. HAHACS bridges this gap by composing formal methods from computer science with control-theoretic verification, then formalizing reactor operations using hybrid automata.

Hybrid system verification faces a fundamental challenge: the interaction between discrete and continuous dynamics. When discrete transitions occur, the governing vector field changes, creating discontinuities in system behavior. Traditional verification techniques—designed for purely discrete or purely continuous systems—cannot handle this interaction directly. Our methodology decomposes the problem: discrete switching logic and continuous mode behavior are verified separately, then composed to reason about the complete hybrid system. This two-layer approach mirrors reactor operations themselves. Discrete supervisory logic determines which control mode is active; continuous controllers govern plant behavior within each mode.

Building a high-assurance hybrid autonomous control system (HAHACS) requires
first establishing a mathematical description of the system. This work draws on
automata theory, temporal logic, and control theory. A hybrid system is a
dynamical system with both continuous and discrete states. The specific type
of system discussed in this proposal is a continuous autonomous hybrid system.
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
for convenience:

\begin{equation}
  H = (\mathcal{Q}, \mathcal{X}, \mathbf{f}, Init, \mathcal{G}, \delta, \mathcal{R}, Inv)
\end{equation}

where:

\begin{itemize}
  \item $\mathcal{Q}$: the set of discrete states (modes) of the system
  \item $\mathcal{X} \subseteq \mathbb{R}^n$: the continuous state space
  \item $\mathbf{f}: \mathcal{Q} \times \mathcal{X} \rightarrow \mathbb{R}^n$:
    vector fields defining the continuous dynamics for each discrete mode $q_i$
  \item $Init \subseteq \mathcal{Q} \times \mathcal{X}$: the set of initial states
  \item $\mathcal{G}$: guard conditions that define when discrete state
    transitions may occur
  \item $\delta: \mathcal{Q} \times \mathcal{G} \rightarrow \mathcal{Q}$: the
    discrete state transition function
  \item $\mathcal{R}$: reset maps that define any instantaneous changes to
    continuous state upon discrete transitions
  \item $Inv$: safety invariants on the continuous dynamics
\end{itemize}

Creating a HAHACS requires constructing such a tuple together with proof artifacts demonstrating that the control system's actual implementation satisfies its intended behavior.

\textbf{What is new:} Each component's infrastructure has matured, but no existing work composes them for end-to-end hybrid system verification. Our novelty lies in the architecture connecting discrete synthesis with continuous verification through well-defined interfaces.

\textbf{Why it will succeed:} Defining entry, exit, and safety conditions at the discrete level first transforms the intractable problem of global hybrid verification into local verification problems with clear interfaces. Verification then operates per mode rather than on the full hybrid system, keeping analysis tractable even for complex reactor operations. Critically, nuclear procedures already define discrete boundaries between operating regimes. This existing structure provides the natural decomposition our methodology requires—making the approach practical for real systems.

\begin{figure} 
  \centering
  \begin{tikzpicture}[
    state/.style={
      circle, draw=black, thick, minimum size=2.2cm,
      fill=blue!10, align=center, font=\small
    },
    trans/.style={
      ->, thick, >=stealth
    },
    guard/.style={
      font=\scriptsize, align=center, fill=white, inner sep=2pt
    },
    dynamics/.style={
      font=\scriptsize\itshape, text=blue!70!black
    }
  ]
    % States
    \node[state] (q0) at (0,0) {$q_0$\\Cold\\Shutdown};
    \node[state] (q1) at (5,0) {$q_1$\\Heatup};
    \node[state] (q2) at (10,0) {$q_2$\\Power\\Operation};
    \node[state, fill=red!15] (q3) at (5,-3.5) {$q_3$\\SCRAM};
    
    % Normal transitions
    \draw[trans] (q0) -- node[guard, above] {$T_{avg} > T_{min}$} (q1);
    \draw[trans] (q1) -- node[guard, above] {$T_{avg} \in [T_{op} \pm \delta]$\\$P > P_{crit}$} (q2);
    
    % Fault transitions
    \draw[trans, red!70!black] (q1) -- node[guard, left, text=red!70!black] {$\neg Inv_1$} (q3);
    \draw[trans, red!70!black] (q2) to[bend left=20] node[guard, right, text=red!70!black] {$\neg Inv_2$} (q3);
    
    % Recovery transition
    \draw[trans, dashed] (q3) to[bend left=30] node[guard, below] {Manual reset} (q0);
    
    % Self-loops indicating staying in mode
    \draw[trans] (q2) to[loop right] node[guard, right] {$Inv_2$} (q2);
    
    % Dynamics labels below states
    \node[dynamics] at (0,-1.4) {$\dot{x} = f_0(x)$};
    \node[dynamics] at (6,-1.2) {$\dot{x} = f_1(x,u)$};
    \node[dynamics] at (10,-1.4) {$\dot{x} = f_2(x,u)$};
    \node[dynamics] at (5,-4.9) {$\dot{x} = f_3(x)$};
    
  \end{tikzpicture}
  \caption{Simplified hybrid automaton for reactor startup. Each discrete state 
    $q_i$ has associated continuous dynamics $f_i$. Guard conditions on 
    transitions (e.g., $T_{avg} > T_{min}$) are predicates over continuous 
    state. Invariant violations ($\neg Inv_i$) trigger transitions to the 
    SCRAM state. The operational level manages discrete transitions; the 
    tactical level executes continuous control within each mode.}
  \label{fig:hybrid_automaton}
\end{figure}

%%% NOTES (Section 1):
% - May want to clarify the "no external input" claim with a footnote about
%   strategic inputs (e.g., remote start/stop commands)
% - The reset map R is often identity for physical systems; clarify if needed

% ----------------------------------------------------------------------------
% 2. SYSTEM REQUIREMENTS AND SPECIFICATIONS
% ----------------------------------------------------------------------------

\subsection{System Requirements, Specifications, and Discrete Controllers}

The preceding section established the mathematical framework for hybrid systems. We now show how to construct such systems from existing operational knowledge. Nuclear operations already possess a natural hybrid structure that maps directly to the automaton formalism.

Human control of nuclear power divides into three scopes: strategic,
operational, and tactical. Strategic control is 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 scheduled maintenance and downtime. The time scale at this level is
long, often spanning months or years. The lowest level of control is the
tactical level. This is the individual control of pumps, turbines, and
chemistry. Tactical control has already been somewhat automated in nuclear power
plants today, and is generally considered ``automatic control'' when autonomous.
These controls are almost always continuous systems with a direct impact on the
physical state of the plant. Tactical control objectives include maintaining
pressurizer level, maintaining core temperature, or adjusting reactivity with a
chemical shim.

The level of control linking these two extremes 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 bridges high-level and low-level goals. 

Consider an example: a strategic goal may be to
perform refueling at a certain time, while the tactical level of the plant is
currently focused on maintaining a certain core temperature. The operational
level issues the shutdown procedure, using several smaller tactical goals along
the way to achieve this objective. 

This structure reveals why the combination of the operational and
tactical levels 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 that determines
which tactical control law to apply. This operational level is precisely what we target for autonomous control.


\begin{figure}
  \centering
  \begin{tikzpicture}[scale=0.8]
    % Pyramid layers
    \fill[blue!30!white] (0,4) -- (2,4) -- (1,5.) -- cycle;
    \fill[blue!20!white] (-1.5,2.5) -- (3.5,2.5) -- (2,4) -- (0,4) -- cycle;
    \fill[blue!10!white] (-3,1) -- (5,1) -- (3.5,2.5) -- (-1.5,2.5) -- cycle;
    
    % Labels inside pyramid
    \node[font=\small\bfseries] at (1,4.5) {Strategic};
    \node[font=\small\bfseries] at (1,3.1) {Operational};
    \node[font=\small\bfseries] at (1,1.6) {Tactical};
    
    % Descriptions to the right
    \node[anchor=west, font=\small, text width=8cm] at (5.5,4.6) 
      {\textit{Long-term planning:} maintenance scheduling, capacity planning, economic dispatch};
    \node[anchor=west, font=\small, text width=8cm] at (5.5,3.1) 
      {\textit{Discrete decisions:} startup/shutdown sequences, power level changes, mode transitions};
    \node[anchor=west, font=\small, text width=8cm] at (5.5,1.6) 
      {\textit{Continuous control:} temperature regulation, pressure control, load following};
    
    % Bracket showing HAHACS scope (simple line with text)
    \draw[thick] (5.0,1.0) -- (-3.5,1) -- (-3.5,4) -- (2.0,4) -- cycle;
    \node[font=\small, align=center, rotate=90] at (-4.2,2.5) {HAHACS scope};
  \end{tikzpicture}
  \caption{Control scope hierarchy in nuclear power operations. Strategic control 
    (long-term planning) remains with human management. HAHACS addresses the 
    operational level (discrete mode switching) and tactical level (continuous 
    control within modes), which together form a hybrid control system.}
  \label{fig:strat_op_tact}
\end{figure}


This operational control level is the main reason for the requirement of human
operators in nuclear control today. The hybrid nature of this control system
makes it difficult to prove that a controller will perform according to
strategic requirements, as unified infrastructure for building and verifying
hybrid systems does not currently exist. Humans have been used for this layer
because their general intelligence has been relied upon as a safe way to manage
the hybrid nature of this system. But these operators use 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.

Constructing a HAHACS leverages two key
observations about current practice. First, operational scope control is
effectively discrete control. Second, operating procedures describe the rules for implementing this control
before implementation. Before
constructing a HAHACS, we must completely describe its intended behavior. The
behavior of any control system originates in requirements: statements about what
the system must do, must not do, and under what conditions. For nuclear systems,
these requirements derive from multiple sources including regulatory mandates,
design basis analyses, and operating procedures. The challenge is formalizing
these requirements with sufficient precision that they can serve as the
foundation for autonomous control system synthesis and verification. We can
build these requirements using temporal logic.

Temporal logic is a powerful set of semantics for building systems with complex
but deterministic behavior. Temporal logic extends classical propositional logic
with operators that express properties over time. Using temporal logic, we can
make statements relating discrete control modes to one another and define all
the requirements of a HAHACS. The guard conditions $\mathcal{G}$ are defined by
determining boundary conditions between discrete states and specifying their
behavior, while continuous mode invariants can also be expressed as temporal
logic statements. These specifications form the basis of any proofs about a
HAHACS and constitute the fundamental truth statements about what the behavior
of the system is designed to be.

Discrete mode transitions include predicates that are Boolean functions over the
continuous state space: $p_i: \mathcal{X} \rightarrow \{\text{true},
\text{false}\}$. These predicates formalize conditions like ``coolant
temperature exceeds 315°C'' or ``pressurizer level is between 30\% and 60\%.''
We do not impose this discrete abstraction artificially. Operating
procedures for nuclear systems already define go/no-go conditions as discrete
predicates. Design basis safety analysis determined these thresholds, and decades of operational experience have
validated them. Our methodology assumes
this domain knowledge exists and provides a framework to formalize it. The approach is feasible for nuclear applications because generations
of nuclear engineers have already done the hard
work of defining safe operating boundaries. 

Linear temporal logic (LTL) is particularly well-suited for
specifying reactive systems. LTL formulas are built from atomic propositions
(our discrete predicates) using Boolean connectives and temporal operators.
The key temporal operators are:
\begin{itemize}
  \item $\mathbf{X}\phi$ (next): $\phi$ holds in the next state
  \item $\mathbf{G}\phi$ (globally): $\phi$ holds in all future states
  \item $\mathbf{F}\phi$ (finally): $\phi$ holds in some future state
  \item $\phi \mathbf{U} \psi$ (until): $\phi$ holds until $\psi$ becomes true
\end{itemize}
These operators allow us to express safety properties (``the reactor never
enters an unsafe configuration''), liveness properties (``the system
eventually reaches operating temperature''), and response properties (``if
coolant pressure drops, the system initiates shutdown within bounded time'').


We will use FRET (Formal Requirements Elicitation Tool) to build these temporal logic statements. NASA developed FRET for high-assurance timed systems. FRET provides an intermediate language between temporal logic and natural language, enabling rigid definitions of temporal behavior through syntax accessible to engineers without formal methods expertise. This accessibility is crucial for industrial feasibility. By reducing required expert knowledge, we make these tools adoptable by the current workforce.

A key feature of FRET is the ability to start with logically imprecise
statements and consecutively refine them into well-posed specifications. We can
use this to our advantage by directly importing operating procedures and design
requirements into FRET in natural language, then iteratively refining them into
specifications for a HAHACS. This has two distinct benefits. First, it allows us
to draw a direct link from design documentation to digital system
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
% ----------------------------------------------------------------------------

Having defined system requirements as temporal logic specifications, we now use
them to build the discrete control system through reactive synthesis.
Reactive synthesis is a field in computer science that deals with
the automated creation of reactive programs from temporal logic specifications.
A reactive program takes an input for a given state and produces
an output. Our systems fit this model: the current discrete state and
status of guard conditions form the input; the next discrete
state is the output. 

Reactive synthesis solves the following problem: given an LTL formula $\varphi$
that specifies desired system behavior, automatically construct a finite-state
machine (strategy) that produces outputs in response to environment inputs such
that all resulting execution traces satisfy $\varphi$. If such a strategy
exists, the specification is called \emph{realizable}. The synthesis algorithm
either produces a correct-by-construction controller or reports that no such
controller can exist. This realizability check is itself valuable: an
unrealizable specification indicates conflicting or impossible requirements in
the original procedures. 

Reactive synthesis offers a decisive advantage: producing the discrete automaton requires no human engineering of the implementation. The resultant automaton is correct by construction. This eliminates human error at the implementation stage entirely. Human designers focus their effort where it belongs: on specifying system behavior itself. This has two critical implications. First, it makes the
creation of the discrete controller tractable. The reasons the controller
changes between modes can be traced back to the specification and thus to any
requirements, which provides a trace for liability and justification of system
behavior. Second, discrete control decisions made by humans are reliant on the
human operator operating correctly. Humans are intrinsically probabilistic
creatures who cannot eliminate human error. By defining the behavior of this
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)

(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
% ----------------------------------------------------------------------------

\subsection{Continuous Control Modes}

The discrete controller is provably correct—synthesis guarantees this. Yet hybrid control requires more than discrete correctness. We now turn to the continuous dynamics executing within each discrete mode.

The discrete operational controller, while provably correct, represents only half of an
autonomous controller. Hybrid control systems require both discrete and
continuous components. This section describes the continuous control modes that
execute within each discrete state, and how we verify that they satisfy the
requirements imposed by the discrete layer. We must clarify this methodology's scope regarding continuous controller design. This work verifies continuous controllers—it does not synthesize them. The distinction parallels model checking in software verification. Model checking does not tell engineers how to write correct software; it verifies whether a given implementation satisfies its specification. We make a similar assumption: continuous controllers can be designed using standard control theory techniques. Our contribution is the verification framework that confirms candidate controllers compose correctly with the discrete layer to produce a safe hybrid system.

The operational control scope defines go/no-go decisions that determine what
kind of continuous control to implement. The entry or exit conditions of a
discrete state are themselves the guard conditions $\mathcal{G}$ that define the
boundaries for each continuous controller's allowed state-space region. These
continuous controllers all share a common state space, but each individual
continuous control mode operates within its own partition defined by the
discrete state $q_i$ and the associated guards. This partitioning of the
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}.

We circumvent these issues by designing our hybrid system from the bottom up
with verification in mind. The discrete transitions define each continuous
control mode's input set and output set clearly \textit{a priori}.

The continuous state space is $\mathcal{X}$. Each discrete mode $q_i$ provides
three key pieces of information for continuous controller design:
\begin{enumerate}
  \item \textbf{Entry conditions:} $\mathcal{X}_{entry,i} \subseteq \mathcal{X}$,
    the set of possible initial states when entering this mode
  \item \textbf{Exit conditions:} $\mathcal{X}_{exit,i} \subseteq \mathcal{X}$,
    the target states that trigger transition to the next mode, or is the region
    in the state space a stabilizing mode remains within.
  \item \textbf{Safety invariants:} $\mathcal{X}_{safe,i} \subseteq \mathcal{X}$,
    the envelope of safe states during operation in this mode. These are derived
    from invariants \(Inv\).
\end{enumerate}
These sets come directly from the discrete controller synthesis and define
precise objectives for continuous control. The continuous controller for mode
$q_i$ 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. 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}

The first mode type, transitory modes, moves
the plant from one discrete operating condition to another. Their purpose is to
execute transitions: starting from entry conditions, reach exit conditions,
and maintain safety invariants throughout. Examples include power ramp-up sequences,
cooldown procedures, and load-following maneuvers.

The control objective for a transitory mode can be stated
formally. Given entry conditions $\mathcal{X}_{entry}$, exit conditions
$\mathcal{X}_{exit}$, safety invariant $\mathcal{X}_{safe}$, and closed-loop
dynamics $\dot{x} = f(x, u(x))$, the controller must satisfy:
\[
\forall x_0 \in \mathcal{X}_{entry}: \exists T > 0: x(T) \in \mathcal{X}_{exit}
\land \forall t \in [0,T]: x(t) \in \mathcal{X}_{safe}
\]
That is, from any valid entry state, the trajectory must eventually reach the
exit condition without ever leaving the safe region.

Verification of transitory modes uses reachability analysis.
Reachability analysis computes the set of all states reachable from a given
initial set under the system dynamics. For a transitory mode to be valid, the
reachable set from $\mathcal{X}_{entry}$ must satisfy two conditions:
\begin{enumerate}
  \item The reachable set eventually intersects $\mathcal{X}_{exit}$ (the mode
    achieves its objective)
  \item The reachable set never leaves $\mathcal{X}_{safe}$ (safety is maintained
    throughout the transition)
\end{enumerate}
Formally, if $\text{Reach}(\mathcal{X}_{entry}, f, [0,T])$ denotes the states
reachable within time horizon $T$:
\[
\text{Reach}(\mathcal{X}_{entry}, f_i, [0,T]) \subseteq \mathcal{X}_{safe} \land
\text{Reach}(\mathcal{X}_{entry}, f_i, [0,T]) \cap \mathcal{X}_{exit} \neq \emptyset
\]

\textcolor{blue}{Because the discrete controller defines clear boundaries in continuous state
space, the verification problem for each transitory mode is well-posed. We know
the possible initial conditions, we know the target conditions, and we know the
safety envelope. The verification task is to confirm that the candidate
continuous controller achieves the objective from all possible starting points.}

Several tools exist for computing reachable sets of hybrid
systems, including CORA, Flow*, SpaceEx, and JuliaReach. The choice of tool
depends on the structure of the continuous dynamics. Linear systems admit
efficient polyhedral or ellipsoidal reachability computations. Nonlinear
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}

Transitory modes drive the system toward exit conditions. Stabilizing modes, the second type,
maintain the system within a desired operating region indefinitely. Rather than driving the system toward an
exit condition, they keep the system within a safe operating region. Examples
include steady-state power operation, hot standby, and load-following at
constant power level. Reachability analysis for stabilizing modes may not be a
suitable approach to validation. Instead, we plan to use barrier certificates.
Barrier certificates analyze the dynamics of the system to determine whether
flux across a given boundary exists. They evaluate whether any trajectory leaves
a given boundary. This definition is exactly what defines the validity of a
stabilizing continuous control mode.

A barrier certificate (or control barrier function) is a
scalar function $B: \mathcal{X} \rightarrow \mathbb{R}$ that certifies forward
invariance of a safe set. The idea is analogous to Lyapunov functions for
stability: rather than computing trajectories explicitly, we find a certificate
function whose properties guarantee the desired behavior. For a safe set
$\mathcal{C} = \{x : B(x) \geq 0\}$ and dynamics $\dot{x} = f(x,u)$, the
barrier certificate condition requires:
\[
\forall x \in \partial\mathcal{C}: \dot{B}(x) = \nabla B(x) \cdot f(x,u(x)) \geq 0
\]
This condition states that on the boundary of the safe set (where $B(x) = 0$),
the time derivative of $B$ is non-negative. Geometrically, this means the
vector field points inward or tangent to the boundary, never outward. If this
condition holds, no trajectory starting inside $\mathcal{C}$ can ever leave.

Because the design of the discrete controller defines careful boundaries in
continuous state space, the barrier is known prior to designing the continuous
controller. This eliminates the search for an appropriate barrier and minimizes
complication in validating stabilizing continuous control modes. The discrete
specifications tell us what region must be invariant; the barrier certificate
confirms that the candidate controller achieves this invariance.

Finding barrier certificates can be formulated as a
sum-of-squares (SOS) optimization problem for polynomial systems, or solved
using satisfiability modulo theories (SMT) solvers for broader classes of
dynamics. The key advantage is that the verification is independent of how
the controller was designed. Standard control techniques can be used to
build continuous controllers, and barrier certificates provide a separate
check that the result satisfies the required invariants. This also allows for
the checking of control modes with different models than they are designed for.
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}

Transitory and stabilizing modes handle nominal operations. Expulsory modes
handle off-nominal conditions.

When the plant deviates from expected behavior, expulsory modes take over. These
continuous controllers ensure safety when failures occur. They are designed for robustness rather
than optimality. The control objective is to drive the plant to a safe shutdown
state from potentially anywhere in the state space, under degraded or uncertain
dynamics. Examples include emergency core cooling, reactor SCRAM sequences, and
controlled depressurization procedures.

Detecting physical failures becomes straightforward because we have already proven our controllers correct through reachability and barrier certificates. Our controller cannot be incorrect for the nominal plant model. Therefore, when an invariant is violated, the plant dynamics must have changed. The HAHACS identifies faults when continuous controllers violate discrete boundary conditions. This direct consequence of verified nominal control modes means: unexpected behavior implies off-nominal conditions.

The mathematical formulation for expulsory mode verification
differs from transitory modes in two key ways. First, the entry conditions may
be the entire state space (or a large, conservatively bounded region) rather
than a well-defined entry set. The failure may occur at any point during
operation. Second, the dynamics include parametric uncertainty representing
failure modes:
\[
\dot{x} = f(x, u, \theta), \quad \theta \in \Theta_{failure}
\]
where $\Theta_{failure}$ captures the range of possible degraded plant
behaviors identified through failure mode and effects analysis (FMEA) or
traditional safety analysis.

We verify expulsory modes using reachability analysis with parametric
uncertainty. The verification condition requires that for all parameter values
within the uncertainty set, trajectories from the expanded entry region reach
the safe shutdown state:
\[
\forall \theta \in \Theta_{failure}:
\text{Reach}(\mathcal{X}_{current}, f_\theta, [0,T]) \subseteq \mathcal{X}_{shutdown}
\]
This is more conservative than nominal reachability, accounting for the fact
that we cannot know exactly which failure mode is active.

Traditional safety analysis techniques inform the construction
of $\Theta_{failure}$. Probabilistic risk assessment, FMEA, and design basis
accident analysis identify credible failure scenarios and their effects on
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
systems using industrial-grade hardware to demonstrate practical feasibility.
This research will leverage the University of Pittsburgh Cyber Energy Center's
partnership with Emerson to implement and test the HAHACS methodology on
production control equipment. Emerson's Ovation distributed control system is widely deployed
in power generation facilities, including nuclear plants. The Ovation platform
provides a realistic target for demonstrating that formally synthesized
controllers can execute on industrial hardware meeting timing and reliability
requirements. The discrete automaton produced by reactive synthesis will be
compiled to run on Ovation controllers, with verification that the implemented
behavior matches the synthesized specification exactly.

For the continuous dynamics, we will use a small modular
reactor simulation. The SmAHTR (Small modular Advanced High Temperature
Reactor) model provides a relevant testbed for startup and shutdown procedures.
The ARCADE (Advanced Reactor Control Architecture Development Environment)
interface will establish communication between the Emerson Ovation hardware and
the reactor simulation, enabling hardware-in-the-loop testing of the complete
hybrid controller.

Working with Emerson on such an implementation is an incredible advantage for
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 excess 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

