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+\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
+% ----------------------------------------------------------------------------
+
+To build a high-assurance hybrid autonomous control system (HAHACS), we must
+first establish a mathematical description of the system. This work draws on
+automata theory, temporal logic, and control theory. The nomenclature across
+these fields is far from homogeneous, and the reviewer of this proposal is not
+expected to be an expert in all of them simultaneously. To present the research
+ideas as clearly as possible, the following definitions are provided.
+
+A hybrid system is a dynamical system that has 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, 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}
+
+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.
+
+\textcolor{blue}{Previous approaches to autonomous control have verified
+discrete switching logic or continuous control behavior, but not both
+simultaneously. Validation of continuous controllers today consists of
+extensive simulation trials. Discrete switching logic for routine operation
+has been driven by human operators, whose evaluation includes simulated
+control room testing and human factors research. Neither method, despite
+being extremely resource intensive, provides rigorous guarantees of control
+system behavior. HAHACS bridges this gap by composing formal methods from
+computer science with control-theoretic verification, formalizing reactor
+operations using the framework of hybrid automata.}
+
+\textcolor{blue}{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.}
+
+\textcolor{blue}{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 of reactor operations themselves: discrete
+supervisory logic determines which control mode is active, while continuous
+controllers govern plant behavior within each mode.}
+
+\textcolor{blue}{This approach is tractable now because the infrastructure for each component
+has matured. Reactive synthesis from temporal logic specifications has
+progressed from theoretical results to practical tools; solvers like Strix
+can synthesize controllers from LTL specifications in seconds. Reachability
+analysis and barrier certificates for continuous systems have decades of
+theoretical foundation and modern computational tools. The novelty is not in
+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}[htbp]
+  \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 (4,0) {$q_1$\\Heatup};
+    \node[state] (q2) at (8,0) {$q_2$\\Power\\Operation};
+    \node[state, fill=red!15] (q3) at (4,-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)$};
+    \node[dynamics] at (4,-1.4) {$\dot{x} = f_1(x,u)$};
+    \node[dynamics] at (8,-1.4) {$\dot{x} = f_2(x,u)$};
+    \node[dynamics] at (4,-4.9) {$\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 and Specifications}
+
+\textcolor{blue}{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 automated controller
+synthesis and verification.}
+
+Autonomous control systems are fundamentally different from automatic control
+systems. The difference lies in the level at which they operate. Automatic
+control systems are purely operational systems that maintain setpoints or track
+references. Autonomous control systems make decisions about which operational
+objectives to pursue.
+
+\begin{figure}[htbp]
+  \centering
+  \begin{tikzpicture}[scale=0.8]
+    % Pyramid layers
+    \fill[blue!60!black] (0,4) -- (2,4) -- (1,5.5) -- cycle;
+    \fill[blue!30!white] (-1.5,2.5) -- (3.5,2.5) -- (2,4) -- (0,4) -- cycle;
+    \fill[blue!15!white] (-3,1) -- (5,1) -- (3.5,2.5) -- (-1.5,2.5) -- cycle;
+    
+    % Labels inside pyramid
+    \node[font=\small\bfseries, white] 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=6cm] at (5.5,4.5) 
+      {\textit{Long-term planning:} maintenance scheduling, capacity planning, economic dispatch};
+    \node[anchor=west, font=\small, text width=6cm] at (5.5,3.1) 
+      {\textit{Discrete decisions:} startup/shutdown sequences, power level changes, mode transitions};
+    \node[anchor=west, font=\small, text width=6cm] at (5.5,1.6) 
+      {\textit{Continuous control:} temperature regulation, pressure control, load following};
+    
+    % Bracket showing HAHACS scope (simple line with text)
+    \draw[thick] (-3.3,1) -- (-3.5,1) -- (-3.5,4) -- (-3.3,4);
+    \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}
+
+Human control of nuclear power can be divided into three different 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. 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. Thus, 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 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.
+
+The method of constructing a HAHACS in this proposal leverages two key
+observations about current practice. First, the operational scope control is
+effectively discrete control. Second, the rules for implementing this control
+are described prior to their implementation in operating procedures. We can
+exploit these facts by formalizing the rules for transitioning between discrete
+states using temporal logic.
+
+\textcolor{blue}{The discrete predicates that trigger mode transitions 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\%.'' Critically, we do not impose this discrete
+abstraction artificially. Operating procedures for nuclear systems already
+define go/no-go conditions as discrete predicates. These thresholds come from
+design basis safety analysis and have been validated over decades of
+operational experience. Our methodology assumes this domain knowledge exists
+and provides a framework to formalize it. This is why the approach is
+feasible for nuclear applications specifically: the hard work of defining
+safe operating boundaries has already been done by generations of nuclear
+engineers. We are formalizing existing practice, not inventing new
+abstractions.}
+
+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.
+
+\textcolor{blue}{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'').}
+
+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 developed by NASA to build high-assurance timed systems. FRET is an
+intermediate language between temporal logic and natural language that allows
+for rigid definitions of temporal behavior while using a syntax accessible to
+engineers without formal methods expertise. This benefit is crucial for the
+feasibility of this methodology in industry. By reducing the expert knowledge
+required to use these tools, their adoption with the current workforce becomes
+easier.
+
+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.
+
+%%% 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
+% ----------------------------------------------------------------------------
+
+\subsection{Discrete Controller Synthesis}
+
+Once system requirements are defined as temporal logic specifications, we use
+them to build the discrete control system. To do this, reactive synthesis tools
+are employed. 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 fit exactly this mold: the current discrete state and
+status of guard conditions are the input, while the output is the next discrete
+state. The output of a reactive synthesis algorithm is a discrete automaton.
+
+\textcolor{blue}{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.}
+
+The main advantage of reactive synthesis is that at no point in the production
+of the discrete automaton is human engineering of the implementation required.
+The resultant automaton is correct by construction. This method of construction
+eliminates the possibility of human error at the implementation stage entirely.
+Instead, the effort on the human designer is directed at the specification of
+system behavior itself.
+
+This has two critical implications. First, it makes the creation of the
+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.
+
+\textcolor{blue}{FRET can export specifications directly to formats compatible
+with reactive synthesis solvers such as Strix, a state-of-the-art LTL
+synthesis tool. The synthesis pipeline proceeds as follows:
+\begin{enumerate}
+  \item Operating procedures are formalized in FRET's structured English
+  \item FRET translates requirements to past-time or future-time LTL
+  \item Realizability analysis checks for specification conflicts
+  \item If realizable, synthesis produces a Mealy machine implementing the
+    discrete controller
+  \item The Mealy machine is compiled to executable code for the target
+    platform
+\end{enumerate}
+This pipeline provides complete traceability from natural language procedures
+to verified implementation, with each step producing artifacts that can be
+independently reviewed and validated.}
+
+%%% 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 Controllers}
+
+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. 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.
+
+\textcolor{blue}{It is important to clarify the scope of this methodology with
+respect to 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, but it verifies whether a given implementation
+satisfies its specification. Similarly, we assume that continuous controllers
+can be designed using standard control theory techniques. Our contribution is
+a 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. 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.
+
+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
+output set clearly defined by our discrete transitions \textit{a priori}.
+Consider that we define the continuous state space as $\mathcal{X}$. Whenever we
+create guard conditions from our design requirements, we are effectively
+creating subsets $\mathcal{X}_{entry,i}$ and $\mathcal{X}_{exit,i}$ for each
+discrete mode $q_i$. These subsets define when state transitions occur between
+discrete modes. More importantly, when building continuous control modes, they
+become control objectives.
+
+\textcolor{blue}{Mathematically, 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
+  \item \textbf{Safety invariants:} $\mathcal{X}_{safe,i} \subseteq \mathcal{X}$,
+    the envelope of safe states during operation in this mode
+\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}
+
+\textcolor{blue}{Transitory modes are continuous controllers designed to move
+the plant from one discrete operating condition to another. Their purpose is to
+execute transitions: starting from entry conditions, reaching exit conditions,
+while maintaining safety throughout. Examples include power ramp-up sequences,
+cooldown procedures, and load-following maneuvers.}
+
+\textcolor{blue}{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.}
+
+\textcolor{blue}{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 $T$:
+\[
+\text{Reach}(\mathcal{X}_{entry}, f, [0,T]) \subseteq \mathcal{X}_{safe} \land
+\text{Reach}(\mathcal{X}_{entry}, f, [0,T]) \cap \mathcal{X}_{exit} \neq \emptyset
+\]}
+
+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.
+
+\textcolor{blue}{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 through the
+Emerson partnership.}
+
+%%% 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}
+
+\textcolor{blue}{Stabilizing modes are continuous controllers with an objective
+of maintaining a particular discrete state 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 the most prudent approach
+to validation. Instead, barrier certificates must be used. 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.
+
+\textcolor{blue}{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.
+
+\textcolor{blue}{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.}
+
+%%% 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}
+
+The validation of transitory and stabilizing modes hinges on an assumption of
+correct plant models. In the case of a mechanical failure, the model will almost
+certainly be invalidated. For this reason, we must also build safe shutdown
+modes, since a human will not be in the loop to handle failures.
+
+\textcolor{blue}{Expulsory modes are continuous controllers responsible for
+ensuring 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.}
+
+We can detect that physical failures exist because our physical controllers have
+been previously proven correct by reachability and barrier certificates. We know
+our controller cannot be incorrect for the nominal plant model, so if an
+invariant is violated, we know the plant dynamics have changed. The HAHACS can
+identify that a fault occurred because a discrete boundary condition was
+violated by the continuous physical controller. This is a direct consequence of
+having verified the nominal continuous control modes: unexpected behavior
+implies off-nominal conditions.
+
+\textcolor{blue}{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.
+
+\textcolor{blue}{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:
+safety margins 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}
+
+\textcolor{blue}{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.}
+
+\textcolor{blue}{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.}
+
+\textcolor{blue}{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.}
+
+\textcolor{blue}{The demonstration will proceed through stages aligned with
+Technology Readiness Levels:
+\begin{enumerate}
+  \item \textbf{TRL 3:} Individual components validated in isolation (synthesized
+    automaton, verified continuous modes)
+  \item \textbf{TRL 4:} Integrated hybrid controller executing complete sequences
+    in pure simulation
+  \item \textbf{TRL 5:} Hardware-in-the-loop testing with Ovation executing the
+    discrete controller and simulation providing plant response
+\end{enumerate}
+Success at TRL 5 demonstrates that the methodology produces deployable
+controllers, not merely theoretical constructs.}
+
+%%% 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
+
diff --git a/goals-only.tex b/goals-only.tex
new file mode 100644
index 0000000..6e29af5
--- /dev/null
+++ b/goals-only.tex
@@ -0,0 +1,17 @@
+\documentclass[12pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage{mathptmx}
+\usepackage{enumitem}
+\setlist{noitemsep}
+
+\title{\textbf{Goals and Outcomes}\\[0.5em]\large Example for Proposal Writing}
+\author{Dane A. Sabo\\PhD Candidate, Mechanical Engineering\\University of Pittsburgh}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+\input{1-goals-and-outcomes/v1}
+
+\end{document}
diff --git a/main.aux b/main.aux
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diff --git a/main.synctex.gz b/main.synctex.gz
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diff --git a/main.tex b/main.tex
index dbd49d8..46ec167 100644
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+++ b/main.tex
@@ -23,7 +23,7 @@
 \input{2-state-of-the-art/v1}
 \newpage
 
-\input{3-research-approach/v2}
+\input{3-research-approach/v3}
 \newpage
 
 \input{4-metrics-of-success/v1}
diff --git a/main.toc b/main.toc
index fd1dea7..60ee273 100644
--- a/main.toc
+++ b/main.toc
@@ -5,17 +5,22 @@
 \contentsline {subsection}{\numberline {2.2}Human Factors in Nuclear Accidents}{3}{}%
 \contentsline {subsection}{\numberline {2.3}HARDENS and Formal Methods}{4}{}%
 \contentsline {section}{\numberline {3}Research Approach}{6}{}%
-\contentsline {subsection}{\numberline {3.1}System Requirement and Specifications}{6}{}%
-\contentsline {subsection}{\numberline {3.2}Continuous Controllers}{8}{}%
-\contentsline {section}{\numberline {4}Metrics for Success}{10}{}%
-\contentsline {paragraph}{TRL 3 \textit {Critical Function and Proof of Concept}}{10}{}%
-\contentsline {paragraph}{TRL 4 \textit {Laboratory Testing of Integrated Components}}{10}{}%
-\contentsline {paragraph}{TRL 5 \textit {Laboratory Testing in Relevant Environment}}{10}{}%
-\contentsline {section}{\numberline {5}Risks and Contingencies}{12}{}%
-\contentsline {subsection}{\numberline {5.1}Computational Tractability of Synthesis}{12}{}%
-\contentsline {subsection}{\numberline {5.2}Discrete-Continuous Interface Formalization}{12}{}%
-\contentsline {subsection}{\numberline {5.3}Procedure Formalization Completeness}{13}{}%
-\contentsline {section}{\numberline {6}Broader Impacts}{15}{}%
-\contentsline {section}{\numberline {7}Schedule, Milestones, and Deliverables}{17}{}%
-\contentsline {subsection}{\numberline {7.1}Milestones and Deliverables}{17}{}%
-\contentsline {section}{References}{18}{}%
+\contentsline {subsection}{\numberline {3.1}System Requirements and Specifications}{7}{}%
+\contentsline {subsection}{\numberline {3.2}Discrete Controller Synthesis}{10}{}%
+\contentsline {subsection}{\numberline {3.3}Continuous Controllers}{10}{}%
+\contentsline {subsubsection}{\numberline {3.3.1}Transitory Modes}{11}{}%
+\contentsline {subsubsection}{\numberline {3.3.2}Stabilizing Modes}{12}{}%
+\contentsline {subsubsection}{\numberline {3.3.3}Expulsory Modes}{13}{}%
+\contentsline {subsection}{\numberline {3.4}Industrial Implementation}{14}{}%
+\contentsline {section}{\numberline {4}Metrics for Success}{15}{}%
+\contentsline {paragraph}{TRL 3 \textit {Critical Function and Proof of Concept}}{15}{}%
+\contentsline {paragraph}{TRL 4 \textit {Laboratory Testing of Integrated Components}}{15}{}%
+\contentsline {paragraph}{TRL 5 \textit {Laboratory Testing in Relevant Environment}}{15}{}%
+\contentsline {section}{\numberline {5}Risks and Contingencies}{17}{}%
+\contentsline {subsection}{\numberline {5.1}Computational Tractability of Synthesis}{17}{}%
+\contentsline {subsection}{\numberline {5.2}Discrete-Continuous Interface Formalization}{17}{}%
+\contentsline {subsection}{\numberline {5.3}Procedure Formalization Completeness}{18}{}%
+\contentsline {section}{\numberline {6}Broader Impacts}{20}{}%
+\contentsline {section}{\numberline {7}Schedule, Milestones, and Deliverables}{22}{}%
+\contentsline {subsection}{\numberline {7.1}Milestones and Deliverables}{22}{}%
+\contentsline {section}{References}{23}{}%