\section{Research Approach} This research will overcome the limitations of current practice to build high-assurance hybrid control systems for critical infrastructure. To achieve this goal, we must accomplish 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 using reachability analysis, barrier certificates, and assume-guarantee contracts \end{enumerate} The following sections discuss how these thrusts will be accomplished. \subsection{Requirements: $(Procedures \wedge FRET) \rightarrow Temporal Specifications$} % COMMENT: Consider strengthening this opening with specific examples or % statistics about nuclear plant automation levels vs other industries The motivation behind this work stems from the fact that commercial nuclear power operations remain manually controlled by human operators, despite significant advances in control systems sophistication. The key insight is that procedures performed by human operators are highly prescriptive and well-documented. This suggests that human operators in nuclear power plants may not be entirely necessary given today's available technology. Written procedures and requirements in nuclear power are sufficiently detailed that we may be able to translate them into logical formulae with minimal effort. If successful, this approach would enable automation of existing procedures without requiring system reengineering—a significant advantage for deployment in existing facilities. The most efficient path to accomplish this translation is through automated tools such as NASA's Formal Requirement Elicitation Tool (FRET). % COMMENT: Consider adding a brief example of a nuclear procedure that would % benefit from this approach FRET employs a specialized requirements language similar to natural language called FRETish. FRETish restricts requirements to easily understood components while eliminating ambiguity. 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} % COMMENT: The realizability discussion could be clearer - consider defining % "realizable" more precisely upfront FRET provides functionality to check the \textit{realizability} of a system. Realizability analysis determines whether written requirements are complete by examining the six structural components. Complete requirements are those that 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 that represent the physical equivalent of software bugs. Using FRET during autonomous controller development allows us to identify and resolve these errors systematically. The second category of realizability issues involves undefined behaviors that are typically left to human judgment during control operations. This ambiguity is undesirable for high-assurance systems, since even well-trained humans remain prone to errors. By addressing these specification gaps in FRET during autonomous controller development, we can deliver controllers free from these vulnerabilities. % COMMENT: This transition could be smoother - consider adding a sentence about % how the temporal logic output connects to the next section FRET also provides the capability to export requirements in temporal logic format. This export functionality enables progression to the next step of our approach: synthesizing discrete mode switching behavior from the formalized requirements. \subsection{Discrete Synthesis: $(TemporalLogic \wedge ReactiveSynthesis) \rightarrow DiscreteAutomata$} This section describes how the discrete component of the hybrid system will be constructed. The formal specifications created in FRET will be processed by reactive synthesis tools to generate mode switching components. These components effectively automate the human decision-making elements of reactor operation by implementing the logic typically specified in written procedures. By eliminating the human component from control decisions, we remove the possibility of human error and advance hybrid system autonomy. Reactive synthesis is an active research field in computer science 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 in size, where each node represents 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}. From the automaton graph, it becomes possible to fully describe the dynamics of the discrete system and develop intuitive understanding of system behavior. Once constructed, the automaton can be straightforwardly implemented using standard programming control flow constructs. % COMMENT: The "correct by construction" concept is important - consider % expanding this explanation slightly 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, we can prove that the automaton—and therefore our hybrid switching behavior—is correct by construction. Correctness of the switching controller is paramount to this work. 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 that extend beyond written procedures. Autonomous control lacks this adaptive advantage. Instead, we must ensure that autonomous controllers replacing human operators will not make switching errors between continuous modes. By synthesizing controllers from logical specifications with guaranteed correctness, we eliminate the possibility of switching errors. \subsection{$(DiscreteAutomata \wedge ControlTheory \wedge Reachability) \rightarrow ContinuousModes$} 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. This section describes how we will develop continuous control modes, verify their correctness, and address the unique verification challenges of hybrid systems. % COMMENT: This "glaring problem" phrasing is a bit dramatic - consider toning % down while maintaining the technical point First, we must address the limitation left unresolved in the previous section. The approach described for producing discrete automata yields physics-agnostic specifications that represent 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. 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} % COMMENT: This paragraph has some word choice issues ("alieviate" -> % "alleviate") - fixed in revision 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. Current techniques struggle with this problem because dynamic discontinuities complicate verification. This work alleviates these problems by designing continuous controllers specifically with transitions in mind. By decomposing continuous modes according to their required behavior at transition points, we avoid solving trajectories through the entire hybrid system. Instead, we can use local behavior information at transition boundaries. To ensure continuous modes satisfy their requirements, we will 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. We will use reachability to define continuous state ranges at discrete transition boundaries and verify that requirements are satisfied within continuous modes. Assume-guarantee contracts will be employed 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 ensures each continuous controller is prepared for its possible input range, enabling subsequent reachability analysis. Finally, we will use barrier certificates to prove that mode transitions are satisfied. Barrier certificates ensure that continuous modes on either side of a transition behave appropriately. 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 mode. Combining these three techniques will enable us to prove that continuous components of our hybrid controller satisfy discrete requirements. % COMMENTS FOR FUTURE REVISION: % 1. Add concrete examples throughout (specific nuclear procedures, requirements) % 2. Include a figure showing the overall workflow/methodology % 3. Consider adding a subsection on validation approach % 4. Strengthen the connections between subsections % 5. Add discussion of limitations and assumptions % 6. Consider reorganizing the continuous modes classification earlier This unified approach addresses a fundamental gap in hybrid system design by bridging formal methods and control theory through a systematic, tool-supported methodology. By translating existing nuclear procedures into temporal logic, synthesizing provably correct discrete switching logic, and developing verified continuous controllers, we create 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.