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