\section{Research Approach}

This research will overcome the limitations of current practice to build 
high-assurance hybrid control systems for critical infrastructure. Hybrid 
systems combine continuous dynamics (flows) with discrete transitions (jumps), 
which can be formally expressed as:

\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.

To build these systems with formal correctness guarantees, 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

\end{enumerate}

The following sections discuss how these thrusts will be accomplished.

\subsection{$(Procedures \wedge FRET) \rightarrow Temporal Specifications$}

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. To formalize these procedures, we will
use temporal logic, which captures system behaviors through temporal relations.
Linear Temporal Logic (LTL) provides four fundamental operators: next ($X$),
eventually ($F$), globally ($G$), and until ($U$). These operators enable
precise specification of time-dependent requirements.

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}
G(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 
global operator $G$ ensures this property holds throughout system operation, 
while the next operator $X$ enforces immediate response. The until operator 
$U$ maintains the state constraint until the reset condition occurs.

The most efficient path to accomplish this translation is through 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. FRETish bridges natural 
language and mathematical specifications through a structured English-like 
syntax that is 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 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.

FRET provides the capability to export requirements in temporal logic format
compatible with reactive synthesis tools. This export functionality enables
progression to the next step of our approach: synthesizing discrete mode
switching behavior from the formalized requirements.

\subsection{$(TemporalLogic \wedge ReactiveSynthesis) \rightarrow
DiscreteAutomata$}

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}. Discrete
automata can be represented graphically as a series of nodes that are discrete
states, with traces indicating transitions between states. From the automaton
graph, it becomes possible to fully describe the dynamics of the discrete system
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. Strix
translates linear temporal logic specifications into deterministic automata
automatically while maximizing generated automata quality. Once constructed, the
automaton can be straightforwardly implemented using standard programming
control flow constructs. The graphical representation provided by the automaton
enables inspection and facilitates communication with controls programmers who
may not have 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, we can prove that the automaton—and
therefore our hybrid switching behavior—is \textit{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, as described by $f(\cdot)$ in Equation 1. This section 
describes how we will develop continuous control modes, verify their 
correctness, and address the unique verification challenges of hybrid systems.

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}

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.

\textbf{Reachability Analysis:} 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.
Recent advances using neural network approximations of Hamilton-Jacobi equations
have demonstrated significant speedups while maintaining safety guarantees for
high-dimensional systems, expanding the practical applicability of these
methods.

\textbf{Assume-Guarantee Contracts:} 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 compositional approach 
ensures each continuous controller is prepared for its possible input range, 
enabling subsequent reachability analysis without requiring global system 
analysis.

\textbf{Barrier Certificates:} 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. Control 
barrier functions provide a method to certify safety by establishing 
differential inequality conditions that guarantee forward invariance of safe 
sets. 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.

Combining these three techniques will enable us to prove that continuous
components of our hybrid controller satisfy discrete requirements, and thus,
complete system behavior. To demonstrate this methodology, we will develop an
autonomous startup controller for a Small Modular Advanced High Temperature
Reactor (SmAHTR). SmAHTR represents an ideal test case as a liquid-salt cooled
reactor design with well-documented startup procedures that must transition
through multiple distinct operational modes: initial cold conditions, controlled
heating to operating temperature, approach to criticality, low-power physics
testing, and power ascension to full operating capacity. 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, which is 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, providing assessment of computational
performance, real-time execution constraints, and communication latency effects.
By demonstrating autonomous startup control on this representative platform, we
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. 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.

% 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
