From 6e10a4db81abc7077070759867bae07af1e60f32 Mon Sep 17 00:00:00 2001 From: Split Date: Mon, 23 Feb 2026 16:00:26 -0500 Subject: [PATCH] Add v3 research approach: clean structure, DICE framing, figures - Created v3.tex with structure matching Thesis.RA tasks - Added pyramid figure (strategic/operational/tactical hierarchy) - Added hybrid automaton figure with guards and dynamics - Wove in DICE abstract framing (gap, why now, two-layer approach) - Split contributions marked in blue for review - Switched main.tex to use v3 --- 3-research-approach/v3.tex | 681 +++++++++++++++++++++++++++++++++++++ goals-only.tex | 17 + main.aux | 47 +-- main.blg | 6 +- main.fdb_latexmk | 264 -------------- main.fls | 564 ------------------------------ main.synctex.gz | Bin 183855 -> 0 bytes main.tex | 2 +- main.toc | 33 +- 9 files changed, 745 insertions(+), 869 deletions(-) create mode 100644 3-research-approach/v3.tex create mode 100644 goals-only.tex delete mode 100644 main.fdb_latexmk delete mode 100644 main.fls delete mode 100644 main.synctex.gz diff --git a/3-research-approach/v3.tex b/3-research-approach/v3.tex new file mode 100644 index 0000000..9f8e12f --- /dev/null +++ b/3-research-approach/v3.tex @@ -0,0 +1,681 @@ +\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. 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