GLM2FSA: Symbolic Abstractions for Control

• GLM2FSA procedure is a method for constructing finite-state symbolic abstractions using Feedback Refinement Relations, applicable to nonlinear, time-delay, and uncertain systems.
• It leverages static quantization and over-approximation techniques to enable automated, correct-by-construction controller synthesis.
• The method ensures robust specification satisfaction through explicit refinement relations and supports scalable, on-the-fly abstraction for complex cyber-physical systems.

The GLM2FSA procedure refers to the construction of symbolic abstractions for control systems using the Feedback Refinement Relation (FRR) framework, underpinning formal synthesis of controllers with correctness guarantees under quantized and uncertain conditions. “GLM2FSA” is not an explicit term in the source literature, but its essential ingredients—generalized system models, layered abstraction, and formal symbolic automata—are situated within the rigorous theory and practice of FRRs, as first defined in (Reissig et al., 2015), expanded for nonlinear and time-delay systems in (Ren et al., 2020), and contextualized through memoryless refinement hierarchies in (Calbert et al., 2024). This procedure enables the transformation of a concrete, possibly infinite-state, nonlinear or hybrid system into a finite-state symbolic abstraction suitable for automated controller synthesis, while guaranteeing robust satisfaction of specifications via quantized state feedback and explicit refinement relations.

1. Mathematical Foundation of Feedback Refinement Relations

The FRR formalism operates on “simple” stateful systems, denoted $S=(X,X,U,U,X,F,\mathrm{id})$, with dynamics

$$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$

Given two systems $S_1=(X_1,\ldots)$ and $S_2=(X_2,\ldots)$ with $U_2\subseteq U_1$, a strict relation $Q\subset X_1\times X_2$ is a Feedback Refinement Relation (written $S_1\preceq_{Q} S_2$) if for every $(x_1,x_2)\in Q$:

• $\textbf{(i)}$ \emph{Input Lifting}: $U_{S_2}(x_2)\subseteq U_{S_1}(x_1)$.
• $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$0 \emph{Uniform Transition Refinement}: $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$1, where $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$2.

These properties ensure that every admissible abstract input and abstract transition can be “lifted” or “refined” on the concrete system, enabling correct-by-construction controller synthesis for the abstracted model.

Key properties include:

• \emph{Reflexivity}: $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$3.
• \emph{Transitivity}: If $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$4 and $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$5 then $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$6.
• \emph{Closed-loop behavioral inclusion}: For any controller $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$7 feedback-composable with $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$8, the closed-loop behaviors satisfy $$x^+ \in F(x,u), \qquad u \text{ admissible at } x \iff F(x,u)\neq\emptyset.$$9, and vice versa for the refined controller (Reissig et al., 2015, Ren et al., 2020).

2. Symbolic Abstraction Construction and Static Quantization

A core step in the GLM2FSA procedure is constructing a symbolic abstraction—a finite-state transition system $S_1=(X_1,\ldots)$0—using static quantization. The quantization is realized via lattices, e.g., a logarithmic scalar quantizer $S_1=(X_1,\ldots)$1 mapping $S_1=(X_1,\ldots)$2 to a finite set $S_1=(X_1,\ldots)$3, inducing quantized state and input sets $S_1=(X_1,\ldots)$4 and $S_1=(X_1,\ldots)$5.

In this abstraction, each cell represents a quantized region, and state transitions are defined via over-approximation: $S_1=(X_1,\ldots)$6 where $S_1=(X_1,\ldots)$7, $S_1=(X_1,\ldots)$8 is the sampling time, $S_1=(X_1,\ldots)$9cell radius, and $S_2=(X_2,\ldots)$0 is a computed growth-bound derived from system local Lipschitz constants (Ren et al., 2020).

This procedure can be extended to delay systems by combining static and dynamic (“zoom”) quantizers and spline-based representations to handle the infinite-dimensional history required for time-delay dynamics.

3. Existence Conditions, Canonical Abstractions, and Refinement Algorithm

The existence of an FRR is characterized by explicit algebraic and combinatorial conditions:

• For cover-based abstractions, $S_2=(X_2,\ldots)$1 defined via $S_2=(X_2,\ldots)$2 covering $S_2=(X_2,\ldots)$3 satisfies the FRR if for all $S_2=(X_2,\ldots)$4, all $S_2=(X_2,\ldots)$5, $S_2=(X_2,\ldots)$6 and for all $S_2=(X_2,\ldots)$7, if $S_2=(X_2,\ldots)$8 then $S_2=(X_2,\ldots)$9 (Reissig et al., 2015).

Canonical abstractions can be constructed by factoring an FRR through the nonempty fibers of $U_2\subseteq U_1$0, yielding a minimal abstract system that preserves specification satisfaction properties. Algorithms implement symbolic abstractions via:

• Choosing a cover $U_2\subseteq U_1$1 (e.g., hyper-intervals).
• Over-approximating reachable sets for each cell/action pair.
• Populating the abstract transition relation with overlap tests.

This construction is computationally efficient and robust, and it supports on-the-fly local refinement to mitigate state explosion (Ren et al., 2020).

4. Robust Controller Synthesis and Handling Uncertainty

The FRR framework admits systematic incorporation of:

• Bounded input disturbances, modeled as set-valued maps $U_2\subseteq U_1$2 on the input.
• Measurement errors, as set-valued $U_2\subseteq U_1$3 on the state.
• Model uncertainty, as additional set-valued dynamics $U_2\subseteq U_1$4.

An auxiliary system $U_2\subseteq U_1$5 and an enlarged quantizer $U_2\subseteq U_1$6 are defined. If a symbolic controller $U_2\subseteq U_1$7 composed with $U_2\subseteq U_1$8 solves the abstract robustified control problem on $U_2\subseteq U_1$9, then $Q\subset X_1\times X_2$0 in the practical setting—post-quantization, with disturbances—guarantees specification satisfaction (Reissig et al., 2015). This connection underpins robustness of the abstraction-based synthesis to practical imperfections.

5. Comparative Frameworks: FRR, MCR, and ASR

The FRR is situated among several abstraction-simulation relations:

• \textbf{Alternating Simulation Relation (ASR)}: Allows for arbitrary (possibly memoryful) concretization, no guarantee of static abstraction.
• \textbf{Memoryless Concretization Relation (MCR)}: Admits state-dependent refinement, i.e., controllers that may be state-dependent inside each cell, critical for reducing abstraction non-determinism in overlapping-cell or non-regular discretizations (Calbert et al., 2024).
• \textbf{Feedback Refinement Relation (FRR)}: Restricts concretization to piecewise-constant (cell-wise constant) controllers that rely only on the symbolic state, enabling static and robust symbolic control (Calbert et al., 2024).

FRR is strictly stronger than MCR, but they coincide on partition-based abstractions. MCR offers greater flexibility in abstraction construction (e.g., state-dependent actions) while preserving memoryless control refinement.

6. Illustrative Case Studies and Computational Aspects

Case studies demonstrate practical application of the GLM2FSA procedure:

• \textbf{Autonomous Vehicle Path Planning}: Abstraction of a sampled bicycle model using 50×50×34 cell cover, growth-bound computations, and synthesis of a patrol specification using a GR(1) fixed-point controller. Construction time for the abstract transition system is 2.3 s for 37 million transitions, with symbolic controller synthesis completing in 0.5 s (Reissig et al., 2015).
• \textbf{Aircraft Landing Maneuver}: DC9 aircraft model under bounded disturbances and quantization. The constructed symbolic transition system comprises $Q\subset X_1\times X_2$1 transitions, assembled in 674 s, and permits robust reach–avoid specification synthesis via Dijkstra-style algorithms (Reissig et al., 2015).

Complexity is governed by the quantization granularity, local Lipschitz constants, and the dimension of state and input spaces. Strategies such as non-uniform quantization, zoom-based local refinement, and on-the-fly abstraction mitigate worst-case exponential blow-up (Ren et al., 2020).

7. Significance, Extensions, and Open Directions

The GLM2FSA procedure, formalized through FRR, offers:

• Automated, correct-by-construction controller synthesis for systems with general (nonlinear, time-delay, uncertain) dynamics.
• Explicit guarantees of closed-loop behavior transfer from symbolic to continuous systems under quantization.
• Efficient and modular construction of symbolic abstractions that scale via local refinement and compositional techniques.

Limitations include the exponential state-space explosion for high-dimensional systems, and the need for judicious selection of quantization and abstraction parameters to balance accuracy and tractability. Extensions to networks of systems, stochastic or hybrid models, and automated parameter tuning are active areas of research. The FRR abstraction-refinement methodology continues to underpin advances in formal methods, robust control synthesis, and compositional verification for complex cyber-physical systems (Reissig et al., 2015, Ren et al., 2020, Calbert et al., 2024).