Uncontrollable Event Admissibility

• Uncontrollable Event Admissibility is a foundational constraint in supervisory control theory that mandates supervisors to accept every plant-enabled uncontrollable event.
• It is characterized through both language-level (Ramadge–Wonham framework) and state-level (automata-based formulations) definitions, ensuring accurate system modeling.
• The incorporation of event forcing refines admissibility by conditionally preempting undesirable uncontrollable events with forcible alternatives, thereby enhancing system realizability.

Uncontrollable event admissibility is a foundational constraint in supervisory control theory for discrete-event systems: if the plant can execute an uncontrollable event, the supervisor is not permitted to disable that event. In the Ramadge–Wonham framework this requirement appears as the core admissibility condition behind language controllability, while in later automata-based formulations it is stated as a local property on reachable states of the supervised plant. Recent work preserves this basic interpretation but also refines it: in the presence of event forcing, admissibility is no longer absolute at every state, because a forcible event may preempt a problematic uncontrollable event and thereby make realizability conditional on forcing capability rather than unconditional closure under all uncontrollable successors (Keiren et al., 7 Aug 2025, Reniers et al., 2024).

1. Classical Ramadge–Wonham meaning

In the standard DES model, the plant is a finite generator

$P = (Q,\Sigma,\delta,q_0,Q_m),$

with generated and marked languages $L(P)$ and $L_m(P)$, and enabled-event set

$$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$

The event alphabet is partitioned as

$\Sigma = \Sigma_c \cup \Sigma_u,$

where $\Sigma_c$ are controllable events and $\Sigma_u$ are uncontrollable events. The classical supervisory interaction consists only of enabling and disabling: controllable events may be disabled by the supervisor, whereas uncontrollable events are “admissible” in the sense that, if enabled in the plant, they must be accepted by the supervisor (Reniers et al., 2024).

This condition is expressed language-theoretically by controllability. Given a sublanguage $F \subseteq L_m(P)$, $F$ is controllable with respect to $P$ if

$L(P)$0

Equivalently,

$L(P)$1

The meaning is exact: if an uncontrollable event $L(P)$2 is enabled by the plant after a supervised string $L(P)$3, then $L(P)$4 must remain in the supervised closed language. The supervisor is therefore not allowed to “forbid” uncontrollable behavior (Reniers et al., 2024).

The same idea appears in supervisor form. For a supervisor

$L(P)$5

the classical admissibility condition is

$L(P)$6

This is the operational statement of uncontrollable event admissibility: every plant-enabled uncontrollable event must remain enabled in closed loop. In classical Ramadge–Wonham theory, this obligation is absolute.

2. Product-state admissibility in automata form

In automata-based supervisory control, uncontrollable event admissibility is often formulated on the synchronous product of supervisor and plant. If

$L(P)$7

denotes the supervised plant, then $L(P)$8-admissibility, denoted $L(P)$9, requires that for all reachable pairs $L_m(P)$0 and all $L_m(P)$1,

$L_m(P)$2

Because a transition in the synchronous product exists exactly when both components can execute it, this is equivalent to

$L_m(P)$3

for every reachable product state. The interpretation given in the survey is direct: in every reachable state of the supervised system, no uncontrollable transition of the plant is disabled (Keiren et al., 7 Aug 2025).

This formulation is stronger than a purely language-level reading because it quantifies over reachable state pairs, not merely over strings. It therefore distinguishes between different supervisor states that may correspond to the same trace under nondeterminism. The same survey relates this condition to Zhou’s nondeterministic state controllability: under the assumption $L_m(P)$4, Zhou’s definition and $L_m(P)$5-admissibility are equivalent (Keiren et al., 7 Aug 2025).

The product-state view is practically significant because it aligns the definition with the supervised behavior that must actually be implemented. Rather than asking only whether the language of the supervisor is closed under uncontrollable events, it asks whether every reachable state of the realized supervised plant still enables every uncontrollable event that the plant itself can perform.

3. Relations among controllability notions

The literature contains multiple notions of controllability, especially once nondeterministic automata are allowed. The survey “Overview of Controllability Definitions in Supervisory Control Theory” compares language controllability, uncontrollable event admissibility, Flordal–Malik controllability, partial bisimulation, control relation, and state controllability, and identifies the relationships that hold in different modeling regimes (Keiren et al., 7 Aug 2025).

Before the table below, two points are central. First, in the fully nondeterministic setting, uncontrollable event admissibility is not merely a reformulation of classical language controllability; it is strictly stronger. Second, in deterministic settings, these distinctions collapse: several notions coincide with Ramadge–Wonham controllability.

Setting	Relations stated in the survey	Consequence
Nondeterministic $L_m(P)$6, nondeterministic $L_m(P)$7	$L_m(P)$8, and $L_m(P)$9	UEA implies language controllability, but AC does not imply UEA
Deterministic $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$0, nondeterministic $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$1	$$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$2	UEA collapses to language controllability
Deterministic $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$3, deterministic $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$4	$$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$5	Classical notions coincide

Here $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$6 denotes Kushi–Takai $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$7-admissibility, $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$8 denotes Flordal–Malik controllability, and $$E_P(s) := \{\sigma \in \Sigma \mid s\sigma \in L(P)\}.$$9 denotes automata language controllability. In the general nondeterministic case, the survey states that Flordal–Malik controllability and uncontrollable event admissibility are equivalent, and that these are also the only notions that imply the traditional notion of language controllability (Keiren et al., 7 Aug 2025).

A recurring misconception is that language controllability already captures every relevant operational aspect of uncontrollable events. The nondeterministic comparison results show otherwise. A language may be controllable in the Ramadge–Wonham sense while a particular reachable supervisor state still disables an uncontrollable transition of the plant. Uncontrollable event admissibility rules out precisely that failure mode by quantifying over reachable product states.

4. Event forcing and conditional admissibility

The event-forcing framework enriches supervisory interaction by introducing a third event class,

$\Sigma = \Sigma_c \cup \Sigma_u,$0

the set of forcible events. Forcible events may be controllable or uncontrollable. Forcing is a mechanism distinct from enabling and disabling: when the supervisor chooses forcing mode, forcible events preempt all other events at that state (Reniers et al., 2024).

The closed-loop supervisor remains a map $\Sigma = \Sigma_c \cup \Sigma_u,$1, but its meaning changes. If no forcible event is enabled, interaction reduces to classical Ramadge–Wonham supervision. If at least one forcible event is enabled, the supervisor can either remain in ordinary enable/disable mode or switch to forcing mode and enable only a subset of forcible events. In forcing mode, all nonforcible events are suppressed, including nonforcible uncontrollable events. The paper explicitly characterizes this as strong preemption (Reniers et al., 2024).

This change leads to the central notion of forcibly-controllability. A sublanguage $\Sigma = \Sigma_c \cup \Sigma_u,$2 is forcibly-controllable if

$\Sigma = \Sigma_c \cup \Sigma_u,$3

The first disjunct is classical uncontrollable admissibility. The second is the forcing alternative: if some uncontrollable event would leave $\Sigma = \Sigma_c \cup \Sigma_u,$4, the specification can still be realizable provided there exists a forcible successor that remains in $\Sigma = \Sigma_c \cup \Sigma_u,$5 and every nonforcible successor is excluded from $\Sigma = \Sigma_c \cup \Sigma_u,$6. In that case the supervisor can force a suitable event and preempt the undesirable uncontrollable one (Reniers et al., 2024).

The conceptual effect is exact and significant. In classical SCT, admissibility of uncontrollable events is rigid. In the forcing framework, admissibility becomes state-dependent and conditional on the existence of a forcible alternative. If $\Sigma = \Sigma_c \cup \Sigma_u,$7, the definition reduces to classical controllability. In general, controllable implies forcibly-controllable, but the converse fails; the paper gives examples of forcibly-controllable languages that are not controllable. This establishes that forcible-controllability strictly generalizes controllability (Reniers et al., 2024).

5. Existence, maximal permissiveness, and synthesis

The forcing supervisory control problem asks for a supervisory control $\Sigma = \Sigma_c \cup \Sigma_u,$8 such that $\Sigma = \Sigma_c \cup \Sigma_u,$9 is nonblocking and

$\Sigma_c$0

for a nonempty specification $\Sigma_c$1. The core solvability result is exact: such a supervisor exists if and only if $\Sigma_c$2 is forcibly-controllable. The proof is constructive and defines a supervisor that switches between ordinary mode and forcing mode depending on whether classical uncontrollable admissibility holds at the current string or must be replaced by preemption (Reniers et al., 2024).

When a given specification is not forcibly-controllable, the relevant object is the family

$\Sigma_c$3

This family is nonempty and closed under unions, so there exists a unique supremal forcibly-controllable sublanguage

$\Sigma_c$4

If $\Sigma_c$5, then there exists a supervisory control $\Sigma_c$6 such that $\Sigma_c$7 is nonblocking and

$\Sigma_c$8

The resulting supervisor is maximally permissive in the event-forcing sense (Reniers et al., 2024).

Algorithmically, the paper gives a fixed-point synthesis procedure. It maintains retained states $\Sigma_c$9, nonblocking states $\Sigma_u$0, bad states $\Sigma_u$1, forcing states $\Sigma_u$2, and the current transition relation $\Sigma_u$3. A state becomes bad when an uncontrollable successor is bad and all forcible successors are also bad; it becomes a forcing state when an uncontrollable successor is bad but at least one forcible successor is not. At the pruning stage, the algorithm removes all nonforcible outgoing transitions from forcing states: $\Sigma_u$4 This implements the second clause of forcibly-controllability by making only forcible events admissible at forcing states (Reniers et al., 2024).

The case studies show how this changes admissibility concretely. In the manufacturing line example, when $\Sigma_u$5, states threatened by uncontrollable $\Sigma_u$6 must be avoided by disabling incoming controllable transitions. When $\Sigma_u$7, some of those states are retained as forcing states: the supervisor can force $\Sigma_u$8 and preempt $\Sigma_u$9. In the small factory example, taking $F \subseteq L_m(P)$0 allows states previously removed under classical admissibility to be retained, because $F \subseteq L_m(P)$1 or $F \subseteq L_m(P)$2 can be forced to preempt uncontrollable breakdown-related behaviors. In both examples, behaviors discarded by classical supervision become admissible under forcing (Reniers et al., 2024).

6. Scope, neighboring uses, and conceptual boundaries

Within supervisory control theory, uncontrollable event admissibility has a specific meaning: a specification or supervised plant must be closed under the plant’s uncontrollable behavior, either absolutely in the classical case or conditionally under a forcing alternative in the event-forcing case. A related but distinct idea appears in temporal-logic control with uncontrollable dynamic agents. There, the uncontrollable behaviors are continuous stochastic agent trajectories, and admissibility is represented by prediction regions

$F \subseteq L_m(P)$3

constructed from conformal prediction. The controller is synthesized so that STL predicates hold for all $F \subseteq L_m(P)$4, yielding satisfaction with probability at least $F \subseteq L_m(P)$5 when the true uncontrollable trajectory remains inside the prediction region (Yu et al., 2023). This suggests a broader interpretation of admissibility as closure under a designated class of uncontrollable behaviors, but the formal object is not a discrete uncontrollable event alphabet.

Another important boundary is terminological. In infinite-dimensional control theory, “admissibility” refers to control operators $F \subseteq L_m(P)$6 for semigroup generators $F \subseteq L_m(P)$7, with infinite-time admissibility requiring the input-state map

$F \subseteq L_m(P)$8

to be $F \subseteq L_m(P)$9-valued and uniformly bounded for all $F$0. The paper on compact perturbations shows that infinite-time admissibility is not preserved under compact perturbations $F$1, even when both $F$2 and $F$3 generate strongly stable semigroups (Schmid, 2019). That notion concerns well-posedness and boundedness of continuous-time input-state maps, not supervisor obligations with respect to uncontrollable events.

In the supervisory-control literature itself, the main conceptual divide is therefore not between “language” and “state” descriptions of the same property, but between unconditional and conditional forms of admissibility. Classical Ramadge–Wonham theory treats uncontrollable events as always admissible when enabled by the plant. The nondeterministic automata literature sharpens this into a reachable-product-state condition. Event forcing then refines the same principle by allowing admissibility to depend on the availability of a forcible preemption. Across these variants, the persistent invariant is that uncontrolled plant behavior cannot simply be ignored: it must either be tolerated in closed loop or preempted by a formally justified forcing mechanism (Keiren et al., 7 Aug 2025, Reniers et al., 2024).