Bounded Fuzzy Logic Control (BFLC)

• BFLC is defined as fuzzy control designs that structurally enforce finite bounds on controller outputs, states, and inference domains.
• It encompasses diverse methodologies, including hardware-based Mamdani systems, state-space controllers with LMIs, and supervisory schedulers for applications like green hydrogen production.
• BFLC techniques yield practical benefits such as stability assurances via Lyapunov methods and efficient implementation in finite precision, quantized environments.

Bounded Fuzzy Logic Control (BFLC) denotes a class of fuzzy-control constructions in which boundedness is enforced structurally in the controller, the signals it generates, or the domains on which inference is realized. In the literature considered here, the explicit label appears in a day-ahead supervisory controller for green hydrogen scheduling under a Hydrogen Purchase Agreement (HPA), where the fuzzy output is bounded by producibility and by time-dependent cumulative delivery corridors (Farah et al., 2 Aug 2025). Closely related work treats boundedness in other technically distinct senses: finite universes of discourse and finite truth-value resolution in hardware Mamdani inference (Chiu et al., 2013), explicit control-amplitude constraints imposed by LMIs in an evolving fuzzy state-space controller (Leite et al., 2021), and full-state constraint preservation via Barrier Lyapunov Functions (BLFs) or integral BLFs in adaptive fuzzy backstepping for pure-feedback nonlinear systems (Wu et al., 2023, Wu et al., 2023). Taken together, these works show that “bounded” in BFLC is not a single semantic category but a family of design commitments linking fuzzy reasoning to finite domains, constrained inputs, constrained states, or bounded supervisory trajectories.

1. Scope and principal interpretations

The available literature supports at least five technically distinct interpretations of boundedness in fuzzy control. In hardware-oriented Mamdani implementations, boundedness is imposed by bounded universes of discourse, finite discretization, finite truth-value resolution, fixed chip formats, finite memory, and output quantization. In evolving state-space fuzzy control, boundedness refers to a prescribed control-amplitude constraint of the form $\|u(k+1)\|_2 \le \zeta$. In constrained adaptive fuzzy control for pure-feedback systems, boundedness is attached to full-state constraints $|x_i| \le k_{ci}$ and to semi-global ultimate boundedness of all closed-loop signals. In the green-hydrogen scheduling formulation, boundedness applies to the daily HPA delivery target and to the cumulative HPA export trajectory over the year, preventing the controller from proceeding “too quickly or too slowly” relative to benchmark-optimal schedules (Chiu et al., 2013, Leite et al., 2021, Wu et al., 2023, Wu et al., 2023, Farah et al., 2 Aug 2025).

Paper	Controller form	Boundedness mechanism
(Chiu et al., 2013)	Mamdani-style fuzzy inference compiled to hardware	finite intervals, 16 levels of resolution, 16 truth levels, 4-bit integer truth values
(Leite et al., 2021)	evolving fuzzy state-space controller with PDC	LMI-enforced $\|u(k+1)\|_2 \le \zeta$
(Wu et al., 2023)	adaptive fuzzy backstepping/DSC with input delay	BLFs, full-state constraints, SGUUB
(Wu et al., 2023)	finite-time adaptive fuzzy tracking for pure-feedback systems	integral BLFs, prescribed state bounds, SGUUB
(Farah et al., 2 Aug 2025)	sequential day-ahead supervisory BFLC	producibility cap and cumulative upper/lower bounds

A common ambiguity in the terminology is that only one of these papers explicitly uses the phrase “Bounded Fuzzy Logic Control.” The others are BFLC-relevant because boundedness is central to the controller synthesis or implementation, not because they adopt a shared formal label. This suggests that BFLC is best treated as an umbrella classification for fuzzy-control designs in which constraint enforcement or finite realizability is intrinsic rather than incidental.

2. Boundedness in inference, representation, and scheduling

The earliest implementation-oriented formulation is the hardware compilation environment of Chiu and Togai, which uses standard Mamdani-style rules such as

$$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$

A fuzzy variable $A$ is represented by a membership function $A(x)$ over a user-specified universe of discourse with explicit numeric limits, such as TEMPERATURE $(0,200)$, PRESSURE $(0,500)$, HEATER.POWER $(0,10)$, and VALVE.OPENING $(0,10)$. In the hardware-oriented internal representation, the universe of discourse is discretized into 16 levels of resolution, each with 16 levels of truth value, and membership functions are stored as vectors of 4-bit integer truth values. The special antecedents ANY and NULL correspond respectively to all maximum truth values and all zero truth values. Boundedness therefore appears simultaneously in the support domain, the truth-value codomain, and the chip architecture itself (Chiu et al., 2013).

In the state-space evolving controller, boundedness is explicit at the control channel. The fuzzy model is a convex combination of local affine state-space consequents,

$|x_i| \le k_{ci}$0

and the controller is a PDC-style convexly blended state-feedback law,

$|x_i| \le k_{ci}$1

The bounded-input condition is imposed as

$|x_i| \le k_{ci}$2

with accompanying LMIs

$|x_i| \le k_{ci}$3

Here boundedness is not post hoc clipping but preventive synthesis: gains are redesigned so that the commanded control signal remains within a prescribed magnitude bound (Leite et al., 2021).

In the pure-feedback tracking papers, boundedness is primarily state-constraint enforcement. One formulation considers

$|x_i| \le k_{ci}$4

with disturbances satisfying

$|x_i| \le k_{ci}$5

while the other uses the same class of full-state constraints and bounded disturbances $|x_i| \le k_{ci}$6. In both, fuzzy logic systems (FLSs) approximate unknown nonlinear functions, but the hard boundedness mechanism is external to the approximator: BLFs or integral BLFs force the transformed coordinates to remain in prescribed sets, and Lyapunov analysis yields semi-global ultimate boundedness of all closed-loop signals (Wu et al., 2023, Wu et al., 2023).

In the explicit BFLC scheduler for green hydrogen, boundedness is attached to the daily HPA target. The fuzzy controller produces a raw daily target $|x_i| \le k_{ci}$7, which is limited by the maximum possible daily hydrogen production: $|x_i| \le k_{ci}$8 A second layer bounds cumulative HPA delivery within time-dependent upper and lower boundaries derived from the convex hull of benchmark-optimal cumulative exports across years. This prevents over-exporting too early or delaying too long relative to the annual target $|x_i| \le k_{ci}$9 tonnes (Farah et al., 2 Aug 2025).

3. Controller architectures and fuzzy mechanisms

Despite sharing a concern with boundedness, the controller architectures are heterogeneous. The hardware compilation paper is a classical Mamdani system with conjunction by minimum, aggregation by maximum, and centroid defuzzification. For two rules and two inputs, the firing strengths are

$\|u(k+1)\|_2 \le \zeta$0

and the preferred hardware output-set construction is

$\|u(k+1)\|_2 \le \zeta$1

An alternative max-product form,

$\|u(k+1)\|_2 \le \zeta$2

is described but rejected for the dedicated inference chip because the min-max method is “more efficient and suited for VLSI implementation, since it only requires min-max comparisons and no multiplications.” Crisp output is obtained from the centroid

$\|u(k+1)\|_2 \le \zeta$3

This is a clipped Mamdani architecture whose boundedness is reinforced by discretized storage and fixed chip formats (Chiu et al., 2013).

The evolving controller of SS-FBeM is structurally different. Its local rules are state-space consequents,

$\|u(k+1)\|_2 \le \zeta$4

augmented to an affine form. Antecedent fuzzy sets are trapezoidal, $\|u(k+1)\|_2 \le \zeta$5, and rule activity is determined by expansion regions

$\|u(k+1)\|_2 \le \zeta$6

and by the Hamming-like similarity

$\|u(k+1)\|_2 \le \zeta$7

The model begins with no rules, creates a new rule if no existing granule is active, and updates local matrices through Recursive Least Squares. This is a Takagi–Sugeno / local linear state-feedback style fuzzy controller rather than a Mamdani controller with linguistic consequents (Leite et al., 2021).

The pure-feedback tracking designs again differ. Their FLSs are function approximators with singleton fuzzifier, product inference, and center-average defuzzifier. For a continuous function $\|u(k+1)\|_2 \le \zeta$8 on a compact set,

$\|u(k+1)\|_2 \le \zeta$9

with normalized basis vector

$$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$0

In the input-delay case, the controller combines adaptive backstepping, dynamic surface control (DSC), BLFs, Padé approximation of the delay, and a Minimal Learning Parameter (MLP) method that reduces adaptation to a single scalar fuzzy-related parameter per subsystem. In the finite-time constrained design, the controller combines fuzzy approximation with a finite-time-stable-like output-envelope transformation and integral BLFs (Wu et al., 2023, Wu et al., 2023).

The explicit BFLC scheduler is supervisory and sequential. Its inputs are the daily mean electricity price $$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$1, daily mean hydrogen price $$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$2, and daily mean wind capacity factor $$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$3. Each input and the output are partitioned into three linguistic fuzzy sets—low, medium, high—represented by parametrised triangular membership functions. The rule base contains 27 rules after conflict resolution from 81 possible input-output assignments, and the numerical output is computed by centroid defuzzification. The resulting daily HPA target is then imposed as a minimum requirement in hourly dispatch optimization (Farah et al., 2 Aug 2025).

4. Constraint enforcement and analytical guarantees

The strongest formal boundedness results in this set of papers come from Lyapunov- and LMI-based designs. In the evolving state-space controller, a fuzzy Lyapunov function

$$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$4

is used together with stability LMIs of the form

$$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$5

which imply asymptotic stability when feasible. The paper explicitly states that local gains are redesigned in real time whenever the corresponding local fuzzy models change, and only for active rules. Boundedness here is therefore both analytical and online: stability is certified by feasible LMIs, and control amplitude is limited by the additional bound-enforcing inequalities (Leite et al., 2021).

The pure-feedback input-delay controller uses BLFs to preserve full-state constraints and proves that all closed-loop signals are semiglobally uniformly ultimately bounded. The final state-boundedness relation is

$$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$6

and the tracking error can be within an arbitrary small neighbor of origin via selecting appropriate parameters of controllers. The controller does not, however, provide a formal hard input-constraint guarantee of the form $$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$7; the reported boundedness is on closed-loop signals and constrained states, while the actual control $$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$8 is generated by a low-pass filter driven by a newly-defined control input (Wu et al., 2023).

The finite-time pure-feedback controller proves a related but distinct result. A finite-time-stable-like envelope $$\text{IF } X_1 \text{ is NS and } X_2 \text{ is PB THEN } Y \text{ is PB}.$$9 is introduced so that the transformed tracking error satisfies $A$0 for all $A$1, where

$A$2

The paper proves that all closed-loop signals are semi-global ultimately uniformly bounded and that the constrained states never violate prescribed limits, using integral BLFs for $A$3, $A$4. This is practical finite-time convergence of the output tracking error to a predefined residual set, not exact finite-time convergence of the whole state (Wu et al., 2023).

By contrast, the hardware compilation paper is explicit about what it does not provide. It does not discuss controller stability, robustness proofs, Lyapunov arguments, boundedness of closed-loop trajectories, or safe operating limits. Its contribution is implementation and compilation, not control-theoretic guarantees. The green-hydrogen BFLC likewise does not present a generic fuzzy-control stability theory; its bounds are operational and economic, derived from daily producibility and benchmark cumulative export corridors rather than from a plant-level Lyapunov analysis (Chiu et al., 2013, Farah et al., 2 Aug 2025).

5. Implementations, workflows, and empirical results

The hardware implementation work offers a direct picture of BFLC as finite realizability. A fuzzy inference chip designed in $A$5 CMOS processes 16 control rules in parallel and produces 250,000 inferences per second at a 16 MHz clock. The alternative memory-chip method precomputes the full input-output map so that real-time execution reduces to address generation plus memory fetch. In a 2-input controller with 4-bit inputs each, the full input state is 8 bits and acts exactly like a memory address. Outputs may be floating-point or quantized to an integer size chosen by the user, for example 8-bit outputs represented as integers between 0 and 255. This realization is especially significant for BFLC because the fuzzy controller becomes a bounded finite map over a quantized input space (Chiu et al., 2013).

The evolving state-space controller is evaluated on the discrete Henon map,

$A$6

with $A$7 and $A$8, and with control added to the first equation. One-step prediction results show the granularity tradeoff: for $A$9, 60 rules and RMSE $A(x)$0; for $A(x)$1, 20 rules and RMSE $A(x)$2; for $A(x)$3, 5 rules and RMSE $A(x)$4. Under bounded-input synthesis with $A(x)$5, the best stabilization result is reported at $A(x)$6 with settling time $A(x)$7 and state energy $A(x)$8; with tighter bound $A(x)$9, performance degrades and for $(0,200)$0 and $(0,200)$1 there is no settling within the test horizon. These results directly support the claim that stricter input bounds can trade off against stabilization performance (Leite et al., 2021).

The constrained pure-feedback designs provide simulation evidence of bounded tracking under explicit state constraints. In the input-delay case, the illustrative system uses $(0,200)$2 s, desired trajectory $(0,200)$3, and state constraints $(0,200)$4, $(0,200)$5. Reported outcomes are that the output tracks $(0,200)$6 while staying within $(0,200)$7, $(0,200)$8 remains inside $(0,200)$9, transformed errors remain within the BLF limits, the control-generation signal $(0,500)$0 and actual control $(0,500)$1 are smooth, and the adaptive parameters remain bounded. The feasibility conditions were checked using MATLAB’s fseminf.m (Wu et al., 2023).

In the finite-time constrained design, one example uses a second-order pure-feedback nonlinear system with $(0,500)$2, $(0,500)$3, desired output $(0,500)$4, and envelope parameters $(0,500)$5, $(0,500)$6, $(0,500)$7, yielding $(0,500)$8 s. The paper reports that $(0,500)$9 remains within the shrinking tube $(0,10)$0 and converges to the predefined set in finite time, both states stay inside their prescribed bounds, adaptive parameters remain bounded, and the virtual control and input remain bounded. A second example addresses an inverted pendulum with unknown control direction and again reports bounded states, bounded Nussbaum variable, and bounded control signals (Wu et al., 2023).

The explicit BFLC application is a Danish wind-to-hydrogen plant with a 2 MW wind farm, a 1 MW electrolyser, no grid import, access to electricity and hydrogen spot markets, and an annual HPA target of $(0,10)$1 tonnes. Historical data cover Denmark, 2015–2023; benchmark results from 2017–2022 are used to train BFLC, and 2015, 2016, and 2023 are out-of-sample evaluation years. The implementation uses Python, with scikit-fuzzy for fuzzy inference, pyswarm for particle swarm optimisation, PyPSA for plant optimisation modelling, and Gurobi for mathematical programming. Revenue comparisons show that BFLC achieves total annual revenues within 9% of optimal revenues based on perfect foresight, with normalized total revenues consistently higher than 92% of benchmark and a lowest BFLC normalized revenue of 92.8% in 2022. Steady control is more than 12% lower than benchmark in 2020–2023, with a worst case in 2022 of 16.8% lower than benchmark. The largest differences are observed under elevated price levels and variability; in 2022 the mean electricity price is 219.0 €/MWh with standard deviation 145.5 €/MWh (Farah et al., 2 Aug 2025).

6. Limitations, misconceptions, and research significance

A recurrent misconception is to treat BFLC as a single controller topology. The cited work does not support that view. One strand uses clipped Mamdani inference on bounded discrete universes; another uses evolving trapezoidal antecedents and local affine state-space consequents with PDC; two use FLSs as adaptive approximators embedded in backstepping; and one uses a fuzzy supervisory scheduler whose output is bounded before dispatch optimization. The commonality is the deliberate enforcement of bounds, not a uniform inference formalism (Chiu et al., 2013, Leite et al., 2021, Wu et al., 2023, Wu et al., 2023, Farah et al., 2 Aug 2025).

A second misconception is to equate boundedness only with actuator saturation. The literature here supports a broader taxonomy. In (Chiu et al., 2013), boundedness is representational and architectural. In (Leite et al., 2021), it is an explicit Euclidean-norm bound on the control signal. In (Wu et al., 2023) and (Wu et al., 2023), it is primarily state-constraint preservation together with bounded internal signals. In (Farah et al., 2 Aug 2025), it is a bounded supervisory target trajectory enforced by physical producibility and cumulative corridor constraints. This suggests that BFLC is better understood as a constraint-oriented perspective on fuzzy control than as a synonym for clipped outputs.

The limitations are equally heterogeneous. The hardware paper does not analyze stability, robustness, quantization effects on closed-loop properties, or plant-level saturation management. The evolving state-space paper bounds control amplitude only; it does not treat input-rate bounds, input-energy bounds, or saturation dynamics explicitly, and its guarantee is best understood as online local or instantaneous bounded-input enforcement within the feasible region of the LMI problem. The input-delay pure-feedback paper relies on smoothness assumptions, bounded disturbances, full-state access, and a constant delay handled by a first-order Padé model; it also contains notation inconsistencies in the delayed-input stage. The finite-time pure-feedback paper requires a feasibility check to ensure recursive virtual controls remain compatible with barrier admissibility, and its finite-time result is to a residual set rather than exact convergence to zero. The explicit BFLC scheduler uses synthetic hydrogen prices, empirical convex-hull bounds derived from benchmark years, and a rule base whose final outputs are mostly “low,” so its boundedness concept is application-specific rather than a general theory (Chiu et al., 2013, Leite et al., 2021, Wu et al., 2023, Wu et al., 2023, Farah et al., 2 Aug 2025).

The significance of BFLC, as evidenced by these papers, lies in making fuzzy control compatible with finite hardware, constrained nonlinear dynamics, and economically constrained supervisory scheduling. A plausible implication is that BFLC should be viewed less as a narrow doctrinal label and more as a research program concerned with how fuzzy reasoning behaves once it must satisfy explicit bounds: finite truth precision, bounded control amplitude, hard state envelopes, or bounded cumulative commitments. Under that interpretation, the literature forms a coherent progression from bounded finite-state realizations in silicon to Lyapunov-certified bounded-input and state-constrained controllers, and finally to application-specific bounded supervisory control in energy systems (Chiu et al., 2013, Leite et al., 2021, Wu et al., 2023, Wu et al., 2023, Farah et al., 2 Aug 2025).