Fractional Order Fuzzy PID Controller

• FOFPID is a hybrid controller that combines fuzzy inference with a fractional-order PID framework, offering additional tuning parameters for control design.
• It employs nonlocal fractional operators and rational approximation methods to capture memory effects and improve transient and steady-state performance.
• Tuning of FOFPID often leverages evolutionary algorithms like GA, PSO, and WOA to optimize gains, fractional orders, and fuzzy rules for handling nonlinear and uncertain dynamics.

Searching arXiv for the cited FOFPID and related FOPID papers to ground the article and citations. Fractional Order Fuzzy PID Controller (FOFPID) denotes a class of hybrid controllers that combine fuzzy inference with fractional-order PID dynamics. In the formulation emphasized in recent control literature, the controller retains the $PI^\lambda D^\mu$-type structure of a fractional-order PID (FOPID), but replaces fixed gains with fuzzy-logic-mediated or fuzzy-generated actions, while preserving non-integer integral and derivative operators (Behboudifar et al., 21 Jul 2025, Das et al., 2012, Das et al., 2013). In this sense, FOFPID extends both classical PID and FOPID by coupling distributed-memory dynamics with nonlinear rule-based adaptation. Across the cited applications—Depth of Anesthesia regulation, nuclear reactor power control, hybrid power systems, delayed nonlinear processes, and oscillatory fractional-order dead-time plants—the recurring rationale is that fractional orders provide additional tuning degrees of freedom, while fuzzy logic provides nonlinear adaptation under uncertainty, nonlinearity, and operating-point variation (Behboudifar et al., 21 Jul 2025, Pan et al., 2016, Das et al., 2013, Das et al., 2013).

1. Formal definition and controller architectures

The canonical continuous-time FOPID law is written as

$$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$

or, in Laplace form,

$C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$

with $\lambda$ and $\mu$ denoting the fractional orders of integration and differentiation (Das et al., 2012, Merrikh-Bayat et al., 2013, Das et al., 2016). A classical PID is recovered when $\lambda=1$ and $\mu=1$ (Merrikh-Bayat et al., 2013, Das et al., 2013).

Within the FOFPID family, two principal architectural patterns recur. In one pattern, fuzzy inference adjusts the gains of an underlying FOPID in real time. The 2025 Depth of Anesthesia study defines FOFPID as a controller in which the proportional, integral, and derivative gains are adjusted online by a fuzzy inference system, while the integral and derivative actions remain fractional-order (Behboudifar et al., 21 Jul 2025). With error $e(t)$ and error derivative $\dot e(t)$ as fuzzy inputs, and $K_p$, $$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$0, $$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$1 as fuzzy outputs, the controller is expressed as

$$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$2

(Behboudifar et al., 21 Jul 2025).

In a second pattern, the fuzzy block itself produces an intermediate control action that is then fractionally integrated or differentiated. The 2012 formulation “A Novel Fractional Order Fuzzy PID Controller and Its Optimal Time Domain Tuning Based on Integral Performance Indices” defines a controller whose inputs are the closed-loop error and its fractional derivative, and whose output includes a fractional integrator (Das et al., 2012). In that architecture,

$$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$3

(Das et al., 2012). The 2013 nuclear-reactor paper adopts the same general pattern, with the fuzzy block producing a signal $$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$4 that is then combined through fractional integration and a direct path (Das et al., 2013).

A further structural refinement is the use of decomposed or hybrid fuzzy-fractional realizations. The family analyzed in “Performance Comparison of Optimal Fractional Order Hybrid Fuzzy PID Controllers for Handling Oscillatory Fractional Order Processes with Dead Time” includes FO fuzzy PID, FO fuzzy PI+PD, FO fuzzy P+ID, FO fuzzy PI+D, and FO fuzzy PD+I structures (Das et al., 2013). This suggests that FOFPID is not a single controller topology but a broader design family whose defining characteristic is the joint use of fuzzy inference and fractional differintegration.

2. Fractional-order operators and memory effects

Fractional-order control replaces integer-order differentiation and integration with nonlocal operators. Several cited works formulate these operators explicitly. The hybrid power system study adopts the Caputo definition

$$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$5

(Pan et al., 2016). Other papers state the operators in transfer-function form and rely on the correspondence $$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$6 (Behboudifar et al., 21 Jul 2025, Das et al., 2012).

The control-theoretic significance of the fractional orders is consistently described as added flexibility for shaping the closed-loop dynamics. Compared with integer-order PID, FOPID and FOFPID introduce extra degrees of freedom through $$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$7 and $$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$8, permitting more flexible shaping of transient response, damping, robustness, and phase behavior (Behboudifar et al., 21 Jul 2025, Merrikh-Bayat et al., 2013, Das et al., 2012, Das et al., 2012). In the PHWR step-back study, this flexibility is directly linked to iso-damped behavior through frequency-domain flattening of the open-loop phase near crossover (Das et al., 2012). In the nonlinear boost-converter study, it is linked to improved transient response, robustness to input-voltage variation, and reduced switching activity (Merrikh-Bayat et al., 2013).

In FOFPID architectures, fractionalization appears at different signal locations. The rate-of-error input to the fuzzy system is often fractional, $$u(t) = K_p e(t) + K_i D^{-\lambda} e(t) + K_d D^\mu e(t),$$9, rather than $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$0 (Das et al., 2012, Das et al., 2013, Das et al., 2013, Pan et al., 2016). The output of the fuzzy block is frequently passed through a fractional integrator $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$1 (Das et al., 2012, Das et al., 2013, Das et al., 2013). This creates richer effective control laws than a simple fuzzy tuning of a conventional PID. In the reactor-power FOFPID derivation, expansion of the fuzzy-plus-fractional structure yields effective proportional, fractional integral, fractional derivative, and mixed $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$2 terms (Das et al., 2013). A plausible implication is that FOFPID designs can realize cross-coupled integro-differential behaviors that are not naturally available in fixed-structure integer-order fuzzy PID schemes.

3. Fuzzy inference mechanisms and rule-based adaptation

The fuzzy component in FOFPID is usually a type-I Mamdani inference system with two inputs and one or more outputs. The most common inputs are the error $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$3 and either the derivative of error $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$4 or a fractional derivative $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$5 (Behboudifar et al., 21 Jul 2025, Das et al., 2012, Das et al., 2013, Das et al., 2013, Pan et al., 2016). Depending on the architecture, the outputs may be the controller gains $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$6, $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$7, $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$8, or a single crisp control signal later mapped through fractional operators and scaling factors (Behboudifar et al., 21 Jul 2025, Pan et al., 2016, Das et al., 2012).

Membership-function design is relatively standardized across the literature. The 2012 FOFPID paper uses seven triangular membership functions for each input and output, with linguistic labels NL, NM, NS, ZR, PS, PM, and PL, and 50% overlap over a normalized universe (Das et al., 2012). The nuclear-reactor FOFPID paper uses the same seven-term triangular partition and center-of-gravity defuzzification (Das et al., 2013). By contrast, the 2025 anesthesia controller uses five membership functions per input and output, with labels such as NM, NS, Z, PS, and PM, and tunes membership-function parameters via Whale Optimization Algorithm (WOA) (Behboudifar et al., 21 Jul 2025). The hybrid power system controller similarly uses seven symmetric triangular membership functions over $C(s)=K_p+K_i s^{-\lambda}+K_d s^\mu,$9 and center-of-gravity defuzzification (Pan et al., 2016).

The rule bases are generally fixed, symmetric, and PID-like. The 2016 hybrid power system study explicitly describes a 7×7 rule table in which large errors with same-sign error-rate produce large outputs, opposite-sign combinations move the output toward zero, and near-zero conditions yield near-zero control (Pan et al., 2016). The 2012 FOFPID paper groups the 49 rules into qualitative categories corresponding to steady-state behavior, motion toward the setpoint, and motion away from the setpoint (Das et al., 2012). The 2025 anesthesia paper does not print the detailed rule table, but states that membership functions, rules, and scaling factors are optimized via WOA (Behboudifar et al., 21 Jul 2025).

An important methodological distinction emerges across the literature. Some FOFPID designs keep the rule base and membership-function shapes fixed and optimize only scaling factors and fractional orders (Das et al., 2012, Das et al., 2013). Others include membership-function parameters and even rules within the optimization variables (Behboudifar et al., 21 Jul 2025). This suggests two distinct design philosophies: one treats fuzzy structure as a fixed nonlinear template whose gains and orders are tuned; the other treats the fuzzy mechanism itself as part of the search space.

4. Numerical realization and digital implementation

Because fractional operators are not directly realizable, FOFPID implementations rely on rational approximations. Oustaloup recursive approximation is the dominant method across the cited literature. The hybrid power system study states that each fractional element $\lambda$0 is approximated by a 5th-order Oustaloup recursive filter over $\lambda$1 rad/s (Pan et al., 2016). The nuclear-reactor FOFPID paper uses a 5th-order Oustaloup approximation in the same band for both controller fractional elements and fractional Gaussian noise generation (Das et al., 2013). The delayed-process FOFPID paper also uses a 5th-order Oustaloup approximation over $\lambda$2 rad/s (Das et al., 2012). The PHWR FOPID step-back paper uses 4th-order Oustaloup approximation over $\lambda$3 rad/s (Das et al., 2012). The AVR FOPID paper uses Oustaloup approximation with $\lambda$4, corresponding to a 5th-order rational approximation over $\lambda$5 rad/s (Das et al., 2013).

Some works focus on alternative realization strategies. “Symbolic Representation for Analog Realization of A Family of Fractional Order Controller Structures via Continued Fraction Expansion” gives continued-fraction-expansion-based analog realizations for FOPID, FO[PD], and FO lead-lag structures, providing parameterized rational approximations useful for analog synthesis (Pakhira et al., 2016). This paper does not address fuzzy logic, but it provides the fractional building blocks that could be embedded within a FOFPID architecture (Pakhira et al., 2016).

A further implementation development is the directly discrete-time formulation of fractional PID. “Discrete-Time Fractional-Order PID Controller: Definition, Tuning, Digital Realization and Experimental Results” defines a long-memory discrete-time fractional PID using weighted sums of past errors and an adjustable memory length $\lambda$6, instead of discretizing a continuous-time FOPID after tuning (Merrikh-Bayat et al., 2014). This suggests a possible route for discrete-time FOFPID design in which the fuzzy layer operates atop directly discrete long-memory operators rather than rational approximants derived from continuous-time models. A plausible implication is that, for ARMA or ARMAX plant models, discrete-time FOFPID may be preferable to continuous-design-then-discretize workflows.

The following table summarizes the realization strategies explicitly described.

Realization approach	Description in the cited works	Representative papers
Oustaloup recursive approximation	Fractional operators approximated by rational filters over a selected frequency band	(Pan et al., 2016, Das et al., 2013, Das et al., 2012, Das et al., 2012, Das et al., 2013)
FOMCON-based implementation	Fractional derivative and integral computed with the Fractional Order Modeling and Control Toolbox for MATLAB	(Behboudifar et al., 21 Jul 2025)
Continued Fraction Expansion	Symbolic rational approximations for analog realization of FO controller structures	(Pakhira et al., 2016)
Direct discrete-time long-memory realization	Native discrete fractional PID defined as weighted sums with adjustable memory length	(Merrikh-Bayat et al., 2014)

5. Tuning methodologies and optimization frameworks

FOFPID tuning is typically nonconvex and multi-parameter, involving scaling factors, gains, fractional orders, and sometimes fuzzy membership functions or rules. As a result, evolutionary and swarm-based optimizers dominate the literature.

Genetic Algorithm (GA) is used in the early generalized FOFPID literature. The 2012 delayed-process study tunes $\lambda$7 for the proposed FOFPID and analogous parameter sets for benchmark PID, fuzzy PID, and FOPID controllers (Das et al., 2012). The optimization criteria are combinations of time-domain error measures with control effort: ITAE + ISCO, ITSE + ISCO, ISTES + ISCO, and ISTSE + ISCO, with equal weights on error and control signal (Das et al., 2012). The 2013 nuclear-reactor FOFPID paper uses real-coded GA to tune scaling factors and fractional orders under a stochastic cost

$\lambda$8

identified as ITSE + ISCO, and later extends this to expected-value optimization under random delay and long-range-dependent noise (Das et al., 2013). The hybrid FO fuzzy PID family for oscillatory FO dead-time processes is also tuned with real-coded GA under a cost combining ISTSE and ISDCO, then analyzed further with NSGA-II for Pareto trade-offs (Das et al., 2013).

Particle Swarm Optimization (PSO) and its chaotic variants appear in power-system applications. The hybrid power system paper tunes $\lambda$9 by minimizing

$\mu$0

with $\mu$1 s and compares standard PSO with logistic-map and Henon-map variants (Pan et al., 2016). The reported best values for the FO fuzzy PID with Henon-map PSO are $\mu$2, $\mu$3, $\mu$4, $\mu$5, $\mu$6, and $\mu$7, with $\mu$8 (Pan et al., 2016).

Whale Optimization Algorithm has been adopted in recent anesthesia-control papers. In “Whale Optimization Algorithms based fractional order fuzzy PID controller for Depth of Anesthesia,” WOA optimizes the fractional orders, membership-function parameters, rules, and scaling factors of the fuzzy controller (Behboudifar et al., 21 Jul 2025). The cost function is

$\mu$9

with

$\lambda=1$0

(Behboudifar et al., 21 Jul 2025). WOA is described in terms of exploration, encircling prey, and bubble-net attacking, with parameter update equations standard to the algorithm (Behboudifar et al., 21 Jul 2025). A later anesthesia paper reports the same broad tuning logic and presents comparative metrics for FOFPID versus FOPID (Shahbandari et al., 17 Aug 2025).

Other tuning paradigms, although not FOFPID-specific, are directly relevant. The AVR study formulates FOPID tuning as a mixed $\lambda=1$1 multi-objective loop-shaping problem, then uses NSGA-II to compute Pareto fronts and fuzzy membership-based decision making to select compromise solutions (Das et al., 2013). The data-driven FOPID study based on fictitious reference signals provides a one-shot model-free tuning method for FOPID that could plausibly serve as a baseline core inside a FOFPID design (Yonezawa et al., 2023). The delayed fractional-order process study uses multi-objective LQR plus NSGA-II to generate Pareto-optimal FOPID parameters and explicit tuning rules in terms of plant delay ratio and fractional order, which could serve as nominal seeds for FOFPID adaptation (Das et al., 2016).

6. Domains of application and comparative performance

FOFPID has been studied primarily in systems that combine nonlinearity, uncertainty, delay, or strong operating-point dependence. The most detailed recent biomedical application is Depth of Anesthesia control via propofol infusion. The 2025 anesthesia paper models the patient by a three-compartment PK subsystem and a nonlinear PD BIS equation, investigates eight patient models, and concludes that “both controllers can control the nonlinearity and uncertainty of the system,” but that “the FOFPID controller has a better performance than the FOPID controller in transient and steady-state behavior due to its more degree of flexibility because of the more tunable parameters” (Behboudifar et al., 21 Jul 2025). It further states that patients generally reach the desired BIS value in about 3 minutes and that the FOFPID shows lower settling time and steady-state error than the FOPID (Behboudifar et al., 21 Jul 2025). The later 2025 DOA paper makes this comparison quantitative: average settling time $\lambda=1$2 min versus $\lambda=1$3 min, average steady-state error $\lambda=1$4 versus $\lambda=1$5, IAE $\lambda=1$6 versus $\lambda=1$7, and ITAE $\lambda=1$8 versus $\lambda=1$9, for FOFPID versus FOPID, respectively (Shahbandari et al., 17 Aug 2025).

In nuclear-reactor power control, the FOFPID literature emphasizes robustness to operating-point variation and stochastic disturbances rather than only nominal tracking. The 2013 reactor paper shows that FOFPID tuned at the highest power model works well across lower power levels and maintains smoother control effort than conventional fuzzy PID under network-induced random delay and sensor noise with long-range dependence (Das et al., 2013). Persistent, white, and anti-persistent noise are compared, and anti-persistent noise is identified as the most detrimental (Das et al., 2013). Under stochastic tuning, FOFPID achieves visibly reduced oscillation bands in both power and control signals relative to integer-order fuzzy PID (Das et al., 2013).

In hybrid power systems with renewable generation, the FO fuzzy PID controller is compared against classical PID and integer-order fuzzy PID. Under nominal UC parameters, the Henon-PSO-tuned FO fuzzy PID achieves $\mu=1$0 versus $\mu=1$1 for fuzzy PID and $\mu=1$2 for PID (Pan et al., 2016). More importantly, the paper reports stronger robustness of FO fuzzy PID to UC parameter variations, component disconnection, and nonlinear rate constraints, with lower or equal ISE and ISDCO across all tested UC perturbation cases (Pan et al., 2016). This suggests that in heavily switched, uncertain networks the main benefit of FOFPID lies less in nominal-performance gains than in robustness.

In delayed nonlinear and unstable process control, the 2012 benchmark study shows that the proposed FOFPID outperforms conventional PID, fuzzy PID, and FOPID in most cases, especially in setpoint tracking and often in control effort, though disturbance rejection superiority depends on the chosen performance index and plant (Das et al., 2012). For the unstable delayed plant, FOFPID is generally best or close to best in setpoint tracking, while fuzzy PID sometimes yields smaller control signals and disturbance rejection can favor different controllers depending on the cost criterion (Das et al., 2012). This cautions against the common misconception that FOFPID uniformly dominates all other controllers on all metrics.

For oscillatory fractional-order plants with dead time, the 2013 comparative study finds that no single hybrid FO fuzzy PID structure is universally best (Das et al., 2013). For lag-dominant oscillatory plants, FO fuzzy P+ID offers the best Pareto trade-off between tracking and control effort, whereas FO fuzzy PD+I is best for tracking versus disturbance rejection (Das et al., 2013). For balanced lag–delay plants, FO fuzzy P+ID and FO fuzzy PD+I intersect in the tracking-versus-effort trade-off, while FO fuzzy PI+D is best for tracking versus disturbance rejection (Das et al., 2013). For delay-dominant plants, FO fuzzy PID is best for tracking versus control effort, and FO fuzzy PD+I is best for tracking versus disturbance rejection (Das et al., 2013). This suggests that “FOFPID” should not be treated as a monolithic controller class; structural decomposition matters materially.

7. Relationship to neighboring controller classes

FOFPID sits at the intersection of several established control paradigms. Relative to classical PID, it adds nonlinear fuzzy inference and fractional orders $\mu=1$3, thereby increasing design dimensionality and expressiveness (Behboudifar et al., 21 Jul 2025, Das et al., 2012). Relative to FOPID, it introduces operating-point-dependent or error-dependent adaptation through fuzzy logic (Behboudifar et al., 21 Jul 2025, Pan et al., 2016). Relative to integer-order fuzzy PID, it introduces nonlocal dynamics and additional shaping parameters through fractional differintegration (Das et al., 2012, Das et al., 2013).

One related but distinct line is adaptive fractional PID without fuzzy logic. The synchronization paper on fractional chaotic systems derives gradient-based adaptation laws for a fractional PID controller,

$\mu=1$4

with a sliding surface $\mu=1$5 and fractional controller orders tied to the plant order (HosseinNia et al., 2012). Although this is not a FOFPID design, the variables used in adaptation—error, error derivative, error integral, and sliding surface—are precisely the kinds of signals often used as fuzzy inputs. This suggests a natural link between Lyapunov-inspired adaptive FOPID design and rule-based FOFPID design.

Another neighboring class is the “fractional order fuzzy control” used in hybrid renewable power systems, where the fuzzy controller does not merely tune a fixed FOPID but generates a PID-like action through fractionalized fuzzy input/output processing (Pan et al., 2016). Similarly, the FO fuzzy hybrid structures for dead-time plants blur the distinction between “fuzzy tuning of FOPID” and “fuzzy realization of fractional PID behavior” (Das et al., 2013). The literature therefore uses FOFPID in both a narrow and broad sense. In the narrow sense, FOFPID means an FOPID with fuzzy-adjusted gains. In the broad sense, it denotes any fuzzy PID-like controller whose I and/or D actions are fractional-order.

A final relation is methodological rather than architectural. Several FOPID studies—on AVR systems, PHWR step-back control, nonlinear boost converters, and delayed fractional-order plants—provide rigorous tuning tools, robustness metrics, or implementation methods directly transferable to FOFPID research (Das et al., 2013, Das et al., 2012, Merrikh-Bayat et al., 2013, Das et al., 2016). This suggests that FOFPID is best understood not as an isolated design idea but as an extension layered atop the substantial existing theory of FOPID tuning, realization, and robustness analysis.

8. Limitations, trade-offs, and recurring misconceptions

A recurring limitation across the literature is design and implementation complexity. FOFPID generally requires more parameters than PID, fuzzy PID, or FOPID: gains, scaling factors, fuzzy membership-function parameters, and fractional orders must all be selected or optimized (Behboudifar et al., 21 Jul 2025, Das et al., 2012, Das et al., 2013). The 2025 anesthesia paper notes that FOFPID is more complex than FOPID because it has more parameters to tune and requires fuzzy inference in addition to fractional filtering, though the added computational burden is largely offline in WOA-based tuning (Behboudifar et al., 21 Jul 2025). The delayed-process study likewise notes that more degrees of freedom do not automatically guarantee better performance; the benefit depends on the performance index and problem structure (Das et al., 2012).

A second limitation is realization burden. Fractional operators require rational approximations or discrete long-memory kernels, and their accuracy depends on approximation order and frequency band (Pan et al., 2016, Das et al., 2013, Merrikh-Bayat et al., 2014). In high-speed or embedded settings, the combined cost of fuzzy inference and FO filtering can be significant. This suggests that practical FOFPID implementations may need reduced-order fractional approximants, fixed rule bases, and bounded adaptation ranges.

A third issue is that FOFPID superiority is problem-dependent. In the 2012 benchmark study, fuzzy PID slightly outperforms the proposed FOFPID numerically for the nonlinear delayed plant under ITAE + ISCO, though the practical differences are small (Das et al., 2012). In the oscillatory dead-time study, no single FO fuzzy structure dominates across all plant classes and objective pairs (Das et al., 2013). A common misconception is therefore that FOFPID is uniformly superior to all PID-family controllers. The literature does not support that claim. Instead, it supports a narrower statement: FOFPID often offers superior robustness or improved trade-offs in nonlinear, uncertain, delayed, or variable-operating-point systems, provided the structure and tuning method are well matched to the application (Das et al., 2013, Pan et al., 2016, Behboudifar et al., 21 Jul 2025).

A fourth issue concerns validation scope. Many reported results are simulation-based and application-specific. The anesthesia papers evaluate eight patient models but do not report hardware or clinical implementation details (Behboudifar et al., 21 Jul 2025, Shahbandari et al., 17 Aug 2025). The nuclear-reactor studies use linearized operating-point models and stochastic simulation environments (Das et al., 2013). The hybrid power study tests extensive uncertainty scenarios but remains simulation-based (Pan et al., 2016). This suggests that, despite strong numerical evidence, FOFPID still requires careful experimental validation in many domains.

9. Research directions and methodological outlook

The literature points toward several coherent directions for further FOFPID development. One is tighter integration with rigorous nominal FOPID design. Frequency-domain mixed $\mu=1$6 loop shaping for FOPID, as developed for AVR systems, offers objective functions and robustness metrics that could be extended to FOFPID tuning by treating fuzzy scaling factors or rule parameters as additional decision variables (Das et al., 2013). Iso-damped FOPID design for PHWR step-back control likewise suggests that a robust backbone FOPID can be designed first and then augmented with a fuzzy adaptation layer (Das et al., 2012).

Another direction is model-free or data-driven FOFPID design. The one-shot fictitious-reference FOPID tuning method provides a practical route to obtaining a nominal FOPID directly from data, without a plant model, while explicitly considering closed-loop stability (Yonezawa et al., 2023). A plausible implication is that such data-driven FOPID cores could serve as nominal starting points for FOFPID architectures, after which fuzzy adaptation handles unmodeled nonlinearities or operating-point variation.

A third direction is discrete-time-native FOFPID design. The discrete-time long-memory PID framework indicates that direct discrete formulations can avoid the distortions introduced by “design in continuous time, then discretize” workflows (Merrikh-Bayat et al., 2014). This is especially relevant for microprocessor-based fuzzy controllers and for plants modeled directly in ARMA or ARMAX form. A discrete FOFPID could combine long-memory weighted-sum operators with fuzzy logic in a unified sampled-data framework.

Finally, multi-objective optimization is likely to remain central. Pareto-based analyses in FOPID and hybrid FO fuzzy PID design repeatedly show that tracking, disturbance rejection, noise attenuation, and control effort are inherently conflicting (Das et al., 2013, Das et al., 2016, Das et al., 2013). FOFPID’s extra flexibility is best interpreted as the ability to navigate these trade-offs more effectively, not as a guarantee of dominance on every metric. This suggests that future FOFPID work will continue to rely on NSGA-II, PSO, WOA, and related optimizers, often coupled with problem-specific robustness or safety constraints.

In aggregate, the cited literature presents FOFPID as a mature but heterogeneous controller family: mathematically rooted in fractional calculus, operationally enabled by fuzzy inference, and practically shaped by metaheuristic optimization and rational approximations. Its main value lies in applications where integer-order linear control is too rigid and where purely fuzzy control lacks the memory and phase-shaping capabilities supplied by fractional-order dynamics (Behboudifar et al., 21 Jul 2025, Das et al., 2012, Das et al., 2013, Pan et al., 2016, Das et al., 2013).