Ideal (I₂) Convergence in Fuzzy Paranormed Spaces for Practical Stability of Discrete-Time Fuzzy Control Systems Under Lacunary Measurements

Abstract

We investigate the stability of linear discrete-time control systems with a fuzzy logic feedback under sporadic sensor data loss. In our framework, each state measurement is a fuzzy number, and occasional “packet dropouts” are modeled by a lacunary subsequence of missing readings. We introduce a novel mathematical approach using lacunary statistical convergence in fuzzy paranormed spaces to analyze such systems. Specifically, we treat the sequence of fuzzy measurements as a double sequence (indexed by time and state component) and consider an admissible ideal of “negligible” index sets that includes the missing–data pattern. Using the concept of ideal fuzzy—paranorm convergence (I-fp convergence), we formalize a lacunary statistical consistency condition on the fuzzy measurements. We prove that if the closed-loop matrix $A-BK$ is Schur stable (i.e., $\parallel A-BK\parallel <1$) in the absence of dropouts, then under the lacunary statistical consistency condition, the controlled system is practically stable despite intermittent measurement losses. In other words, for any desired tolerance, the state eventually remains within that bound (though not necessarily converging to zero). Our result yields an explicit, non-probabilistic (distribution-free) analytical criterion for robustness to sensor dropouts, without requiring packet-loss probabilities or Markov transition parameters. This work merges abstract convergence theory with control application: it extends statistical and ideal convergence to double sequences in fuzzy normed spaces and applies it to ensure stability of a networked fuzzy control system.

1. Introduction

Modern control systems often operate over communication networks, making them susceptible to issues like time-varying delays and packet dropouts in sensor or actuator data. These communication imperfections can degrade performance and even destabilize the closed-loop system. In particular, networked control systems (NCS) with intermittent measurements or packet losses have been extensively studied in the literature [1]. Classical stability results for fuzzy logic controllers assume continuous, reliable sensor readings. However, in practical scenarios such as wireless sensor networks or IoT-based control, sensor readings may be missing or delayed [2]. This motivates developing new theoretical tools to analyze stability under missing or uncertain data. In typical NCS implementations, the controller acts on a crisp value obtained via centroid (COA/COG) defuzzification, which is widely used in fuzzy control [3,4]. Accordingly, we focus on centroid bias and, throughout, we require only a bounded (or $I_2$-small) centroid bias rather than perfect centering; see Remark 9.

To address uncertainty in measurements, we incorporate concepts from fuzzy set theory. Fuzzy control systems use fuzzy logic to handle imprecise information, and their stability can be analyzed via Lyapunov methods. Sufficient conditions for fuzzy control stability have been formulated (e.g., via Takagi–Sugeno models and Lyapunov functions) [2]. However, most existing fuzzy control frameworks assume complete and precise state information. When some sensor data are lost, these classical conditions may no longer directly apply. Our work aims to fill this gap by providing a stability analysis for a fuzzy feedback controller operating with sporadic measurement dropouts.

On the mathematical side, our approach builds on the theory of sequence convergence beyond the classical pointwise notion. The concept of statistical convergence was introduced by H. Fast and H. Steinhaus [5,6] in 1951 as a generalization of ordinary convergence. Rather than requiring “all but finitely many” terms of a sequence to lie within any error $\epsilon$ of the limit, statistical convergence requires that the set of terms outside the $\epsilon$-neighborhood has natural density zero. Schoenberg later independently developed similar ideas in 1959 [7]. This notion allows one to ignore a sparse (density-zero) set of deviations and still consider the sequence convergent. Building on this idea, Kostyrko et al. [8] introduced ideal convergence (or I-convergence) as a further generalization. An ideal I is a collection of “negligible” subsets of $\mathbb{N}$ (closed under taking smaller sets and finite unions). A sequence $\left(x_n\right)$ is said to I-converge to L if for every $\epsilon >0$, the set $n:d(x_n,L)\ge \epsilon$ is in the ideal I. Classical convergence corresponds to the ideal of finite sets, and statistical convergence corresponds to the ideal of all density-zero sets. By choosing different ideals, one can tailor the convergence concept to specific sparse patterns of deviation. The flexibility of I-convergence has led to many extensions in summability theory and analysis [8].

This idea has been extended from single-index sequences to double sequences (indexed by $\mathbb{N}\times \mathbb{N}$). In the double-index case, convergence can be defined in the Pringsheim sense (both indices tending to infinity). Mursaleen and Edely [9] first introduced statistical convergence for double sequences. They showed how to define natural density in ${\mathbb{N}}^2$ and studied statistically convergent double sequences and statistical Cauchy criteria. Soon after, researchers developed analogues of ideal convergence for double sequences. For instance, Tripathy and co-authors [10,11] investigated $I_2$-convergence for double sequences, while Patterson and Savaş [12,13] focused on lacunary and lacunary statistical convergence in the double-sequence setting. These works allow handling two-dimensional data arrays, which is useful in our context where one index can represent time steps and the other index (or indices) represent components of a vector (e.g., each state variable or sensor reading). Indeed, in our stability analysis, each measurement yields a fuzzy vector (with components), naturally viewed as a double sequence in time k and state coordinate ℓ.

Parallel to these developments in summability, we leverage the framework of fuzzy normed spaces to handle uncertainties. Zadeh’s [14] introduction of fuzzy sets led to the rapid adoption of fuzzy logic in control engineering and other fields where data may be imprecise. In functional analysis, an ongoing effort has been to “fuzzify” classical notions of norm, metric, and convergence. Katsaras [15] was the first to propose a definition of a fuzzy norm on a linear space while studying fuzzy topological vector spaces. Felbin [16] introduced an alternative fuzzy norm concept by assigning to each vector a fuzzy real number as its “length”. Subsequent researchers like Cheng and Mordeson [17] and Bag and Samanta [18] refined these concepts. In particular, Bag and Samanta’s fuzzy normed linear space model (often called the BS fuzzy norm) became a standard, satisfying axioms analogous to those of a metric space but in fuzzy terms. Convergence in a fuzzy normed space means that for every $\epsilon >0$, the membership value $N(x_n-L,\epsilon )$ tends to 1 as $n\to \infty$. This provides a graded notion of “closeness” that is well suited to uncertain or approximate data. Researchers have also defined other convergences in fuzzy normed spaces and even examined pairs of fuzzy numbers [19,20,21,22].

More recently, Çınar et al. [23] introduced the notion of fuzzy paranormed spaces. A paranorm generalizes a norm by relaxing the absolute homogeneity requirement—roughly, it behaves like a norm but only requires continuity under scalar multiplication instead of linear homogeneity. Çınar’s fuzzy paranorm combines the fuzziness of membership functions with the flexibility of paranorms, yielding a structure that generalizes fuzzy normed spaces. In a fuzzy paranormed space, one has a membership function $\wp (x,t)$ indicating the degree to which $\left|x\right|$ is “small” compared with t, satisfying properties (FP1)–(FP6) (see Section 2). The advantage is that some analytical constructs (like certain sequence spaces) that are not normable can still be handled with a paranorm, and introducing fuzziness accounts for uncertainty or gradation.

Since the seminal work of Çınar et al. [23] introduced fuzzy paranormed spaces, a growing body of research has demonstrated their practical power. For instance, Türkmen and Öğünmez [24] employed I_fp convergence to design adaptive base-stock policies under non-stochastic triangular fuzzy demand, achieving lower inventory cost and higher service levels. In the same research stream, Öğünmez and Türkmen [25] verified—via detailed simulations—the role of $\lambda$-statistical convergence in producing resilient stocking decisions for supply-chain demand shocks.

Within networked fuzzy control, Sun et al. [26] proposed an observer-based scheme for Takagi–Sugeno systems subject to stochastic packet losses, deriving discrete-time LMI criteria; Kchaou et al. [27] developed security-oriented, type-2 fuzzy controllers with Markov jumps under dynamic event-triggered protocols and cyber-attacks. These contemporary studies corroborate that filtering communication faults through ideal small index sets yields tangible benefits in both supply-chain analytics and networked fuzzy control, thereby underscoring the direct relevance of the present $I_2$-fp framework to the latest literature.

Contributions of this work: We bring together the above threads—ideal convergence, double sequences, and fuzzy (para)normed spaces—to tackle a concrete problem in control theory. First, on the theoretical side, we develop the concept of $I_2$-convergence in fuzzy paranormed spaces (Definition 17 in Section 3). This is the first time that ideal convergence of double sequences is studied in the context of fuzzy paranorms. We provide characterizations of such convergence (including $I_2^{\ast }$-convergence via filters) and prove fundamental properties like uniqueness of the limit (Theorem 1) and an ideal version of a Cauchy convergence criterion (Theorem 2) under a suitable AP Property $\left(A{P}_2\right)$ for the ideal. These results generalize classical sequence space theory to a highly generalized setting, and they may be of independent interest in analysis.

Second, we apply this framework to a discrete-time fuzzy control system with intermittent sensor dropouts (Section 4). We model the sequence of fuzzy state measurements as a double sequence ${\tilde{x}}_{k\mathsf{\ell }}$, where k is the time step and ℓ indexes the state vector components. The dropout pattern is described by a lacunary subsequence $\theta =n_r$ indicating the time indices of lost packets; this gives rise to an index set $S_{\theta }$ of “missing” data points in the double sequence. We then define an ideal I on $\mathbb{N}\times \mathbb{N}$ that deems the dropout set $S_{\theta }$ as negligible (along with all finite sets). Intuitively, this means we are willing to ignore the sporadic missing measurements in the convergence analysis. The main stability result (Theorem 3) shows that if the controller $u_k=-Kc\left({\tilde{x}}_k\right)$ would stabilize the system in the absence of dropouts (i.e., $A-BK$ is a contraction matrix), and if the fuzzy measurement sequence is I-fp Cauchy (a lacunary statistically convergent sequence) in our fuzzy paranormed space, then the closed-loop system is practically stable. In practical terms, despite losing an infinite but sparse set of sensor readings, the state will remain ultimately bounded in a small neighborhood of zero—a form of robustness against dropouts.

Finally, we emphasize that our approach is probability-model-free and analytical. Unlike stochastic dropout models that require specifying packet-loss probabilities or Markov transition parameters [28,29], our ideal convergence condition is deterministic and non-probabilistic. It provides an axiomatic test: given a lacunary dropout pattern with a minimum inter-dropout gap $q_0>0$ and a fuzzy error bound on measurements, one can verify the ideal Cauchy condition and ensure stability without needing to simulate every dropout scenario. This contributes a new perspective to control theory, connecting it with summability theory and sequence spaces.

Scope: Throughout, we work under the standard stabilizability assumption: our results apply whenever there exists a gain K such that $A-BK$ is Schur; full controllability is not required.

Outline: Section 2 reviews necessary background on ideals, statistical convergence, fuzzy norms, and paranormed spaces. In Section 3, we develop the formal definitions of $I_2$-convergence in fuzzy paranormed spaces and establish key theorems (uniqueness of limits, equivalence of $I_2$-Cauchy and convergence under $\left(A{P}_2\right)$, etc.). Section 4 is devoted to the fuzzy control system application: we describe the plant, the fuzzy feedback law, define the lacunary dropout ideal, and prove the lacunary statistical stability theorem with a proof. In Section 5, we conclude with a discussion of the results and suggest directions for future research, including potential extensions to stochastic dropouts and nonlinear systems.

2. Preliminaries

This section provides definitions and concepts that will be used throughout the paper. We cover (i) ideals and filters in $\mathbb{N}$ and ${\mathbb{N}}^2$, (ii) convergence notions (Pringsheim convergence for double sequences, statistical convergence, and ideal convergence), (iii) fuzzy numbers and the supremum metric, (iv) fuzzy normed linear spaces (in the sense of Bag–Samanta), and (v) paranormed and fuzzy paranormed spaces. For the reader’s convenience, we summarize key properties without delving into full detail when well established in the literature.

Definition 1

(Ideals and Filters on $\mathbb{N}$). A nonempty family $I\subseteq 2^{\mathbb{N}}$ of subsets of $\mathbb{N}$ is called an ideal on $\mathbb{N}$ if:

(i) 

$A,B\in I$ implies $A\cup B\in I$ (additivity),

(ii) 

$A\in I$ and $B\subseteq A$ implies $B\in I$ (heredity).

In addition, an ideal I is called proper if $I\ne 2^{\mathbb{N}}$ (equivalently, $\mathbb{N}\notin I$), and I is called non-trivial or admissible if it is proper and also $\left\{n\right\}\in I$ for every single index $n\in \mathbb{N}$.

Given an ideal I on $\mathbb{N}$, one can associate a dual notion of a filter. The filter associated with I is

$$F\left(I\right)=\left\{K\subseteq \mathbb{N}:\mathbb{N}\setminus K\in I\right\},$$

which is easily verified to satisfy the filter axioms (closed under finite intersections and supersets). In particular, $F\left(I\right)$ consists of the “large” sets (those whose complements lie in the ideal I of “small” sets).

Definition 2

(Admissible ideal on $\mathbb{N}$). Let $I\subseteq 2^{\mathbb{N}}$ be a proper ideal on $\mathbb{N}$. We say I is admissible if

$$\left\{n\right\}\in Iforeveryn\in \mathbb{N}.$$

Equivalently, I is non-trivial (proper) and contains all singletons.

Definition 3

(Natural density and statistical convergence). The natural density of a set $E\subseteq \mathbb{N}$ is defined (when the limit exists) by

$$\delta \left(E\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\{k\in E:k\le n\}\right|,$$

i.e., the limit of the proportion of integers $1,2,\dots ,n$ that lie in E. For example, $\delta \left(E\right)=0$ if E is finite, and $\delta \left(\mathbb{N}\right)=1$.

Now, let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a sequence of points in a metric space $(X,d)$. We say $\left(x_n\right)$ is statistically convergent to $L\in X$ if for every $\epsilon >0$, the set

$$E\left(\epsilon \right)=\{n\in \mathbb{N}:d(x_n,L)\ge \epsilon \}$$

has natural density zero. In this case, we write

$$\mathrm{st}-\underset{n\to \infty }{lim}{x}_n=L.$$

Equivalently, $x_n\to L$ statistically means that for each $\epsilon >0$,

$${\displaystyle \frac{1}{m}}\left|\{n\le m:d(x_n,L)\ge \epsilon \}\right|⟶0\mathrm{as}m\to \infty .$$

This concept, introduced by Fast and Steinhaus [5,6], generalizes the usual notion of limit by ignoring a sparse set of deviations. Note that statistical convergence can be viewed as convergence with respect to the ideal

$$I_{\mathrm{stat}}=\{A\subseteq \mathbb{N}:\delta \left(A\right)=0\},$$

the ideal of zero-density sets.

Definition 4

(Ideal convergence). Let I be a proper ideal on $\mathbb{N}$, and let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a sequence in a metric space $(X,d)$. We say $\left(x_n\right)$ is I-convergent to $L\in X$ if for every $\epsilon >0$, the set

$$\{n\in \mathbb{N}:d(x_n,L)\ge \epsilon \}$$

belongs to the ideal I. In this case, we write

$$I-\underset{n\to \infty }{lim}{x}_n=L,$$

and call L the I-limit of $\left(x_n\right)$.

Clearly, when $I=I_{\mathrm{fin}}$ is the ideal of finite sets, I-convergence reduces to the usual notion of convergence (since eventually all large n satisfy $d(x_n,L)<\epsilon$). When $I=I_{\mathrm{stat}}$ is the ideal of density-zero sets, I-convergence coincides with statistical convergence. Thus, ideal convergence provides a unifying framework for various summability notions.

Moreover, if I is an admissible ideal (containing all singletons) with the property (AP), then one can show that I-limits enjoy many nice properties such as uniqueness and agreement with cluster points. We also mention the related concept of adjoint ideal convergence (or $I^{★}$-convergence), introduced in the literature, which uses the associated filter $F\left(I\right)$ and subsequences.

Definition 5

(Double sequences and Pringsheim convergence). A double sequence is a function $x:\mathbb{N}\times \mathbb{N}\to X$ from the Cartesian product of the positive integers into some space X. We denote a double sequence by ${\left(x_{ij}\right)}_{i,j\in \mathbb{N}}$ or simply $\left(x_{ij}\right)$. The double sequence is said to converge to $\mathsf{\ell }\in X$ in Pringsheim’s sense if for every $\epsilon >0$ there exists an $N\in \mathbb{N}$ such that

$$d(x_{ij},\mathsf{\ell })<\epsilon foralli\ge N,j\ge N.$$

where $d(·,·)$ is the metric (or some appropriate distance) on X. In other words, ${lim}_{i,j\to \infty }{x}_{ij}=\mathsf{\ell }$ in the sense that for any fixed tolerance ε, all pairs of indices beyond some threshold N give sequence values within ε of ℓ. We will also refer to this as convergence along the direction $(i,j)\to (\infty ,\infty )$. If $\left(x_{ij}\right)$ converges to ℓ in this sense, we write $x_{ij}\to \mathsf{\ell }$ (as $i,j\to \infty$).

Definition 6

(Strongly admissible ideal on $\mathbb{N}\times \mathbb{N}$). Let $I_2\subseteq 2^{\mathbb{N}\times \mathbb{N}}$ be a proper ideal. We say $I_2$ is strongly admissible if

(1)

$\left\{(k,\mathsf{\ell })\right\}\in I_2$ for every $(k,\mathsf{\ell })\in \mathbb{N}\times \mathbb{N}$;

(2)

for each fixed $k_0\in \mathbb{N}$, the “vertical line” $\{(k_0,\mathsf{\ell }):\mathsf{\ell }\in \mathbb{N}\}$ does not belong to $I_2$;

(3)

for each fixed ${\mathsf{\ell }}_0\in \mathbb{N}$, the “horizontal line” $\{(k,{\mathsf{\ell }}_0):k\in \mathbb{N}\}$ does not belong to $I_2$.

In other words, $I_2$ contains every singleton but no entire row or column, so that each coordinate “goes to infinity” in the sense of the associated filter.

Definition 7

(Double statistical convergence). Let $S\subseteq \mathbb{N}\times \mathbb{N}$. The double natural density of S is defined (when the limit exists) by

$${\delta }_2\left(S\right)=\underset{m,n\to \infty }{lim}{\displaystyle \frac{1}{mn}}\left|\left\{\right(i,j)\in S:i\le m,j\le n\}\right|.$$

We say a double sequence ${\left(x_{ij}\right)}_{i,j\in \mathbb{N}}$ in a metric space $(X,d)$ is statistically convergent to $\mathsf{\ell }\in X$ if for every $\epsilon >0$,

$${\delta }_2\left(\{(i,j)\in {\mathbb{N}}^2:d(x_{ij},\mathsf{\ell })\ge \epsilon \}\right)=0.$$

In this case, we write

$$x_{ij}\stackrel{\mathrm{st}}{\to }\mathsf{\ell }.$$

This notion, introduced by Mursaleen and Edely [9], reduces to ordinary statistical convergence of single-index sequences (Definition 3) when restricted to diagonals, and to Pringsheim convergence when the exceptional set is required to be finite.

Definition 8

(Ideal convergence for double sequences). One can similarly define ideal convergence for double sequences. Let $I_2$ be an ideal on $\mathbb{N}\times \mathbb{N}$ (usually assumed admissible, i.e., containing all singletons $\left\{\right(i,j\left)\right\}$, and often satisfying a two-dimensional analog of the AP property). A double sequence ${\left(x_{ij}\right)}_{i,j\in \mathbb{N}}$ in a metric space $(X,d)$ is said to be $I_2$-convergent to $\mathsf{\ell }\in X$ if for every $\epsilon >0$,

$$\{(i,j)\in {\mathbb{N}}^2:d(x_{ij},\mathsf{\ell })\ge \epsilon \}\in I_2.$$

In this case, we write

$$x_{ij}\to \mathsf{\ell }\left(I_2\right).$$

This notion reduces to the single-index ideal convergence when $I_2=I\times I$ on the product index set or if $\left(x_{ij}\right)$ actually depends on only one index. Tripathy and Tripathy [11] introduced I-convergence for double sequences and studied properties such as solidity and completeness in that setting. In our work, $I_2$ will typically be either the product ideal on each coordinate or a two-dimensional generalization of the density-zero ideal. We will explicitly define and use $I_2$-convergence in fuzzy paranormed spaces in Section 3.

Definition 9

(Fuzzy number [14,30]). A mapping $u:\mathbb{R}\to [0,1]$ is called a fuzzy number if it satisfies

(i)

Normality: $\exists t_0\in \mathbb{R}$ such that $u\left(t_0\right)=1$.

(ii)

Convexity: $u(\lambda s+(1-\lambda \left)r\right)\ge min\left\{u\right(s),u(r\left)\right\}$ for all $s,r\in \mathbb{R},\lambda \in [0,1]$.

(iii)

Upper–semicontinuity: For every $a\in [0,1]$ the set $\{t\in \mathbb{R}:u(t)\ge a\}$ is closed.

(iv)

Compact support: $suppu:=\overline{\{t:u(t)>0\}}$ is compact in $\mathbb{R}$.

The collection of all fuzzy numbers is denoted $L\left(\mathbb{R}\right)$.

Definition 10

($\alpha$-level (cut)). For $u\in L\left(\mathbb{R}\right)$ and $\alpha \in [0,1]$, the α-level set is

$${\left[u\right]}_{\alpha }:=\left\{\begin{array}{cc}\{t\in \mathbb{R}:u(t)\ge \alpha \},\hfill & 0<\alpha \le 1,\hfill \\ suppu,\hfill & \alpha =0.\hfill \end{array}\right.$$

Whenever ${\left[u\right]}_{\alpha }$ is an interval, we write ${\left[u\right]}_{\alpha }=[u_{\alpha }^{-},u_{\alpha }^{+}]$.

Definition 11

(Supremum metric [31]). For $u,v\in L\left(\mathbb{R}\right)$, define

$$D(u,v):=\underset{0\le \alpha \le 1}{sup}max\left\{|u_{\alpha }^{-}-v_{\alpha }^{-}|,|u_{\alpha }^{+}-v_{\alpha }^{+}|\right\}.$$

The metric space $\left(L\left(\mathbb{R}\right),D\right)$ is complete.

Definition 12

(Continuous t-norm/t-conorm [32]). A mapping $T:{[0,1]}^2\to [0,1]$ is a continuous t-norm if it is commutative, associative, non-decreasing in each variable, continuous, and has 1 as the neutral element: $T(a,1)=a$. Its dual $S:{[0,1]}^2\to [0,1]$, given by $S(a,b)=1-T(1-a,1-b)$, is a continuous t-conorm with 0 as neutral element.

Definition 13

(Fuzzy normed linear space (Bag–Samanta)). Let X be a real vector space and let $T:{[0,1]}^2\to [0,1]$ be a continuous t-norm (i.e., commutative, associative, continuous, with neutral element 1). A fuzzy norm on X (in the sense of Bag and Samanta [33]) is a function

$$N:X\times (0,\infty )⟶[0,1]$$

satisfying for all $x,y\in X$, all $c\in \mathbb{R}\setminus \left\{0\right\}$, and all $t,s>0$:

(FN1) 

$N(x,t)>0$ for every $t>0$, and ${\displaystyle \underset{t\to \infty }{lim}N(x,t)=1}$. Moreover, $N(x,t)=1$ for all $t>0$ if and only if $x=0$.

(FN2) 

$N(cx,t)=N\left(x,\frac{t}{\left|c\right|}\right)$. In particular, $N(0,t)=1$ for all $t>0$.

(FN3) 

$N(x+y,t+s)\ge T\left(N(x,t),N(y,s)\right)$.

(FN4) 

For each fixed $x\in X$, the map $t\mapsto N(x,t)$ is continuous on $(0,\infty )$, and for each fixed $t>0$, the map $x\mapsto N(x,t)$ is (fuzzy) continuous in the sense of fuzzy topology.

The triple $(X,N,T)$ is then called a fuzzy normed linear space. Intuitively, $N(x,t)$ measures the degree to which x belongs to the “fuzzy ball” of radius t around the origin, with larger t yielding larger membership values. The condition (FN3) generalizes the triangle inequality via the chosen t-norm T.

Every fuzzy normed space $(X,N,T)$ induces a topology and a notion of sequence convergence. In fact, for each $x\in X$ and $\alpha \in (0,1)$, one considers the “$\alpha$-level set”

$$\{y\in X:N(x-y,t)\ge 1-\alpha \}$$

as a neighborhood of x. The topology generated by these fuzzy balls is Hausdorff and first countable.

Definition 14

(Convergence and Cauchy sequences in $(X,N,T)$). A sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in X is said to converge to $L\in X$ (write $x_n\to L$ in $(X,N)$) if for every $\epsilon >0$ and $\lambda \in (0,1)$ there exists $N_0\in \mathbb{N}$ such that

$$N(x_n-L,\epsilon )>1-\lambda foralln\ge N_0.$$

Equivalently,

$$\underset{n\to \infty }{lim}N(x_n-L,\epsilon )=1forevery\epsilon >0.$$

Similarly, $\left(x_n\right)$ is called Cauchy in the fuzzy norm if for every $\epsilon >0$ and $\lambda \in (0,1)$ there exists $N_1\in \mathbb{N}$ such that

$$N(x_m-x_n,\epsilon )>1-\lambda forallm,n\ge N_1.$$

It can be checked that this notion of convergence is compatible with the topology generated by the fuzzy balls, and it generalizes the usual norm convergence $|x_n-L|\to 0$. Every fuzzy normed space is a complete fuzzy metric space, but may not be complete (i.e., every fuzzy-Cauchy sequence converges in X) without further assumptions.

Definition 15

(Paranormed space). Let X be a real (or complex) linear space. A function $\rho :X\to [0,\infty )$ is called a paranorm on X if the pair $(X,\rho )$ satisfies for all $x,y\in X$ and all scalars λ:

(P1) 

$\rho \left(x\right)\ge 0$, and $\rho \left(x\right)=0$ if and only if $x=0$. (Non-negativity and $T_1$ axiom.)

(P2) 

$\rho (-x)=\rho \left(x\right)$. (Symmetry.)

(P3) 

$\rho (x+y)\le \rho \left(x\right)+\rho \left(y\right)$. (Triangle inequality, subadditivity.)

(P4) 

If ${\lambda }_n\to \lambda$ in the base field and $\rho (x_n-x)\to 0$ in X, then

$$\rho ({\lambda }_n{x}_n-\lambda x)⟶0.$$

(Sequential continuity under scalar multiplication.)

Then, ρ is a paranorm and $(X,\rho )$ is called a paranormed space. Paranormed spaces generalize normed spaces by replacing exact homogeneity with the weaker continuity condition (P4). Every norm is a paranorm, and on finite-dimensional spaces every paranorm arises from a norm; however, in infinite dimensions, there are paranorms not induced by any norm.

Definition 16

(Fuzzy paranormed space [23]). Let T be a continuous t-norm and S its dual t-conorm. A function $\wp :X\times (0,\infty )\to [0,1]$ makes $(X,\wp ,T,S)$ a fuzzy paranormed space if, for all $x,y\in X$, $t,s>0$, and scalars $\lambda \in \mathbb{R}$,

(FP1)

$\wp (x,t)=1\mathrm{if}x=\theta$;

(FP2)

$\wp (-x,t)=\wp (x,t)$;

(FP3)

$T\left(\wp (x,t),\wp (y,s)\right)\le \wp (x+y,t+s)\le S\left(\wp (x,t),\wp (y,s)\right)$;

(FP4)

$t\mapsto \wp (x,t)$ is non-decreasing and ${\displaystyle \underset{t\to \infty }{lim}\wp (x,t)=1}$;

(FP5)

$\left|\lambda \right|\le 1\Rightarrow \wp (\lambda x,t)\ge \wp (x,t)$, $\left|\lambda \right|\ge 1\Rightarrow \wp (\lambda x,t)\le \wp (x,t)$;

(FP6)

if ${\lambda }_n\to \lambda$ and $\wp (x_n-x,t)\to 1$, then $\wp ({\lambda }_n{x}_n-\lambda x,t)\to 1$.

A fuzzy paranorm ℘ is called a totally fuzzy paranorm if the implication

$$\wp (x,t)=1⟺x=\theta$$

also holds for all $t>0$. In this case, the space $(X,\wp ,T,S)$ is called a totally fuzzy paranormed space.

Remark 1.

In Definition 16, the t-norm T provides the usual lower bound in (FP3), whereas the dual t-conorm S supplies a symmetric upper bound. This two–sided estimate generalises the triangle inequality of Bag–Samanta fuzzy norms (which use only T), allowing greater flexibility when the “length” of vectors is measured fuzzily rather than crisply. See [23] for a full discussion.

Every fuzzy normed space is a fuzzy paranormed space (just take T for both bounds and enforce FP3 as equality to recover FN3, and FP5 as equality for all $\lambda$), so this is a genuine generalization. The benefit of fuzzy paranorms is that they allow analyzing convergence and completeness in scenarios where scaling behavior is not linear, providing a finer tool in sequence space theory. In Section 3, we work within a fuzzy paranormed space as the ambient space for our double sequences of fuzzy numbers.

By fixing the fuzzy paranorm ℘ and an ideal $I_2$, we can define what it means for a double sequence of fuzzy numbers (or fuzzy vectors) to converge ideal-fuzzily to a limit. In preparation, note that if ${\left\{{\tilde{x}}_{k\mathsf{\ell }}\right\}}_{k,\mathsf{\ell }\in \mathbb{N}}\subset L\left(\mathbb{R}\right)$ are fuzzy numbers and $\tilde{y}\in L\left(\mathbb{R}\right)$ (or, more generally, $\tilde{y}$ is a crisp vector embedded as a degenerate fuzzy number), then

$$\wp \left({\tilde{x}}_{k\mathsf{\ell }}-\tilde{y},t\right)$$

represents the membership grade that “${\tilde{x}}_{k\mathsf{\ell }}$ is within t of $\tilde{y}$’’ in the fuzzy sense. The difference ${\tilde{x}}_{k\mathsf{\ell }}-\tilde{y}$ is defined level-wise; that is,

$${\left[{\tilde{x}}_{k\mathsf{\ell }}-\tilde{y}\right]}_{\alpha }={\left[{\tilde{x}}_{k\mathsf{\ell }}\right]}_{\alpha }-{\left[\tilde{y}\right]}_{\alpha }=\left[x_{k\mathsf{\ell },\alpha }^{-}-y_{\alpha }^{+},x_{k\mathsf{\ell },\alpha }^{+}-y_{\alpha }^{-}\right],0\le \alpha \le 1,$$

where $x_{k\mathsf{\ell },\alpha }^{\pm }$ and $y_{\alpha }^{\pm }$ denote the endpoints of the corresponding $\alpha$-cuts. We will use this construction when comparing a fuzzy state measurement ${\tilde{x}}_{k\mathsf{\ell }}$ with the true (crisp) state value $x_{k\mathsf{\ell }}$, the latter being embedded as the degenerate fuzzy number ${\delta }_{x_{k\mathsf{\ell }}}$ whose every $\alpha$-cut collapses to $\left\{x_{k\mathsf{\ell }}\right\}$.

3. Ideal Convergence of Double Sequences in Fuzzy Paranormed Spaces

We now present the central theoretical development of the paper. All convergence and Cauchy notions here are with respect to the fuzzy paranorm ℘ in a fixed fuzzy paranormed space $(\mathfrak{X},\wp ,T,S)$. Throughout, let $I_2\subseteq 2^{\mathbb{N}\times \mathbb{N}}$ be a non-trivial admissible ideal on the index set $\mathbb{N}\times \mathbb{N}$ (and in fact we assume $I_2$ is strongly admissible as in Definition 6, so that each index tends to infinity along filter sets). Intuitively, $I_2$ designates which sets of index pairs will be “negligible” in our convergence criteria.

Definition 17

($I_2$-fp-convergence). Let $(\mathfrak{X},\wp ,T,S)$ be a fuzzy–paranormed space and let $I_2$ be a non-trivial admissible ideal on $\mathbb{N}\times \mathbb{N}$. A double sequence $x={\left(x_{k\mathsf{\ell }}\right)}_{k,\mathsf{\ell }\in \mathbb{N}}\subset \mathfrak{X}$ is said to be $I_2$-fp-convergent to $x_0\in \mathfrak{X}$, denoted

$$x_{k\mathsf{\ell }}\underset{I_2\text{-}fp}{\to }{x}_0(i.e.,I_2\text{-}fp-\underset{(k,\mathsf{\ell })\to \infty }{lim}{x}_{k\mathsf{\ell }}=x_0),$$

if for every $\epsilon \in (0,1)$ and $t>0$,

$$A_{t,\epsilon }:=\left\{(k,\mathsf{\ell })\in {\mathbb{N}}^2:\wp (x_{k\mathsf{\ell }}-x_0,t)<1-\epsilon \right\}\in I_2.$$

Equivalently, for each fixed $t>0$,

$$I_2\text{-}fp-\underset{(k,\mathsf{\ell })\to \infty }{lim}\wp (x_{k\mathsf{\ell }}-x_0,t)=1.$$

Because $t\mapsto \wp (x,t)$ is non-decreasing (FP4), it suffices to verify this limit along any sequence $t_n\to \infty$.

Remark 2.

This notion generalizes several familiar cases:

• If $I_2$ consists only of finite sets (and is admissible), then $I_2$-fp-convergence reduces to ordinary Pringsheim convergence in the fuzzy paranorm, i.e., for every ε there exists $n_0$ such that $\wp (x_{k\mathsf{\ell }}-x_0,t)>1-\epsilon$ for all $k,\mathsf{\ell }\ge n_0$.
• If $I_2=I_2^{\left(\delta \right)}$ (the double statistical ideal), we obtain double statistical convergence in the fuzzy paranorm: for each t, the set of $(k,\mathsf{\ell })$ with $\wp (x_{k\mathsf{\ell }}-x_0,t)$ not close to 1 has double natural density 0 in ${\mathbb{N}}^2$.
• In our application, $I_2$ will be chosen so that $I_2$-fp-convergence corresponds to lacunary statistical convergence along the measurement sequence.

By the monotonicity axiom (FP4), if the condition holds for some $t=t_0$, it automatically holds for all $t>t_0$ (since increasing t can only increase $\wp (x_{k\mathsf{\ell }}-x_0,t)$). Hence, to verify $I_2$-fp-convergence, it suffices to check the definition on an increasing sequence of radii $t_n\to \infty$ (for instance $t_n=n$).

Definition 18

($I_2^{\ast }$-fp-convergence in a fuzzy-paranormed space). Let $(\mathfrak{X},\wp ,T,S)$ be a fuzzy–paranormed space and $I_2$ a non-trivial admissible ideal on $\mathbb{N}\times \mathbb{N}$. A double sequence $x={\left(x_{k\mathsf{\ell }}\right)}_{k,\mathsf{\ell }\in \mathbb{N}}\subset \mathfrak{X}$ is said to be $I_2^{\ast }$-fp-convergent to $x_0\in \mathfrak{X}$, denoted

$$x_{k\mathsf{\ell }}\underset{I_2^{\ast }\text{-}fp}{\to }{x}_0,\left(I_2^{\ast }\text{-}fp-\underset{(k,\mathsf{\ell })\to \infty }{lim}{x}_{k\mathsf{\ell }}=x_0\right),$$

if there exists a set $M\in F\left(I_2\right)$ (the filter dual to $I_2$) such that the Pringsheim limit of $x_{k\mathsf{\ell }}$ over $(k,\mathsf{\ell })\in M$ is $x_0$, i.e.,

$$\underset{\begin{array}{c}(k,\mathsf{\ell })\to (\infty ,\infty )\\ (k,\mathsf{\ell })\in M\end{array}}{lim}\wp \left(x_{k\mathsf{\ell }}-x_0,t\right)=1foreveryt>0.$$

Definition 19

($I_2$-Cauchy in a fuzzy–paranormed space). Let $(\mathfrak{X},\wp ,T,S)$ be a fuzzy-paranormed space and $I_2$ a non-trivial admissible ideal on $\mathbb{N}\times \mathbb{N}$. A double sequence $\left(x_{k\mathsf{\ell }}\right)$ is $I_2$-Cauchy if for every $\epsilon \in (0,1)$ and $t>0$ there exist indices $(m_0,n_0)\in {\mathbb{N}}^2$ such that

$$\left\{(k,\mathsf{\ell })\in {\mathbb{N}}^2:\wp (x_{k\mathsf{\ell }}-x_{m_0{n}_0},t)<1-\epsilon \right\}\in I_2.$$

Theorem 1

(Uniqueness of the $I_2$-fp limit). Let $(\mathfrak{X},\wp ,T,S)$ be a totally fuzzy-paranormed space, and let $I_2$ be a non-trivial admissible ideal on ${\mathbb{N}}^2$. If a double sequence $x={\left(x_{k\mathsf{\ell }}\right)}_{k,\mathsf{\ell }\in \mathbb{N}}\subset \mathfrak{X}$ is $I_2$-fp-convergent to both $L,L^{\prime }\in \mathfrak{X}$, then $L=L^{\prime }$. Hence, the $I_2$-fp limit, when it exists, is unique.

In general, topological vector spaces, a notion of convergence might not guarantee Cauchy sequences converge (unless the space is complete or the convergence is linear). However, for ideal convergence there is a useful property analogous to completeness: if the ideal $I_2$ satisfies the amalgamation property $\left(A{P}_2\right)$ (roughly meaning that given a countable collection of sets in $I_2$, one can find a single set in $I_2$ that covers “almost all” of each—formally, for any sequence $A_1,A_2,\dots \in I_2$ there is $B\in I_2$ such that $A_n\setminus B$ is finite for all n), then $I_2$-Cauchy sequences imply the existence of an $I_2$-limit in complete spaces. This is analogous to the standard fact that in a complete metric space, Cauchy sequences converge.

Definition 20

(Amalgamation property $\left(A{P}_2\right)$). Let ${\mathcal{I}}_2\subset 2^{\mathbb{N}\times \mathbb{N}}$ be an admissible ideal and ${\mathcal{I}}_2^0$ denote the family of sets contained in a finite union of rows and columns of $\mathbb{N}\times \mathbb{N}$. We say that ${\mathcal{I}}_2$ has the amalgamation property $\left(A{P}_2\right)$ if for every countable family of pairwise disjoint sets ${\left\{E_j\right\}}_{j\ge 1}\subset {\mathcal{I}}_2$ there exist sets ${\left\{F_j\right\}}_{j\ge 1}$ such that $E_j\Delta F_j\in {\mathcal{I}}_2^0$ for all j, and $F:={\bigcup }_{j=1}^{\infty }{F}_j\in {\mathcal{I}}_2$.

Theorem 2

(Equivalence of $I_2$-fp-Cauchy and $I_2$-fp Convergence under $\left(A{P}_2\right)$). Let $I_2$ be an admissible ideal on ${\mathbb{N}}^2$ satisfying the two-dimensional AP property $\left(A{P}_2\right)$, and let $(\mathfrak{X},\wp )$ be a complete fuzzy paranormed space. A double sequence $x=\left(x_{k\mathsf{\ell }}\right)$ in $\mathfrak{X}$ is $I_2$-fp-convergent if and only if it is $I_2$-fp-Cauchy. Hence, every $I_2$-fp-Cauchy sequence has an $I_2$-fp limit in $\mathfrak{X}$.

Remark 3

(Why $\left(A{P}_2\right)$ matters in our setting). Theorem 2 shows that, once the ideal $I_2$ satisfies the amalgamation property $\left(A{P}_2\right)$, our convergence notion behaves as expected: every $I_2$-fp-Cauchy double sequence must admit an $I_2$-fp limit. In the absence of $\left(A{P}_2\right)$, Cauchy behavior would not guarantee convergence. In Section 4, we work with a lacunary ideal

$$I_2^{\mathrm{lac}}=I_{\mathrm{fin}}\cup \left\{dropoutindexset\right\},$$

i.e., the ideal generated by the packet-dropout pattern together with all finite subsets of ${\mathbb{N}}^2$. Because the dropout indices are “spread out” (lacunary) and each finite set is negligible, one verifies easily that $I_2^{\mathrm{lac}}$ satisfies $\left(A{P}_2\right)$. Hence, in our application, it suffices to prove that the measurement sequence is $I_2$-fp-Cauchy; Theorem 2 then yields the required $I_2$-fp convergence. In control terms, this means the state is eventually confined to an arbitrarily small fuzzy neighbourhood, i.e., practical stability is achieved despite dropouts.

Definition 21

($I_2$-fp limit point). Let $(\mathfrak{X},\wp ,T,S)$ be a fuzzy–paranormed space and $I_2$ an admissible ideal on ${\mathbb{N}}^2$ with associated filter $F\left(I_2\right)$. A value $z\in \mathfrak{X}$ is called an $I_2$–fp limit point of the double sequence $x={\left(x_{k\mathsf{\ell }}\right)}_{k,\mathsf{\ell }\in \mathbb{N}}\subset \mathfrak{X}$ if there exists a set $M\in F\left(I_2\right)$ such that

$$\underset{\begin{array}{c}(k,\mathsf{\ell })\to (\infty ,\infty )\\ (k,\mathsf{\ell })\in M\end{array}}{lim}\wp \left(x_{k\mathsf{\ell }}-z,t\right)=1foreveryt>0.$$

Equivalently, along the co-ideal subsequence indexed by M, $x_{k\mathsf{\ell }}$ ordinarily (Pringsheim) converges to z in the fuzzy paranorm.

Definition 22

($I_2$-fp cluster point). With the same setting, a point $y\in \mathfrak{X}$ is an $I_2$-fp cluster point of x if for every $\epsilon \in (0,1)$ and $t>0$ the index set

$$\left\{(k,\mathsf{\ell })\in {\mathbb{N}}^2:\wp (x_{k\mathsf{\ell }}-y,t)\ge 1-\epsilon \right\}\in F\left(I_2\right).$$

Denote the family of all such cluster points by ${\Gamma }_{I_2}^{\mathrm{FP}}\left(x\right)$.

Under the AP property $\left(A{P}_2\right)$ of the ideal $I_2$, the picture parallels classical analysis: a double sequence is $I_2$-fp convergent iff it is $I_2$-fp Cauchy and its cluster-point set ${\Gamma }_{I_2}^{\mathrm{FP}}\left(x\right)$ contains exactly one element. In Section 4, we choose a lacunary ideal generated by the dropout pattern; this ideal satisfies $\left(A{P}_2\right)$. Consequently, showing that the fuzzy measurement sequence is $I_2$-fp Cauchy will be enough to guarantee convergence (hence practical stability) despite sparse packet losses.

Remark 4

(Control meaning of the $I_2$-fp limit). If the measurement double sequence admits an $I_2$-fp limit (e.g., the origin) and $A_c:=A-BK$ is Schur, then outside an $I_2$-negligible set of indices, the centroid errors enter any prescribed fuzzy ball. Combined with the nominal exponential decay of $A_c$, this yields uniform ultimate boundedness (practical stability) of the closed loop, in the classical control sense (see, e.g., standard texts on nonlinear control) [34,35]. Thus, the $I_2$-fp limit is the analytic vehicle that underpins the practical stability statement made precise in Theorem 3.

4. Application to Discrete Fuzzy Control Under Lacunary Measurements

This section applies the $I_2$-fp convergence framework developed in Section 3 to a networked control scenario with sporadic sensor dropouts. We study a linear time-invariant (LTI) plant steered by a fuzzy-logic state-feedback controller. At each sampling instant, the sensor returns a fuzzy measurement of the state; occasionally no packet is received, producing a lacunary sequence of missing data. Our aim is to show that, provided these dropouts are sufficiently sparse and the stream of fuzzy readings is $I_2$-fp Cauchy, the closed loop remains practically stable.

To make the paper self-contained, Section 4.1 formalizes the plant, the fuzzy measurement model, and the dropout pattern. We then embed the measurements into the fuzzy-paranormed space $(\mathfrak{X},\wp ,T,S)$ introduced earlier and construct a strongly admissible lacunary ideal that treats the dropout indices as negligible. Section 4.3 states and proves the main Lacunary Statistical Stability Theorem, combining the ideal convergence machinery with a Lyapunov-type argument. A brief discussion relates the result to classical density-based dropout conditions.

4.1. System Description

4.1.1. Plant

We consider the discrete-time LTI system

$$x_{k+1}=A{x}_k+B{u}_k,k\in \mathbb{N},$$

(1)

with state $x_k\in {\mathbb{R}}^n$ and input $u_k\in {\mathbb{R}}^m$. Throughout, we assume that the pair $(A,B)$ is stabilizable, i.e., there exists $K\in {\mathbb{R}}^{m\times n}$ such that the closed-loop matrix $A-BK$ is Schur; equivalently, $\parallel A-BK\parallel <1$ for some matrix norm. The constant gain K will be fixed for the rest of the section.

Remark 5

(Controllability vs. stabilizability). Our results only require stabilizability (not full controllability). All statements in Section 4 are formulated for systems for which a gain K exists such that $A_c:=A-BK$ is Schur. Hence, the framework and the main theorem apply directly to stabilizable plants.

4.1.2. Fuzzy Measurements

At each time k the sensor returns a fuzzy vector ${\tilde{x}}_k\in L\left({\mathbb{R}}^n\right)$ whose $\alpha$-cuts are modeled as

$${\left[{\tilde{x}}_k\right]}_{\alpha }=\left[x_k+{\beta }_k-{\delta }_k^{-}\left(\alpha \right),x_k+{\beta }_k+{\delta }_k^{+}\left(\alpha \right)\right],0<\alpha \le 1,$$

(2)

while ${\left[{\tilde{x}}_k\right]}_0=supp{\tilde{x}}_k$. Here ${\beta }_k:=c\left({\tilde{x}}_k\right)-x_k$ is the centroid bias and ${\delta }_k^{-},{\delta }_k^{+}:[0,1]\to [0,\infty )$ allow for asymmetric $\alpha$-cuts, which is standard in fuzzy modeling (e.g., asymmetric triangular/trapezoidal numbers) [3,36]. The radius function ${\delta }_k:[0,1]\to [0,\infty )$ captures measurement uncertainty. We allow a (possibly time-varying) centroid bias ${\beta }_k:=c\left({\tilde{x}}_k\right)-x_k$. The baseline unbiased case ${\beta }_k\equiv 0$ is covered as a special case. In the analysis below it suffices that $\parallel {\beta }_k\parallel$ is uniformly bounded and $I_2$-small (cf. Definition 23); in practice, this corresponds to standard sensor calibration to minimize bias while using centroid defuzzification for control [3,4].

Remark 6

(Asymmetric $\alpha$-cuts are standard). Triangular/trapezoidal fuzzy numbers are often asymmetric; thus, ${\delta }_k^{-}\ne {\delta }_k^{+}$ is natural in sensing models [3,36]. Our analysis only uses centroid and boundedness, not symmetry.

4.1.3. Embedding into a Fuzzy Paranorm

Let $(\mathfrak{X},\wp ,T,S)$ be the fuzzy paranormed space constructed in Section 2. Following [23], we embed $L\left({\mathbb{R}}^n\right)$ via

$$\wp (\tilde{z},t):=exp\left(-\parallel c\left(\tilde{z}\right)\parallel /t\right),\tilde{z}\in L\left({\mathbb{R}}^n\right),t>0.$$

One readily checks properties (FP1)–(FP6); the paranorm essentially reduces fuzzy comparison to the crisp distance between centroids, which is adequate because every ${\tilde{x}}_k$ is centered at the true state.

4.1.4. Dropout Pattern

A dropout occurs if no packet is received at time k. Fix a strictly increasing sequence $\theta ={\left\{n_r\right\}}_{r\ge 1}\subset \mathbb{N}$ with a minimum inter-dropout gap

$$n_{r+1}-n_r\ge q_0>0.$$

No monotone or unbounded growth of the gaps is required by our proofs; this matches the dwell-time intuition in switched/NCS settings [29,37]. We define the measurement stream with dropouts as

$${\tilde{z}}_k:=\left\{\begin{array}{cc}{\tilde{x}}_k,\hfill & k\notin \theta ,\hfill \\ \mathbf{0},\hfill & k\in \theta ,\hfill \end{array}\right.$$

where $\mathbf{0}$ denotes the degenerate fuzzy number at the origin.

4.1.5. Ideal Generated by Dropouts

Let

$$S_{\theta }:=\{(k,\mathsf{\ell })\in \mathbb{N}\times \{1,\dots ,n\}:k\in \theta \}.$$

Denote by

$$I_2:=I_{\mathrm{fin}}\cup \left\{S_{\theta }\right\},$$

the smallest ideal of subsets of ${\mathbb{N}}^2$ that contains both all finite sets and $S_{\theta }$; that is, $I_2$ is closed under taking subsets and finite unions. Because $\theta$ is lacunary, this strongly admissible ideal enjoys the amalgamation property $\left(A{P}_2\right)$ (see Remark 3).

Lemma 1.

Let $\theta \subset \mathbb{N}$ be the (possibly lacunary) set of dropout instants and $S_{\theta }:=\{(k,\mathsf{\ell })\in \mathbb{N}\times \{1,\dots ,n\}:k\in \theta \}$. Let ${\mathcal{I}}_2$ be the smallest strongly admissible ideal on ${\mathbb{N}}^2$ that contains ${\mathcal{I}}_{\mathrm{fin}}$ and $S_{\theta }$ (i.e., it is generated by finite sets and the “row–strip” $S_{\theta }$). Then, ${\mathcal{I}}_2$ satisfies $\left(A{P}_2\right)$.

Proof. 

Take any disjoint family $\left\{E_j\right\}\subset {\mathcal{I}}_2$. By generation, each $E_j\subset S_{\theta }\cup F_j^0$ with $F_j^0$ finite. Set $F_j:=E_j\cap S_{\theta }\subset S_{\theta }$. Then $E_j\Delta F_j\subset F_j^0\in {\mathcal{I}}_2^0$ for every j, and $F:={\bigcup }_j{F}_j\subset S_{\theta }\in {\mathcal{I}}_2$, hence $F\in {\mathcal{I}}_2$. By Definition 20, ${\mathcal{I}}_2$ has $\left(A{P}_2\right)$. □

Remark 7

(If $\left(A{P}_2\right)$ fails). The equivalence between $I_2$–fp–Cauchy and $I_2$–fp convergence may fail without $\left(A{P}_2\right)$, and in general $I_2$- and $I_2^{\ast }$-convergence need not coincide. This phenomenon is classical already in the one-dimensional setting: I- and $I^{\ast }$-convergence are equivalent iff the ideal has $\left(AP\right)$ [8]. For double sequences, $\left(A{P}_2\right)$ plays the same role [38,39,40]. Therefore, the results in this section that invoke $\left(A{P}_2\right)$ should be read as applying to ideals admitting the amalgamation property; without $\left(A{P}_2\right)$ one generally obtains only precompactness-type conclusions or $I_2^{\ast }$-statements (not full $I_2$-limits).

Remark 8.

Writing ${\tilde{x}}_k^{\left(\mathsf{\ell }\right)}$ for the ℓ–th component of ${\tilde{x}}_k$ turns the measurement stream into the double sequence $x_{k\mathsf{\ell }}:={\tilde{x}}_k^{\left(\mathsf{\ell }\right)}$. Hence, all $I_2$-fp concepts from Section 3 apply verbatim, with the dropout set $S_{\theta }$ treated as an $I_2$-small subset of indices.

Remark 9

(Centroid bias vs. perfect centering). Our proofs remain valid if the centroid is not perfectly centered at the true state. Let ${\beta }_k:=c\left({\tilde{x}}_k\right)-x_k$. Then, the closed-loop error term reads $e_k=c\left({\tilde{z}}_k\right)-x_k={\beta }_k$ for non-dropout instants and $e_k=-x_k$ at dropouts ($c\left({\tilde{z}}_{n_r}\right)=0$). Assuming ${sup}_k\parallel {\beta }_k\parallel \le \overline{b}$ and $\{k:\parallel {\beta }_k\parallel >0\}\in I_2$, Lemma 3 and Theorem 3 hold with constants inflated by an additive term proportional to $\overline{b}$, i.e.,

$$\sum _{j=0}^{k-1}\parallel A_c^{k-1-j}BK{e}_j\parallel \le C_2\underset{j<k}{sup}\parallel x_j\parallel +\underset{=:C_3}{\underbrace{\parallel BK\parallel {\displaystyle \frac{c}{1-\gamma }}}}\overline{b}.$$

Thus, practical stability follows after shrinking the admissible radius ρ accordingly.

4.1.6. Control Law

We employ the static state-feedback

$$u_k=-Kc\left({\tilde{z}}_k\right),$$

(3)

so that, in the absence of dropouts, $u_k=-K{x}_k$. If $k\in \theta$, then $c\left({\tilde{z}}_k\right)=0$, and the controller applies no input.

4.1.7. Closed-Loop Dynamics

Substituting (3) into (1) yields

$$x_{k+1}=(A-BK)x_k+BK{e}_k,e_k:=c\left({\tilde{z}}_k\right)-x_k.$$

Note that $e_k=0$ whenever $k\notin \theta$, and $e_k=-x_k$ at a dropout instant.

The subsequent subsection establishes stability under the additional assumption that the fuzzy measurement sequence $\left\{{\tilde{z}}_k\right\}$ is $I_2$-fp Cauchy (lacunary statistical consistency).

4.1.8. Implementation Note (Real-Time Footprint)

The double index $(k,\mathsf{\ell })$ is an analysis device (time step and state coordinate); the controller requires no 2-D buffering or look-ahead. Online, at each time k, we compute the centroid $c\left({\tilde{z}}_k\right)$ and apply $u_k=-Kc\left({\tilde{z}}_k\right)$, which is $O\left(n\right)$ per step for an n-state vector. Dropouts are detected via timestamps/flags or simple timeouts, as is standard in NCS [29,41]. The $I_2$-fp consistency and the lacunary gap bound are design/verification conditions; online one may optionally monitor inter-arrival gaps to ensure $q_0$ is respected with negligible overhead.

4.1.9. Dropout Detection and Multi-Sensor Identification (Practical Note)

In practice, occasional packet dropouts can be detected in a protocol-agnostic way using (i) sequence numbers/timestamps and timeouts, and (ii) residual-based fault detection for plausibility checks. For (i), each sensor stream carries a monotonically increasing sequence number; missing indices, stale timestamps, or exceeded inter-arrival timeouts flag a dropout on that specific channel (standardized in real-time transport protocols) [42]. For (ii), an observer (or parity relation) generates residuals $r_k=y_k-{\widehat{y}}_k$; persistent threshold crossings indicate an unreliable sensor even when packets arrive (bias/spikes) [43,44]. With two or more sensors, we maintain per-sensor counters and residual statistics; a dropout and residual escalation localized to channel s marks sensor s as the dropout/failed channel. Ties may be broken by majority voting or by selecting the channel with the smallest innovation variance. These detection steps are orthogonal to the ideal convergence analysis: the controller uses whichever channels are currently marked as valid, while the lacunary index set collects the times at which any sensor is missing.

In practice, COG defuzzification admits efficient table-/scan-based implementations on MCUs [45] and hardware acceleration on FPGAs [46], so the added run-time over a pure linear state feedback is negligible.

4.1.10. Checking the $I_2$–fp Cauchy Condition in Practice

We outline two minimal, testable routes that avoid full-blown identification:

• Sufficient check via bounds (offline). Assume $A_c:=A-BK$ is Schur with $\parallel A_c^k\parallel \le c{\gamma }^k$, the dropout gaps satisfy $n_{r+1}-n_r\ge q_0>0$, centroid bias is uniformly bounded $\parallel {\beta }_k\parallel \le \overline{b}$, and spreads are bounded (cf. Remark 10). Then there exists $t_{★}>0$ such that for every $\epsilon \in (0,1)$ one can pick $(m_0,{\mathsf{\ell }}_0)$ with

  $$\left\{(k,\mathsf{\ell }):\wp (x_{k\mathsf{\ell }}-x_{m_0{\mathsf{\ell }}_0},t_{★})<1-\epsilon \right\}\in I_2,$$

  i.e., the measurements are $I_2$–fp Cauchy. (Proof sketch: combine Lemma 3 with the bounded ${\beta }_k$ term, yielding a geometric bound across blocks of length $q_0$.)
• Empirical density test (online/offline). Fix $(\epsilon ,t)$ and a reference pair $(m_0,{\mathsf{\ell }}_0)$ (e.g., most recent available index). Compute the empirical proportion

  $${\widehat{\delta }}_N(\epsilon ,t):={\displaystyle \frac{1}{N}}\left|\{1\le j\le N:\wp (x_{k_j{\mathsf{\ell }}_j}-x_{m_0{\mathsf{\ell }}_0},t)<1-\epsilon \}\right|$$

  over a running window excluding dropout indices. If ${\widehat{\delta }}_N\to 0$ while inter-arrival gaps respect $q_0$, then the $I_2$–fp Cauchy criterion is satisfied to any prescribed tolerance. This is a standard density-style check adapted to ideal convergence.

Lemma 2.

Let $\mathfrak{X}=L\left({\mathbb{R}}^n\right)$ and define

$$\wp (\tilde{z},t):=exp\left(-\parallel c\left(\tilde{z}\right)\parallel /t\right),\tilde{z}\in \mathfrak{X},t>0,$$

where $c\left(\tilde{z}\right)\in {\mathbb{R}}^n$ denotes the centroid of the fuzzy vector $\tilde{z}$. Then, $(\mathfrak{X},\wp ,T,S)$ is a fuzzy—paranormed space in the sense of Definition 16; i.e., ℘ satisfies properties (FP1)–(FP6) with any continuous t-norm T and its dual t-conorm S.

Proof. 

(FP1) $\wp (\tilde{z},t)=1$ for all $t>0$ iff $\parallel c\left(\tilde{z}\right)\parallel =0$, hence iff $\tilde{z}$ is the degenerate fuzzy zero vector. (FP2) $\wp (-\tilde{z},t)=\wp (\tilde{z},t)$ because $\parallel c(-\tilde{z})\parallel =\parallel -c\left(\tilde{z}\right)\parallel$. (FP3) For ${\tilde{z}}_1,{\tilde{z}}_2$ and $t,s>0$,

$$\wp ({\tilde{z}}_1+{\tilde{z}}_2,t+s)=exp\left(-\parallel c\left({\tilde{z}}_1\right)+c\left({\tilde{z}}_2\right)\parallel /(t+s)\right)\le S\left(\wp ({\tilde{z}}_1,t),\wp ({\tilde{z}}_2,s)\right),$$

with the left-hand bound obtained analogously via T, since $T(a,b)\le min\{a,b\}\le S(a,b)$. (FP4) The map $t\mapsto \wp (\tilde{z},t)$ is increasing and ${lim}_{t\to \infty }\wp (\tilde{z},t)=1$ by definition. (FP5) For $\left|\lambda \right|\le 1$, $\parallel \lambda c\left(\tilde{z}\right)\parallel \le \parallel c\left(\tilde{z}\right)\parallel$ so $\wp (\lambda \tilde{z},t)\ge \wp (\tilde{z},t)$; the converse for $\left|\lambda \right|\ge 1$ is similar. (FP6) Sequential continuity under scalar multiplication follows from continuity of the norm and of $exp(·)$.

Full details are routine and left to the reader. □

Remark 10

(On ignoring the spread). The paranorm $\wp (\tilde{z},t)=exp\left(-\parallel c\left(\tilde{z}\right)\parallel /t\right)$ measures only the centroid bias. This is deliberate, because the control law in (3) and the error term $e_k=c\left({\tilde{z}}_k\right)-x_k$ depend solely on the centroid. If one wishes to penalise the spread of a fuzzy number, one may use the augmented paranorm

$${\wp }_{\sigma }(\tilde{z},t)=exp\left(-\frac{\parallel c\left(\tilde{z}\right)\parallel +\sigma \left(\tilde{z}\right)}{t}\right),\sigma \left(\tilde{z}\right):=\underset{0<\alpha \le 1}{sup}\frac{1}{2}\left({\tilde{z}}_{\alpha }^{+}-{\tilde{z}}_{\alpha }^{-}\right),$$

which still satisfies(FP1)–(FP6). All results of this paper remain valid after replacing the constant c in the exponential bound $\parallel A_c^k\parallel \le c{\gamma }^k$ by $c_{+}:=c(1+{sup}_k\sigma \left({\tilde{z}}_k\right)/\parallel c\left({\tilde{z}}_k\right)\parallel )$, provided the spreads are $I_2$-fp-bounded. For clarity and alignment with industrial practice, we retain the centroid-only form.

Definition 23

(Lacunary statistical consistency). Let ${\left\{{\tilde{z}}_k\right\}}_{k\in \mathbb{N}}\subset L\left({\mathbb{R}}^n\right)$ be the measurement stream defined in Section 4.1, and let $I_2:=I_{\mathrm{fin}}\cup \left\{S_{\theta }\right\}$ be the lacunary ideal generated by the dropout set $S_{\theta }$. We say that the measurements satisfy $I_2$–fp lacunary statistical consistency if the double sequence

$$x_{k\mathsf{\ell }}:={\tilde{z}}_k^{\left(\mathsf{\ell }\right)},(k,\mathsf{\ell })\in \mathbb{N}\times \{1,\dots ,n\},$$

is $I_2$-fp–Cauchy in the fuzzy paranorm ℘; that is, for every $\epsilon \in (0,1)$ and $t>0$ there exists an index pair $(m_0,{\mathsf{\ell }}_0)$ with

$$\left\{(k,\mathsf{\ell })\in {\mathbb{N}}^2:\wp (x_{k\mathsf{\ell }}-x_{m_0{\mathsf{\ell }}_0},t)<1-\epsilon \right\}\in I_2.$$

Informally, outside an $I_2$-negligible set of sampling instants, the fuzzy readings eventually cluster within any prescribed fuzzy ball.

4.2. Numerical Illustration (Convergence Under Lacunary Dropouts)

We provide a minimal, reproducible example that mirrors the hypotheses of Theorem 3. Let $n=2$, $A=\left[\begin{array}{cc}1.1& 0.2\\ 0& 1.0\end{array}\right]$, $B=I_2$, and choose the desired closed-loop $A_c=diag(0.6,0.5)$; set $K:=A-A_c=\left[\begin{array}{cc}0.5& 0.2\\ 0& 0.5\end{array}\right]$ so that $A-BK=A_c$ is Schur. We impose a lacunary dropout pattern with a fixed gap $q_0=7$ (i.e., a packet is missing every 7 sampling steps), and define fuzzy measurements by triangular numbers centered at the true state (zero centroid bias) with half-width $0.08$ at $\alpha =0$ tapering linearly to 0 at $\alpha =1$; dropped packets are modeled as ${\tilde{z}}_k=\mathbf{0}$.

For this setup, $\parallel A_c^k{\parallel }_2\le {\gamma }^k$ with $\gamma =0.6$. By Lemma 3, the dropout error satisfies $\parallel e_{n_r}\parallel \le C_1{sup}_{j<n_r}\parallel x_j\parallel$ with $C_1={\gamma }^{q_0-1}=0.6^6\approx 4.67\times {10}^{-2}$, and the convolution term is bounded by ${\sum }_{j<k}\parallel A_c^{k-1-j}BK{e}_j\parallel \le C_2{sup}_{j<k}\parallel x_j\parallel$, where $C_2=\parallel K\parallel {\displaystyle \frac{{\gamma }^{q_0-1}}{1-{\gamma }^{q_0}}}\approx 0.029$ (spectral norm). Since $C_2\ll 1$, the closed loop is practically stable and the state norm decays geometrically between rare dropouts. A short script (MATLAB R2023b/Python 3.11) that simulates $x_{k+1}=A_c{x}_k+BK{e}_k$ with the above ${\tilde{z}}_k$ shows $\parallel x_k\parallel$ entering and staying in a small neighborhood of the origin (consistent with Theorem 3) (Any equivalent stable pair $(A,B,K)$ with Schur $A_c$ and $q_0\ge 5$ yields qualitatively identical behavior).

4.3. Stability Under Lacunary Statistical Consistency

Lemma 3

(Lacunary error bound). Let $\theta =\left\{n_r\right\}$ be a θ-lacunary dropout sequence with gaps $n_{r+1}-n_r\ge q_0>0$, and suppose the closed-loop matrix $A_c:=A-BK$ satisfies $\parallel A_c^k\parallel \le c{\gamma }^k$ for some $c>0$ and $\gamma \in (0,1)$. Then, the measurement error $e_k:=c\left({\tilde{z}}_k\right)-x_k$ obeys the uniform bound

$$\parallel e_k\parallel \le \underset{=:C_1}{\underbrace{c{\gamma }^{q_0-1}}}\underset{j<k}{sup}\parallel x_j\parallel ,k\in \theta .$$

Consequently,

$$\sum _{j=0}^{k-1}\parallel A_c^{k-1-j}BK{e}_j\parallel \le \parallel BK\parallel {\displaystyle \frac{c{C}_1}{1-{\gamma }^{q_0}}}\underset{0\le j\le k-1}{sup}\parallel x_j\parallel .$$

Proof. 

Fix a dropout instant $k=n_r\in \theta$. Between two consecutive dropouts, the loop evolves without measurement error, so $x_{n_r}=A_c^{n_r-n_{r-1}}{x}_{n_{r-1}}.$ Because $n_r-n_{r-1}\ge q_0$ we have $\parallel x_{n_r}\parallel \le c{\gamma }^{q_0}\parallel x_{n_{r-1}}\parallel \le C_1{sup}_{j<n_r}\parallel x_j\parallel ,$ with $C_1=c{\gamma }^{q_0-1}$ after one additional step. Since $e_{n_r}=-x_{n_r}$ by construction, the first claim follows. For the series bound observe $\parallel A_c^{k-1-j}\parallel \le c{\gamma }^{k-1-j},$ and split the sum over blocks of length at least $q_0$; the geometric series with ratio ${\gamma }^{q_0}$ yields the stated factor. □

Theorem 3

(Lacunary Statistical Stability). Let the plant (1) be stabilizable, i.e., there is $K\in {\mathbb{R}}^{m\times n}$ with $\rho (A-BK)<1$. Let the dropout sequence $\theta =\left\{n_r\right\}$ be θ-lacunary with minimal gap $q_0>0$ and form the strongly admissible ideal $I_2=I_{\mathrm{fin}}\cup \left\{S_{\theta }\right\}$. If the measurement stream $\left\{{\tilde{z}}_k\right\}$ is $I_2$-fp lacunary statistically consistent, then for every $\epsilon >0$ there exists $\rho >0$ such that $\parallel x_0\parallel <\rho$ implies $\parallel x_k\parallel <\epsilon$ for all $k\in \mathbb{N}$.

Proof. 

Step 1 (Nominal exponential decay). Set $A_c:=A-BK$. Because $\rho \left(A_c\right)<1$, there exist $c>0$ and $\gamma \in (0,1)$ with $\parallel A_c^k\parallel \le c{\gamma }^k$.

Step 2 (Error decomposition).

$$x_k=A_c^k{x}_0+\sum _{j=0}^{k-1}{A}_c^{k-1-j}BK{e}_j.$$

The first term decays exponentially by Step 1.

Step 3 (Bounding the convolution term). Lemma 3 yields

$$\sum _{j=0}^{k-1}\parallel A_c^{k-1-j}BK{e}_j\parallel \le C_2\underset{j<k}{sup}\parallel x_j\parallel ,C_2:=\parallel BK\parallel {\displaystyle \frac{c^2{\gamma }^{q_0-1}}{1-{\gamma }^{q_0}}},$$

where the second factor c comes from $\parallel A_c^{k-1-j}\parallel \le c{\gamma }^{k-1-j}$. Because the measurement stream is $I_2$-fp-Cauchy, ${sup}_{j\ge N}\parallel x_j\parallel$ can be made arbitrarily small outside an $I_2$-negligible set; hence, the convolution term is kept below $\epsilon /2$.

Step 4 (Choice of $\rho$). Pick $\rho$ so that the exponentially decaying homogeneous part plus the bounded convolution term stay within $\epsilon$. Completeness of $(\mathfrak{X},\wp )$ and the amalgamation property $\left(A{P}_2\right)$ guarantee that the $I_2$-small exceptional indices do not violate the bound.

Thus, $\parallel x_k\parallel <\epsilon$ for all k, proving practical stability. □

Remark 11.

Theorem 3 extends the classical density-zero dropout result to fuzzy measurements: it tolerates any lacunary pattern whose index set is $I_2$-small, provided the measurements are $I_2$-fp-Cauchy. In particular, fixed-periodic dropouts or exponentially growing gaps are admissible.

5. Conclusions and Future Work

In this paper, we developed a unified ideal convergence theory for double sequences in fuzzy paranormed spaces and illustrated its power through a networked fuzzy control application with lacunary packet dropouts.

5.1. Main Theoretical Contributions

• We introduced the notions of $I_2$-fp-convergence, $I_2$-fp-Cauchy sequences and $I_2^{\ast }$-fp-con vergence, extending classical ideal and statistical convergence simultaneously to double indices, fuzzy-valued elements and the relaxed homogeneity of paranorms.
• Under a two-dimensional amalgamation property $\left(A{P}_2\right)$ we proved (i) uniqueness of the $I_2$-fp limit, and (ii) the equivalence between $I_2$-fp-Cauchy and $I_2$-fp-convergence in complete fuzzy paranormed spaces.

5.2. Control-Theoretic Contribution

We modeled a discrete-time LTI plant regulated by a fuzzy state–feedback controller that occasionally receives no measurement. Treating the missing-data pattern as an $I_2$-small set, we formulated a lacunary statistical consistency condition on the fuzzy measurements and proved that it ensures practical stability of the closed loop whenever the nominal system is asymptotically stable. Unlike stochastic dropout models, our criterion is deterministic and depends only on the ideal nature of the dropout indices, thereby complementing probability–based robustness results in the NCS literature.

5.3. Future Research Directions

(i)

Nonlinear fuzzy systems. Extend the ideal convergence framework to Takagi–Sugeno or interval type-2 fuzzy models where classical Lyapunov tools can be conservative.

(ii)

Hybrid deterministic–probabilistic analysis. Combine $I_2$-fp convergence with almost-sure convergence to treat packet losses that are both sparse in the ideal sense and random in occurrence.

(iii)

Multi-agent consensus under sparse fuzzy communication. Indexing by (agent, time) would allow $I_2$-fp limits to capture agreement despite intermittent, imprecise exchanges.

(iv)

Experimental validation. Implement the proposed controller on a real test-bed (e.g., an inverted pendulum over a lossy network) to verify boundedness under prescribed lacunary dropouts.

We believe these extensions will further bridge the gap between abstract ideal convergence theory and the practical design of resilient fuzzy control systems.

5.4. Feasibility on Embedded Hardware

The online control law is $u_k=-Kc\left({\tilde{z}}_k\right)$; the fuzzy–paranorm framework is used only for analysis and does not run on the controller. Hence, the real-time workload reduces to (i) centroid (COG) defuzzification and (ii) a matrix–vector multiply. Efficient, discretized COG implementations with quantified accuracy–cost trade-offs are well documented [47,48]. Numerous microcontroller deployments demonstrate that closed-loop fuzzy control at kHz rates is practical on low-power MCUs [45], while FPGA realizations of Takagi–Sugeno/fuzzy inference routinely achieve ∼1–1.5 Msamples/s when higher throughput is required [46,49]. Consequently, implementing our controller on COTS MCUs/SoCs or FPGAs is straightforward: for moderate state dimension ($n\le 10$) and typical rule bases, the per-sample cost is dominated by the $O\left(nm\right)$ multiply–accumulation of $Kc\left({\tilde{z}}_k\right)$, with memory footprints well within on-chip SRAM.