Stability Analysis and Stabilization of General Conformable Polynomial Fuzzy Models with Time Delay

Abstract

This paper introduces a sum-of-squares (S-O-S) approach to Stability Analysis and Stabilization (SAS) of nonlinear dynamical systems described by General Conformable Polynomial Fuzzy (GCPF) models with a time delay. First, we present GCPF models, which are more general compared to the widely recognized Takagi–Sugeno Fuzzy (T-SF) models. Then, we establish SAS conditions for these models using a Lyapunov–Krasovskii functional and the S-O-S approach, making the SAS conditions in this work less conservative than the Linear Matrix Inequalities (LMI)-based approach to the T-SF models. In addition, the SAS conditions are found by satisfying S-O-S conditions dependent on membership functions that are reliant on the polynomial fitting approximation algorithm. These S-O-S conditions can be solved numerically using the recently developed SOSTOOLS. To demonstrate the effectiveness and practicality of our approach, two numerical examples are provided to demonstrate the effectiveness and practicality of our approach.

1. Introduction

Since 1985 [1], there has been growing interest in the T-SF model because of its ability to effectively represent nonlinear systems and generate numerous standard theoretical results. By using LMIs along with the Lyapunov theory, T-SF models have been widely used to analyze and achieve control objectives for nonlinear systems, such as Asymptotic Stability (AS) [2,3], Exponential Stability (ES) [4], Observer-Based Control (O-BC) [5,6], and Fault-Tolerant Control (F-TC) [7]. In 2007, Tanaka and colleagues expanded T-SF models into Polynomial Fuzzy (PF) models [8]. They accomplished this by utilizing a software tool named SOSTOOLS, which had emerged in 2002 [9]. PF models offer two main advantages over T-SF models: they provide better accuracy when it comes to representing complex nonlinear systems and they require fewer fuzzy rules [10]. Furthermore, the method using LMIs is not suitable for dealing with PF models because these models involve polynomials in their matrices. In this scenario, the software tool SOSTOOLS [11] replaces the LMI toolbox [12], providing researchers with the ability to effectively manage polynomial inequalities. Significant research efforts have been directed towards tackling analysis and control issues in PF models. The emphasis has been placed on presenting design solutions using S-O-S conditions. This includes areas such as AS [10,11,12,13], OB-C [14], F-TC [15].

It is important to note that all the results mentioned earlier focused on nonlinear systems without delays and with integer-order derivatives. In the following sections, we will delve into two main topics: the common presence of delays in systems and the use of a General Conformable Derivative (GCD) to describe their dynamics.

On one hand, it is important to include time delay factors when modeling and controlling real-world systems like long transmission lines, mechanical setups, chemical processes, and so on. In this situation, Razumikhin [16] and Krasovskii [17] have put forward key theories to handle equations involving time delays. They have expanded upon Liapunov’s theory, which dates back to 1892. These foundational works, especially the contributions made by Krasovskii, have been widely used to analyze and synthesize control for various models that include time delays.

Various techniques have been used in the literature to investigate the stability analysis problem of delayed Linear Systems (LSs). For instance, by considering multiple time delays, the issue of AS has been studied for a general class of retarded LSs via the bounds of its imaginary spectra [18,19]. By adopting a novel Lyapunov–Krasovskii functional with new double integral terms, the authors of [20] have proposed using relaxed sufficient LMIs to test the AS of a delayed LS with bounded and periodic time-varying delays. Using the Kalman–Yakubovich–Popov lemma, the Strong Delay-Independent Stability (SD-IS) problem is addressed for delayed LSs with a single delay [21]. Recently, the Generalized Dixon resultant theory is adopted to investigate the SD-IS problem for LSs with multiple time delays [22].

In their publication, the authors of [23] present the first attempt to extend T-SF models to include systems with time delays. Since this work was published in 2000, researchers have expressed considerable interest in T-SF models with time delays. This attention has led to many important works, such as those related to AS [24], ES [25], OB-C [26], FTC [27]. Similarly, researchers have enhanced PF models to handle time delays. This has resulted in the discovery of many significant findings in the field in relation to AS [28,29], ES [30], O-BC [31] and FTC [32].

On the other hand, in 2014, Khalil et al. [33] introduced a novel derivative called the Conformable Derivative (CD). This derivative was further developed by T. Abdeljawad in [34], and it is currently being extensively explored in [35,36,37]. A new way of understanding control systems has emerged with the advent of CD (see [38,39]). Alharbi et al. [40] have demonstrated that the CD, which serves as a weighted-time analog of classical derivatives, has applications in physics. The CD is generalized, along with its physical interpretation, by Zhao and Luo in [41]. Furthermore, in their work, Li et al. [42] have demonstrated that the diffusion equation can be solved with a GCD. Recent studies ( [43] and [44]) illustrate how GCD can be used to analyze and understand control systems.

This paper examines the issue of SAS for GCPF models. The principal contributions of this paper are outlined below:

• The PF models are widely used in the literature to represent nonlinear dynamics of systems with integer-order derivative. In this paper, we present the first attempt to apply the PF models to describe delayed systems with GCD. Furthermore, instead of relying on T-SF models, we employ the PF models due to their broader accuracy and generality, as established in prior research [10].
• A new Lyapunov–Krasovskii functional is developed to establish the exponential SAS for GCPF systems. This function is specifically designed to effectively address the challenges of exponential SAS in such systems.
• By utilizing the constructed Lyapunov–Krasovskii functional, S-O-S conditions are derived to ensure the SAS of GCPF systems. In fact, the LMI approach is not applicable in such systems due to the presence of polynomial matrices instead of constant matrices in local models. Furthermore, the SOS approach, as discussed in the literature [10], yields less conservative results compared to the LMI approach.
• In order to take into account the overall behavior of the PF model, the S-O-S conditions contain not only information about the polynomial local models but also information about the Fuzzy Membership Functions (FMFs). Due to their nonlinear dynamics, a polynomial curve fitting method is used to approximate these FMFs as S-O-Ss, enabling these approximations to be incorporated into the S-O-S conditions.

The paper is structured as follows: Section 2 delves into the preliminary concepts, while Section 3 presents the problem formulation. In Section 4, we present our theoretical results. In Section 5, we present numerical examples to showcase the practical applicability of the theoretical results. Section 6 concludes the paper.

Notations: ℑ and $\Delta$ denote the sets of polynomial matrices and S-O-Ss, respectively. For a matrix $\mathcal{N}$, ${\left[\mathcal{N}\right]}_T=\mathcal{N}+{\mathcal{N}}^T$.

2. Preliminaries

In this section, we begin by revisiting pertinent definitions, lemmas, and theorems, as cited in references [8,33,34,41,42].

Definition 1.

Let the function ϕ be defined on the interval $[a,b)$. The General Conformable Derivative (GCD) of ϕ at the initial real value a is defined as follows:

$$T_a^{\theta ,{\psi }_a}\varphi \left(t\right)=\underset{\epsilon \to 0}{lim}{\displaystyle \frac{\varphi \left(t+\epsilon {\psi }_a(t,\theta )\right)-\varphi \left(t\right)}{\epsilon }},\mathit{for}\mathit{every}t>a.$$

(1)

The parameter θ belongs to $(0,1]$, and ${\psi }_a(t,\theta )$ represents a continuous non-negative function that is dependent on t, meeting the following condition:

$${\psi }_a(t,1)=1,{\psi }_a(·,{\theta }_1)\ne {\psi }_a(·,{\theta }_2),\mathit{where}{\theta }_1\ne {\theta }_2\mathit{and}\left({\theta }_1,{\theta }_2\right)\in {(0,1]}^{.}$$

If $T_a^{\theta ,{\psi }_a}\varphi \left(t\right)$ exists, for all $t\in (a,c)$, for some $c>a$ and ${\displaystyle \underset{t⟶a^{+}}{lim}{T}_a^{\theta ,{\psi }_a}\varphi \left(t\right)}$ exists; therefore,

$${\displaystyle T_a^{\theta ,{\psi }_a}\varphi \left(a\right):=\underset{t⟶a^{+}}{lim}{T}_a^{\theta ,{\psi }_a}\varphi \left(t\right).}$$

(2)

Remark 1.

The GCD serves as a generalization that encompasses both the classical derivative case, when $\theta =1$, and the Conformable Derivative, denoted by ${\psi }_a(t,\theta )={(t-a)}^{1-\theta }$ (refer to [34] and [33] for the case when $a=0$).

Remark 2.

For an in-depth exploration of the properties of the GCD, we make the following assumptions:

• The function ${\psi }_a(t,\theta )$ is increasing and satisfies ${\psi }_a(t,\theta )>0$ for all $t>a$.
• The function ${\displaystyle \frac{1}{{\psi }_a}}(.,\theta )$ is locally integrable.
• The integral from a to ∞ of ${\displaystyle \frac{1}{{\psi }_a(x,\theta )}}dx$ diverges, as indicated by ${\int }_a^{\infty }{\displaystyle \frac{1}{{\psi }_a(x,\theta )}}dx=\infty$.

Definition 2.

For $0<\theta <1$, the Conformable Integral of a function ϕ is defined as follows:

$$I_a^{\theta ,{\psi }_a}\varphi \left(t\right)={\int }_a^t{\displaystyle \frac{\varphi \left(x\right)}{{\psi }_a(x,\theta )}}dx.$$

(3)

Lemma 1.

Consider a function ϕ defined on $[a,b]$. Then, for every t, such that $t\ge a$, it follows that

$$T_a^{\theta ,{\psi }_a}{I}_a^{\theta ,{\psi }_a}\varphi \left(t\right)=\varphi \left(t\right).$$

(4)

Lemma 2.

If ϕ is an absolutely continuous function defined on the interval $[a,b]$, then for all t, such that $t\ge a$, the following holds:

$$I_a^{\theta ,{\psi }_a}{T}_a^{\theta ,{\psi }_a}\varphi \left(t\right)=\varphi \left(t\right)-\varphi \left(a\right).$$

(5)

Lemma 3.

Let ${\nu }_1,{\nu }_2,{\nu }_3\in \mathbb{R}$, and the functions ${\varphi }_1,{\varphi }_2:[a,b)⟶\mathbb{R}$ such that $T_a^{\theta ,{\psi }_a}{\varphi }_1\left(t\right)$ and $T_a^{\theta ,{\psi }_a}{\varphi }_2\left(t\right)$ exist on $(a,b)$. Then,

• $T_a^{\theta ,{\psi }_a}({\nu }_1{\varphi }_1+{\nu }_2{\varphi }_2)\left(t\right)={\nu }_1{T}_a^{\theta ,{\psi }_a}{\varphi }_1\left(t\right)+{\nu }_2{T}_a^{\theta ,{\psi }_a}{\varphi }_2\left(t\right)$;
• $T_a^{\theta ,{\psi }_a}{\nu }_3=0$;
• $T_a^{\theta ,{\psi }_a}\left({\varphi }_1{\varphi }_2\right)\left(t\right)={\varphi }_1\left(t\right)T_a^{\theta ,{\psi }_a}{\varphi }_2\left(t\right)+{\varphi }_2\left(t\right)T_a^{\theta ,{\psi }_a}{\varphi }_1\left(t\right)$;
• ${\displaystyle T_a^{\theta ,{\psi }_a}\left(\frac{{\varphi }_1}{{\varphi }_2}\right)\left(t\right)=\frac{{\varphi }_2\left(t\right)T_a^{\theta ,{\psi }_a}{\varphi }_1\left(t\right)-{\varphi }_1\left(t\right)T_a^{\theta ,{\psi }_a}{\varphi }_2\left(t\right)}{{\varphi }_2^2\left(t\right)}}$, for each $t\in (a,b)$, such that ${\varphi }_2\left(t\right)\ne 0$.

Remark 3.

Let $\zeta \in {\mathbb{R}}^{\ast }$. If $g\left(t\right):={\mathbb{E}}_{\theta }^{{\psi }_a}\left(\zeta ,t,a\right)=e^{\zeta {\int }_a^t{\displaystyle \frac{1}{{\psi }_a(x,\theta )}}dx}$, then $T_a^{\theta ,{\psi }_a}g\left(t\right)=\zeta g\left(t\right)$ and $I_a^{\theta ,{\psi }_a}g\left(t\right)={\displaystyle \frac{1}{\zeta }}\left(g\left(t\right)-1\right)$.

Let consider the nonlinear system:

$$\begin{array}{ccc}\hfill T_a^{\theta ,{\psi }_a}\xi \left(\nu \right)& =& F(\nu ,\xi (\nu ),\xi (\nu -\beta \left)\right),\nu \ge a,\hfill \\ \hfill \xi \left(\nu \right)& =& \varrho \left(\nu \right),\nu \in [a-\beta ,a]\hfill \end{array}$$

(6)

where $F\in C\left([a,\infty )\times {\mathbb{R}}^n\times {\mathbb{R}}^n,{\mathbb{R}}^n\right)$ and $\varrho \in C\left([a-\beta ,a),{\mathbb{R}}^n\right).$

Definition 3.

The system (6) is referred to exponentially stable if we have positive scalars M and δ, such that

$$\parallel \xi \left(\nu \right)\parallel \le M\parallel {\xi }_a\parallel {\mathbb{E}}_{\theta }^{{\psi }_a}\left(-\delta ,\nu ,a\right),\forall \nu \ge a,$$

(7)

where ${\displaystyle \parallel {\xi }_a\parallel =\underset{\nu \in [a-\beta ,a]}{sup}\parallel \xi \left(\nu \right)\parallel .}$

Definition 4.

Let $f\left(\xi \right(\rho \left)\right)\in \Im$. If $\exists \{f_1\left(\xi \left(\rho \right)\right),f_2\left(\xi \left(\rho \right)\right),\dots ,f_m\left(\xi \left(\rho \right)\right)\}\in \Im$, such that

$$\begin{array}{c}\hfill f\left(\xi \left(\rho \right)\right)=\sum _{j=1}^m{f}_j^2\left(\xi \left(\rho \right)\right),\end{array}$$

(8)

then $f\left(\xi \right(\rho \left)\right)\in \Delta$, which implies that $f\left(\xi \right(\rho \left)\right)>0$.

3. Problem Formulation

Consider a nonlinear system with delay described by the delayed PF model, which consists of the following r rules:

Plant Rule $i(i=1,2,\cdots ,r)$: If ${\alpha }_1\left(\xi \left(\rho \right)\right)$ is ${\varsigma }_{i1}$ and ⋯ and ${\alpha }_p\left(\xi \left(\rho \right)\right)$ is ${\varsigma }_{ip}$ then,

$$\begin{array}{c}\hfill T_a^{\theta ,{\psi }_a}\xi \left(\rho \right)={\mathfrak{A}}_i\left(\xi \left(\rho \right)\right)\xi \left(\rho \right)+{\mathfrak{A}}_{\beta i}\left(\xi \left(\rho \right)\right)\xi (\rho -\beta )+{\mathfrak{B}}_i\left(\xi \left(\rho \right)\right)u\left(\rho \right)\end{array}$$

(9)

where the measurable premise variables are represented by ${\alpha }_j\left(\xi \left(\rho \right)\right)$, $j=1\dots p$. The fuzzy sets are ${\varsigma }_{ij}$, $i=1\dots r$ and $j=1\dots p$, s and the rules’ number is r. $\xi \left(\rho \right)$ represents the current state vector, $\xi (\rho -\beta )$ represents the delayed state vector, and $u\left(\rho \right)$ is the vector for control inputs. The sets $\{{\mathfrak{A}}_i\left(\xi \left(\rho \right)\right),{\mathfrak{A}}_{\beta i}\left(\xi \left(\rho \right)\right),{\mathfrak{B}}_i\left(\xi \left(\rho \right)\right)\}$ belong to the set ℑ. The delay $\beta$ is presumed to remain constant and be known.

After the defuzzication process of model (9), we obtain the following result:

$$\begin{array}{c}\hfill {\displaystyle T_a^{\theta ,{\psi }_a}\xi \left(\rho \right)=\sum _{i=1}^r{\iota }_i\left(\xi \left(\rho \right)\right)\left({\mathfrak{A}}_i\left(\xi \left(\rho \right)\right)\xi \left(\rho \right)+{\mathfrak{A}}_{\beta i}\left(\xi \left(\rho \right)\right)\xi (\rho -\beta )+{\mathfrak{B}}_i\left(\xi \left(\rho \right)\right)u\left(\rho \right)\right)}\end{array}$$

(10)

where ${\iota }_i\left(\xi \left(\rho \right)\right)={\displaystyle \frac{{\displaystyle \prod _{i=1}^r{\varsigma }_{ij}\left({\alpha }_j\left(\xi \left(\rho \right)\right)\right)}}{{\displaystyle \sum _{i=1}^r\prod _{j=1}^p{\varsigma }_{ij}\left({\alpha }_j\left(\xi \left(\rho \right)\right)\right)}}}$ are the FMFs.

In the remaining equations, ${\iota }_i\left(\xi \left(\rho \right)\right)$ and $\xi \left(\rho \right)$ are denoted as ${\iota }_i\left(\xi \right)$ and $\xi$, respectively.

To incorporate the information regarding FMFs, we employ the polynomial curve fitting technique to approximate each FMF ${\iota }_i\left(\xi \right)$ as a square of a polynomial $p_i\left(\xi \right)$. To enhance the precision of this approximation, $\xi$ is partitioned into m sub-segments $s_e$, in which $e\in \mathcal{M}=\{1,2,\dots ,m\}$. Within each of these segments $s_e$, an approximation by a square of polynomial $p_{i,s_e}\left(\xi \right)$ is employed for every ${\iota }_i\left(\xi \right)$:

$$\begin{array}{c}\hfill {\iota }_i\left(\xi \right)\approx p_{i,s_e}\left(\xi \right),\mathrm{for}\xi \in s_e.\end{array}$$

(11)

4. Main Results

4.1. Stability Analysis

Theorem 1.

For positive scalars $\sigma ,\beta$ and a, the system (10), with $u=0$, is exponentially stable if there are symmetric matrices P and Q, such that the following S-O-S conditions are satisfied:

$$\begin{array}{c}{ϵ}_1^T(P-n_{1}I){ϵ}_1\in \Delta ,\hfill \end{array}$$

(12)

$$\begin{array}{c}{ϵ}_1^T(Q-n_{2}I){ϵ}_1\in \Delta ,\hfill \end{array}$$

(13)

$$\begin{array}{c}-{ϵ}_2^T(\sum _{i=1}^r{p}_{i,s_e}\left(\xi \right){\Lambda }_i\left(\xi \right)+n_3\left(\xi \right)I){ϵ}_1\in \Delta ,\forall s_e,\end{array}$$

(14)

where ${\kappa }^T={\left[\begin{array}{cc}{\xi }^T& {\xi }^T(\rho -\beta )\end{array}\right]}^T$, ${\Lambda }_i\left(\xi \right)=\left[\begin{array}{cc}{[P{\mathfrak{A}}_i\left(\xi \right)+\sigma P]}_T+Q& P{\mathfrak{A}}_{\beta i}\left(\xi \right)\\ \ast & -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}Q\end{array}\right]$, ${ϵ}_1$ and ${ϵ}_2$ are vectors such that ${ϵ}_2$ is independent of ξ, $n_1$, $n_2$ are strictly positive scalars, and $n_3\left(\xi \right)$ is a non-negative polynomial, such that $n_3\left(\xi \right)>0$ for $\xi \ne 0$.

Proof. 

Let the Lyapunov–Krasovskii functional candidate presented below,

$$\begin{array}{c}\hfill V\left(\rho \right)=V_1\left(\rho \right)+V_2\left(\rho \right)\end{array}$$

(15)

where

$$\begin{array}{ccc}\hfill V_1\left(\rho \right)& =& {\xi }^{T}P\xi ,\hfill \\ \hfill V_2\left(\rho \right)& =& {\int }_{\rho }^{\rho +\beta }{\displaystyle \frac{1}{{\psi }_a(r,\theta )}}{e}^{2\sigma ({\int }_a^r{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{\rho }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}})}{\xi }^T(r-\beta )Q\xi (r-\beta )dr.\hfill \end{array}$$

(29) and (13) imply that $P>0$ and $Q>0$.

For $\rho >a$, we obtain the following GCD of $V_1\left(\rho \right)$ and $V_2\left(\rho \right)$

$$\begin{array}{ccc}\hfill T_a^{\theta ,{\psi }_a}{V}_1\left(\rho \right)& =& 2{\left(T_a^{\theta ,{\psi }_a}\xi \right)}^{T}P\xi \hfill \\ \hfill T_a^{\theta ,{\psi }_a}{V}_2\left(\rho \right)& =& {\psi }_a(\rho ,\theta )[e^{2\sigma ({\int }_a^{\rho +\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{\rho }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}})}{\displaystyle \frac{{\xi }^T\left(\rho \right)Q\xi \left(\rho \right)}{{\psi }_a(\rho +\beta ,\theta )}}\hfill \\ & & -e^{2\sigma ({\int }_a^{\rho }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{\rho }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}})}{\displaystyle \frac{{\xi }^T(\rho -\beta )Q\xi (\rho -\beta )}{{\psi }_a(\rho ,\theta )}}]\hfill \\ & & -{\displaystyle \frac{2\sigma }{{\psi }_a(\rho ,\theta )}}{V}_2\left(\rho \right)\hfill \end{array}$$

(16)

Since $\rho ⟼{\psi }_a(\rho ,\theta )$ is increasing, then

$${\displaystyle \frac{{\psi }_a(\rho ,\theta )}{{\psi }_a(\rho +\beta ,\theta )}}\le 1$$

and

$${\int }_a^{\rho +\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{\rho }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}-{\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}\le 0.$$

Hence,

$$\begin{array}{c}\hfill T_a^{\theta ,{\psi }_a}{V}_2\left(\rho \right)\le {\xi }^T\left(\rho \right)Q\xi -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}{\xi }^T(\rho -\beta )Q\xi (\rho -\beta )-2\sigma V_2\left(\rho \right)\end{array}$$

(17)

From (16) and (17), we obtain the following result:

$$\begin{array}{ccc}\hfill T_a^{\theta ,{\psi }_a}{V}_1\left(\rho \right)+2\sigma V_1\left(\rho \right)& =& 2{\left(T_a^{\theta ,{\psi }_a}\xi \right)}^{T}P\xi +2\sigma {\xi }^{T}P\xi \hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill T_a^{\theta ,{\psi }_a}{V}_2\left(\rho \right)+2\sigma V_2\left(\rho \right)& \le & {\xi }^T\left(\rho \right)Q\xi -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}{\xi }^T(\rho -\beta )Q\xi (\rho -\beta )\hfill \end{array}$$

(19)

From (15), (18) and (19), we obtain

$$\begin{array}{ccc}\hfill T_a^{\theta ,{\psi }_a}V\left(\rho \right)+2\sigma V\left(\rho \right)& \le & 2{\left(T_a^{\theta ,{\psi }_a}\xi \right)}^{T}P\xi +2\sigma {\xi }^{T}P\xi +{\xi }^T\left(\rho \right)Q\xi -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}{\xi }^T(\rho -\beta )Q\xi (\rho -\beta )\hfill \\ \hfill & \le & {\displaystyle 2\sum _{i=1}^r{\iota }_i\left(\xi \right)\left({\xi }^T{\mathfrak{A}}_i^T\left(\xi \right)+{\xi }^T(\rho -\beta ){\mathfrak{A}}_{\beta i}^T\left(\xi \right)\right)P\xi +2\sigma {\xi }^{T}P\xi +{\xi }^T\left(\rho \right)Q\xi }\hfill \\ \hfill & & -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}{\xi }^T(\rho -\beta )Q\xi (\rho -\beta ).\hfill \end{array}$$

(20)

The GCD of $V\left(\rho \right)$ satisfies

$$\begin{array}{c}\hfill T_a^{\theta ,{\psi }_a}V\left(\rho \right)+2\sigma V\left(\rho \right)\le \sum _{i=1}^r{\iota }_i\left(\xi \right){\kappa }^T{\Lambda }_i\left(\xi \right)\kappa \end{array}$$

(21)

By employing the polynomial curve fitting technique, we can approximate the FMF ${\iota }_i\left(\xi \right)$ as square polynomials. So, according to (11), we can rewrite the previous inequality as follows:

$$\begin{array}{c}\hfill T_a^{\theta ,{\psi }_a}V\left(\rho \right)+2\sigma V\left(\rho \right)\le \sum _{i=1}^r{p}_{i,s_r}\left(\xi \right){\kappa }^T{\Lambda }_i\left(\xi \right)\kappa \end{array}$$

(22)

Therefore, if (14) holds, then

$$T_a^{\theta ,{\psi }_a}V\left(\rho \right)+2\sigma V\left(\rho \right)\le 0.$$

It follows from Theorem 2 in [43] that

$$V\left(\rho \right)\le V\left(a\right){\mathbb{E}}_{\theta }^{{\psi }_a}\left(-2\sigma ,\rho ,a\right),\forall \rho \ge a.$$

(23)

We have

$$\begin{array}{ccc}\hfill V\left(a\right)& \le & K_1{\parallel \xi \parallel }^2,\hfill \\ \hfill V\left(\rho \right)& \ge & K_2{\parallel \xi \left(\rho \right)\parallel }^2,\hfill \end{array}$$

(24)

where

$$K_1=\left({\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right){\int }_a^{a+\beta }{\displaystyle \frac{1}{{\psi }_a(r,\theta )}}dr\right)$$

and

$$K_2={\lambda }_{mim}\left(P\right).$$

Therefore,

$$\begin{array}{c}\hfill \parallel \xi \left(\rho \right)\parallel \le \sqrt{{\displaystyle \frac{K_1}{K_2}}}\parallel {\xi }_a\parallel {\mathbb{E}}_{\theta }^{{\psi }_a}\left(-\sigma ,\rho ,a\right),\forall \rho \ge a.\end{array}$$

(25)

□

4.2. Stabilization via SOS

Since the PDC replicates the structure of the fuzzy model of a system, it is possible to build a fuzzy controller with polynomial rule consequences using the provided PF model (9):

Plant Rule $i(i=1,2,\cdots ,r)$: If ${\alpha }_1\left(\rho \right)$ is ${\varsigma }_{i1}$ and ⋯ and ${\alpha }_p\left(\rho \right)$ is ${\varsigma }_{ip}$ then,

$$\begin{array}{c}\hfill u\left(\rho \right)=-{\mathfrak{F}}_i\left(\xi \right)\xi \end{array}$$

(26)

We can determine the complete fuzzy controller by

$$\begin{array}{c}\hfill {\displaystyle u\left(\rho \right)=-\sum _{i=1}^r{\iota }_i\left(\xi \right){\mathfrak{F}}_i\left(\xi \right)\xi }\end{array}$$

(27)

Using (10) and (27), we can express the closed-loop system as

$$\begin{array}{c}\hfill {\displaystyle T_a^{\theta ,{\psi }_a}\xi =\sum _{i=1}^r\sum _{j=1}^r{\iota }_i\left(\xi \right){\iota }_j\left(\xi \right)\left(\{{\mathfrak{A}}_i\left(\xi \right)-{\mathfrak{B}}_i\left(\xi \right){\mathfrak{F}}_j\left(\xi \right)\}\xi +{\mathfrak{A}}_{\beta i}\left(\xi \right)\xi (\rho -\beta )\right)}\end{array}$$

(28)

Theorem 2.

For positive scalars $\sigma ,\beta$ and a, the system (28) is exponentially stable if we have symmetric matrices P and $\tilde{Q}$, such that the following S-O-Ss are satisfied:

$$\begin{array}{ccc}& & {ϵ}_1^T(P-n_{1}I){ϵ}_1\in \Delta ,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & {ϵ}_1^T(\tilde{Q}-n_{2}I){ϵ}_1\in \Delta ,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & -{ϵ}_2^T(\sum _{i=1}^r\sum _{j=1}^r{p}_{i,s_e}\left(\xi \right)p_{j,s_e}\left(\xi \right)\{{\Gamma }_{ij}\left(\xi \right)+{\Gamma }_{ji}\left(\xi \right)\}+n_3\left(\xi \right)I){ϵ}_2\in \Delta ,\forall s_e,\hfill \end{array}$$

(31)

where ${\Gamma }_{ij}\left(\xi \right)=\left[\begin{array}{cc}{[{{\mathfrak{A}}_i\left(\xi \right)P-{\mathfrak{B}}_i\left(\xi \right)\mathfrak{M}}_j\left(\xi \right)+\sigma P]}_T+\tilde{Q}& {\mathfrak{A}}_{\beta i}\left(\xi \right)P\\ \ast & -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}\tilde{Q}\end{array}\right]$, ${ϵ}_1$ and ${ϵ}_2$ are vectors such that ${ϵ}_2$ is independent of ξ, $n_1$, $n_2$ are strictly positive scalars, and $n_3\left(\xi \right)$ is a non-negative polynomial, such that $n_3\left(\xi \right)>0$ for $\xi \ne 0$. In this case, the controller’s gains are as follows:

$$\begin{array}{c}\hfill {\mathfrak{F}}_j\left(\xi \right)={\mathfrak{M}}_j\left(\xi \right)P_1^{-1}.\end{array}$$

(32)

Proof. 

The Lyapunov–Krasovskii functional candidate is presented below:

$$\begin{array}{c}\hfill V\left(\rho \right)=V_1\left(\rho \right)+V_2\left(\rho \right),\end{array}$$

(33)

where $V_1\left(\rho \right)={\xi }^T{P}^{-1}\xi$ and $V_2\left(\rho \right)$ is the same as (16).

By following the same steps as outlined in the proof of Theorem 1, we obtain

$$\begin{array}{ccc}\hfill T_a^{\theta ,{\psi }_a}V\left(\rho \right)+2\sigma V\left(\rho \right)& \le & 2{\left(T_a^{\theta ,{\psi }_a}\xi \right)}^T{P}^{-1}\xi +2\sigma {\xi }^T{P}^{-1}\xi +{\xi }^T\left(\rho \right)Q\xi -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}{\xi }^T(\rho -\beta )Q\xi (\rho -\beta )\hfill \\ & \le & {\displaystyle 2\sum _{i=1}^r\sum _{j=1}^r{\iota }_i\left(\xi \right){\iota }_j\left(\xi \right)\left({\xi }^T{\{{\mathfrak{A}}_i\left(\xi \right)-{\mathfrak{B}}_i\left(\xi \right){\mathfrak{F}}_j\left(\xi \right)\}}^T+{\xi }^T(\rho -\beta ){\mathfrak{A}}_{\beta i}^T\left(\xi \right)\right)P^{-1}\xi }\hfill \\ & & +2\sigma {\xi }^T{P}^{-1}\xi +{\xi }^T\left(\rho \right)Q\xi -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}{\xi }^T(\rho -\beta )Q\xi (\rho -\beta )\hfill \end{array}$$

(34)

The GCD of $V\left(\rho \right)$ satisfies

$$\begin{array}{c}\hfill T_a^{\theta ,{\psi }_a}V\left(\rho \right)+2\sigma V\left(\rho \right)\le \sum _{i=1}^r\sum _{i=j}^r{\iota }_i\left(\xi \right){\iota }_j\left(\xi \right){\kappa }^T{\nabla }_{ij}\left(\xi \right)\kappa ={\displaystyle \frac{1}{2}}\sum _{i=1}^r\sum _{i=j}^r{\iota }_i\left(\xi \right){\iota }_j\left(\xi \right){\kappa }^T\{{\nabla }_{ij}\left(\xi \right)+{\nabla }_{ji}\left(\xi \right)\}\kappa \end{array}$$

(35)

where ${\nabla }_{ij}\left(\xi \right)=\left[\begin{array}{cc}{[P^{-1}\{{\mathfrak{A}}_i\left(\xi \right)-{\mathfrak{B}}_i\left(\xi \right){\mathfrak{F}}_j\left(\xi \right)\}+\sigma P^{-1}]}_T+Q& P^{-1}{\mathfrak{A}}_{\beta i}\left(\xi \right)\\ \ast & -e^{-2\sigma {\int }_a^{a+\beta }{\displaystyle \frac{du}{{\psi }_a(u,\theta )}}}Q\end{array}\right]$.

The condition (31) implies that

$$\begin{array}{c}\hfill \sum _{i=1}^r\sum _{j=1}^r{p}_{i,s_e}\left(\xi \right)p_{j,s_e}\left(\xi \right)\{{\Gamma }_{ij}\left(\xi \right)+{\Gamma }_{ji}\left(\xi \right)\}⩾0\forall s_e\end{array}$$

(36)

Define ${\mathfrak{M}}_j\left(\xi \right)={\mathfrak{F}}_j\left(\xi \right)P$, $\tilde{Q}=PQP$, and if we multiply the last expression on both sides, $diag\{P^{-1},P^{-1}\}$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^r\sum _{j=1}^r{p}_{i,s_e}\left(\xi \right)p_{j,s_e}\left(\xi \right)\{{\nabla }_{ij}\left(\xi \right)+{\nabla }_{ji}\left(\xi \right)\}⩾0\forall s_e}\end{array}$$

(37)

Based on Equation (11), we can express (37) as follows:

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^r\sum _{j=1}^r{\iota }_i\left(\xi \right){\iota }_j\left(\xi \right)\{{\nabla }_{ij}\left(\xi \right)+{\nabla }_{ji}\left(\xi \right)\}⩾0\forall i\le j}\end{array}$$

(38)

Therefore, if (31) holds, then

$$T_a^{\theta ,{\psi }_a}V\left(\rho \right)+2\sigma V\left(\rho \right)\le 0.$$

Similar to the proof of Theorem 1, we achieve the exponential stability of the GCD systems (28).  □

Remark 4.

In [45], we utilized a conformable derivative within the context of LS. In contrast, this work expands upon this by employing a GCD and introducing PF models. These models provide a more general framework, surpassing both LS and T-SF models in terms of applicability and flexibility.

Remark 5.

The use of FP models instead of T-SF models allows a reduction in the number of if–then rules, which in turn decreases the overall system complexity. An example is given in the next section to illustrate this point.

5. Illustrative Examples

5.1. Example 1

Consider the following PF model with GCD and time delay:

$$\begin{array}{c}\hfill {\displaystyle T_a^{\theta ,{\psi }_a}\xi =\sum _{i=1}^2{\iota }_i\left(\xi \right)\{{\mathfrak{A}}_i\left(\xi \right)\xi +{\mathfrak{A}}_{\beta i}\xi (\rho -\beta )\}}\end{array}$$

(39)

where ${\mathfrak{A}}_1\left(\xi \right)=\left[\begin{array}{cc}-1-{\xi }_1^2& 3\\ -10& -35\end{array}\right]$, ${\mathfrak{A}}_2\left(\xi \right)=\left[\begin{array}{cc}-{\xi }_1^2& 3\\ -10& -35\end{array}\right]$,${\mathfrak{A}}_{{\beta }_1}\left(\xi \right)=\left[\begin{array}{cc}-0.1{\xi }_1^2& 0\\ 1& 0\end{array}\right]$, ${\mathfrak{A}}_{{\beta }_2}\left(\xi \right)=\left[\begin{array}{cc}-0.1{\xi }_1^2& 0\\ 0.2& 0\end{array}\right].$ The FMFs are defined as follows:

$$\begin{array}{c}\hfill {\iota }_1\left(\xi \right)={\displaystyle \frac{1}{2}}(1-sin\left({\xi }_2\right));{\iota }_2\left(\xi \right)=1-{\iota }_1\left({\xi }_2\right).\end{array}$$

We consider three sub-regions $s_1=\{{\xi }_2,-2\le {\xi }_2\le 0\}$, $s_2=\{{\xi }_2,0\le {\xi }_2\le 2\}$ and $s_3=\{{\xi }_2,2\le {\xi }_2\le 4\}$. Using the polynomial approximation method, we approximate the FMFs with fourth-degree square polynomials as follows:

$$\begin{array}{ccc}\hfill p_{1,s_1}\left({\xi }_2\right)& =& {(-0.1174{\xi }_2^2-0.3714{\xi }_2+0.7053)}^2\hfill \\ \hfill p_{1,s_2}\left({\xi }_2\right)& =& {(0.1932{\xi }_2^2-0.7079{\xi }_2+0.7666)}^2\hfill \\ \hfill p_{1,s_3}\left({\xi }_2\right)& =& {(-0.0802{\xi }_2^2+0.8476{\xi }_2-1.1659)}^2\hfill \\ \hfill p_{2,s_1}\left({\xi }_2\right)& =& {(0.1932{\xi }_2^2+0.7079{\xi }_2+0.7666)}^2\hfill \\ \hfill p_{2,s_2}\left({\xi }_2\right)& =& {(-0.1174{\xi }_2^2+0.3714{\xi }_2+0.7053)}^2\hfill \\ \hfill p_{2,s_3}\left({\xi }_2\right)& =& {(-0.0924{\xi }_2^2+0.2368{\xi }_2+0.8768)}^2\hfill \end{array}$$

The approximation of FMF by eight-degree square polynomials are given as follows:

$$\begin{array}{ccc}\hfill p_{1,s_1}\left({\xi }_2\right)& =& {(0.0024{\xi }_2^4+0.0156{\xi }_2^3-0.0878{\xi }_2^2-0.3534{\xi }_2+0.7071)}^2\hfill \\ \hfill p_{1,s_2}\left({\xi }_2\right)& =& {(0.1871{\xi }_2^4-0.4635{\xi }_2^3+0.2714{\xi }_2^2-0.4286{\xi }_2+0.7075)}^2\hfill \\ \hfill p_{1,s_3}\left({\xi }_2\right)& =& {(0.0016{\xi }_2^4-0.0357{\xi }_2^3+0.1484{\xi }_2^2+0.2689{\xi }_2-0.660)}^2\hfill \\ \hfill p_{2,s_1}\left({\xi }_2\right)& =& {(0.1871{\xi }_2^4+0.4635{\xi }_2^3+0.2714{\xi }_2^2+0.4286{\xi }_2+0.7075)}^2\hfill \\ \hfill p_{2,s_2}\left({\xi }_2\right)& =& {(0.0024{\xi }_2^4+0.0156{\xi }_2^3-0.0878{\xi }_2^2-0.3534{\xi }_2+0.7071)}^2\hfill \\ \hfill p_{2,s_3}\left({\xi }_2\right)& =& {(0.0019{\xi }_2^4-0.0098{\xi }_2^3-0.1104{\xi }_2^2+0.3918{\xi }_2+0.6830)}^2\hfill \end{array}$$

In Figure 1 and Figure 2, we present FMFs and their estimates within the areas labeled $s_1$, $s_2$, and $s_3$. It is clear that when we increase the degree of the polynomial, the accuracy of the polynomial approximation method improves.

Figure 1. FMF and their fourth-degree square polynomials approximations in ${\xi }_2\in [-2,4]$.

Figure 2. FMF and their eighth-degree square polynomials approximations in ${\xi }_2\in [-2,4]$.

The S-O-S design conditions in Theorem 1 are feasible for $\sigma =0.8$, $\beta =0.9$ and $n_1=n_2=n_3={10}^{-6}$. Figure 3 illustrates the temporal progression of $\xi \left(\rho \right)$, for $\rho \in [-0.9,0]$ and we can see from this figure that the system becomes stable.

Figure 3. Temporal progression of $\xi \left(\rho \right)$.

5.2. Example 2

Consider the following delayed NS:

$$\begin{array}{c}\hfill T_a^{\theta ,{\psi }_a}\xi =\mathfrak{A}\left(\xi \right)\xi +{\mathfrak{A}}_{\beta }\xi (\rho -\beta )+\mathfrak{B}\left(\xi \right)u,\end{array}$$

(40)

where $\mathfrak{A}\left(\xi \right)=\left[\begin{array}{cc}-1+{\xi }_1+{\xi }_1^2+{\xi }_1{\xi }_2-{\xi }_2^2& 1\\ -sin\left({\zeta }_1\right)& -1\end{array}\right]$, ${\mathfrak{A}}_{\beta }\left(\xi \right)=\left[\begin{array}{cc}{\xi }_1& 0\\ 0.2& 0\end{array}\right]$, $\mathcal{B}\left(\xi \right)=\left[\begin{array}{c}{\xi }_1^2+5\\ 0\end{array}\right]$.

Let $y_{min}=\underset{\left|{\xi }_1\right|<q_1,\left|{\xi }_2\right|<q_2}{min}\left(y\right)$ and $y_{max}=\underset{\left|{\xi }_1\right|<q_1,\left|{\xi }_2\right|<q_2}{max}\left(y\right)$, where $y=-1+{\xi }_1+{\xi }_1^2+{\xi }_1{\xi }_2-{\xi }_2^2$.

By using the concept of sector nonlinearity, the NS (40) is represented by the following T-SF model for ${\xi }_1\in \left[\begin{array}{cc}-q_1& q_1\end{array}\right]$, ${\xi }_2\in \left[\begin{array}{cc}-q_2& q_2\end{array}\right]$:

$$\begin{array}{c}\hfill {\displaystyle T_a^{\theta ,{\psi }_a}\xi =\sum _{i=1}^8{\iota }_i\{{\mathfrak{A}}_i\xi +{\mathfrak{A}}_{\beta i}\xi (\rho -\beta )+{\mathfrak{B}}_{i}u\},}\end{array}$$

(41)

where ${\mathfrak{A}}_1={\mathfrak{A}}_2=\left[\begin{array}{cc}{y}_{max}& 1\\ -1& -1\end{array}\right]$, ${\mathfrak{A}}_3={\mathfrak{A}}_4=\left[\begin{array}{cc}{y}_{max}& 1\\ -{\displaystyle \frac{sin{q}_1}{q_1}}& -1\end{array}\right]$, ${\mathfrak{A}}_5={\mathfrak{A}}_6\left[\begin{array}{cc}{y}_{min}& 1\\ -1& -1\end{array}\right]$, ${\mathfrak{A}}_7={\mathfrak{A}}_8=\left[\begin{array}{cc}{y}_{min}& 1\\ -{\displaystyle \frac{sin{q}_1}{q_1}}& -1\end{array}\right]$, ${\mathfrak{A}}_{{\beta }_1}={\mathfrak{A}}_{{\beta }_3}={\mathfrak{A}}_{{\beta }_5}={\mathfrak{A}}_{{\beta }_7}=\left[\begin{array}{cc}{q}_1& 0\\ 0.2& 0\end{array}\right],{\mathfrak{A}}_{{\beta }_2}={\mathfrak{A}}_{{\beta }_4}={\mathfrak{A}}_{{\beta }_6}={\mathfrak{A}}_{{\beta }_8}=\left[\begin{array}{cc}-q_1& 0\\ 0.2& 0\end{array}\right]$, ${\mathcal{B}}_1={\mathcal{B}}_3={\mathcal{B}}_5={\mathcal{B}}_7\left[\begin{array}{c}{q}_1\\ 0\end{array}\right]$, ${\mathcal{B}}_2={\mathcal{B}}_4={\mathcal{B}}_6={\mathcal{B}}_8\left[\begin{array}{c}-q_1\\ 0\end{array}\right]$.

The FMFs are defined as

$$\begin{array}{c}\hfill \begin{array}{c}{\iota }_1\left(\xi \right)={\displaystyle \frac{y-y_{min}}{y_{max}-y_{min}}}.{\displaystyle \frac{sin{\xi }_1-(sin{q}_1/q_1){\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{{\xi }_1+q_1}{2q_1}}\hfill \\ {\iota }_2\left(\xi \right)={\displaystyle \frac{y-y_{min}}{y_{max}-y_{min}}}.{\displaystyle \frac{sin{\xi }_1-(sin{q}_1/q_1){\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{q_1-{\xi }_1}{2q_1}}\hfill \\ {\iota }_3\left(\xi \right)={\displaystyle \frac{y-y_{min}}{y_{max}-y_{min}}}.{\displaystyle \frac{{\xi }_1-sin{\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{{\xi }_1+q_1}{2q_1}}\hfill \\ {\iota }_4\left(\xi \right)={\displaystyle \frac{y-y_{min}}{y_{max}-y_{min}}}.{\displaystyle \frac{{\xi }_1-sin{\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{q_1-{\xi }_1}{2q_1}}\hfill \\ {\iota }_5\left(\xi \right)={\displaystyle \frac{y_{max}-y}{y_{max}-y_{min}}}.{\displaystyle \frac{sin{\xi }_1-(sin{q}_1/q_1){\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{{\xi }_1+q_1}{2q_1}}\hfill \\ {\iota }_6\left(\xi \right)={\displaystyle \frac{y_{max}-y}{y_{max}-y_{min}}}.{\displaystyle \frac{sin{\xi }_1-(sin{q}_1/q_1){\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{q_1-{\xi }_1}{2q_1}}\hfill \\ {\iota }_7\left(\xi \right)={\displaystyle \frac{y_{max}-y}{y_{max}-y_{min}}}.{\displaystyle \frac{{\xi }_1-sin{\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{{\xi }_1+q_1}{2q_1}}\hfill \\ {\iota }_8\left(\xi \right)={\displaystyle \frac{y_{max}-y}{y_{max}-y_{min}}}.{\displaystyle \frac{{\xi }_1-sin{\xi }_1}{(1-(sin{q}_1/q_1)){\xi }_1}}.{\displaystyle \frac{q_1-{\xi }_1}{2q_1}}\hfill \end{array}\end{array}$$

In opposition to the T-SF model, the consequence part of the PF model is represented by a polynomial equation rather than a linear one. Then, the NS (40) is represented by the following PF model:

$$\begin{array}{c}\hfill {\displaystyle T_a^{\theta ,{\psi }_a}\xi =\sum _{i=1}^2{\iota }_i\{{\mathfrak{A}}_i\left(\xi \right)\xi +{\mathfrak{A}}_{\beta i}\xi (\rho -\beta )+{\mathfrak{B}}_i\left(\xi \right)u\}}\end{array}$$

(42)

where ${\mathfrak{A}}_1\left(\xi \right)=\left[\begin{array}{cc}-1+{\xi }_1+{\xi }_1^2+{\xi }_1{\xi }_2-{\xi }_2^2& 1\\ -1& -1\end{array}\right]$, ${\mathfrak{A}}_2\left(\xi \right)=\left[\begin{array}{cc}-1+{\xi }_1+{\xi }_1^2+{\xi }_1{\xi }_2-{\xi }_2^2& 1\\ 0.2172& -1\end{array}\right]$, ${\mathfrak{A}}_{{\beta }_1}\left(\xi \right)={\mathfrak{A}}_{{\beta }_2}\left(\xi \right)\left[\begin{array}{cc}{\xi }_1& 0\\ 0.2& 0\end{array}\right]$, ${\mathcal{B}}_1\left(\xi \right)={\mathcal{B}}_2\left(\xi \right)=\left[\begin{array}{c}{\xi }_1^2+5\\ 0\end{array}\right]$.

The FMFs are defined as

$$\begin{array}{c}\hfill {\iota }_1={\displaystyle \frac{sin\left({\xi }_1\right)+0.2172{\xi }_1}{1.2172{\xi }_1}};{\iota }_2={\displaystyle \frac{{\xi }_1-sin\left({\xi }_1\right)}{1.2172{\xi }_1}}\end{array}$$

Figure 4 shows the behavior of (42) with $u=0$ for these initial states: $\xi \left(\rho \right)={\left[\begin{array}{cc}8& -3\end{array}\right]}^T$, ${\left[\begin{array}{cc}-5& -5\end{array}\right]}^T$, ${\left[\begin{array}{cc}5& 5\end{array}\right]}^T$ and ${\left[\begin{array}{cc}-8& 3\end{array}\right]}^T$ for $\rho \in [-0.9,0]$. From this figure, we can see that the system (42) is unstable.

Figure 4. Behaviors in ${\xi }_1\left(\rho \right)$–${\xi }_2\left(\rho \right)$ plane (without feedback).

We assume that ${\xi }_1\in [0.01,4]$ and the interval divides into two segments as $s_1=[0.01,2]$ and $s_2=[2,4]$. Using the polynomial approximation method, we approximate the fuzzy MFs using the following fourth-degree square polynomials:

$$\begin{array}{ccc}\hfill p_{1,s_1}\left({\xi }_1\right)& =& {(-0.0617{\xi }_1^2-0.0066{\xi }_1+1.0010)}^2\hfill \\ \hfill p_{1,s_2}\left({\xi }_1\right)& =& {(-0.0186{\xi }_1^2-0.1869{\xi }_1+1.1944)}^2\hfill \\ \hfill p_{2,s_1}\left({\xi }_1\right)& =& {(-0.0253{\xi }_1^2+0.3885{\xi }_1-0.0026)}^2\hfill \\ \hfill p_{2,s_2}\left({\xi }_1\right)& =& {(-0.0560{\xi }_1^2+0.4958{\xi }_1-0.0989)}^2\hfill \end{array}$$

Figure 5 displays the FMFs and their approximations in regions $s_1$ and $s_2$. From this figure, it is evident that the polynomial approximation method works effectively. This polynomial approximation allows us to include the precise information of the FMFs in the stabilization conditions.

Figure 5. FMFs and their square polynomials approximations in ${\xi }_1\in [0.01,4]$.

In our example, we solved the S-O-S conditions in Theorem 2 for $\sigma =0.9$, $\beta =0.9$ and $n_1=n_2=n_3={10}^{-4}$. We obtained the following solution:

$$\begin{array}{ccc}\hfill {\mathfrak{F}}_1\left({\xi }_2\right)& =& \left[\begin{array}{cc}0.1964{\xi }_2^2+0.0416{\xi }_2+2.6623& 3.2530\times {10}^{-5}{\xi }_2^2+4.9540\times {10}^{-5}{\xi }_2-0.00408\end{array}\right]\hfill \\ \hfill {\mathfrak{F}}_2\left({\xi }_2\right)& =& \left[\begin{array}{cc}2.7990{\xi }_2^2+0.0681{\xi }_2+4.2348& 2.3139\times {10}^{-4}{\xi }_2^2-4.1918\times {10}^{-5}{\xi }_2+0.00568\end{array}\right]\hfill \end{array}$$

(43)

Figure 6 shows the behavior of (42) for the same initial states as in Figure 4. It is important to highlight that the proposed controller effectively stabilizes the system’s states across different initial conditions.

Figure 6. Behaviors in ${\xi }_1\left(\rho \right)$–${\xi }_2\left(\rho \right)$ plane.

6. Conclusions

This paper has focused on the SAS of delayed nonlinear dynamic systems with GCD. We have presented a novel framework for GCPF modeling. PF models have been used extensively in the literature to describe delayed nonlinear systems with integer-order derivatives. However, these models were utilized for the first time in this work, representing the dynamic of delayed nonlinear systems with GCD. We have derived the exponential SAS conditions of these systems using a novel Lyapunov–Krasovskii function. Notably, these conditions are represented in terms of S-O-Ss, allowing for numerical (and partially symbolic) solutions using the recently developed SOSTOOLS. Further improvement is introduced by taking the FMFs into account. In fact, these functions are estimated as square polynomials, through a polynomial approximation method, in order to derive the SAS conditions dependent on FMFs. Two design examples are provided to demonstrate the effectiveness and applicability of our approach.