Finite-Time Stability Analysis for Delayed Fuzzy Hadamard Fractional-Order Systems

Abstract

This paper focuses on the Finite-Time Stability Analysis (FTSA) problem for a Hadamard fractional-order system with time delay represented by a Delayed Takagi–Sugeno Fuzzy Model (DTSFM). Based on the Linear Matrix Inequality (LMI) approach, we propose two methods for FTSA. The first procedure is accomplished in two steps, while the second one is provided in only one step. The proposed results are extended to the case of DTSFM with uncertainties. An example is proposed to validate these results and to demonstrate the advantages of the one-step results compared to the two-step procedure.

1. Introduction

The stability theory of Fractional Differential Equations (FDEs), with and without time delay, has received significant attention in recent decades (see [1,2,3,4,5]). Therefore, effective techniques and methodologies have been used to determine stability criteria for FDEs, e.g., the Lyapunov method [6,7,8], Gronwall inequalities [9] and the Fixed-Point Approach (FPA) [10]. Note that the feasibility of stability criteria, proposed in the works cited previously, implies the stability of FDEs over an infinite time interval. However, it is more suitable, in practical applications, that FDEs are stable within a finite time interval, the Finite-Time Stability (FTS) concept. So, there is significant research interest in studying FTS for FDEs. The authors in [11] have studied the FTS of linear stochastic FDEs with time delay using the Gronwall inequality. Ben Makhlouf et al. have proven, in [12], FTS using FPA. The authors in [6] have investigated the FTS of FDEs with the Caputo derivative using Lyapunov methods.

Due to its ability to model nonlinear systems effectively, the Takagi–Sugeno fuzzy model (TSFM) [13] is of critical importance in a range of fields. The FTS of integer-order TSFM is a well-explored topic in the literature, with various works presenting different techniques to address this issue. Research by Ruan et al. [14] focuses on designing a robust controller for delayed TSFM (DTSFM), emphasizing FTS. The challenge of designing an input–output FTS of DTSFM is addressed in [15]. The authors in [16] have investigated the FTS problem of a Hydraulic Turbine System (HTS) represented by a DTSFM. The issue of $H_{\infty }$ FTS is addressed for singular TSFM in the standard case [17] and in interval type-2 case [18]. All the results previously cited concerned the integer-order case of TSFM with and without delay. In the literature, very few studies have addressed the problem of FTS for fractional-order TSFM. For example, in [19], a generalized TSFM with the Caputo derivative is used to examine the FTS for HTS. However, it is noted that there are no works in the related literature on FTS in the field of fractional-order DTSFM.

Taking into account the aforementioned discussions, the following work aims to study the FTSA for a class of DTSFMs using the Caputo–Hadamard derivative. The main highlights of the paper are as follows:

• The FTSA problem of delayed fractional-order systems with the Caputo–Hadamard derivative has not yet been addressed in the literature. Accordingly, this issue is resolved in this work.
• The issue of FTSA is similarly explored in [20] for delayed fractional-order systems utilizing the Caputo derivative. Like this work, an interval time-varying delay is considered in this paper. The LMI conditions in [20] are formulated with a tuning parameter that must represent the lower bound of the time delay. In this paper, this restriction is overcome and the tuning parameter can be any positive scalar.
• Unlike [6,20,21], which did not account for the nonlinearities present in the state and input matrices of the system, the TSFM is used in this work to effectively approximate these nonlinearities by a set of local linear models presented in the form of IF–THEN rules.
• In the existing results that consider the Caputo derivative, the problem of FTS is solved in two steps [6,20,21,22]. Firstly, the LMI conditions are solved. Then, conditions containing Mittag-Leffler functions are verified. In contrast to these works, we address the drawback of the two-step results by proposing one-step results that require solving LMIs and verifying the conditions containing Mittag-Leffler functions in only a single step.

The content of the paper is as follows: In Section 2, we give some preliminary results. Section 3 is devoted to investigating the FTS results. Finally, we present in Section 4 an illustrative example to show the effectiveness of our results.

Notations: For a matrix M, ${\left\{M\right\}}_{Sym}=M+M^T$, $\varkappa$ is employed to designate a matrix block prompted by symmetry.

2. Preliminaries

In this section, we provide specific definitions and lemmas, as outlined in [8].

Definition 1 

([8]). The Hadamard integral of a locally integrable function g of order $\alpha >0$ is given by

$$\begin{array}{c}\hfill I^{\alpha }g\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_1^t{\left(log{\displaystyle \frac{t}{s}}\right)}^{\alpha -1}{\displaystyle \frac{g\left(s\right)}{s}}ds,t\ge 1.\end{array}$$

(1)

Definition 2 

([8]).  The Caputo–Hadamard derivative with order $0<\alpha <1$ for an absolutely continuous function $g:[1,\infty )\to \mathbb{R}$ is as follows:

$$\begin{array}{c}\hfill {}_C{}^{H}D_1^{\alpha }g\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_1^t{\left(log{\displaystyle \frac{t}{s}}\right)}^{-\alpha }{g}^{\prime }\left(s\right)ds,t\ge 1.\end{array}$$

(2)

Lemma 1 

([8]).  Let $\alpha \in (0,1)$ and $S\in {\mathbb{R}}^{n\times n}$ is a Symmetric Positive Definite (SPD) matrix. Then,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{}_C{}^{H}D_1^{\alpha }\left(x^T\left(t\right)Sx\left(t\right)\right)\le x^T\left(t\right)S{}_C{}^{H}D_1^{\alpha }x\left(t\right),t\ge 1.\end{array}$$

(3)

Definition 3 

([8]).  The Mittag-Leffler function is given by

$$\begin{array}{c}\hfill {\displaystyle E_{c_1,c_2}\left(z\right)=\sum _{k=0}^{+\infty }\frac{z^k}{\mathrm{\Gamma }(k{c}_1+c_2)},}\end{array}$$

(4)

where $c_1,c_2>0$, $z\in \mathbb{C}$.

• The solution of the FOS

$$\begin{array}{ccc}\hfill {}_C{}^{H}D_1^{\alpha }y\left(t\right)& =& \theta y+m\left(t\right),t\ge 1,\hfill \\ \hfill y\left(1\right)& =& y_0,\hfill \end{array}$$

(5)

is given by [23]

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\displaystyle E_{\alpha }\left(\theta {\left(logt\right)}^{\alpha }\right)y_0+{\int }_1^t{\left(log\frac{t}{l}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\theta {\left(log\frac{t}{l}\right)}^{\alpha }\right)m\left(l\right)\frac{dl}{l}.}\hfill \end{array}$$

(6)

Lemma 2. 

Let $T\ge 1$, $\iota >0$ $M_1\ge 1$, $M_2\ge 0$ and a function $H:[1-\iota ,T]⟶{\mathbb{R}}_{+}$ be non-decreasing. Then,

$$H\left(t\right)\le M_{1}H\left(1\right)+M_{2}H(t-\iota ),\forall t\in [1-\iota ,T]$$

implies

$${\displaystyle H\left(t\right)\le M_{1}H\left(1\right)\sum _{\kappa =0}^{\left[\frac{T}{\iota }\right]+1}{M}_2^{\kappa }.}$$

Proof. 

The proof is similar to the proof of Lemma $2.7$ in [6]. □

In the following, let us consider the DTSFM:

• Plant Rule $\mu$: If ${\beta }_1\left(t\right)$ is ${\xi }_{\mu 1}$ and … and ${\beta }_p\left(t\right)$ is ${\xi }_{\mu p}$, then

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_C{}^{H}D_1^{\alpha }x\left(t\right)=A_{\mu }x\left(t\right)+D_{\mu }x(t-\tau \left(t\right)),1\le t\le T,\hfill \\ x\left(t\right)=\phi \left(t\right),t\in [1-t_{max},1],\hfill \end{array}\right.\end{array}$$

(7)

where ${\beta }_l\left(t\right)(l=1,2,\dots ,p)$ are the premise variables associated with the fuzzy sets ${\xi }_{\mu l},(\mu =1,2,\dots ,r)$; $x\left(t\right)$ and $x(t-\tau (t\left)\right)$ are, respectively, the state and the delayed state at the instant t; $\{A_{\mu },D_{\mu }\}\in {\mathbb{R}}^{n_1\times n_1}$; $\tau \left(t\right)$ is an interval delay satisfying $0<t_{min}\le \tau \left(t\right)\le t_{max},\forall t\ge 1$; and $\phi \in C\left([1-t_{max},1],{\mathbb{R}}^{n_1}\right)$.

• The global DTSFM is

$$\begin{array}{c}\hfill {\displaystyle {}_C{}^{H}D_1^{\alpha }x\left(t\right)=\sum _{\mu =1}^r{s}_{\mu }\left(\beta \left(t\right)\right)\left(A_{\mu }x\left(t\right)+D_{\mu }x(t-\tau \left(t\right))\right)}\end{array}$$

(8)

where

$$\begin{array}{c}\hfill s_{\mu }\left(\beta \left(t\right)\right)\ge 0,\sum _{\mu =1}^r{s}_{\mu }\left(\beta \left(t\right)\right)=1,\mathrm{in}\mathrm{which}\beta \left(t\right)=\left[\begin{array}{ccc}{\beta }_1\left(t\right)& & \dots {\beta }_p\left(t\right)\end{array}\right].\end{array}$$

(9)

Due to uncertainties, $A_{\mu }$ and $B_{\mu }$ can be decomposed into a nominal part and an uncertain part as follows:

$$\begin{array}{c}\hfill A_{\mu }=a_{\mu }+\Delta A_{\mu },D_{\mu }=d_{\mu }+\Delta D_{\mu }\end{array}$$

(10)

where $a_{\mu }$, $d_{\mu }$ are constant matrices, and $\Delta A_{\mu }$ and $\Delta D_{\mu }$ are expressed as follows:

• $\Delta A_{\mu }=F_{\mu }{R}_{\mu }\left(t\right)G_{\mu },\Delta D_{\mu }=H_{\mu }{S}_{\mu }\left(t\right)K_{\mu }$, in which ${\left\{R_{\mu }\left(t\right)\right\}}_{Sym}\le I$ and ${\left\{S_{\mu }\left(t\right)\right\}}_{Sym}\le I$.

Definition 4 

([6]).  The FOS (8) is FTS with respect to $\{{\epsilon }_1,{\epsilon }_2,T\}$, ${\epsilon }_1<{\epsilon }_2$ if

$${\parallel \phi \parallel }^2\le {\epsilon }_1$$

which implies

$${\parallel x\left(t\right)\parallel }^2\le {\epsilon }_2,\forall t\in [1,T],$$

with ${\displaystyle \parallel \phi \parallel =\underset{t\in [1-t_{max},1]}{sup}\parallel \phi \left(t\right)\parallel }$.

Lemma 3 

([24]).  Consider matrices $\mathcal{M}$ and $\mathcal{Q}$; then, $\forall w\in {\mathbb{R}}^{+}$

$$\begin{array}{c}\hfill {\left\{\mathcal{M}\mathcal{T}\left(t\right)\mathcal{Q}\right\}}_{Sym}\le w\mathcal{M}{\mathcal{M}}^T+{\displaystyle \frac{1}{w}}{\mathcal{Q}}^T\mathcal{Q}\end{array}$$

(11)

is satisfied for any matrices $\mathcal{T}\left(t\right)$ such that $\mathcal{T}{\left(t\right)}^T\mathcal{T}\left(t\right)\le I$.

3. Main Results

Theorem 1. 

For a given scalar $\theta >0$, the FOS (8) is FTS with respect to $\{{\epsilon }_1,{\epsilon }_2,T\}$, ${\epsilon }_1<{\epsilon }_2$ if there $\exists P=P^T\in {\mathbb{R}}^{n_1\times n_1}$ such that the following conditions are met:

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{\Theta }}_{\mu }=\left[\begin{array}{cc}{\left\{P{A}_{\mu }\right\}}_{Sym}-\theta P& P{D}_{\mu }\\ \varkappa & -\theta P\end{array}\right]& <& 0\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(P\right)}{E}_{\alpha ,1}\left(\theta {(logT)}^{\alpha }\right)\sum _{\kappa =0}^{\left[\frac{T}{t_{min}}\right]+1}{\left[\theta {(logT)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {(logT)}^{\alpha }\right)\right]}^{\kappa }{\epsilon }_1}& <& {\epsilon }_2.\hfill \end{array}$$

(13)

Proof. 

Let

$$\vartheta \left(x\right)=x{\left(t\right)}^{T}Px\left(t\right).$$

Using Lemma 1, we obtain

$${}_C{}^{H}D_1^{\alpha }\vartheta \left(t\right)\le 2x{\left(t\right)}^{T}P{}_C{}^{H}D_1^{\alpha }x\left(t\right)$$

Replacing ${}_C{}^{H}D_1^{\alpha }x\left(t\right)$ by its expression, we obtain

$$\begin{array}{ccc}\hfill {}_C{}^{H}D_1^{\alpha }\vartheta \left(t\right)& \le & {\displaystyle \sum _{\mu =1}^r{s}_{\mu }\left(\beta \left(t\right)\right)2x{\left(t\right)}^{T}P\left(A_{\mu }x\left(t\right)+D_{\mu }x(t-\tau \left(t\right))\right)}\hfill \end{array}$$

(14)

Taking into account (9), we obtain

$$\begin{array}{ccc}\hfill {}_C{}^{H}D_1^{\alpha }\vartheta \left(t\right)& \le & {\displaystyle \sum _{\mu =1}^r{s}_{\mu }\left(\beta \left(t\right)\right)\eta {\left(t\right)}^T{\mathrm{\Theta }}_{\mu }\eta \left(t\right)+\mathcal{V}\left(t\right)}\hfill \end{array}$$

(15)

where

$$\begin{array}{c}\hfill \eta {\left(t\right)}^T=\left[\begin{array}{cc}x{\left(t\right)}^T,& x{(t-\tau \left(t\right))}^T\end{array}\right],\mathcal{V}\left(t\right)=\theta \vartheta \left(t\right)+\theta x{\left(t-\tau \left(t\right)\right)}^{T}Px\left(t-\tau \left(t\right)\right)\end{array}$$

Then, from (12), we obtain

$${}_C{}^{H}D_1^{\alpha }\vartheta \left(t\right)\le \mathcal{V}\left(t\right).$$

(16)

Let

$$g\left(t\right)={}_C{}^{H}D_1^{\alpha }\vartheta \left(t\right)-\theta \vartheta \left(t\right).$$

Using (5) and (16), we obtain

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& =& {\displaystyle E_{\alpha ,1}\left(\theta {(logt)}^{\alpha }\right)\vartheta \left(1\right)+\theta {\int }_1^t{\left(log\frac{t}{l}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\theta {\left(log\frac{t}{l}\right)}^{\alpha }\right)g\left(l\right)\frac{dl}{l}}\hfill \\ & \le & E_{\alpha ,1}\left(\theta {(logt)}^{\alpha }\right)\vartheta \left(1\right)\hfill \\ & +& {\displaystyle \theta {\int }_1^t{\left(log\frac{t}{l}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\theta {\left(log\frac{t}{l}\right)}^{\alpha }\right)x{\left(l-\tau \left(l\right)\right)}^{T}Px\left(l-\tau \left(l\right)\right)\frac{dl}{l}}\hfill \\ & \le & E_{\alpha ,1}\left(\theta {(logt)}^{\alpha }\right)\vartheta \left(1\right)\hfill \\ & +& {\displaystyle \theta \underset{1\le l\le t}{sup}\left(x{\left(l-\tau \left(l\right)\right)}^{T}Px\left(l-\tau \left(l\right)\right)\right){\int }_1^t{\left(log\frac{t}{l}\right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\theta {\left(log\frac{t}{l}\right)}^{\alpha }\right)\frac{dl}{l}.}\hfill \end{array}$$

(17)

Using the change in variable $u=logl$, we obtain from Lemma 7 in [25]

$$\begin{array}{ccc}\hfill & & {\int }_1^t{{\displaystyle \left(log{\displaystyle \frac{t}{l}}\right)}}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\theta {\left(log{\displaystyle \frac{t}{l}}\right)}^{\alpha }\right){\displaystyle \frac{dl}{l}}\hfill \\ & =& {\int }_0^{logt}{{\displaystyle \left(logt-u\right)}}^{\alpha -1}{E}_{\alpha ,\alpha }\left(\theta {\left(logt-u\right)}^{\alpha }\right)du\hfill \\ & =& {\left(logt\right)}^{\alpha }{E}_{\alpha ,\alpha +1}{\displaystyle (}\theta {\left(logt\right)}^{\alpha }{\displaystyle )}.\hfill \end{array}$$

(18)

Therefore,

$$\begin{array}{ccc}\hfill \vartheta \left(t\right)& \le & {\displaystyle E_{\alpha ,1}\left(\theta {(logt)}^{\alpha }\right)\vartheta \left(1\right)+\theta \underset{1-t_{max}\le l\le t-t_{min}}{sup}\vartheta \left(l\right){\left(logt\right)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {\left(logt\right)}^{\alpha }\right)}\hfill \\ & \le & {\displaystyle E_{\alpha ,1}\left(\theta {(logt)}^{\alpha }\right){\phi }^T\left(1\right)P\phi \left(1\right)+\theta \underset{1-t_{max}\le l\le t-t_{min}}{sup}\vartheta \left(l\right){\left(logt\right)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {\left(logt\right)}^{\alpha }\right).}\hfill \end{array}$$

(19)

For $1\le \upsilon \le t\le T$, we obtain

$${\displaystyle \vartheta \left(s\right)\le E_{\alpha ,1}\left(\theta {(logT)}^{\alpha }\right){\phi }^T\left(1\right)P\phi \left(1\right)+\theta \underset{1-t_{max}\le l\le t-t_{min}}{sup}\vartheta \left(l\right){\left(logT\right)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {\left(logT\right)}^{\alpha }\right).}$$

(20)

Therefore,

$${\displaystyle \underset{1-t_{max}\le l\le t}{sup}\vartheta \left(l\right)\le E_{\alpha ,1}\left(\theta {(logT)}^{\alpha }\right)\underset{1-t_{max}\le l\le 1}{sup}{\phi }^T\left(l\right)P\phi \left(l\right)+\theta \underset{1-t_{max}\le l\le t-t_{min}}{sup}\vartheta \left(l\right){\left(logT\right)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {\left(logT\right)}^{\alpha }\right).}$$

(21)

Let ${\displaystyle H\left(t\right)=\underset{1-t_{max}\le l\le t}{sup}\vartheta \left(l\right)}$, $M_1=E_{\alpha ,1}\left(\theta {(logT)}^{\alpha }\right)$ and $M_2=\theta {\left(logT\right)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {\left(logT\right)}^{\alpha }\right).$

• Then,

$$H\left(t\right)\le M_{1}H\left(1\right)+M_{2}H(t-t_{min}),\forall t\in [1,T].$$

• It follows from Lemma 2 that

$${\displaystyle H\left(t\right)\le M_{1}H\left(1\right)\sum _{\kappa =0}^{\left[\frac{T}{t_{min}}\right]+1}{M}_2^{\kappa }.}$$

Thus,

$${\displaystyle {\parallel x\left(t\right)\parallel }^2\le \frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(P\right)}{E}_{\alpha ,1}\left(\theta {(logT)}^{\alpha }\right)\sum _{\kappa =0}^{\left[\frac{T}{t_{min}}\right]+1}{\left[\theta {(logT)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {(logT)}^{\alpha }\right)\right]}^{\kappa }\underset{1-t_{max}\le l\le 1}{sup}{\parallel \phi \left(l\right)\parallel }^2.}$$

Hence, if

$${\displaystyle \underset{1-t_{max}\le l\le 1}{sup}{\parallel \phi \left(l\right)\parallel }^2<{\epsilon }_1}$$

and

$${\displaystyle \frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(P\right)}{E}_{\alpha ,1}\left(\theta {(logT)}^{\alpha }\right)\sum _{\kappa =0}^{\left[\frac{T}{t_{min}}\right]+1}{\left[\theta {(logT)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {(logT)}^{\alpha }\right)\right]}^{\kappa }{\epsilon }_1<{\epsilon }_2,}$$

then

$${\parallel x\left(t\right)\parallel }^2\le {\epsilon }_2.$$

Then, the FOS is FTS with respect to $\{{\epsilon }_1,{\epsilon }_2,T\}$. □

3.1. Nominal Case

In this subsection, we consider the nominal case in which $\Delta A_{\mu }=0$ and $\Delta D_{\mu }=0$. It is clear that the conditions in Theorem 1 cannot be solved directly by using LMI Toolbox. Additionally, to ameliorate the performance of FTS, minimizing ${\epsilon }_2$ (or maximizing T) is crucial. This problem is solved by utilizing the following two-step algorithm:

Theorem 2. 

The FOS (8) without uncertainties is FTS with respect to $\{{\epsilon }_1,{\epsilon }_2,T\}$, ${\epsilon }_1<{\epsilon }_2$ if the following hold:

• Step 1: For a given scalar $\theta >0$, there $\exists P=P^T\in {\mathbb{R}}^{n_1\times n_1}$ such that LMIs (12) are satisfied.
• Step 2: P consists of a known matrix obtained in step 1. Minimize ${\epsilon }_2$ subject to (13).

Remark 1.

The main disadvantage of this theorem is that the conditions are solved via a two-step algorithm, which increases the conservatism. We mitigate this drawback by proposing the following single-step method:

Theorem 3. 

The FOS (8) without uncertainties is FTS with respect to $\{{\epsilon }_1,{\epsilon }_2,T\}$, ${\epsilon }_1<{\epsilon }_2$ if a feasible solution exists for the following problem.

• Minimize ${\epsilon }_2$ subject to the following:
• $\exists \{P=P^T\in {\mathbb{R}}^{n_1\times n_1},{\zeta }_m\in \mathbb{R},{\zeta }_M\in \mathbb{R}\}$ such that (12) and the following LMIs are satisfied:

$$\begin{array}{ccc}\hfill & & 0<{\zeta }_{m}I<P<{\zeta }_{M}I\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill & & {\displaystyle \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\zeta }_M{E}_{\alpha ,1}\left(\theta {(logT)}^{\alpha }\right)\sum _{\kappa =0}^{\left[\frac{T}{t_{min}}\right]+1}{\left[\theta {(logT)}^{\alpha }{E}_{\alpha ,\alpha +1}\left(\theta {(logT)}^{\alpha }\right)\right]}^{\kappa }{\epsilon }_1-{\zeta }_m{\epsilon }_2<0.}\hfill \end{array}$$

(23)

3.2. Uncertain Case

Now, we focus on the challenge of uncertain system. We propose two theorems. The first one is given in two steps, while the second is provided in only one step:

Theorem 4. 

The FOS (8) is FTS with respect to $\{{\epsilon }_1,{\epsilon }_2,T\}$, ${\epsilon }_1<{\epsilon }_2$ if the following hold:

• Step 1: For a given scalar $\theta >0$, there $\exists \{P=P^T\in {\mathbb{R}}^{n_1\times n_1},w_{1\mu }\in {\mathbb{R}}^{+},w_{2\mu }\in {\mathbb{R}}^{+}\}$ such that these LMIs are satisfied:

$$\begin{array}{ccc}\hfill {\Omega }_{\mu }=\left[\begin{array}{cccc}{\left\{P{a}_{\mu }\right\}}_{Sym}-\theta P+w_{1\mu }{G}_{\mu }^T{G}_{\mu }+w_{2\mu }{K}_{\mu }^T{K}_{\mu }& P{d}_{\mu }& P{F}_{\mu }& P{H}_{\mu }\\ \varkappa & -\theta P& 0& 0\\ \varkappa & \varkappa & -w_{1\mu }I& 0\\ \varkappa & \varkappa & \varkappa & -w_{2\mu }I\end{array}\right]& <& 0\hfill \end{array}$$

(24)

• Step 2: P consists of a known matrix obtained in step 1. Minimize ${\epsilon }_2$ subject to (13).

Proof. 

In this uncertain case, we obtain the following inequalities instead of (12):

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_{\mu }+\Delta {\mathrm{\Theta }}_{\mu }<0\end{array}$$

(25)

where

$$\begin{array}{ccc}\hfill \Delta {\mathrm{\Theta }}_{\mu }& =& \left[\begin{array}{cc}{\{P\Delta A_{\mu }\}}_{Sym}& P\Delta D_{\mu }\\ \varkappa & 0\end{array}\right]\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=& {\{\left[\begin{array}{c}P{F}_{\mu }\\ 0\end{array}\right]R_{\mu }\left(t\right)\left[\begin{array}{cc}{G}_{\mu }& 0\end{array}\right]\}}_{Sym}+{\{\left[\begin{array}{c}P{H}_{\mu }\\ 0\end{array}\right]S_{\mu }\left(t\right)\left[\begin{array}{cc}0& K_{\mu }\end{array}\right]\}}_{Sym}\hfill \end{array}$$

(27)

Applying Lemma 3, we obtain

$$\begin{array}{ccc}\hfill \Delta {\mathrm{\Theta }}_{\mu }& \le & w_{1\mu }\left[\begin{array}{c}{G}_{\mu }^T\\ 0\end{array}\right]\left[\begin{array}{cc}{G}_{\mu }& 0\end{array}\right]+{\displaystyle \frac{1}{w_{1\mu }}}\left[\begin{array}{c}P{F}_{\mu }\\ 0\end{array}\right]\left[\begin{array}{cc}{F}_{\mu }^{T}P& 0\end{array}\right]\hfill \end{array}$$

(28)

$$\begin{array}{ccc}& \hspace{1em}\hspace{1em} & +w_{2\mu }\left[\begin{array}{c}{K}_{\mu }^T\\ 0\end{array}\right]\left[\begin{array}{cc}{K}_{\mu }& 0\end{array}\right]+{\displaystyle \frac{1}{w_{2\mu }}}\left[\begin{array}{c}P{H}_{\mu }\\ 0\end{array}\right]\left[\begin{array}{cc}{H}_{\mu }^{T}P& 0\end{array}\right]\hfill \end{array}$$

(29)

Using Schur’s complement, we derive (24). □

Similar to Theorem 3, we obtain the following result for the uncertain case:

Theorem 5. 

The FOS (8) is FTS with respect to $\{{\epsilon }_1,{\epsilon }_2,T\}$, ${\epsilon }_1<{\epsilon }_2$ if a feasible solution exists for the following problem:

Minimize ${\epsilon }_2$ subject to

$\exists \{P=P^T\in {\mathbb{R}}^{n_1\times n_1},{\zeta }_m\in {\mathbb{R}}^{+},{\zeta }_M\in {\mathbb{R}}^{+},w_{1\mu }\in {\mathbb{R}}^{+},w_{2\mu }\in {\mathbb{R}}^{+}\}$ such that (22)–(24) are satisfied.

4. Illustrative Example

Consider an example in the form of (8) with $r=2$ in which

$$\begin{array}{ccc}\hfill & & A_1=\left[\begin{array}{cc}-30& 0\\ 0& -2.1\end{array}\right],A_2=\left[\begin{array}{cc}-15& 0\\ 0& -2.1\end{array}\right],D_1=\left[\begin{array}{cc}1& 0.9\\ 0& -2\end{array}\right],D_2=\left[\begin{array}{cc}2& 0.9\\ 0& -0.7\end{array}\right],\hfill \\ \hfill & & \beta \left(t\right)=x_1\left(t\right),s_1\left(\beta \left(t\right)\right)=({\displaystyle \frac{1-sin\left(x_1\left(t\right)\right)}{2}}),s_2\left(\beta \left(t\right)\right)=1-s_1\left(\beta \left(t\right)\right).\hfill \end{array}$$

We choose $T=2.8,\theta =0.9,\tau \left(t\right)=2.3+0.3sin\left(t\right)$ and ${\epsilon }_1=0.01$. It is clear that $t_{min}=2$ and $t_{max}=2.6.$

By applying the two-step LMI conditions from Theorem 2, we obtain the following solution:

$$\begin{array}{c}\hfill P=\left[\begin{array}{cc}0.0212& 4\times {10}^{-4}\\ 4\times {10}^{-4}& 0.2147\end{array}\right]\end{array}$$

and the minimum of ${\epsilon }_2$ is $8.1$.

Now, by utilizing Theorem 3 to address the corresponding only one-step LMIs, we obtain

$$\begin{array}{c}\hfill P=\left[\begin{array}{c}0.8533-5\times {10}^{-4}\\ -5\times {10}^{-4}0.8522\end{array}\right],{\zeta }_m=0.8511,{\zeta }_M=0.8540.\end{array}$$

and the minimum of ${\epsilon }_2$ is $0.80$, which numerically demonstrates the that the one-step procedure is less conservative than the two-step LMI conditions.

Figure 1 shows the time evolution of $x\left(t\right)$ for initial conditions $x\left(t\right)=\phi \left(t\right)=[0.03,-0.03],t\in [1-t_{max},1].$

Figure 1. Time evolution of $x\left(t\right)$.

Now, we consider the DTSFM with uncertainties, in which

$$\begin{array}{c}\hfill \Delta A_1=\Delta A_2=\left[\begin{array}{c}0.2\\ 0.2\end{array}\right]sin\left(t\right)\left[\begin{array}{cc}0.1& 0\end{array}\right],\Delta D_1=\Delta D_2=\left[\begin{array}{c}0.3\\ 0.1\end{array}\right]sin\left(t\right)\left[\begin{array}{cc}0& 0.2\end{array}\right]\end{array}$$

(30)

Firstly, we obtain that the minimum of ${\epsilon }_2$ is $10.36$ by using Theorem 4. Secondly, we use the one-step result proposed in Theorem 5 and we find that the optimal value of ${\epsilon }_2$ is $0.80$, which confirms Remark 1.

5. Conclusions

We have investigated the issue of the FTSA of nonlinear delayed fractional-order systems, nominal and uncertain, with the Caputo–Hadamard derivative. The TSFM is used to approximate the nonlinearities using a set of local linear models expressed in the form of IF–THEN rules. In each case, nominal and uncertain, two results are presented. The first one allows us to solve, in two steps, sufficient LMI conditions and verify conditions in terms of Mittag-Leffler functions. The second one allows us to solve the problem in only one step by solving a set of LMI conditions. A numerical example is provided to demonstrate the advantage of the one-step procedure. Hence, this work presents the first attempt to address the FTSA problem of delayed fractional-order nonlinear systems with the Caputo–Hadamard derivative, which is significantly more challenging than the case of integer-order derivatives extensively studied in the literature. We should point out that this paper focuses on the case of a single time-varying delay, $\tau \left(t\right)$. Extending the proposed method to handle multiple delay cases would be a highly interesting direction for future research. Also notice that the Gronwall inequality approach serves as an effective alternative to the Lyapunov approach. While it has been extensively applied to the FTS of fractional order systems with the Caputo derivative [26,27,28,29,30], no studies in the current literature have explored its application to the case with the Caputo–Hadamard derivative. The adaptation of this approach to the FTSA of delayed TSFM remains an open problem.