**Hydrodynamics Interactions of Metachronal Waves on Particulate-Liquid Motion through a Ciliated Annulus: Application of Bio-Engineering in Blood Clotting and Endoscopy**

#### **Muhammad Mubashir Bhatti 1, Asmaa F. Elelamy 2, Sadiq M. Sait <sup>3</sup> and Rahmat Ellahi 4,5,6,\***


Received: 17 February 2020; Accepted: 26 March 2020; Published: 3 April 2020

**Abstract:** This study deals with the mass transport phenomena on the particle-fluid motion through an annulus. The non-Newtonian fluid propagates through a ciliated annulus in the presence of two phenomenon, namely (i) endoscopy, and (ii) blood clot. The outer tube is ciliated. To examine the flow behavior we consider the bi-viscosity fluid model. The mathematical modeling has been formulated for small Reynolds number to examine the inertia free flow. The purpose of this assumption is that wavelength-to-diameter is maximal, and the pressure could be considerably uniform throughout the entire cross-section. The resulting equations are analytically solved, and exact solutions are given for particle- and fluid-phase profiles. Computational software Mathematica has been used to evaluate both the closed-form and numerical results. The graphical behavior across each parameter has been discussed in detail and presented with graphs. The trapping mechanism is also shown across each parameter. It is noticed clearly that particle volume fraction and the blood clot reveal converse behavior on fluid velocity; however, the velocity of the fluid reduced significantly when the fluid behaves as a Newtonian fluid. Schmidt and Soret numbers enhance the concentration mechanism. Furthermore, more pressure is required to pass the fluid when the blood clot appears.

**Keywords:** cilia motion; blood clot; endoscopy; mass transport; particle-fluid

#### **1. Introduction**

Flagella and cilia are two distinct names, but are used interchangeably for similar structure of eukaryotic cells. In animals, cilia, which are hair-like appendages, are prominent in the digestive system, respiratory system, reproductive tracts of human beings, and the nervous systems. The movement of cilia plays an essential part in physiological systems, i.e., circulation, respiration, locomotion, alimentation, spermatic fluid propagation, reproduction, etc. It is well-known that ciliary and flagellar movements consist of active sliding, similar to the peristaltic flow of fluid in smooth muscles, whereas flagellar is more complicated. Cilia can be split into two categories, i.e., non-motile and motile. When cilia and flagella are close to each other, they manifest a propagation of

waves on a large scale known as *Metachronal waves*. Beating cilia produce metachronal waves over the surface in large numbers, on the ciliated surface of protozoa, and the adjoining activity of cilia coordinates via the hydrodynamics interactions. It is worth mentioning here that metachronal waves are self-organized. Cilia produce bending waves to derive single cells through a medium or to push the fluid over the surface of a fixed cell. The standard form of the ciliary model, motivated by the cilium structure or axoneme, is known as sliding filament model. As shown in Figure 1, the axoneme structure has nine microtubule doublets around the outside, and two are located in the center. Most of the cilia beat at approximately 10–40 Hz, but the form of beating varies. Their length starts from 2 μm to several millimeters, and their diameters are approximatley 0.2 μm. As a result, a low Reynolds number (Re→0) approximation can be applied.

**Figure 1.** Strutcure and size of cilia [1] (**a**) typical dimension, (**b**) cross-section, (**c**) cilia stroking.

Nadeem et al. [2] considered the Carreau fluid model to examine the cilia motion through a symmetric channel using the perturbation method. Nadeem and Sadaf [3] discussed the cilia motion of viscous nanofluid through the curvy compliant channel. They used a homotopy analysis method to examine the closed-form solution against the temperature and velocity profile. Maiti and Pandey [4] presented a theoretical study on the nonlinear cilia motion using the Power-law fluid model. Abo-Elkhair et al. [5] used the Adomian decomposition scheme to simluate the cilia motion of magneto-fluid through a ciliated channel. Bhatti et al. [6] discussed the impact of the magnetic field on a ciliated channel using the particle-fluid mechanism. Ashraf et al. [7] examined the peristaltic cilia motion through a human fallopian tube using a Newtonian fluid model. And finally, Ramesh et al. [8] used the behavior of magnetized couple stress fluid model moving through a ciliated channel

Particles in fluid appear in multifarious applications, including biology, geology, chemical engineering, and fluid mechanics [9] to name a few. Several industrial processes include fluidized catalyst beds, pneumatic propagation, and sedimentation. Further, in the biological systems, it involves the flow of blood in the cardiovascular system. The collisions among the particles and the fluids may influence the rheological and the viscosity behavior of the suspension. Particle-wall and particle-particle interactions produce the migration of particles, which causes the anisotropic particle micro-structures and clusters [10,11]. At the mesoscopic scale, a well-known example of the particle-fluid interaction is the movement of the red blood cells (RBC). The flow behavior of the RBC plays a pivotal role in the different pathological and physiological mechanisms. For instance, the rotation and random transverse propagation of RBC in a shear flow plays an essential role in thrombogenesis [12]. These types of movements are firmly associated with the interaction of RBC to RBC and fluid (i.e., plasma) to RBC since one RBC is obstructed by another coming towards it from below or above. RBC is the essential determinant of the blood characteristics in micro-circulation due to their large volume fraction in blood and their aggregability. Mekheimer and Abd Elmaboud [13] investigated the peristaltic motion of fluid having solid particles through different forms of annulus and showed the exact solutions. Mekheimer and Mohamed [14] presented an application of

a clot blood model using particle-fluid flow through an annulus. Further, they considered the pulsatile flow and obtained analytical solutions. Bhatti et al. [15] discussed the behavior of slip effects using a non-Newtonian fluid model that contains spherical particles. Bhatti and Zeeshan [16] explained the blood flow through an annulus filled with particles and fluid in the presence of a variable magnetic field. Some critical analysis of multiphase simulations are given in the following references [17–21].

Mass transfer with heat on Particle-fluid through fixed and fluidized beds play an essential role and provide necessary information for the development and design of numerous mass and heat transfer operation and chemical reactors including a system of particle and fluid. Gireesha et al. [22] investigated the particle-fluid suspension mechanism through a non-isothermal stretching plate in the presence of a magnetic field and radiative heat flux. They applied a numerical method to obtain the solutions. Bhatti et al. [23] presented a mathematical model of particle-fluid motion induced by a peristaltic wave with thermal radiation and electromagnetohydrodynamics effects. Kumar et al. [24] considered a particle-fluid motion with a nonlinear Williamson fluid model towards a stretching sheet with heat transfer effects. Bhatti et al. [25] explored the particle-fluid motion with heat and mass transfer using Sisko fluid model through a Darcy–Brinkman–Forchheimer porous medium. Some relevant studies on particle-fluid with mass and heat transfer are given in the references [26–28].

The main goal of the present study is to examine the mass transport on the particle-fluid suspension through a ciliated annulus with endoscopy and blood clot effects. Endoscopy plays an essential role in exploring the problems in human organs. In the mentioned studies, mostly work has been done with endoscopy and blood clot with simple Newtonian and non-Newtonian fluid models. In contrast, the present study deals with mass transport on particle-fluid motion through a ciliated annulus under different effects. Cilia motion plays a critical part, i.e., ciliary imperfections tend to create several human diseases. A genetic change compromises an appropriate function of the ciliopathies, cilia, which results in chronic disorders, i.e., primary ciliary dyskinesia and Senior–Loken syndrome or nephronophthisis. Furthermore, a flaw in primary cilium in renal tubule cells causes polycystic kidney disease. Ectopic pregnancy can occur due to a lack of functional cilia in a fallopian tube. If the cilia fail to move, then a fertilized ovum is unable to reach the uterus, which results in the ovum implant in a fallopian tube and tubal pregnancy will occur, which is the most usual type of ectopic pregnancy [29]. Therefore, the present study is essential to fill this gap and also beneficial to overcome the difficulties. Bi-viscosity fluid model is considered to examine the flow behavior. The governing mathematical modeling is performed under low Reynolds number approximation. Exact solutions are given for the fluid- and particulate-phase. The physical action of all the leading parameters is discussed against velocity, concentration, temperature profile, and the trapping mechanism is also presented through streamlines.

#### **2. Problem Formulation**

Consider two-dimensional co-axial infinite tubes. The outer tube is ciliated. The cylindrical coordinate system is selected, i.e., *r*˜ lies toward the radial direction, and *z*˜ lies toward the middle of an inner and the outer tube as given in Figure 2. The inner area between both the tubes is filled with bi-viscosity fluid. The flow is irrotational and the fluid is incompressible having constant viscosity. The fluid contains small spherical particles. The stress tensor for bi-viscosity fluid model [30] is defined as:

$$\chi = \begin{cases} 2\left[y\_s / \sqrt{2\pi} + \bar{\mu}\_B\right] \xi\_{ij\prime} & \pi \ge \pi\_{Y\prime} \\ 2\left[y\_s / \sqrt{2\pi} + \bar{\mu}\_B\right] \xi\_{ij\prime} & \pi \le \pi\_{Y\prime} \end{cases} \tag{1}$$

where *μ*˜*<sup>B</sup>* the plastic viscosity, Υ the volume fraction density, *ξij* the deformation rate of component, and *ys* the yield stress, *π* denotes the second invariant tensor of *ξij*, *π*<sup>Υ</sup> represents the critical value comprises on the non-Newtonian fluid model.

**Figure 2.** Blood flow structure through an ciliated annulus.

The Mathematical expression for the envelope of the cilia tips reads as [31,32]:

$$
\vec{r} = \hbar\_1 = f\_1(\vec{t}, \vec{z}) = b\_0 + b\_0 \phi \cos\left[k(\vec{z} - c\vec{t})\right],\tag{2}
$$

$$\overline{z} = \hbar\_2 = g\_1(\overline{t}, \overline{z}) = \overline{z}\_0 + b\_0 a \phi \sin\left[k(\overline{z} - c\overline{t})\right],\tag{3}$$

where *k* = 2*π*/*λ*, *z*<sup>0</sup> describes the reference location of the cilia, the non-dimensional parameter *φ*, which combines with *b*<sup>0</sup> (mean radius of the outer tube) in the form of *b*0*φ* and represents the amplitude of metachronal wave, *λ* is the metachronal wavelength, *c* the velocity, and *α* describes the measure of the eccentricity of the elliptical motion.

The vertical and axial velocities are evaluated as [31,32]:

$$
\vec{u} = \frac{\partial \vec{r}}{\partial \vec{t}} = \frac{\partial f\_1}{\partial \vec{t}} + \frac{\partial f\_1}{\partial \vec{z}} \frac{\partial \vec{z}}{\partial \vec{t}} = \frac{\partial f\_1}{\partial \vec{t}} + \frac{\partial f\_1}{\partial \vec{z}} \vec{u}, \ \vec{z} = \vec{z}\_{0\prime} \tag{4}
$$

$$
\vec{\sigma} = \frac{\partial \vec{z}}{\partial \vec{t}} = \frac{\partial g\_1}{\partial \vec{t}} + \frac{\partial g\_1}{\partial \vec{z}} \frac{\partial \vec{z}}{\partial \vec{t}} = \frac{\partial g\_1}{\partial \vec{t}} + \frac{\partial g\_1}{\partial \vec{z}} \vec{u}, \ \vec{z} = \vec{z}\_{0\prime} \tag{5}
$$

After some mathematical manipulation, Equations (4) and (5) read as:

$$\vec{n} = -\frac{b\_0 \kappa ck \phi \cos\left[k(\vec{z} - c\vec{t})\right]}{1 - b\_0 \kappa k \phi \cos\left[k(\vec{z} - c\vec{t})\right]},\tag{6}$$

$$\vec{v} = \frac{b\_0 ck\phi \sin\left[k(\vec{z} - c\vec{t})\right]}{1 - b\_0 ak\phi \cos\left[k(\vec{z} - c\vec{t})\right]}.\tag{7}$$

The above boundary conditions help us to discriminate between the effective stroke and less effective recovery stroke of the cilia by considering the shortening of the cilia.

In view of above frame work, the mathematical modeling for the fluid- and particulate-phase is as follows [33]:

#### (i) *Fluid phase*

The continuity and momentum equations are proposed as:

$$
\varphi \frac{\partial \vec{v}\_f}{\partial \vec{r}} + \varphi \frac{\vec{v}\_f}{\vec{r}} + \varphi \frac{\partial \vec{u}\_f}{\partial \vec{z}} = 0,\tag{8}
$$

$$\rho \frac{\partial \vec{p}}{\partial \vec{r}} - CS(\vec{v}\_{\mathcal{P}} - \vec{v}\_{f}) = \rho \left[ \frac{1}{\vec{r}} \frac{\partial}{\partial \vec{r}} r \chi r + \frac{\partial}{\partial \vec{z}} \chi r - \frac{\chi\_{\Phi \Psi}}{\vec{r}} \right],\tag{9}$$

$$\boldsymbol{\varrho} \frac{\partial \tilde{p}}{\partial \tilde{z}} - \boldsymbol{\mathrm{C}} \boldsymbol{S} (\boldsymbol{\uphat{n}}\_{\mathcal{V}} - \boldsymbol{\uphat{n}}\_{f}) = \boldsymbol{\varrho} \left[ \frac{\partial}{\partial \tilde{z}} \boldsymbol{\chi}\_{\mathcal{Z}\tilde{z}} + \frac{1}{\tilde{r}} \frac{\partial}{\partial \tilde{r}} \tilde{r} \boldsymbol{\chi}\_{\mathcal{V}\tilde{z}} \right],\tag{10}$$

The energy equation for the current flow is described as

$$\rho \rho\_f \overline{\varepsilon} \left[ \frac{\partial}{\partial \overline{t}} + \mathbf{V}\_f \cdot \nabla \right] T\_f = \kappa \rho \nabla^2 T\_f + \eta \chi\_{f^2} \left[ \frac{\partial u\_f}{\partial \overline{r}} \right] + \frac{\rho\_p c\_p \mathbf{C}}{\omega\_T} (T\_p - T\_f), \tag{11}$$

The concentration equation for the current flow is described as

$$\oint \boldsymbol{\varrho} \left[ \frac{\partial}{\partial \boldsymbol{f}} + \mathbf{V}\_f \cdot \boldsymbol{\nabla} \right] \boldsymbol{K}\_f = D\_m \boldsymbol{\varrho} \boldsymbol{\nabla}^2 \boldsymbol{K}\_f + \frac{\rho\_p \mathbf{C}}{\rho\_f \omega\_c} (\boldsymbol{\varrho}\_p - \boldsymbol{\varrho}\_f) + \frac{D\_m}{T\_m} \boldsymbol{\varrho} \boldsymbol{K}\_T \boldsymbol{\nabla}^2 T\_f. \tag{12}$$

where *ϕ* = 1 − *C*.

#### (ii) *Particulate phase*

The continuity and momentum equation for this case read as

$$\mathcal{C}\frac{\partial\tilde{\upsilon}\_p}{\partial\tilde{r}} + \mathcal{C}\frac{\tilde{\upsilon}\_p}{\tilde{r}} + \mathcal{C}\frac{\partial\tilde{u}\_p}{\partial\tilde{z}} = 0,\tag{13}$$

$$\mathcal{C}\frac{\partial \vec{p}}{\partial \vec{r}} - \mathcal{S}\mathcal{C}(\tilde{v}\_f - \tilde{v}\_p) = 0,\tag{14}$$

$$\mathcal{C}\frac{\partial \vec{p}}{\partial \vec{z}} - \mathcal{S}\mathcal{C}(\tilde{u}\_f - \tilde{u}\_\mathbb{P}) = 0,\tag{15}$$

In this case, the energy equation is described as

$$
\rho\_p \mathbb{C} \mathbf{c}\_p \left[ \frac{\partial}{\partial \mathbf{f}} + \mathbf{V}\_p \cdot \nabla \right] T\_p = \frac{\rho\_p \mathbf{c}\_p \mathbf{C}}{\omega\_T} (T\_f - T\_p), \tag{16}
$$

The concentration equation is described as

$$\left[\frac{\partial}{\partial \tilde{t}} + \mathbf{V}\_p \cdot \nabla\right] K\_p = \frac{1}{\omega\_c} (K\_f - K\_p)\_\prime \tag{17}$$

where *S* the drag coefficient, *ρ* the density of the fluid, *C* the particle volume fraction density, *T* the temperature, *ω<sup>T</sup>* the thermal equilibrium time, *ω<sup>c</sup>* is the required time period by a particle to regulate its concentration associated to the fluid, *Dm* the mass diffusivity coefficient, *KT* is the thermal diffusion ratio, *Tm* the mean temperature, *κ* the thermal conductivity, *cp* particle-phase specific heat, and *c*˜ the specific heat at constant volume.

The mathematical form of drag coefficient is expressed as [23]

$$S = \frac{9}{2} \frac{\mu\_0}{B\_0^2} \Phi(\mathbb{C}),\\\Phi(\mathbb{C}) = \frac{3\mathbb{C} + 4 + [\mathbb{C}(8 - 3\mathbb{C})]^{1/2}}{(2 - 3\mathbb{C})^2},\tag{18}$$

where *B*<sup>0</sup> is the radius of each particle, and *μ<sup>o</sup>* the fluid viscosity. The empirical relation for the viscosity suspension is expressed as [23]

$$
\mu \tilde{m} = 0.70e^{\left[\frac{11\text{k}\Omega}{\text{l}}e^{(-1.69\text{C})} + 2.49\text{C}\right]}, \\
\mu\_{\text{s}} = \frac{\mu\_{0}}{1 - \tilde{m}\text{C}}.\tag{19}
$$

where *μ<sup>s</sup>* denotes the viscosity of particle fluid mixture.

It is noted here that the results reduced for dusty-gas flows for small particle volume fraction as presented by Marble [34].

Defining the following non-dimensional variables

$$\begin{split} \dot{\tau} &= \frac{\ddot{r}}{b\_0}, u\_{f,p} = \frac{\ddot{u}\_{f,p}}{c}, z = \frac{\ddot{z}}{\lambda}, v\_{f,p} = \frac{\lambda \ddot{v}\_{f,p}}{b\_0 c}, t = \frac{\dddot{t}c}{\lambda}, p = \frac{b\_0^2}{\lambda \mu\_0 c} \ddot{p}, \dot{p} = \frac{\mu\_s}{\mu\_0}, \\\ v\_0 &= \frac{\ddot{v}\_0}{c}, r\_1 = \frac{\ddot{r}\_1}{b\_0}, r\_2 = \frac{\ddot{r}\_2}{b\_0}, \theta\_{f,p} = \frac{T\_{f,p} - T\_0}{T\_1 - T\_0}, \theta\_{f,p} = \frac{K\_{f,p} - K\_0}{K\_1 - K\_0}. \end{split} \tag{20}$$

Applying Equation (20) in Equations (8)–(18), and applying the approximation of low Reynolds number and ignoring the inertial forces. The resulting equations are found as

$$0 = \frac{\partial p}{\partial r}'\tag{21}$$

$$\frac{\partial p}{\partial z} = \frac{CSb\_0^2}{q\mu\_0} \left( u\_p - u\_f \right) + \frac{\bar{\mu}}{r} \eta \frac{\partial}{\partial r} \left( r \frac{\partial u\_f}{\partial r} \right) \,. \tag{22}$$

It is noted here that the results for Newtonian fluid model can be recovered by taking *ζ* → ∞. The temperature and concentration equations read as

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \theta\_f}{\partial r}\right) + B\_n\bar{\mu}\eta \left(\frac{\partial u\_f}{\partial r}\right)^2 = 0,\tag{23}$$

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \theta\_f}{\partial r}\right) + S\_c S\_r \frac{1}{r} \frac{\partial}{\partial r}\left(r\frac{\partial \theta\_f}{\partial r}\right) = 0,\tag{24}$$

where *η* = (1 + 1/*ζ*), *Bn* the Brinkman number, *δ* defines the wave number, *Sc* the Schmidt number, Pr the Prandtl number, *Sr* the Soret number, Ec the Eckert number, and *ζ* the fluid parameter. These parameters are defined as

$$B\_{n} = \text{EcPr}, \text{Pr} = \frac{\vec{\pi} \nu \rho\_f}{\kappa}, \zeta = \frac{\tilde{\mu}\_B \sqrt{2 \pi \kappa\_Y}}{y\_s}, \text{Ec} = \frac{c^2}{\tilde{\varepsilon} (T\_1 - T\_0)}, S\_c = \frac{\nu}{D\_m}, \delta = \frac{b\_0}{\lambda},$$

$$S\_r = \frac{D\_m K\_T}{\nu T\_m} \left(\frac{\theta\_1 - \theta\_0}{\theta\_1 - \theta\_0}\right). \tag{25}$$

The particulate-phase equations are found as

$$\frac{\partial p}{\partial z} + \frac{Sb\_0^2}{\mu\_0} \left(\mu\_p - \mu\_f\right) = 0,\tag{26}$$

$$
\theta\_f = \theta\_{p\prime} \tag{27}
$$

$$
\theta\_f = \theta\_p.\tag{28}
$$

From Equation (21), it is found that *p* cannot be the function of *r*. The relevant boundary conditions read as

$$u\_f(r\_1) = v\_{0\prime} \theta\_f(r\_1) = 1,\qquad\qquad r\_1 = a(z) = a\_0 + h\_c e^{-\pi^2 (z - z\_d - 0.5)^2},\tag{29}$$

$$\mu\_f(r\_2) = -\frac{2\pi a\phi\delta\cos 2\pi\Xi}{1 - 2\pi a\phi\delta\cos 2\pi\Xi}, \theta\_f(r\_2) = 0,\qquad r\_2 = 1 + \frac{\lambda\Gamma z}{b\_0} + \phi\sin 2\pi\Xi,\tag{30}$$

where Ξ = (*z* − *t*), Γ is a constant which represents the magnitude that relies on the annulus length and its exit inlet dimensions, maximum height of the clot denoted by *hc*, *v*<sup>0</sup> typify the velocity of the inner tube, the axial displacement of the clot is denoted by *zd*, and the radius of the inner tube which makes the clot in the appropriate place is denoted by *a*0. The results for endoscopy can be reduced by considering *hc* = 0 in Equation (31) as a particular case of the present study.

#### **3. Solutions of the Proposed Problem**

Equations (22)–(24) are solved analytically using a computational software "Mathematica 10.3v", and the exact solutions are presented below:

$$
\mu\_f = \mathcal{C}\_1 + r\mathcal{C}\_2 + \mathcal{C}\_3 r \log r,\tag{31}
$$

$$
\mu\_{\mathcal{P}} = \mathcal{C}\_1 + r\mathcal{C}\_2 + \mathcal{C}\_3 r \log r - \frac{\mu\_0}{S b\_0^2} \frac{dp}{dz},\tag{32}
$$

The solutions for the temperature profile for particulate- and fluid-phase are found as

$$
\theta\_{f,p} = \theta\_0 + r^2 \theta\_1 + r^4 \theta\_2 + \theta\_3 \log r + \theta\_4 \log^2 r,\tag{33}
$$

The solutions for the concentration profile for particulate- and fluid-phase are found as

$$
\theta\_{f,p} = \theta\_0 + r^2 \theta\_1 + r^4 \theta\_2 + \vartheta\_3 \log r + \vartheta\_4 \log^2 r + \vartheta\_5 \log^3 r,\tag{34}
$$

and the constants appearing in above Equations (31)–(34) i.e. *Cn*, *θn*, *ϑ<sup>n</sup>* (*n* = 1, 2 ...) are given Appendix A.

The instantaneous volume flow rate for the present flow read as

$$Q(t,z) = 2\pi\mathfrak{q}\int\_{r\_1}^{r\_2} ru\_f \mathrm{d}r + 2\pi\mathbb{C} \int\_{r\_1}^{r\_2} ru\_p \mathrm{d}r.\tag{35}$$

The pressure gradient can be obtained after solving the above equation.

The pressure rise along the whole ciliated annulus can be determined as

$$
\Delta p = \int\_0^{L/\lambda} \wp \mathrm{d}z,\tag{36}
$$

where ℘ represents the pressure gradient.

#### **4. Graphical Analysis**

In thissection, using the obtained numerical results we analyze the behavior of all the physical parameter. Particularly, we determine the behavior of velocity profile, concentration, and the temperature profile, against the height of the clot *hc*, particle volume fraction *C*, wave number *δ*, Soret number *Sr*, Brinkmann number *Bn*, Schmidt number *Sc*, and eccentricity of the elliptic path of cilia *α*. Following parametric values [1] are chosen to analyze the graphical performance of all the leading parameters, i.e., *b*<sup>0</sup> = 1.25 cm, *φ* = 0.1 − 0.5, *C* = 0 − 0.6, *α* = 0.3 − 1, Γ = 3*b*0/*λ*, *L* = *λ* = 8.01 cm , *δ* = 0.05 − 0.2. Furthermore, the results for single-phase model can be recovered by considering *C* = 0 in the governing equations (see Equations (21)–(28)). Assume that the instantaneous volume flow rate is periodic in Ξ, i.e.,

$$\frac{Q}{\pi} = -\frac{\phi^2}{2} + \frac{Q}{\pi} + 2\phi\sin 2\pi\Sigma + \frac{2\phi\lambda z}{b\_0}\Gamma\sin 2\pi\Sigma + \phi^2\sin^2 2\pi\Sigma,\tag{37}$$

where *Q*¯ denotes the average time flow rate over one period of wavelength.

Figure 3 depicts the behavior of blood clots and particle volume fraction on the velocity profile. We can observe from this figure that an increment in particle volume fraction *C* significantly suppresses the velocity profile. The velocity profile shows a decreasing behavior for endoscopic case, i.e., *hc* = 0, whereas it increases due to the blood clot *hc* = 0.15. It can be observed from Figure 4 that both parameters *α* and *δ* cause a positive impact on the velocity profile while its trend becomes reverse when *r* > 1.35. Figure 5 shows a plot of velocity profile against numerous values of *φ*. It can be seen from this figure that the velocity profile is remarkably suppressed with increments in *φ*. Furthermore, we also noticed that as compared with non-Newtonian case *ζ* = 0.1, the fluid velocity lessen more when the fluid behaves as a Newtonian model *ζ* → ∞.

**Figure 3.** Velocity curves for different values of *hc* and *C*.

**Figure 4.** Velocity curves for different values of *α* and *δ*. Solid line: *δ* = 0.05, dashed line: *δ* = 0.2.

**Figure 5.** Velocity curves for different values of *φ* and *ζ*.

In Figures 6–8 we see the mechanism of temperature profile plotted against the multiple leading parameters. From Figure 6, we can see that the temperature profile rises due to the increment in particle volume fraction *C*. Further, we noticed that for the blood clot case *hc* = 0.15, the temperature profile is increasing and has a higher magnitude as compared with the endoscopic case *hc* = 0. It is analyzed from Figure 7 that the parameters *α* and *δ* restrain the temperature profile. Unfortunately, both parameters have small effects, especially when the wavenumber is very small at *δ* = 0.05. Figure 8 shows plots with multiple values of Brinkman number *Bn*. Brinkman number represents the product of Eckert and Prandtl number Ec × Pr. Generally, it is the ratio between heat generated due to viscous dissipation and transport of heat due to molecular conduction. It can noticed that the temperature profile remarkably increases for higher values of Brinkman number. However, a similar behavior is observed against the higher values of *φ*.

**Figure 6.** Temperature distribution for different values of *hc* and *C*.

**Figure 7.** Temperature distribution for different values of *α* and *δ*. Solid line: *δ* = 0.05, dashed line: *δ* = 0.2.

**Figure 8.** Temperature distribution for different values of *φ* and *Bn*.

Figure 9 is illustrated to analyze the mechanisms of Schmidt number *Sc* and Soret number *Sr* on the concentration profile. We can see from this figure that the concentration profile shows a decreasing mechanism against both parameters and remains uniform throughout the entire region. An increment in Schmidt number indicates that the viscous diffusion rate is more dominant as compared with the molecular diffusion rate, whose results tend to decline the concentration profile. Similarly, when the Soret number increases, the Thermophoresis forces generated, which oppose the concentration profile.

**Figure 9.** Concentration distribution for different values of *Sc* and *Sr*.

Figures 10–12 depict the variation of pressure rise versus time against different values of emerging parameters. It can be observed from Figure 10 that by enhancing the particle volume fraction *C*, the pressure rise is significantly decreasing, while due to the presence of blood clot, more pressure is required to pass the fluid. Further, we can see that the pressure rise is maximum in the region when *t* ∈ (0.3, 0.7). It is clear from the Figure 11 that both parameters *α* and *δ* reveal versatile behavior on the pressure rise. We can also see that there are two critical points, for instance, at *t* = 0.4 and *t* = 0.9. In the region *t* ∈ (0.4, 0.9) the pressure rise acts as an increasing function whereas in the other area it decreases. Similarly, we can observe that the pressure rise increases due to the increment in *φ*, as shown in Figure 12.

Trapping mechanism is presented in Figures 13–15 for different values of *α*, *δ* and *hc*. It can be noticed from Figure 13 that by increasing the values of *α*, the trapping bolus reduces, and a number of boluses disappear. Similarly, in Figure 14, we can see that the higher values of *δ* tend to diminish the immensity of the trapping bolus, whereas the number of trapping bolus increase and streamlines increases. It is seen in Figure 15 that when the height of the blood clot increases, then streamlines shrink , and trapping bolus increase significantly.

**Figure 10.** Pressure rise for different values of *hc* and *C*.

**Figure 11.** Pressure rise for different values of *α* and *δ*.

**Figure 12.** Pressure rise for different values of *φ*.

**Figure 13.** Trapping mechanism for different values of *α*.

**Figure 14.** Trapping mechanism for different values of *δ*.

**Figure 15.** Trapping mechanism for different values of *hc*.

#### **5. Conclusions**

In this study, we explained the behavior of particle-fluid with mass and heat transfer through a ciliated annulus. The effects of endoscopy and blood clot are also taken into account. To analyze the behavior of fluid in an annulus, we considered the bi-viscosity fluid model. The mathematical formulation is undertaken for low Reynolds number approximation. The formulated differential equations are analytically solved, and closed-form solutions are presented. The main observations of the present study are followed as:


Furthermore, in this study, several effects have been ignored, i.e., magnetic field, porosity, chemical reaction and activation energy, respectively, which can be considered in future research.

**Author Contributions:** Investigation, M.M.B.; Methodology & Conceptualization, R.E.; Validation, A.F.E.; Writing—review & editing, S.M.S. All authors have read and agreed to the published version of the manuscript.

**Acknowledgments:** Authors thanks to those who are working in front line to save the humanity from corona pandemic in China and across the globe. We appreciate their devotion and services.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

$$\mathbf{C}\_{1} = \frac{1}{4\rho\eta} \frac{dp}{dz'}\tag{A1}$$

$$\mathcal{C}\_2 = \frac{1}{4\rho\eta} \left[ \frac{dp}{dz} (r\_2^2 - r\_1^2) - 4\xi\rho\eta \right]\_{\prime\prime} \tag{A2}$$

$$C\_3 = \frac{\theta\_5}{4q\eta\zeta\log\frac{r\_1}{r\_2}\prime\prime}\,,\tag{A3}$$

$$\zeta = -\frac{2\pi\varkappa\epsilon\delta\cos 2\pi\Xi}{1 - 2\pi\varkappa\epsilon\delta\cos 2\pi\Xi}.\tag{A4}$$

$$\begin{split} \theta\_0 &= \frac{1}{16\theta\_7^2 \log \frac{r\_1}{r\_2}} \left[ -\left\{ 16\theta\_2^2 + B\_n \theta\_6 r\_1^2 (8\theta\_5 + \theta\_6 r\_1^2) \eta \right\} - 8B\_n \theta\_5^2 \eta \log r\_1^2 \log r\_2 \right. \\ &+ B\_n \eta \log r\_1 \left( \theta\_6 r\_2^2 (8\theta\_5 + \theta\_6 r\_2^2) + 8\theta\_5^2 \log^2 r\_2 \right) \right], \tag{A5} \end{split} \tag{A5}$$

$$\theta\_1 = -\frac{8B\_n\theta\_5\theta\_6\eta}{16\theta\_7^2},\tag{A6}$$

$$\theta\_2 = -\frac{B\_n \theta\_6^2 \eta}{16 \theta\_7^2},\tag{A7}$$

$$\theta\_3 = \frac{1}{16\theta\_7^2 \log \frac{r\_1}{r\_2}} \left[ 16\theta\_7^2 + B\_n \theta\_6 (r\_1^2 - r\_2^2) \{ 8\theta\_5 + \theta\_6 (r\_1^2 + r\_2^2) \} \eta + 8B\_n \theta\_5^2 \eta \log^2 \frac{r\_1}{r\_2} \right],\tag{A8}$$

$$\theta\_4 = -\frac{B\_\text{n}\theta\_5^2\eta}{2\theta\_7^2},\tag{A9}$$

$$\theta\_5 = \zeta \left( \frac{dp}{dz} (r\_1^2 - r\_2^2) + (\zeta - v\_0) 4q\eta \right),\tag{A10}$$

$$\theta\_{\theta} = 2\zeta \frac{dp}{dz} \log \frac{r\_2}{r\_1} \tag{A11}$$

$$
\theta\_7 = 4q\eta\tilde{\chi}\log\frac{r\_2}{r\_1}.\tag{A12}
$$

$$\begin{split} \theta\_{0} &= \frac{1}{192\theta\_{7}^{2}\log\frac{r\_{1}}{r\_{2}}} \left[ 3(-64\theta\_{7}^{2} + B\_{n}\theta\_{6}r\_{1}^{2}(16\theta\_{5} + 3\theta\_{6}r\_{1}^{2})S\_{c}S\_{r}\eta)\log r\_{2} \\ &+ 96B\_{n}\theta\_{5}^{2}S\_{c}S\_{r}\eta\log^{2}r\_{1}\log r\_{2} + 16B\_{n}\theta\_{5}^{2}S\_{c}S\_{r}\eta\log r\_{1}^{3}\log r\_{2} \\ &+ S\_{c}S\_{r}\log r\_{1} \left\{-3B\_{n}\theta\_{5}r\_{2}^{2}(16\theta\_{5} + 3\theta\_{6}r\_{2}^{2})\eta + 2\log r\_{2}(48\theta\_{7}^{2} + 3B\_{n} \\ &\times \theta\_{6}(r\_{1}^{2} - r\_{2}^{2})(8\theta\_{5} + \theta\_{6}(r\_{1}^{2} + r\_{2}^{2}))\eta - 8B\_{n}\theta\_{7}^{2}\eta\log r\_{2}(6 + \log r\_{2})) \right\} \right], \tag{A13} \end{split}$$

$$
\theta\_1 = \frac{B\_n S\_c S\_r \theta\_5 \theta\_6 \eta}{\theta\_7^2},
\tag{A14}
$$

$$
\theta\_2 = \frac{3B\_n S\_c S\_r \theta\_6^2 \eta}{64\theta\_7^2},
\tag{A15}
$$

$$\begin{split} \theta\_{3} &= \frac{1}{192\theta\_{7}^{2}\log\frac{r\_{1}}{r\_{2}}} \Big[ 192\theta\_{7}^{2} - 3B\_{n}\theta\_{5}(r\_{1}^{2} - r\_{2}^{2})(16\theta\_{5} + 3\theta\_{6}(r\_{1}^{2} + r\_{2}^{2}))S\_{c}S\_{r}\eta \\ &+ 2S\_{c}S\_{7} \Big( \log r\_{1}(-48\theta\_{7}^{2}\zeta - 3B\_{n}\theta\_{6}(r\_{1}^{2} - r\_{2}^{2})(8\theta\_{5} + \theta\_{6}(r\_{1}^{2} + r\_{2}^{2}))\eta - 8B\_{n} \\ \theta\_{5}^{2}\eta\log r\_{1}(6 + \log r\_{1}))3\zeta(16\theta\_{7}^{2} + B\_{n}\theta\_{6}(r\_{1}^{2} - r\_{2}^{2})(8\theta\_{5} + \theta\_{6}(r\_{1}^{2} + r\_{2}^{2}))\eta \\ &+ 8B\_{n}\theta\_{5}^{2}\eta\log r\_{1}^{2})\log r\_{2} + 24B\_{n}\theta\_{5}^{2}\eta(2 + \log r\_{1})\log r\_{2}^{2} + 8B\_{n}\theta\_{5}^{2}\eta\log r\_{2}^{3})\Big], \tag{A16} \end{split}$$

$$\theta\_4 = \frac{1}{32\theta\_7^2 \log \frac{r\_1}{r\_2}} \left[ S\_c S\_r \{ 16\theta\_7^2 + B\_n \theta\_6 (r\_1^2 - r\_2^2)(8\theta\_5 + \theta\_6 (r\_1^2 + r\_2^2)) \eta \right]$$

$$+ 8B\_n \theta\_5^2 \eta \log \frac{r\_1}{r\_2} (2 + \log r\_1 + \log r\_2) \},\tag{A17}$$

$$\theta\_5 = -\frac{B\_n \theta\_5^2 S\_c S\_r}{6\theta\_7^2} \eta. \tag{A18}$$

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Finite-Time Control for Nonlinear Systems with Time-Varying Delay and Exogenous Disturbance**

**Yanli Ruan 1,\* and Tianmin Huang <sup>2</sup>**


Received: 31 December 2019; Accepted: 24 February 2020; Published: 11 March 2020

**Abstract:** This paper is concerned with the problem of finite-time control for nonlinear systems with time-varying delay and exogenous disturbance, which can be represented by a Takagi–Sugeno (T-S) fuzzy model. First, by constructing a novel augmented Lyapunov–Krasovskii functional involving several symmetric positive definite matrices, a new delay-dependent finite-time boundedness criterion is established for the considered T-S fuzzy time-delay system by employing an improved reciprocally convex combination inequality. Then, a memory state feedback controller is designed to guarantee the finite-time boundness of the closed-loop T-S fuzzy time-delay system, which is in the framework of linear matrix inequalities (LMIs). Finally, the effectiveness and merits of the proposed results are shown by a numerical example.

**Keywords:** finite-time boundedness; T-S fuzzy systems; time-varying delay; Lyapunov–Krasovskii functional (LKF)

#### **1. Introduction**

During the past several decades, the control problem of nonlinear systems has attracted considerable attention [1–6] as various practical systems are essentially nonlinear and cannot be easily simplified into a linear model. Up to now, many fuzzy logic control approaches have been proposed for the control problem of nonlinear systems. In particular, the Takagi–Sugeno (T-S) fuzzy model, developed in [7], is an important tool to approximate complex nonlinear systems by combining the fruitful linear system theory and the flexible fuzzy logic approach. Additionally, time-delay is unavoidably encountered in many dynamic systems, such as power systems, network control systems, neural networks, etc., which often results in chaos, oscillation, and even instability. Therefore, the study of T-S fuzzy time-delay systems has become more and more popular in recent years. In particular, many significant and interesting results on stability analysis and the control synthesis of T-S fuzzy time-delay systems have been developed in the literature [8–15].

Much attention has been paid to obtain the delay-dependent stability criteria for T-S fuzzy time-delay systems over the last few decades. It is well-known that the conservativeness of the stability criteria mainly has two sources: the choice of the Lyapunov–Krasovskii functional (LKF) and the estimation of its derivative. It is of great importance to construct an appropriate Lyapunov–Krasovskii functional for deriving less conservative stability conditions. In recent years, delay-partitioning Lyapunov–Krasovskii functionals and augmented Lyapunov–Krasovskii functionals have been developed to reduce the conservativeness of simple LKFs and have attracted growing attention. A delay-partitioning approach was applied to study the Lyapunov asymptotic stability of T-S fuzzy time-delay systems and some less conservative stability conditions were obtained in [16–18].

In [19], the authors introduced the triple-integral terms into the LKFs to derive the stability conditions for T-S fuzzy time-delay systems. In addition, various approaches have been proposed

to estimate the derivatives of LKFs when dealing with stability analysis and control synthesis of time-delay systems, such as the free weighting matrix approach [20], Jensen inequality approach [21], Wirtinger-based integral inequality approach [10], reciprocally convex combination approach [22], auxiliary function-based inequality approach [23], and free-matrix-based integral inequality approach [24].

By applying the Wirtinger-based integral inequality approach and reciprocally convex combination approach, Zeng et al. [18] derived some less conservative stability criteria for uncertain T-S fuzzy systems with time-varying delays. An improved free weighting matrix approach was employed to obtain several new delay-dependent stability conditions in terms of the linear matrix inequalities for T-S fuzzy systems with time-varying delays in [25]. In [26], the authors investigated Lyapunov asymptotic stability analysis problems for T-S fuzzy time-delay systems by constructing a new augmented Lyapunov–Krasovskii functional and employing the free-matrix-based integral inequality approach.

The aforementioned results regarding the stability analysis of T-S fuzzy systems mainly focus on Lyapunov asymptotic stability, in which the states of systems converge asymptotically to equilibrium in an infinite time interval. However, in many practical engineering applications, the main concern may be the transient performances of the system trajectory during a specified finite-time interval. Unlike Lyapunov asymptotic stability, finite-time stability, introduced in [27], is another stability concept, which requires that the states of dynamical systems do not exceed a certain threshold in a fixed finite-time interval with a given bound for the initial condition. Up to now, the problem of finite-time stability, finite-time boundedness, and finite-time stabilization of dynamical systems has attracted growing attention, and many significant results have been reported in [28–32].

Several results on finite-time stability and stabilization of T-S fuzzy systems can also be found. The problem of finite-time stability and finite-time stabilization for T-S fuzzy time-delay systems was investigated in [28]. Sakthivel et al. [31] studied finite-time dissipative based fault-tolerant control problem for a class of T-S fuzzy systems with a constant delay. However, to the best of our knowledge, until now there have been few results on finite-time boundedness and finite-time stabilization of T-S fuzzy systems with time-varying delay and exogenous disturbance. Furthermore, it should be mentioned that most of the existing works on finite-time control for T-S fuzzy time-delay systems are fairly conservative. Motivated by the above discussions, in this paper, we deal with the problem of finite-time control for a class of nonlinear systems with time-varying delay and exogenous disturbance, which can be described by a T-S fuzzy model.

The main contributions of this paper are summarized as follows. First, a new augmented Lyapunov–Krasovskii functional is constructed, which makes full use of the information about time-varying delay. Based on the proposed Lyapunov–Krasovskii functional, a less conservative finite-time boundedness condition is obtained for T-S fuzzy time-delay systems by utilizing an improved reciprocally convex combination technique. Second, based on parallel distributed compensation schemes, a memory state feedback controller is designed to finite-time stabilize the T-S fuzzy time-delay system, which can be derived by solving a series of linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the advantages and validity of the developed results.

The rest of this paper is organized as follows: the problem statement is given in Section 2. The main results on the finite-time boundedness and finite-time stabilization of nonlinear systems with time-varying delay and exogenous disturbance are presented in Section 3. In Section 4, a numerical example is proposed to show the effectiveness of the developed results. Finally, our conclusions are drawn in Section 5.

*Notations:* Throughout this paper, **R***<sup>n</sup>* denotes the n-dimensional Euclidean space; **R***n*×*<sup>m</sup>* stands for the set of all *n* × *m* real matrices; the superscripts *T* and −1 denote the transpose and inverse of a matrix, respectively; *I* and 0 represent the identity matrix and zero matrix, respectively, with compatible dimensions; *diag*{· · · } denotes a block-diagonal matrix; the notation *P* > 0(≥ 0) means that the matrix

*P* is real symmetric and positive definite (semi-positive definite); ∗ stands for the symmetric terms in a symmetric matrix; for any matrix *<sup>X</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*n*, *Sym*{*X*} is defined as *<sup>X</sup>* <sup>+</sup> *<sup>X</sup>T*.

#### **2. Problem Formulation**

Consider a class of nonlinear systems with time-varying delay and exogenous disturbance, which can be represented by the following T-S fuzzy model:

Plant Rule i: IF *ξ*1(*t*) is *Ni*1, ··· , and *ξ <sup>p</sup>*(*t*) is *Nip*, THEN *x*˙(*t*) = *Aix*(*t*) + *Adix*(*t* − *d*(*t*)) + *Biu*(*t*) + *Giω*(*t*) *x*(*t*) = *φ*(*t*), *t* ∈ [−*h*, 0]

where *<sup>i</sup>* ∈ {1, 2, ... ,*r*}, *<sup>r</sup>* is the number of IF-THEN rules, *<sup>x</sup>*(*t*) <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is the state vector, *<sup>u</sup>*(*t*) <sup>∈</sup> **<sup>R</sup>***<sup>p</sup>* is the control input, *<sup>ω</sup>*(*t*) <sup>∈</sup> **<sup>R</sup>***<sup>l</sup>* is the exogenous disturbance, which satisfies *Tf* <sup>0</sup> *<sup>ω</sup>*(*t*)*Tω*(*t*)*dt* ≤ *<sup>δ</sup>*; *δ* ≥ 0 is a given scalar. *Ai*, *Adi*, *Bi*, and *Gi* are known constant matrices with appropriate dimensions. *ξ*1(*t*), *ξ*2(*t*), ... , *ξ <sup>p</sup>*(*t*) are premise variables, *Ni*1, *Ni*2, ... , *Nip* are fuzzy sets. The time delay *d*(*t*) is a time-varying function that satisfies

$$0 \le d(t) \le h \quad \text{and} \quad \mu\_1 \le \dot{d}(t) \le \mu\_2 \tag{1}$$

where *h* > 0 and *μ*1, *μ*<sup>2</sup> are constants. The initial condition *φ*(*t*) is a continuous vector-valued function for all *t* ∈ [−*h*, 0].

Let *ξ*(*t*)=[*ξ*1(*t*), *ξ*2(*t*), ... , *ξ <sup>p</sup>*(*t*)]*T*, by employing a singleton fuzzifier, product inference, and center-average defuzzifer, the input–output form of the above T-S fuzzy time-delay system can be represented by

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} \rho\_i(\mathbf{\dot{s}}(t)) \left\{ A\_i \mathbf{x}(t) + A\_{di} \mathbf{x}(t - d(t)) + B\_i u(t) + \mathbf{G}\_l \omega(t) \right\} \tag{2}$$

where *<sup>ρ</sup>i*(*ξ*(*t*)) = *<sup>θ</sup>i*(*ξ*(*t*)) ∑*r <sup>i</sup>*=<sup>1</sup> *<sup>θ</sup>i*(*ξ*(*t*)), *<sup>θ</sup>i*(*ξ*(*t*)) = <sup>∏</sup>*<sup>p</sup> <sup>j</sup>*=<sup>1</sup> *Nij*(*ξj*(*t*)), *Nij*(*ξj*(*t*)) is the grade of membership of *<sup>ξ</sup>j*(*t*) in the fuzzy set *Nij*. We note that *<sup>θ</sup>i*(*ξ*(*t*)) <sup>≥</sup> 0, <sup>∑</sup>*<sup>r</sup> <sup>i</sup>*=<sup>1</sup> *θi*(*ξ*(*t*)) > 0 for all *t*, and we can obtain *<sup>ρ</sup>i*(*ξ*(*t*)) <sup>≥</sup> 0, <sup>∑</sup>*<sup>r</sup> <sup>i</sup>*=<sup>1</sup> *ρi*(*ξ*(*t*)) = 1.

In this paper, for simplicity, we denote *S*(*t*) = ∑*<sup>r</sup> <sup>i</sup>*=<sup>1</sup> *ρi*(*ξ*(*t*))*Si* for any matrix *Si*. Therefore, the T-S fuzzy time-delay system (2) can be rewritten as follows:

$$\dot{\mathbf{x}}(t) = A(t)\mathbf{x}(t) + A\_d(t)\mathbf{x}(t-d(t)) + B(t)u(t) + G(t)\omega(t). \tag{3}$$

Now, the definition of finite-time boundedness (FTB) for the T-S fuzzy time-delay system (3) with *u*(*t*) ≡ 0 is given as follows:

**Definition 1** ([31])**.** *The T-S fuzzy time-delay system (3) with u*(*t*) ≡ 0 *is said to be finite-time bounded with respect to* (*c*1, *<sup>c</sup>*2, *Tf* , *<sup>R</sup>*, *<sup>δ</sup>*, *<sup>h</sup>*)*, where* <sup>0</sup> <sup>&</sup>lt; *<sup>c</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>c</sup>*2*, Tf* <sup>&</sup>gt; <sup>0</sup>*, R* <sup>∈</sup>*Rn*×*<sup>n</sup> and R* <sup>&</sup>gt; <sup>0</sup>*, if*

$$\sup\_{\|\mathbf{x}\|\_{\ast} \leq \theta \leq 0} \{\mathbf{x}^T(\theta)\mathbf{R}\mathbf{x}(\theta), \dot{\mathbf{x}}^T(\theta)\mathbf{R}\dot{\mathbf{x}}(\theta)\} \leq c\_1 \Rightarrow \mathbf{x}^T(t)\mathbf{R}\mathbf{x}(t) < c\_{2\ast}$$

∀*t* ∈ [0, *Tf* ], ∀*ω*(*t*) : *Tf* <sup>0</sup> *<sup>ω</sup>*(*t*)*Tω*(*t*)*dt* ≤ *<sup>δ</sup>.* *Symmetry* **2020**, *12*, 447

Based on the parallel distributed compensation scheme, we aim to design the following memory state feedback controller, which can guarantee the corresponding closed-loop T-S fuzzy time-delay system finite-time bounded:

$$\mathbf{u}(t) = \mathbb{K}\_1(t)\mathbf{x}(t) + \mathbb{K}\_2(t)\mathbf{x}(t - d(t)),\tag{4}$$

where *K*1(*t*) = ∑*<sup>r</sup> <sup>j</sup>*=<sup>1</sup> *<sup>ρ</sup>j*(*ξ*(*t*))*K*1*j*, *<sup>K</sup>*2(*t*) = <sup>∑</sup>*<sup>r</sup> <sup>j</sup>*=<sup>1</sup> *ρj*(*ξ*(*t*))*K*2*j*, and *K*1*j*, *K*2*j*, *j* = 1, 2, ... ,*r* are the state feedback gain matrices to be determined.

By substituting (4) into (3), the corresponding closed-loop T-S fuzzy time-delay system can be represented as follows:

$$\dot{\mathbf{x}}(t) = \left[A(t) + B(t)K\_1(t)\right]\mathbf{x}(t) + \left[A\_d(t) + B(t)K\_2(t)\right]\mathbf{x}(t - d(t)) + G(t)\omega(t). \tag{5}$$

In order to derive the main results in this paper, the following lemma, i.e., the improved reciprocally convex combination inequality approach will be utilized in finite-time boundedness analysis and controller design of T-S fuzzy time-delay systems.

**Lemma 1** ([33])**.** *Let <sup>R</sup>*1, *<sup>R</sup>*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>m*×*<sup>m</sup> be real symmetric positive definite matrices,* 1, <sup>2</sup> <sup>∈</sup> *<sup>R</sup><sup>m</sup> and a scalar <sup>α</sup>* <sup>∈</sup> (0, 1)*. Then for any matrices Y*1,*Y*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>m*×*m, the following inequality holds*

$$\begin{aligned} &\frac{1}{\alpha}\boldsymbol{\phi}\_1^T \boldsymbol{R}\_1 \boldsymbol{\phi}\_1 + \frac{1}{1-\alpha} \boldsymbol{\phi}\_2^T \boldsymbol{R}\_2 \boldsymbol{\phi}\_2 \\ \geq & \boldsymbol{\phi}\_1^T [\boldsymbol{R}\_1 + (1-\alpha)(\boldsymbol{R}\_1 - \boldsymbol{Y}\_1 \boldsymbol{R}\_2^{-1} \boldsymbol{Y}\_1^T)] \boldsymbol{\phi}\_1 \\ &+ \boldsymbol{\phi}\_2^T [\boldsymbol{R}\_2 + \alpha(\boldsymbol{R}\_2 - \boldsymbol{Y}\_2^T \boldsymbol{R}\_1^{-1} \boldsymbol{Y}\_2)] \boldsymbol{\phi}\_2 \\ &+ 2\boldsymbol{\phi}\_1^T [\boldsymbol{\alpha} \boldsymbol{Y}\_1 + (1-\alpha)\boldsymbol{Y}\_2] \boldsymbol{\phi}\_2. \end{aligned}$$

#### **3. Main Results**

#### *3.1. Finite-Time Boundedness Analysis*

In this subsection, our aim is to develop a new delay-dependent finite-time boundedness criterion for T-S fuzzy systems with time-varying delay and norm-bounded disturbance. Before deriving the main results, the nomenclature simplifying the representations for matrices and vectors is given as follows:

$$\begin{split} \varepsilon\_{1}(t) &= \begin{pmatrix} \mathbf{x}(t) \\ \mathbf{x}(t-d(t)) \\ \mathbf{x}(t-h) \\ \dot{\mathbf{x}}(t-d(t)) \\ \dot{\mathbf{x}}(t-h) \end{pmatrix}, \ \varepsilon\_{2}(t) = \begin{pmatrix} \frac{1}{d(t)} \int\_{t-d(t)}^{t} \mathbf{x}(s)ds \\ \frac{1}{h-d(t)} \int\_{t-h}^{t} \mathbf{x}(s)ds \\ \frac{1}{d^{2}(t)} \int\_{t-h}^{t} \int\_{\theta}^{t} \mathbf{x}(s)dsd\theta \\ \frac{1}{(h-d(t))^{2}} \int\_{t-h}^{t-d(t)} \int\_{\theta}^{t-d(t)} \mathbf{x}(s)dsd\theta \\ \frac{1}{h} \int\_{t-h}^{t} \mathbf{x}(s)ds \\ \frac{1}{h^{2}} \int\_{t-h}^{t} \int\_{\theta}^{t} \mathbf{x}(s)dsd\theta \end{pmatrix}, \\\\ \varepsilon(t) &= \begin{pmatrix} \varepsilon\_{1}^{T}(t) & \varepsilon\_{2}^{T}(t) & \varepsilon\_{1}^{T}(t) & \mathbf{x}^{T}(t) \end{pmatrix}^{T}, \end{split}$$

$$\mathfrak{e}\_i = \begin{bmatrix} \mathbf{0}\_{n \times (i-1)n} & I\_n & \mathbf{0}\_{n \times (12-i)n} \end{bmatrix}, \quad i = 1, 2, \dots, 12,$$

*Symmetry* **2020**, *12*, 447

$$\vec{\sigma}\_i = \begin{bmatrix} \mathbf{0}\_{n \times (i-1)n} & I\_n & \mathbf{0}\_{n \times (13-i)n} \end{bmatrix}, \quad i = 1, 2, \dots, 13.$$

**Theorem 1.** *For given scalars h* > 0 *and μ*1, *μ*2*, the T-S fuzzy system (3) with u*(*t*) = 0 *and a time-varying delay d*(*t*) *satisfying (1) is finite-time bounded with respect to* (*c*1, *c*2, *Tf* , *R*, *δ*, *h*)*, if there exists a scalar β* > 0*, symmetric positive definite matrices <sup>P</sup>* <sup>∈</sup> *<sup>R</sup>*5*n*×<sup>5</sup>*n, <sup>S</sup>*1, *<sup>S</sup>*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>*2*n*×<sup>2</sup>*n, <sup>Q</sup>*1, *<sup>Q</sup>*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>*3*n*×<sup>3</sup>*n, <sup>W</sup>*, *<sup>Z</sup>*, *<sup>U</sup>* <sup>∈</sup> *<sup>R</sup>n*×*n, and any matrices Y*1,*Y*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>*3*n*×<sup>3</sup>*n, such that the following conditions hold:*

$$
\begin{pmatrix}
\Sigma\_{1i}(0,\mu\_1) & \Lambda\_1^T Y\_1 \\
Y\_1^T \Lambda\_1 & -\mathcal{W}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{6}
$$

$$
\begin{pmatrix}
\Sigma\_{1i}(0, \mu\_2) & \Lambda\_1^T Y\_1 \\
\chi\_1^T \Lambda\_1 & -\mathcal{W}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{7}
$$

$$
\begin{pmatrix}
\Sigma\_{1i}(h,\mu\_1) & \Lambda\_2^T \mathcal{Y}\_2^T \\
\mathcal{Y}\_2 \Lambda\_2 & -\mathcal{W}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{8}
$$

$$
\begin{pmatrix}
\Sigma\_{1i}(h,\mu\_2) & \Lambda\_2^T \boldsymbol{\Upsilon}\_2^T \\
\boldsymbol{\Upsilon}\_2 \Lambda\_2 & -\boldsymbol{\mathcal{W}}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{9}
$$

$$
\kappa\_1 \Pi + \lambda\_{37} \delta \preccurlyeq \lambda\_{36} \mathcal{C}\_2 e^{-\beta T\_f},
\tag{10}
$$

*where*

*P* = ⎛ ⎜⎜⎜⎜⎜⎝ *P*<sup>11</sup> *P*<sup>12</sup> *P*<sup>13</sup> *P*<sup>14</sup> *P*<sup>15</sup> ∗ *P*<sup>22</sup> *P*<sup>23</sup> *P*<sup>24</sup> *P*<sup>25</sup> ∗ ∗ *P*<sup>33</sup> *P*<sup>34</sup> *P*<sup>35</sup> ∗∗∗ *P*<sup>44</sup> *P*<sup>45</sup> ∗∗∗∗ *P*<sup>55</sup> ⎞ ⎟⎟⎟⎟⎟⎠ , *Q*<sup>1</sup> = ⎛ ⎜⎝ *Q*<sup>11</sup> *Q*<sup>12</sup> *Q*<sup>13</sup> ∗ *Q*<sup>22</sup> *Q*<sup>23</sup> ∗ ∗ *Q*<sup>33</sup> ⎞ ⎟⎠ , *<sup>Q</sup>*<sup>2</sup> <sup>=</sup> ⎛ ⎜⎝ *q*<sup>11</sup> *q*<sup>12</sup> *q*<sup>13</sup> ∗ *q*<sup>22</sup> *q*<sup>23</sup> ∗ ∗ *q*<sup>33</sup> ⎞ ⎟⎠ , *S*<sup>1</sup> = *S*<sup>11</sup> *S*<sup>12</sup> <sup>∗</sup> *<sup>S</sup>*<sup>22</sup> , *S*<sup>2</sup> = *s*<sup>11</sup> *s*<sup>12</sup> <sup>∗</sup> *<sup>s</sup>*<sup>22</sup> ,

$$\begin{split} \Sigma\_{li}(d(t),\dot{d}(t)) &= \text{Sym}\{\Xi\_{1}^{T}P\Xi\_{2i}\} + \dot{d}(t)\Xi\_{3}^{T}S\_{1}\Xi\_{3} - \dot{d}(t)\Xi\_{4}^{T}S\_{2}\Xi\_{4} + \text{Sym}(\Xi\_{3}^{T}S\_{1}\Xi\_{5} + \Xi\_{4}^{T}S\_{2}\Xi\_{6}) \\ &+ \text{Sym}(\Xi\_{2}^{T}Q\_{1}\Xi\_{8i}) + \Xi\_{9i}^{T}Q\_{1}\Xi\_{9i} - (1 - \dot{d}(t))\Xi\_{10}^{T}Q\_{1}\Xi\_{10} + \text{Sym}(\Xi\_{11}^{T}Q\_{2}\Xi\_{12}) \\ &+ (1 - \dot{d}(t))\Xi\_{13}^{T}Q\_{2}\Xi\_{13} - \Xi\_{14}^{T}Q\_{2}\Xi\_{14} + h^{2}e\_{\text{si}}^{T}\mathcal{W}e\_{\text{si}} + \frac{h^{4}}{4}e\_{\text{si}}^{T}Ze\_{\text{si}} - h^{2}\Xi\_{15}^{T}Z\Xi\_{15} \\ &- 2h^{2}\Xi\_{16}^{T}Z\Xi\_{16} - e\_{12}^{T}L\epsilon\_{12} + (a - 2)\Lambda\_{1}^{T}W\_{0}\Lambda\_{1} - (a + 1)\Lambda\_{2}^{T}W\_{0}\Lambda\_{2} \\ &- \text{Sym}\{\Lambda\_{1}^{T}[aY\_{1} + (1 - a)Y\_{2}]\Lambda\_{2}\}, \end{split}$$

$$\begin{array}{llllll} \mathsf{a} & = & \frac{d(t)}{h}, & \mathsf{W}\_{0} = \text{diag}\{\mathsf{W}, \mathsf{W}, \mathsf{W}\}, & \Sigma\_{1} = \begin{bmatrix} e\_{1}^{T} & e\_{2}^{T} & e\_{3}^{T} & d(t)e\_{6}^{T} & (h-d(t))e\_{7}^{T} \end{bmatrix}^{\top}, \\\ \Sigma\_{2i} & = & \begin{bmatrix} e\_{si}^{T} & (1-\dot{d}(t))e\_{4}^{T} & e\_{5}^{T} & e\_{1}^{T} - (1-\dot{d}(t))e\_{2}^{T} & (1-\dot{d}(t))e\_{2}^{T} - e\_{3}^{T} \end{bmatrix}^{\top}, \\\ \Sigma\_{3} & = & \begin{bmatrix} e\_{1}^{T} & e\_{6}^{T} \end{bmatrix}^{\top}, & \Sigma\_{4} = \begin{bmatrix} e\_{1}^{T} & e\_{7}^{T} \end{bmatrix}^{\top}, & \Sigma\_{5i} = \begin{bmatrix} d(t)e\_{si}^{T} & -\dot{d}(t)e\_{6}^{T} + e\_{1}^{T} - (1-\dot{d}(t))e\_{2}^{T} \end{bmatrix}^{\top}, \end{array}$$

Ξ6*<sup>i</sup>* = (*<sup>h</sup>* <sup>−</sup> *<sup>d</sup>*(*t*))*e<sup>T</sup> si* ˙*d*(*t*)*e<sup>T</sup>* + (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))*e<sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup> T* , Ξ<sup>7</sup> = *d*(*t*)*e<sup>T</sup> <sup>e</sup><sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup> <sup>d</sup>*(*t*)(*e<sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup>* ) *T* , Ξ8*<sup>i</sup>* = 0 0 *e<sup>T</sup> siT* , Ξ9*<sup>i</sup>* = *eT <sup>e</sup><sup>T</sup> si* 0 *T* , Ξ<sup>10</sup> = *eT <sup>e</sup><sup>T</sup> <sup>e</sup><sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup> T* , <sup>Ξ</sup><sup>11</sup> = [(*<sup>h</sup>* <sup>−</sup> *<sup>d</sup>*(*t*))*e<sup>T</sup> <sup>e</sup><sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup>* (*<sup>h</sup>* <sup>−</sup> *<sup>d</sup>*(*t*))(*e<sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup>* )]*T*, <sup>Ξ</sup><sup>12</sup> = 0 0 (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))*e<sup>T</sup> T* , Ξ<sup>13</sup> = *eT <sup>e</sup><sup>T</sup>* 0 *T* , Ξ<sup>14</sup> = *eT <sup>e</sup><sup>T</sup> <sup>e</sup><sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup> T* , Ξ<sup>15</sup> = *e*<sup>1</sup> − *e*10, Ξ<sup>16</sup> = *e*<sup>1</sup> + 2*e*<sup>10</sup> − 6*e*11, Λ<sup>1</sup> = [*e<sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup> <sup>e</sup><sup>T</sup>* + *<sup>e</sup><sup>T</sup>* <sup>−</sup> <sup>2</sup>*e<sup>T</sup> <sup>e</sup><sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup>* + <sup>6</sup>*e<sup>T</sup>* <sup>−</sup> <sup>12</sup>*e<sup>T</sup>* ] *T*, Λ<sup>2</sup> = [*e<sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup> <sup>e</sup><sup>T</sup>* + *<sup>e</sup><sup>T</sup>* <sup>−</sup> <sup>2</sup>*e<sup>T</sup> <sup>e</sup><sup>T</sup>* <sup>−</sup> *<sup>e</sup><sup>T</sup>* + <sup>6</sup>*e<sup>T</sup>* <sup>−</sup> <sup>12</sup>*e<sup>T</sup>* ] *T*, *esi* = *Aie*<sup>1</sup> + *Adie*<sup>2</sup> + *Gie*12, Π = *λ*<sup>1</sup> + *λ*<sup>2</sup> + *λ*<sup>3</sup> + *h*2(*λ*<sup>4</sup> + *λ*<sup>5</sup> + *λ*<sup>26</sup> + *λ*<sup>27</sup> + *λ*<sup>32</sup> + *λ*33) + 2(*λ*<sup>6</sup> + *λ*<sup>7</sup> + *λ*10) +2*h*(*λ*<sup>8</sup> + *λ*<sup>9</sup> + *λ*<sup>11</sup> + *λ*<sup>12</sup> + *λ*<sup>13</sup> + *λ*<sup>14</sup> + *λ*<sup>18</sup> + *λ*<sup>21</sup> + *λ*<sup>25</sup> + *λ*31) + 2*h*2*λ*<sup>15</sup> <sup>+</sup>*h*(*λ*<sup>16</sup> <sup>+</sup> *<sup>λ</sup>*<sup>17</sup> <sup>+</sup> *<sup>λ</sup>*<sup>19</sup> <sup>+</sup> *<sup>λ</sup>*<sup>20</sup> <sup>+</sup> *<sup>λ</sup>*<sup>22</sup> <sup>+</sup> *<sup>λ</sup>*<sup>23</sup> <sup>+</sup> *<sup>λ</sup>*<sup>28</sup> <sup>+</sup> *<sup>λ</sup>*29) + *<sup>h</sup>*<sup>3</sup> (*λ*<sup>24</sup> <sup>+</sup> *<sup>λ</sup>*30) + *h*3 *<sup>λ</sup>*<sup>34</sup> <sup>+</sup> *h*5 *λ*35, *λ*<sup>1</sup> = *λmax*(*P*¯ ), *λ*<sup>2</sup> = *λmax*(*P*¯ ), *λ*<sup>3</sup> = *λmax*(*P*¯ ), *λ*<sup>4</sup> = *λmax*(*P*¯ ), *λ*<sup>5</sup> = *λmax*(*P*¯ ), *λ*<sup>6</sup> = *λmax*(*P*¯ ), *λ*<sup>7</sup> = *λmax*(*P*¯ ), *λ*<sup>8</sup> = *λmax*(*P*¯ ), *λ*<sup>9</sup> = *λmax*(*P*¯ ), *λ*<sup>10</sup> = *λmax*(*P*¯ ), *λ*<sup>11</sup> = *λmax*(*P*¯ ), *λ*<sup>12</sup> = *λmax*(*P*¯ ), *λ*<sup>13</sup> = *λmax*(*P*¯ ), *λ*<sup>14</sup> = *λmax*(*P*¯ ), *λ*<sup>15</sup> = *λmax*(*P*¯ ), *λ*<sup>16</sup> = *λmax*(*S*¯ ), *λ*<sup>17</sup> = *λmax*(*S*¯ ), *λ*<sup>18</sup> = *λmax*(*S*¯ ), *λ*<sup>19</sup> = *λmax*(*s*¯11), *λ*<sup>20</sup> = *λmax*(*s*¯22), *λ*<sup>21</sup> = *λmax*(*s*¯12), *λ*<sup>22</sup> = *λmax*(*Q*¯ <sup>11</sup>), *λ*<sup>23</sup> = *λmax*(*Q*¯ <sup>22</sup>), *λ*<sup>24</sup> = *λmax*(*Q*¯ <sup>33</sup>), *λ*<sup>25</sup> = *λmax*(*Q*¯ <sup>12</sup>), *λ*<sup>26</sup> = *λmax*(*Q*¯ <sup>13</sup>), *λ*<sup>27</sup> = *λmax*(*Q*¯ <sup>23</sup>), *λ*<sup>28</sup> = *λmax*(*q*¯11), *λ*<sup>29</sup> = *λmax*(*q*¯22), *λ*<sup>30</sup> = *λmax*(*q*¯33), *λ*<sup>31</sup> = *λmax*(*q*¯12), *λ*<sup>32</sup> = *λmax*(*q*¯13), *λ*<sup>33</sup> = *λmax*(*q*¯23), *λ*<sup>34</sup> = *λmax*(*W*¯ ), *λ*<sup>35</sup> = *λmax*(*Z*¯), *λ*<sup>36</sup> = *λmin*(*P*¯ ), *λ*<sup>37</sup> = *λmax*(*U*), *P*¯ *<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>P</sup>*1*jR*<sup>−</sup> <sup>1</sup> , *j* = 1, 2, 3, 4, 5, *P*¯ *<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>P</sup>*2*jR*<sup>−</sup> <sup>1</sup> , *j* = 2, 3, 4, 5, *P*¯ *<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>P</sup>*3*jR*<sup>−</sup> <sup>1</sup> , *j* = 3, 4, 5, *P*¯ *<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>P</sup>*4*jR*<sup>−</sup> <sup>1</sup> , *j* = 4, 5, *P*¯ <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>P</sup>*55*R*<sup>−</sup> <sup>1</sup> , *S*¯ *<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>S</sup>*1*jR*<sup>−</sup> <sup>1</sup> , *j* = 1, 2, *S*¯ <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>S</sup>*22*R*<sup>−</sup> <sup>1</sup> , *<sup>s</sup>*¯1*<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>s</sup>*1*jR*<sup>−</sup> <sup>1</sup> , *<sup>j</sup>* <sup>=</sup> 1, 2, *<sup>s</sup>*¯22 <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>s</sup>*22*R*<sup>−</sup> <sup>1</sup> , *<sup>Q</sup>*¯ <sup>1</sup>*<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>Q</sup>*1*jR*<sup>−</sup> <sup>1</sup> , *<sup>j</sup>* <sup>=</sup> 1, 2, 3, *<sup>Q</sup>*¯ <sup>2</sup>*<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>Q</sup>*2*jR*<sup>−</sup> <sup>1</sup> , *<sup>j</sup>* <sup>=</sup> 2, 3, *<sup>Q</sup>*¯ <sup>33</sup> <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>Q</sup>*33*R*<sup>−</sup> <sup>1</sup> , *<sup>q</sup>*¯1*<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>q</sup>*1*jR*<sup>−</sup> <sup>1</sup> , *<sup>j</sup>* <sup>=</sup> 1, 2, 3, *<sup>q</sup>*¯2*<sup>j</sup>* <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>q</sup>*2*jR*<sup>−</sup> <sup>1</sup> , *<sup>j</sup>* <sup>=</sup> 2, 3, *<sup>q</sup>*¯33 <sup>=</sup> *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *<sup>q</sup>*33*R*<sup>−</sup> <sup>1</sup> , *<sup>W</sup>*¯ = *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *WR*<sup>−</sup> <sup>1</sup> , *<sup>Z</sup>*¯ = *<sup>R</sup>*<sup>−</sup> <sup>1</sup> *ZR*<sup>−</sup> <sup>1</sup> .

**Proof.** We construct the following Lyapunov–Krasovskii functional candidate for the T-S fuzzy time-delay system (3):

$$V(\mathbf{x}(t)) = V\_1(\mathbf{x}(t)) + V\_2(\mathbf{x}(t)) + V\_3(\mathbf{x}(t)) + V\_4(\mathbf{x}(t)) + V\_5(\mathbf{x}(t)) + V\_6(\mathbf{x}(t)),\tag{11}$$

where

$$\begin{aligned} V\_1(\mathbf{x}(t)) &= \eta\_1^T(t) P \eta\_1(t), \\\\ V\_2(\mathbf{x}(t)) &= d(t) \eta\_2^T(t) S\_1 \eta\_2(t) + (h - d(t)) \eta\_3^T(t) S\_2 \eta\_3(t), \\\\ V\_3(\mathbf{x}(t)) &= \int\_{t-d(t)}^t \eta\_4^T(s) Q\_1 \eta\_4(s) ds, \\\\ V\_4(\mathbf{x}(t)) &= \int\_{t-h}^{t-d(t)} \eta\_5^T(s) Q\_2 \eta\_5(s) ds, \end{aligned}$$

$$\begin{split} &V\_{5}(\mathbf{x}(t)) = h \int\_{t-h}^{t} \int\_{\theta}^{t} \mathbf{x}^{T}(s) \mathcal{W} \dot{\mathbf{x}}(s) ds d\theta, \\ &V\_{6}(\mathbf{x}(t)) = \frac{h^{2}}{2} \int\_{t-h}^{t} \int\_{\sigma}^{t} \int\_{\theta}^{t} \mathbf{x}^{T}(s) \mathcal{Z} \dot{\mathbf{x}}(s) ds d\theta d\sigma, \\ &\qquad \text{and} \qquad \qquad \qquad \eta\_{1}(t) = [\mathbf{x}^{T}(t) \quad \mathbf{x}^{T}(t-d(t)) \quad \mathbf{x}^{T}(t-h) \quad \int\_{t-d(t)}^{t} \mathbf{x}^{T}(s) ds \quad \int\_{t-h}^{t-d(t)} \mathbf{x}^{T}(s) ds]^{T} \boldsymbol{\mu}, \\ &\qquad \qquad \eta\_{2}(t) = [\mathbf{x}^{T}(t) \quad \frac{1}{d(t)} \int\_{t-d(t)}^{t} \mathbf{x}^{T}(s) ds]^{T} , \\ &\qquad \eta\_{3}(t) = [\mathbf{x}^{T}(t) \quad \frac{1}{h-d(t)} \int\_{t-h}^{t-d(t)} \mathbf{x}^{T}(s) ds]^{T}, \\ &\qquad \eta\_{4}(s) = [\mathbf{x}^{T}(s) \quad \mathbf{x}^{T}(s) \quad \int\_{s}^{t} \mathbf{x}^{T}(\theta) d\theta]^{T}, \\ &\qquad \eta\_{5}(s) = [\mathbf{x}^{T}(s) \quad \mathbf{x}^{T}(s) \quad \int\_{s}^{t-d(t)} \mathbf{x}^{T}(\theta) d\theta]^{T}. \end{split}$$

Then, the time derivatives of *Vi*(*x*(*t*))(*i* = 1, 2, 3, 4, 5, 6) along the trajectory of the T-S fuzzy system (3) are obtained as follows:

$$\begin{aligned} \dot{V}\_{1}(\mathbf{x}(t)) &= 2 \begin{pmatrix} \mathbf{x}(t) \\ \mathbf{x}(t-d(t)) \\ \mathbf{x}(t-h) \\ \int\_{t-d(t)}^{t} \mathbf{x}(s)ds \\ \int\_{t-h}^{t-d(t)} \mathbf{x}(s)ds \end{pmatrix} P \begin{pmatrix} \dot{\mathbf{x}}(t) \\ (1-\dot{d}(t))\dot{\mathbf{x}}(t-d(t)) \\ \dot{\mathbf{x}}(t-h) \\ (1-\dot{d}(t))\mathbf{x}(t-d(t)) - \mathbf{x}(t-h) \end{pmatrix} \\ &= \sum\_{i=1}^{r} \rho\_{i}(\xi(t))\varepsilon^{T}(t)[\text{Sym}(\Xi\_{1}^{T}P\Sigma\_{2i})]\varepsilon(t). \end{aligned} \tag{12}$$

Similarly, we can also obtain

$$\mathcal{W}\_2(\mathbf{x}(t)) = \sum\_{i=1}^r \rho\_i(\xi(t)) \varepsilon^T(t) [d(t) \Xi\_3^T \mathcal{S}\_1 \Xi\_3 - d(t) \Xi\_4^T \mathcal{S}\_2 \Xi\_4 + \mathcal{S}ym(\Xi\_3^T \mathcal{S}\_1 \Xi\_{5i} + \Xi\_4^T \mathcal{S}\_2 \Xi\_{6i})] \varepsilon(t), \tag{13}$$

$$\dot{W}\_3(\mathbf{x}(t)) = \sum\_{i=1}^r \rho\_i(\ddot{\xi}(t)) \varepsilon^T(t) \left[ \text{Sym}(\Xi\_7^T Q\_1 \Xi\_{8i}) + \Xi\_9^T Q\_1 \Xi\_{9i} - (1 - \dot{d}(t)) \Xi\_{10}^T Q\_1 \Xi\_{10} \right] \varepsilon(t), \tag{14}$$

$$\dot{W}\_4(\mathbf{x}(t)) = \sum\_{i=1}^r \rho\_i(\xi(t)) \varepsilon^T(t) [\text{Sym}(\Xi\_{11}^T Q\_2 \Xi\_{12}) + (1 - \dot{d}(t)) \Xi\_{13}^T Q\_2 \Xi\_{13} - \Xi\_{14}^T Q\_2 \Xi\_{14}] \varepsilon(t), \tag{15}$$

$$\begin{split} \dot{V}\_5(\mathbf{x}(t)) &= h^2 \dot{\mathbf{x}}^T(t) W \dot{\mathbf{x}}(t) - h \int\_{t-h}^t \dot{\mathbf{x}}^T(s) W \dot{\mathbf{x}}(s) ds \\ &= \sum\_{i=1}^r \rho\_i(\mathbf{\dot{x}}(t)) \varepsilon^T(t) (h^2 e\_{si}^T W e\_{si}) \varepsilon(t) \\ &- h \int\_{t-h}^t \dot{\mathbf{x}}^T(s) W \dot{\mathbf{x}}(s) ds, \end{split} \tag{16}$$

$$\begin{split} \mathcal{V}\_{6}(\mathbf{x}(t)) &= \frac{\hbar^{4}}{4} \dot{\mathbf{x}}^{T}(t) Z \dot{\mathbf{x}}(t) - \frac{\hbar^{2}}{2} \int\_{t-h}^{t} \int\_{\theta}^{t} \dot{\mathbf{x}}^{T}(s) Z \dot{\mathbf{x}}(s) ds d\theta \\ &= \sum\_{i=1}^{r} \rho\_{i}(\xi(t)) \varepsilon^{T}(t) (\frac{\hbar^{4}}{4} e\_{si}^{T} Z e\_{si}) \varepsilon(t) \\ &- \frac{\hbar^{2}}{2} \int\_{t-h}^{t} \int\_{\theta}^{t} \dot{\mathbf{x}}^{T}(s) Z \dot{\mathbf{x}}(s) ds d\theta. \end{split} \tag{17}$$

Now, we split −*h t <sup>t</sup>*−*<sup>h</sup> <sup>x</sup>*˙ *<sup>T</sup>*(*s*)*Wx*˙(*s*)*ds*into two integrals, i.e., −*<sup>h</sup> t <sup>t</sup>*−*<sup>h</sup> <sup>x</sup>*˙ *<sup>T</sup>*(*s*)*Wx*˙(*s*)*ds* =−*<sup>h</sup> t <sup>t</sup>*−*d*(*t*) *<sup>x</sup>*˙ *<sup>T</sup>*(*s*)*Wx*˙(*s*)*ds*

−*h <sup>t</sup>*−*d*(*t*) *<sup>t</sup>*−*<sup>h</sup> <sup>x</sup>*˙ *<sup>T</sup>*(*s*)*Wx*˙(*s*)*ds*. Then, utilizing the integral inequality (24) in Lemma 5.1 of [23] for each of them yields

$$-h\int\_{t-d(t)}^{t} \dot{\mathbf{x}}^T(\mathbf{s}) \mathcal{W} \dot{\mathbf{x}}(\mathbf{s}) ds \le -\frac{h}{d(t)} \varepsilon^T(t) \Lambda\_1^T \mathcal{W}\_0 \Lambda\_1 \varepsilon(t) \tag{18}$$

and

$$-h\int\_{t-h}^{t-d(t)} \dot{\mathbf{x}}^T(\mathbf{s}) \mathcal{W} \dot{\mathbf{x}}(\mathbf{s}) d\mathbf{s} \le -\frac{h}{h - d(t)} \varepsilon^T(t) \Lambda\_2^T \mathcal{W}\_0 \Lambda\_2 \varepsilon(t),\tag{19}$$

where *W*<sup>0</sup> = *diag*{*W*, 3*W*, 5*W*}, Λ<sup>1</sup> = ⎛ ⎜⎝ *e*<sup>1</sup> − *e*<sup>2</sup> *e*<sup>1</sup> + *e*<sup>2</sup> − 2*e*<sup>6</sup> *e*<sup>1</sup> − *e*<sup>2</sup> + 6*e*<sup>6</sup> − 12*e*<sup>8</sup> ⎞ ⎟⎠ and Λ<sup>2</sup> = ⎛ ⎜⎝ *e*<sup>2</sup> − *e*<sup>3</sup> *e*<sup>2</sup> + *e*<sup>3</sup> − 2*e*<sup>7</sup> *e*<sup>2</sup> − *e*<sup>3</sup> + 6*e*<sup>7</sup> − 12*e*<sup>9</sup> ⎞ ⎟⎠.

According to Lemma 1, let *α* = *<sup>d</sup>*(*t*) *<sup>h</sup>* , *R*<sup>1</sup> = *R*<sup>2</sup> = *W*0, <sup>1</sup> = Λ1*ε*(*t*), <sup>2</sup> = Λ2*ε*(*t*), from inequalities (18) and (19), then we can obtain

$$\begin{split} & -\hbar \int\_{t-d(t)}^{t} \dot{\mathbf{x}}^{T}(\mathbf{s}) \mathcal{W} \dot{\mathbf{x}}(\mathbf{s}) d\mathbf{s} - \hbar \int\_{t-h}^{t-d(t)} \dot{\mathbf{x}}^{T}(\mathbf{s}) \mathcal{W} \dot{\mathbf{x}}(\mathbf{s}) d\mathbf{s} \\ & \leq \varepsilon^{T}(t) [(a-2)\Lambda\_{1}^{T}\mathcal{W}\_{0}\Lambda\_{1} - (a+1)\Lambda\_{2}^{T}\mathcal{W}\_{0}\Lambda\_{2} - \mathcal{Sym}\{\Lambda\_{1}^{T}[aY\_{1} + (1-a)Y\_{2}] \Lambda\_{2}\} \\ & + (1-a)\Lambda\_{1}^{T}Y\_{1}\mathcal{W}\_{0}^{-1}Y\_{1}^{T}\Lambda\_{1} + a\Lambda\_{2}^{T}Y\_{2}^{T}\mathcal{W}\_{0}^{-1}Y\_{2}\Lambda\_{2}\} \varepsilon(t). \end{split} \tag{20}$$

Applying the integral inequality (25) in Lemma 5.1 of [23] to the double integral −*h*2 2 *t t*−*h t θ x*˙ *<sup>T</sup>*(*s*)*Zx*˙(*s*)*dsdθ* in inequality (17) leads to

$$-\frac{h^2}{2} \int\_{t-h}^t \int\_{\theta}^t \dot{\mathbf{x}}^T(s) Z \dot{\mathbf{x}}(s) ds d\theta \le \varepsilon(t)^T (-h^2 \Xi\_{15}^T Z \Xi\_{15} - 2h^2 \Xi\_{16}^T Z \Xi\_{16}) \varepsilon(t),\tag{21}$$

where Ξ<sup>15</sup> = *e*<sup>1</sup> − *e*10, Ξ<sup>16</sup> = *e*<sup>1</sup> + 2*e*<sup>10</sup> − 6*e*11.

Notice that ∑*<sup>r</sup> <sup>i</sup>*=<sup>1</sup> *ρi*(*ξ*(*t*)) = 1, and we can derive the following result from (12)–(17), (20) and (21):

$$\dot{V}(\mathbf{x}(t)) \le \sum\_{i=1}^{r} \rho\_i(\ddot{\mathbf{y}}(t)) \varepsilon^T(t) \Sigma\_i(d(t), \dot{d}(t)) \varepsilon(t) + \omega^T(t) \mathcal{U} \omega(t), \tag{22}$$

where Σ*i*(*d*(*t*), ˙*d*(*t*)) = Σ1*i*(*d*(*t*), ˙*d*(*t*)) + Σ2(*d*(*t*)),

<sup>Σ</sup>1*i*(*d*(*t*), ˙*d*(*t*)) =*Sym*{Ξ*<sup>T</sup>* <sup>1</sup> *<sup>P</sup>*Ξ2*i*} <sup>+</sup> ˙*d*(*t*)Ξ*<sup>T</sup>* <sup>3</sup> *<sup>S</sup>*1Ξ<sup>3</sup> <sup>−</sup> ˙*d*(*t*)Ξ*<sup>T</sup>* <sup>4</sup> *<sup>S</sup>*2Ξ<sup>4</sup> + *Sym*(Ξ*<sup>T</sup>* <sup>3</sup> *<sup>S</sup>*1Ξ5*<sup>i</sup>* + <sup>Ξ</sup>*<sup>T</sup>* <sup>4</sup> *S*2Ξ6*i*) + *Sym*(Ξ*<sup>T</sup>* <sup>7</sup> *<sup>Q</sup>*1Ξ8*i*) + <sup>Ξ</sup>*<sup>T</sup>* <sup>9</sup>*iQ*1Ξ9*<sup>i</sup>* <sup>−</sup> (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))Ξ*<sup>T</sup>* <sup>10</sup>*Q*1Ξ<sup>10</sup> + *Sym*(Ξ*<sup>T</sup>* <sup>11</sup>*Q*2Ξ12) + (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))Ξ*<sup>T</sup>* <sup>13</sup>*Q*2Ξ<sup>13</sup> <sup>−</sup> <sup>Ξ</sup>*<sup>T</sup>* <sup>14</sup>*Q*2Ξ<sup>14</sup> + *<sup>h</sup>*2*e<sup>T</sup> siWesi* + *h*4 4 *eT siZesi* <sup>−</sup> *<sup>h</sup>*2Ξ*<sup>T</sup>* <sup>15</sup>*Z*Ξ<sup>15</sup> <sup>−</sup> <sup>2</sup>*h*2Ξ*<sup>T</sup>* <sup>16</sup>*Z*Ξ<sup>16</sup> <sup>−</sup> *<sup>e</sup><sup>T</sup>* <sup>12</sup>*Ue*<sup>12</sup> + (*<sup>α</sup>* <sup>−</sup> <sup>2</sup>)Λ*<sup>T</sup>* <sup>1</sup> *<sup>W</sup>*0Λ<sup>1</sup> <sup>−</sup> (*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)Λ*<sup>T</sup>* <sup>2</sup> *W*0Λ<sup>2</sup> <sup>−</sup> *Sym*{Λ*<sup>T</sup>* <sup>1</sup> [*αY*<sup>1</sup> + (1 − *α*)*Y*2]Λ2},

$$
\Sigma \varphi(d(t)) = (1 - \mathfrak{a}) \Lambda\_1^T \mathcal{Y}\_1 \mathcal{W}\_0^{-1} \mathcal{Y}\_1^T \Lambda\_1 + \mathfrak{a} \Lambda\_2^T \mathcal{Y}\_2^T \mathcal{W}\_0^{-1} \mathcal{Y}\_2 \Lambda\_2.
$$

Assuming <sup>Σ</sup>*i*(*d*(*t*), ˙*d*(*t*)) <sup>&</sup>lt; 0 for *<sup>i</sup>* <sup>=</sup> 1, 2, ··· ,*r*, we have

$$
\dot{\mathcal{V}}(\mathbf{x}(t)) < \beta \mathcal{V}(\mathbf{x}(t)) + \omega^T(t) l \mathcal{U} \omega(t), \tag{23}
$$

where *β* > 0 is a constant.

However, Σ*i*(*d*(*t*), ˙*d*(*t*)) depends on the time-varying delay *d*(*t*) and its derivative ˙*d*(*t*). Therefore, Σ*i*(*d*(*t*), ˙*d*(*t*)) < 0 cannot be solved directly by applying an LMI tool. Noting that Σ*i*(*d*(*t*), ˙*d*(*t*)) is a linear function of *d*(*t*) and ˙*d*(*t*), it is obvious that Σ*i*(*d*(*t*), ˙*d*(*t*)) < 0 can be satisfied if the following inequalities (24)–(27) hold,

$$
\Sigma\_i(0, \mu\_1) < 0,\tag{24}
$$

$$
\Sigma\_i(0, \mu\_2) < 0,\tag{25}
$$

$$
\Sigma\_i(\hbar, \mu\_1) < 0,\tag{26}
$$

$$
\Sigma\_i(h, \mu\_2) < 0.\tag{27}
$$

According to Schur complement lemma, the inequalities (24)–(27) are equivalent to inequalities (6)–(9), respectively. Thus, the inequalities (6)–(9) can ensure Σ*i*(*d*(*t*), ˙*d*(*t*)) < 0 holds. Furthermore, the inequalities (6)–(9) can also guarantee that the inequality (23) holds.

Multiplying (23) by *e*−*β<sup>t</sup>* , we can obtain

$$e^{-\beta t}\dot{V}(\mathbf{x}(t)) - \beta e^{-\beta t}V(\mathbf{x}(t)) \prec e^{-\beta t}\omega^T(t)l\mathcal{U}\omega(t),$$

i.e.,

$$\frac{d}{dt}(e^{-\beta t}V(\mathbf{x}(t))) < e^{-\beta t}\omega^T(t)\mathcal{U}\omega(t). \tag{28}$$

Integrating (28) from 0 to t with *t* ∈ [0, *Tf* ], we have

$$e^{-\beta t}V(\varkappa(t)) - V(\varkappa(0)) < \int\_0^t e^{-\beta s} \omega^T(s) l l \omega(s) ds.$$

Noting that *β* > 0, we can derive

$$\begin{aligned} V(\mathbf{x}(t)) &< \varepsilon^{\beta t} V(\mathbf{x}(0)) + \varepsilon^{\beta t} \int\_0^t \varepsilon^{-\beta s} \omega^T(s) \mathcal{U} \omega(s) ds \\ &\le \varepsilon^{\beta t} V(\mathbf{x}(0)) + \varepsilon^{\beta t} \lambda\_{\max}(\mathcal{U}) \int\_0^t \omega^T(s) \omega(s) ds. \end{aligned}$$

Therefore, we have

$$V(\mathbf{x}(t)) < e^{\delta T\_f} \left[ V(\mathbf{x}(0)) + \lambda\_{\max}(\mathcal{U}) \delta \right]. \tag{29}$$

In addition, it can be easily obtained that

*<sup>V</sup>*(*x*(*t*)) <sup>≥</sup> *<sup>x</sup>T*(*t*)*P*11*x*(*t*) = *<sup>x</sup>T*(*t*)*R*<sup>1</sup> 2 *P*¯ 11*R*<sup>1</sup> <sup>2</sup> *<sup>x</sup>*(*t*) <sup>≥</sup> *<sup>λ</sup>min*(*P*¯ <sup>11</sup>)*xT*(*t*)*Rx*(*t*) = *λ*36*xT*(*t*)*Rx*(*t*), *V*(*x*(0)) = *η<sup>T</sup>* <sup>1</sup> (0)*Pη*1(0) + *<sup>d</sup>*(0)*η<sup>T</sup>* <sup>2</sup> (0)*S*1*η*2(0)+(*<sup>h</sup>* <sup>−</sup> *<sup>d</sup>*(0))*η<sup>T</sup>* <sup>3</sup> (0)*S*2*η*3(0) + <sup>0</sup> −*d*(0) *ηT* <sup>4</sup> (*s*)*Q*1*η*4(*s*)*ds* + <sup>−</sup>*d*(0) −*h ηT* <sup>5</sup> (*s*)*Q*2*η*5(*s*)*ds* + *h* <sup>0</sup> −*h* <sup>0</sup> *θ x*˙ *<sup>T</sup>*(*s*)*Wx*˙(*s*)*dsdθ* + *h*2 2 <sup>0</sup> −*h* <sup>0</sup> *σ* <sup>0</sup> *θ x*˙ *<sup>T</sup>*(*s*)*Zx*˙(*s*)*dsdθdσ* <sup>≤</sup> [*λmax*(*P*¯ <sup>11</sup>) + *λmax*(*P*¯ <sup>22</sup>) + *λmax*(*P*¯ <sup>33</sup>) + *h*2*λmax*(*P*¯ <sup>44</sup>) + *h*2*λmax*(*P*¯ <sup>55</sup>) + 2*λmax*(*P*¯ <sup>12</sup>) + 2*λmax*(*P*¯ <sup>13</sup>) + 2*hλmax*(*P*¯ <sup>14</sup>) + 2*hλmax*(*P*¯ <sup>15</sup>) + 2*λmax*(*P*¯ <sup>23</sup>) + 2*hλmax*(*P*¯ <sup>24</sup>) + 2*hλmax*(*P*¯ <sup>25</sup>)

+ 2*hλmax*(*P*¯ <sup>34</sup>) + 2*hλmax*(*P*¯ <sup>35</sup>) + 2*h*2*λmax*(*P*¯ <sup>45</sup>) + *hλmax*(*S*¯ <sup>11</sup>) + *hλmax*(*S*¯ <sup>22</sup>) + 2*hλmax*(*S*¯ <sup>12</sup>) <sup>+</sup> *<sup>h</sup>λmax*(*s*¯11) + *<sup>h</sup>λmax*(*s*¯22) + <sup>2</sup>*hλmax*(*s*¯12) + *<sup>h</sup>λmax*(*Q*¯ <sup>11</sup>) + *<sup>h</sup>λmax*(*Q*¯ <sup>22</sup>) + *<sup>h</sup>*<sup>3</sup> <sup>3</sup> *<sup>λ</sup>max*(*Q*¯ <sup>33</sup>) <sup>+</sup> <sup>2</sup>*hλmax*(*Q*¯ <sup>12</sup>) + *<sup>h</sup>*2*λmax*(*Q*¯ <sup>13</sup>) + *<sup>h</sup>*2*λmax*(*Q*¯ <sup>23</sup>) + *<sup>h</sup>λmax*(*q*¯11) + *<sup>h</sup>λmax*(*q*¯22) + *<sup>h</sup>*<sup>3</sup> <sup>3</sup> *<sup>λ</sup>max*(*q*¯33) <sup>+</sup> <sup>2</sup>*hλmax*(*q*¯12) + *<sup>h</sup>*2*λmax*(*q*¯13) + *<sup>h</sup>*2*λmax*(*q*¯23) + *<sup>h</sup>*<sup>3</sup> <sup>2</sup> *<sup>λ</sup>max*(*W*¯ ) + *<sup>h</sup>*<sup>5</sup> <sup>12</sup>*λmax*(*Z*¯)] <sup>×</sup> *sup*−*h*≤*θ*≤0{*xT*(*θ*)*Rx*(*θ*), *<sup>x</sup>*˙ *<sup>T</sup>*(*θ*)*Rx*˙(*θ*)} <sup>≤</sup>[*λ*<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*<sup>2</sup> <sup>+</sup> *<sup>λ</sup>*<sup>3</sup> <sup>+</sup> *<sup>h</sup>*2(*λ*<sup>4</sup> <sup>+</sup> *<sup>λ</sup>*<sup>5</sup> <sup>+</sup> *<sup>λ</sup>*<sup>26</sup> <sup>+</sup> *<sup>λ</sup>*<sup>27</sup> <sup>+</sup> *<sup>λ</sup>*<sup>32</sup> <sup>+</sup> *<sup>λ</sup>*33) + <sup>2</sup>(*λ*<sup>6</sup> <sup>+</sup> *<sup>λ</sup>*<sup>7</sup> <sup>+</sup> *<sup>λ</sup>*10) + <sup>2</sup>*h*(*λ*<sup>8</sup> <sup>+</sup> *<sup>λ</sup>*<sup>9</sup> + *λ*<sup>11</sup> + *λ*<sup>12</sup> + *λ*<sup>13</sup> + *λ*<sup>14</sup> + *λ*<sup>18</sup> + *λ*<sup>21</sup> + *λ*<sup>25</sup> + *λ*31) + 2*h*2*λ*<sup>15</sup> + *h*(*λ*<sup>16</sup> + *λ*<sup>17</sup> + *λ*<sup>19</sup> + *λ*<sup>20</sup> <sup>+</sup> *<sup>λ</sup>*<sup>22</sup> <sup>+</sup> *<sup>λ</sup>*<sup>23</sup> <sup>+</sup> *<sup>λ</sup>*<sup>28</sup> <sup>+</sup> *<sup>λ</sup>*29) + *<sup>h</sup>*<sup>3</sup> <sup>3</sup> (*λ*<sup>24</sup> <sup>+</sup> *<sup>λ</sup>*30) + *<sup>h</sup>*<sup>3</sup> <sup>2</sup> *<sup>λ</sup>*<sup>34</sup> <sup>+</sup> *h*5 <sup>12</sup>*λ*35]*c*1.

We substitute the above two inequalities into (29) and assume the inequality (10) holds, we can easily derive that *<sup>x</sup>T*(*t*)*Rx*(*t*) ≤ *<sup>c</sup>*<sup>2</sup> for all *<sup>t</sup>* ∈ [0, *Tf* ]. Thus, the proof is completed.

**Remark 1.** *The novel augmented Lyapunov–Krasovskii functional constructed in (11) takes advantage of information regarding the time-varying delay, which can make the obtained new finite-time boundedness condition less conservative. In addition, the Lyapunov–Krasovskii functional (11) is more general due to the introduction of several augmented vectors and two delay-product-type terms, such as η*1(*t*)*, η*4(*s*)*, η*5(*s*)*, d*(*t*)*η<sup>T</sup>* <sup>2</sup> (*t*)*S*1*η*2(*t*) *and* (*<sup>h</sup>* − *<sup>d</sup>*(*t*))*η<sup>T</sup>* <sup>3</sup> (*t*)*S*2*η*3(*t*)*. When several subblocks of the partitioned matrices P*, *S*1, *S*2, *Q*1, *Q*<sup>2</sup> *are zero matrices with appropriate dimensions and W* = 0*, Z* = 0*, the augmented Lyapunov–Krasovskii functional V*(*x*(*t*)) *reduces to the simpler Lyapunov functions in some literature [24,26,34]. Additionally, to the best of our knowledge, the chosen Lyapunov–Krasovskii functional is a simple LKF instead of an augmented LKF in most existing studies regarding finite-time boundedness of dynamical systems, which is because the augmented LKF increases the difficulty of deriving finite-time boundedness criteria in terms of LMIs. However, this problem has been successfully solved in Theorem 1.*

**Remark 2.** *In Theorem 1, the improved reciprocally convex combination inequality and the auxiliary function-based integral inequalities are utilized to estimate the bound of the derivative of the constructed LKF. The auxiliary function-based integral inequalities are more general, as they can reduce to some other integral inequalities by appropriately choosing the auxiliary functions [23], such as the Jensen inequality, Bessel–Legendre inequality and Wirtinger-based integral inequality. In addition, the improved reciprocally convex combination inequality can provide a maximum lower bound with less slack matrix variables for several reciprocally convex combinations, which plays a critical role in reducing the conservativeness and the calculation complexity of the delay-dependent finite-time boundedness conditions for T-S fuzzy systems with time-varying delay and norm-bounded disturbance.*

#### *3.2. Controller Design*

Based on the delay-dependent finite-time boundedness criterion proposed in Theorem 1, we develop a memory state feedback controller to ensure the finite-time boundness of the resulting closed-loop T-S fuzzy time-delay system in the following theorem, which can be derived by solving a feasibility problem in terms of the linear matrix inequalities.

**Theorem 2.** *For the given scalars h* > 0*, μ*1*, and μ*2*, the T-S fuzzy system (5) with a time-varying delay d*(*t*) *satisfying (1) is finite-time bounded with respect to* (*c*1, *c*2, *Tf* , *R*, *δ*, *h*)*, if there exist scalars β* > 0*, γ, symmetric positive definite matrices <sup>P</sup>*˜ <sup>∈</sup> *R*5*n*×<sup>5</sup>*n, <sup>S</sup>*˜ 1, *S*˜ <sup>2</sup> <sup>∈</sup> *<sup>R</sup>*2*n*×<sup>2</sup>*n, <sup>Q</sup>*˜ 1, *<sup>Q</sup>*˜ <sup>2</sup> <sup>∈</sup> *<sup>R</sup>*3*n*×<sup>3</sup>*n, <sup>W</sup>*˜ , *<sup>Z</sup>*˜, *<sup>U</sup>*˜ <sup>∈</sup> *<sup>R</sup>n*×*n, any matrices Y*˜ 1,*Y*˜ <sup>2</sup> <sup>∈</sup> *<sup>R</sup>*3*n*×<sup>3</sup>*n, X* <sup>∈</sup> *<sup>R</sup>n*×*<sup>n</sup> and L*1*j*, *<sup>L</sup>*2*<sup>j</sup>* <sup>∈</sup> *<sup>R</sup>p*×*n*(*<sup>j</sup>* <sup>=</sup> 1, 2, . . . ,*r*)*, such that the following conditions hold:*

$$
\begin{pmatrix}
\Sigma\_{ii}(0,\mu\_1) & \tilde{\Lambda}\_1^T \tilde{Y}\_1 \\
\tilde{Y}\_1^T \tilde{\Lambda}\_1 & -\tilde{W}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{30}
$$

$$
\begin{pmatrix}
\tilde{\Sigma}\_{ii}(0, \mu\_2) & \tilde{\Lambda}\_1^T \tilde{Y}\_1 \\
\tilde{Y}\_1^T \tilde{\Lambda}\_1 & -\tilde{W}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{31}
$$

$$
\begin{pmatrix}
\Sigma\_{ii}(h\_r\mu\_1) & \tilde{\Lambda}\_2^T \tilde{Y}\_2^T \\
\tilde{Y}\_2 \tilde{\Lambda}\_2 & -\tilde{W}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{32}
$$

$$
\begin{pmatrix}
\Sigma\_{ii}(h,\mu\_2) & \bar{\Lambda}\_2^T \bar{Y}\_2^T \\
\bar{Y}\_2 \bar{\Lambda}\_2 & -\bar{W}\_0
\end{pmatrix} < 0, \quad i = 1, 2, \dots, r
\tag{33}
$$

$$
\begin{pmatrix}
\tilde{\Psi}\_{ij}(0, \mu\_1) & \tilde{\Lambda}\_1^T \tilde{Y}\_1 \\
\tilde{Y}\_1^T \tilde{\Lambda}\_1 & \frac{1-r}{r} \tilde{W}\_0
\end{pmatrix} < 0, \quad i, j = 1, 2, \dots, r, \ i \neq j \tag{34}
$$

$$\begin{pmatrix} \Psi\_{ij}(0,\mu\_2) & \bar{\Lambda}\_1^T \bar{Y}\_1\\ \bar{Y}\_1^T \bar{\Lambda}\_1 & \frac{1-r}{r} \tilde{W}\_0 \end{pmatrix} < 0, \quad i,j = 1,2,\dots,r, \ i \neq j \tag{35}$$

$$\left(\begin{array}{c} \Psi\_{ij}(h\_{\boldsymbol{r}}\mu\_{1}) \\ \Psi\_{2}\bar{\Lambda}\_{2} \end{array} \begin{array}{c} \bar{\Lambda}\_{2}^{T}\bar{Y}\_{2}^{T} \\ \frac{1-r}{r}\bar{W}\_{0} \end{array} \right) < 0, \quad i,j = 1,2,...,r, \; i \neq j \tag{36}$$

$$
\begin{pmatrix}
\Psi\_{ij}(\mathsf{h}\_{r}\mu\_{2}) & \mathsf{A}\_{2}^{T}\mathsf{Y}\_{2}^{T} \\
\tilde{\mathsf{Y}}\_{2}\tilde{\mathsf{A}}\_{2} & \frac{1-r}{r}\tilde{\mathsf{W}}\_{0}
\end{pmatrix} < 0, \quad \mathsf{i}, \mathsf{j} = 1, 2, \dots, r, \ \mathsf{i} \neq \mathsf{j} \tag{37}
$$

$$
\mathcal{L}\_1 \mathbf{\tilde{I}} + \mathbf{\tilde{A}}\_{\mathfrak{J}\mathfrak{T}} \boldsymbol{\delta} < \mathbf{\tilde{A}}\_{\mathfrak{M}\mathfrak{G}} \mathbf{e}^{-\mathfrak{G}T\_f},\tag{38}
$$

*where*

*P*˜ = ⎛ ⎜⎜⎜⎜⎜⎝ *P*˜ <sup>11</sup> *P*˜ <sup>12</sup> *P*˜ <sup>13</sup> *P*˜ <sup>14</sup> *P*˜ 15 <sup>∗</sup> *<sup>P</sup>*˜ <sup>22</sup> *P*˜ <sup>23</sup> *P*˜ <sup>24</sup> *P*˜ 25 ∗ ∗ *<sup>P</sup>*˜ <sup>33</sup> *P*˜ <sup>34</sup> *P*˜ 35 ∗∗∗ *<sup>P</sup>*˜ <sup>44</sup> *P*˜ 45 ∗∗∗∗ *<sup>P</sup>*˜ 55 ⎞ ⎟⎟⎟⎟⎟⎠ , *Q*˜ <sup>1</sup> = ⎛ ⎜⎝ *Q*˜ <sup>11</sup> *Q*˜ <sup>12</sup> *Q*˜ <sup>13</sup> <sup>∗</sup> *<sup>Q</sup>*˜ <sup>22</sup> *<sup>Q</sup>*˜ <sup>23</sup> ∗ ∗ *<sup>Q</sup>*˜ <sup>33</sup> ⎞ ⎟⎠ , *<sup>Q</sup>*˜ <sup>2</sup> <sup>=</sup> ⎛ ⎜⎝ *q*˜11 *q*˜12 *q*˜13 ∗ *q*˜22 *q*˜23 ∗ ∗ *q*˜33 ⎞ ⎟⎠ , *S*˜ <sup>1</sup> = *S*˜ <sup>11</sup> *S*˜ 12 <sup>∗</sup> *<sup>S</sup>*˜ 22 , *S*˜ <sup>2</sup> = *s*˜11 *s*˜12 <sup>∗</sup> *<sup>s</sup>*˜22 ,

$$\begin{split} \Sigma\_{ij}(\boldsymbol{d}(t),\dot{\boldsymbol{d}}(t)) &= \text{Sym}\{\mathfrak{S}\_1^T \boldsymbol{P} \mathfrak{S}\_2\} + \dot{\boldsymbol{d}}(t) \mathfrak{S}\_3^T \tilde{\boldsymbol{S}}\_1 \mathfrak{S}\_3 - \dot{\boldsymbol{d}}(t) \mathfrak{S}\_4^T \tilde{\boldsymbol{S}}\_2 \mathfrak{S}\_4 + \text{Sym}(\mathfrak{S}\_3^T \tilde{\boldsymbol{S}}\_1 \mathfrak{S}\_5 + \mathfrak{S}\_4^T \tilde{\boldsymbol{S}}\_2 \mathfrak{S}\_6) \\ &+ \text{Sym}(\tilde{\boldsymbol{\Xi}}\_T^T \tilde{\boldsymbol{Q}}\_1 \tilde{\boldsymbol{\Xi}}\_8) + \tilde{\boldsymbol{\Xi}}\_9^T \tilde{\boldsymbol{Q}}\_1 \tilde{\boldsymbol{\Xi}}\_9 - (1 - \dot{\boldsymbol{d}}(t)) \tilde{\boldsymbol{\Xi}}\_{10}^T \tilde{\boldsymbol{Q}}\_1 \tilde{\boldsymbol{\Xi}}\_{10} + \text{Sym}(\tilde{\boldsymbol{\Xi}}\_{11}^T \tilde{\boldsymbol{Q}}\_2 \tilde{\boldsymbol{\Xi}}\_{12}) \end{split}$$

$$\begin{split} &+ (1 - \dot{d}(t)) \mathfrak{S}\_{13}^{T} \mathsf{Q}\_{2} \mathfrak{S}\_{13} - \mathfrak{S}\_{14}^{T} \mathsf{Q}\_{2} \mathfrak{S}\_{14} + h^{2} \bar{\varepsilon}\_{13}^{T} \mathsf{W} \ddot{\varepsilon}\_{13} + \frac{h^{4}}{4} \bar{\varepsilon}\_{13}^{T} \mathsf{Z} \dot{\varepsilon}\_{13} - h^{2} \mathfrak{S}\_{15}^{T} \mathsf{Z} \dot{\mathfrak{S}}\_{15} \\ &- 2h^{2} \mathfrak{S}\_{16}^{T} \mathsf{Z} \dot{\mathfrak{S}}\_{16} - \bar{\varepsilon}\_{12}^{T} \mathsf{Q} \dot{\varepsilon}\_{12} + (a - 2) \bar{\Lambda}\_{1}^{T} \mathsf{W}\_{0} \bar{\Lambda}\_{1} - (a + 1) \bar{\Lambda}\_{2}^{T} \mathsf{W}\_{0} \bar{\Lambda}\_{2} \\ &- \mathrm{Sym} \{ \bar{\Lambda}\_{1}^{T} [a \bar{\chi}\_{1} + (1 - a) \bar{\chi}\_{2}] \bar{\Lambda}\_{2} \} + \mathrm{Sym} \{ (\bar{\varepsilon}\_{1}^{T} + \gamma \bar{\epsilon}\_{13}^{T}) [A\_{i} X \bar{\varepsilon}\_{1} + B\_{i} L\_{1j} \bar{\varepsilon}\_{1} \\ &+ A\_{di} X \bar{\varrho}\_{2} + B\_{i} L\_{2j} \bar{\varepsilon}\_{2} + G\_{i} X \bar{\varepsilon}\_{12} - X \bar{\varepsilon}\_{13} \}, \end{split}$$

$$\Psi\_{i\bar{j}}(d(t), \dot{d}(t)) = \frac{1}{r-1} \Sigma\_{i\bar{i}}(d(t), \dot{d}(t)) + \frac{1}{2} \Sigma\_{i\bar{j}}(d(t), \dot{d}(t)) + \frac{1}{2} \Sigma\_{\bar{j}i}(d(t), \dot{d}(t)),$$

$$\begin{array}{llllllll}\hline\\l&\omega&\omega&\mathbb{Z}\_{1}&\operatorname{\bf{f}}&\operatorname{\bf{div}}(\operatorname{\bf{V}},\operatorname{\bf{div}},\operatorname{\bf{F}})&\operatorname{\bf{\widetilde{\omega}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}&\operatorname{\widetilde{\mathsf{\mathcal{E}}}}\\\hline\\l&\omega&\mathbb{Z}\_{2}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}\\\hline\\l&\omega&\mathbb{Z}\_{3}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}&\operatorname{\bf{\mathcal{E}}}\\\hline\\l&\omega&\mathbb{Z}\_{2}&\operatorname{\bf$$

*In this case, the memory state feedback controller gains are given by K*1*<sup>j</sup>* = *L*1*jX*−1*, K*2*<sup>j</sup>* = *L*2*jX*−1*, j* = 1, 2, . . . ,*r.*

**Proof.** Choose the Lyapunov–Krasovskii functional candidate (11) again for the resulting closed-loop T-S fuzzy time-delay system (5).

From the proof of Theorem 1, we obtain the inequality (22):

$$\dot{V}(\mathbf{x}(t)) \le \sum\_{i=1}^r \rho\_i(\dot{\xi}(t)) \varepsilon^T(t) \Sigma\_i(d(t), \dot{d}(t)) \varepsilon(t) + \omega^T(t) l L \omega(t).$$

Furthermore, it can be easily obtained that

$$\dot{V}(x(t)) \le \varepsilon^T(t)\hat{\Sigma}(d(t), \dot{d}(t))\overline{\varepsilon}(t) + \omega^T(t)l\mathcal{U}\omega(t),\tag{39}$$

where Σˆ(*d*(*t*), ˙*d*(*t*)) = Σˆ <sup>1</sup>(*d*(*t*), ˙*d*(*t*)) + Σˆ <sup>2</sup>(*d*(*t*)),

<sup>Σ</sup><sup>ˆ</sup> <sup>1</sup>(*d*(*t*), ˙*d*(*t*)) =*Sym*{Ξ˜ *<sup>T</sup>* <sup>1</sup> *<sup>P</sup>*Ξ˜ <sup>2</sup>} <sup>+</sup> ˙*d*(*t*)Ξ˜ *<sup>T</sup>* <sup>3</sup> *<sup>S</sup>*1Ξ˜ <sup>3</sup> <sup>−</sup> ˙*d*(*t*)Ξ˜ *<sup>T</sup>* <sup>4</sup> *<sup>S</sup>*2Ξ˜ <sup>4</sup> + *Sym*(Ξ˜ *<sup>T</sup>* <sup>3</sup> *<sup>S</sup>*1Ξ˜ <sup>5</sup> + <sup>Ξ</sup>˜ *<sup>T</sup>* <sup>4</sup> *<sup>S</sup>*2Ξ˜ <sup>6</sup>) + *Sym*(Ξ˜ *<sup>T</sup>* <sup>7</sup> *<sup>Q</sup>*1Ξ˜ <sup>8</sup>) + <sup>Ξ</sup>˜ *<sup>T</sup>* <sup>9</sup> *<sup>Q</sup>*1Ξ˜ <sup>9</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))Ξ˜ *<sup>T</sup>* <sup>10</sup>*Q*1Ξ˜ <sup>10</sup> + *Sym*(Ξ˜ *<sup>T</sup>* <sup>11</sup>*Q*2Ξ˜ <sup>12</sup>) + (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))Ξ˜ *<sup>T</sup>* <sup>13</sup>*Q*2Ξ˜ <sup>13</sup> <sup>−</sup> <sup>Ξ</sup>˜ *<sup>T</sup>* <sup>14</sup>*Q*2Ξ˜ <sup>14</sup> + *<sup>h</sup>*2*e*˜ *T* <sup>13</sup>*We*˜13 + *h*4 4 *e*˜ *T* <sup>13</sup>*Ze*˜13 <sup>−</sup> *<sup>h</sup>*2Ξ˜ *<sup>T</sup>* <sup>15</sup>*Z*Ξ˜ <sup>15</sup> <sup>−</sup> <sup>2</sup>*h*2Ξ˜ *<sup>T</sup>* <sup>16</sup>*Z*Ξ˜ <sup>16</sup> <sup>−</sup> *<sup>e</sup>*˜ *T* <sup>12</sup>*Ue*˜12 + (*<sup>α</sup>* <sup>−</sup> <sup>2</sup>)Λ˜ *<sup>T</sup>* <sup>1</sup> *<sup>W</sup>*0Λ˜ <sup>1</sup> <sup>−</sup> (*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)Λ˜ *<sup>T</sup>* <sup>2</sup> *<sup>W</sup>*0Λ˜ <sup>2</sup> <sup>−</sup> *Sym*{Λ˜ *<sup>T</sup>* <sup>1</sup> [*αY*<sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)*Y*2]Λ˜ <sup>2</sup>},

$$
\hat{\Sigma}\_2(d(t)) = (1 - \kappa)\tilde{\Lambda}\_1^T \mathcal{Y}\_1 \mathcal{W}\_0^{-1} \mathcal{Y}\_1^T \tilde{\Lambda}\_1 + \kappa \tilde{\Lambda}\_2^T \mathcal{Y}\_2^T \mathcal{W}\_0^{-1} \mathcal{Y}\_2 \tilde{\Lambda}\_2.
$$

According to the inequality (39), we have *V*˙ (*x*(*t*)) < *βV*(*x*(*t*)) + *ωT*(*t*)*Uω*(*t*), if the following inequality holds,

$$
\mathbb{E}^T(t)\hat{\Sigma}(d(t), \dot{d}(t))\mathbb{E}(t) < 0. \tag{40}
$$

Then, similar to Theorem 1, if the inequalities (10) and (40) hold, we can easily obtain that the closed-loop T-S fuzzy time-delay system (5) is finite-time bounded with respect to (*c*1, *c*2, *Tf* , *R*, *δ*, *h*).

Now, the closed-loop T-S fuzzy time-delay system (5) can be rewritten as:

$$
\Theta(t)\overline{\varepsilon}(t) = 0,
$$

where Θ(*t*)=[*A*(*t*) + *B*(*t*)*K*1(*t*) *Ad*(*t*) + *B*(*t*)*K*2(*t*) 000000000 *G*(*t*) − *I*].

According to Finsler lemma, from (40) and (41), it can be obtained that the closed-loop T-S fuzzy time-delay system (5) is finite-time bounded with respect to (*c*1, *c*2, *Tf* , *R*, *δ*, *h*) if there exists a matrix <sup>Φ</sup> <sup>∈</sup> **<sup>R</sup>**13*n*×*n*, such that:

$$
\hat{\Sigma}(d(t), \dot{d}(t)) + \text{Sym}\{\Phi \Theta(t)\} < 0. \tag{42}
$$

Let Φ = [*X*−<sup>1</sup> 00000000000 *γX*−1] *<sup>T</sup>*, where *γ* is an arbitrary scalar. Then, we have the inequality (42) is equivalent to

$$\begin{split} \hat{\Sigma}(d(t), d(t)) &+ \text{Sym}\{ (\overline{\varepsilon}\_1^T X^{-T} + \gamma \overline{\varepsilon}\_{13}^T X^{-T}) [A(t)\overline{\varepsilon}\_1 + B(t)K\_1(t)\overline{\varepsilon}\_1 \\ &+ A\_d(t)\overline{\varepsilon}\_2 + B(t)K\_2(t)\overline{\varepsilon}\_2 + G(t)\overline{\varepsilon}\_{12} - \overline{\varepsilon}\_{13}] \} < 0. \end{split} \tag{43}$$

Let Γ<sup>1</sup> = *diag*(*X*, *X*, *X*, *X*, *X*, *X*, *X*, *X*, *X*, *X*, *X*, *X*, *X*). Multiplying (43) left by Γ*<sup>T</sup>* <sup>1</sup> and right by Γ1, we can obtain the equivalent condition of (43) as follows:

$$\begin{aligned} &\Gamma\_1^T \Sigma(d(t), d(t)) \Gamma\_1 + Sym\{ (\overline{e}\_1^T + \gamma \overline{e}\_{13}^T) [A(t)X\overline{e}\_1 + B(t)K\_1(t)X\overline{e}\_1 \\ &+ A\_d(t)X\overline{e}\_2 + B(t)K\_2(t)X\overline{e}\_2 + G(t)X\overline{e}\_{12} - X\overline{e}\_{13} \} < 0. \end{aligned}$$

Let Γ<sup>2</sup> = *diag*(*X*, *X*, *X*, *X*, *X*), Γ<sup>3</sup> = *diag*(*X*, *X*), Γ<sup>4</sup> = *diag*(*X*, *X*, *X*), *P*˜ = Γ*<sup>T</sup>* <sup>2</sup> *<sup>P</sup>*Γ2, *<sup>S</sup>*˜ <sup>1</sup> = Γ*<sup>T</sup>* <sup>3</sup> *<sup>S</sup>*1Γ3, *<sup>S</sup>*˜ <sup>2</sup> = Γ*T* <sup>3</sup> *<sup>S</sup>*2Γ3, *<sup>Q</sup>*˜ <sup>1</sup> = <sup>Γ</sup>*<sup>T</sup>* <sup>4</sup> *<sup>Q</sup>*1Γ4, *<sup>Q</sup>*˜ <sup>2</sup> <sup>=</sup> <sup>Γ</sup>*<sup>T</sup>* <sup>4</sup> *<sup>Q</sup>*2Γ4, *<sup>W</sup>*˜ <sup>=</sup> *<sup>X</sup>TWX*, *<sup>Z</sup>*˜ <sup>=</sup> *<sup>X</sup>TZX*, *<sup>U</sup>*˜ <sup>=</sup> *<sup>X</sup>TUX*, *<sup>W</sup>*˜ <sup>0</sup> <sup>=</sup> <sup>Γ</sup>*<sup>T</sup>* <sup>4</sup> *W*0Γ4, *Y*˜ <sup>1</sup> = Γ*<sup>T</sup>* <sup>4</sup> *<sup>Y</sup>*1Γ4, *<sup>Y</sup>*˜ <sup>2</sup> = Γ*<sup>T</sup>* <sup>4</sup> *<sup>Y</sup>*2Γ4. It can be easily derived that <sup>Γ</sup>*<sup>T</sup>* <sup>1</sup> <sup>Σ</sup>ˆ(*d*(*t*), ˙*d*(*t*))Γ<sup>1</sup> <sup>=</sup> <sup>Σ</sup>˜ <sup>1</sup>(*d*(*t*), ˙*d*(*t*)) + <sup>Σ</sup>˜ <sup>2</sup>(*d*(*t*)), where

<sup>Σ</sup>˜ <sup>1</sup>(*d*(*t*), ˙*d*(*t*)) =*Sym*{Ξ˜ *<sup>T</sup>* <sup>1</sup> *<sup>P</sup>*˜Ξ˜ <sup>2</sup>} <sup>+</sup> ˙*d*(*t*)Ξ˜ *<sup>T</sup>* 3 *S*˜ <sup>1</sup>Ξ˜ <sup>3</sup> <sup>−</sup> ˙*d*(*t*)Ξ˜ *<sup>T</sup>* 4 *S*˜ <sup>2</sup>Ξ˜ <sup>4</sup> + *Sym*(Ξ˜ *<sup>T</sup>* 3 *S*˜ <sup>1</sup>Ξ˜ <sup>5</sup> + Ξ˜ *<sup>T</sup>* 4 *S*˜ <sup>2</sup>Ξ˜ <sup>6</sup>) + *Sym*(Ξ˜ *<sup>T</sup>* <sup>7</sup> *<sup>Q</sup>*˜ <sup>1</sup>Ξ˜ <sup>8</sup>) + <sup>Ξ</sup>˜ *<sup>T</sup>* <sup>9</sup> *<sup>Q</sup>*˜ <sup>1</sup>Ξ˜ <sup>9</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))Ξ˜ *<sup>T</sup>* <sup>10</sup>*Q*˜ <sup>1</sup>Ξ˜ <sup>10</sup> + *Sym*(Ξ˜ *<sup>T</sup>* <sup>11</sup>*Q*˜ <sup>2</sup>Ξ˜ <sup>12</sup>) + (<sup>1</sup> <sup>−</sup> ˙*d*(*t*))Ξ˜ *<sup>T</sup>* <sup>13</sup>*Q*˜ <sup>2</sup>Ξ˜ <sup>13</sup> <sup>−</sup> <sup>Ξ</sup>˜ *<sup>T</sup>* <sup>14</sup>*Q*˜ <sup>2</sup>Ξ˜ <sup>14</sup> + *<sup>h</sup>*2*e*˜ *T* <sup>13</sup>*W*˜ *<sup>e</sup>*˜13 + *h*4 4 *e*˜ *T* <sup>13</sup>*Z*˜*e*˜13 <sup>−</sup> *<sup>h</sup>*2Ξ˜ *<sup>T</sup>* <sup>15</sup>*Z*˜Ξ˜ <sup>15</sup> <sup>−</sup> <sup>2</sup>*h*2Ξ˜ *<sup>T</sup>* <sup>16</sup>*Z*˜Ξ˜ <sup>16</sup> <sup>−</sup> *<sup>e</sup>*˜ *T* <sup>12</sup>*U*˜ *<sup>e</sup>*˜12 + (*<sup>α</sup>* <sup>−</sup> <sup>2</sup>)Λ˜ *<sup>T</sup>* <sup>1</sup> *<sup>W</sup>*˜ <sup>0</sup>Λ˜ <sup>1</sup> <sup>−</sup> (*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)Λ˜ *<sup>T</sup>* <sup>2</sup> *<sup>W</sup>*˜ <sup>0</sup>Λ˜ <sup>2</sup> <sup>−</sup> *Sym*{Λ˜ *<sup>T</sup>* <sup>1</sup> [*αY*˜ <sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)*Y*˜ <sup>2</sup>]Λ˜ <sup>2</sup>},

$$
\Sigma\_2(d(t)) = (1 - \alpha)\tilde{\Lambda}\_1^T \tilde{Y}\_1 \tilde{W}\_0^{-1} \tilde{Y}\_1^T \tilde{\Lambda}\_1 + \alpha \tilde{\Lambda}\_2^T \tilde{Y}\_2^T \tilde{W}\_0^{-1} \tilde{Y}\_2 \tilde{\Lambda}\_2.
$$

Now, let *L*1*<sup>j</sup>* = *K*1*jX*, *L*2*<sup>j</sup>* = *K*2*jX*, *j* = 1, 2, ... ,*r*, then we can easily derive that (42) is equivalent to <sup>Σ</sup>˜ *ij*(*d*(*t*), ˙*d*(*t*)) + <sup>Σ</sup>˜ <sup>2</sup>(*d*(*t*)) <sup>&</sup>lt; 0, where <sup>Σ</sup>˜ *ij*(*d*(*t*), ˙*d*(*t*)) = <sup>Σ</sup>˜ <sup>1</sup>(*d*(*t*), ˙*d*(*t*)) + *Sym*{(*e*˜ *T* <sup>1</sup> + *γe*˜ *T* <sup>13</sup>)[*AiXe*˜1 + *BiL*1*je*˜1 + *AdiXe*˜2 + *BiL*2*je*˜2 + *GiXe*˜12 − *Xe*˜13]}, *i*, *j* = 1, 2, ... ,*r*. Additionally, it is clear that the condition (38) is the equivalent condition of (10).

According to the Schur complement lemma and Lemma 2 in [35], similar to the proof of Theorem 1, the conditions (30)–(38) can ensure the closed-loop T-S fuzzy time-delay system (5) finite-time bounded with respect to (*c*1, *c*2, *Tf* , *R*, *δ*, *h*), and we can obtain the memory state feedback controller gains *K*1*<sup>j</sup>* = *L*1*jX*<sup>−</sup>1, *K*2*<sup>j</sup>* = *L*2*jX*<sup>−</sup>1, *j* = 1, 2, . . . ,*r*. Thus, this completes the proof of the theorem.

**Remark 3.** *It is well known that the concept of finite-time boundedness reduces to the concept of finite-time stability when ω*(*t*) = 0*. Thus, finite-time stability is a special case of finite-time boundedness. The authors in [28] discuss the problem of finite-time stability and stabilization for a class of T-S fuzzy systems with time-varying delay. However, this paper is concerned with finite-time boundness analysis and the finite-time stabilization problem for T-S fuzzy systems with a time-varying delay and norm-bounded disturbance. Therefore, the developed results in this paper are more general.*

**Remark 4.** *The Finsler lemma is employed to design the memory state feedback controller for a T-S fuzzy time-delay system in the proof of Theorem 2. In order to derive a finite-time stabilization condition in the form of LMIs, the matrix* Φ = [*X*−<sup>1</sup> 00000000000 *γX*−1] *<sup>T</sup> is defined, which may introduce some conservativeness. However, it should be mentioned that this is indeed an effective approach to obtain a finite-time stabilization criterion. To reduce the aforementioned conservativeness, an appropriate parameter γ can be obtained by applying several powerful optimization algorithms.*

**Remark 5.** *Based on the parallel distributed compensation scheme, the memory state feedback controller is designed to ensure finite-time boundness of the corresponding closed-loop T-S fuzzy time-delay system in Theorem 2. For fixed* (*c*1, *c*2, *Tf* , *R*, *δ*, *h*)*, the optimal minimum values of c*<sup>2</sup> *for guaranteeing the closed-loop T-S fuzzy system finite-time bounded can be obtained by solving a series of LMIs, namely,*

$$\min\_{((30)-(38))} c\_2.$$

*The memory state feedback controller gains are given by K*1*<sup>j</sup>* = *L*1*jX*<sup>−</sup>1, *K*2*<sup>j</sup>* = *L*2*jX*<sup>−</sup>1, *j* = 1, 2, . . . ,*r.*

*Symmetry* **2020**, *12*, 447

#### **4. Numerical Example**

In this section, a numerical example is given to illustrate the effectiveness of the proposed results. This example deals with a truck-trailer system with time-varying delay. The dynamic model is described as follows:

$$\begin{cases}
\dot{\mathbf{x}}\_{1}(t) = -a \frac{v\mathbb{I}}{Lt\_{0}} \mathbf{x}\_{1}(t) - (1-a) \frac{v\mathbb{I}}{Lt\_{0}} \mathbf{x}\_{1}(t-d(t)) + \frac{v\mathbb{I}}{Lt\_{0}} u(t) + \omega\_{1}(t) \\
\dot{\mathbf{x}}\_{2}(t) = a \frac{v\mathbb{I}}{Lt\_{0}} \mathbf{x}\_{1}(t) + (1-a) \frac{v\mathbb{I}}{Lt\_{0}} \mathbf{x}\_{1}(t-d(t)) \\
\dot{\mathbf{x}}\_{3}(t) = \frac{v\mathbb{I}}{t\_{0}} \sin[\mathbf{x}\_{2}(t) + a \frac{v\mathbb{I}}{L} \mathbf{x}\_{1}(t) + (1-a) \frac{v\mathbb{I}}{2L} \mathbf{x}\_{1}(t-d(t))]
\end{cases}$$

where *x*1(*t*) is the angle difference between the truck and the trailer, *x*2(*t*) is the angle of the trailer, *x*3(*t*) represents the vertical position of the rear end of the trailer, *u*(*t*) denotes the steering angle, *ω*(*t*) = *ω<sup>T</sup>* <sup>1</sup> (*t*) *<sup>ω</sup><sup>T</sup>* <sup>2</sup> (*t*) *<sup>ω</sup><sup>T</sup>* <sup>3</sup> (*t*) *<sup>T</sup>* is the exogenous disturbance.

Let *<sup>σ</sup>*(*t*) = *<sup>x</sup>*2(*t*) + *<sup>a</sup> <sup>v</sup>*¯*<sup>t</sup> <sup>L</sup> <sup>x</sup>*1(*t*)+(<sup>1</sup> <sup>−</sup> *<sup>a</sup>*) *<sup>v</sup>*¯*<sup>t</sup>* <sup>2</sup>*<sup>L</sup> x*1(*t* − *d*(*t*)), the T-S fuzzy time-delay system that represents the above truck-tailer model is as follows:

Plant Rule 1: If *σ*(*t*) is about 0, then

$$\dot{x}(t) = A\_1 x(t) + A\_{d1} x(t - d(t)) + B\_1 u(t) + B\_{\omega 1} \omega(t);$$

Plant Rule 2: If *σ*(*t*) is about ±*π*, then

$$\dot{\mathbf{x}}(t) = A\_2 \mathbf{x}(t) + A\_{d2} \mathbf{x}(t - d(t)) + B\_2 \boldsymbol{\mu}(t) + B\_{\omega 2} \boldsymbol{\omega}(t),$$

where

$$\begin{aligned} A\_{1} &= \begin{pmatrix} -a\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ a\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ a\frac{\overline{v}\overline{l}^{2}}{2I\overline{l}\_{0}} & \frac{\overline{v}\overline{l}}{I\_{0}} & 0 \end{pmatrix}, \ A\_{d1} = \begin{pmatrix} -b\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ b\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ b\frac{\overline{v}\overline{l}^{2}}{2I\overline{l}\_{0}} & 0 & 0 \end{pmatrix}, \\\\ A\_{2} &= \begin{pmatrix} -a\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ a\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ a\frac{\overline{v}\overline{v}\overline{l}^{2}}{2I\overline{l}\_{0}} & \frac{\overline{v}\overline{v}}{I\overline{l}} & 0 \end{pmatrix}, \ A\_{d2} = \begin{pmatrix} -b\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ b\frac{\overline{v}\overline{l}}{I\overline{l}\_{0}} & 0 & 0\\ b\frac{\overline{v}\overline{v}^{2}\overline{l}}{2I\overline{l}\_{0}} & 0 & 0 \end{pmatrix}, \\\\ B\_{1} &= B\_{2} = \begin{pmatrix} \frac{\overline{v}\overline{l}}{I\overline{l}\_{0}}\\ 0\\ 0 \end{pmatrix}, \ B\_{\omega 1} = B\_{\omega 2} = \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}. \end{aligned}$$

with *a* + *b* = 1.

In order to illustrate the developed results, we borrow the model parameters from [36], such as *<sup>a</sup>* = 0.7, *<sup>v</sup>* = −1.0, *<sup>L</sup>* = 5.5, *<sup>l</sup>* = 2.8, ¯*<sup>t</sup>* = 2.0, *<sup>t</sup>*<sup>0</sup> = 0.5, and *<sup>d</sup>* = <sup>10</sup>*t*0/*π*. The membership functions are defined as *ρ*1(*x*(*t*)) = 1/ (1 + *exp*(*x*1(*t*) + 0.5)), *ρ*2(*x*(*t*)) = 1 − *ρ*1(*x*(*t*)). Additionally, the other parameters involved in the simulation are chosen as *c*<sup>1</sup> = 1, *δ* = 0.3, *β* = 0.01, *γ* = 0.8, *Tf* = 10, *R* = *I*, *μ*<sup>1</sup> = −0.1, *μ*<sup>2</sup> = 0.1, and *h* = 0.6. We aim to design a memory state feedback controller such that the resulting closed-loop T-S fuzzy time-delay system is finite-time bounded. By solving the LMI-based finite-time stabilization criterion proposed in Theorem 2 using the Matlab LMI toolbox, we can derive the feasible solutions for the optimal minimum value of *c*<sup>2</sup> = 2.8830. Furthermore, all the control gain matrices are obtained as follows:

$$K\_{11} = L\_{11}X^{-1} = \begin{pmatrix} 6.9875 & -13.5452 & 1.4041 \end{pmatrix}, \ K\_{12} = L\_{12}X^{-1} = \begin{pmatrix} 6.9871 & -13.7540 & 1.3966 \end{pmatrix}, \ K\_{12} = L\_{12}X^{-1} = \begin{pmatrix} 6.9871 & -13.5452 \end{pmatrix}, \ K\_{12} = L\_{12}X^{-1} = \begin{pmatrix} 6.9871 & -13.5452 \end{pmatrix}, \ K\_{12} = L\_{12}X^{-1} = \begin{pmatrix} 6.9871 & -13.5452 \end{pmatrix}, \ K\_{12} = L\_{12}X^{-1} = \begin{pmatrix} 6.9871 & -13.5452 \end{pmatrix}, \ K\_{12} = L\_{12}X^{-1} = \begin{pmatrix} 6.9871 & -13.5452 \end{pmatrix}$$

$$K\_{21} = L\_{21}X^{-1} = \begin{pmatrix} 0.3679 & 0.0085 & -0.0010 \end{pmatrix}, \quad K\_{22} = L\_{22}X^{-1} = \begin{pmatrix} 0.3861 & -0.0013 & 0.0001 \end{pmatrix}.$$

For the simulation framework, the exogenous disturbance is selected as *ω*(*t*)=(0.06 sin *t* 0 0)*T*, and the time-varying delay is assumed to be *d*(*t*) = 0.25 + 0.25 sin(0.3*t*). For the initial condition *<sup>x</sup>*(0)=(0.8 − 0.5 0.2)*T*, the state response of the corresponding closed-loop T-S fuzzy time-delay system is depicted in Figure 1, and the evolution of *xT*(*t*)*Rx*(*t*) is shown in Figure 2. From the simulation results, it is obvious that the closed-loop T-S fuzzy time-delay system is finite-time bounded with respect to (1, 2.8830, 10, *I*, 0.3, 0.6) via the above memory state feedback controller. In addition, for different *h*, the optimal minimum values of *c*<sup>2</sup> for ensuring the closed-loop T-S fuzzy system finite-time bounded are summarized in Table 1. This proves the effectiveness of our developed results in Theorem 2.

**Figure 1.** The state response of the closed-loop Takagi–Sugeno fuzzy system.

**Figure 2.** The time history of *xT*(*t*)*Rx*(*t*).

**Table 1.** The optimum bound values of *c*<sup>2</sup> for different *h*.


#### **5. Conclusions**

In this paper, the problem of finite-time boundedness and finite-time stabilization for a class of T-S fuzzy time-delay systems was discussed. First, based on a new augmented LKF and by applying an improved reciprocally convex combination technique, a novel delay-dependent finite-time boundedness sufficient condition has been derived for an open-loop T-S fuzzy time-delay system. Secondly, a memory state feedback controller has been developed to ensure the finite-time boundedness of the corresponding closed-loop T-S fuzzy time-delay system. Finally, the effectiveness and advantages of the presented methods were demonstrated by a numerical example. Our future research work will focus on the problem of robust finite-time control for uncertain T-S fuzzy systems with time-varying delay and exogenous disturbance.

**Author Contributions:** Conceptualization, Y.R. and T.H.; methodology, Y.R.; software, Y.R.; validation, Y.R. and T.H.; formal analysis, Y.R.; investigation, Y.R. and T.H.; resources, T.H.; writing—original draft preparation, Y.R.; writing—review and editing, T.H.; visualization, Y.R. and T.H.; supervision, T.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Natural Science Foundation of China under Grant 61372187, Grant 61473239, and Grant 61702317.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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