On the boundedness of solutions of some fuzzy dynamical control systems

1.   Introduction

Fuzzy systems have become more and more essential in control engineering and industry since Zadeh ^[40] originally presented them in 1965. In the last twenty years, extensive research has been conducted on nonlinear systems using Takagi-Sugeno (T-S) fuzzy systems ^{[4,5,6,8,16,18,21,22,23,24,26,27,28,29,31]}. Fuzzy controllers or models utilize fuzzy rules, which consist of linguistic if-then statements integrating fuzzy sets, fuzzy logic, and fuzzy inference. These rules are crucial for establishing a connection between the input and output variables in fuzzy controllers and models. They serve the purpose of encapsulating knowledge and expertise derived from expert control and modeling ^{[13,14,15,17,19,20,31,32]}. The stability and performance criteria of T-S-fuzzy dynamical systems have been the subject of numerous techniques in the literature ^{[29,30,31,32]}. Essentially, the configuration that is most often taken into consideration is known as parallel-distributed compensation (PDC), wherein the plant that needs to be regulated shares membership functions with the controller. The principle of approaches has been to create less conservative LMI conditions, like those found in ^[33,34]. T-S fuzzy models ^[35,36] are nonlinear systems that offer local linear representations of an underlying system by employing an if-then rule set. Numerous nonlinear systems may be accurately simulated by this family of models. As such, it is imperative to study their stability or design stabilizing controllers. Nonlinear system stability analysis has received a lot of attention ^{[1,2,7,9,10,11,12]}. The authors in ^[41] have constructed a unified control rule in order to improve the accuracy of both the disturbance estimates and stabilization of nonlinear T-S fuzzy semi-Markovian jump systems, which are modelled by using an equivalent-input-disturbance technique based on proportional-integral observers.

In the presence of perturbations such as unmodeled dynamics, measurement noises, and disturbances ^[3,12,37], asymptotic stability tends to be conservative. For such systems, ultimate boundedness implies the solution's capacity to stay within a specific vicinity around the origin for a prolonged duration, which is commonly the most achievable property of the solutions. Certain constraints are typically imposed on the perturbation terms to establish this behavior ^[12]. It may be challenging to find a Lyapunov function for nonlinear systems, but it is simple for linear systems. Actually, by linearizing the equations and calculating a local linear Lyapunov function that should be valid in the region of a fixed point, this technique is frequently used to propose candidate Lyapunov functions for nonlinear systems. This paper presents sufficient conditions to ensure the ultimate exponential convergence of solutions for a certain category of uncertain fuzzy control systems. Furthermore, a numerical example is included to illustrate the validity of the primary outcome.

The remainder of this paper is organized as follows. In section two, we present the T-S fuzzy system where we present the conventional T-S fuzzy model and issues about stability. In section three, we study the boundedness and convergence analysis for uncertain fuzzy systems where some feedback controllers are constructed. The section four treats an example as an application of the obtained results.

2.   T-S fuzzy dynamical system

The majority of physical systems have complex and ambiguous mathematical models that are challenging to obtain. For small range motion, these systems dynamics could exhibit linear or nonlinear characteristics. In order to study the asymptotic behaviors of the solutions via local approximation, Lyapunov's linearization method is frequently used. Takagi and Sugeno have suggested utilizing fuzzy inferences to provide an efficient method of aggregating these models. To design a T-S fuzzy controller, one needs a T-S fuzzy model for a nonlinear system. As a result, creating a fuzzy model is a crucial and fundamental step in this process. Let us examine a particular kind of continuous-time T-S fuzzy control system that is characterized by the following fuzzy rules,

where $x\left(t\right)$ is the state of the system, $u\left(t\right)\in {\mathbb{R}}^{m}$ represents the inputs and r is the number of fuzzy rules.

${A}_{i}\in {\mathbb{R}}^{n\times n}$ and ${B}_{i}\in {\mathbb{R}}^{n\times n}$ represent the input matrices, where $i = \mathrm{1, 2}, ..., r$ serves as the index of the fuzzy rules. ${F}_{ij}$ represents the inputs fuzzy sets, while $z\left(t\right) = \left[{z}_{1}\right(t), ..., {z}_{p}(t){]}^{T}$ constitutes measurable variables, termed as premise variables. Utilizing weighted average defuzzifiers, an aggregated fuzzy model can be derived:

The membership functions, denoted as ${\mu }_{i}\left(z\right)$, belong to the class C¹, implying that they are continuously differentiable and defined as:

where ${\mu }_{i} = {\mu }_{i}\left(z\right) > 0$ for all $i = \mathrm{1, 2}, ..., r$ and satisfy $\sum _{i = 1}^{r}{\mu }_{i}\left(z\right) = 1$. Then the fuzzy system has the state-space form:

The PDC concept is used in many published results related to fuzzy system control. The design of the fuzzy controller incorporates a linear state feedback control in its consequent part, sharing the same antecedent as the fuzzy system. The controller for each local dynamic is defined as:

The local state feedback gain is denoted as ${K}_{i}$. Consequently, the defuzzified outcome is:

The system (2.2) in closed-loop with the fuzzy controller (2.4) yields the following fuzzy system,

For the asymptotic behaviors of the solutions, where some adequate requirements for the stability are inferred, we shall here employ the Lyapunov technique. Assume that the following criteria are met in order for there to be a common positive definite matrix P:

and

Or, without loss of generality,

and

The matrices ${Q}_{i}, i = \mathrm{1, 2}, ..., r,$ and ${Q}_{ij}$, i < j ≤ r, are symmetric positive definite. In such case, the fuzzy system (2.5) is asymptotically stable when these requirements are met. Thus, this may be converted into a convex problem ^[29,30,31], which can then be resolved using the optimization of linear matrix inequalities. Feedback controllers can be built if the solution is viable, which means that the stabilization restrictions are satisfied. The fuzzy system stability precondition theorems are consistent with Lyapunov's definition of stability. Numerous studies have demonstrated (see, for example, ^[30,31,32]) that identifying a common positive definite matrix P ensures the stability of the T-S system as a whole.

Choosing $V\left(x\right) = {x}^{T}Px$ as the Lyapunov function of the T-S system, the following stability conditions for ensuring stability is derived by using the Lyapunov approach.

3.   Convergence analysis of uncertain fuzzy systems

Inspired by the findings from the previous section regarding the fuzzy-model concept of control, our objective is to expand the T-S fuzzy system to include external disturbances. Let's look at the perturbed model below:

${\mathrm{\Lambda }}_{i}:{\mathbb{R}}^{+}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ are smooth functions, ${\mathrm{\Gamma }}_{i}:{\mathbb{R}}^{+}\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{n}$ represents the perturbations terms for $i = 1, ..., r$.

In order to improve the stability analysis condition by placing limitations on the term of uncertain- ties, the first section of this work focuses on utilizing the property of the system's shape, which consists of using the controller connected to the linear part. The first portion of this work focuses on employing the form property of the system, i.e., using the controller associated with the linear part, to improve the stability analysis condition by limiting the term of uncertainty, $u\left(t\right) = \sum _{i = 1}^{r}{\mu }_{i}\left(z\right){K}_{i}x\left(t\right)$.

The T-S fuzzy model is given by:

where $x\left(t\right)\in {\mathbb{R}}^{n}$ is the state, $u\left(t\right)\in {\mathbb{R}}^{m}$ is the input, ${A}_{i}(n, n)$ constant matrix, ${B}_{i}(n, n)$ matrix control input and the functions ${\mathrm{\Lambda }}_{i}$ represent the uncertainties of each fuzzy subsystem and are time-varying are of appropriate dimension. The perturbed terms are the functions ${\mathrm{\Gamma }}_{i}$ with $\varepsilon$ > 0, for $i = 1, ..., r.$ F_ik is the fuzzy set (k = 1, 2, ..., p), z(t) = T (z₁ (t), ..., z_p (t)) is the premise variable vector associated with the system states and inputs and r is the number of fuzzy rules. This yields the output of fuzzy system:

where ${w}_{i}\left(z\right) = \prod _{j = 1}^{p}{F}_{ij}\left({z}_{i}\right)$ and ${F}_{ij}\left({z}_{i}\right)$ denotes the grade of the membership function ${F}_{ij}$, corresponding to ${z}_{i}\left(t\right)$.

Let ${\mu }_{i}\left(z\right)$ defines as:

Then the fuzzy system has the state-space form:

Clearly, $\sum _{i = 1}^{r}{\mu }_{i}\left(z\right) = 1$ and ${\mu }_{i}\left(z\right)\ge 0$ for $i = 1, ..., r$.

Regarding the T-S fuzzy system (3.2), we assume that the pairs ${(A}_{i}, {B}_{i}), i = 1, ..., r$ are controllable. Based on this assumption, a stabilizing controller gain ${K}_{i}$ can be obtained by using the pole placement design or Ackerman's formula, such that each local dynamic is stably controlled. The global control input matrix, represented by B, is represented as follows:

This indicates that the control performance is dominated by the global control input matrix. The fuzzy controller's design can be seen as a linear state feedback control for the system (3.2), which is defined as follows:

where ${K}_{i}$ is the local state feedback gain, where the defuzzified result is $u\left(t\right) = \sum _{i = 1}^{r}{\mu }_{i}\left(z\right){K}_{i}x\left(t\right).$

For dynamic systems, we must first remember what uniformly eventually (ultimate) boundedness and exponential convergence in the sense of ^[1,12,38]. Let consider the following system described by:

with ${F}^{\varepsilon }$ are smooth functions, $t\in$ ${\mathbb{R}}_{+}$ is the time and $x\in {\mathbb{R}}^{n}$ is the state, $\varepsilon > 0$.

The system (2.3) is said uniformly ultimately bounded if their exists $R\left(\varepsilon \right) > 0$, such that for all ${R}_{1} > 0$, there exists a $T = Tx\left({R}_{1}, \varepsilon \right) > 0$ such that:

The system (2.3) is said to be globally uniformly ultimately bounded if the above property holds for arbitrarily large.

We consider the case where ${t}_{0} = 0$, the solution denotes $x\left(t\right)$ of (2.3) with respect the initial condition $(0, x\left(t\right))$. System (2.3) is said to be globally exponentially ultimately bounded if:

with $\text{θ}\left(\varepsilon \right)> 0, \upsilon \left(\varepsilon \right)> 0.$

Noting that, one can use Lyapunov analysis to prove the ultimate boundedness in the case of quadratic Lyapunov function of the form $V\left(x\right) = {x}^{T}Px$, if the derivative of $V$ along the trajectories satisfies $\dot{V}\le -a\Vert x{\Vert }^{2}+b\Vert x\Vert$, with $\mathrm{a}> 0$ and $\mathrm{a}> 0$ which implies that $\dot{V}\le -a\Vert x{\Vert }^{2}+a\theta \Vert x{\Vert }^{2}-a\theta \Vert x{\Vert }^{2}+b\Vert x\Vert$, for $0 < \theta < 1$.

Then one gets the following estimation: $\dot{V}\le -a(1-\theta )\Vert x{\Vert }^{2}-(a\theta \Vert x\Vert -b)\Vert x\Vert$, thus $\dot{V}\le -a(1-\theta )\Vert x{\Vert }^{2} < 0for\Vert x\Vert > \frac{b}{a\theta }$, which gives the ultimate boundedness of solutions. Moreover, one can reach an estimation as (3.10) via a differential inequality of the form: $\dot{Z}\le -\alpha Z+\beta \sqrt{Z},$ where $\alpha > 0, \beta > 0$ and $Z\ge 0.$ In this case, one can obtain the following estimation:

Therefore, for a quadratic Lyapunov function of the form $V\left(x\right) = {x}^{T}Px$, one can take $Z\left(t\right) = V\left(x\right(t\left)\right) = {x}^{T}\left(t\right)Px\left(t\right)$ to get:

Thus, using the fact that ${\lambda }_{min}\left(P\right)\Vert x{\Vert }^{2}\le V(t, x) = {x}^{T}Px\le {\lambda }_{max}\left(P\right)\Vert x{\Vert }^{2},$ one obtains $\sqrt{{\lambda }_{min}\left(P\right)}\Vert x\left(t\right)\Vert \le \sqrt{V\left(x\right(t\left)\right)}\le (\sqrt{{\lambda }_{max}\left(P\right)}\Vert x(0)\Vert -\frac{\beta }{\alpha }){e}^{-\frac{\alpha }{2}t}+\frac{\beta }{\alpha }.$

Note that, we must have $\sqrt{{\lambda }_{max}\left(P\right)}\Vert x\left(0\right)\Vert -\frac{\beta }{\alpha } > 0$, this is because the initial conditions are taken outside the set $\{x\in {\mathbb{R}}^{n}/\Vert x\Vert \le \frac{\beta }{\sqrt{{\lambda }_{max}\left(P\right)}\alpha }\}.$ Thus, an estimation as in (3.10) can be obtained:

Remark: It is well known that most plants in industry show significant nonlinearities, which usually make the analysis and controller design difficult. However, exact mathematical models of most physical systems are difficult to obtain because of the existence of complexities and uncertainties. In order to overcome such difficulties, various schemes have been developed in the past two decades, among which a successful approach is the fuzzy control. The dynamics of these systems may include linear or nonlinear behaviors for small range motion. Lyapunov's linearization method is often implemented to deal with the local dynamics of nonlinear systems and to formulate local linearized approximation. So, the complex system can be divided into a set of local mathematical models. Takagi and Sugeno have proposed an effective means of aggregating these models by using the fuzzy inferences to construct the system. Therefore, an interesting line of research which gives promising prospects in several applications is the study of the stability of solutions. The asymptotic stability is conservative or difficult to obtain in many applications. So, the asymptotic behavior of the solutions can be studied throughout the boundedness property. The advantage of the exponential ultimate boundedness of solutions is that the solutions convergence toward a small neighborhood of the origin, which can be supposed not necessarily equilibrium point of the system.

Next, we will consider the following fuzzy systems based fuzzy controller:

Since the inception of fuzzy controllers, considerable interest has been shown in this field. The highly positive experimental results show that the fuzzy controller performed consistently better than the traditional controllers and was less susceptible to noise and parameters' changes. Our objective is to find some conditions on the bound of the term of perturbation such that the fuzzy system (3.2) is globally exponentially ultimately bounded in closed-loop with a fuzzy controller. In that case the solutions converge toward a neighborhood of the origin which is characterized by a small ball centered at the origin. This stability criterion enable the selection of the common positive-definite matrix P where the quadratic function ${x}^{T}Px$ can be used as a Lyapunov function candidate for the fuzzy system.

3.1.   Control analysis of fuzzy systems under perturbations

Let consider the T-S fuzzy model (3.2):

The function ${f}_{l}$ represent the terms of uncertainties of each fuzzy subsystem that satisfy the following assumption:

$\left({\mathcal{H}}_{1}\right)$ for all $i = 1, \dots , r,$

where ${\varrho }_{i}$ are some nonnegative continuous functions, such that ${\varrho }_{i}\left(0\right) = 0, i = 1, \dots , r.$ The representation of the nonlinearities associated with the following bound which is a positive continuous function, named $\varrho :{\mathbb{R}}^{n}\to {\mathbb{R}}_{+},$ such that $\varrho \left(0\right) = 0,$ which has the form:

$\left({\mathcal{H}}_{2}\right)$ assume that,

for all $t\ge 0$ and $x\in {\mathbb{R}}^{n}$ where ${\upsilon }_{i}^{\varepsilon }$ are known nonnegative continuous functions for $i = \mathrm{1, 2}, ..., r,$ with

where

$\widetilde{\upsilon }\left(\varepsilon \right)$ is a nonnegative constant depending on the parameter $\varepsilon$.

We will use the following fuzzy controller:

The closed-loop system is given by

where

The stability of the local fuzzy dynamics is taken into account first in the controller synthesis. That is, each subsystem's steady feedback gains are identified. Assume that certain matrices ${K}_{i}$, $i = \mathrm{1, 2}, ..., r,$ and a symmetric positive definite matrix P that met the following stability criteria:

where ${Q}_{i}$ is a positive define matrix.

Considering these assumptions, each subsystem is locally controllable and therefore a stabilizing feedback gain is obtained. Let consider the function $V\left(x\right) = {x}^{T}Px$ as a Lyapunov candidate. The derivative of $V\left(x\right)$ with respect to time is given by,

Regarding each matrix $({G}_{ii}^{T}P+P{G}_{ii})$, one has

where ${\lambda }_{min}(.)(resp.{\lambda }_{max}(.\left)\right)$ denotes the smallest (resp. the largest) eigenvalue of the matrix.

Define

For $1\le i < j\le r$. A relaxed condition concerning the coupling effect is expressed as:

where $k = \frac{r(r-1)}{2}\alpha$. Indeed, one has

It follows that,

Hence,

Then one can state the following theorem.

Theorem 3.1. If the conditions $\left({\mathcal{H}}_{1}\right)$ and $\left({\mathcal{H}}_{2}\right)$ are verified for the fuzzy system (3.2) and there exist a common define positive matrix P and feedback gain matrices ${\mathrm{K}}_{\mathrm{i}}$, $\mathrm{i} = \mathrm{1, 2}, ..., \mathrm{r},$ such that the conditions givin in (3.26) hold, then the closed-loop system is globally exponentially ultimately bounded with the control law (3.21) where $\mathrm{\varrho }$ verifies

with $\theta \in \left[\mathrm{0, 1}\right]$ and ${\lambda }_{0} = :\underset{i = 1, ..., r}{inf}{\lambda }_{min}\left({Q}_{i}\right)\sum _{i = 1}^{r}{\mu }_{i}^{2}-2k > 0.$

Proof. Using the Lyapunov function $V\left(x\right) = {x}^{T}Px$. The derivative of $V\left(x\right)$ along the trajectories of (3.2) in closed-loop with (2.6) with respect to time is given by,

Since, $\sum _{i = 1}^{r}\sum _{j = 1}^{r}{\mu }_{i}{\mu }_{j} = 1,$ then, $\left({\mathcal{H}}_{1}\right)$ and $\left({\mathcal{H}}_{2}\right)$, from we have:

and

Taking into account the last expressions, it follows that

Thus,

Using Cauchy-Schwartz inequality

Since

then by taking $\Vert P\Vert = {\lambda }_{max}\left(P\right)$, yields:

Let

and

Given the earlier notations, it can be inferred that:

The variable in the last expression is changed as follows: $\mathrm{{\rm Y}}\left(t\right) = V(t, x{)}^{\frac{1}{2}}$. The derivative with respect to time is given by:

This implies that,

Thus,

It follows that,

So,

One gets,

Hence,

Then,

We obtain an estimation on the trajectories as the one given in (3.3), with

and

Hence, the system is globally exponentially ultimately bounded. It follows that ${\mathcal{B}}_{r\left(\varepsilon \right)}$, with $r\left(\varepsilon \right) = 2\frac{{\lambda }_{max}^{2}\left(P\right)}{{\lambda }_{min}^{\frac{3}{2}}\left(P\right)}\frac{\widetilde{\upsilon }\left(\varepsilon \right)}{{\zeta }^{\frac{1}{2}}},$ is globally uniformly exponentially stable.

Remark. Note that the last inequality implies that system (3.2) in closed loop is uniformly ultimately bounded in the sense that the trajectories satisfy the inequality (3.10). Moreover, in the case when $r = r\left(\varepsilon \right)$ goes to zero as $\varepsilon$ tends to zero, then the error approaches the origin exponentially as t tends to infinity.

The design process for T-S fuzzy systems can be summed up as follows using the last method.

Step 1: Verify the controllability of $({A}_{i}, {B}_{i})$, for $i = \mathrm{1, 2}, ..., r.$

Step 2: The conditions $\left({\mathcal{H}}_{1}\right)$ and $\left({\mathcal{H}}_{2}\right)$ are satisfied.

Step 3: Solve the equations (3.26) to obtain P, ${K}_{i}$, ${Q}_{i}$ for $i = \mathrm{1, 2}, \dots , r.$

Note that, for simplicity one can choose ${Q}_{i} = I.$

Step 4: By using Matlab (control-toolbox), consider the nonlinear program based on equations:

To ascertain the value of k, the nonlinear programming is formulated in the following manner: Determine ${\lambda }_{max}\left({G}_{ii}^{T}P+P{G}_{ii}\right)$ and then for $1\le i < j\le r$.

Step 5: Use the fuzzy controller (3.21) for (3.2).

Step 6: Verify the condition (3.34) imposed on $\varrho \left(x\right)$ that holds for a suitable choice of $\varepsilon \in ]0, {\varepsilon }^{*}[, {\varepsilon }^{*} > 0$ small enough.

4.   Simulation examples

4.1.   Example 1

Let's consider a dynamical system representing a translational oscillator coupled with an eccentric rotational proof mass actuator, as described in ^[3,25]. Here, ${x}_{1}$ and ${x}_{2}$ denote the translational position and velocity of the cart, with ${x}_{1}$ being the derivative of ${x}_{2}$. Additionally, ${x}_{3}$ and ${x}_{4}$ signify the angular position and velocity for the rotational proof mass model, where ${x}_{3}$ corresponds to the derivative of ${x}_{4}$. The system's dynamics can be described by the nonlinear equation:

In this equation:

● u represents the torque applied to the eccentric mass,

● d represents disturbances chosen to encompass the system's perturbation terms (3.2),

● $A\left(x\right) = {\left[\begin{array}{cccc}{x}_{2}& \frac{-{x}_{1}+\varepsilon {x}_{4}^{2}\mathit{sin}{x}_{3}}{1-{\varepsilon }^{2}{\mathit{cos}}^{2}{x}_{3}}& {x}_{4}& \frac{\varepsilon \mathit{cos}{x}_{3}\left({x}_{1}-\varepsilon {x}_{4}^{2}\mathit{sin}{x}_{3}\right)}{1-{\varepsilon }^{2}{\mathit{cos}}^{2}{x}_{3}}\end{array}\right]}^{T},$ and

● $B\left(x\right) = {\left[\begin{array}{cccc}0& \frac{-\varepsilon cos{x}_{3}}{1-{\varepsilon }^{2}{cos}^{2}{x}_{3}}& 0& \frac{1}{1-{\varepsilon }^{2}{cos}^{2}{x}_{3}}\end{array}\right]}^{T}$, $x\left(t\right) = {\left[\begin{array}{cccc}{x}_{1}\left(t\right)& {x}_{2}\left(t\right)& {x}_{3}\left(t\right)& {x}_{4}\left(t\right)\end{array}\right]}^{T}\varepsilon \in ]0, \frac{1}{2}[.$

The corresponding T-S fuzzy model associated to this nonlinear system is given by:

where,

We use the following fuzzy controller: $\text{u}\left(t\right) = \sum _{j = 1}^{4}{\mu }_{j}\left(z\right){K}_{j}x\left(t\right)$ with $\varepsilon = 0.01$ to stabilize the system, in the sense that all trajectories of the system are ultimately bounded (see Figure 1).

The Figure 1 shows the time evolution of the states (${x}_{1}\left(t\right);{x}_{2}\left(t\right);{x}_{3}\left(t\right);{x}_{4}\left(t\right)$) of the system with the initial states $\left({x}_{1}\left(0\right);{x}_{2}\left(0\right);{x}_{3}\left(0\right);{x}_{4}\left(0\right)\right) = (0.5;0.5;0.5;0.5)$. One can see that the trajectories are bounded and converge to a small neighborhood of the origin.

We need to select a positive definite matrix Q, such as Q = I, to address the algebraic Lyapunov equation (3.26) and determine the matrix P. For $\varrho \left(x\right) < 1$ we can take $, \varrho \left(x\right) = 0.1$ where ${\varepsilon }^{*} = \frac{1}{2}$ and by using MATLAB, we get the following solutions, for ${K}_{1}, {K}_{2}, {K}_{3}, {K}_{4}$, and the matrix P:

It should be noted that when $t\to +\mathrm{\infty },$ the trajectory tends to the origin exponentially if the bounds of the nonlinearities go to zero when t approaches to infinity.

Consequently, $\underset{\varepsilon \to 0}{lim}r\left(\varepsilon \right) = 0$ indicates that the state can converge to the origin exponentially as t goes to infinity if the perturbation term's bound can be made as small as desired.

4.2.   Example 2

Consider the following nonlinear fuzzy planar system.

where $x\left(t\right) = \left[{x}_{1}\left(t\right){x}_{2}\right(t){]}^{T}\in {\mathbb{R}}^{2}$, the state vector and $u\left(t\right)$ is the input vector.

One can represent exactly the system by the following two-rule fuzzy model:

Rule 1: $\mathrm{I}\mathrm{f}{x}_{1}\;\mathrm{i}\mathrm{s}\;{\mathrm{M}}_{11}\;\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\;\dot{x}\left(t\right) = {A}_{1}x\left(t\right)+{B}_{1}u\left(t\right).$

Rule 2: $\mathrm{I}\mathrm{f}{x}_{1}\;\mathrm{i}\mathrm{s}\;{M}_{21}\;\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\;\dot{x}\left(t\right) = {A}_{2}x\left(t\right)+{B}_{2}u\left(t\right),$

where: ${A}_{1} = \left[\begin{array}{cc}-2& 0\\ -1& 0\end{array}\right], {B}_{1} = \left[\begin{array}{c}-1\\ 1\end{array}\right], {A}_{2} = \left[\begin{array}{cc}-2& 0\\ 1& 0\end{array}\right], {B}_{2} = \left[\begin{array}{c}1\\ 1\end{array}\right].$

We define the membership functions as:

Using an LMI optimization algorithm, yields:

The following feedback gains: ${K}_{1} = \left[\begin{array}{cc}-0.0452& 0.7962\end{array}\right]\text{}\;\text{and}\;{K}_{2} = \left[\begin{array}{cc}0.0452& 0.7962\end{array}\right],$ and the matrices:

Then, we have ${\lambda }_{min}\left(P\right) = 0.0183, {\lambda }_{max}\left(P\right) = \Vert P\Vert = 0.0377,$ and ${\lambda }_{0} = 0.0001$ within $f\left\{\right({\lambda }_{min}\left({Q}_{i}\right);i = \mathrm{1, 2}), ({\lambda }_{min}\left({Q}_{12}\right)\left)\right\} = 0.0139.$ Now, we introduce parametric perturbations and external disturbances and we approximate the system by the following fuzzy models:

Rule 1: $\mathrm{I}\mathrm{f}{x}_{1}\;\mathrm{i}\mathrm{s}\;{F}_{11}\;\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\;$

Rule 2: $\mathrm{I}\mathrm{f}{x}_{1}\;\mathrm{i}\mathrm{s}\;{F}_{21}\;\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\;$

where, ${\mathrm{\Lambda }}_{1}(t, x) = {\mathrm{\Lambda }}_{2}(t, x) = \frac{1}{5}\frac{\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}}{1+{x}_{1}^{2}},$ and ${\mathrm{\Gamma }}_{1}(t, x, \varepsilon ) = {\mathrm{\Gamma }}_{2}(t, x, \varepsilon ) = \frac{\varepsilon }{5}{e}^{-t}, t\ge 0.$ One can take,

with $\varepsilon = 0.1, \upsilon \left(\varepsilon \right) < \frac{0.1}{5}0.95 = 0.019.$ All the assumptions of Theorem 3.1 are satisfied, by using $\rho$ as in (4.5), it follows that the trajectories of the system are bounded and converge toward a small neighborhood of the origin. Hence, the solutions of system (4.3) with a fuzzy controller of the form (2.6), under some restrictions on the perturbations as they are given in the system to be studied, are globally exponentially ultimately bounded.

5.   Conclusions

In this paper, the feedback controller problem is treated for perturbed Takagi-Sugeno fuzzy models. In order to guarantee the exponential ultimate boundedness of solutions for fuzzy control systems with uncertainties related to a tiny parameter, certain new adequate criteria are provided. The applicability of the main result is demonstrated with an example and simulation results.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research has been funded by Scientific Research Deanship at University of Ha'il-Saudi Arabia through project number (RG-23 070).

Conflict of interest

The authors declare no conflict of interest.