On the stability of fuzzy linear dynamical systems

doi:10.1016/j.jfranklin.2020.02.023

Journal of the Franklin Institute

Volume 357, Issue 9, June 2020, Pages 5502-5522

https://doi.org/10.1016/j.jfranklin.2020.02.023 Get rights and content

Highlights

•
Investigating the stability of fully fuzzy linear dynamical systems governed by fuzzy differential equations.
•
Presenting granular improper fuzzy integral and granular fuzzy Laplace transform.
•
Introducing some new notions as like as fuzzy zeros, fuzzy poles, fuzzy transfer function, and granular fuzzy Routh–Hurwitz matrix.
•
Defining the concepts of fuzzy equilibrium points, fuzzy marginally stability and fuzzy asymptotically stability.

Abstract

This paper deals with the investigation of the stability of fuzzy linear dynamical systems using the new notion called granular fuzzy Laplace transform. In order to analyzing the stability, some new notions have been introduced such as granular improper fuzzy integral, granular fuzzy Laplace transform, equilibrium points, granular fuzzy transfer function, and etc.. Based on the concept of granular metric, the fuzzy marginal and asymptotic stability of fuzzy dynamical systems are defined. Moreover, using the granular fuzzy Laplace transform, the concept of fuzzy poles and fuzzy zeros are presented. The findings shed light on the advantages and efficiency of the granular fuzzy Laplace transform in comparison with the previous definition of fuzzy Laplace transform. Furthermore, using a theorem proved in this paper we show that the stability of fuzzy linear dynamical systems can also been investigated by a matrix called the granular fuzzy Routh–Hurwitz matrix.

Introduction

The analysis of the behavior of most phenomena are often based on mathematical models in the form of differential equations. Due to uncertainties in many cases of phenomena, considering uncertainties in differential equations may be inevitable. Uncertainties can be categorized into two types: the uncertainty associated with words handled by fuzzy sets (possibility sets); and the uncertainty associated with unpredictability handled by probability theory.

Fuzzy differential equations are differential equations in which uncertainties are modeled by fuzzy sets (possibilities sets). In recent years, analyzing the behavior of dynamical systems whose mathematical models have been considered as fuzzy differential equations has captured more attentions. So far, extensive research has already been conducted on fuzzy differential equations. Obtaining solutions of fuzzy differential equations have been investigated under the concepts of H-derivative and SGH-derivative [1], gH-derivative and g-derivative [2], H₂-differentiability [3], and gr-derivative [4]. Using SGH-derivative, gH-derivative, and gr-derivative, designing an optimal controller for dynamical systems modeled by fuzzy differential equations proposed in [5], [6], [7], [8], [9]. In [10], the authors have studied on fuzzy bang-bang control systems under gr-differentiability. Existence and uniqueness of the solution of fuzzy differential equations were studied in [11], [12]. Moreover, in [13], [14], [15], [16], [17], fuzzy differential equations have been examined in the context of fractional calculus. One of the interesting subject in control system is to control the system in an optimal manner, i.e. to design an optimal controller, e.g. see [18], [19], [20], [20], [21], [22], [23], [24].

Although, the enormous volume of the literature has been carried out on fuzzy dynamical systems, little research has been performed in analyzing inherent features of dynamical systems modeled by fuzzy differential equations. Stability, controllability, observability and etc. can be regarded as some of the inherent features of a dynamical system, among which the stability analysis is most important.

In [25], by differential inclusions, the author studied Lyapunov stability of fuzzy differential equations and the periodicity of the fuzzy solution set. The asymptotic equilibrium for fuzzy evolution equations and the stability properties of the trivial fuzzy solution of the perturbed semilinear fuzzy evolution equations were investigated under the concept of Hukuhara derivative in [26]. In [27], using a generalization of Kharitonov theorem in the context of fuzzy sets, stability of fuzzy linear dynamical systems has been studied. With a Lyapunov function presented, stability analysis of fuzzy differential equations under the concept of strongly generalized Hukuhara derivative was investigated in [28]. Practical stabilities of fuzzy differential equations with the second type of Hukuhara derivative has been considered in [29].

The approach proposed in [28] can be utilized for the stability analysis as long as the underlying fuzzy functions are strongly generalized Hukuhara differentiable. Therefore, the restriction associated with the mentioned approach concerns with the existence of strongly generalized Hukuhara derivative. In addition, since the approach proposed in [28] is based on the so-called Fuzzy Standard Interval Arithmetic (FSIA), then it suffers from a shortcoming called UBM phenomenon - see [4] for more details. Although the approach presented in [27] is a useful way for investigating the stability of fuzzy linear dynamical systems, no definition of fuzzy derivative concept has been considered in the fuzzy linear dynamical systems.

In order to address the shortcomings come with the FSIA-based approaches, a new concept of fuzzy derivative called granular derivative based on Relative-Distance-Measure Fuzzy Interval Arithmetic (RDM-FIA) was proposed in [4]. RDM-FIA was established using a new version of the ordinary membership functions called Horizontal Membership Functions (HMFs) presented in [30]. On the basis of the results obtained in [31], [32], [33], [34], [35], it has been shown that RDM-FIA is a powerful tool than the FSIA in applications.

In this paper, fuzzy dynamical systems are considered under the concept of granular differentiability. With the concept of granular differentiability considered, this paper aims at analysis of stability of fuzzy linear dynamical system by some new proposed notions. The new notions introduced in this paper are granular fuzzy Laplace transform, fuzzy zeros, fuzzy poles, fuzzy transfer function, and granular fuzzy Routh–Hurwitz matrix. Using these new notions, the concepts of fuzzy marginally and asymptotically stable systems are presented. Furthermore, some theorems concerning with the concepts of fuzzy stability are given. The findings shed light on the advantages and efficiency of the granular fuzzy Laplace transform in comparison with the previous definition of fuzzy Laplace transform. It has been shown that the negative real poles cannot be determined using the previous definition of fuzzy Laplace transform.

Briefly, in this paper the stability of dynamical system based on uncertain information is studied. The point should be highlighted is that as far as we know this is the first time in the literatures that a complete stability analysis of fuzzy linear dynamical systems, whose model has been presented by fuzzy differential equations, is investigated. In these systems, all the coefficients and initial conditions can be uncertain. Therefore, based on the aforementioned explanations, a list of novelty and contributions of the paper can be made as follows:

1.
Investigating the stability of fuzzy linear dynamical systems, whose model has been presented by fuzzy differential equations, for the first time in the literature.
2.
Considering the fuzzy dynamical system structure as a fully fuzzy dynamical system.
3.
Defining the granular improper fuzzy integral and granular fuzzy Laplace transform.
4.
Presenting the granular fuzzy transfer function.
5.
Defining the fuzzy equilibrium points and fuzzy marginal and asymptotic stability of fuzzy dynamical systems.
6.
Presenting the concepts of fuzzy poles and fuzzy zeros and granular fuzzy Routh–Hurwitz matrix.
7.
Presenting the advantages and efficiency of the granular fuzzy Laplace transform in comparison with the previous definition of fuzzy Laplace transform.
8.
Excluding the following drawbacks:
- (a)
  Technical difficulties in the process of studying the stability of fuzzy linear dynamical systems.
- (b)
  Multiplicity of the fuzzy transfer function for the fuzzy linear dynamical system.
9.
Finally, the applications of the fuzzy stability of fuzzy dynamical system in the mechanical engineering and aircraft power system are illustrated.

Section snippets

Basic concepts

Throughout this paper, the set of fuzzy numbers defined on the real numbers $R$ is denoted by E¹ and $E^{n} = {\underset{︸}{E^{1} \times E^{1} \times . . . \times E^{1}}}_{n}$ . The μ-level sets of $\tilde{A} \in E^{1}$ is ${\tilde{A}}^{μ} = {[\tilde{A}]}^{μ} ≜ [{\underset{̲}{A}}^{μ}, {\bar{A}}^{μ}]$ .

Definition 1

[1]

Let $\tilde{u}, \tilde{v} \in E^{1}$ . If there exists a $\tilde{w} \in E^{1}$ such that $\tilde{u} = \tilde{v} + \tilde{w}$ , then $\tilde{w}$ is called H-difference of $\tilde{u}$ and $\tilde{v}$ ; and is denoted by $\tilde{u} \frac{H}{} \tilde{v}$ .

Definition 2

[1]

Let $\tilde{f} : [a, b] \subseteq R \to E^{1}$ and t ∈ [a, b]. If there is a fuzzy number ${\tilde{f}}_{S G H}^{^{'}} (t)$ such that

(1)
there exist the H-differences $\tilde{f} (t + h) \frac{H}{} \tilde{f} (t),$ $\tilde{f} (t) \frac{H}{} \tilde{f} (t - h),$ and limits $\lim_{h \to 0} \frac{\tilde{f} (t + h) \frac{H}{} \tilde{f} (t)}{h} = \lim_{h \to 0} \frac{\tilde{f} (t) \frac{H}{} \tilde{f} (t - h)}{h} = {\tilde{f}}_{S G H}^{^{'}} (t)$

Some new definitions and notes

In this section, we define granular fuzzy polynomial and fuzzy root. Moreover, the horizontal membership function of a fuzzy complex number and a note related to fuzzy roots are also given.

Definition 12

[39]

A fuzzy complex number $\tilde{z}$ is defined as $\tilde{a} + i \tilde{b},$ where $\tilde{a}, \tilde{b} \in E^{1}$ . $\tilde{a}$ and $\tilde{b}$ are called real and imaginary parts of $\tilde{z}$ and shown by $R e (\tilde{z})$ and $I m (\tilde{z}),$ respectively.

Fig. 1 illustrates the fuzzy complex number $\tilde{z} = \tilde{a} + i \tilde{b},$ where $\tilde{a} = (1, 2, 3)$ and $\tilde{b} = (3, 4, 5)$ .

Note 5

The horizontal membership function of the fuzzy complex number

Granular fuzzy Laplace transform

In this section, we define granular fuzzy Laplace transform. Moreover, the linear property of the granular fuzzy Laplace transform is presented. Additionally, the granular fuzzy Laplace transforms of the gr-derivative is given.

Definition 15

Let $\tilde{f} : [a, \infty) \to E^{1}$ . For any fixed μ, α_f ∈ [0, 1], assume $H (\tilde{f} (t)) = f^{g r} (t, μ, α_{f})$ is integrable on [a, b] for every b ≥ a. Moreover, suppose there is a positive constant M(μ, α_M) such that $\int_{a}^{b} | f^{g r} (t, μ, α_{f}) | d t \leq M (μ, α_{M})$ . Then, $\tilde{f} (t)$ is called granular improper fuzzy integrable on [a

Stability of fuzzy linear dynamical systems

This section presents a definition of fuzzy equilibrium points for fuzzy dynamical systems. Moreover, the concepts of fuzzy marginally and fuzzy asymptotically stability of fuzzy equilibrium points are given. The concepts are an extension of those introduced in the sense of Lyapunov stability for crisp dynamical systems.

Consider the following fuzzy dynamical system: ${\begin{matrix} ^{g r} D \tilde{x} (t) = \tilde{f} (\tilde{x} (t), \tilde{u} (t)) \\ \tilde{x} (t_{0}) = {\tilde{x}}_{0}, \end{matrix}$ where $\tilde{x} : [t_{0}, t_{f}] \to E^{n}$ is the gr-differentiable states vector, $\tilde{u} : [t_{0}, t_{f}] \to E^{m}$ is the control inputs

Advantages of granular fuzzy Laplace transform

In this section, the advantages of granular fuzzy Laplace transform in comparison with the fuzzy Laplace transform defined in Definition 4 are highlighted. The highlights are expressed from the view point of applying the Laplace transforms to fuzzy dynamical systems and the investigation of its stability.

1. Existence of fuzzy Laplace transform

Based on Theorem 1, the fuzzy Laplace transform defined in Definition 4 is restricted to the cases in which the fuzzy functions are SGH-differentiable.

Examples

In this section we give some examples to show the efficiency of the proposed approach in analyzing the stability of fuzzy linear dynamical systems. Moreover, using a comparative example the superiority of the proposed approach in comparison with the previous one is shown.

Example 1

Consider the following fuzzy dynamical system: ${\begin{matrix} \dot{\tilde{x}} (t) = - \tilde{x} (t) + \tilde{u} (t) \\ \tilde{y} (t) = \tilde{x} (t) \\ \tilde{x} (0) = {\tilde{x}}_{0} . \end{matrix}$

(I)
Using Theorem 1, if $\tilde{x} (t)$ is (1)-SGH-differentiable function, then $L [\dot{\tilde{x}} (t)] = s L [\tilde{x} (t)] \frac{H}{} \tilde{x} (0)$ . Therefore, we have: ${\begin{matrix} s L [\tilde{x} (t)] \frac{H}{} \tilde{x} (0) = - L [\tilde{x} (t)] + L \end{matrix}$

Conclusions

This paper aims at studying of the stability of fuzzy linear dynamical systems, whose model has been presented by fuzzy differential equations. To reach such a purpose, some new definitions and concepts such as the granular fuzzy Laplace transform, fuzzy marginal and asymptotic stability of fuzzy equilibrium points, granular fuzzy transfer function, fuzzy poles, and fuzzy zeros are defined to complete theoretical proof. The new concepts enable us to obtain the conditions of the stability of

Acknowledgment

The authors acknowledge support from National Natural Science Foundation of China (NSFC) under Project No. 61573119 and a Fundamental Research Project of Shenzhen under Projects No. JCYJ20170307151312215, No. JCYJ20150403161923533, and JCYJ20150625142543468.

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Citation Excerpt :
According to the results obtained in [5–7,12], and [22–26], RDM-FIA is a much more powerful and efficient tool than the FSIA in applications. However, the research works that have been conducted on fuzzy dynamical systems governed by fuzzy differential equations are limited to [13,14,17–21]. Furthermore, some conceptual interpretations that might be extracted from the information conveyed by fuzzy differential equations are presented in [11].
This paper is assigned to study the stability and controllability of fuzzy singular dynamical systems. Some new notions such as granular fuzzy matrix norm, the algebraic operations on the space of fuzzy matrices, fuzzy equilibrium point, and the granular fuzzy transfer function of fuzzy singular dynamical systems are introduced. Furthermore, by presenting some theorems proved in this paper, the fuzzy solutions of fully fuzzy singular dynamical systems are obtained. Moreover, some new notions regarding the analysis of the stability of fuzzy singular dynamical systems are given. The stability analysis underlies the concepts of fuzzy stable, fuzzy critical stable, and fuzzy unstable singular dynamical systems. Besides using the notions of controllability of the fuzzy slow and fast subsystems, the concept of granular controllability of the fuzzy singular dynamical system is investigated.

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On the stability of fuzzy linear dynamical systems

Highlights

Abstract

Introduction

Section snippets

Basic concepts

[1]

[1]

Some new definitions and notes

[39]

Granular fuzzy Laplace transform

Stability of fuzzy linear dynamical systems

Advantages of granular fuzzy Laplace transform

Examples

Conclusions

Acknowledgment

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Commun. Nonlinear Sci. Numer. Simul.

J. Frankl. Inst.

J. Comput. Appl. Math.

ISA Trans.

J. Frankl. Inst.

Comput. Math. Appl.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Commun. Nonlinear Sci. Numer. Simul.

Comput. Math. Appl.

Pattern Recogn.

Granular differentiability of fuzzy-number-valued functions

J. IEEE Trans. Fuzzy Syst.

Optimal control of fuzzy linear controlled system with fuzzy initial conditions

Iran. J. Fuzzy Syst.

Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives

IEEE Trans. Fuzzy Syst.

Existence and uniqueness theorem for a solution of fuzzy differential equations

Int. J. Math. Math. Sci.

Ulam stability for fractional partial integro-differential equation with uncertainty

Acta Math.

Fuzzy fractional partial differential equations in partially ordered metric spaces

Iran. J. Fuzzy Syst.

Multi-objective fault-tolerant control for fuzzy switched systems with persistent dwell-time and its application in electric circuits

IEEE Trans. Fuzzy Syst.