T–S fuzzy observed based design and synchronization of chaotic and hyper-chaotic dynamical systems

Abstract

In this paper, we present the Takagi–Sugeno (T–S) fuzzy observed based design and synchronization of chaotic and hyper chaotic dynamical systems. Using T–S fuzzy observed based design, the fuzzy logic controllers for chaotic synchronization are constructed via linear matrix inequality (LMI). Lyapunov exponents of 3D and 4D Chen systems have been calculated. This represents the chaoticity and hyper chaoticity of Chen systems. Analytical and computational studies of chaotic and hyper chaotic Chen systems have been performed by using LMI toolbox. The qualitative and simulated results show the validity and the performance of the proposed observer in an excellent agreement.

Appendix A:

Proposition

: The continuous-time linear system in the form of

$$\begin{aligned} \dot{x}(t)= A x(t) \end{aligned}$$

(22)

with $A\in R^{n\times n}$ is Hurwitz stable if and only if there exists a matrix $P\in R^{n\times n}$, such that

$$\begin{aligned} \begin{bmatrix} P>0 \\ A^{T}P + PA <0. \end{bmatrix} \end{aligned}$$

(23)

Theorem A.1

[10]: The equilibrium of the augmented system described by  (10) is globally asymptotically stable if there exists a common positive definite matrix P such that

$$\begin{aligned}&G_{ii}^{T} P + P G_{ii} < 0, \end{aligned}$$

(24)

$$\begin{aligned}&\left( \frac{G_{ij} + G_{ji}}{2}\right) ^{T} P + P \left( \frac{G_{ij} + G_{ji}}{2}\right) \le 0,\nonumber \\&i < j \quad \hbox { such}\hbox { that }\quad h_{i}\cap h_{j}\ne \phi . \end{aligned}$$

(25)

Theorem A.2

[10]: If the number of the rules that fire for all t is less than or equal to s, where $1 < s\le r$, then

$$\begin{aligned} \sum _{i=1} ^{r} h_{i}^{2}(s(t)) - \frac{1}{s-1} \sum _{i=1} ^{r} \sum _{i<j} ^{r} 2 h_{i} (s(t))h_{j} (s(t)) \ge 0 \end{aligned}$$

(26)

where $\sum _{i=1}^{r} h_{i}(s(t))=1 , \quad h_{i} (s(t)) \ge 0$,    for all i.