Robust Adaptive Terminal Fixed Time Sliding-Mode Control for a Secure Communication of T-S Fuzzy Systems

Abstract

This paper presents a robust adaptive boundary layer thickness (BLT) of sliding mode control (SMC) for a secure communication of chaos-based system. First, chaotic system was remodeled into the Takagi–Sugeno (T-S) fuzzy format. Second, a novel SMC with an adaptive finite time reaching phase and an adaptive BLT of fixed time sliding phase was proposed to synchronize the master and slave systems (MSSs). The matched disturbances on three channels of the secure communication system (SCS) were rejected by the proposed method. To prove that the proposed theories are corrected, the Lyapunov condition was used to meet the goal. Furthermore, the simulation study was used to verify the correction and effectiveness of the proposed algorithm. The tracking errors and reaching times are very small. The sent and received data were precisely tracked each other in the short time.

Appendix

Proof of Lemma 2

In Tian et al. (2020), the Lyapunov condition should be selected as below.

$$\begin{aligned} V(\sigma )= 0.5\sigma ^2 \end{aligned}$$

(30)

If $V(\sigma )=0$ is a special case. If $V(\sigma )\ne 0$, we have $\square$

Case 1: $\mid \sigma (t)\mid \ge 1$ Giap et al. (2021a)

$$\begin{aligned} T_{max}<\frac{1}{\alpha _1}\frac{b_1}{a_1-b_1}+\frac{1}{\alpha _2}\frac{b_2}{b_2-a_2} \end{aligned}$$

(31)

Case 2: If $\mid \sigma (t)\mid <1$, the settling time is limited as

$$\begin{aligned} T_{max}=\frac{1}{\alpha _1+\alpha _2} \end{aligned}$$

(32)

Proof

The Lyapunov candidate once again should be selected as follows:

$$\begin{aligned} V(\sigma )=\frac{1}{2}\sigma ^2 \end{aligned}$$

(33)

or

$$\begin{aligned} V(\sigma )&= \sigma {{\dot{\sigma }}} \nonumber \\&= \sigma [-\alpha _1 sign(\sigma ^{\frac{a_1}{b_1}})-\alpha _2 sign(\sigma ^{\frac{a_2}{b_2}})] \nonumber \\&\le [-\alpha _1 \mid \sigma ^{\frac{a_1}{b_1}}\mid -\alpha _2 \mid \sigma ^{\frac{a_2}{b_2}}\mid ] \nonumber \\&< -2\alpha _1 -2\alpha _2 \end{aligned}$$

(34)

Taking integration for both side of (33) yields

$$\begin{aligned} \int _{V(0)}^{V(T_{max})}dV(\sigma )=\int _{0}^{T_{max}}(-2\alpha _1 -2\alpha _2) \end{aligned}$$

(35)

or

$$\begin{aligned} V(T_{max})-V(0)&=T_{max}(-2\alpha _1 -2\alpha _2) \nonumber \\&\quad \;-0(-2\alpha _1 -2\alpha _2) \end{aligned}$$

(36)

or

$$\begin{aligned} T_{max}<\frac{V(0)}{\alpha _1+\alpha _2} \end{aligned}$$

(37)

Since $\mid \gamma (0)\mid <1,$ it leads to

$$\begin{aligned} T_{max}<\frac{1}{\alpha _1+\alpha _2} \end{aligned}$$

(38)

The stability of fixed time for (3) was proved as (30) and (37).

The operation of T-S fuzzy modeling of Lendek et al. (2011) is slightly written as below.

$$\begin{aligned} {{\dot{\chi }}}&=g(\chi ,u)\chi +h(\chi ,u)\nonumber \\ y&=l(\chi ,u)\chi \end{aligned}$$

(39)

where g, h, and l are the smooth functions. y is output vector. The scheduling $\chi _j \in [\chi _{min};\chi _{max}]$ where $j \in [1,..,p].$ The weighting function is

$$\begin{aligned} n^j_0(.)&= \frac{\chi _{max}-\chi _j}{\chi _{max}-\chi _{min}}\nonumber \\ n^j_1(.)&=1- n^j_0(.) \end{aligned}$$

(40)

The fuzzy membership is

$$\begin{aligned} \varphi _j(\chi )=\sum \limits _{j=1}^{p}\varphi _{ij}(\chi _i) \end{aligned}$$

(41)

where $\varphi _ij$ is either $n^j_0(.)$ or $n^j_1(.)$ Using these concepts, system (7) becomes

$$\begin{aligned} {{\dot{\chi }}}&=\sum \limits _{i=1}^{m}\varphi _i(\chi _j)(A_i\chi +B_iu) \nonumber \\ y&=\sum \limits _{i=1}^{m}\varphi _i(\chi _j)C_i\chi \end{aligned}$$

(42)

$\square$