Improved Delay-Range-Dependent Stability Condition for T–S Fuzzy Systems with Variable Delays Using New Extended Affine Wirtinger Inequality

Abstract

In this paper, a new integral inequality lemma along with an appropriate Lyapunov–Krasovskii functional (LKF) is proposed for fuzzy time-delay system to enhance the delay upper bound estimate. The proposed lemma is referred to hereafter as Extended Affine Wirtinger inequality. The novelty of the lemma is twofold, it has the ability to integrate uncertain delay information utilizing the convex combination of certain and uncertain delay intervals involved in the proposed LKF, and it is compatible to derive delay-range-dependent (DRD) stability conditions for continuous time Takagi–Sugeno (T–S) fuzzy time-delay system. One noteworthy advantage of the proposed stability condition in a linear matrix inequality (LMI) framework is that it requires less number of matrix variables compared to the existing integral inequalities of the similar type, thus reducing the computational burden too. The efficacy of the proposed DRD stability condition over existing conditions is validated numerically by solving three examples related to fuzzy time-delay system.

Appendix 1

Proof of Lemma 3:

Based on Schur’s complement, for positive definite symmetric matrix R and free matrices $N_{1},\,N_{2},\,N_{3},\,N_{4}$ with appropriate dimension, the following inequality holds

$$\begin{aligned} \mho _{k}=\begin{bmatrix} \begin{bmatrix} N_{2k-1}\\N_{2k} \end{bmatrix}R^{-1}\begin{bmatrix} N_{2k-1}\\N_{2k} \end{bmatrix}^\mathrm{T}&\begin{bmatrix} N_{2k-1}\\N_{2k} \end{bmatrix}\\ {*}&R \end{bmatrix}\ge 0,\quad k=1,2 \end{aligned}$$

which implies that

$$\begin{aligned} \int _{a}^{c(t)}\vartheta _{1}^\mathrm{T}\mho _{1}\vartheta _{1} \,\mathrm{d}s+\int _{c(t)}^{b}\vartheta _{2}^\mathrm{T}\mho _{2}\vartheta _{2} \,\mathrm{d}s\ge 0,\quad a<c(t)<b, \end{aligned}$$

(48)

where $\vartheta _{1},\,\vartheta _{2}$ are any vectors. Next, for the function $\gamma (s,a,b)=\dfrac{2s-b-a}{b-a}$, the following integration is calculated as

$$\begin{aligned} \int _{a}^{b}\gamma (s,a,b)\,\mathrm{d}s&=0,\int _{a}^{b}\gamma ^{2}(s,a,b)\,\mathrm{d}s=\dfrac{b-a}{3}. \end{aligned}$$

(49)

$$\begin{aligned} \int _{a}^{b}\dot{x}(s)\,\mathrm{d}s&=\left[ x(b)-x(a) \right] .\end{aligned}$$

(50)

$$\begin{aligned} \int _{a}^{b}\gamma (s,a,b)\dot{x}(s)\,\mathrm{d}s&=\left[ x(b)+x(a)-\dfrac{2}{b-a}\int _{a}^{b}x(s)\,\mathrm{d}s \right] . \end{aligned}$$

(51)

Define the vectors $\vartheta _{1}$ and $\vartheta _{2}$ as follows

$$\begin{aligned} \vartheta _{1}&=\left[ -\zeta ^\mathrm{T}(t)\,\,-\gamma _{1}\zeta ^\mathrm{T}(t)\,\,\dot{x}^\mathrm{T}(s) \right] ^\mathrm{T},\,\gamma _{1}=\gamma (s,a,c(t)),\\ \vartheta _{2}&=\left[ -\zeta ^\mathrm{T}(t)\,\,-\gamma _{2}\zeta ^\mathrm{T}(t)\,\,\dot{x}^\mathrm{T}(s) \right] ^\mathrm{T},\,\gamma _{2}=\gamma (s,c(t),b). \end{aligned}$$

Thus, recalculating (48) based on the above definitions, yields

$$\begin{aligned}&\int _{a}^{c(t)}\vartheta _{1}^\mathrm{T}\mho _{1}\vartheta _{1} \,\mathrm{d}s+\int _{c(t)}^{b}\vartheta _{2}^\mathrm{T}\mho _{2}\vartheta _{2} \,\mathrm{d}s\nonumber \\&\quad =\int _{a}^{c(t)}\dot{x}^\mathrm{T}(s)R\dot{x}(s)\,\mathrm{d}s+\int _{c(t)}^{b}\dot{x}^\mathrm{T}(s)R\dot{x}(s)\,\mathrm{d}s\nonumber \\&\qquad +\,\zeta ^\mathrm{T}(t)(N_{1}R^{-1}N^\mathrm{T}_{1})\zeta (t)\int _{a}^{c(t)}\,\mathrm{d}s\nonumber \\&\qquad +\,\zeta ^\mathrm{T}(t)(N_{2}R^{-1}N^\mathrm{T}_{2})\zeta (t)\int _{a}^{c(t)}\gamma ^{2}(s,a,c(t))\,\mathrm{d}s\nonumber \\&\qquad +\,\zeta ^\mathrm{T}(t)(N_{3}R^{-1}N^\mathrm{T}_{3})\zeta (t)\int _{c(t)}^{b}\,\mathrm{d}s\nonumber \\&\qquad +\,\zeta ^\mathrm{T}(t)(N_{4}R^{-1}N^\mathrm{T}_{4})\zeta (t)\int _{c(t)}^{b}\gamma ^{2}(s,c(t),b)\,\mathrm{d}s\nonumber \\&\qquad -\,2\zeta ^\mathrm{T}(t)N_{1}\int _{a}^{c(t)}\dot{x}(s)\,\mathrm{d}s\nonumber \\&\qquad -\,2\zeta ^\mathrm{T}(t)N_{2}\int _{a}^{c(t)}\gamma (s,a,c(t))\dot{x}(s)\,\mathrm{d}s\nonumber \\&\qquad -\,2\zeta ^\mathrm{T}(t)N_{3}\int _{c(t)}^{b}\dot{x}(s)\,\mathrm{d}s\nonumber \\&\qquad -\,2\zeta ^\mathrm{T}(t)N_{4}\int _{c(t)}^{b}\gamma (s,c(t),b)\dot{x}(s)\,\mathrm{d}s\end{aligned}$$

(52)

$$\begin{aligned}&\quad =\int _{a}^{c(t)}\dot{x}^\mathrm{T}(s)R\dot{x}(s)\,\mathrm{d}s+\int _{c(t)}^{b}\dot{x}^\mathrm{T}(s)R\dot{x}(s)\,\mathrm{d}s\nonumber \\&\qquad +\,\zeta ^\mathrm{T}(t)\left[ \left( c(t)-a\right) \left( N_{1}R^{-1}N^\mathrm{T}_{1}+\dfrac{1}{3}N_{2}R^{-1}N^\mathrm{T}_{2} \right) \right. \nonumber \\&\qquad \left. +\,\left( b-c(t)\right) \left( N_{3}R^{-1}N^\mathrm{T}_{3}+\dfrac{1}{3}N_{4}R^{-1}N^\mathrm{T}_{4} \right) \right. \nonumber \\&\qquad \left. -\,2N_{1}\left( \overline{e}_{2}-\overline{e}_{3}\right) -2N_{2}\left( \overline{e}_{2}+\overline{e}_{3}-2\overline{e}_{4}\right) \right. \nonumber \\&\qquad \left. -\,2N_{3}\left( \overline{e}_{1}-\overline{e}_{2}\right) -2N_{4}\left( \overline{e}_{1}+\overline{e}_{2}-2\overline{e}_{5}\right) \right] \zeta (t)\nonumber \\&\quad =\int _{a}^{c(t)}\dot{x}^\mathrm{T}(s)R\dot{x}(s)\,\mathrm{d}s+\int _{c(t)}^{b}\dot{x}^\mathrm{T}(s)R\dot{x}(s)\,\mathrm{d}s\nonumber \\&\qquad +\,\zeta ^\mathrm{T}(t)\left[ \left( c(t)-a\right) \left( N_{1}R^{-1}N^\mathrm{T}_{1}+\dfrac{1}{3}N_{2}R^{-1}N^\mathrm{T}_{2} \right) \right. \nonumber \\&\qquad \left. +\,\left( b-c(t)\right) \left( N_{3}R^{-1}N^\mathrm{T}_{3}+\dfrac{1}{3}N_{4}R^{-1}N^\mathrm{T}_{4} \right) \right. \nonumber \\&\qquad \left. -\,\left( NL+L^\mathrm{T}N^\mathrm{T}\right) \right] \zeta (t)\ge 0. \end{aligned}$$

(53)

Using the concept of convex combination of LMIs in [21] and doing some algebraic manipulation, (53) can be written as

$$\begin{aligned}&\int _{a}^{c(t)}\dot{x}^\mathrm{T}(s)\textit{R}\dot{x}(s)\,\mathrm{d}s+\int _{c(t)}^{b}\dot{x}^\mathrm{T}(s)\textit{R}\dot{x}(s)\,\mathrm{d}s\nonumber \\&\quad \ge \zeta ^\mathrm{T}(t)\left[ NL+L^\mathrm{T}N^\mathrm{T}\right. \nonumber \\&\qquad \left. -\,h\alpha _{1}\left( N_{1}R^{-1}N^\mathrm{T}_{1}+\dfrac{1}{3}N_{2}R^{-1}N^\mathrm{T}_{2} \right) \right. \nonumber \\&\qquad \left. -\,h(1-\alpha _{1})\left( N_{3}R^{-1}N^\mathrm{T}_{3}+\dfrac{1}{3}N_{4}R^{-1}N^\mathrm{T}_{4} \right) \right] \zeta (t), \end{aligned}$$

(54)

where $\alpha _{1}=\dfrac{c(t)-a}{h},\,h=(b-a)$. This complete the proof of Lemma 3. $\square$