Novel robust stability analysis method for uncertain systems with interval time-varying delay

Abstract

Stability analysis of interval time-varying delay is of great significance to ensure the reliable control of industrial processes. Aim to improve the robust stability analysis performance for a class of linear systems with norm-bounded uncertainty and interval time-varying delay. In this paper, less conservative robust stability criterion is proposed based on augmented Lyapunov–Krasovskii functional (LKF) method and reciprocally convex combination. Firstly, the delay interval is divided into multiple non-equidistant subintervals by two-stage segmenting strategies and a new LKF comprising quadruple integral term is introduced for each subinterval. Secondly, a novel delay-dependent stability criterion in terms of linear matrix inequalities is proved by less conservative Wirtinger-based integral inequality approach. Two numerical comparative examples and an IEEE four-generator 11-bus power system are selected to verify the superiority of the proposed method. For the two numerical examples about closed-loop control systems and uncertain system with interval time-varying delays, the proposed robust stability criterion can enlarge the maximum allowable delay bound (MADB) about 28.4% and 67.4% than the best results in the previous literatures, respectively, and for the power system with four-generator 11-bus, The MADB in this paper is about 4.4% better than the best result in the previous literatures. All the above results show the effectiveness of the proposed approach.

Appendix

Proof

For the sake of simplicity, Theorem 1 holds when $h(t) \in \left[ {h_{2} ,h_{3} } \right]$, and then Theorem 1 is extended to be suitable for use when $h(t) \in \left[ {h_{i} ,h_{i + 1} } \right]$, i = 1,3,4,…,N₁.

For $h(t) \in \left[ {h_{2} ,h_{3} } \right]$, the LKF is constructed as follows:

$$ V_{2} (x(t)) = V_{21} (x(t)) + V_{22} (x(t)) + V_{23} (x(t)) + V_{24} (x(t)) + V_{25} (x(t)) $$

(7)

where

$$ \begin{aligned} V_{21} (x(t)) & = x^{T} (t)P_{1} x(t) + \int_{{t - h_{2} }}^{t} {x^{T} (s){\text{d}}s} P_{2} \int_{{t - h_{2} }}^{t} {x(s){\text{d}}s} \\ & \quad + \,\int_{{t - h_{3} }}^{{t - h_{2} }} {x^{T} (s)} {\text{d}}sP_{3} \int_{{t - h_{3} }}^{{t - h_{2} }} {x(s)} {\text{d}}s + \int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {x^{T} (s){\text{d}}s{\text{d}}\beta } } P_{4} \int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {x(s){\text{d}}s{\text{d}}\beta } } \\ & \quad + \,\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {x^{T} (s){\text{d}}s{\text{d}}\beta } } P_{5} \int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {x(s){\text{d}}s{\text{d}}\beta } } , \\ \end{aligned} $$

$$ V_{22} (x(t)) = \int_{{t - h_{2} }}^{t} {x^{T} (s)Q_{1} x(s){\text{d}}s} + \int_{{t - h_{3} }}^{{t - h_{2} }} {x^{T} (s)} Q_{2} x(s){\text{d}}s $$

$$ \begin{aligned} V_{23} (x(t)) & = h_{2} \int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {x^{T} (s)} } X_{1} x(s){\text{d}}s{\text{d}}\beta + h_{2} \int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)} } X_{2} \dot{x}(s){\text{d}}s{\text{d}}\beta \\ & \quad + \,(h_{3} - h_{2} )\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {x^{T} (s)} } X_{3} x(s){\text{d}}s{\text{d}}\beta + (h_{3} - h_{2} )\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)} } X_{4} \dot{x}(s){\text{d}}s{\text{d}}\beta , \\ \end{aligned} $$

$$ \begin{aligned} V_{24} (x(t)) & = \left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{2} }}^{0} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {x^{T} (s)} } R_{1} x(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta \\ & \quad + \left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{2} }}^{0} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} } R_{2} \dot{x}(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta \\ & \quad + \left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {x^{T} (s)} } R_{3} x(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta \\ & \quad + \left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} } R_{4} \dot{x}(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta , \\ \end{aligned} $$

$$ \begin{aligned} V_{25} (x(t)) & = \left( {{{h_{2}^{3} } \mathord{\left/ {\vphantom {{h_{2}^{3} } 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{2} }}^{0} {\int_{\beta }^{0} {\int_{\lambda }^{0} {\int_{t + \varphi }^{t} {\dot{x}^{T} (s)U_{1} \dot{x}(s)} } {\text{d}}s{\text{d}}\varphi } } {\text{d}}\lambda {\text{d}}\beta \\ & \quad + \left( {{{(h_{3}^{3} - h_{2}^{3} )} \mathord{\left/ {\vphantom {{(h_{3}^{3} - h_{2}^{3} )} 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{\beta }^{0} {\int_{\lambda }^{0} {\int_{t + \varphi }^{t} {\dot{x}^{T} (s)U_{2} \dot{x}(s)} } {\text{d}}s{\text{d}}\varphi } } {\text{d}}\lambda {\text{d}}\beta , \\ \end{aligned} $$

The derivative of LKF V(t) along the nominal system (3) is calculated as Eq. (8).

$$ \dot{V}_{2} (t) = \dot{V}_{21} (t) + \dot{V}_{22} (t) + \dot{V}_{23} (t) + \dot{V}_{24} (t) + \dot{V}_{25} (t) $$

(8)

where

$$ \begin{aligned} \dot{V}_{21} (t) & = 2x^{T} (t)A^{T} P_{1} x(t) + 2x^{T} (t - h(t))B^{T} P_{1} x(t) + 2x^{T} (t)P_{2} \int_{{t - h_{2} }}^{t} {x(s)} {\text{d}}s \\ & \quad - \,2x^{T} (t - h_{2} )P_{2} \int_{{t - h_{2} }}^{t} {x(s)} {\text{d}}s + 2x^{T} (t - h_{2} )P_{3} \int_{{t - h_{3} }}^{{t - h_{2} }} {x(s)} {\text{d}}s \\ & \quad - \,2x^{T} (t - h_{3} )P_{3} \int_{{t - h_{3} }}^{{t - h_{2} }} {x(s)} {\text{d}}s - 2\int_{{t - h_{2} }}^{t} {x^{T} (s)} {\text{d}}sP_{4} \int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {x(s)} {\text{d}}s} {\text{d}}\beta \\ & \quad + \,2h_{2} x^{T} (t)P_{4} \int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {x(s)} {\text{d}}s} {\text{d}}\beta + 2(h_{3} - h_{2} )x^{T} (t)P_{5} \int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {x(s)} {\text{d}}s} {\text{d}}\beta \\ & \quad - \,2\int_{{t - h_{3} }}^{{t - h_{2} }} {x^{T} (s)} dsP_{5} \int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {x(s)} {\text{d}}s} {\text{d}}\beta , \\ \end{aligned} $$

$$ \begin{aligned} \dot{V}_{22} (t) & = x^{T} (t)Q_{1} x(t) - x^{T} (t - h_{2} )Q_{1} x(t - h_{2} ) + x^{T} (t - h_{2} )Q_{2} x(t - h_{2} ) \\ & \quad - x^{T} (t - h_{3} )Q_{2} x(t - h_{3} ), \\ \end{aligned} $$

$$ \begin{aligned} \dot{V}_{23} (t) & = h_{2}^{2} x^{T} (t)X_{1} x(t) - h_{2} \int_{{t - h_{2} }}^{t} {x^{T} (s)} X_{1} x(s){\text{d}}s - h_{2} \int_{{t - h_{2} }}^{t} {\dot{x}^{T} (s)} X_{2} \dot{x}(s){\text{d}}s \\ & \quad + h_{2}^{2} \dot{x}^{T} (t)X_{2} \dot{x}(t) + (h_{3} - h_{2} )^{2} x^{T} (t)X_{3} x(t) + (h_{3} - h_{2} )^{2} \dot{x}^{T} (t)X_{4} \dot{x}(t) \\ & \quad - (h_{3} - h_{2} )\int_{{t - h_{3} }}^{{t - h_{2} }} {x^{T} (s)} X_{3} x(s){\text{d}}s - (h_{3} - h_{2} )\int_{{t - h_{3} }}^{{t - h_{2} }} {\dot{x}^{T} (s)} X_{4} \dot{x}(s){\text{d}}s, \\ \end{aligned} $$

$$ \begin{aligned} \dot{V}_{24} (t) & = \left( {{{h_{2}^{4} } \mathord{\left/ {\vphantom {{h_{2}^{4} } 4}} \right. \kern-\nulldelimiterspace} 4}} \right)x^{T} (t)R_{1} x(t) - \left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {x^{T} (s)} R_{1} x(s)} {\text{d}}s{\text{d}}\beta \\ & \quad + \left( {{{h_{2}^{4} } \mathord{\left/ {\vphantom {{h_{2}^{4} } 4}} \right. \kern-\nulldelimiterspace} 4}} \right)\dot{x}^{T} (t)R_{2} \dot{x}(t) - \left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)R_{2} \dot{x}(s)} } {\text{d}}s{\text{d}}\beta \\ & \quad + \left( {{{(h_{3}^{2} - h_{2}^{2} )^{2} } \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )^{2} } 4}} \right. \kern-\nulldelimiterspace} 4}} \right)x^{T} (t)R_{3} x(t) - \left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {x^{T} (s)R_{3} x(s)} } {\text{d}}s{\text{d}}\beta \\ & \quad + \left( {{{(h_{3}^{2} - h_{2}^{2} )^{2} } \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )^{2} } 4}} \right. \kern-\nulldelimiterspace} 4}} \right)\dot{x}^{T} (t)R_{4} \dot{x}(t) - \left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)R_{4} \dot{x}(s)} } {\text{d}}s{\text{d}}\beta , \\ \end{aligned} $$

$$ \begin{aligned} \dot{V}_{25} (t) & = \left( {{{h_{2}^{6} } \mathord{\left/ {\vphantom {{h_{2}^{6} } {36}}} \right. \kern-\nulldelimiterspace} {36}}} \right)\dot{x}^{T} (t)U_{1} \dot{x}(t) + \left( {{{(h_{3}^{3} - h_{2}^{3} )^{2} } \mathord{\left/ {\vphantom {{(h_{3}^{3} - h_{2}^{3} )^{2} } {36}}} \right. \kern-\nulldelimiterspace} {36}}} \right)\dot{x}^{T} (t)U_{2} \dot{x}(t) \\ & \quad - \left( {{{h_{2}^{3} } \mathord{\left/ {\vphantom {{h_{2}^{3} } 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{2} }}^{0} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} U_{1} \dot{x}(s){\text{d}}s} {\text{d}}\lambda } {\text{d}}\beta \\ & \quad - \left( {{{(h_{3}^{3} - h_{2}^{3} )} \mathord{\left/ {\vphantom {{(h_{3}^{3} - h_{2}^{3} )} 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} U_{2} \dot{x}(s){\text{d}}s} {\text{d}}\lambda } {\text{d}}\beta , \\ \end{aligned} $$

From Lemmas 1 and 2, we can obtain the follow inequalities.

$$ - h_{2} \int_{{t - h_{2} }}^{t} {x^{T} (s)} X_{1} x(s){\text{d}}s \le - \zeta^{T} (t)e_{5} X_{1} e_{5}^{T} \zeta (t) $$

(9)

$$ \begin{aligned} & - h_{2} \int_{{t - h_{2} }}^{t} {\dot{x}^{T} (s)} X_{2} \dot{x}(s){\text{d}}s \le - \zeta^{T} (t)(e_{1} - e_{3} )X_{2} (e_{1}^{T} - e_{3}^{T} )\zeta (t) \\ & - 3\zeta^{T} (t)(e_{1} + e_{3} - \left( {{2 \mathord{\left/ {\vphantom {2 {h_{2} }}} \right. \kern-\nulldelimiterspace} {h_{2} }}} \right)e_{5} )X_{2} (e_{1}^{T} + e_{3}^{T} - \left( {{2 \mathord{\left/ {\vphantom {2 {h_{2} }}} \right. \kern-\nulldelimiterspace} {h_{2} }}} \right)e_{5}^{T} )\zeta (t) \\ \end{aligned} $$

(10)

where $\varsigma \left( t \right)$ is consistent with i = 2 in Lemma 3.

From Lemma 3, we can obtain as follows:

$$ \begin{aligned} - & (h_{3} - h_{2} )\int_{{t - h_{3} }}^{{t - h_{2} }} {x^{T} (s)} X_{3} x(s){\text{d}}s \le - \zeta^{T} (t)e_{7} X_{3} e_{7}^{T} \zeta (t) \\ - & \zeta^{T} (t)e_{6} X_{3} e_{6}^{T} \zeta (t) - \alpha \zeta^{T} (t)e_{7} X_{3} e_{7}^{T} \zeta (t) - (1 - \alpha )\zeta^{T} (t)e_{6} X_{3} e_{6}^{T} \zeta (t) \\ \end{aligned} $$

(11)

Similarly, according to Lemma 3, we can obtain the follow inequalities.

$$ \begin{aligned} & - (h_{3} - h_{2} )\int_{{t - h_{3} }}^{{t - h_{2} }} {\dot{x}^{T} (s)} X_{4} \dot{x}(s)ds \le - \zeta^{T} (t)(e_{2} - e_{4} )X_{4} (e_{2}^{T} - e_{4}^{T} )\zeta (t) \\ & - \zeta^{T} (t)(e_{3} - e_{2} )X_{4} (e_{3}^{T} - e_{2}^{T} )\zeta (t) - \alpha \zeta^{T} (t)(e_{2} - e_{4} )X_{4} (e_{2}^{T} - e_{4}^{T} )\zeta (t) \\ & - (1 - \alpha )\zeta^{T} (t)(e_{3} - e_{2} )X_{4} (e_{3}^{T} - e_{2}^{T} )\zeta (t) \\ \end{aligned} $$

(12)

$$ - \left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {x^{T} (s)R_{1} x(s)} } {\text{d}}s{\text{d}}\beta \le - \zeta^{T} (t)e_{8} R_{1} e_{8}^{T} \zeta (t) $$

(13)

$$ - \left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{2} }}^{0} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)R_{2} \dot{x}(s)} } {\text{d}}s{\text{d}}\beta \le - \zeta^{T} (t)(h_{2} e_{1} - e_{5} )R_{2} (h_{2} e_{1}^{T} - e_{5}^{T} )\zeta (t) $$

(14)

$$ \begin{aligned} & - \left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {x^{T} (s)R_{3} x(s)} } {\text{d}}s{\text{d}}\beta \le - \zeta^{T} (t)e_{10} R_{3} e_{10}^{T} \zeta (t) \\ & - \zeta^{T} (t)e_{9} R_{3} e_{9}^{T} \zeta (t) - \varepsilon \zeta^{T} (t)e_{10} R_{3} e_{10}^{T} \zeta (t) - (1 - \varepsilon )\zeta^{T} (t)e_{9} R_{3} e_{9}^{T} \zeta (t) \\ \end{aligned} $$

(15)

$$ \begin{aligned} & - \left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)R_{4} \dot{x}(s)} } dsd\beta \\ & \le - \zeta^{T} (t)((h_{3} - h_{2} )e_{1} - e_{7} )R_{4} ((h_{3} - h_{2} )e_{1}^{T} - e_{7}^{T} )\zeta (t) \\ & \quad - \zeta^{T} (t)((h_{3} - h_{2} )e_{1} - e_{6} )R_{4} ((h_{3} - h_{2} )e_{1}^{T} - e_{6}^{T} )\zeta (t) \\ & \quad - \varepsilon \zeta^{T} (t)((h_{3} - h_{2} )e_{1} - e_{7} )R_{4} ((h_{3} - h_{2} )e_{1}^{T} - e_{7}^{T} )\zeta (t) \\ & \quad - (1 - \varepsilon )\zeta^{T} (t)((h_{3} - h_{2} )e_{1} - e_{6} )R_{4} ((h_{3} - h_{2} )e_{1}^{T} - e_{6}^{T} )\zeta (t) \\ \end{aligned} $$

(16)

$$ \begin{aligned} & - \left( {{{h_{2}^{3} } \mathord{\left/ {\vphantom {{h_{2}^{3} } 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{2} }}^{0} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} U_{1} \dot{x}(s){\text{d}}s} {\text{d}}\lambda } {\text{d}}\beta \\ & \le - \zeta^{T} (t)(\left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)e_{1} - e_{8} )U_{1} (\left( {{{h_{2}^{2} } \mathord{\left/ {\vphantom {{h_{2}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)e_{1}^{T} - e_{8}^{T} )\zeta (t) \\ \end{aligned} $$

(17)

$$ \begin{aligned} & - \left( {{{(h_{3}^{3} - h_{2}^{3} )} \mathord{\left/ {\vphantom {{(h_{3}^{3} - h_{2}^{3} )} 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{3} }}^{{ - h_{2} }} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} U_{2} \dot{x}(s){\text{d}}s} {\text{d}}\lambda } {\text{d}}\beta \\ & \le - \zeta^{T} (t)(\left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)e_{1} - e_{9} - e_{10} )U_{2} (\left( {{{(h_{3}^{2} - h_{2}^{2} )} \mathord{\left/ {\vphantom {{(h_{3}^{2} - h_{2}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)e_{1}^{T} - e_{9}^{T} - e_{10}^{T} )\zeta (t) \\ \end{aligned} $$

(18)

Substituting (9)–(18) to (8), then we can obtain $\dot{V}_{2} (x(t))$ as the following inequality.

$$ \dot{V}_{2} (x(t)) \le \zeta^{T} (t)\left[ {\alpha \Gamma_{1} + (1 - \alpha )\Gamma_{2} + \varepsilon \Gamma_{3} + (1 - \varepsilon )\Gamma_{4} } \right]\zeta (t) $$

(19)

where

$$ \Gamma_{1} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{7} X_{3} e_{7}^{T} - (e_{2} - e_{4} )X_{4} (e_{2}^{T} - e_{4}^{T} ) $$

$$ \Gamma_{2} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{6} X_{3} e_{6}^{T} - (e_{3} - e_{2} )X_{4} (e_{3}^{T} - e_{2}^{T} ) $$

$$ \Gamma_{3} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{10} R_{3} e_{10}^{T} - ((h_{3} - h_{2} )e_{1} - e_{7} )R_{4} ((h_{3} - h_{2} )e_{1}^{T} - e_{7}^{T} ) $$

$$ \Gamma_{4} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{9} R_{3} e_{9}^{T} - ((h_{3} - h_{2} )e_{1} - e_{6} )R_{4} ((h_{3} - h_{2} )e_{1}^{T} - e_{6}^{T} ) $$

For 0 ≤ α, ε ≤ 1, using the convex combination technique, we can obtain the following two inequalities.

$$ \alpha (\Gamma_{1} + \lambda_{1} I) + (1 - \alpha )(\Gamma_{2} + \lambda_{1} I) < 0 $$

(20)

$$ \varepsilon (\Gamma_{3} - \lambda_{2} I) + (1 - \varepsilon )(\Gamma_{4} - \lambda_{2} I) < 0 $$

(21)

Further, we can obtain the follow inequalities conveniently

$$ \alpha \Gamma_{1} + (1 - \alpha )\Gamma_{2} < - \lambda_{1} I $$

(22)

$$ \varepsilon \Gamma_{3} + (1 - \varepsilon )\Gamma_{4} < \lambda_{2} I $$

(23)

Due to λ₁ > λ₂, combining (22) and (23), the following inequality is available.

$$ \alpha \Gamma_{1} + (1 - \alpha )\Gamma_{2} + \varepsilon \Gamma_{3} + (1 - \varepsilon )\Gamma_{4} < (\lambda_{2} - \lambda_{1} )I < 0 $$

(24)

If $\alpha \Gamma_{1} + (1 - \alpha )\Gamma_{2} + \varepsilon \Gamma_{3} + (1 - \varepsilon )\Gamma_{4} < 0$, according to the L–K stability theorem, there exists a sufficient small positive number δ₂ makes $\dot{V}_{2} (t) < - \delta_{2} \left\| {x(t)} \right\|^{2}$ hold, and then the system (3) is asymptotically stable.

Without loss of generality, when $h(t) \in \left[ {h_{i},h_{i + 1} } \right]$, i = 1,3,4,…,N₁, the LKF is constructed as follows:

$$ V_{i} (x(t)) = V_{i1} (x(t)) + V_{i2} (x(t)) + V_{i3} (x(t)) + V_{i4} (x(t)) + V_{i5} (x(t)) $$

(25)

where

$$ \begin{aligned} V_{i1} (x(t)) & = x^{T} (t)P_{1} x(t) + \int_{{t - h_{i} }}^{t} {x^{T} (s){\text{d}}s} P_{2} \int_{{t - h_{i} }}^{t} {x(s){\text{d}}s} \\ & \quad + \int_{{t - h_{i + 1} }}^{{t - h_{i} }} {x^{T} (s)} dsP_{3} \int_{{t - h_{i + 1} }}^{{t - h_{i} }} {x(s)} {\text{d}}s + \int_{{ - h_{i} }}^{0} {\int_{t + \beta }^{t} {x^{T} (s){\text{d}}s{\text{d}}\beta } } P_{4} \int_{{ - h_{i} }}^{0} {\int_{t + \beta }^{t} {x(s){\text{d}}s{\text{d}}\beta } } \\ & \quad + \int_{{ - h_{i + 1} }}^{{ - h_{i} }} {\int_{t + \beta }^{t} {x^{T} (s){\text{d}}s{\text{d}}\beta } } P_{5} \int_{{ - h_{i + 1} }}^{{ - h_{i} }} {\int_{t + \beta }^{t} {x(s){\text{d}}s{\text{d}}\beta } } , \\ \end{aligned} $$

$V_{i2} (x(t)) = \int_{{t - h_{i} }}^{t} {x^{T} (s)Q_{1} x(s){\text{d}}s} + \int_{{t - h_{i + 1} }}^{{t - h_{i} }} {x^{T} (s)Q_{2} x(s){\text{d}}s}$,

$$ \begin{aligned} V_{i3} (x(t)) & = h_{i} \int_{{ - h_{i} }}^{0} {\int_{t + \beta }^{t} {x^{T} (s)} } X_{1} x(s){\text{d}}s{\text{d}}\beta + h_{i} \int_{{ - h_{i} }}^{0} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)} } X_{2} \dot{x}(s){\text{d}}s{\text{d}}\beta \\ & \quad + (h_{i + 1} - h_{i} )\int_{{ - h_{i + 1} }}^{{ - h_{i} }} {\int_{t + \beta }^{t} {x^{T} (s)} } X_{3} x(s){\text{d}}s{\text{d}}\beta \\ & \quad + (h_{i + 1} - h_{i} )\int_{{ - h_{i + 1} }}^{{ - h_{i} }} {\int_{t + \beta }^{t} {\dot{x}^{T} (s)} } X_{4} \dot{x}(s){\text{d}}s{\text{d}}\beta , \\ \end{aligned} $$

$$ \begin{aligned} V_{i4} (x(t)) & = \left( {{{h_{i}^{2} } \mathord{\left/ {\vphantom {{h_{i}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{i} }}^{0} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {x^{T} (s)} } R_{1} x(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta \\ & \quad + \left( {{{h_{i}^{2} } \mathord{\left/ {\vphantom {{h_{i}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{i} }}^{0} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} } R_{2} \dot{x}(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta \\ & \quad + \left( {{{(h_{i + 1}^{2} - h_{i}^{2} )} \mathord{\left/ {\vphantom {{(h_{i + 1}^{2} - h_{i}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{i + 1} }}^{{ - h_{i} }} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {x^{T} (s)} } R_{3} x(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta \\ & \quad + \left( {{{(h_{i + 1}^{2} - h_{i}^{2} )} \mathord{\left/ {\vphantom {{(h_{i + 1}^{2} - h_{i}^{2} )} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\int_{{ - h_{i + 1} }}^{{ - h_{i} }} {\int_{\beta }^{0} {\int_{t + \lambda }^{t} {\dot{x}^{T} (s)} } R_{4} \dot{x}(s){\text{d}}s{\text{d}}\lambda } {\text{d}}\beta, \\ \end{aligned} $$

$$ \begin{aligned} V_{i5} (x(t)) & = \left( {{{h_{i}^{3} } \mathord{\left/ {\vphantom {{h_{i}^{3} } 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{i} }}^{0} {\int_{\beta }^{0} {\int_{\lambda }^{0} {\int_{t + \varphi }^{t} {\dot{x}^{T} (s)U_{1} \dot{x}(s)} } {\text{d}}s{\text{d}}\varphi } } {\text{d}}\lambda {\text{d}}\beta \\ & \quad + \left( {{{(h_{i + 1}^{3} - h_{i}^{3} )} \mathord{\left/ {\vphantom {{(h_{i + 1}^{3} - h_{i}^{3} )} 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\int_{{ - h_{i + 1} }}^{{ - h_{i} }} {\int_{\beta }^{0} {\int_{\lambda }^{0} {\int_{t + \varphi }^{t} {\dot{x}^{T} (s)U_{2} \dot{x}(s)} } {\text{d}}s{\text{d}}\varphi } } {\text{d}}\lambda {\text{d}}\beta. \\ \end{aligned} $$

where ζ(t) is the same as in Lemma 3. P_i(i = 1,2,3,4,5), Q₁, Q₂, U₁, U₂, X_j, R_j (j = 1,2,3,4) are the matrices as the same as in inequality (4). The same method is available, and then, the following conclusions can be achieved.

$$ \dot{V}_{i} (x(t)) \le \zeta^{T} (t)\left[ {\alpha \Gamma_{i1} + (1 - \alpha )\Gamma_{i2} + \varepsilon \Gamma_{i3} + (1 - \varepsilon )\Gamma_{i4} } \right]\zeta (t) $$

(26)

where

$$ \Gamma_{i1} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{7} X_{3} e_{7}^{T} - (e_{2} - e_{4} )X_{4} (e_{2}^{T} - e_{4}^{T} ) $$

$$ \Gamma_{i2} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{6} X_{3} e_{6}^{T} - (e_{3} - e_{2} )X_{4} (e_{3}^{T} - e_{2}^{T} ) $$

$\Gamma_{i3} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{10} R_{3} e_{10}^{T} - ((h_{i + 1} - h_{i} )e_{1} - e_{7} )R_{4} ((h_{i + 1} - h_{i} )e_{1}^{T} - e_{7}^{T} )$

$\Gamma_{i4} = {\Phi \mathord{\left/ {\vphantom {\Phi 2}} \right. \kern-\nulldelimiterspace} 2} - e_{9} R_{3} e_{9}^{T} - ((h_{i + 1} - h_{i} )e_{1} - e_{6} )R_{4} ((h_{i + 1} - h_{i} )e_{1}^{T} - e_{6}^{T} )$

In the same way, there also exists a sufficient small positive number δ_i to make $\dot{V}_{i} (t) < - \delta_{i} \left\| {x(t)} \right\|^{2}$ hold, and then, the nominal system (3) is asymptotically stable.

The combination of the inequalities (19) and (26) is equivalent to (4). This fulfills the proof.