# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 5 Numerical Examples  
In this section, three numerical examples are given to verify the improvement of the proposed approaches in reduction of conservatism. The conservatism is measured by the maximum allowable delay bounds (MADBs). **Example 1**. : Consider the following system with time delay:  
\[\dot{x}(t)=\left[ \begin{array}{cc}-2&0\\ 0&-0.9\end{array} \right]x(t)+\left[ \begin{array}{cc}-1&0\\ -1&-1\end{array} \right]x(t-d(t)) \tag{45}\]  
Case 1: For \(\mu=0.9\) and given lower bounds of delay, the MADBs guaranteeing the stability of system (45) are listed in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & \(\mathbf{d_{1}=2}\) & \(\mathbf{d_{1}=3}\) & \(\mathbf{d_{1}=4}\) & \(\mathbf{d_{1}=5}\) & **Number of Variables** \\ \hline [PERSON] [30] & 2.5048 & 3.2591 & 4.0744 & - & \(3n^{2}+3n\) \\ [PERSON] et al. [14] & 2.5663 & 3.3408 & 4.1690 & 5.0275 & \(18n^{2}+8n\) \\ [PERSON] and [PERSON] [31] & 2.5213 & 3.3311 & 4.1880 & 5.0722 & \(6n^{2}+4n\) \\ [PERSON] et al. [32] & 2.7241 & 3.4580 & 4.2576 & 5.0977 & \(11.5n^{2}+3.5n\) \\ [PERSON] et al. [25] & 2.9525 & 3.5837 & 4.3490 & 5.1623 & \(25n^{2}+5.5n\) \\ [PERSON] [33] & 3.3465 & 4.1461 & 5.1012 & \(17n^{2}+8n\) \\ [PERSON] and [PERSON] [17] & 3.1634 & 3.6648 & 4.4467 & 5.2147 & \(7n^{2}+5n\) \\ Theorem 1 & 3.9038 & 4.2741 & 4.8907 & 5.6759 & \(12n^{2}+6n\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MADBs \(d_{2}\) for various \(d_{1}\) and \(\mu=0.9\) (Example 1). The common feature of the methods in [14; 30; 31] is the use of Jensen inequality for the estimation task, which introduces some conservatism. In [17], a new integral inequality is developed for single integral functional, which cannot be proved to include the Wirtinger inequality theoretically but could make use of more information on triple integrals of state, while in Theorem 1, the improved integral inequalities are utilized, through which more accurate bounds and more system information are obtained. From Table 1, one can see clearly that Theorem 1 leads to more desirable performance than the approaches in [14; 17; 25; 30; 31; 32; 33] and involves fewer decision variables than those in [14; 33], which demonstrate the potential of the proposed integral inequalities in reduction of conservatism. Case 2: When the derivative of the delay \(\mu\) is unavailable, for different values of \(d_{1}\), the comparisons of Corollary 1 with some recent ones are listed in Table 2.  
From Table 2, the methods in [27; 30; 34; 35] are inapplicable for \(d_{1}=5\), since the used LKF terms contain no '\(t-d_{1}\)' in the integral upper limit. Thus, the information on lower bound of delay is not used adequately. In Corollary 1, the quadratic functions in single, double and triple integral forms including '\(t-d_{1}\)' are introduced into LKF. Moreover, \(\int_{t-d_{1}}^{t}\int_{\theta}^{t}x(s)dsd\theta\) and \(\int_{t-d_{1}}^{t}\int_{\sigma}^{t}\int_{\sigma}^{t}(s)dsd\theta d\sigma\) are utilized to form the augmented vector. As a result, Corollary 1 achieves improvements, especially for the larger lower bounds cases. In [11], the cross term in derivative of triple integral is converted to a quadratic function multiplied by a scalar and the relations among the state, delayed state and single integral of state are established by applying slack matrices. By Lemma 3, more relations among the double and triple integrals are obtained in the criterion without requiring extra slack variables. As seen from Table 2, compared to some existing ones including the refined Jensen inequality [23], Corollary 1 results in larger feasible stability regions with fewer variables than [11; 23]. This verifies that the developed integral inequalities and construction of LKF are effective in deriving less conservative criterion.