# Improved Integral Inequalities for Stability Analysis of Interval Time-Delay Systems
## 5 Numerical Examples
Case 3: For the system with constant delay \(\overline{d}\), the MADBs are calculated using the methods in [16; 19; 20; 21; 36] and Corollary 1 (setting \(d_{1}=d_{2}=\overline{d}\)) are listed in Table 3.  
In [36], a free-matrix-based integral inequality is developed that includes the Wirtinger inequality as a special case. Moreover, this approach has been improved in [20] by employing extra information on double integral of state. However, as discussed in [21], many slack variables are required in [20; 36], bringing heavy computational burden. For two single integral terms with time delay, a novel \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Methods** & \(\mathbf{d_{1}=1}\) & \(\mathbf{d_{1}=2}\) & \(\mathbf{d_{1}=3}\) & \(\mathbf{d_{1}=4}\) & \(\mathbf{d_{1}=5}\) & **Number of Variables** \\ \hline [PERSON] [30] & 1.87 & 2.50 & 3.25 & 4.07 & - & \(2.5n^{2}+2.5n\) \\ [PERSON] et al. [34] & 2.02 & 2.59 & 3.30 & 4.08 & - & \(2.5n^{2}+2.5n\) \\ [PERSON] et al. [35] & 2.04 & 2.60 & 3.30 & 4.08 & - & \(2.5n^{2}+2.5n\) \\ [PERSON] et al. [27] & 2.06 & 2.61 & 3.31 & 4.09 & - & \(3.5n^{2}+3.5n\) \\ [PERSON] et al. [11] & 2.09 & 2.67 & 3.38 & 4.16 & - & \(27.5n^{2}+5.5n\) \\ [PERSON] et al. [32] & 2.12 & 2.72 & 3.45 & 4.25 & 5.09 & \(11.5n^{2}+3.5n\) \\ [PERSON] and [PERSON] [23] & 2.31 & 2.80 & 3.50 & 4.30 & 5.14 & \(19.5n^{2}+4.5n\) \\ Corollary 1 & 2.96 & 3.75 & 4.19 & 4.73 & 5.51 & \(11.5n^{2}+5.5n\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum allowable delay bounds (MADBs) \(d_{2}\) for various \(d_{1}\) and unknown \(\mu\) (Example 1). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & \(\overline{d}\) & **Number of Variables** \\ \hline [PERSON] [16] & 4.975 & \(49n^{2}+5n\) \\ [PERSON] et al. [36] & 6.059 & \(54n^{2}+9n\) \\ [PERSON] and [PERSON] [19] & 6.059 & \(10n^{2}+3n\) \\ [PERSON] et al. [21] & 6.165 & \(23n^{2}+4n\) \\ [PERSON] et al. [20] & 6.166 & \(17.5n^{2}+2.5n\) \\ Corollary 1 & 6.170 & \(11.5n^{2}+5.5n\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MADB for constant delay system (Example 1). integral inequality is proposed in [21], which provides smaller bounding gap than the combination of auxiliary function-based inequality and reciprocally convex lemma. Considering the constant delay case, the approach in [21] will reduce to the auxiliary function-based one. However, the estimation gaps of those inequalities have been further reduced in the Lemmas 2 and 3, which are independent on slack matrices. As seen from Table 3, Corollary 1 delivers better result and fewer variables than some recent integral inequality approach. **Example 2. ** Consider the following time-delay system:  
\[\dot{x}(t)=\left[ \begin{array}{cc}0&1\\ -1&-2\end{array} \right]x(t)+\left[ \begin{array}{cc}0&0\\ -1&1\end{array} \right]x(t-d(t)) \tag{46}\]  
Case 1: For \(d_{1}=0\), the MADBs for various \(\mu\) are presented in Table 4.  
From Table 4, it can be found that Corollary 2 outperforms some existing ones including delay decomposition [13], triple-integral term [14] and convex combination with delay partitioning [34; 35]. It is noted that the LKFs in [34; 35] do not contain any triple integrals and their derivatives are estimated by Jensen inequality. In [13], the lower bound of delay is divided into two uniform segments to reduce the conservatism by larger lower bounds, while the computational burden will increase. In Corollary 2, the same aim is achieved by augmented vector including information on the lower bound of delay, which yields less conservatism and fewer variables than [13]. Case 2: When \(\mu=0.3\), Table 5 gives the comparative results of the MADBs for different \(d_{1}\). Different from [11; 31], as discussed in Remark 4, the relation between time delay and delay range is fully exploited in Theorem 1, which benefits the reduction of conservatism. This may explain that the methods of [11; 31] are more conservative than Theorem 1 in spite of having some triple-integral terms.