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Theory and Modern Applications

Table 1 Comparing our results in this paper with some published ones: The delay upper bound and \(\pmb{H_{\infty} }\) performance for switched system ( 1a )-( 1h ) with ( 17 )

From: \({H} _{ {\infty}} \) analysis and switching control for uncertain discrete switched time-delay systems by discrete Wirtinger inequality

Results

Number of LMI variable elements

[4]

Fail to justisfy the stability (when zero state feedback and perturbations)

[16]

Fail to justisfy the stability (when zero state feedback and \(\Xi_{i} = 0\), \(\Xi_{zi} = 0\), i = 1,2)

[21]

Fail to justisfy the stability (when zero state feedback and \(\Xi_{i} = 0\), \(\Xi_{zi} = 0\), i = 1,2)

[18]

τ = 12, \(H_{\infty} \) performance γ = 0.2546, \(\bar{\Omega}_{1} = \{ [ x_{1} \ x_{2} ]^{T} \in R^{2}:0.244x_{1}^{2} - 0.0038x_{1}x_{2} - 0.4426x_{2}^{2} \le 0 \}\)

339 (Program running time about 1 minute)

Our results (Theorem 1)

τ = 12, \(H_{\infty} \) performance γ = 0.2512, \(\bar{\Omega}_{1} = \{ [ x_{1} \ x_{2} ]^{T} \in \Re^{2}: - 0.0327x_{1}^{2} + 0.000594x_{1}x_{2} + 0.07687x_{2}^{2} \ge 0 \}\)

31 (Program running time about 1 second)

τ = 316, \(H_{\infty} \) performance γ = 0.254, \(\bar{\Omega}_{1} = \{ [ x_{1} \ x_{2} ]^{T} \in \Re^{2}: - 0.0487x_{1}^{2} - 0.00194x_{1}x_{2} + 0.0865x_{2}^{2} \ge 0 \}\)