New sufficient conditions for the stability of continuous and discrete time-delay interval systems
Stability for time-delay interval systems is investigated. Several new delay-independent sufficient conditions are given both for the continuous and the discrete case. The results are illustrated by some numerical examples. A typical continuous time delay system is \( \dot x(t)=Ax(t)+Bx(t-\tau) \) where \(x(\cdot)\in \mathbb{R}^n\) and \(\tau>0\). The interval property means that the matrices \(A\) and \(B\) satisfy: \[ A^-=[a_{ij}^-]\leq A=[a_{ij}]\leq A^+=[a_{ij}^+]\quad (i.e.\quad a_{ij}^-\leq a_{ij}\leq a_{ij}^+) \] \[ B^-=[b_{ij}^-]\leq B=[b_{ij}]\leq B^+=[b_{ij}^+]\quad (i.e.\quad b_{ij}^-\leq b_{ij}\leq b_{ij}^+) \] The stability is investigated by means of the roots of the characteristic equation \( \text{det}(sI-A-Be^{-s\tau})=0 \) and uses the so called Gerschgorin theorem. A typical result is the following one. The considered system is stable if the following condition holds: \[ \min\left\{ \max_i\sum_{j=1}^n(g_{ij}+f_{ij}), \max_i\sum_{j=1}^n(g_{ji}+f_{ji})\right\}<0 , \] with \(G=[g_{ij}]\), \(g_{ii}=a_{ii}^+\), \(g_{ij}=\max(|a_{ij}|,|a_{ij}^+|)\) for \(i\neq j\) and \(F=[f_{ij}]\), \(f_{ij}=\max(|b_{ij}|,|b_{ij}^+|)\).
scientific article; zbMATH DE number 997657
scientific article