\(\mathcal D\)-stability for linear continuous-time systems with multiple time delays. (English) Zbl 1133.34365
Summary: This paper addresses the \(\mathcal D\)-stability problem of linear continuous-time systems with multiple time delays. A sufficient condition to guarantee all poles of a given linear time-delay system located in the specified region is proposed. This condition can be solved using LMI toolbox. The result is also extended to the robust \(\mathcal D\)-stability problem of perturbed linear time-delay systems. The illustrative example shows that this method is effective to check the \(\mathcal D\)-stability of linear time-delay systems.
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 34K06 | Linear functional-differential equations |
| 93D09 | Robust stability |
Keywords:
linear continuous-time systems; multiple time delays; \(\mathcal D\)-stability; unstructured perturbation; linear matrix inequality (LMI)Software:
LMI toolboxReferences:
| [1] | Bellman, R.; Cooke, K. L., Differential-difference equations (1963), Academic Press: Academic Press London · Zbl 0115.30102 |
| [2] | Chilali, M.; Gahinet, P., \(H_\infty\) design with pole placement constraints: An LMI approach, IEEE Transactions on Automatic Control, 41, 3, 358-367 (1996) · Zbl 0857.93048 |
| [3] | Chilali, M.; Gahinet, P.; Apkarian, P., Robust pole placement in LMI regions, IEEE Transactions on Automatic Control, 44, 12, 2257-2270 (1999) · Zbl 1136.93352 |
| [4] | Lee, C.-H., D-stability of continuous time-delay systems subjected to a class of highly structured perturbations, IEEE Transactions on Automatic Control, 40, 10, 1803-1807 (1995) · Zbl 0841.93062 |
| [5] | Niculescu, S.-I. (2001). Delay effects on stability: A robust control approach. In Lecture notes in control and information sciences; Niculescu, S.-I. (2001). Delay effects on stability: A robust control approach. In Lecture notes in control and information sciences · Zbl 0997.93001 |
| [6] | Niculescu, S.-I.; de Souza, C. E.; Dugard, L.; Dion, J.-M., Robust exponential stability of uncertain systems with time-varying delays, IEEE Transactions on Automatic Control, 43, 5, 743-748 (1998) · Zbl 0912.93053 |
| [7] | Su, J.-H.; Fong, I.-K.; Tseng, C.-L., Stability analysis of linear systems with time delay, IEEE Transactions on Automatic Control, 39, 6, 1341-1344 (1994) · Zbl 0812.93061 |
| [8] | Sun, Y.-J.; Lien, C.-H.; Hsieh, J.-G., Comments on “D-stability of continuous time-delay systems subjected to a class of highly structured perturbations”, IEEE Transactions on Automatic Control, 43, 5, 689 (1998) · Zbl 0912.93059 |
| [9] | Tissir, E.; Hmamed, A., Further results on stability of \(\dot{x}(t) = Ax(t) + Bx(t - \tau)\), Automatica, 32, 12, 1723-1726 (1996) · Zbl 0869.34061 |
| [10] | Zhang, Y.; Wang, Q.-G.; Astrom, K. J., Dominant pole placement for multi-loop control systems, Automatica, 38, 7, 1213-1220 (2002) · Zbl 1006.93034 |
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