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Delayed feedback control of uncertain systems with time-varying input delay. (English) Zbl 1072.93023

A stabilization problem is investigated for the linear control system \[ \dot{x}(t)=(A+\Delta A(t)) x(t)+(B_{0}+\Delta B_{0}(t) )u(t)+(B_{1}+\Delta B_{1}(t) )u(t-\tau(t)),\quad t\geq 0, \]
\[ x(0)=x_{0}\in \mathbb R^{n},\qquad u(t)=\Phi(t)\in C([-\tau,0]; \mathbb R^{m}), \quad t\leq 0, \] with a feedback control \[ u(t)=K[x(t)+\int_{t-\tau_{0}}^{t} \exp\{A(t-s-\tau_{0}) \}B_{1}u(s)\,ds] \in \mathbb R^{m}. \] Here \(\Delta A(t),\Delta B_{0}(t)\) , \(\Delta B_{1}(t)\) are uncertain matrices satisfying \[ [\Delta A(t)\;\Delta B_{0}(t)\;\Delta B_{1}(t)]=DF(t)[E\;E_{0}\;E_{1}], \] where \(F(t)\) is an unknown measurable matrix function and satisfies \(F^{T}(t)F(t)\leq I\). \(\tau (t)\in [0,\tau] \) is the delay in controls. Stabilization criteria are established for cases when \(\tau(t)\in C([0,\infty);[\tau_{0}-\delta ,\tau_{0}+\delta]) \) or \(\tau(t)\in C^{1}([0,\infty);[\tau_{0}-\delta ,\tau_{0}+\delta])(\dot{\tau}(t)\leq d<1 )\), where \(\tau_{0},\delta \) and \(d\) are known constants, moreover \([\tau_{0}-\delta ,\tau_{0}+\delta]\subset [0,\tau] \).

MSC:

93D15 Stabilization of systems by feedback
93C23 Control/observation systems governed by functional-differential equations
93D21 Adaptive or robust stabilization
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[1] Cheres, E.; Palmor, Z. J.; Gutman, S., Min-max predictor control for uncertain systems with input delays, IEEE Transactions on Automatic Control, 35, 210-214 (1990) · Zbl 0705.93059
[2] Choi, H. H.; Chung, M. J., Observer-based \(H_\infty\) controller design for state delayed linear systems, Automatica, 32, 1073-1075 (1996) · Zbl 0850.93215
[3] Fiagbedzi, Y. A.; Pearson, A. E., Feedback stabilization of linear autonomous time lag systems, IEEE Transactions on Automatic Control, 31, 855-857 (1986) · Zbl 0601.93045
[4] Gu, K.; Niculescu, S.-I., Additional dynamics in transformed time-delay systems, IEEE Transactions on Automatic Control, 45, 497-500 (2000)
[5] Hollot, C. V.; Misra, V.; Towsley, D.; Gong, W. B., Analysis and design of controllers for AQM routers supporting TCP flows, IEEE Transactions on Automatic Control, 47, 945-959 (2002) · Zbl 1364.93279
[6] Kim, J. H.; Jeung, E. T.; Park, H. B., Robust control for parameter uncertain delay systems in state and control input, Automatica, 32, 1337-1339 (1996) · Zbl 0855.93075
[7] Kim, D. S.; Lee, Y. S.; Kwon, W. H.; Park, H. S., Maximum allowable delay bounds of networked control systems, Control Engineering Practice, 11, 1301-1313 (2003)
[8] Kolmanovskii, V.; Myshkis, A., Applied theory of functional differential equations (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0917.34001
[9] Kwon, W. H.; Pearson, A. E., Feedback stabilization of linear systems with delayed control, IEEE Transactions on Automatic Control, 25, 266-269 (1980) · Zbl 0438.93055
[10] Luo, R. C.; Chung, L.-Y., Stabilization for linear uncertain system with time latency, IEEE Transactions on Industrial Electronics, 49, 905-910 (2002)
[11] Moon, Y. S.; Park, P.; Kwon, W. H., Robust stabilization of uncertain input-delayed systems using reduction method, Automatica, 37, 307-312 (2001) · Zbl 0969.93035
[12] Niculescu, S.-I., Delay effects on stability—a robust control approach (2001), Springer: Springer London
[13] 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, 743-748 (1998) · Zbl 0912.93053
[14] Park, H. S.; Kim, Y. H.; Kim, D. S.; Kwon, W. H., A scheduling method for network based control systems, IEEE Transactions on Control Systems Technology, 10, 318-330 (2002)
[15] Yue, D., Robust stabilization of uncertain systems with unknown input delay, Automatica, 40, 331-336 (2004) · Zbl 1034.93058
[16] Zhang, W.; Branicky, M. S.; Phillips, S. M., Stability of networked control systems, IEEE Control Systems Magazine, 21, 84-99 (2001)
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