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The effect of small delays in the feedback loop on the stability of neutral systems. (English) Zbl 0866.93089

Summary: It is well-known that exponential stabilization of a neutral system with unstable difference operator is only possible by allowing for control laws containing derivative feedback. We show that closed-loop stability of a neutral system with unstable open-loop difference operator obtained by applying a derivative feedback scheme is extremely sensitive to arbitrarily small time delays in the feedback loop.

MSC:

93D15 Stabilization of systems by feedback
34K35 Control problems for functional-differential equations
Full Text: DOI

References:

[1] Barman, J. F.; Callier, F. M.; Desoer, C. A., \(L^2\)-stability and \(L^2\)-instability of linear time-invariant distributed feedback systems perturbed by a small delay in the loop, IEEE Trans. Automat. Control, 18, 479-484 (1973) · Zbl 0303.93049
[2] Byrnes, C. I.; Spong, M. W.; Tarn, T. J., A several complex variables approach to feeback stabilization of neural-delay differential equations, Math. Systems Theory, 17, 97-133 (1984) · Zbl 0539.93064
[3] Carvalho, L. A.V., An analysis of the characteristic equation of the scalar linear difference equation with two delays, (Izé, A. F., Functional Differential Equations and Bifurcation (1980), Springer: Springer Berlin), 69-81 · Zbl 0438.34060
[4] Corduneanu, C., Almost Periodic Functions (1968), Interscience Publishers: Interscience Publishers New York · Zbl 0175.09101
[5] Datko, R., Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26, 697-713 (1988) · Zbl 0643.93050
[6] Datko, R., Two examples of ill-posedness with respect to small time delays in stabilized elastic systems, IEEE Trans. Automat. Control, 38, 163-166 (1993) · Zbl 0775.93184
[7] Datko, R.; Lagnese, J.; Polis, M. P., An example of the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24, 152-156 (1986) · Zbl 0592.93047
[8] Desch, W.; Wheeler, R. L., Destabilization due to delay in one-dimensional feedback, (Kappel, F.; Kunisch, K.; Schappacher, W., Control and Estimation of Distributed Parameter Systems (1989), Birkhäuser: Birkhäuser Boston), 61-83 · Zbl 0686.93074
[9] Hale, J. K., Parametric stability in difference equations, Bolle. U.M.I., 11, 209-214 (1975), (Suppl. fasc. 3) · Zbl 0318.39004
[10] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer: Springer New York · Zbl 0787.34002
[11] Henry, D., Linear autonomous neutral functional differential equations, J. Differential Equations, 15, 106-128 (1974) · Zbl 0294.34047
[12] Kamen, E. W.; Khargonekar, P. P.; Tannenbaum, A., Pointwise stability and feedback control of linear systems with noncommensurate time-delays, Acta Appl. Math., 2, 159-184 (1984) · Zbl 0561.93045
[13] Kolmogorov, A. N.; Fomin, S. V., Introductory Real Analysis (1975), Dover: Dover New York · Zbl 0213.07305
[14] Levin, B. J., Distribution of Zeros of Entire Functions, (Translations of Math. Monographs, Vol. 5 (1964), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0022.43106
[15] Logemann, H., On the existence of finite-dimensional compensators for retarded and neutral systems, Int. J. Control, 43, 109-121 (1986) · Zbl 0641.93045
[16] Logemann, H., On the transfer matrix of a neutral system: characterizations of exponential stability in input-output terms, Systems Control Lett., 9, 393-400 (1987) · Zbl 0629.93054
[17] Logemann, H.; Pandolfi, L., A note on stability and stabilizability of neutral systems, IEEE Trans. Automat. Control, 39, 138-143 (1994) · Zbl 0825.93676
[18] Logemann, H.; Rebarber, R., The effect of small time-delays on the stability of feedback systems described by partial differential equations, (Mathematics \(Preprint^3, 95 (1995)\), University of Bath), submitted · Zbl 0928.93023
[19] Logemann, H.; Rebarber, R.; Weiss, G., Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop, SIAM J. Control Optim., 34, 572-600 (1996) · Zbl 0853.93081
[20] Melvin, W. R., Stability properties of functional difference equations, J. Math. Analysis Appl., 48, 749-763 (1974) · Zbl 0311.39002
[21] O’Connor, D.; Tarn, T. J., On stabilization by state feedback for neutral differential difference equations, IEEE Trans. Automat. Control, 28, 615-618 (1983) · Zbl 0527.93049
[22] Salamon, D., Control and Observation of Neutral Systems (1984), Pitman: Pitman London · Zbl 0546.93041
[23] Salamon, D., Realization theory in Hilbert space, Math. Systems Theory, 21, 147-164 (1989) · Zbl 0668.93018
[24] Spong, M. W., A theorem on neutral delay systems, Systems Control Lett., 6, 291-294 (1985) · Zbl 0573.93048
[25] Weiss, G., Transfer functions of regular linear systems, part I: characterization of regularity, Trans. Amer. Math. Soc., 342, 827-854 (1994) · Zbl 0798.93036
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