zbMATH — the first resource for mathematics

An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type. (English) Zbl 1091.93026
Summary: An eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented, which extends the recently developed continuous pole placement method for delay equations of retarded type. The approach consists of two steps. First the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots, which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation. No restrictions are made on the dimension of the system and the number of delays. Also the effect of small delay perturbations is explicitly taken into account. As a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots. As a second step, stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi-continuous way, by applying small changes to the controller parameters, and meanwhile monitoring other characteristic roots with a large real part. A numerical example is presented.

93D15 Stabilization of systems by feedback
93C23 Control/observation systems governed by functional-differential equations
34K35 Control problems for functional-differential equations
93B55 Pole and zero placement problems
Full Text: DOI
[1] Bellen, A.; Guglielmi, N.; Ruehli, A.E., Methods for linear systems of circuit delay differential equations of neutral type, IEEE transactions on circuits and systems I, 76, 1, 212-215, (1999) · Zbl 0952.94015
[2] Emre, E.; Khargonekar, P.P., Regulation of split linear systems over ringscoefficient assignment and observers, IEEE transactions on automatic control, 27, 1, 104-113, (1982) · Zbl 0502.93019
[3] Engelborghs, K.; Dambrine, M.; Roose, D., Limitations of a class of stabilizing methods for delay equations, IEEE transactions on automatic control, 46, 2, 336-339, (2001) · Zbl 1056.93607
[4] Engelborghs, K.; Roose, D., Bifurcation analysis of periodic solutions of neutral functional differential equationsa case study, International journal of bifurcation and chaos, 8, 10, 1889-1905, (1998) · Zbl 0941.34070
[5] Hale, J.K., Effects of delays on dynamics, (), 191-238 · Zbl 0834.34084
[6] Hale, J. K., & Verduyn Lunel, S. M. (1993). Introduction to functional differential equations, Applied Mathematical Sciences, Vol. 99. New York: Springer. · Zbl 0787.34002
[7] Hale, J.K.; Verduyn Lunel, S.M., Strong stabilization of neutral functional differential equations, IMA journal of mathematical control and information, 19, 5-23, (2002) · Zbl 1005.93026
[8] Hale, J.K.; Verduyn Lunel, S.M., Effects of small delays on stability and control, (), 275-301 · Zbl 0983.34070
[9] Hale, J.K.; Verduyn Lunel, S.M., Stability and control of feedback systems with time delays, International journal of systems science, 34, 8-9, 497-504, (2003) · Zbl 1052.93028
[10] Henry, D., Linear autonomous neutral functional differential equations, Journal of differential equations, 15, 106-128, (1974) · Zbl 0294.34047
[11] Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, (1986), Academic Press London · Zbl 0593.34070
[12] Lu, W.-S.; Lee, E.; Zak, S., On the stabilization of linear neutral delay-difference systems, IEEE transactions on automatic control, 31, 1, 65-67, (1986) · Zbl 0595.93050
[13] Luzyanina, T.; Roose, D., Equations with distributed delaysbifurcation analysis using computational tools for discrete delay equations, Functional differential equations, 11, 1-2, 87-92, (2004) · Zbl 1064.34057
[14] Michiels, W.; Engelborghs, K.; Roose, D.; Dochain, D., Sensitivity to infinitesimal delays in neutral equations, SIAM journal of control and optimization, 40, 4, 1134-1158, (2002) · Zbl 1016.34079
[15] Michiels, W.; Engelborghs, K.; Vansevenant, P.; Roose, D., Continuous pole placement method for delay equations, Automatica, 38, 5, 747-761, (2002) · Zbl 1034.93026
[16] Morse, A.S., Ring models for delay differential equations, Automatica, 12, 5, 529-531, (1976) · Zbl 0345.93023
[17] Murray, R. M., Jacobson, C. A., Casas, R., Khibnik, A. I., Johnson Jr., C. R., Bitmead, R., Peracchio, A. A., & Proscia, W. M. (1998). System identification for limit cycling systems: A case study for combustion instabilities. In Proceedings of the 1998 American control conference (pp. 2004-2008).
[18] Niculescu, S.-I.; Brogliato, B., Fore measurements time-delays and contact instability phenomenon, European journal of control, 5, 279-289, (1999) · Zbl 0936.93031
[19] Pandolfi, L., Stabilization of neutral functional differential equations, Journal of optimization theory and applications, 20, 191-204, (1976) · Zbl 0313.93023
[20] Sename, O.; Lafay, J.-F.; Rabah, R., Controllability indices of linear systems with delays, Kybernetika, 6, 559-580, (1995) · Zbl 0864.93023
[21] Vyhlídal, T., & Zítek, P. (2003). Quasipolynomial mapping based rootfinder for analysis of time delay systems. In 4th IFAC workshop on time delay systems, Rocquencourt, France (pp. 146-151).
[22] Watanabe, K., Finite spectrum assignment and observer for multivariable systems with commensurate delays, IEEE transactions on automatic control, 31, 6, 543-550, (1995) · Zbl 0596.93009
[23] Zítek, P., & Vyhlídal, T. (2000). State feedback control of time delay system: Conformal mapping aided design. In 2nd IFAC workshop on linear time delay systems, Ancona, Italy (pp. 146-151).
[24] Zítek, P., & Vyhlídal, T. (2002). Dominant eigenvalue placement for time delay systems. In 5th Portuguese conference on automatic control, Aveiro (pp. 605-610).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.