A multistage reduction technique for feedback stabilizing distributed time-lag systems. (English) Zbl 0629.93046

This paper presents a method which can be used to stabilize a system with state and control delays of the form: \[ \dot x=\int^{0}_{-h}d\alpha (s)x(t+s)+\int^{0}_{-h}d\beta (s)u(t+s). \] A preliminary requirement for this method is that the equation \(A=\int^{0}_{-h}\exp (As)d\alpha (s)\) can be solved with respect to the unknown matrix A. Let \(\Gamma\) be the set of the matrices A which solve this last equation. Then, it is proved that \(\sigma\) (A) is contained in the spectrum of the delay system for each \(A\in \Gamma\). Moreover, a special transformation can be applied to the pairs (x,u) (which solve the delay system) so that the transformed pair is solution of an ordinary differential equation. It is shown, under suitable assumptions, that a feedback which stabilizes this last equation can be used for the task of stabilizing the delay system.
Reviewer: L.Pandolfi


93D15 Stabilization of systems by feedback
34K35 Control problems for functional-differential equations
93C05 Linear systems in control theory
93B17 Transformations
93B55 Pole and zero placement problems
93C25 Control/observation systems in abstract spaces
Full Text: DOI


[1] Balas, M. J., Toward a more practical control theory for distributed parameter systems, (Leondes, C. T., Control and Dynamic Systems, Advances in Theory and Applications (1982), Academic Press: Academic Press New York) · Zbl 0523.93051
[2] Banks, H. T.; Rosen, I. G.; Ito, K., A spline based technique for computing Riccati operators and feedback controls in regulator problems for delay equations, SIAM J. Sci. Stat. Comput., 5, 830 (1984) · Zbl 0553.65047
[3] Bhat, K. P.M.; Koivo, N. H., Modal characterizations of controllability and observability in time delay systems, IEEE Trans. Aut. Control, AC-21, 292-293 (1976) · Zbl 0325.93005
[4] Crocco, L., Aspects of combustion stability in liquid propellant rocket motors. Part I: Fundamentals—low frequency instability with monopropellants, J. Amer. Rocket Soc., 21, 163-178 (1951)
[5] Curtain, R. F.; Pritchard, A. J., (Infinite Dimensional Linear Systems Theory (1978), Springer: Springer New York) · Zbl 0389.93001
[6] Davis, P. J., (Circulant Matrices (1979), Wiley: Wiley New York) · Zbl 0418.15017
[7] Delfour, M. C., Linear optimal control of systems with state and control variable delays, Automatica, 20, 69 (1984) · Zbl 0541.93042
[8] El’sgol’ts, L. E.; Norkin, S. B., (Introduction to the Theory and Application of Differential Equations with Deviating Arguments (1973), Academic Press: Academic Press New York) · Zbl 0287.34073
[9] Fiagbedzi, Y. A., Stabilization of a class of autonomous differential delay systems, (Ph.D. Thesis (1985), Brown University: Brown University Providence, Rhode Island) · Zbl 1002.93027
[10] Fiagbedzi, Y. A.; Pearson, A. E., Feedback stabilization of state delayed systems via a reducing transformation, (Proc. IEEE Conf. on Decision and Control, 1 (1985)), 128-129 · Zbl 0709.93059
[11] Fiagbedzi, Y. A.; Pearson, A. E., Feedback stabilization of autonomous time lag systems, IEEE Trans. Aut. Control, AC-31, 847-855 (1986) · Zbl 0601.93045
[12] Fiagbedzi, Y. A.; Pearson, A. E., A finite dimensional approach to the feedback stabilization of linear distributed time lag systems, (Preprint, 1986 IFAC Symposium on Control of Distributed Parameter Systems (1986), UCLA: UCLA California) · Zbl 0629.93046
[13] Gibson, J. S., Linear quadratic optimal control of hereditary differential systems: infinite dimensional Riccati equations and numerical approximations, SIAM J. Control Optim., 21, 95 (1983) · Zbl 0557.49017
[14] Hale, J. K., (Theory of Functional Differential Equations (1977), Springer: Springer New York) · Zbl 0352.34001
[15] Ito, K., On the approximation of eigenvalues associated with functional differential equations, (ICASE Report (1983), NASA Langley Research Center), 82-89
[16] Kailath, T., (Linear Systems (1980), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey) · Zbl 0458.93025
[17] Krasovskii, N. N., (Stability of Motion (1963), Translation, Stanford University Press: Translation, Stanford University Press Stanford, California) · Zbl 0109.06001
[18] SIAM J. Sci. Stat. Comput., 8, 3 (1987), May 1987 · Zbl 0624.65078
[19] Olbrot, A. W., Stabilizability, detectability and spectrum assignment for linear autonomous systems with general time delays, IEEE Trans. Aut. Control, AC-23, 887-890 (1978) · Zbl 0399.93008
[20] Pandolfi, L., Stabilization of neutral functional differential equations, J. Opt. Theory Application, 20, 191 (1976) · Zbl 0313.93023
[21] Pandolfi, L., On the zeroes of transfer functions of delayed systems, System Control Lett., 1, 204 (1981) · Zbl 0472.93018
[22] Sikora, A.; Kociecki, M., Numerical evaluation of roots of quasi-polynomials, (Abstracts of the International Conference on Functional Differential Systems and Related. Abstracts of the International Conference on Functional Differential Systems and Related, Topics, Blazejewko, Poland (1979), Institute of Mathematics, Polish Academy of Sciences) · Zbl 0438.65072
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