×

zbMATH — the first resource for mathematics

Observability and related structural results for linear hereditary systems. (English) Zbl 0531.93015
The class of systems considered in this paper is assumed to be modeled by the equations \(\dot x(t)=A(d)x(t)+B(d)u(t),\quad y(t)=C(d)x(t)\) where A(d), B(d), and C(d) are polynomial matrices in the delay operator d with the property that \(dx(t)=x(t-h),\) where \(h>0\) is the delay duration.
For this class of systems a number of different concepts of observability and observers are discussed. The paper successfully unifies results obtained by applying three different approaches towards the observability problem for linear time-invariant systems with delays. Thus the results which arose from a purely algebraic approach, results which were generated by an application of functional analysis techniques, and results which arose from the abstract semigroup approach are nicely bridged together in this paper. A natural consequence of the analysis of the observability problem is the observer design methodology. Design algorithms for different types of observers for time delay systems are developed. These are: finite-time observers, asymptotic observers, and optimal observers. The results have been established only for the case of systems with commensurable delays; however, most of the results of this paper can be generalized for systems with non-commensurable delays and for systems with distributed delays.
Reviewer: S.H.Żak

MSC:
93B07 Observability
34K35 Control problems for functional-differential equations
93B25 Algebraic methods
34K30 Functional-differential equations in abstract spaces
93C05 Linear systems in control theory
93C25 Control/observation systems in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] ASMYKOVICH I. K., Avtomatika Telemekh 7 pp 5– (1976)
[2] BHAT K. P. M., I.E.E.E. Trans. autom. Control 21 pp 233– (1976) · doi:10.1109/TAC.1976.1101167
[3] BHAT , K. P. M. , and WONHAM , W. M. , 1976 , Report Dept. of Elec. Eng. , Univ. of Toronto .
[4] DELFOUR M., SIAM J. Control 10 pp 298– (1972) · Zbl 0242.93011 · doi:10.1137/0310023
[5] GABASOV R. F., Control Inf. Theory 1 pp 217– (1972)
[6] GRESSANG R. V., Proc. 12th Allerton Conf. Circ. Syst. Theory (1974)
[7] GRESSANG R. V., I.E.E.E. Trans. autom. Control 20 pp 523– (1975) · Zbl 0315.93010 · doi:10.1109/TAC.1975.1101021
[8] HALE J., Functional Differential Equations (1971) · Zbl 0222.34003 · doi:10.1007/978-1-4615-9968-5
[9] HAUTUS M. L. J., Proc. Harvard Conf. on Algebraic Methods (1979)
[10] HENRY D., J. Diff. Eqs. 8 pp 494– (1970) · Zbl 0228.34044 · doi:10.1016/0022-0396(70)90021-5
[11] HEWER G. A., Int. J. Control 18 pp 1– (1973) · Zbl 0266.93006 · doi:10.1080/00207177308932482
[12] KAMEN E. W., Math. Syst. Theory 9 pp 57– (1975) · Zbl 0318.93003 · doi:10.1007/BF01698126
[13] KOCIECKI M., Control and. Cyber (1981)
[14] KOIVO H., Inf. Control 19 pp 232– (1971) · Zbl 0239.93021 · doi:10.1016/S0019-9958(71)90115-X
[15] KOPEIKINA T. B., Differen. Uravnenia 10 pp 933– (1974)
[16] KRASOVSKII N. N., Theory of Control Motion (1968)
[17] KRASOVSKII N. N., Differen. Uravnenia 2 pp 299– (1966)
[18] KWONG , R. , 1975 , Doctoral Dissertation , Massachussetss Institute of Technology ; 1980,SIAM J. Control and Optim.,18, 49 .
[19] LEE , E. B. , 1976 ,Calculus of Variations and Control Theory( London : Academic Press ), p. 47 ; 1979,Class Notes on Infinite Dimensional Systems(Univ. of Minnesota) .
[20] LEE E. B., Foundations of Optimal Control Theory (1967) · Zbl 0159.13201
[21] LEE E. B., Proc. 19th I.E.E.E. CDC pp 744– (1980)
[22] LEE , E. B. , and OLBROT , A. , 1980 , Proc. 19th Conf. Inf. Sci. and Syst. , p. 344 .
[23] LINDQUIST A., J. Math. Anal. Applic. 37 pp 516– (1972) · Zbl 0208.17603 · doi:10.1016/0022-247X(72)90293-4
[24] MANITIUS A., Proc. 15th I.E.E.E. CDC (1976)
[25] MANITIUS A., I.E.E.E. Trans. autom. Control 24 pp 541– (1979) · Zbl 0425.93029 · doi:10.1109/TAC.1979.1102124
[26] MANITIUS A., Proc. 15th I.E.E.E. CDC (1976)
[27] METELSKII A. W., Differen. Uravnenia 14 pp 624– (1978)
[28] MORSE A. S., Automatica 12 pp 529– (1976) · Zbl 0345.93023 · doi:10.1016/0005-1098(76)90013-3
[29] OLBROT A. W., Control and Cyber 4 pp 71– (1975)
[30] OLBROT A., W., Proc. Conf. Functional Differential Systems and Related Topics (1979)
[31] PANDOLFI L., J. Optim. Theory Appl. 20 pp 191– (1976) · Zbl 0313.93023 · doi:10.1007/BF01767451
[32] ROLEWICZ S., Studia Mathematics 44 pp 411– (1972)
[33] SIKORA A., Int. Conf. on Functional-Differential Systems and Related Topics (1979)
[34] SALAMON D., I.E.E.E. Trans. autom. Control (1981)
[35] SONTAG E. D., Ricerche di Automatica 7 pp 1– (1976)
[36] VINTER R. B., SIAM J. Control and Optimization 19 pp 139– (1981) · Zbl 0465.93043 · doi:10.1137/0319011
[37] ZAKIAN V., Control Centre Report (1972)
[38] ZHERNIAK R. M., Differen. Uravnenia 6 pp 1519– (1970)
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.