Robust stability of uncertain time-delay systems. (English) Zbl 0629.93051

The paper presents some problems connected with robustness of time delay nominally linear uncontrolled systems perturbed by nonlinear conic bounded uncertain functions. The authors present four theorems which give sufficient conditions for robust stability of these systems. The first condition is derived for single systems with a single time delay and is expressed in terms of norms of the known system matrices and bounds for uncertainties. The line of reasoning is based on the Bellman-Gronwall inequality, and an exponential type bound for the trajectory of the system is given. The condition is independent of the delay, thus it is applicable to perturbed time delay systems without exact knowledge of the delay. The second theorem deals with uncertain composite time delay systems consisting of subsystems with the same delay. It expresses robust stability conditions in terms of norms as well as other matrix measures. The condition is once more delay-independent and, as in the first theorem, reveals an intuitive relation between the degree of stability of each subsystem and the magnitudes of the delay and perturbation terms. The third theorem presents a delay dependent condition for the single time delay system subjected to parametric perturbations. As before the condition permits the assessment of the transient behaviour of the trajectory. The last theorem extends the results of the first theorem to the case of uncertain systems with multiple noncommensurate time delays. The condition is once more delay-independent, thus may be applied to systems with time delays not known exactly. This condition requires the index of stability for the nominal systems without delay to be larger than the sum of the magnitudes of the delay terms and the perturbation values. The results presented in the paper are simple enough to be applied as introductory tests in the design of robust control systems.
Reviewer: A.Swierniak


93D20 Asymptotic stability in control theory
34K20 Stability theory of functional-differential equations
93B35 Sensitivity (robustness)
26D10 Inequalities involving derivatives and differential and integral operators
34D10 Perturbations of ordinary differential equations
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
Full Text: DOI


[1] DOI: 10.1109/TAC.1986.1104106 · Zbl 0603.93036
[2] COPPEL W. A., Stability and Asymptotic Behaviour of Differential Equations (1965) · Zbl 0154.09301
[3] DESOER C. A., Feedback System: Input–Output Properties (1975) · Zbl 0327.93009
[4] DOI: 10.1109/TAC.1981.1102555 · Zbl 0462.93027
[5] DOI: 10.1109/TAC.1981.1102653 · Zbl 0477.93048
[6] DOI: 10.1109/TAC.1980.1102444 · Zbl 0493.93044
[7] DOI: 10.1109/TAC.1979.1102025 · Zbl 0399.93037
[8] DOI: 10.1109/TAC.1980.1102288 · Zbl 0438.93055
[9] DOI: 10.1109/TAC.1981.1102755 · Zbl 0544.93052
[10] DOI: 10.1080/00207178608933535 · Zbl 0588.93056
[11] DOI: 10.1080/00207178408933243
[12] DOI: 10.1109/TAC.1985.1103901 · Zbl 0557.93058
[13] DOI: 10.1080/00207178108922590 · Zbl 0471.93054
[14] DOI: 10.1016/0005-1098(83)90013-4 · Zbl 0544.93055
[15] DOI: 10.1080/0020718508961174 · Zbl 0566.93048
[16] DOI: 10.1080/00207178208932894 · Zbl 0482.93046
[17] DOI: 10.1016/0005-1098(79)90021-9 · Zbl 0407.93033
[18] DOI: 10.1109/TAC.1985.1103804 · Zbl 0565.93047
[19] DOI: 10.1080/00207728108963833 · Zbl 0504.93049
[20] DOI: 10.1109/TAC.1983.1103258 · Zbl 0506.93027
[21] DOI: 10.1080/00207178308933069 · Zbl 0511.93040
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.