×

Stability of periodically switched linear systems and the switching frequency. (English) Zbl 0619.93057

Several propositions are made concerning the stability of a repetitively switched linear time-invariant system with periodically variable structure. Particularly, relationships between the system’s stability and the one of a composite matrix and the switching frequency are shown. It has been proved that at large values of the switching frequency the stability of the composite matrix determines the stability for the linear system.
In practical applications it is of interest to estimate the interval of the switching frequency inside of which the system is stable or not. Unfortunately, the appropriate results are limited to diagonalizable composite matrices only. A more precise estimation of the interval of the switching frequency may be obtained by taking into account higher order terms with respect to frequency. However, this procedure requires complicated calculations.
The problem of finding for the system practical stability criteria without evaluation of exponential matrices remains as a challenge for future research. Such criteria should also be applicable at low and middle switching frequencies.
The paper closes with illustrative examples one of which shows a behaviour of a sampled-data system the continuous transition of the system from a stable trajectory onto an unstable one caused only by a step change of the input and of the switching frequency.
Reviewer: H.Fischer

MSC:

93D20 Asymptotic stability in control theory
93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C35 Multivariable systems, multidimensional control systems
93C57 Sampled-data control/observation systems
93D15 Stabilization of systems by feedback

Keywords:

time-dependent
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] ACKERMANN J., Abtastregelung Sampled-Data Control (1972) · Zbl 0254.93026
[2] ARUT A. O., Theory of Group Representations and Applications (1977)
[3] DOI: 10.1007/BF01386217 · Zbl 0101.25503 · doi:10.1007/BF01386217
[4] BELLMAN R., Stability Theory of Differential Equations (1953)
[5] ENGELKING R., General Topology (1975)
[6] HELGASON S., Differential Geometry and Symmetric Spaces (1962) · Zbl 0111.18101
[7] DOI: 10.1137/0316047 · Zbl 0388.49025 · doi:10.1137/0316047
[8] DOI: 10.1002/cta.4490050208 · Zbl 0379.94045 · doi:10.1002/cta.4490050208
[9] LANCASTER R., Theory of Matrices (1969) · Zbl 0186.05301
[10] DOI: 10.1109/TAES.1973.309702 · doi:10.1109/TAES.1973.309702
[11] LEE F. C., I.E.E.E. Trans. Jnd. Applic 15 pp 511– (1972)
[12] LIOU M. L., I.E.E.E. Trans. Circuit Theory 19 pp 142– (1972) · doi:10.1109/TCT.1972.1083422
[13] DOI: 10.1109/TCS.1979.1084633 · Zbl 0399.94018 · doi:10.1109/TCS.1979.1084633
[14] MIDDLEBROOK , H. , and CUK , S. , 1977 ,I.E.E.E. Int. Semiconductor Power Conf. Records, 90 .
[15] POSTNIKOW M. M., Lie Groups and Lie Algebras (1982)
[16] DOI: 10.1109/TAES.1983.309438 · doi:10.1109/TAES.1983.309438
[17] SIKORSKI R., Real Functions I (1957)
[18] DOI: 10.1109/TCS.1977.1084268 · doi:10.1109/TCS.1977.1084268
[19] DOI: 10.1137/0317029 · Zbl 0439.93041 · doi:10.1137/0317029
[20] STOER J., Einfurung in die Numerische Mathematik, II (1974)
[21] DOI: 10.1109/TCS.1977.1084274 · Zbl 0364.94062 · doi:10.1109/TCS.1977.1084274
[22] TOKARZEWSKI J., Arch, autom. Telemech 35 pp 145– (1983)
[23] WILKINSON J. H., The Algebraic Eigenvalue Problem (1965) · Zbl 0258.65037
[24] WONHAM W. M., Linear Multivariate Control a Geometric Approach (1979) · Zbl 0424.93001 · doi:10.1007/978-1-4684-0068-7
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.