Robust control by two Lyapunov functions. (English) Zbl 0751.93019

Summary: A new method of designing a robust control law is proposed for a general class of nonlinear or linear systems with bounded uncertainties. The method uses the property that the Lyapunov function is not unique for a stable or stabilizable system. It is shown that the proposed control law normally guarantees the stability of the system if there are two Lyapunov functions whose null sets have a trivial intersection. The null set of a Lyapunov function is defined to be the set in state space in which the product of the transpose of the system input matrix and the gradient of the Lyapunov function is equal to zero. The robust control results impose no restriction on the structure and size of the input-unrelated uncertainties. Moreover, it is shown that asymptotic stabilization of the nominal system is not necessarily required in this method.


93B35 Sensitivity (robustness)
93D30 Lyapunov and storage functions
Full Text: DOI


[1] AUBIN J. B., Differential Inclusions (1984) · Zbl 0538.34007 · doi:10.1007/978-3-642-69512-4
[2] BARMISH B. R., SIAM Journal on Control and Optimization 21 pp 246– (1983) · Zbl 0503.93049 · doi:10.1137/0321014
[3] BARMISH B. R., I.E.E.E. Transaction on Automatic Control 27 pp 153– (1982) · Zbl 0469.93043 · doi:10.1109/TAC.1982.1102862
[4] CHEN Y. H., Proceedings of the American Control Conference pp 229– (1986)
[5] CORLESS M. J., I.E.E.E. Transactions on Automatic Control 26 pp 1139– (1981) · Zbl 0473.93056 · doi:10.1109/TAC.1981.1102785
[6] GUTMAN S., I.E.E.E. Transactions on Automatic Control 24 pp 437– (1979) · Zbl 0416.93076 · doi:10.1109/TAC.1979.1102073
[7] HAHN W., Theory and Application of Lyapunov’s Direct Method (1963) · Zbl 0119.07403
[8] HALE J. K., Ordinary Differential Equations (1980) · Zbl 0433.34003
[9] HOLLOT C. V., Proceedings of the 1987 American Control Conference pp 496– (1987)
[10] LASALLE J. P., Bulletin of the Institute of Mathematics, Academia Sinica 3 pp 139– (1975)
[11] PETERSEN I. R., SIAM Journal of Control and Optimization 26 pp 1257– (1988) · Zbl 0667.93087 · doi:10.1137/0326069
[12] PETERSEN I. R., Automatica 22 pp 397– (1986) · Zbl 0602.93055 · doi:10.1016/0005-1098(86)90045-2
[13] Qu Z., International Journal of Robust and Nonlinear Control submitted for publication (1991)
[14] SCHMITENDORF W. E., I.E.E.E. Transactions on Automatic Control 33 pp 376– (1988) · Zbl 0643.93052 · doi:10.1109/9.192193
[15] STALFORD H. L., Proceedings of the 26th Conference on Decision and Control pp 1298– (1987) · doi:10.1109/CDC.1987.272622
[16] THORP J. S., Journal of Optimization Theory and Applications 35 pp 559– (1981) · Zbl 0446.93040 · doi:10.1007/BF00934932
[17] VIDYASAGAR M., Nonlinear Systems Analysis (1978) · Zbl 0407.93037 · doi:10.1115/1.3426360
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.