Sira-Ramirez, Hebertt Differential geometric methods in variable-structure control. (English) Zbl 0658.93022 Int. J. Control 48, No. 4, 1359-1390 (1988). The theory of variable structure control of nonlinear systems is presented in a differential geometric framework. In particular, conditions for the existence of sliding regimes and the associated ideal sliding dynamics are given. The author studies also regular canonical forms and disturbance invariant properties of sliding regimes. Both single-input and multi-input cases are considered, and several examples are worked out. Reviewer: A.Bacciotti Cited in 25 Documents MSC: 93B27 Geometric methods 93C10 Nonlinear systems in control theory 57R27 Controllability of vector fields on \(C^\infty\) and real-analytic manifolds 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 93C15 Control/observation systems governed by ordinary differential equations 93B10 Canonical structure Keywords:variable structure control; nonlinear systems; sliding regimes; regular canonical forms; disturbance invariant properties PDFBibTeX XMLCite \textit{H. Sira-Ramirez}, Int. J. Control 48, No. 4, 1359--1390 (1988; Zbl 0658.93022) Full Text: DOI References: [1] BOOTHBY W. M., An Introduction to Differentiable Manifolds and Riemannian Geometry (1975) · Zbl 0333.53001 [2] DOI: 10.1016/0005-1098(69)90071-5 · Zbl 0182.48302 · doi:10.1016/0005-1098(69)90071-5 [3] DOI: 10.1109/TAC.1984.1103665 · Zbl 0542.93032 · doi:10.1109/TAC.1984.1103665 [4] DOI: 10.1080/00207178308933100 · Zbl 0538.93035 · doi:10.1080/00207178308933100 [5] EMEL’YANOV S. V., Variable Structure Control Systems (1967) [6] DOI: 10.1109/TAC.1985.1104079 · Zbl 0574.93027 · doi:10.1109/TAC.1985.1104079 [7] DOI: 10.1109/TAC.1985.1103927 · Zbl 0562.93041 · doi:10.1109/TAC.1985.1103927 [8] DOI: 10.1109/TAC.1983.1103137 · Zbl 0502.93036 · doi:10.1109/TAC.1983.1103137 [9] DOI: 10.1109/TAC.1987.1104616 · Zbl 0611.93039 · doi:10.1109/TAC.1987.1104616 [10] DOI: 10.1007/BFb0006368 · doi:10.1007/BFb0006368 [11] ITKIS U., Control Systems of Variable Structure (1976) · Zbl 0256.93033 [12] KOKOTOVIC , P. , and MARINO , R. , 1987 ,10th IFAC World Congress, Preprints , Munich . [13] LUK’YANOV A. G., Automn remote Control 44 pp 5– (1981) [14] DOI: 10.1109/TAC.1984.1103504 · Zbl 0529.93033 · doi:10.1109/TAC.1984.1103504 [15] MARINO R., Proc. I.E.E.E. Conf. on Robotics and Automation pp 1030– (1986) [16] SIRA-RAMIREZ H., Int. J. Systems Sci. 19 pp 875– (1988) · Zbl 0653.93029 · doi:10.1080/00207728808547171 [17] DOI: 10.1109/TAC.1987.1104627 · Zbl 0611.93049 · doi:10.1109/TAC.1987.1104627 [18] DOI: 10.1080/00207178408933284 · Zbl 0541.93034 · doi:10.1080/00207178408933284 [19] DOI: 10.1080/00207178308933088 · Zbl 0519.93036 · doi:10.1080/00207178308933088 [20] SPONG , M. , and SIRA-RAMIREZ , H. , 1986 , Robust control design techniques for a class of non-linear systems .Proc. American Control Conf., Seattle , Washington , p. 1515 . [21] STEELE R., Delta Modulation Systems (1975) [22] UTKIN V. I., Automn remote Control 32 pp 1897– (1972) [23] WOOD , R. , 1974 , Power conversion in electrical networks . NASA Report, CR-120830 , Harvard University , Cambridge , U.S.A . [24] DOI: 10.1109/TSMC.1978.4309907 · Zbl 0369.93002 · doi:10.1109/TSMC.1978.4309907 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.