# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Variable structure exponential stabilization of chained systems based on the extended nonholonomic integrator. (English) Zbl 0980.93067
Summary: A new variable structure control strategy for exponentially stabilizing chained systems is presented based on the extended nonholonomic integrator model, the discontinuous coordinate transformation and the “reaching law method” in variable structure control design. The proposed approach converts the stabilization problem of an $n$-dimensional chained system into the pole-assignment problem of an $(n-3)$-dimensional linear time-invariant system and consequently simplifies the stabilization controller design of nonholonomic chained systems.

##### MSC:
 93D15 Stabilization of systems by feedback 70F25 Nonholonomic systems (particle dynamics) 93B12 Variable structure systems 93B17 System transformation 93B55 Pole and zero placement problems
Full Text:
##### References:
 [1] A. Astolfi, Exponential stabilization of a car-like vehicle, Proceedings of the 1995 IEEE International Conference on Robotics and Automation, Nagoya, Japan, 1995, pp. 1391--1396. [2] Astolfi, A.: Discontinuous control of nonholonomic systems. Systems control lett. 27, 31-45 (1996) · Zbl 0877.93107 [3] A. Astofi, Discontinuous control of the Brockett integrator, Proceedings of the 36th Conference on Decision and Control, San Diego, CA, 1997, pp. 4334--4339. [4] A.M. Bloch, S. Drakunov, Stabilization of a nonholonomic system via sliding modes, Proceedings of the 33rd Conference on Decision and Control, Orlando, FL, 1994, pp. 2961--2963. [5] A.M. Bloch, S. Drakunov, Tracking in nonholonomic dynamic systems via sliding modes, Proceedings of the 34th Conference on Decision and Control, 1995, pp. 2103--2106. [6] Bloch, A. M.; Drakunov, S.: Stabilization and tracking in the nonholonomic integrator via sliding modes. Systems control lett. 29, 91-99 (1996) · Zbl 0866.93085 [7] Bloch, A. M.; Reyhanoglu, M.; Mcclamroch, N. H.: Control and stabilization of nonholonomic dynamic systems. IEEE trans. Automat. control 37, 1746-1757 (1992) · Zbl 0778.93084 [8] R.W. Brockett, Asymptotic stability and feedback stabilization, in: R.W. Brockett, R.S. Hillman, H.J. Sussmann (Eds.), Differential Geometry Control Theory, Birkhäuser, Boston, 1983. · Zbl 0528.93051 [9] C. Canudas de Wit, H. Berghuis, H. Nijmeijer, Practical stabilization of nonlinear systems in chained form, Proceedings of the 33rd Conference on Decision and Control, Orlando, FL, 1994, pp. 3475--3480. [10] Gao, W.; Hung, J. C.: Variable structure control of nonlinear systems: a new approach. IEEE trans. Ind. electron. 40, 45-55 (1993) [11] H. Gartner, A. Astolfi, Stability study of a fuzzy controlled mobile robot, Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, 1996, pp. 1121--1126. [12] J. Guldner, V.I. Utkin, Stabilization of nonholonomic mobile robots using Lyapunov functions for navigation and sliding mode control, Proceedings of the 33rd Conference on Decision and Control, Orlando, FL, 1994, pp. 2967--2972. [13] J.P. Hespanha, Stabilization of nonholonomic integrators via logic-based switching, Proceedings of the 1996 IFAC World Congress, 1996, Sydney, Australia, pp. 467--472. [14] Hung, J. Y.; Gao, W.; Hung, J. C.: Variable structure control: a survey. IEEE trans. Ind. electron. 40, 2-22 (1993) [15] Jiang, Z. P.: Iterative design of time-varying stabilizers for multi-input systems in chained form. Systems control lett. 28, 255-262 (1996) · Zbl 0866.93084 [16] H. Khnennouf, C. Canudas de Wit, Quasi-exponential stabilizers for nonholonomic systems, Proceedings of the 1996 IFAC World Congress, Sydney, Australia, 1996, pp. 49--54. [17] Kolmanovsky, I.; Mcclamroch, N. H.: Development in nonholonomic control problems. IEEE control system mag. 15, 20-36 (1995) [18] M’closkey, R. T.; Murray, R.: Exponential stabilization of driftless nonlinear control systems via time-varying homogeneous feedback. IEEE trans. Automat. control 42, 614-628 (1997) [19] Murray, R. M.; Sastry, S. S.: Nonholonomic motion planning: steering using sinusoids. IEEE trans. Automat. control 38, 700-716 (1993) · Zbl 0800.93840 [20] G. Oriolo, S. Panzieri, G. Ulivi, Cyclic learning control of chained-form systems with application to car-like robotics, Proceedings of the 1996 IFAC World Congress, Sydney, Australia, 1996, pp. 187--192. [21] Pomet, J. B.: Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift. Systems control lett. 18, 147-158 (1992) · Zbl 0744.93084 [22] Samson, C.: Control of chained systems: application to path following and time-varying point-stabilization of mobile robots. IEEE trans. Automat. control 40, 64-77 (1995) · Zbl 0925.93631 [23] H.S. Shim, J.H. Kim, K. Koh, Variable structure control of nonholonomic wheeled mobile robot, Proceedings of the 1995 IEEE International Conference on Robotics and Automation, Nagoya, Japan, 1995, pp. 1694--1699. [24] Sordalen, O. J.; Egeland, O.: Exponential stabilization of nonholonomic chained systems. IEEE trans. Automat. control 40, 35-49 (1995) · Zbl 0828.93055 [25] A. Tayebi, M. Tadjine, A. Rachid, Invariant manifold approach for the stabilization of nonholonomic systems in chained form: application to a car-like mobile robot, Proceedings of the 36th Conference on Decision and Control, San Diego, CA, 1997, pp. 4038--4043. [26] Teel, A. R.; Murray, R.; Walsh, G. C.: Nonholonomic control systems: from steering to stabilization with sinusoids. Internat. J. Control 62, 849-870 (1995) · Zbl 0837.93062 [27] Utkin, V. I.: Sliding modes and their application in variable structure systems. (1978) · Zbl 0398.93003 [28] S.Y. Wang, W. Huo, W.L. Xu, Order-reduced stabilization design of nonholonomic chained systems based on new canonical forms, Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, 1999, pp. 3464--3469.