### “Stabilization of Stable Manifold of Upright Position of the Spherical Pendulum”

**Authors:**Hallgeir Ludvigsen, Anton Shiriaev and Olav Egeland,

**Affiliation:**Cybernetica and NTNU, Department of Engineering Cybernetics

**Reference:**2001, Vol 22, No 1, pp. 3-14.

**Keywords:**Passivity, stabilization of the upright position of the spherical pendulum

**Abstract:**The stabilization problem of the upright position of the sherical pendulum is treated in detail. This problem is reduced to the stabilization of the stable manifold Omega_st of the upright position of the unforced spherical pendulum. It is shown that for any smooth feedback control derived by the speed-gradient algorithm with the objective to stabilize Omega_st the closed loop system has a limit cycle Gamma, which does not belong to the desired attractor Omega_st. It is shown that Gamma is hyperbolic.

PDF (1311 Kb) DOI: 10.4173/mic.2001.1.1

**References:**

[1] BYRNES, C. I., ISIDORI A. WILLEMS, J. C. (1991). Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems, Transactions on AC, Vol. 36, N 11, pp. 1228-1240.

[2] FRADKOV, A.L. (1990). Adaptive control of complex systems, Moscow: Nauka..Russian.

[3] FRADKOV, A.L. (1996). Swinging control of nonlinear oscillations, Internat. J of Control, Vol. 64, N 6, pp. 1189-1202 doi:10.1080/00207179608921682

[4] FRADKOV, A.L. MAKAROV, I.A., SHIRIAEV, A.S. TOMPCHINA, O.P. (1997). Control of oscillations in Hamiltonian systems, Proceedings of 4th European Control Conference, Brussels.

[5] SHIRIAEV, A.S. FRADKOV, A.L. (1998). Stabilization of invariant set of nonlinear systems, In: Nonlinear dynamical systems. St. Petersburg: St. Petersburg State University..Russian.

[6] SHIRIAEV, A.S. (1998). The notion of V-detectability and stabilization of invariant set of nonlinear systems, Proceedings of 37th Conference on Decision and Control, Tampa.

[7] SHIRIAEV, A.S. FRADKOV, A.L. (1998). Stabilization of invariant manifolds for nonlinear nonaffine systems, Proceedings of IFAC Conference NOLCOSÂ´98, Enschede, pp. 215-220.

[8] SHIRIAEV, A.S., LUDVIGSEN, H. EGELAND, O. (1999). Swinging up of the spherical pendulum, Proceedings of the 14th IFAC World Congess, Beijing, pp. 65-70.

[9] SHIRIAEV, A.S., LUDVIGSEN, H., EGELAND, O. FRADKOV, A.L. (1999). Swinging up of non-affine in control pendulum, Proceeding of American Control Conference, ACCÂ´99, San Diego, pp. 4040-4044.

**BibTeX:**

@article{MIC-2001-1-1,

title={{Stabilization of Stable Manifold of Upright Position of the Spherical Pendulum}},

author={Ludvigsen, Hallgeir and Shiriaev, Anton and Egeland, Olav},

journal={Modeling, Identification and Control},

volume={22},

number={1},

pages={3--14},

year={2001},

doi={10.4173/mic.2001.1.1},

publisher={Norwegian Society of Automatic Control}

};