Stationary bifurcations control with applications. (English) Zbl 1190.93045
Summary: Given a family of nonlinear control systems, where the Jacobian of the driver vector field at one equilibrium has a simple zero eigenvalue, with no other eigenvalues on the imaginary axis, we split it into two parts, one of them being a generic family, where it is possible to control the stationary bifurcations: saddle-node, transcritical and pitchfork bifurcations, and the other one being a non-generic family, where it is possible to control the transcritical and pitchfork bifurcations. The polynomial control laws designed are given in terms of the original control system. The center manifold theory is used to simplify the analysis to dimension one. Finally, the results obtained are applied to two underactuated mechanical systems: the pendubot and the pendulum of Furuta.
MSC:
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 70Q05 | Control of mechanical systems |
| 34H20 | Bifurcation control of ordinary differential equations |
Keywords:
saddle-node; transcritical and pitchfork bifurcations; bifurcation control; center manifold theorem; pendubot; pendulum of FurutaReferences:
| [1] | Abed, E.H., Fu, J.H.: Local feedback stabilization and bifurcation control, II. Stationary bifurcation. Syst. Control Lett. 8, 467–473 (1987) · Zbl 0626.93058 · doi:10.1016/0167-6911(87)90089-2 |
| [2] | Abed, E.H., Wang, H.O., Tesi, A.: Control of bifurcation and chaos. In: Levine, W.S. (ed.) The Control Handbook, pp. 951–966. CRC Press, Boca Raton (1995) |
| [3] | Chen, G. (ed.): Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, Boca Raton (2000) · Zbl 0929.00012 |
| [4] | Chen, G., Moiola, J.L., Wang, H.O.: Bifurcation control: theories, methods, and applications. Int. J. Bif. Chaos 10(3), 511–548 (2000) · Zbl 1090.37552 |
| [5] | Chen, G., Hill, D.J., Yu, X. (eds.): Bifurcation Control, Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 293. Springer, New York (2003) · Zbl 1024.00022 |
| [6] | Colonius, F., Grüne, L. (eds.): Dynamics, Bifurcations and Control. Lecture Notes in Control and Information Sciences, vol. 273. Springer, New York (2002) |
| [7] | Fu, J.H., Abed, E.H.: Linear feedback stabilization of nonlinear systems with an uncontrollable critical mode. Automatica 29(4), 999–1010 (1993) · Zbl 0782.93078 · doi:10.1016/0005-1098(93)90102-Y |
| [8] | Furuta, K., Yamakita, M., Kobayashi, S.: Swing-up control of inverted pendulum using pseudostate feedback. Proc. Inst. Mech. Eng. 206, 263–269 (1992) |
| [9] | Gu, G., Chen, X., Sparks, A.G., Banda, S.S.: Bifurcation stabilization with local output feedback. SIAM J. Control Optim. 37(3), 934–956 (1999) · Zbl 0960.93017 · doi:10.1137/S0363012997320924 |
| [10] | Kang, W.: Invariants and stability of control systems with transcritical and saddle-node bifurcations. In: Proc. 36th IEEE CDC, San Diego, CA, USA, 1997 |
| [11] | Kang, W.: Bifurcation and normal form of nonlinear control systems. Part I and II. SIAM J. Control Optim. 36, 193–232 (1998) · Zbl 0965.93022 · doi:10.1137/S0363012995290288 |
| [12] | Kang, W., Liang, K.: The stability and invariants of control systems with pitchfork or cusp bifurcations. In: Proc. 36th IEEE CDC, San Diego, CA, USA, 1997 |
| [13] | Kim, T., Abed, E.H.: Stationary bifurcation control of systems with uncontrollable linearization. Int. J. Control 74(5), 445–452 (2001) · Zbl 1030.93021 · doi:10.1080/00207170010003450 |
| [14] | Pagano, D.J.: Bifurcations in nonlinear control systems. Ph.D. dissertation (in Spanish). Dept. of Automatic Control and Systems Engineering, University of Seville, Spain |
| [15] | Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (1998) · Zbl 0854.34001 |
| [16] | Verduzco, F.: Controlling from the center manifold codimension one bifurcations. In: Proc. Asian Control Conference 2004, Melbourne, Australia, 2004 |
| [17] | Verduzco, F.: Control of the saddle-node and transcritical bifurcations. In: Proc. NOLCOS’04, Stuttgart, Germany, 2004 |
| [18] | Verduzco, F.: Control of codimension one stationary bifurcations. Int. J. Bif. Chaos 17(2), 575–582 (2007) · Zbl 1140.37348 · doi:10.1142/S0218127407017434 |
| [19] | Verduzco, F., Alvarez, J.: Bifurcation analysis of a 2-DOF robot manipulator driven by constant torques. Int. J. Bif. Chaos 9(4), 617–627 (1999) · Zbl 0979.70008 · doi:10.1142/S0218127499000432 |
| [20] | Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics, vol. 2. Springer, New York (1990) · Zbl 0701.58001 |
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.
