DocumentCode :
3310009
Title :
On the periodically driven inverted pendulum
Author :
Bailey, Robert ; Hauser, John
Author_Institution :
Dept. of Electr. & Comput. Eng., Univ. of Colorado, Boulder, CO, USA
fYear :
2009
fDate :
15-18 Dec. 2009
Firstpage :
3142
Lastpage :
3148
Abstract :
We study the solution properties of a family of inverted pendulum systems driven by odd periodic forcing. Using the Schauder fixed point theorem, we show that the inverted pendulum with an odd periodic driving acceleration at the pivot always possesses an odd periodic solution. Fundamental to the production of good estimates is the development of a Green´s function for an unstable harmonic oscillator with Dirichlet boundary conditions. We also show that it is sometimes possible to use the Banach fixed point theorem to ensure that there is a unique solution within an invariant region of the space of possible solution curves. Using these results, we characterize the solutions of periodically driven inverted pendulum systems such as that given by ¿¿ = ¿2 sin ¿ + ß sin (¿ - ¿t), which describes a pendubot with constant inner arm velocity. These results are important as the driven inverted pendulum is a common subsystem in systems ranging from motorcycles and bicycles to rockets and aircraft.
Keywords :
Banach spaces; Green´s function methods; boundary-value problems; nonlinear systems; pendulums; periodic control; Banach fixed point theorem; Dirichlet boundary condition; Green function; Schauder fixed point theorem; constant inner arm velocity; odd periodic driving acceleration; odd periodic solution; pendubot; periodically driven inverted pendulum; unstable harmonic oscillator; Acceleration; Aircraft; Bicycles; Boundary conditions; Green´s function methods; Motorcycles; Nonlinear dynamical systems; Oscillators; Production; Rockets;
fLanguage :
English
Publisher :
ieee
Conference_Titel :
Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on
Conference_Location :
Shanghai
ISSN :
0191-2216
Print_ISBN :
978-1-4244-3871-6
Electronic_ISBN :
0191-2216
Type :
conf
DOI :
10.1109/CDC.2009.5400433
Filename :
5400433
Link To Document :
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