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Stabilization of the Furuta pendulum around its homoclinic orbit. (English) Zbl 1017.93087

The authors present the Lyapunov technique to rise and control the uppermost unstable equilibrium position of the Furuta pendulum. The Furuta pendulum system consists of a direct-drive motor as its actuator source and inverted pendulum attached to a motor arm rotating in the horizontal plane. The Furuta pendulum having the torque moment of the motor as input and the angular velocity of the motor shaft as output is a passive system. The passivity property of the system permits the total energy to be used in controller design. Here, the authors use the positive-definite properties and the skew-symmetry of corresponding matrices. This permits them to elaborate a control low which leads to a positive semi-definite Lyapunov function whose derivative is negative semi-definite. Convergence of the trajectories of the system to a homoclinic orbit is proved by using LaSalle’s invariance theorem. Such an approach guarantees the stability of the uppermost unstable equilibrium position of the Furuta pendulum.

MSC:

93D15 Stabilization of systems by feedback
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70Q05 Control of mechanical systems
93D30 Lyapunov and storage functions
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