In this paper, we propose a new swing-up strategy for cart inverted pendulums with restricted rail length. The proposed swing-up strategy is derived from a new Lyapunov function. The Lyapunov function is defined as the sum of the square of the pendulum energy and the weighted square of the cart's velocity. The resulting swing-up strategy is represented in a compact form and has two design parameters. By adjusting these design parameters, we can affect the swing-up strategy such that the restriction on the rail length is satisfied. We also provide a state-dependent transformation to obtain voltage input to a DC motor required to generate the cart's acceleration obtained from the proposed swing-up strategy. Finally, we illustrate the performance of the proposed swing-up law through simulation and experiments. It is shown that there is quite good correspondence between theory and experiments.

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In this paper, we consider the modeling of the sensor and actuator characteristics of a magnetic levitation system. The sensor part consists of an infra-red emitting diode (IRED) and a strip of photo-transistors to widen the control range of the maglev. Actuator is a lab-built electromagnet made of coil. In order to levitate a load through small amount of current, we attach a strong magnet block to the top of the levitated object. The sensor used in this paper has nonlinear characteristics because the photo-transistors are not accurately positioned. Furthermore, the actuator also has some sort of nonlinearity. We propose to use a screw-based height gauge system for modeling the actuator characteristics and the relationship between the height of the levitated object and the sensor output. We adopt two methods to model the sensor and actuator characteristics : look-up table and segmented linear approximation. The merit of the proposed modeling method is that it is very cost-effective because the screw-based measurement system can be built at a reasonable cost. Finally we show the usefulness of the proposed modeling approach through a real control experiment.

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