The organization of the paper is as follows. First, we obtain the dynamical equations of the pole-cart system by applying the Lagrange equations for the two variables of interest: the cart position and the deviation angle of the pendulum with respect to the vertical line. The corresponding system of two differential equations is highly nonlinear, with a rather restricted analytical treatment. After obtaining the approximate, linear equations of the pole-cart couple, we discuss its control and stabilization throughout an external force that depends on the deviation angle of the pendulum. The type of feedback control schemes that stabilize the pendulum on its unstable, vertical position are discussed in the paper. We also illustrate such stabilization with several simulated cases, for both the complete, nonlinear version and the approximate, linear version. Conventional P and PD control schemes are applied, with excellent results. Finally, we approach the evaluation of the computer-based implementation by means of computer simulations and obtain the critical design parameters for the control and the stabilization of the polecart system.
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