A novel branch of Computational Cybernetics was formerly developed for the adaptive control of approximately and partially modeled Classical Mechanical, Electro-mechanical and Hydro-mechanical systems. It essentially was based on the Modified Renormalization Transformation for the convergence of which the positive definite nature of the inertia matrix was needed. In this paper the extension of this method is formulated for negative definite, Single Input, Single Output (SISO) systems that may also occur in Classical Mechanics when a multiple Degree Of Freedom (DOF) system has only one driven axis and this drive is used to control a different axis utilizing the nonlinear dynamic coupling between the axes. As an example the inverted pendulum-cart system is considered in which the drive of the cart’s translational DOF is used for controlling the rotational axis of the pendulum. The extension results in a whole family of parametric transformations in which two parameters, a “multiplicative” and a “shift” parameter are present. It is shown that the fact of the convergence is robust against the variation of these parameters and their actual value mainly concerns only the speed of convergence. Simulation results illustrate these statements.
Adaptive Control; Robust Control; Computational Cybernetics; Soft Computing; Renormalization Transformation; Modified Renormalization Transformation; Extended Modified Renormalization Transformation.
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