Adaptive control of a dynamic system having unmodeled and unconstrained internal degree of freedom

A new branch of computational cybernetics based on principles akin to that of the traditional soft computing (SC) was recently developed for the control of inaccurately modeled dynamic systems under external disturbances. In the present paper the operation of this controller is studied in the case of an incompletely modeled dynamic system, that is when the system to be controlled contains internal degree of freedom not modeled by the controller. As starting point the method uses a simple, incomplete dynamic model to predict the propagation of the state of the modeled degrees of freedom also influenced by that of the unmodeled internal ones by nonlinear coupling. The controller is restricted to the observation of the behavior of the generalized coordinates the models of which are available for it. By the use of a priori known, uniform, lucid structure of reduced size, simple and short explicit algebraic procedures especially fit to real-time applications the controller is able to learn the behavior of the observed system. Simulation examples are presented for the control of a double pendulum-cart system in which the first pendulum and the linear degree of freedom of the cart has drives only. The second pendulum can move freely and serves as the unmodeled component. Rotation of the second pendulum influences the inertia matrix of the whole system. It can obtain potential energy via the inertial and gravitational forces. It is found that the adaptive controller can successfully cope with the problem of imperfect modeling.