Abstract

Although the inverted pendulum remains prevalent as a case study control system, linear controllability theory affords in itself little in the way of pure research until it is used in conjunction with symbolic computation. It is, therefore, the purpose of this paper to highlight an application of this now established research tool, currently being utilized on an increasing scale and enabling the user to perform automated algebraic manipulation which would otherwise prove too arduous a manual task. The paper - written with the non-specialist in control in mind - details theoretical controllability constraints arising from such computation, and demonstrates their validity in a variety of balancing control assignments indigenous to the system of two links which is damped by (linear) viscous friction and operates under an assumed single input. The necessary theory required of such an investigation is given, and, for each problem, the resulting symbolic analysis is presented in relation to the "friction space" associated with the system. Support for correctness of results is provided by an anticipated pole-zero cancellation found in the system transfer functions along the whole of the so called "curve of non-controllability" in one of the friction planes, through which it is possible to both identify those aspects of motion which become detached from the influence of the input and to explore the nature of latent instabilities which accompany the pendulum when uncontrollable. There is also included a discussion concerning the possibility of further validation, through simulation of the model, of the type of theoretical constraints found here, and the authors highlight a hypothesized "fuzzy" element in the pendulum's controllability characteristics which incorporates the notion of a system possessing a "level" of controllability whereby the effect of an input is diminished as a state of non-controllability is approached.