Gain selection criterion for the swing-up control of the Acrobot using collocated partial feedback linearisation

A novel set of feedback gain selection conditions pertaining to the swing-up control of the Acrobot using collocated partial feedback linearisation is derived and presented in this research. This set of conditions, collectively known as the gain selection criterion, highlights a region of possible feedback gain combinations, known as the region of appropriate gains (RAG), that will guarantee the unstable response of the Acrobot when initialised approximately near the fully-pendant equilibrium point. The criterion is derived using the Routh-Hurwitz stability criterion applied on the characteristic equation of the system linearised about the fully-pendant equilibrium point. Only one left-column coefficient, known as the critical Routh coefficient, demonstrates the potential of influencing the stability of the system. The boundaries of the RAG are defined from the analysis of the critical Routh coefficient. The gain selection criterion will prevent the unintended selection of a combination of feedback gains that will cause a stable response about the fully-pendant equilibrium point, preventing the satisfactory execution of swing-up control on the Acrobot from the fully-pendant equilibrium point. This criterion may, more conveniently, be applied to specific types of multi-link pendulum system that approximate the behaviour of the partially linearised Acrobot.