Calculation of the critical delay for the double inverted pendulum

Abstract

Single and double inverted pendulum systems subjected to delayed state feedback are analyzed in terms of stabilizability. The maximum (critical) delay that allows a stable closed-loop system is determined via the multiplicity-induced-dominancy property of the characteristic roots, that is the dominant (rightmost) roots are associated with higher multiplicity under certain conditions of the system parameters. Other methods such as tracking the changes of the D-curves with increasing delay and the Walton–Marshall method are also demonstrated for the example of the single pendulum. For the double inverted pendulum subjected to full state feedback, the number of control gains is four, and application of numerical methods requires therefore high computational effort (i.e. optimization in a four-dimensional space). It is shown that, with the multiplicity-induced-dominancy–based approach, the critical delay and the associated control gains can be determined directly using the characteristic equation and its derivatives.