Stabilization of an inertia wheel pendulum using an implicit controller design

Abstract

Control design for an underactuated system is complicated, as feedback linearization cannot be applied directly. This issue can be addressed by using partial feedback linearization with appropriate coordinate transformations. Unfortunately, this often results in leaving the core system non-affine. Design of a control law for a non-affine system is quiet difficult due to lack of mathematical tools needed. The Inertia Wheel Pendulum is a benchmark example of such non-affine systems. The under actuation property, posing problems in exact feedback linearization makes design of the control law for this a challenging task. Although Partial feedback linearization reduces a part of the system to linear but leaves the core system non-affine in nature. A novel nonlinear controller design fusing recently introduced Sliding Surface Control technique with Implicit Control of the nonlinear core is presented to tackle the issue. The task of the nonlinear controller is not only to stop the wheel but also to stabilize the pendulum at its unstable upright equilibrium in such a way that the inertial wheel stops rotating. The design procedure is simpler and more intuitive than currently available sliding surfaces, integrator backstepping or energy shaping designs. Stability is analyzed by decomposing the system into a cascade of linear and nonlinear sub-systems. Stability and advantages over existing controller designs are analyzed theoretically and verified using numerical simulations.