Sigmoid Generated Fixed Point Transformation Control Scheme for Stabilization of Kapitza’s Pendulum System

In adaptive nonlinear control Lyapunov's 2nd or Direct method became a fundamental tool in control design. Recently the application of the "Sigmoid Generated Fixed Point Transformation (SGFPT)" has been introduced for replacing the Lyapunov technique. This systematic method has been presented for the generation of whole families of Fixed Point Transformations and has been extended from Single Input Single Output (SISO) to Multiple Input Multiple Output (MIMO) systems. Furthermore, the Stretched Sigmoid Functions have been introduced. In this paper a new function of this family have been investigated in order to obtain a more precise positioning of the function in the vicinity of the solution of the control task. The applicability and effectiveness of the proposed control method have been confirmed by the adaptive control of the inverted pendulum with vertical vibration of the pivot, i.e. the so-called Kapitza's pendulum. Results of numerical simulations have revealed that the proposed control design ensures performance enhancement.