Fuzzy parameter tuning in the stabilization of an RFPT-based adaptive control for an underactuated system

In this paper a “Robust Fixed Point Transformation (RFPT)” based adaptive control of an underactuated physical system is stabilized by adaptively tuning only one parameter of a single fuzzy membership function. This approach serves as an alternative to Lyapunov´s “direct” method that suffers from mathematical difficulties when a Lyapunov function candidate has to be found for the control of a dynamically singular or badly conditioned system as an underactuated cart plus double pendulum. In our case the reaction forces of the directly driven two rotary axles are used for controlling the linear motion of the cart within possible physical limits. As a modification of the common TORA (Translational Oscillations with an Eccentric Rotational Proof Mass Actuator) that has only one counterweight, the present solution applies two counterweights: one of them is actively used and the other one is kept in a “safe” position while the system is far from the dynamic singularity. When the singularity is approached the reserved axle takes the active role and the previously used weight is moved back to the nonsingular region. This session results in oscillatory motion that precisely has to be implemented by the controller. Instead developing complete and generally useful system model this approach extracts information on the recent behavior of the controlled system only in the given control situation by using only three adaptive control parameters. Two of them can be kept fixed but the third one may need fine tuning for stable control. Now a formerly proposed intricate tuning strategy is replaced by simple fuzzy tuning. It is illustrated by simulations that the controller can precisely track the prescribed trajectory even in the presence of considerable modeling errors and badly conditioned dynamics.