Revisiting Lyapunov's Technique in the Fixed Point Transformation-based Adaptive Control

In the adaptive control of strongly nonlinear systems the prevailing design methodology is based on the creation of a Lyapunov function then proving that its time-derivative is non-positive. The use of this approach requires skillful designers with creative capacities and cannot be carried out as the application of a simple "algorithm". Furthermore, it works on meeting satisfactory conditions instead of necessary and satisfactory conditions, therefore its application often requires "too much". This approach concentrates on the global and asymptotic stability that are met in the case of an ample set of arbitrary control parameters that partly come into the control law as the "fragments" of the Lyapunov function. However, they considerably affect the transient behavior of the controlled motion for which further optimization may be necessary. To evade these problems the Fixed Point Transformation-based adaptive control was introduced as the alternative of the Lyapunov function-based technique. It directly concentrates on the control of the details of the transient phase of the controlled motion. For transforming the control task into a Fixed Point problem and subsequently solving it via an iteration various Fixed Point Transformations were invented. In the present paper it is shown that this novel approach can be also formulated by the use of a Lyapunov function. This statement is illustrated via simulation results belonging to the adaptive control of two nonlinearly coupled mass-points and spring system with nonlinear damping.