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

Continuous time Robust Fixed Point Transformations based control

“…As one can see, this equation is a complicated nonlinear expression, which prevents the use of linear controllers. The goal is to force the ρth derivative of the output y (ρ) to match exactly the ITC output u itc ; if the model is known exactly then this is carried out by the inverse dynamics (12). However, if the model is not known the inverse dynamics transformation cannot ensure this equivalence and we require a modified inputũ itc .…”

“…Despite of the obvious practical advantages, one must be aware that results concerning stability and robustness of the method has not been established due to its novelty. Rudimentary results can be found on stability and auto tuning, however, a more rigorous treatment must be developed in order to be a vastly applicable control technique [11], [12]. The aim of this paper is to formalize continuous counterpart of the discrete time RFPT by connecting it to the feedback linearization principle.…”

On the Alternatives of Lyapunov’s Direct Method in Adaptive Control Design

“…As it can be seen in Figure 7, the rotational speed was kept at almost constant (in spite of the very noisy measurement data), and the adaptive deformation and the control signal were well adapted to the external braking forces in harmony with the simulation results belonging to the "Illustrative Example" in subsection 2.3. The nominal and the realized rotational speed (the average of the whole data set was 59:9383rpm, the nominal constant value was 60rpm); The "Desired" and adaptively "Deformed" 2 nd timederivatives of the rotational speed; The control signal (from [67], courtesy of Tamás Faitli) In [68] the novel adaptive control approach was considered from the side of the Lyapunov function-based technique and it was found that it can be interpreted as a novel methodology that is able to drive the Lyapunov function near zero and keeping it in its vicinity afterwards. On this basis a new MRAC controller design was suggested in [69] that has similarity with the idea of the "Backstepping Controller" [70,71].…”

“…Figure7: The experimental setup used for the verification of the FPI-based adaptive control in the case of a pulse-width modulated brushless electric DC motor; The nominal and the realized rotational speed (the average of the whole data set was 59:9383rpm, the nominal constant value was 60rpm); The "Desired" and adaptively "Deformed" 2 nd timederivatives of the rotational speed; The control signal (from[67], courtesy of Tamás Faitli) In[68] the novel adaptive control approach was considered from the side of the Lyapunov function-based technique and it was found that it can be interpreted as a novel methodology that…”

A Simple Soft Computing Structure for Modeling and Control

“…Its mathematical basis is Banach's Fixed-Point Theorem [26], which is far simpler than the Lyapunov function-based technique and has many theoretical mathematical applications, too. The successful combination of this adaptive method with the classic parameter tuning-based approaches was reported in [27,28], and its relationship with the Lyapunov function was clarified in [29].…”