VS-type stabilization of MRAC controllers using robust fixed point transformations

Abstract

Nowadays, control of dynamical systems with uncer tainties is a common problem. Many sulutions can be found in the literature, one of these methods is the family of Robust Fixed Point Transformations (RFPT) with local basin of attraction.The method is based on the idea that if someone has to use an approximate model in a control task, there is a function which, locally converging to the right solution, can reduce the disadvantages of the approximation. In this paper, authors show that though RFPT can loose its local convergensity, it can still improve a simple controller's results and this improvement makes the controller's behavior very similar to that of a sliding mode controller. The similarity includes the so called chattering effect, but a simple smoothing algorithm is also introuced to minimize the fluctuation of the control signal. I. INTRODUC TIONIn the 19th century stability of dynamical systems was a problematic subject for the scientists. It was sometimes very elaborate to determine whether a system was stable or not and only a few results were at hand to answer these questions. One of the first breakthroughs was made by Aleksandr Lyapunov in 1892. In his PhD dissertation [1] he introduced an approach, called Lyapunov's "direct" method, in which on the basis of relatively simple estimations, stability (either global or local, common, exponential, or asymptotic) could be determined without obtaining and studying the solutions of the equations of motion. (It is well known that the most of the practically occurring problems do not have analytical solutions in closed form, while the numerical solutions are normally valid only for the limited time-span of investigations and without deeper mathematical background their results cannot be extrapolated.) Lyapunov's work was translated to English approximately 75 years later, in the 20th century [2] and since that it has become the possibly most significant mathematical tool for designing stable controllers for linear and nonlinear systems (for some examples [3]-[11]).The advantage of the Sliding Mode Control (SMC) is its simplicity, characterized by robustness which makes the con troller able to handle dynamical systems with heavy uncertain conditions. In sliding mode, SMC does not just behave as a reduced order system with respect to the original system, but is insensitive to uncertainities and disturbances. On the contrary it has a main disadvantage, the so-called chattering effect, which demands the system to fluctuate so quick that it might get damaged in short time because of the stress [12].Robust Fixed Point Transformation is a method based on an idea that if someone has to use an approximate model in a control task, there is a function which, locally converging to the right solution, can reduce the disadvantages of the approximation. As included in its name, RFPT is characterized by robustness and it has the ability to handle rough approxima tions also, without increasing the performance of the controller considerably. These properties make RFPT very si...

Robust Fixed Point Transformation based design for Model Reference Adaptive Control of a modified TORA system

“…The behavior of the sequence {Q Def n } outside the region of convergence depends on the global properties of G. In [19], a chattering-like behavior was accidentally found and was further analyzed in [20]. A more detailed analysis in [21] revealed that the following cases may happen: a) monotone or non-monotone convergence to the solution of the control task, b) bounded chaotic behavior that can be treated by the method outlined in [19], c) divergence to infinity, d) convergence to the trivial fixed point of G i.e., to −K c .…”

“…A more detailed analysis in [21] revealed that the following cases may happen: a) monotone or non-monotone convergence to the solution of the control task, b) bounded chaotic behavior that can be treated by the method outlined in [19], c) divergence to infinity, d) convergence to the trivial fixed point of G i.e., to −K c .…”

Cognitive Control initiative

“…[72]) that help keeping the controlled system in the region of convergence. It have been also shown that quitting the region of convergence cannot result in catastrophic aftermaths: the controller can still work with considerable chattering that can be reduced and evaded [73], [74]. 7) Finally, these controllers can behave in a cognitive way that besides applying iterative corrections to a given approximate model they can select various approximate models by observing the behavior of the controlled systems.…”

Observation-based data driven adaptive control of an electromechanical device

“…At the beginning, only the local properties of this transformation were investigated in the vicinity of the fixed point, and various tuning methods were suggested for keeping the control signal in the basin of attraction of the useful fixed point [11], [12]. Later its global properties were investigated in [13], [14], [15] and [16]. The conclusion of these investigations was that for a wide class of physical systems it was possible to so tune only one of the altogether three adaptive control parameters that monotone convergence to the useful fixed point could be maintained.…”