Nowadays, dynamical systems are getting increasingly complex because they have to fulfill more expectations and further they have to handle more and more uncertainties. The controllers built in these systems have to face with the same difficulties. Luckily, the applied new controllers are designed to take into account the grown claims and they are able to fit the extreme conditions. The method called Robust Fixed Point Transformations (RFPT) is used to ameliorate existing controllers’ results when an approximate model is used to predict the dynamical system’s behavior. To achieve the stability of RFPT many efforts have been taken in the recent past. In this paper, a new algorithm is proposed that prevents the occurrence of short temporal unstable periods of the RFPT-based controllers, i.e., ensures the continuous stability.
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...
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.
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