Improvement of a Fixed Point Transformations and SVD-based Adaptive Controller

In this paper some refinement of a novel control approach is reported that fits to the “traditional line of thinking” according to which in the most practical cases neither very precise, nor even complete system model is needed for obtaining precise control for dynamical systems. The validity of this statement is briefly pointed out in the most popular approaches as the main idea of the “Robust Sliding Mode / Variable Structure Controllers”, in the Adaptive Inverse Dynamics and in the Slotine-Li Adaptive Controllers based on Lyapunov's 2nd Method, and in a recently published problem tackling using the simple geometric interpretation of the Singular Value Decomposition (SVD). In the present approach the originally proposed convergent, iterative Cauchy sequences are nonlinearly moderated to adaptively control a coupled nonlinear system, the cart plus double pendulum serving as popular paradigm of dynamicall not very well conditioned systems. It is shown that the proposed moderation removes the small sharp fluctuation in the control torque that inherently belonged to the original solution without significantly degrading the control quality. This statement is substantiated by simulation results.N/

Evasion of Instabilities Caused by Neglected Subsystems and Saturations in the Control of a Cart of Asynchronous Electric Drives

The task of solving the adaptive control of a partially and imprecisely modeled electrical vehicle driven by three omnidirectional wheels together with the torque and/or power limits of their electric driving motors is considered. Thevehicle is very roughly modeled as a rigid body while the a part of the burden carried by it through elastic connection is completely neglected in the controller’s model. Instead parameter estimation techniques a simple, kinematically designed, PID– type trajectory tracking is formulated that is implemented via robust fixed point transformations. It is shown that if the nominal trajectory does not significantly excite the vehicle– burden connection precise and stable control can be achieved by the adaptivity, while the pure PID–type control may considerably excite this degree of freedom and can be corrupted by achieving either the torque or the power limits of the motors. The motors are supposed to be voltage controlled asynchronous drives with constant frequency excitation. Our statement is substantiated by numerical simulations. The main advantage of the proposed control is that it operates with local basin of attraction developed for convergent iterative Cauchy sequences that is easy to design by setting only a few parameters. Its disadvantage is that it cannot guarantee global stability therefore its application must be preceded by numerical tests. Its use may be especially useful in applications in industrial workshops when modeling the dynamics of the coupled subsystem technically is very difficult, e.g. when it is a tank partially containing heavy liquid.info:eu-repo/semantics/publishedVersio

Possible Adaptive Control by Tangent Hyperbolic Fixed Point Transformations Used for Controlling the Φ6-Type Van der Pol Oscillator

In this paper a further step towards a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years is reported. Its main advantage in comparison with the complicated Lyapunov function based techniques is that its fundament is some simple geometric consideration allowing to formulate the control task as a Fixed Point Problem for the solution of which various Contractive Mappings can be created that generate Iterative Cauchy Sequences for Single Input - Single Output (SISO) systems. These sequences can converge to the fixed points that are the solutions of the control tasks. Recently alternative potential solutions were proposed and sketched by the use of special functions built up of the “response function” of the excited system under control. These functions have almost constant values apart from a finite region in which they have a “wrinkle” in the vicinity of the desired solution that is the “proper” fixed point of these functions. It was shown that at one of their sides these fixed points were repulsive, while at the opposite side they were attractive. It was shown, too, that at the repulsive side another, so called “false” fixed points were present that were globally attractive, with the exception of the basins of attraction of the “proper” ones. This structure seemed to be advantageous because no divergences could occur in the iterations, the convergence to the “false” values could easily be detected, and by using some ancillary tricks in the most of the cases the solutions could be kicked from the wrong fixed points into the basins of attraction of the “proper ones”. It was expected that via adding simple rules to the application of these transformations good adaptive control can be developed. However, due to certain specialties of these functions practical problems arose. In the present paper novel transformations are presented that seem to evade these difficulties. Their applicability is illustrated via simulations in the adaptive control of the popular nonlinear paradigm, the Φ6 Van der Pol oscillator.info:eu-repo/semantics/publishedVersio