scholarly journals The Significant and Profound Impacts of Chou’s Invariance Theorem

2020 ◽  
Vol 12 (09) ◽  
pp. 659-660
Author(s):  
Kuo-Chen Chou
Keyword(s):  
Invariance Theorem

scholarly journals On the domain invariance theorem

2004 ◽  
Vol 145 (1-3) ◽  
pp. 205-208
Author(s):  
G. De Marco
Keyword(s):  
Invariance Theorem

scholarly journals The strong Suslin reciprocity law

2021 ◽  
Vol 157 (4) ◽  
pp. 649-676
Author(s):  
Daniil Rudenko
Keyword(s):  
Rational Functions ◽  
Projective Curve ◽  
Main Ingredient ◽  
Homotopy Invariance ◽  
Reciprocity Law ◽  
Hyperbolic Volume ◽  
Scissors Congruence ◽  
Dehn Invariant ◽  
Contracting Homotopy ◽  
Invariance Theorem

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$ -theory. The Milnor $K$ -groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

scholarly journals Modeling and energy-based sway reduction control for tower crane systems with double-pendulum and spherical-pendulum effects

2019 ◽  
Vol 53 (1-2) ◽  
pp. 141-150 ◽  
Author(s):  
Menghua Zhang ◽  
Yongfeng Zhang ◽  
Bing Ji ◽  
Changhui Ma ◽  
Xingong Cheng
Keyword(s):  
Closed Loop ◽  
Double Pendulum ◽  
Control Methods ◽  
Spherical Pendulum ◽  
Closed Loop System ◽  
Tower Crane ◽  
Lyapunov Techniques ◽  
Payload Mass ◽  
System States ◽  
Invariance Theorem

As typical underactuated systems, tower crane systems present complicated nonlinear dynamics. For simplicity, the payload swing is traditionally modeled as a single-pendulum in existing works. Actually, when the hook mass is close to the payload mass, or the size of the payload is large, a tower crane may exhibit double-pendulum effects. In addition, existing control methods assume that the hook and the payload only swing in a plane. To tackle the aforementioned practical problems, we establish the dynamical model of the tower cranes with double-pendulum and spherical-pendulum effects. Then, on this basis, an energy-based controller is designed and analyzed using the established dynamic model. To further obtain rapid hook and payload swing suppression and elimination, the swing part is introduced to the energy-based controller. Lyapunov techniques and LaSalle’s invariance theorem are provided to demonstrate the asymptotic stability of the closed-loop system and the convergence of the system states. Simulation results are illustrated to verify the correctness and effectiveness of the designed controller.

Energy-Based Approach for Controller Design of Overhead Cranes: A Comparative Study

2013 ◽  
Vol 365-366 ◽  
pp. 784-787 ◽  
Author(s):  
Nguyen Quang Hoang ◽  
Soon Geul Lee
Keyword(s):  
Closed Loop ◽  
Controller Design ◽  
Asymptotically Stable ◽  
Overhead Crane ◽  
Overhead Cranes ◽  
Swing Angle ◽  
Optimal Linear ◽  
Linear And Nonlinear ◽  
Invariance Theorem ◽  
Lyapunov Technique

In this paper, five controllers including linear and nonlinear ones for an underactuated overhead crane are derived based on the passivity of the system. The total energy of the system and its square are used in Lyapunov candidate function to design controllers. The equilibrium point of the closed loop is proven to be asymptotically stable by the Lyapunov technique and LaSalle invariance theorem. In addition, the optimal linear controller is also combined to force the swing angle to converge fast to zero by reaching destination of the trolley. Numerical simulations are carried out to evaluate the controllers.

Mechanical Representation and Stability of Dynamical Systems Containing Fractional Springpot Elements

2018 ◽  
Author(s):  
Matthias Hinze ◽  
André Schmidt ◽  
Remco I. Leine
Keyword(s):  
Dynamical System ◽  
Potential Energy ◽  
Lyapunov Functional ◽  
Analogue Model ◽  
Infinite Dimensional ◽  
Mechanical Analogue ◽  
Stability Of Dynamical Systems ◽  
Functional Differential ◽  
Invariance Theorem ◽  
Simple Dynamical System

In this article we consider the Lyapunov stability of mechanical systems containing fractional springpot elements. We obtain the potential energy of a springpot by an infinite dimensional mechanical analogue model. Furthermore, we consider a simple dynamical system containing a springpot as a functional differential equation and use the potential energy of the springpot in a Lyapunov functional to prove uniform stability and discuss asymptotic stability of the equilibrium with the help of an invariance theorem.

Controlling Chaos by Bounded Self-Controlling Function Using Butterworth Filter Feedback

2014 ◽  
Vol 541-542 ◽  
pp. 1216-1221
Author(s):  
Lin Du ◽  
Fei Lei Jia
Keyword(s):  
Maximum Amplitude ◽  
Sigmoid Function ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Input Signals ◽  
Low Pass ◽  
Large Maximum ◽  
Effectiveness And Efficiency ◽  
Feedback Control Strategy ◽  
Invariance Theorem

Using the bounded sigmoid function and two-order Butterworth low-pass filter, a self-controlling feedback method for regulate the motion of a chaotic system is presented in this paper. It is shown that such controller has the advantage of being easy to implement based on the measurable input signals. A rigorous stability proof is provided from LaSalle Invariance theorem. Furthermore, the effectiveness and efficiency of the proposed feedback control strategy is illustrated by means of the numerical simulations of two-well Duffing Vander Pol oscillator. Finally, the result reveals that the enough large maximum amplitude results in a more possible regular domain in parameter space of the controlled oscillator.

scholarly journals Boom motion trajectory generation approach for load sway rejection in rotary cranes considering double-pendulum effect

2020 ◽  
pp. 002029402094496
Author(s):  
Huimin Ouyang ◽  
Xiang Xu ◽  
Guangming Zhang
Keyword(s):  
Simple Structure ◽  
Double Pendulum ◽  
Motion Trajectory ◽  
Robust Performance ◽  
Excellent Performance ◽  
Simulation Data ◽  
Barbalat’S Lemma ◽  
Rotary Crane ◽  
Control Research ◽  
Invariance Theorem

In the control research on the rotary crane systems with double-pendulum effect, a motion trajectory with both simple structure and excellent robust performance is proposed to achieve the positioning of the boom and the suppression of the load sway. The presented trajectory consists of an anti-swing component and a boom positioning component, where the first part is used to achieve the sway angle elimination without affecting boom positioning; the second one is used to move the boom to the desired location precisely. The Lyapunov technique, LaSalle’s invariance theorem, and Barbalat’s lemma are used to prove the excellent performance of the method. Eventually, the effectiveness of the proposed method was verified through a large amount of simulation data analysis.

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