scholarly journals Ein systematischer Backstepping-Zugang zur Regelung gekoppelter ODE-PDE-ODE-Systeme

2020 ◽  
Vol 68 (8) ◽  
pp. 654-666
Author(s):  
Nicole Gehring
Keyword(s):  
Integrator Backstepping ◽  
Ode System

ZusammenfassungDer systematische Zugang zum Backstepping-Entwurf von Zustandsrückführungen für gekoppelte ODE-PDE-ODE-Systeme, die sich zumeist für verteilt-parametrische Prozesse unter Berücksichtigung der Dynamik von Aktoren und Sensoren ergeben, erlaubt nicht nur die sukzessive und vereinfachte Herleitung von bereits bekannten Backstepping-Reglern sondern ermöglicht auch deren Entwurf für bisher nicht betrachtete Systemklassen. Der Zugang nutzt dabei die diesen Systemen inhärente strenge Rückkopplungsform aus. Wie beim klassischen Integrator-Backstepping wird das ODE-PDE-ODE-System schrittweise durch virtuelle Zustandsrückführungen stabilisiert und der Zustand in Fehlerkoordinaten überführt. Der vorgeschlagene, modulare Entwurf ist dabei im Wesentlichen unabhängig davon, ob der verteilt-parametrische Teil des Systems parabolisch oder hyperbolisch ist.

scholarly journals Lie symmetry reductions and conservation laws for fractional order coupled KdV system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hong Guang Sun ◽  
Marzieh Azadi
Keyword(s):  
Conservation Laws ◽  
Fractional Order ◽  
Lie Symmetry ◽  
Theoretical Physics ◽  
Lie Symmetry Analysis ◽  
Kdv Equations ◽  
Ode System ◽  
Coupled Kdv Equations ◽  
Scientific Phenomena ◽  
New System

AbstractLie symmetry analysis is achieved on a new system of coupled KdV equations with fractional order, which arise in the analysis of several problems in theoretical physics and numerous scientific phenomena. We determine the reduced fractional ODE system corresponding to the governing factional PDE system.In addition, we develop the conservation laws for the system of fractional order coupled KdV equations.

scholarly journals Stability analysis of the coexistence equilibrium of a balanced metapopulation model

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Shodhan Rao ◽  
Nathan Muyinda ◽  
Bernard De Baets
Keyword(s):  
Equilibrium Point ◽  
Mathematical Proof ◽  
Mean Field ◽  
Patch Dynamics ◽  
System Modelling ◽  
Metapopulation Model ◽  
Ode System ◽  
Homogeneous Network ◽  
The Stability ◽  
Coexistence Equilibrium

AbstractWe analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.

scholarly journals Invariant tracking control design via integrator backstepping

PAMM ◽  
2010 ◽  
Vol 10 (1) ◽  
pp. 603-604 ◽  
Author(s):  
Carsten Collon ◽  
Joachim Rudolph
Keyword(s):  
Tracking Control ◽  
Control Design ◽  
Integrator Backstepping

scholarly journals Numerical Method of Fabric Dynamics Using Front Tracking and Spring Model

2013 ◽  
Vol 14 (5) ◽  
pp. 1228-1251 ◽  
Author(s):  
Yan Li ◽  
I-Liang Chern ◽  
Joung-Dong Kim ◽  
Xiaolin Li
Keyword(s):  
Upper Bound ◽  
Mesh Size ◽  
Front Tracking ◽  
Time Step ◽  
Mass System ◽  
Ode System ◽  
Spring System ◽  
Fabric Surface ◽  
Mass Spring ◽  
Eigen Frequency

AbstractWe use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the “Impulse method” which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.

scholarly journals Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160798 ◽  
Author(s):  
Wenjun Cai ◽  
Huai Zhang ◽  
Yushun Wang
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Acoustic Waves ◽  
Conservation Law ◽  
Heterogeneous Media ◽  
Numerical Dispersion ◽  
Discrete Version ◽  
Symplectic Method ◽  
Ode System ◽  
Strang Splitting

We propose a novel stable and efficient dissipation-preserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude. Through introducing the conformal multi-symplectic structure, the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation. The proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law. More specifically, two subsystems in conjunction with the original damped wave equation are derived. One is actually the conservative Hamiltonian wave equation and the other is a dissipative linear ordinary differential equation (ODE) system. Standard symplectic method is devoted to the conservative system, whereas the analytical solution is obtained for the ODE system. An explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the Strang splitting technique. Stability analysis and convergence tests are given thereafter. A benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation. Further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media.

scholarly journals Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Shengmao Fu ◽  
Lina Zhang
Keyword(s):  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Sex Structure ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Ode System ◽  
Prey Model ◽  
Weakly Coupled ◽  
Coupled Reaction

In this paper, we consider a cross-diffusion predator-prey model with sex structure. We prove that cross-diffusion can destabilize a uniform positive equilibrium which is stable for the ODE system and for the weakly coupled reaction-diffusion system. As a result, we find that stationary patterns arise solely from the effect of cross-diffusion.

scholarly journals Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cross Section ◽  
Circular Cross Section ◽  
Complex Form ◽  
Real Form ◽  
Element Analysis ◽  
Numerical Studies ◽  
Ode System ◽  
Symmetrical Deformation

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

Theoretical Analysis of Activation Energy Effect on Prandtl–Eyring Nanoliquid Flow Subject to Melting Condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ikram Ullah ◽  
Rashid Ali ◽  
Hamid Nawab ◽  
Abdussatar ◽  
Iftikhar Uddin ◽  
...  
Keyword(s):  
Differential Equation ◽  
Activation Energy ◽  
Convective Flow ◽  
Skin Friction Coefficient ◽  
System Of Equations ◽  
Energy Effect ◽  
Melting Condition ◽  
Ode System ◽  
Physical Importance ◽  
Controlling Variables

Abstract This study models the convective flow of Prandtl–Eyring nanomaterials driven by a stretched surface. The model incorporates the significant aspects of activation energy, Joule heating and chemical reaction. The thermal impulses of particles with melting condition is addressed. The system of equations is an ordinary differential equation (ODE) system and is tackled numerically by utilizing the Lobatto IIIA computational solver. The physical importance of flow controlling variables to the temperature, velocity and concentration is analyzed using graphical illustrations. The skin friction coefficient and Nusselt number are examined. The results of several scenarios, mesh-point utilization, the number of ODEs and boundary conditions evaluation are provided via tables.

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