scholarly journals Estimation and Global Stability Analysis of PDAEs with Singular Time Derivative Matrix and Wetland Conservation Application

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Yushan Jiang ◽  
Qingling Zhang
Keyword(s):  
Global Stability ◽  
Singular Systems ◽  
Time Derivative ◽  
Estimation Method ◽  
Differential Algebraic Equations ◽  
Wetland Conservation ◽  
Descriptor System ◽  
Algebraic Equations ◽  
Energy Estimation ◽  
Singular Time

Some partial differential algebraic equations (PDAEs) system with singular time derivative matrices is analyzed. First, by PDE spectrum theory, this system is formulated as infinite-dimensional singular systems. Second, the state space and its properties of the system are built according to descriptor system theory. Third, the admissible property of the PDAEs is given via LMIs. Finally, the developed energy estimation method is proposed to investigate the global stability of PDAEs. The proposed approach is evaluated by an application in numerical simulations on some wetland conservation system with social behavior.

Towards an Energy-Based Linear Analysis of Nonholonomic Systems

1999 ◽  
Author(s):  
Marwan Bikdash ◽  
Richard A. Layton
Keyword(s):  
Steady State ◽  
Singular Systems ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Linear Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Indirect Approach ◽  
Physical Systems ◽  
Future Work

Abstract Guidelines toward an energy-based, linear analysis of discrete physical systems are presented, based on previous work in systematic modeling using Lagrangian differential-algebraic equations (DAEs). Recent work in this area is extended by accommodating nonholonomic constraints and explicit inputs. An equilibrium postulate is proposed and equilibrium is characterized for static and steady-state conditions. Lagrangian DAEs are linearized using a local, indirect approach. Alternate descriptor formulations leading to different linear singular systems are compared and one formulation is determined to be a good foundation for future work in linear analysis using Lagrangian DAEs.

Implicit Integration of the Equations of Multibody Dynamics in Descriptor Form

1997 ◽  
Author(s):  
Edward J. Haug ◽  
Mirela Iancu ◽  
Dan Negrut
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Kinetic Equations ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Integration Formulas ◽  
Multibody Dynamic ◽  
Constrained Equations

Abstract An implicit numerical integration approach, based on generalized coordinate partitioning of the descriptor form of the differential-algebraic equations of motion of multibody dynamics, is presented. This approach is illustrated for simulation of stiff mechanical systems using the well known Newmark integration method from structural dynamics. Second order Newmark integration formulas are used to define independent generalized coordinates and their first time derivative as functions of independent accelerations. The latter are determined as the solution of discretized equations obtained using the descriptor form of the equations of motion. Dependent variables in the formulation, including Lagrange multipliers, are determined to satisfy all the kinematic and kinetic equations of multibody dynamics. The approach is illustrated by solving the constrained equations of motion for mechanical systems that exhibit stiff behavior. Results show that the approach is robust and has the capability to integrate differential-algebraic equations of motion for stiff multibody dynamic systems.

scholarly journals Identifiability for a Class of Discretized Linear Partial Differential Algebraic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Begoña Cantó ◽  
Carmen Coll ◽  
Elena Sánchez
Keyword(s):  
Singular Systems ◽  
Mathematical Physics ◽  
Differential Algebraic Equations ◽  
Drazin Inverse ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Partial Differential Algebraic Equations ◽  
Discrete Singular Systems ◽  
The Difference ◽  
Identifiability Problem

This paper presents the use of an iteration method to solve the identifiability problem for a class of discretized linear partial differential algebraic equations. This technique consists in replacing the partial derivatives in the PDAE by differences and analyzing the difference algebraic equations obtained. For that, the theory of discrete singular systems, which involves Drazin inverse matrix, is used. This technique can also be applied to other differential equations in mathematical physics.

scholarly journals Maximal Regularity for Non-autonomous Evolutionary Equations

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Sascha Trostorff ◽  
Marcus Waurick
Keyword(s):  
Differential Equations ◽  
Maximal Regularity ◽  
Normal Operator ◽  
Time Derivative ◽  
Parabolic Type ◽  
Regularity Result ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Evolutionary Equations

AbstractWe discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

Solvability conditions in the analytical form of a descriptor system of partial differential equations

2021 ◽  
pp. 130-142
Author(s):  
Abdulftah H. Mohamad
Keyword(s):  
Original System ◽  
Analytical Form ◽  
Differential Algebraic Equations ◽  
Operator Pencil ◽  
Descriptor System ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Solvability Conditions ◽  
Partial Differential Algebraic Equations ◽  
Degenerate Operators

A system of first-order partial differential-algebraic equations in a Banach space with constant degenerate operators in the case of a regular operator pencil is considered. In this case, under some additional condition, the original system splits into two subsystems in disjoint subspaces in order to search for the projections of the original unknown function in the subspaces. The matching conditions for the parameters of the systems are identified. A solution of the considered system of differential-algebraic equations is constructed.

A Non-Redundant Formulation for the Dynamics Simulation of Multibody Systems in Terms of Unit Dual Quaternions

2016 ◽  
Author(s):  
Andreas Müller ◽  
Zdravko Terze ◽  
Viktor Pandza
Keyword(s):  
Rigid Body ◽  
Time Derivative ◽  
Natural Extension ◽  
Differential Algebraic Equations ◽  
Dynamics Simulation ◽  
Integration Scheme ◽  
Integration Step ◽  
Algebraic Equations ◽  
Dual Quaternions ◽  
Rigid Body Motions

Quaternions are favorable parameters to describe spatial rotations of rigid bodies since they give rise to simple equations governing the kinematics and attitude dynamics in terms of simple algebraic equations. Dual quaternions are the natural extension to rigid body motions. They provide a singularity-free purely algebraic parameterization of rigid body motions, and thus serve as global parameters within the so-called absolute coordinate formulation of MBS. This attractive feature is owed to the inherent redundancy of these parameters since they must satisfy two quadratic conditions (unit condition and Plcker condition). Formulating the MBS kinematics in terms of dual quaternions leads to a system of differential-algebraic equations (DAE) with index 3. This is commonly transformed to an index 1 DAE system by replacing the algebraic constraints with their time derivative. This leads to the well-known problem of constraint violation. A brute force method, enforcing the unit constraint of quaternions, is to normalize them after each integration step. Clearly this correction affects the overall solution and the dynamic consistency. Moreover, for unit dual quaternions the two conditions cannot simply be enforced in such a way. In this paper a non-redundant formulation of the motion equations in terms of dual quaternions is presented. The dual quaternion constraints are avoided by introducing a local canonical parameterization. The key to this formulation is to treat dual unit quaternions as Lie group. The formulation can be solved with any standard integration scheme. Examples are reported displaying the excellent performance of this formulation regarding the constraint satisfaction as well as the solution accuracy.

Stability of Linear Mechanical Systems With Holonomic Constraints

1993 ◽  
Vol 46 (11S) ◽  
pp. S160-S164 ◽  
Author(s):  
Peter C. Mu¨ller
Keyword(s):  
Singular Systems ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Descriptor Systems ◽  
Algebraic Equations ◽  
Holonomic Constraints ◽  
Lyapunov Matrix Equation ◽  
Lyapunov Matrix ◽  
Complete Set ◽  
And Control

Singular systems (descriptor systems, differential-algebraic equations) are a recent topic of research in numerical mathematics, mechanics and control theory as well. But compared with common methods available for investigating regular systems many problems still have to be solved making also available a complete set of tools to analyze, to design and to simulate singular systems. In this contribution the aspect of stability is considered. Some new results for linear singular systems are presented based on a generalized Lyapunov matrix equation. Particularly, for mechanical systems with holonomic constraints the well-known stability theorem of Thomson and Tait is generalized.

Markov Jump Linear System Analysis of a Microgrid Operating in Islanded and Grid Tied Modes

2020 ◽  
Author(s):  
Gilles Mpembele ◽  
Jonathan Kimball
Keyword(s):  
Monte Carlo ◽  
Power Systems ◽  
Power System ◽  
System Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Numerical Application ◽  
Conditional Moments ◽  
Markov Jump Linear Systems

<div>The analysis of power system dynamics is usually conducted using traditional models based on the standard nonlinear differential algebraic equations (DAEs). In general, solutions to these equations can be obtained using numerical methods such as the Monte Carlo simulations. The use of methods based on the Stochastic Hybrid System (SHS) framework for power systems subject to stochastic behavior is relatively new. These methods have been successfully applied to power systems subjected to</div><div>stochastic inputs. This study discusses a class of SHSs referred to as Markov Jump Linear Systems (MJLSs), in which the entire dynamic system is jumping between distinct operating points, with different local small-signal dynamics. The numerical application is based on the analysis of the IEEE 37-bus power system switching between grid-tied and standalone operating modes. The Ordinary Differential Equations (ODEs) representing the evolution of the conditional moments are derived and a matrix representation of the system is developed. Results are compared to the averaged Monte Carlo simulation. The MJLS approach was found to have a key advantage of being far less computational expensive.</div>

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