scholarly journals Non-autonomous higher-order Moreau's sweeping process: Well-posedness, stability and Zeno trajectories

2018 ◽  
Vol 29 (5) ◽  
pp. 941-968 ◽  
Author(s):  
BERNARD BROGLIATO
Keyword(s):  
Dynamical Systems ◽  
Differential Algebraic Equations ◽  
Higher Order ◽  
Well Posedness ◽  
Sweeping Process ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Uniqueness Of Solutions ◽  
Autonomous Case ◽  
Moreau’S Sweeping Process

In this article, we study the higher-order Moreau's sweeping process introduced in [1], in the case where an exogenous time-varying functionu(·) is present in both the linear dynamics and in the unilateral constraints. First, we show that the well-posedness results (existence and uniqueness of solutions) obtained in [1] for the autonomous case, extend to the non-autonomous case whenu(·) is smooth and piece-wise analytic, after a suitable state transformation is done. Stability issues are discussed. The complexity of such non-smooth non-autonomous dynamical systems is illustrated in a particular case named the higher-order bouncing ball, where trajectories with accumulations of jumps are exhibited. Examples from mechanics and circuits illustrate some of the results. The link with complementarity dynamical systems and with switching differential-algebraic equations is made.

scholarly journals Condensed Forms for Linear Port-Hamiltonian Descriptor Systems

2019 ◽  
Vol 35 ◽  
pp. 65-89 ◽  
Author(s):  
Lena Scholz
Keyword(s):  
Dynamical Systems ◽  
Existence And Uniqueness ◽  
Hamiltonian Formulation ◽  
Differential Algebraic Equations ◽  
Descriptor Systems ◽  
Constant Rank ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
Structure Preserving ◽  
Strangeness Index

Motivated by the structure which arises in the port-Hamiltonian formulation of constraint dynamical systems, structure preserving condensed forms for skew-adjoint differential-algebraic equations (DAEs) are derived. Moreover, structure preserving condensed forms under constant rank assumptions for linear port-Hamiltonian differential-algebraic equations are developed. These condensed forms allow for the further analysis of the properties of port-Hamiltonian DAEs and to study, e.g., existence and uniqueness of solutions or to determine the index. It can be shown that under certain conditions for regular port-Hamiltonian DAEs the strangeness index is bounded by $\mu\leq1$.

An implicit class of continuous dynamical system with data-sample outputs: a robust approach

2019 ◽  
Vol 37 (2) ◽  
pp. 589-606
Author(s):  
Raymundo Juarez ◽  
Vadim Azhmyakov ◽  
A Tadeo Espinoza ◽  
Francisco G Salas
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Models ◽  
Differential Algebraic Equations ◽  
Linear Feedback ◽  
Continuous Systems ◽  
Algebraic Equations ◽  
Nonlinear Dynamical ◽  
Ellipsoid Method ◽  
Invariant Ellipsoid

Abstract This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the continuous time domain. We deal with nonlinear models described by differential-algebraic equations (DAEs) in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear sampling-type outputs. The linear feedback control design proposed in this manuscript is created by application of an extended version of the conventional invariant ellipsoid method. Moreover, we also apply some specific Lyapunov-based descriptor techniques from the stability theory of continuous systems. The above combination of the modified invariant ellipsoid approach and descriptor method makes it possible to obtain the robustness of the designed control and to establish some well-known stability properties of dynamical systems under consideration. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.

scholarly journals Mathematical Formulation of Soft-Contact Problems for Various Rheological Models of Damper

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wiesław Grzesikiewicz ◽  
Artur Zbiciak
Keyword(s):  
Mathematical Formulation ◽  
Contact Problems ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Soft Contact ◽  
Rheological Models ◽  
Model Interaction ◽  
Time Histories

The paper deals with analysis of selected soft-contact problems in discrete mechanical systems. Elastic-dissipative rheological schemes representing dampers as well as the notion of unilateral constraints were used in order to model interaction between colliding bodies. The mathematical descriptions of soft-contact problems involving variational inequalities are presented. The main finding of the paper is a method of description of soft-contact phenomenon between rigid object and deformable rheological structure by the system of explicit nonlinear differential-algebraic equations easy for numerical implementation. The results of simulations, that is, time histories of displacements and contact forces as well as hysteretic loops, are presented.

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

1999 ◽  
Vol 121 (4) ◽  
pp. 594-598 ◽  
Author(s):  
V. Radisavljevic ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Algebraic Equations ◽  
The Stability ◽  
The Mathematical Model

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

Adding Non-Smooth Analysis Capabilities to General-Purpose Multibody Dynamics by Co-Simulation

2013 ◽  
Author(s):  
Matteo Fancello ◽  
Pierangelo Masarati ◽  
Marco Morandini
Keyword(s):  
Differential Algebraic Equations ◽  
General Purpose ◽  
Rigid Body Dynamics ◽  
Nonsmooth Dynamics ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Frictional Contacts ◽  
Continuous Contact ◽  
Hard Constraints ◽  
Dynamics Problems

Multi-rigid-body dynamics problems with unilateral constraints, like frictionless and frictional contacts, are characterized by nonsmooth dynamics. The issue of nonsmoothness can be addressed with methods that apply a mathematical regularization, called continuous contact methods; alternatively, hard constraints with complementarity approaches can be proficiently used. This work presents an attempt at integrating consistently modeled unilateral constraints in a general purpose multibody formulation and implementation originally designed to address intrinsically smooth problems. The focus is on the analysis of generally smooth problems, characterized by significant multidisciplinarity, with the need to selectively include nonsmooth events localized in time and in specific components of the model. A co-simulation approach between the smooth Differential-Algebraic Equations solver and the classic Moreau-Jean timestepping approach is devised as an alternative to entirely redesigning a monolithic nonsmooth solver, in order to provide elements subject to frictionless and frictional contact in the general-purpose, free multibody solver MBDyn. The implementation uses components from the INRIA’s Siconos library for the solution of Complementarity Problems. The proposed approach is applied to several problems of increasing complexity to empirically evaluate its properties and versatility. The applicability of the family of second-order accurate, A/L stable multistep integration algorithms used by MBDyn to nonsmooth dynamics is also discussed and assessed.

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