A Method for Solving Large-Scale Multiloop Constrained Dynamical Systems Using Structural Decomposition

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Tao Xiong ◽  
Jianwan Ding ◽  
Yizhong Wu ◽  
Liping Chen ◽  
Wenjie Hou
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
The State ◽  
State Variables ◽  
Structural Decomposition ◽  
Multiple Degrees Of Freedom ◽  
Constraint Equations ◽  
Constrained Dynamical Systems ◽  
Algebraic Constraint

A structural decomposition method based on symbol operation for solving differential algebraic equations (DAEs) is developed. Constrained dynamical systems are represented in terms of DAEs. State-space methods are universal for solving DAEs in general forms, but for complex systems with multiple degrees-of-freedom, these methods will become difficult and time consuming because they involve detecting Jacobian singularities and reselecting the state variables. Therefore, we adopted a strategy of dividing and conquering. A large-scale system with multiple degrees-of-freedom can be divided into several subsystems based on the topology. Next, the problem of selecting all of the state variables from the whole system can be transformed into selecting one or several from each subsystem successively. At the same time, Jacobian singularities can also be easily detected in each subsystem. To decompose the original dynamical system completely, as the algebraic constraint equations are underdetermined, we proposed a principle of minimum variable reference degree to achieve the bipartite matching. Subsequently, the subsystems are determined by aggregating the strongly connected components in the algebraic constraint equations. After that determination, the free variables remain; therefore, a merging algorithm is proposed to allocate these variables into each subsystem optimally. Several examples are given to show that the proposed method is not only easy to implement but also efficient.

Nonlinear Observers for Constrained Systems

2009 ◽  
Author(s):  
Giscard A. Kfoury ◽  
Nabil G. Chalhoub
Keyword(s):  
Degrees Of Freedom ◽  
Sliding Mode ◽  
The State ◽  
Constrained Systems ◽  
State Variables ◽  
Nonlinear Observers ◽  
Constraint Equations ◽  
Single Cylinder Engine ◽  
Robust Observers ◽  
Selection Of

Three procedures for designing robust observers to estimate the state variables of nonlinear constrained systems have been developed in this work. All observers are based on the sliding mode methodology and assume that the number of transducers matches that of the degrees of freedom of the system. The conceptual differences between the proposed observer designs are in the number and selection of the sliding surfaces along with the formulations pertaining to their nominal models. The observers have been applied to estimate the state variables of a crank-slider mechanism of a single cylinder engine. The simulation results demonstrate the capabilities of the observers in accurately estimating the state variables of the system, including the superfluous ones, in the presence of significant structured and unstructured uncertainties. In addition, the results show that the nominal constraint equations are satisfied by the estimated state variables.

Observability, Reconstructibility and Observer Design for Linear Mechanical Systems Unobservable in Absence of Impacts†

2003 ◽  
Vol 125 (4) ◽  
pp. 549-562 ◽  
Author(s):  
Francesco Martinelli ◽  
Laura Menini ◽  
Antonio Tornambe`
Keyword(s):  
Structural Properties ◽  
Degrees Of Freedom ◽  
Mechanical Systems ◽  
The State ◽  
Observer Design ◽  
State Variables ◽  
Multiple Degrees Of Freedom ◽  
Position Variable

This paper deals with a class of mechanical systems, constituted by two composite bodies which interact with each other only during collisions: since only one position variable of only one body is measured, the whole system would not be observable without impacts. Each body is possibly constituted by several masses connected by linear springs, so that the internal deformations can be taken into account (the so-called multiple-degrees-of-freedom impacts). For two relevant and different cases, the structural properties of observability and reconstructibility are studied, and observers are proposed in order to estimate all the state variables, including those that would be unobservable in absence of impacts.

TLISMNI/Adams algorithm for the solution of the differential/algebraic equations of constrained dynamical systems

2017 ◽  
Vol 232 (1) ◽  
pp. 129-149
Author(s):  
Ahmed A Shabana ◽  
Dayu Zhang ◽  
Gengxiang Wang
Keyword(s):  
Dynamical Systems ◽  
Numerical Integration ◽  
Sparse Matrix ◽  
Differential Algebraic Equations ◽  
Variable Step Size ◽  
Step Size ◽  
Constraint Equations ◽  
Adams Method ◽  
Constrained Dynamical Systems ◽  
Algebraic Constraint

This paper examines the performance of the 3rd and 4th order implicit Adams methods in the framework of the two-loop implicit sparse matrix numerical integration method in solving the differential/algebraic equations of heavily constrained dynamical systems. The variable-step size two-loop implicit sparse matrix numerical integration/Adams method proposed in this investigation avoids numerical force differentiation, ensures satisfying the nonlinear algebraic constraint equations at the position, velocity, and acceleration levels, and allows using sparse matrix techniques for efficiently solving the dynamical equations. The iterative outer loop of the two-loop implicit sparse matrix numerical integration/Adams method is aimed at achieving the convergence of the implicit integration formulae used to solve the independent differential equations of motion, while the inner loop is used to ensure the convergence of the iterative procedure used to satisfy the algebraic constraint equations. To solve the independent differential equations, two different implicit Adams integration formulae are examined in this investigation; a 3rd order implicit Adams-Moulton formula with a 2nd order explicit predictor Adams Bashforth formula, and a 4th order implicit Adams-Moulton formula with a 3rd order explicit predictor Adams Bashforth formula. A standard Newton–Raphson algorithm is used to satisfy the nonlinear algebraic constraint equations at the position level. The constraint equations at the velocity and acceleration levels are linear, and therefore, there is no need for an iterative procedure to solve for the dependent velocities and accelerations. The algorithm used for the error check and step-size change is described. The performance of the two-loop implicit sparse matrix numerical integration/Adams algorithm developed in this investigation is evaluated by comparison with the explicit predictor-corrector Adams method which has a variable-order and variable-step size. Simple and heavily constrained dynamical systems are used to evaluate the accuracy, robustness, damping characteristics, and effect of the outer-loop iterations of the proposed implicit schemes. The results obtained in this investigation show that the two-loop implicit sparse matrix numerical integration methods proposed in this study can be more efficient for stiff systems because of their ability to damp out high-frequency oscillations. Explicit integration methods, on the other hand, can be more efficient in the case of non-stiff systems.

Coordinate Reduction in the Dynamics of Constrained Multibody Systems—A New Approach

1988 ◽  
Vol 55 (4) ◽  
pp. 899-904 ◽  
Author(s):  
S. K. Ider ◽  
F. M. L. Amirouche
Keyword(s):  
Null Space ◽  
Computationally Efficient ◽  
New Approach ◽  
Constraint Equations ◽  
Kane’S Equations ◽  
Linearly Independent ◽  
Kane's Equations ◽  
Constrained Dynamical Systems ◽  
Coordinate Reduction ◽  
Algebraic Constraint

In this paper a new theorem for the generation of a basis for the null space of a rectangular matrix, with m linearly independent rows and n (n > m) columns is presented. The method is based on Gaussian row operations to transform the constraint Jacobian matrix to an uptriangular matrix. The Gram-Schmidt process is then utilized to identify basis vectors orthogonal to the uptriangular matrix. A complement orthogonal array which forms the basis for the null space for which the algebraic constraint equations are satisfied is then formulated. An illustration of the theorem application to constrained dynamical systems for both Lagrange and Kane’s equations is given. A numerical computer algorithm based on Kane’s equations with embedded constraints is also presented. The method proposed is well conditioned and computationally efficient and inexpensive.

scholarly journals A new dynamical systems perspective on atmospheric predictability: Eastern Mediterranean weather regimes as a case study

2019 ◽  
Vol 5 (6) ◽  
pp. eaau0936 ◽  
Author(s):  
Assaf Hochman ◽  
Pinhas Alpert ◽  
Tzvi Harpaz ◽  
Hadas Saaroni ◽  
Gabriele Messori
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Climate Model ◽  
Eastern Mediterranean ◽  
Cmip5 Models ◽  
Local Dimension ◽  
Geographical Regions ◽  
Recent Developments ◽  
New Perspective

The atmosphere is a chaotic system displaying recurrent large-scale configurations. Recent developments in dynamical systems theory allow us to describe these configurations in terms of the local dimension—a proxy for the active number of degrees of freedom—and persistence in phase space, which can be interpreted as persistence in time. These properties provide information on the intrinsic predictability of an atmospheric state. Here, this technique is applied to atmospheric configurations in the eastern Mediterranean, grouped into synoptic classifications (SCs). It is shown that local dimension and persistence, derived from reanalysis and CMIP5 models’ daily sea-level pressure fields, can serve as an extremely informative qualitative method for evaluating the predictability of the different SCs. These metrics, combined with the SC transitional probability approach, may be a valuable complement to operational weather forecasts and effective tools for climate model evaluation. This new perspective can be extended to other geographical regions.

The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems

2011 ◽  
Vol 10 (5) ◽  
pp. 1241-1256 ◽  
Author(s):  
Guo-Kang Er ◽  
Vai Pan Iu
Keyword(s):  
Large Scale ◽  
Joint Probability ◽  
Approximate Solutions ◽  
The State ◽  
Stochastic Dynamic ◽  
State Variables ◽  
High Dimensions ◽  
Polynomial Type ◽  
Fpk Equation ◽  
Stochastic Dynamic Systems

AbstractThe probabilistic solutions of the nonlinear stochastic dynamic (NSD) systems with polynomial type of nonlinearity are investigated with the subspace-EPC method. The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure method. Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases.

TIME DOMAIN DIAKOPTIC ANALYSIS BASED ON REDUCED-ORDER STATE EQUATIONS

2007 ◽  
Vol 17 (10) ◽  
pp. 3625-3631 ◽  
Author(s):  
MIHAI IORDACHE ◽  
LUCIA DUMITRIU
Keyword(s):  
Analog Circuits ◽  
Large Scale ◽  
State Equation ◽  
The State ◽  
State Variables ◽  
State Equations ◽  
Integration Algorithm ◽  
Reduced Order ◽  
Decomposition Procedure ◽  
Minimum Number

In this paper we present some new tearing techniques to systematically formulate the state equations in symbolic normal-form for linear and/or nonlinear time-invariant large-scale analog circuits. The excess elements of the first and of the second kind are unitarily treated in order to allow a symbolic representation of the circuit with a minimum number of state variables. A procedure to reduce the state equation number of each subcircuit is also presented. The reduced-order is based on an implicit integration algorithm and on the successive elimination of the selected state variables. Examples are given to illustrate the decomposition procedure, the assignment of the connection sources and the reduced-order technique.

Modeling and Predictive Control of an Unmanned Underwater Vehicle

2019 ◽  
Author(s):  
Renato Rodriguez Nunez ◽  
Damoon Soudbakhsh
Keyword(s):  
Feedback Linearization ◽  
Degrees Of Freedom ◽  
Travel Speed ◽  
The State ◽  
Operating Conditions ◽  
State Variables ◽  
Unmanned Underwater Vehicles ◽  
Six Degrees Of Freedom ◽  
State Space System ◽  
Predictive Controller

Abstract This paper presents a model and optimal controller for Unmanned Underwater Vehicles (UUVs). We present a nonlinear six degrees of freedom model of the UUV that includes hydrodynamic and hydrostatic terms. To design the controller, we simplify the model using the geometry of the UUV as well as its operating conditions such as the depth and expected travel speed. Instead of designing a controller for the state space system, we used feedback linearization technique to decouple the motions. Then, a set of controllers were designed for each motion. To incorporate the constraints on the input and the state variables, we designed a fast Model Predictive Controller (MPC) for the UUV and compared its performance with a conventional controller.

Dynamics of Large Scale Mechanical Models Using Multilevel Substructuring

2006 ◽  
Vol 2 (1) ◽  
pp. 40-51 ◽  
Author(s):  
C. Papalukopoulos ◽  
S. Natsiavas
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Equations Of Motion ◽  
Numerical Results ◽  
Ground Vehicles ◽  
Response Characteristics ◽  
Sparsity Pattern ◽  
Mechanical Models ◽  
Set Up

An appropriate substructuring methodology is applied in order to study the dynamic response of very large scale mechanical systems. The emphasis is put on enabling a systematic study of dynamical systems with nonlinear characteristics, but the method is equally applicable to systems possessing linear properties. The accuracy and effectiveness of the methodology are illustrated by numerical results obtained for example vehicle models, having a total number of degrees of freedom lying in the order of a million or even bigger. First, the equations of motion of each component are set up by applying the finite element method. The order of the resulting models is so high that the classical substructuring methodologies become numerically ineffective or practically impossible to apply. However, the method developed overcomes these difficulties by imposing a further, multilevel substructuring of each component, based on the sparsity pattern of the stiffness matrix. In this way, the number of the equations of motion of the complete system is substantially reduced. Consequently, the numerical results presented demonstrate that besides the direct computational savings, this reduction in the dimensions enables the application of numerical codes, which capture response characteristics of dynamical systems sufficiently accurate up to a prespecified level of forcing frequencies. The study concludes by investigating biodynamic response of passenger-seat subsystem models coupled with complex mechanical models of ground vehicles resulting from deterministic or random road excitation.

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