An Efficient Approximated Algorithm for Balancing and Order Reduction of Lightly Damped Systems

1995 ◽  
Author(s):  
Yoram Halevi
Keyword(s):  
Substantial Reduction ◽  
Mechanical Systems ◽  
Order Reduction ◽  
Transformation Matrix ◽  
Special Structure ◽  
Reduced Order ◽  
Controllability Gramian ◽  
Balancing Transformation ◽  
Damped Systems ◽  
Observability Gramian

Abstract A method of approximating the controllability gramian, observability gramian and the balancing transformation for lightly damped mechanical systems is presented, the approximation uses the special structure of the system and the fact that the damping is small to reduce the amount of computation considerably. Furthermore, one can avoid the calculation of the entire balancing transformation matrix and calculate only the parts that are required for order reduction. In cases where the reduced order is much smaller than the original that leads to another substantial reduction of computation effort.

scholarly journals Multimodeling Control via System Balancing

2010 ◽  
Vol 2010 ◽  
pp. 1-20
Author(s):  
Nada Ratković Kovačević ◽  
Dobrila Škatarić
Keyword(s):  
Power Plants ◽  
Order Reduction ◽  
Reduced Order Models ◽  
Linear Quadratic ◽  
Linear Quadratic Gaussian ◽  
Reduced Order ◽  
Reduction Techniques ◽  
Balanced Model ◽  
Balancing Transformation ◽  
Different Order

A new approach in multimodeling strategy is proposed. Multimodel strategies in which control agents use different simplified models of the same system are being developed using balancing transformation and the corresponding order reduction concepts. Traditionally, the multimodeling concept was studied using the ideas of multitime scales (singular perturbations) and weak subsystem coupling. For all reduced-order models obtained, a Linear Quadratic Gaussian (LQG) control problem was solved. Different order reduction techniques were compared based on the values of the optimized criteria for the closed-loop case where the full-order balanced model utilizes regulators calculated to be the optimal for various reduced-order models. The results obtained were demonstrated on a real-world example: a multiarea power system consisting of two identical areas, that is, two identical power plants.

scholarly journals Order Reduction based on Coulomb’s and Franklin’s laws Algorithm

2021 ◽  
Author(s):  
Ram Kumar ◽  
Afzal Sikander
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Order Reduction ◽  
Absolute Error ◽  
Search Space ◽  
Engineering Problem ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Linear Time Invariant

Abstract The Coulomb and Franklin laws (CFL) algorithm is used to construct a lower order model of higher-order continuous time linear time-invariant (LTI) systems in this study. CFL is quite easy to implement in obtaining reduced order model of large scale system in control engineering problem as it employs the combined effect of Coulomb’s and Franklin’s laws to find the best values in search space. The unknown coefficients are obtained using the CFLA methodology, which minimises the integral square error (ISE) between the original and proposed ROMs. To achieve the reduced order model, five practical systems of different orders are considered. Finally, multiple performance indicators such as the ISE, integral of absolute error (IAE), and integral of time multiplied by absolute error were calculated to determine the efficacy of the proposed methodology. The simulation results were compared to previously published well-known research.

Order Reduction of Controller Based on LMIs Improving Error Bounds

1997 ◽  
Author(s):  
Roberd Saragih ◽  
Yoshida Kazuo
Keyword(s):  
Projection Method ◽  
Order Reduction ◽  
Feasibility Problem ◽  
Closed Loop System ◽  
Reduced Order ◽  
Bounded Real Lemma ◽  
Convex Feasibility ◽  
Alternating Projection ◽  
Alternating Projection Method ◽  
Order Reduction Method

Abstract In this paper, we propose an order reduction method of controller based on combination of the alternating projection method and the balanced truncation. In this method both the errors of controller and the closed-loop system caused by the reduced-order controller can be improved simultaneously. By using a generalized Bounded Real Lemma, a feasible reduced-order controller can be derived. The sufficient condition for the existence of a reduced-order controller leads to a non-convex feasibility problem. To solve the problem, we can use an improved computational scheme based on the alternating projection method. But it is needed so much time to solve the problem if compared by the other methods. To validate the proposed method, some numerical calculations and simulations are carried out.

scholarly journals Toward Optimality of Proper Generalised Decomposition Bases

2019 ◽  
Vol 24 (1) ◽  
pp. 30 ◽  
Author(s):  
Shadi Alameddin ◽  
Amélie Fau ◽  
David Néron ◽  
Pierre Ladevèze ◽  
Udo Nackenhorst
Keyword(s):  
Greedy Algorithms ◽  
Order Reduction ◽  
Nonlinear Material ◽  
Proper Generalized Decomposition ◽  
Model Order ◽  
Reduced Order ◽  
Structural Problems ◽  
Low Dimensional ◽  
Value Decomposition ◽  
The Greedy Algorithm

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method.

scholarly journals Study of methods for dimension reduction of complex dynamic linear systems models

2018 ◽  
Vol 226 ◽  
pp. 04036
Author(s):  
Yuriy M. Manatskov ◽  
Torsten Bertram ◽  
Danil V. Shaykhutdinov ◽  
Nikolay I. Gorbatenko
Keyword(s):  
Linear Systems ◽  
Main Idea ◽  
Computation Time ◽  
Order Reduction ◽  
Complex Dynamic ◽  
Model Order ◽  
Reduced Order ◽  
Advantages And Disadvantages ◽  
Optimization Stage ◽  
Linear Systems Of Equations

Complex dynamic linear systems of equations are solved by numerical iterative methods, which need much computation and are timeconsuming ones, and the optimization stage requires repeated solution of these equation systems that increases the time on development. To shorten the computation time, various methods can be applied, among them preliminary (estimated) calculation or oversimple models calculation, however, while testing and optimizing the full model is used. Reduced order models are very popular in solving this problem. The main idea of a reduced order model is to find a simplified model that may reflect the required properties of the original model as accurately as possible. There are many methods for the model order reduction, which have their advantages and disadvantages. In this article, a method based on Krylov subspaces and SVD methods is considered. A numerical experiments is given.

Order Reduction of Nonlinear Systems Subjected to an External Periodic Excitation

2005 ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
State Space ◽  
Invariant Manifold ◽  
Large Scale ◽  
Order Reduction ◽  
Reduced Order Models ◽  
Periodic Excitation ◽  
External Excitation ◽  
Reduced Order ◽  
Linear Approach

In this work, the basic problem of order reduction nonlinear systems subjected to an external periodic excitation is considered. This problem deserves attention because the modes that interact (linearly or nonlinearly) with the external excitation dominate the response. A linear approach like the Guyan reduction does not always guarantee accurate results, particularly when nonlinear interactions are strong. In order to overcome limitations of the linear approach, a nonlinear order reduction methodology through a generalization of the invariant manifold technique is proposed. Traditionally, the invariant manifold techniques for unforced problems are extended to the forced problems by ‘augmenting’ the state space, i.e., forcing is treated as an additional degree of freedom and an invariant manifold is constructed. However, in the approach suggested here a nonlinear time-dependent relationship between the dominant and the non-dominant states is assumed and the dimension of the state space remains the same. This methodology not only yields accurate reduced order models but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. Following this approach, various ‘reducibility conditions’ are obtained that show interactions among the eigenvalues, the nonlinearities and the external excitation. One can also recover all ‘resonance conditions’ commonly obtained via perturbation or averaging techniques. These methodologies are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control of large-scale externally excited nonlinear systems.

scholarly journals A Broadband Enhanced Structure-Preserving Reduced-Order Interconnect Macromodeling Method for Large-Scale Equation Sets of Transient Interconnect Circuit Problems

Energies ◽  
2020 ◽  
Vol 13 (21) ◽  
pp. 5746
Author(s):  
Ning Wang ◽  
Huifang Wang ◽  
Shiyou Yang
Keyword(s):  
Large Scale ◽  
Krylov Subspace ◽  
Equivalent Circuit Model ◽  
Order Reduction ◽  
Wide Band ◽  
Circuit Model ◽  
Band Frequency ◽  
Reduced Order ◽  
Structure Preserving ◽  
Expansion Point

In the transient analysis of an engineering power electronics device, the order of its equivalent circuit model is excessive large. To eliminate this issue, some model order reduction (MOR) methods are proposed in the literature. Compared to other MOR methods, the structure-preserving reduced-order interconnect macromodeling (SPRIM) based on Krylov subspaces will achieve a higher reduction radio and precision for large multi-port Resistor-Capacitor-Inductor (RCL) circuits. However, for very wide band frequency transients, the performance of a Krylov subspace-based MOR method is not satisfactory. Moreover, the selection of the expansion point in this method has not been comprehensively studied in the literature. From this point of view, a broadband enhanced structure-preserving reduced-order interconnect macromodeling (SPRIM) method is proposed to reduce the order of equation sets of a transient interconnect circuit model. In addition, a method is introduced to determine the optimal expansion point at each frequency in the proposed method. The proposed method is validated by the numerical results on a transient problem of an insulated-gate bipolar transistor (IGBT)-based inverter busbar under different exciting conditions.

scholarly journals A Reduced Order Modeling Technique for Mistuned Bladed Disks

1997 ◽  
Vol 119 (3) ◽  
pp. 439-447 ◽  
Author(s):  
M. P. Castanier ◽  
G. O´ttarsson ◽  
C. Pierre
Keyword(s):  
Monte Carlo ◽  
Degrees Of Freedom ◽  
Order Reduction ◽  
Reduced Order Modeling ◽  
Modeling Technique ◽  
Mode Analysis ◽  
Bladed Disks ◽  
Engineering Structures ◽  
Element Analysis ◽  
Reduced Order

The analysis of the response statistics of mistuned turbomachinery rotors requires an expensive Monte Carlo simulation approach. Simple lumped parameter models capture basic localization effects but do not represent well actual engineering structures without a difficult parameter identification. Current component mode analysis techniques generally require a minimum number of degrees of freedom which is too large for running Monte Carlo simulations at a reasonable cost. In the present work, an order reduction method is introduced which is capable of generating reasonably accurate, very low order models of tuned or mistuned bladed disks. This technique is based on component modes of vibration found from a finite element analysis of a single disk-blade sector. It is shown that the phenomenon of mode localization is well captured by the reduced order modeling technique.

Model Simplification and Stability Robustness With Elastic Flight Vehicles

1995 ◽  
Vol 117 (3) ◽  
pp. 336-342
Author(s):  
Brett Newman ◽  
David K. Schmidt
Keyword(s):  
Closed Loop ◽  
Order Reduction ◽  
Higher Order ◽  
Reduced Order Model ◽  
Control Law ◽  
Model Simplification ◽  
Closed Loop System ◽  
Order Model ◽  
Reduced Order ◽  
Stability Robustness

Quantitative criteria are presented for model simplification, or order reduction, such that the reduced order model may be used to synthesize and evaluate a control law, and the stability and stability robustness obtained using the reduced order model will be preserved when controlling the higher order system. The error introduced due to model simplification is treated as modeling uncertainty, and some of the results from multivariable robustness theory are brought to bear on the model simplification problem. Also, the importance of the control law itself, in meeting the modeling criteria, is underscored. A weighted balanced order reduction technique is shown to lead to results that meet the necessary criteria. The procedure is applied to an aeroelastic vehicle model, and the results are used for control law development. Critical robustness properties designed into the lower order closed-loop system are shown to be present in the higher order closed-loop system.

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