Order Reduction of Parametrically Excited Nonlinear Systems Subjected to External Periodic Excitations

2009 ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Fundamental Solution ◽  
Invariant Manifold ◽  
Large Scale ◽  
Order Reduction ◽  
Reduced Order Model ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
And Control

In this work, some techniques for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and/or nonlinearity takes the form of quasiperiodic functions. The techniques proposed here; construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system. Specifically, two methods are outlined to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’, the second novel technique proposed here, utilizes the concept of ‘invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on invariant manifold technique yields unique ‘reducibility conditions’. If these ‘reducibility conditions’ are satisfied only then an accurate order reduction via ‘invariant manifold’ is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. One can also recover ‘resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handing systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. These methodologies are applied to a typical problem 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 system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.

Order Reduction of Nonlinear Systems With Periodic-Quasiperiodic Coefficients

2005 ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Invariant Manifold ◽  
Large Scale ◽  
Evolution Equations ◽  
Linear Method ◽  
Dynamical Evolution ◽  
Order Reduction ◽  
Reduced Order Models ◽  
Reduced Order ◽  
And Control

In this work, some techniques for order reduction of nonlinear systems involving periodic/quasiperiodic coefficients are presented. The periodicity of the linear terms is assumed non-commensurate with the periodicity of either the nonlinear terms or the forcing vector. The dynamical evolution equations are transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of the resulting equations become time-invariant while the nonlinear parts and forcing take the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-modulated functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states. Three methods are proposed to carry out this model order reduction (MOR). First type of MOR technique is a linear method similar to the ‘Guyan reduction’, the second technique is a nonlinear projection method based on singular perturbation while the third method utilizes the concept of ‘quasiperiodic invariant manifold’. Order reduction approach based on invariant manifold technique yields a unique ‘generalized reducibility condition’. If this ‘reducibility condition’ is satisfied only then an accurate order reduction via invariant manifold is possible. Next, the proposed methodologies are extended to solve the forced problem. All order reduction approaches except the invariant manifold technique can be applied in a straightforward way. The invariant manifold formulation is modified to take into account the effects of forcing and nonlinear coupling. This approach not only yields accurate reduced order models but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. One can also recover all ‘resonance conditions’ obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handing systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. 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 system design of large-scale parametrically excited nonlinear systems.

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.

A Direct Approach to Order Reduction of Nonlinear Systems Subjected to External Periodic Excitations

2008 ◽  
Vol 3 (3) ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
State Space ◽  
Invariant Manifold ◽  
Large Scale ◽  
Reduction Process ◽  
Order Reduction ◽  
The State ◽  
Reduced Order Models ◽  
External Excitation ◽  
Reduced Order

In this work, the basic problem of order reduction of nonlinear systems subjected to an external periodic excitation is considered. This problem deserves special attention because modes that interact (linearly or nonlinearly) with external excitation dominate the response. These dominant modes are identified and chosen as the “master” modes to be retained in the reduction process. The simplest idea could be to use a linear approach such as the Guyan reduction and choose those modes whose natural frequencies are close to that of external excitation as the master modes. However, this technique does not guarantee accurate results when nonlinear interactions are strong and a nonlinear approach must be adopted. Recently, the invariant manifold technique has been extended to forced problems by “augmenting” the state space, i.e., forcing is treated as an additional state and an invariant manifold is constructed. However, this process does not provide a clear picture of possible resonances and conditions under which an order reduction is possible. In a direct innovative approach suggested here, a nonlinear time-dependent relationship between the dominant and nondominant 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. One obtains various reducibility conditions in a closed form, which show interactions among eigenvalues, nonlinearities and the external excitation. One can also recover all “resonance conditions” obtained via perturbation or averaging techniques. The “linear” as well as the “extended invariant manifold” techniques 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.

Dynamic Analysis of a Protein-Ligand Molecular Chain Attached to an Atomic Force Microscope

2004 ◽  
Vol 126 (4) ◽  
pp. 496-513 ◽  
Author(s):  
Deman Tang ◽  
Earl H. Dowell
Keyword(s):  
Nonlinear Systems ◽  
Atomic Force Microscope ◽  
Reduced Order Model ◽  
Molecular Chain ◽  
Static Equilibrium ◽  
Reduced Order Models ◽  
Base Excitation ◽  
Order Model ◽  
Reduced Order ◽  
Atomic Force

Dynamic numerical simulation of a protein-ligand molecular chain connected to a moving atomic force microscope (AFM) has been studied. A sinusoidal base excitation of the cantilevered beam of the AFM is considered in some detail. A comparison between results for a single molecule and those for multiple molecules has been made. For a small number of molecules, multiple stable static equilibrium positions are observed and chaotic behavior may be generated via a period-doubling cascade for harmonic base excitation of the AFM. For many molecules in the chain, only a single static equilibrium position exists. To enable these calculations, reduced-order (dynamic) models are constructed for fully linear, combined linear/nonlinear and fully nonlinear systems. Several distinct reduced-order models have been developed that offer the option of increased computational efficiency at the price of greater effort to construct the particular reduced-order model. The agreement between the original and reduced-order models (ROM) is very good even when only one mode is included in the ROM for either the fully linear or combined linear/nonlinear systems provided the excitation frequency is lower than the fundamental natural frequency of the linear system. The computational advantage of the reduced-order model is clear from the results presented.

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.

scholarly journals Dominant Pole Based Approximation for Discrete Time System

2019 ◽  
Vol 4 (1) ◽  
pp. 56-65 ◽  
Author(s):  
Richa ◽  
Awadhesh Kumar
Keyword(s):  
Discrete Time ◽  
Model Order Reduction ◽  
Large Scale ◽  
Order Reduction ◽  
Reduced Order Model ◽  
Discrete Time System ◽  
Order Model ◽  
Model Order ◽  
Reduced Order ◽  
Dominant Pole

This paper presents an effective procedure for model order reduction of discrete time control system. The exact model derived from complex dynamic systems proves to be very complicated for analysis, control and design. This necessity brings about using a tool known as model order reduction technique or model simplification. A novel mixed method has been implemented in this paper for reducing the order of the large scale dynamic discrete system. Dominant pole based pole clustering method has been used to derive the coefficients of denominator polynomial while Padé approximation has been applied to obtain the coefficients of numerator polynomial of the reduced order model. The proposed method is quite simple and able to generate a stable reduced order model from high order stable discrete systems. The dominancy of poles has been decided by values of the ratio of residue to its pole. The pole is considered dominant which have larger ratio value. An illustrative example has been considered to show the various reduction steps. The result obtained confirms the effectiveness of the approach.

Order Reduction of Bilinear Systems Containing a Clearance

2001 ◽  
Author(s):  
Eric A. Butcher
Keyword(s):  
Arbitrary Dimension ◽  
Order Reduction ◽  
Bilinear Systems ◽  
Reduced Order Model ◽  
Reduced Order Models ◽  
Full Model ◽  
Order Model ◽  
Reduced Order ◽  
Present Technique ◽  
Frequency Relation

Abstract A technique for order reduction of nonsmooth bilinear systems of arbitrary dimension containing a clearance is presented. First, a linear-based order reduction transformation is applied. When only one master coordinate is retained, the frequency of the reduced order model may be used to approximate the bilinear normal mode (BNM) frequency of the full model. By utilizing other approximation methods previously developed via the well-known bilinear frequency relation (BFR), this result is in turn used to construct improved reduced order models whose frequencies are even better approximations to the BNM frequencies for the full model. The technique is applied to two and four degree-of-freedom bilinear systems with nonvanishing clearance or interference. The resulting approximate frequencies are compared with those obtained from numerical simulations. The advantages of the present technique include a reduced order model which uses a subset of the original physical coordinates, contains the form of the nonsmooth nonlinearity of the full model, and whose time series closely tracks that of the corresponding BNM in the full model.

Model Order Reduction of Transmission Line Model

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Transmission Line ◽  
Large Scale ◽  
Original System ◽  
Reduced Order Model ◽  
Benchmark Problems ◽  
Transmission Line Model ◽  
Order Model ◽  
Line Model ◽  
Reduced Order ◽  
Numerical Effort

Transmission Line model are an important role in the electrical power supply. Modeling of such system remains a challenge for simulations are necessary for designing and controlling modern power systems.In order to analyze the numerical approach for a benchmark collection Comprehensive of some needful real-world examples, which can be utilized to evaluate and compare mathematical approaches for model reduction. The approach is based on retaining the dominant modes of the system and truncation comparatively the less significant once.as the reduced order model has been derived from retaining the dominate modes of the large-scale stable system, the reduction preserves the stability. The strong demerit of the many MOR methods is that, the steady state values of the reduced order model does not match with the higher order systems. This drawback has been try to eliminated through the Different MOR method using sssMOR tools. This makes it possible for a new assessment of the error system Offered that the Observability Gramian of the original system has as soon as been thought about, an H∞ and H2 error bound can be calculated with minimal numerical effort for any minimized model attributable to The reduced order model (ROM) of a large-scale dynamical system is essential to effortlessness the study of the system utilizing approximation Algorithms. The response evaluation is considered in terms of response constraints and graphical assessments. the application of Approximation methods is offered for arising ROM of the large-scale LTI systems which consist of benchmark problems. The time response of approximated system, assessed by the proposed method, is also shown which is excellent matching of the response of original system when compared to the response of other existing approaches .

Decentralized Control of a Tower Crane

1999 ◽  
Author(s):  
Kiyoshi Takagi ◽  
Hidekazu Nishimura
Keyword(s):  
Decentralized Control ◽  
Three Dimensional ◽  
Reduced Order Model ◽  
Flexible Structure ◽  
Order Model ◽  
Modeling And Control ◽  
Reduced Order ◽  
Tower Crane ◽  
Three Dimensional Models ◽  
And Control

Abstract This paper deals with modeling and control of a crane mounted on a tower-like flexible structure. A fast transfer of the load causes the sway of the load rope and the vibration of the flexible structure. Our object is to control both the sway and the vibration by the inherent capability of the tower crane. This paper makes its three-dimensional models for simulation and reduced-order-model in order to design the decentralized control system. Then, we design the decentralized H∞ compensator and verify the efficiency by simulations and experiments.

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