Some Techniques for Order Reduction of Nonlinear Time Periodic Systems

2003 ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha ◽  
Eric A. Butcher
Keyword(s):  
Nonlinear Systems ◽  
Invariant Manifold ◽  
Equations Of Motion ◽  
Linear Method ◽  
Order Reduction ◽  
Reduced Order ◽  
Reduction Techniques ◽  
Nonlinear Projection ◽  
Short Period ◽  
Time Periodic

In this paper, some techniques for order reduction of nonlinear systems with time periodic coefficients are introduced. The equations of motion are first trasformed using the Lyapunov-Floquet transformation such that the linear parts of the new set of equations are time-invariant. To reduce the order of this transformed system three model reduction techniques are suggested. The first approach is simply an application of the well-known linear method to nonlinear systems. In the second technique, the idea of singular perturbation and noninear projection are employed, whereas the concept of invariant manifold for time-periodic system forms the basis for the third method. A discussion of nonlinear projection method and time periodic invariant manifold technique is included. The invariant manifold based technique yields a ‘reducibility condition’. This is an important result due to the fact that various types of resonance are present in such systems. If the ‘reducibility condition’ is satisfied only then a nonlinear order reduction is possible. In order to compare the results obtained from various reduced order modeling techniques, an example consisting of two parametrically excited coupled pendulums is included. Reduced order results and full-scale dynamics are used to construct approximate and exact Poincare´ maps, respectively, because it portrays the long-term behavior of system dynamics. This measure is more convincing than just comparing the time traces over a short period of time. It is found that the invariant manifold yields the most accurate results followed by the nonlinear projection and the linear techniques.

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.

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.

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.

scholarly journals Invariant Manifold Based Reduced-Order Observer Design for Nonlinear Systems

2008 ◽  
Vol 53 (11) ◽  
pp. 2602-2614 ◽  
Author(s):  
Dimitrios Karagiannis ◽  
Daniele Carnevale ◽  
Alessandro Astolfi
Keyword(s):  
Nonlinear Systems ◽  
Invariant Manifold ◽  
Observer Design ◽  
Reduced Order ◽  
Reduced Order Observer

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.

Frequency Domain Updating of Thin-Walled Workpiece Dynamics Using Reduced Order Substructuring Method in Machining

2017 ◽  
Vol 139 (7) ◽  
Author(s):  
Oguzhan Tuysuz ◽  
Yusuf Altintas
Keyword(s):  
Frequency Domain ◽  
Equations Of Motion ◽  
Selection Criterion ◽  
Order Reduction ◽  
Fe Model ◽  
Computationally Efficient ◽  
Peripheral Milling ◽  
Reduced Order ◽  
Dynamic Substructuring ◽  
Thin Walled

The structural dynamics of thin-walled parts vary as the material is removed during machining. This paper presents a new, computationally efficient reduced order dynamic substructuring method to predict the frequency response function (FRF) of the workpiece as the material is removed along the toolpath. The contribution of the removed mass to the dynamics of the workpiece is canceled by adding a fictitious substructure having the opposite dynamics of the removed material. The equations of motion of the workpiece are updated, and workpiece FRFs are evaluated by solving the hybrid set of assembled equations of motion in frequency domain as the tool removes the material between two consecutive dynamics update steps. The orders of the initial workpiece structure and the removed substructures are reduced using a model order reduction method with a newly introduced automatic master set selection criterion. The reduced order FRF update model is validated with peripheral milling tests and FRF measurements on a plate-shaped workpiece. It is shown that the proposed model provides ∼20 times faster FRF predictions than the full order finite element (FE) model.

Order Reduction of Nonlinear Time Periodic Systems Using Invariant Manifolds

2003 ◽  
Author(s):  
S. C. Sinha ◽  
Sangram Redkar ◽  
Eric A. Butcher ◽  
Venkatesh Deshmukh
Keyword(s):  
Center Manifold ◽  
Invariant Manifolds ◽  
Basic Problem ◽  
Equations Of Motion ◽  
Order Reduction ◽  
Center Manifold Theory ◽  
Manifold Theory ◽  
Time Invariant ◽  
Time Periodic ◽  
Linear And Nonlinear Systems

The basic problem of order reduction of linear and nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via Time Periodic Center Manifold Theory. A ‘reducibility condition’ is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.

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