Order Reduction of Discrete-Time System Via Bilinear Routh Approximation

1990 ◽  
Vol 112 (2) ◽  
pp. 292-297 ◽  
Author(s):  
Chyi Hwang ◽  
Ching-Shieh Hsieh
Keyword(s):  
Transfer Function ◽  
Approximation Method ◽  
Order Reduction ◽  
Reduced Order Models ◽  
Discrete Time System ◽  
Initial Value ◽  
Time System ◽  
Bilinear Transformation ◽  
Reduced Order ◽  
Canonical Expansion

In this paper, a method of combining the Routh approximation method with the bilinear transformation is presented for deriving stable reduced-order models of a strictly proper z-transfer function Gn(z). It is based on applying the bilinear transformation to the (z+1)Gn(z), and then deriving a new bilinear Routh γ − δ canonical expansion for Gn(z). The proposed bilinear Routh approximation method has all the advantages of the Routh approximation method [5] while without having initial-value problem caused by the bilinear transformation. A numerical example is included to illustrate the procedure.

Suboptimal Model Reduction by Multi-Point Padé Approximation

1994 ◽  
Vol 208 (2) ◽  
pp. 131-134 ◽  
Author(s):  
T N Lucas
Keyword(s):  
Transfer Function ◽  
Model Reduction ◽  
Approximation Method ◽  
Padé Approximation ◽  
Order Reduction ◽  
Pade Approximation ◽  
Reduced Order ◽  
Numerical Example ◽  
Integral Square Error ◽  
Reduction Methods

A novel Padé approximation method is used to obtain a reduced-order transfer function, with a predetermined denominator, such that the integral square error between the time responses of the full and reduced models is minimized. The method is seen to be easy to apply compared with existing suboptimal order reduction methods. A numerical example is given to illustrate its application.

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 Frequency Weighted Discrete-Time Controller Order Reduction Using Bilinear Transformation

2011 ◽  
Vol 62 (1) ◽  
pp. 44-48 ◽  
Author(s):  
Paknosh Karimaghaee ◽  
Navid Noroozi
Keyword(s):  
Discrete Time ◽  
Closed Loop ◽  
Linear Time ◽  
Order Reduction ◽  
Bilinear Transformation ◽  
Reduced Order ◽  
Linear Time Invariant ◽  
The Stability ◽  
Controllability And Observability ◽  
Time Invariant

Frequency Weighted Discrete-Time Controller Order Reduction Using Bilinear TransformationThis paper addresses a new method for order reduction of linear time invariant discrete-time controller. This method leads to a new algorithm for controller reduction when a discrete time controller is used to control a continuous time plant. In this algorithm, at first, a full order controller is designed ins-plane. Then, bilinear transformation is applied to map the closed loop system toz-plane. Next, new closed loop controllability and observability grammians are calculated inz-plane. Finally, balanced truncation idea is used to reduce the order of controller. The stability property of the reduced order controller is discussed. To verify the effectiveness of our method, a reduced controller is designed for four discs system.

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.

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 New Results On Partial Fraction Based Frequency Weighted Balanced Truncation Technique For Discrete–Time System

2012 ◽  
Author(s):  
Shafishuhaza Sahlan ◽  
Victor Sreeram
Keyword(s):  
Discrete Time ◽  
A Priori ◽  
Partial Fraction ◽  
Discrete Time System ◽  
Time System ◽  
Model Order ◽  
Reduced Order ◽  
Partial Fraction Expansion ◽  
Weighted Model ◽  
Model Reduction Technique

Artikel in membentangkan keputusan baru bagi kaedah pengurangan model frekuensi tertimbang berdasarkan kaedah pengembangan fraksi separa didalam masa diskrit. Rangka pengurangan model bagi kaedah baru yang dicadangkan diperolehi melalui pemotongan langsung, menghasilkan kesalahan yang lebih rendah berbanding kaedah–kaedah lain yang sudah ada. Kaedah baru ini dijamin akan stabil bahkan untuk tertimbang bersisi ganda. Sebuah batas kesalahan a priori yang mudah dan senang dihitung juga diperoleh. Contoh berangka dengan perbandingan dengan teknik yang ada menunjukkan keberkesanan kaedah yang dicadangkan Kata kunci: Rangka pengurangan model; batas kesalahan; kaedah pengembangan fraksi spara In this paper, we present some new results on frequency weighted model reduction technique based on partial fraction expansion idea in discrete–time system. The reduced order models of the newly proposed method obtained by direct truncation, produces lower errors when compared to existing techniques. The new method is guaranteed to be stable even for double sided weightings. A simple and easily computable a priori error bound is also derived. Numerical examples with comparisons to the existing techniques show the effectiveness of the proposed method. Key words: Model order reductions; error bounds; partial fraction expansion

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 in electrical power systems using singular perturbation in different coordinate systems

2018 ◽  
Vol 37 (4) ◽  
pp. 1525-1534
Author(s):  
Hauke Huisinga ◽  
Lutz Hofmann
Keyword(s):  
Singular Perturbation ◽  
Power Systems ◽  
Order Reduction ◽  
Equation System ◽  
Electric Power System ◽  
Reduced Order Models ◽  
Differential Algebraic Equation ◽  
Singular Perturbation Method ◽  
Content Type ◽  
Reduced Order

Purpose Efficient calculations of the transient behaviour after disturbances of large-scale power systems are complex because of, among other things, the non-linearity and the stiffness of the overall state equation system (SES). Because of the rising amount of flexible transmission system elements, there is an increasing need for reduced order models with a negligible loss of accuracy. With the Extended Nodal Approach and the application of the singular perturbation method, it is possible to reduce the order of the SES adapted to the respective setting of the desired tasks and accuracy requirements. Design/methodology/approach Based on a differential-algebraic equation for the electric power system which is formulated with the Extended Nodal Approach, the automatic decomposition into reduced order models is shown in this paper. The paper investigates the effects of different coordinate systems for an automatic order reduction with the singular perturbation method, as well as a comparison of results calculated with the full and reduced order models. Findings The eigenvalues of the full system are approximated sufficiently by the three subsystems. A simulation example demonstrates the good agreement between the reduced order models and the full model independent of the choice of the coordinate system. The decomposed subsystems in rotating coordinates have benefits as compared to those in static coordinates. Originality/value The paper presents a systematic decomposition based only on a differential-algebraic equation system of the electric power system into three subsystems.

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