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.

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.

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.

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.

Orthogonal Polynomials Approximation and Balanced Truncation for a Lowpass Filter

2013 ◽  
Vol 11 (4) ◽  
pp. 2-16
Author(s):  
K. Perev
Keyword(s):  
Orthogonal Polynomials ◽  
Large Scale ◽  
Computational Effort ◽  
Balanced Truncation ◽  
Reduced Order Models ◽  
Lowpass Filter ◽  
Analog Filter ◽  
Reduced Order ◽  
System Properties ◽  
Balancing Transformation

Abstract This paper considers the problem of orthogonal polynomial approximation based balanced truncation for a lowpass filter. The proposed method combines the system properties of balanced truncation, the computational effectiveness of proper orthogonal decomposition and the approximation capability of the orthogonal polynomials approximation. Orthogonal polynomials series expansion of the reachability and observability gramians is used in order to avoid solving large-scale Lyapunov equations and thus, significantly reducing the computational effort for obtaining the balancing transformation. The proposed method is applied for model reduction of a lowpass analog filter. Different sets of orthonormal functions are obtained from Legendre, Laguerre and Chebyshev orthogonal polynomials and the corresponding reduced order models are compared. The approximation precision is measured by the relative mean square error between the outputs of the full order model and the obtained reduced order models.

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.

Time-limited Gramians-based model order reduction for second-order form systems

2018 ◽  
Vol 41 (8) ◽  
pp. 2310-2318 ◽  
Author(s):  
Shafiq Haider ◽  
Abdul Ghafoor ◽  
Muhammad Imran ◽  
Fahad Mumtaz Malik
Keyword(s):  
Model Order Reduction ◽  
Large Scale ◽  
Order Reduction ◽  
Second Order ◽  
Reduced Order Models ◽  
Infinite Time ◽  
Model Order ◽  
Reduced Order ◽  
Order Form ◽  
Second Order Systems

A new scheme for model order reduction of large-scale second-order systems in time-limited intervals is presented. Time-limited Gramians that are solutions of continuous-time algebraic Lyapunov equations for second-order form systems are introduced. Time-limited second-order balanced truncation procedures with provision of balancing position and velocity Gramians are formulated. Stability conditions for reduced-order models are stated and algorithms that preserve stability in reduced-order models are discussed. Numerical examples are presented to validate the superiority of the proposed scheme compared with the infinite-time Gramians technique for time-limited applications.

LQG Control for Nonstandard Singularly Perturbed Discrete-Time Systems

2004 ◽  
Vol 126 (4) ◽  
pp. 860-864 ◽  
Author(s):  
Beom-Soo Kim ◽  
Young-Joong Kim ◽  
Myo-Taeg Lim
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Discrete Time ◽  
Singularly Perturbed ◽  
Linear Quadratic ◽  
Linear Quadratic Gaussian ◽  
Reduced Order ◽  
Discrete Time Systems ◽  
Time Systems

In this paper we present a control method and a high accuracy solution technique in solving the linear quadratic Gaussian problems for nonstandard singularly perturbed discrete time systems. The methodology that exists in the literature for the solution of the standard singularly perturbed discrete time linear quadratic Gaussian optimal control problem cannot be extended to the corresponding nonstandard counterpart. The solution of the linear quadratic Gaussian optimal control problem is obtained by solving the pure-slow and pure-fast reduced-order continuous-time algebraic Riccati equations and by implementing the pure-slow and pure-fast reduced-order Kalman filters. In order to show the effectiveness of the proposed method, we present the numerical result for a one-link flexible robot arm.

scholarly journals Frequency Weighted Model Order Reduction Technique and Error Bounds for Discrete Time Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Imran ◽  
Abdul Ghafoor ◽  
Victor Sreeram
Keyword(s):  
Frequency Response ◽  
Model Reduction ◽  
Discrete Time ◽  
Error Bounds ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Reduction Techniques ◽  
Discrete Time Systems ◽  
Weighted Model ◽  
Time Systems

Model reduction is a process of approximating higher order original models by comparatively lower order models with reasonable accuracy in order to provide ease in design, modeling and simulation for large complex systems. Generally, model reduction techniques approximate the higher order systems for whole frequency range. However, certain applications (like controller reduction) require frequency weighted approximation, which introduce the concept of using frequency weights in model reduction techniques. Limitations of some existing frequency weighted model reduction techniques include lack of stability of reduced order models (for two sided weighting case) and frequency response error bounds. A new frequency weighted technique for balanced model reduction for discrete time systems is proposed. The proposed technique guarantees stable reduced order models even for the case when two sided weightings are present. Efficient technique for frequency weighted Gramians is also proposed. Results are compared with other existing frequency weighted model reduction techniques for discrete time systems. Moreover, the proposed technique yields frequency response error bounds.

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