Balanced structures and model reduction of unstable systems

Zilouchian, Ali

doi:10.1109/secon.1991.147956

Cited by 21 publications

(22 citation statements)

References 9 publications

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“…Even the optimal rational interpolation-based model reduction methods have been extended to unstable systems; see, e.g., [143,165]. Balanced truncation has been also generalized to reducing unstable systems, by either appropriately redefining the system Gramians [25,232,233] or by using different balancing techniques, such as LQG balancing [185]. For the POD, the frequency domain formulation will be the appropriate choice for unstable systems since the time-domain snapshots will grow exponentially in this case.…”

Section: 7mentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: 7mentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…In the second approach (directly OR method for unstable system), the order of the unstable system is directly reduced under the extended balanced truncation algorithm of Zhou [7], algorithm of LQG [6], elemental analysis [2], applied balanced truncation algorithm of Boess [5], and extended balanced truncation of Zilochian [3]. The advantage of this approach is that order of OR system does not depend on order of the unstable system, which means order of this OR system can be lower than the unstable one's.…”

Section: Introductionmentioning

confidence: 99%

The extended balanced truncation algorithm

Nguyen¹

2016

ijco

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Model order reduction is a research direction which has attracted more interest to scientists in recent years. Actually there are many researches on model reduction for higher-order linear system but those on model reduction models for unstable higher-order linear system are still limited and existed many disadvantages. This paper presents extended balanced truncation algorithm for unstable linear system. At the same time, to complete extended balanced truncation algorithm for unstable linear system, the author gives one definition and two new theorems with adequate proofs to determine the upper bound formula of order reduction error thereby the algorithm can automatically reduce order of linear unstable system based on that formula. The illustration shows the correctness of the model order algorithm.

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“…An approach of adaptation is to shift all RHP poles or eigenvalues of an unstable system into the stable region. This Pole-shifting approach has been utilized along with the Routh approximation method, the Wilson optimization formulation, or the balancing reduction method, to derive reduced models for unstable systems [11][12][13]. However, it can not guarantee that the number of RHP poles of the reduced model is equal to that of the original unstable system.…”

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confidence: 99%

A simple method for L<inf>2</inf> approximation of unstable and nonminimum-phase systems

Zhou

Chai

Wu³

2010

2010 Chinese Control and Decision Conference

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Model approximation is very important in many areas of engineering sciences, especially in model-based control and optimization. The problem of model approximation can be briefly described as follows: Given a high-order model, find a reduced-order approximation which is optimal in some sense. Over the last several decades, model approximation problems have received considerable attention [1, 2]; numerous approaches have been proposed, and may be grouped into two major categories, the performance-oriented and the non-performance-oriented approaches [3]. In using a model reduction method that is non-performance-oriented, the original model is first transformed into a canonical form, e.g., the balanced state-space representation, upon which a direct truncation procedure is performed in order to obtain an order-reduced model [3]. Examples of this category include the classical Padé approximation method [4], the Routh approximation method [5], the balanced truncation method [6], and many of their variants. For a performance-oriented model reduction approach, reduced-order models are obtained by minimizing certain performance criterion, such as the H 2 -norm, the L 2 -norm, and the L -norm [2, 3,7-9]. The above mentioned model approximation techniques were originally propose for stable systems. However, there are many situations where the system is inherently unstable, nonminimum-phase and of relatively high dynamic Abstract: A simple performance-oriented method is proposed for approximation the unstable and nonminimum-phase systems. The performance index is taken to be a frequency-domain L 2 -norm of the error between the step responses of the original and the reduced model. A frequency-domain weighted recursive least squares (RLS) algorithm is deduced and adopted to search for the optimal parameters of the approximation model. A numerical example is used to show the effectiveness of the proposed scheme. It is shown that the approximation models obtained by the proposed scheme yield zero steady-state gain errors and have the same RHP pole and RHP zero as the original system, resulting in well-matched dynamic behavior.

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Balanced structures and model reduction of unstable systems

Cited by 21 publications

References 9 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

The extended balanced truncation algorithm

A simple method for L<inf>2</inf> approximation of unstable and nonminimum-phase systems

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