On properties of gramians of continuous control systems

Yadykin, Igor B.

doi:10.1134/s0005117910060032

Cited by 40 publications

(9 citation statements)

References 4 publications

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“…The study of the properties of these forms can be conveniently transferred to the complex domain using the Laplace transform of these functions [22,23] (i) if the spectrum of the matrix A contains multiple eigenvalues s δ of multiplicity m δ , then the following relations are valid:…”

Section: Formulation Of the Problemmentioning

confidence: 99%

On the methods for calculation of grammians and their use in analysis of linear dynamic systems

Yadykin

Galyaev

2013

Autom Remote Control

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Consideration was given to the methods for solution of the differential and algebraic Lyapunov and Sylvester equations in the time and frequency domains. Their solutions are represented as various finite and infinite grammians. The proposed approach to calculation of the grammians lies in expanding them as the sums of the matrix bilinear or quadratic forms generated with the use of the Faddeev matrices and representing each the solution of the linear matrix algebraic equation corresponding to an individual matrix eigenvalue. A lemma was proved representing explicitly the finite and infinite grammians as the matrix exponents depending on the combined spectrum of the original matrices. This result is generalized to the cases where the spectrum of one matrix contains an eigenvalue of the multiplicity two. Examples illustrating calculation of the finite and infinite grammians were discussed.

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Section: Formulation Of the Problemmentioning

confidence: 99%

On the methods for calculation of grammians and their use in analysis of linear dynamic systems

Yadykin

Galyaev

2013

Autom Remote Control

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“…Here we briefly remind the basic results on eigenmode decomposition of Gramians obtained in (Yadykin (2010), ). We consider the following matrix differential Sylvester equation with real or complex matrices: dP (t) dt = AP (t) + P (t)Ã + Q…”

Section: Preliminariesmentioning

confidence: 99%

“…In this paper we extend the method of eigenmode decomposition of Gramians proposed in (Yadykin (2010), Ahmetzyanov et al (2012), ) for the small-signal stability analysis of weakly stable power systems. The proposed method is related with the development of Lyapunov's direct method for the finitedimensional dynamical systems (Lyapunov (1893), Sylvester (1884), Polyak, Shcherbakov (2002), Kundur (1994)).…”

Section: Introductionmentioning

confidence: 98%

Stability analysis of electric power systems using finite Gramians**This work was supported by research project No.14-08-01098-a of the Russian Foundation for Basic Research (RFBR).

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We present an extension of the method of eigenmode decomposition of Gramians proposed earlier for the the small-signal stability analysis of dynamical systems. In this paper we derive the spectral decomposition for finite Gramians on any time interval and with arbitrary initial conditions. These expansions allow both a stability analysis of non-stationary systems and a monitoring of instability development in unstable systems. Eigen components in the expansion of Gramians on a finite interval of time we called finite sub-Gramians. Because each sub-Gramian is associated with a particular eigenvector, the sources of instability can be easily localized and tracked in real time. We perform a simulation experiment in which finite subGramians are used to analyze the development of instability arising in an actual power grid on Russky Island when it is disconnected from the mainland network.

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“…Because each term is associated with a certain characteristic state vector, the potential sources of instability can easily be located and tracked over time. The proposed estimate of the system energy through its sub-Gramians (28) is not redundant, because for some definitions of the Gramian, this inequality becomes the equality (29).…”

Section: Definitionmentioning

confidence: 99%

Stability analysis of large‐scale dynamical systems by sub‐Gramian approach

Yadykin

Iskakov

Akhmetzyanov

2013

Intl J Robust & Nonlinear

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SUMMARY In this paper, we consider two methods for solving differential and algebraic Lyapunov equations in the time and frequency domains. Solutions of these equations are finite and infinite Gramians. In the first approach, we use the Laplace transform to solve the equations, and we apply the expansion of the matrix resolvent of the dynamical system. The expansions are bilinear and quadratic forms of the Faddeev matrices generated by resolvents of the original matrices. The second method allows computation of an infinite Gramian of a stable system as a sum of sub‐Gramians, which characterize the contribution of eigenmodes to the asymptotic variation of the total system energy over an infinite time interval. Because each sub‐Gramian is associated with a particular eigenvector, the potential sources of instability can easily be localized and tracked in real time. When solutions of Lyapunov equations have low‐rank structure typical of large‐scale applications, sub‐Gramians can be represented in low‐rank factored form, which makes them convenient in the stability analysis of large systems. Our numerical tests for Kundur's four‐machine two‐area system confirm the suitability of using Gramians and sub‐Gramians for small‐signal stability analyses of electric power systems. Copyright © 2013 John Wiley & Sons, Ltd.

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On properties of gramians of continuous control systems

Cited by 40 publications

References 4 publications

On the methods for calculation of grammians and their use in analysis of linear dynamic systems

On the methods for calculation of grammians and their use in analysis of linear dynamic systems

Stability analysis of electric power systems using finite Gramians**This work was supported by research project No.14-08-01098-a of the Russian Foundation for Basic Research (RFBR).

Stability analysis of large‐scale dynamical systems by sub‐Gramian approach

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