On the solution of the continuous-time Lyapunov matrix equation in two canonical forms

Xiao, Chengshan; Feng, Zhao; Shan, Xiuming

doi:10.1049/ip-d.1992.0038

Cited by 28 publications

(37 citation statements)

References 10 publications

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“…For example, Lyapunov and Sylvester matrix equations are, respectively, encountered in continuous-time systems and discrete-time systems [4,5]. The coupled Lyapunov matrix equations…”

Section: Introductionmentioning

confidence: 99%

Matrix GPBiCG algorithms for solving the general coupled matrix equations

Hajarian

2015

IET Control Theory & Applications

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Linear matrix equations have important applications in control and system theory. In the study, we apply Kronecker product and vectorisation operator to extend the generalised product bi-conjugate gradient (GPBiCG) algorithms for solving the general coupled matrix equations l j=1 (including the (coupled) Sylvester, the second-order Sylvester and coupled Markovian jump Lyapunov matrix equations). We propose four effective matrix algorithms for finding solutions of the matrix equations. Numerical examples and comparison with other well-known algorithms demonstrate the effectiveness of the proposed matrix algorithms.

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“…For example, Lyapunov and Sylvester matrix equations are, respectively, encountered in continuous-time systems and discrete-time systems [4,5]. The coupled Lyapunov matrix equations…”

Section: Introductionmentioning

confidence: 99%

Matrix GPBiCG algorithms for solving the general coupled matrix equations

Hajarian

2015

IET Control Theory & Applications

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“…It also appears in certain zero optimization problems, either in a PID like controller setting or in a model reduction setting [15]- [18]. Thus it is important to make use of the special zero-plaid Hankel like structure that it turns out to have, also referred to as alternating Hankel in [6] and a Xiao matrix in [12].…”

Section: Introductionmentioning

confidence: 98%

“…Within this area there are in fact some recent papers that present closed form expressions for symbolic computation [6], [7] or parametric presentation of solutions [8], [9]. Three pioneering papers [10], [11], [12] deal with numerical algorithmic aspects. While the work in these papers was followed up in [13] and [14], it seems however to have received relatively little attention.…”

Section: Introductionmentioning

confidence: 98%

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The continuous closed form controllability Gramian and its inverse

Hauksdottir

Sigurdsson

2009

2009 American Control Conference

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The continuous controllability Gramian is the solution of an input Lyapunov equation in the controller (companion) form or equivalently the infinite integral of an outer product of a vector containing the impulse response and its derivatives corresponding to a unity numerator transfer function. In this paper we make use of both these viewpoints in order to derive the simple zero plaid structure of this Gramian and present the interesting links that the entries of the Gramian have to the entries of the Routh table. Moreover, an expression for the inverse of the Gramian is derived as a simple function of the coefficients of the characteristic polynomial from the fact that it is the solution of a Riccati equation.We show how the controllability Gramian forms the core part of closed form expressions of Gramians of more general MIMO systems as well as solutions of general Sylvester equations. The controllability Gramian also appears in certain zero optimization problems, either in a PID like controller setting or in a model reduction setting. The inverse of the controllability Gramian is a key element in such zero optimization.While much of the results presented can be found in closely related forms in published papers, we believe that they deserve more attention as an effective tool in numerical computations of small to mid-size systems.

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“…The problem of searching for analytical and numerical solutions to linear matrix equations such as Equations (1.1)-(1.11) plays a very vital role in engineering and the applied sciences (LaSalle and Lefschetz 1961;Xiao, Feng, and Shan 1992;Betser, Cohen, and Zeheb 1995;Wu, Duan, and Fu 2006;Wu, Duan, and Xue 2007;Wu, Fu, and Duan 2008;Wu, Sun, and Feng 2010). For example, linear matrix equations play very important roles in the theory of linear dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Finite algorithms for solving the coupled Sylvester-conjugate matrix equations over reflexive and Hermitian reflexive matrices

Hajarian

2013

International Journal of Systems Science

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It is known that solving coupled matrix equations with complex matrices can be very difficult and it is sufficiently complicated. In this work, we propose two iterative algorithms based on the Conjugate Gradient method (CG) for finding the reflexive and Hermitian reflexive solutions of the coupled Sylvester-conjugate matrix equations (including Sylvester and Lyapunov matrix equations as special cases). The iterative algorithms can automatically judge the solvability of the matrix equations over the reflexive and Hermitian reflexive matrices, respectively. When the matrix equations are consistent over reflexive and Hermitian reflexive matrices, for any initial reflexive and Hermitian reflexive matrices, the iterative algorithms can obtain reflexive and Hermitian reflexive solutions within a finite number of iterations in the absence of roundoff errors, respectively. Finally, two numerical examples are presented to illustrate the proposed algorithms.

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On the solution of the continuous-time Lyapunov matrix equation in two canonical forms

Cited by 28 publications

References 10 publications

Matrix GPBiCG algorithms for solving the general coupled matrix equations

Matrix GPBiCG algorithms for solving the general coupled matrix equations

The continuous closed form controllability Gramian and its inverse

Finite algorithms for solving the coupled Sylvester-conjugate matrix equations over reflexive and Hermitian reflexive matrices

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