Stability of multidimensional linear time-varying systems

Shrivastava, S. K.; Pradeep, S.

doi:10.2514/3.20025

Cited by 27 publications

(11 citation statements)

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“…Many dynamical systems can be described by matrix second‐order time‐varying differential equations of the form :

M (t) ẍ (t) MathClass-bin+ Q (t) ẋ (t) MathClass-bin+ K (t) x (t) MathClass-rel= 0 MathClass-punc,

where

x (t) MathClass-rel\in {double-struckR}^{n}

is the state vector, M ( t ) Q ( t )

K (t) MathClass-rel\in {double-struckR}^{n MathClass-bin\times n}

are time‐varying matrices, and M ( t ) is invertible. Examples can be found in many engineering fields, such as systems with time‐varying mass distribution (e.g., a space station carrying a mobile crane, and a space structure extended in orbit being typical examples from space engineering) and helicopter structure dynamics with periodic excitation caused by propeller motion . Stability of such matrix second‐order time‐varying systems has attracted considerable attention (see and the references therein).…”

Section: Introductionmentioning

confidence: 99%

“…The most commonly used method for analyzing the stability of the matrix second‐order time‐varying systems with the time‐varying coefficient matrix M ( t ) is the Lyapunov function approach because some eigenvalue‐based criteria (such as Routh–Hurwitz criterion) used for the matrix second‐order time‐invariant systems (see ) cannot be applied for the global stability analysis of the time‐varying cases. For example, in , some sufficient conditions were derived based on the Lyapunov function approach. In and , the coefficient matrices M ( t ) Q ( t ), and K ( t ) are assumed to be differentiable.…”

Section: Introductionmentioning

confidence: 99%

“…Many dynamical systems can be described by matrix second-order time-varying differential equations of the form [1][2][3][4][5]:…”

Section: Introductionmentioning

confidence: 99%

“…Examples can be found in many engineering fields, such as systems with time-varying mass distribution (e.g., a space station carrying a mobile crane, and a space structure extended in orbit being typical examples from space engineering) and helicopter structure dynamics with periodic excitation caused by propeller motion [1][2][3][4][5]. Stability of such matrix second-order time-varying systems has attracted considerable attention (see [3][4][5][6][7][8][9][10][11] and the references therein). Most of them, however, assume that the coefficient matrix M.t/ is constant, except [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

“…Stability of such matrix second-order time-varying systems has attracted considerable attention (see [3][4][5][6][7][8][9][10][11] and the references therein). Most of them, however, assume that the coefficient matrix M.t/ is constant, except [3][4][5]. In [3][4][5], stability criteria were obtained with the time-varying coefficient matrix M.t/.…”

Section: Introductionmentioning

confidence: 99%

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Quadratic stability and stabilization of matrix second‐order time‐varying systems

Xiao

Chen

2011

Intl J Robust & Nonlinear

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SUMMARY This paper investigates the quadratic stability and stabilization of a class of matrix second‐order time‐varying systems. All the system matrices including the second‐order differential coefficient matrix are assumed to have the time‐varying norm‐bounded parameters. Necessary and sufficient conditions for the quadratic stability and stabilization of such time‐varying systems are derived. All the results are obtained in terms of linear matrix inequalities. Two illustrative examples are given to show that our results are effective and less conservative than the results obtained by other researchers. Copyright © 2011 John Wiley & Sons, Ltd.

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“…Many dynamical systems can be described by matrix second‐order time‐varying differential equations of the form :

M (t) ẍ (t) MathClass-bin+ Q (t) ẋ (t) MathClass-bin+ K (t) x (t) MathClass-rel= 0 MathClass-punc,

where

x (t) MathClass-rel\in {double-struckR}^{n}

is the state vector, M ( t ) Q ( t )

K (t) MathClass-rel\in {double-struckR}^{n MathClass-bin\times n}

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Many dynamical systems can be described by matrix second-order time-varying differential equations of the form [1][2][3][4][5]:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quadratic stability and stabilization of matrix second‐order time‐varying systems

Xiao

Chen

2011

Intl J Robust & Nonlinear

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Stability Robustness Analysis Maneuverability, and Sensitivity in the Large

Eslami

1994

Theory of Sensitivity in Dynamic Systems

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Stability of Time Varying Linear Systems

Shrivastava

Pradeep

1986

Astrophysics and Space Science Library

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ABSTRACT. The concept of stability is one of the most fundamental in the study of natural sciences. The complexity of the stability problem depends on the degree of mathematical idealization of the system. In general, time invariant systems are much easier to handle than time varying systems. Here, first a bird's eye view is presented on the important methods in constant parameter linear and nonlinear systems. Following this is a quick review of stability analysis of time varying systems touching upon periodic systems, second order systems, arbitrarily time varying systems, the absolute stability problem and the functional analytic approach. The paper also includes a summary of some important results of the authors' work on the stability of second order multidimensional linear arbitrarily time varying systems. The theorems derived generalize some existing theorems. Application of one of them to the damped Mathieu equation yields better results than those existing in literature for some values in the parameter space. The asymptotic behaviour and boundedness of almost constant coefficient systems have also been studied and some new theorems derived.

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Stability of multidimensional linear time-varying systems

Cited by 27 publications

References 1 publication

Quadratic stability and stabilization of matrix second‐order time‐varying systems

Quadratic stability and stabilization of matrix second‐order time‐varying systems

Stability Robustness Analysis Maneuverability, and Sensitivity in the Large

Stability of Time Varying Linear Systems

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