Solutions of matrix equations in problems of mechanics and control

Larin, V. B.

doi:10.1007/s10778-009-0232-5

Cited by 15 publications

(13 citation statements)

References 32 publications

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“…In the general case, it is necessary to set up an augmented functional based on expressions (15), (16) …”

Section: Problem Formulationmentioning

confidence: 99%

“…Using solution (20), we can easily construct the analytic solution. It is inexpedient to construct a numerical solution subject to constraints (17) and (19) because the conditional extremum cannot give a solution that would be better than that obtained based on the unconditional minimization of function (15) in view of constraint (6). Numerical tests for the segments 0 2 …”

Section: Problem Formulationmentioning

confidence: 99%

“…The cited studies dealt only with one mode of elastic vibrations. Some relevant issues were analyzed in [12][13][14][15].The papers [4, 27] outline a method of designing an optimal slewing program that minimizes the relative accelerations of the flexible appendages of a spacecraft. The method allows for the infinite number of modes of elastic vibrations, not going beyond the finite-dimensional mathematical model.…”

mentioning

confidence: 99%

“…The cited studies dealt only with one mode of elastic vibrations. Some relevant issues were analyzed in [12][13][14][15].…”

mentioning

confidence: 99%

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Optimal limited-rate slewing of a flexible spacecraft

Zakrzhevskii

2010

Int Appl Mech

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The paper presents an efficient method of finding the optimal program control for the reorientation of a spacecraft with flexible appendages at a limited slewing rate. The appendages are incorporated into the mathematical model based on the quasistatic approximation. The problem is solved analytically by parametrizing the functional of a multipoint boundary-value problem. The optimal solution is illustrated graphically for different parts of the attainability domains Introduction. Optimal rest-to-rest maneuvers of spacecraft with flexible appendages have long attracted the attention of researchers and are still of interest because new spacecraft designs, new effectors, new control systems are developed.A common practice is to combine feedforward and feedback controls for such systems. The deformation of flexible appendages can only be measured with distributed sensors. If a spacecraft is not equipped with such sensors, there can be no acceptable quality of feedback. Therefore, whether maneuvers of such spacecraft are successful is strongly dependent on the quality of feedforward control. In this field, a number of fundamental results have been obtained. They allow optimization based on various goal functions. For long-lived spacecraft with a low power-to-weight ratio, it is important to minimize the consumption of energy [17,20,25,26]. For observation satellites that maneuver much, it is important to minimize the maneuver time [5,6,8,9,16,18,19,21,23,24]. The most typical result in this field is bang-bang control, which may appear extremely sensible to simulation errors. Sometimes these goal functions are combined into one with weighted components [11].In some cases, it is important to minimize not the fuel consumption and maneuver time, but rather the moment perturbations of the spacecraft because of the vibrations of flexible appendages. The slewing of such spacecraft should be controlled in such a manner that no vibrations of the flexible appendages occur and affect the operation of the spacecraft equipment. First attempts to solve this problem using goal functions with various combinations of generalized coordinates of elastic displacements and their derivatives were made in the 1970s [2, 3, 10]. Later, it was proposed that feedforward control did not include spectral components of the spacecraft as a whole [7]. Modifications of this approach were applied to elastic nonlinear systems [22], but the residual vibrations were nevertheless significant [11]. The cited studies dealt only with one mode of elastic vibrations. Some relevant issues were analyzed in [12][13][14][15].The papers [4, 27] outline a method of designing an optimal slewing program that minimizes the relative accelerations of the flexible appendages of a spacecraft. The method allows for the infinite number of modes of elastic vibrations, not going beyond the finite-dimensional mathematical model. This is possible if the first several modes are described in the usual formulation and all the higher modes in the quasistatic formulation. Of separat...

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“…In the general case, it is necessary to set up an augmented functional based on expressions (15), (16) …”

Section: Problem Formulationmentioning

confidence: 99%

Section: Problem Formulationmentioning

confidence: 99%

mentioning

confidence: 99%

“…The cited studies dealt only with one mode of elastic vibrations. Some relevant issues were analyzed in [12][13][14][15].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Optimal limited-rate slewing of a flexible spacecraft

Zakrzhevskii

2010

Int Appl Mech

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“…We will also show that in some cases Eq. (1.1) can be solved with the doubling method [6,21,22], which, unlike the algorithm from [10], does not require the matrix A 1 in (1.1) to be invertible. If the matrix A 2 is invertible, Eq.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for Solving a Unilateral Quadratic Matrix Equation and the Model Updating Problem

Larin

2014

Int Appl Mech

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The Schur and doubling methods, which are usually used to solve the algebraic Riccati equation, are generalized to the case of unilateral quadratic matrix equations. The efficiency of the algorithms proposed to solve the unilateral quadratic matrix equation is demonstrated by way of examples. The algorithms are compared with well-known ones. It is shown that the solutions of the unilateral quadratic matrix equation can be used to update model parameters.

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On Normalization of Quadratic Hamiltonian

Titova

2023

Lecture Notes in Networks and Systems

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Solutions of matrix equations in problems of mechanics and control

Cited by 15 publications

References 32 publications

Optimal limited-rate slewing of a flexible spacecraft

Optimal limited-rate slewing of a flexible spacecraft

Algorithms for Solving a Unilateral Quadratic Matrix Equation and the Model Updating Problem

On Normalization of Quadratic Hamiltonian

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