Quadratic inverse eigenvalue problem for damped gyroscopic systems

Qian, Jing; Cheng, Mingsong

doi:10.1016/j.cam.2012.11.026

Cited by 12 publications

(9 citation statements)

References 12 publications

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“…Therefore, it follows that if (38) holds; then, (40) holds for every λ. Additionally, note that λ ∈ R + since D d > 0; hence, the spectrum of A is positive. Remark 7: Another convenient inequality for implementation purposes corresponds to 2 .…”

Section: On the Behavior Of The Closed-loop System Near The Equilibriummentioning

confidence: 97%

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Tuning Rules for a Class of Passivity-Based Controllers for Mechanical Systems

Chan-Zheng

Borja

Scherpen

2021

IEEE Control Syst. Lett.

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This manuscript describes several approaches to tune the parameters of a class of passivity-based controllers for standard nonlinear mechanical systems. In particular, we are interested in controllers that preserve the mechanical system structure in closed-loop. To this end, first, we provide tuning rules for stabilization, i.e., the rate of convergence (exponential stability) and stability margin (input-to-state stability). Then, we provide guidelines to remove the overshoot while prescribing the rise time. Additionally, we propose a methodology to tune the gyroscopic-related parameters. We also provide remarks on the damping phenomena to facilitate the practical implementation of our approaches. We conclude this paper with experimental results obtained from applying our tuning rules to an underactuated and a fully-actuated mechanical configuration.

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Section: On the Behavior Of The Closed-loop System Near The Equilibriummentioning

confidence: 97%

“…In the following proposition, we exploit this tuple for assigning a "worst" (or upper bound) value for the rise time of systems that verify (38). To this end, we define the rise time as the time taken by the system to reach 98% of its steady state value.…”

Section: On the Behavior Of The Closed-loop System Near The Equilibriummentioning

confidence: 99%

Tuning Rules for a Class of Passivity-Based Controllers for Mechanical Systems

Chan-Zheng

Borja

Scherpen

2021

IEEE Control Syst. Lett.

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“…where , , , , 1 , 1 , 1 , 1 , , , , and are, respectively, given by (23), (25), (27), (28), and (29), with s, t ∈ R being arbitrary vectors.…”

Section: The Solution Of Problemmentioning

confidence: 99%

“…It is observed that model updating problems for linear viscously damped elastic systems (or gyroscopic systems) have been considered by many authors [21][22][23][24][25][26][27][28][29]. However, problems for updating hysteretic damping models have not got enough attention in these years.…”

Section: Introductionmentioning

confidence: 99%

Updating Stiffness and Hysteretic Damping Matrices Using Measured Modal Data

Jiang

Yuan

2018

Shock and Vibration

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A new direct method for the finite element (FE) matrix updating problem in a hysteretic (or material) damping model based on measured incomplete vibration modal data is presented. With this method, the optimally approximated stiffness and hysteretic damping matrices can be easily constructed. The physical connectivity of the original model is preserved and the measured modal data are embedded in the updated model. The numerical results show that the proposed method works well.

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“…Inverse eigenvalue problems with Jacobi, Toeplitz, and nonnegative matrices occur in a large number of applications ranging from applied mechanics and physics to numerical analysis, for more details see . The eigenvalues λ and the corresponding eigenvectors x of the quadratic eigenvalue problem Q ( λ ) x : = ( λ 2 A + λ B + C ) x = 0 can interpret the dynamical behavior of the second order differential system

A x ¨ false(t false) + B \overset{x}{true} false(t false) + C x false(t false) = f false(t false)

with mass, damping and stiffness matrices which arises in the acoustic simulation of poro‐elastic materials, the elastic deformation of anisotropic materials, and finite element discretization in structural analysis . The quadratic inverse eigenvalue problem is to construct the matrices A , B , and C such that the quadratic eigenvalue problem holds, that is, for given

X = false(x_{1}, x_{2}, ⋯, x_{n} false) \in R^{n \times m}

and

normalΛ = diag false(λ_{1}, λ_{2}, ⋯, λ_{m} false) \in R^{m \times m}

, we find A , B , C ∈ R n × n such that A X Λ 2 + B X Λ + C X = 0.…”

Section: Introductionmentioning

confidence: 99%

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

Hajarian

2018

Asian Journal of Control

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The inverse eigenvalue problem appears repeatedly in a variety of applications. The aim of this paper is to study a quadratic inverse eigenvalue problem of the form AXΛ2 + BXΛ + CX = 0 where A, B and C should be partially bisymmetric under a prescribed submatrix constraint. We derive an efficient matrix method based on the Hestenes‐Stiefel (HS) version of biconjugate residual (BCR) algorithm for solving this constrained quadratic inverse eigenvalue problem. The theoretical results demonstrate that the matrix method solves the constrained quadratic inverse eigenvalue problem within a finite number of iterations in the absence of round‐off errors. Finally we validate the accuracy and efficiency of the matrix method through the numerical results.

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Quadratic inverse eigenvalue problem for damped gyroscopic systems

Cited by 12 publications

References 12 publications

Tuning Rules for a Class of Passivity-Based Controllers for Mechanical Systems

Tuning Rules for a Class of Passivity-Based Controllers for Mechanical Systems

Updating Stiffness and Hysteretic Damping Matrices Using Measured Modal Data

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

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