scholarly journals Multiple-Input Multiple-Output (MIMO) Linear Systems Extreme Inputs/Outputs

2007 ◽  
Vol 14 (2) ◽  
pp. 107-131 ◽  
Author(s):  
David O. Smallwood
Keyword(s):  
Density Matrix ◽  
Random Process ◽  
Multiple Input Multiple Output ◽  
Linear Structure ◽  
Multiple Points ◽  
Critical Response ◽  
Special Cases ◽  
Multiple Input ◽  
Input Multiple Output ◽  
Phase Relationships

A linear structure is excited at multiple points with a stationary normal random process. The response of the structure is measured at multiple outputs. If the autospectral densities of the inputs are specified, the phase relationships between the inputs are derived that will minimize or maximize the trace of the autospectral density matrix of the outputs. If the autospectral densities of the outputs are specified, the phase relationships between the outputs that will minimize or maximize the trace of the input autospectral density matrix are derived. It is shown that other phase relationships and ordinary coherence less than one will result in a trace intermediate between these extremes. Least favorable response and some classes of critical response are special cases of the development. It is shown that the derivation for stationary random waveforms can also be applied to nonstationary random, transients, and deterministic waveforms.

Complex Coherence Constraint for the Definition of the Spectral Density Matrix

2018 ◽  
Vol 61 (1) ◽  
pp. 7-19
Author(s):  
Zhihua Liu ◽  
Chenguang Cai ◽  
Yan Xia ◽  
Ming Yang
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Random Vibration ◽  
Multiple Input Multiple Output ◽  
New Approach ◽  
Spectral Density Matrix ◽  
Multiple Input ◽  
Input Multiple Output ◽  
Recursive Formulas ◽  
Definition Of

Abstract The cross spectral density (CSD) for a multiple-input/multiple-output (MIMO) random vibration is typically defined by the complex coherence consisting of the modulus and the phase. The purpose of this paper is to present a constraint for the complex coherence to allow the CSD to be defined more easily. The study of the complex coherence constraint is based on Cholesky decomposition of the spectral density matrix (SDM). The complex coherence must be bounded in the interior or on the boundary of a constraint circle to ensure a physically realizable random vibration. This paper proposes a new approach to define the complex coherences of the SDM by using recursive formulas based on the constraint circle.

scholarly journals Matrix Methods for Estimating the Coherence Functions from Estimates of the Cross-Spectral Density Matrix

1996 ◽  
Vol 3 (4) ◽  
pp. 237-246 ◽  
Author(s):  
D.O. Smallwood
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Multiple Input Multiple Output ◽  
Spectral Density Matrix ◽  
Multiple Input ◽  
Cross Spectral Density ◽  
Input Multiple Output ◽  
The Cross ◽  
Value Decomposition ◽  
Physical Order

It is shown that the usual method for estimating the coherence functions (ordinary, partial, and multiple) for a general multiple-input! multiple-output problem can be expressed as a modified form of Cholesky decomposition of the cross-spectral density matrix of the input and output records. The results can be equivalently obtained using singular value decomposition (SVD) of the cross-spectral density matrix. Using SVD suggests a new form of fractional coherence. The formulation as a SVD problem also suggests a way to order the inputs when a natural physical order of the inputs is absent.

Minimum Input Trace for Multiple Input Multiple Output Linear Systems

2013 ◽  
Vol 56 (2) ◽  
pp. 57-67 ◽  
Author(s):  
David Smallwood
Keyword(s):  
Density Matrix ◽  
Minimum Energy ◽  
Multiple Input Multiple Output ◽  
Positive Definite ◽  
Vibration Test ◽  
Control Points ◽  
Spectral Density Matrix ◽  
Multiple Input ◽  
Input Multiple Output ◽  
Drive Signals

Specification of the cross spectra for a multiple-input/multiple-output (MIMO) vibration test is challenging. This paper presents a method for tests where the specifications of the output (the control points) autospectra are available. The autospectra of the outputs are specified and cross spectra between the outputs are derived that will minimize the trace of the autospectra of the inputs (the drive signals) with the constraint that the input spectral density matrix is positive definite. The hypothesis is that nature likes a minimum energy solution.

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