Singular values can be used to extend bode analysis to the singular value bode plots for MIMO systems.[^1] The singular value of matrix \(A\) is the non negative eigenvalue of \(A^HA\) where \(H\) is the complex conjugate of the transpose of \(A\).[^1]
Singular values are ordered in \(\sigma_1\geq\sigma_2\geq …\geq\sigma_n\)n
If the matrix is complex the singular values are the non-negative roots of the eigenvalues of \(A^HA\).[^1] The greatest singular value is the maximum gain of the matrix over all possible directions of the vector x.[^1] The band between minimum and maximum singular values as a function of frequency shows the degree of disturbance rejection, stability, and performance, as a gain with respect to frequency.[^1] Noise suppression at lower frequencies give a smaller sensitivity to measurement noise. [^1]
Singular values can be used to extend bode analysis to the singular value bode plots for MIMO systems.[^1] The singular value of matrix \(A\) is the non negative eigenvalue of $A^HA$ where $H$ is the complex conjugate of the transpose of \(A\).[^1]
Singular values are ordered in \(\sigma_1\geq\sigma_2\geq …\geq\sigma_n\)n
If the matrix is complex the singular values are the non-negative roots of the eigenvalues of \(A^HA\).[^1] The greatest singular value is the maximum gain of the matrix over all possible directions of the vector x.[^1] The band between minimum and maximum singular values as a function of frequency shows the degree of disturbance rejection, stability, and performance, as a gain with respect to frequency.[^1] Noise suppression at lower frequencies give a smaller sensitivity to measurement noise. [^1]
[[Singular Values of a Return Difference Matrix]] – used as a conservative stability criterion for MIMO systems
Steady State Transfer Function Matrix – matrix is singular is the rates and accelerations are output variables.
[[Sensitivity analysis]] – the singular values of the sensitivity matrix determine attenuation of the system
[[Mu-Synthesis]] – robustness is determined with the singular values
Singular Value Decomposition – the diagonal entries of $\sum$ are singular values
[[Matrix Rank]] – given by the number of nonzero singular values
[[Orthogonal Sinusoids]] – can be used to identify the singular value of a matrix
[[F-18 Open Loop Singular Values]]
[[X-29 Minimum Singular Values]]
F-14 Flight Control System – a controller was designed using the structured singular value framework
Simulink – can be used for singular value analysis
[[VAAC H-Infinity Loop Shaping]] – used roll-off filters to implement transfer functions without changing the plant singular values
[[Align Algorithm – aligns the singular values at a chosen cross-over frequency with a constant real matrix
Backlinks
[[Bode Plot]]
[[Gain Margin]]
[[Matrices]]
[[Matrix Transpose]]
[[MIMO]]
Phase Margin
[[Sensitivity Function]]
Sources
[1] “F-18 Robust Control Design Using H2 and H-Infinity Methods”.
[2] “HartreeTutorialPart1”.
[3] W. L. Garrard and B. S. Liebst, “Design of a Multivariable Helicopter Flight Control System for Handling Qualities Enhancement”.
[4] D. M. R. Anderson and A. B. Page, “UNIFIED PILOT-INDUCED OSCILLATION THEORY VOLUME III: PIO ANALYSIS USING MULTIVARIABLE METHODS”.