sigma

Control System Toolbox

Singular value response of LTI systems

sigma(sys)
sigma(sys,w)
sigma(sys,w,type)

sigma(sys1,sys2,...,sysN)
sigma(sys1,sys2,...,sysN,w)
sigma(sys1,sys2,...,sysN,w,type)
sigma(sys1,'PlotStyle1',...,sysN,'PlotStyleN')

[sv,w] = sigma(sys)

Description

sigma calculates the singular value response of an LTI system. For continuous-time systems with transfer function

, sigma computes the singular values of

as a function of the frequency

. For discrete-time systems with transfer function

and sample time

, sigma computes the singular values of

for frequencies

between 0 and the Nyquist frequency

The singular value response is an extension of the Bode magnitude response for MIMO systems and is useful in robustness analysis. The singular value response of a SISO system is identical to its Bode magnitude response. When invoked without left-hand arguments, sigma produces a singular value plot on the screen.

sigma(sys) plots the singular value response of an arbitrary LTI model sys. This model can be continuous or discrete, and SISO or MIMO. The frequency points are chosen automatically based on the system poles and zeros.

sigma(sys,w) explicitly specifies the frequency range or frequency points to be used for the plot. To focus on a particular frequency interval [wmin,wmax], set w = {wmin,wmax}. To use particular frequency points, set w to the corresponding vector of frequencies. Use logspace to generate logarithmically spaced frequency vectors. Frequencies should be specified in radians/sec.

sigma(sys,[],type) or sigma(sys,w,type) plots the following modified singular value responses:


`type = 1`	Singular value response of where is the transfer function of `sys`.
`type = 2`	Singular value response of .
`type = 3`	Singular value response of .

These options are available only for square systems, i.e., with the same number of inputs and outputs.

To superimpose the singular value plots of several LTI models on a single figure:

sigma(sys1,sys2,...,sysN)
sigma(sys1,sys2,...,sysN,[],type)  % modified SV plot
sigma(sys1,sys2,...,sysN,w)         % specify frequency range/grid

All systems must have the same number of inputs and outputs, but may otherwise be a mix of continuous and discrete systems. You can also specify a distinctive color, linestyle, and/or marker for each system plot with the syntax

sigma(sys1,'PlotStyle1',...,sysN,'PlotStyleN')

See bode for an example.

When invoked with left-hand arguments,

[sv,w] = sigma(sys)
sv = sigma(sys,w)

return the singular values sv of the frequency response at the frequencies w. For a system with Nu input and Ny outputs, the array sv has min(Nu,Ny) rows and as many columns as frequency points (length of w). The singular values at the frequency w(k) are given by sv(:,k).

Example

Plot the singular value responses of

and

H = [0 tf([3 0],[1 1 10]) ; tf([1 1],[1 5]) tf(2,[1 6])]

subplot(211)
sigma(H)
subplot(212)
sigma(H,[],2)

Algorithm

sigma uses the svd function in MATLAB to compute the singular values of a complex matrix.