Chapter 5 Classical Feedback Analysis
Xmath Control Design Module 5-20 ni.com
-0.52263
0.00336213 + 3.75217 j
0.00336213 - 3.75217 j
-11.4841
Two of the poles of the closed-loop system are now unstable.
Linear Systems and Power Spectral Density
A key characteristic of the linear, time-invariant systems represented in
Xmath is that the transfer function between a system input and a system
output is just the Fourier transform of the response at that output to a delta
impulse at that input. The power spectral density of a time series is defined
as the Fourier transform of the autocorrelation function of the series.
Given these two concepts, you can obtain the power spectral density of the
output of a linear, time-invariant system just by knowing the power spectral
density of the input and the system’s transfer function [Leo89], [GrD86].
Representing the transfer function by H(q) and the power spectral densities
of the input and output as SU(q) and SY(q), respectively:
You also can obtain the cross-power spectral densities:
These results indicate that you can shape the spectrum of a linear system’s
output by using an input with an appropriate spectrum. Alternatively, you
can choose a system to give you the output spectrum you want, given a
fixed set of input data. When you use linear systems in transfer-function
form for such applications, you generally refer to them as filters rather than
systems.

psd( )

[Ypsd,Yspec] = psd(Sys,{Uspec})
The psd( ) function computes the power spectral density and
cross-spectral density of a system’s outputs as a function of frequency,
given the frequency-dependent input power spectral density matrices. The
input parameter Uspec is a PDM where domain contains the frequency
SYq() Hq()
2SUq()=
SYU q() Hq()SUq()=
SUY q() H∗q()SUq()=