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Chapter 5 Classical Feedback Analysis
Xmath Control Design Module 5-6 ni.com
For discrete-time state-space systems with a sampling interval of T, the
frequency response for each frequency point ω is shown in the following
equation:
Algorithm
The algorithm, based on [Lau86], uses a Hessenberg decomposition to
simplify the previous equations and is quite robust. It finds matrices P and
H such that A = PHP', where PP' = P'P = I and H is a Hessenberg matrix,
and substitutes for A. Because H is zero only below the first subdiagonal,
the number of operations needed to evaluate the response expression is
proportional to the square of the size of A.
freq( ) allows you to prespecify frequency ranges of interest, or it can
generate a representative frequency range from minimum and maximum
frequencies you specify. It then evaluates the complex frequency response
over those frequencies, using specialized algorithms to do this efficiently.
You can specify either a complete set of frequency points (the optional
input
F) or a range of frequency points (the keyword pair Fmin and Fmax)
at which to evaluate the response. The
track keyword indicates that phase
tracking will be used to determine the values of the frequencies between
Fmin and Fmax. The number of intermediate frequency points produced
using
track varies depending on the system and the Fmin and Fmax you
choose. Alternately, you can use the
npts keyword to specify the exact
number of logarithmically-spaced frequency points you want computed.
Specifying
track invokes an algorithm which tracks the phase of the
frequency response to make sure that all peaks and valleys are included in
the computed response. The
delta keyword indicates the amount of phase
change (measured in degrees) to which the response evaluation should be
sensitive. If phase change between two adjacent frequency points exceeds
this delta, closer frequencies are used until either the phase change is less
than
delta or a maximum number of iterations is reached. Evaluation is
forced at key frequency points which include the poles and the points lying
halfway between adjacent poles.
freq( ) returns a PDM having the frequency range as its domain. The
dependent matrices of the frequency response PDM have as many rows as
the system has outputs, and as many columns as the system has inputs. For
MIMO systems, the (i,j) element of a dependent matrix is thus interpreted
as the frequency response from input j to output i. This frequency response
forms the core of the classical control design tools discussed in this chapter.
Hjw() Ce
jwT
IA–()
1–
BD+=