Modern Control Synthesis and Analysis in a
Mathematica/Control System Professional Environment
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For a given plant, an important objective of control theory is to design a controller
whose corresponding closed-loop system satisfies a given set of specifications. One means
to accomplish this task is to transform the specifications into a performance index,
minimize the index with respect to some measure, and compute the resulting controller. In
1996, Wolfram Research, Inc. introduced Control System Professional (CSP) to
solve a wide range of problems, including classical control design and analysis and
introductory modern control design. This paper describes significant improvements and
additions the author has made to CSP, all of which harmonize with the current CSP
conventions and usage. One major improvement is incorporation of a numerical time-domain
simulation routine that is considerably more flexible, efficient, and compact than the
corresponding CSP routine. The routine takes advantage of the form of the input
corresponding to the response requested to cut the time to compute the response. The
routine transparently allows the user to output only a uniformly spaced fraction of the
points computed, saving the computer both time and space. The routine also obviates the
need to add states to compute the response to a time-delayed input, thus avoiding the
commonplace problem inherent in CSP of output spread over many screens'
worth of space. All told, the user may choose from 60 different types of responses using a
single command. Two major additions are full- and reduced-order H2, H(infinity), and
H2/H(infinity) controller design routines and H2 and H2/H(infinity) model reduction
routines. Both sets of routines, based on the optimal projection equations of Haddad and
Bernstein, are about as close to "no-guesswork" as possible.
Compactness of the routines results from using the so-called augmented plant approach,
which transparently allows cross-coupling and obviates the need for separate computation
of estimator and regulator gains and assembly of a controller or a reduced-order model.
Auxiliary routines allow the user to analyze a system for which individual states of a
system are controllable/uncontrollable, observable/unobservable and/or to prove that a
result satisfies two-norm and/or infinity-norm specifications.