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Modern Control Synthesis and Analysis in a Mathematica/Control System Professional Environment

David Dana-Bashian

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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.