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Sampled data design by log gain diagrams

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2 Author(s)
M. Pastel ; U.S. Naval Postgraduate School, Monterey, CA, USA ; G. Thaler

The bilinear transformation z = (1+w)/(1-w) converts a z -transform function G(z) of a sampled-data system into a new function G(w) , called the w -transform function, which is a rational function in variable w . This bilinear transformation maps the unit circle on the z - plane onto the imaginary axis of the w -plane. Consequently, it is now possible to readily draw log magnitude and phase diagrams against a frequency scale of the open-loop w -transform function of a sampled-data system by use of asymptotic techniques. Then, by use of a Nichols chart and correlation information available from continuous systems, it is possible to predict the approximate time domain performance. Design by modification of the open-loop transfer function can be made on the diagram in the same manner as employed for continuous systems on the Bode diagram. The resulting w -transform can be converted to its equivalent Laplace transform. The ratio of this transform function and the original Laplace transform function of the system's equipment gives the required compensator. Remote s-plane poles may have to be added to have the compensator physically realizable. Restricting the modifying w -plane poles to lie between (0) and (-1) permits the compensator to be realizable as an RC network.

Published in:

IRE Transactions on Automatic Control  (Volume:4 ,  Issue: 2 )