Sampled data design by log gain diagrams

Abstract

The bilinear transformation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z = (1+w)/(1-w)</tex> converts a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transform function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(z)</tex> of a sampled-data system into a new function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(w)</tex> , called the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> -transform function, which is a rational function in variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> . This bilinear transformation maps the unit circle on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> - plane onto the imaginary axis of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> -plane. Consequently, it is now possible to readily draw log magnitude and phase diagrams against a frequency scale of the open-loop <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> -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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> -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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> -plane poles to lie between (0) and (-1) permits the compensator to be realizable as an RC network.