Reliable Computing 9: 373–389, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Reliable Computation of Frequency Response
Plots for Nonrational Transfer Functions to
PALURI S. V. NATARAJ and JAYESH J. BARVE
Systems and Control Engineering Group, Department of Electrical Engineering, Indian Institute of
Technology, Bombay 400 076, India, e-mail: email@example.com
(Received: 1 July 2002; accepted: 28 January 2003)
Abstract. We propose algorithms to compute the well known Bode, Nyquist, and Nichols fre-
quency response plots for nonrational transfer functions. The proposed algorithms are very widely
applicable—the magnitude and phase functions need to be only bounded and continuous in frequency.
The proposed algorithms guarantee that the magnitude and phase plots are reliably computed to a
prescribed accuracy, in a ﬁnite number of iterations. Through several practical nonrational examples,
we demonstrate the superior performance of the proposed algorithm over the widely used routines in
MATLAB’s control system toolbox and over the conventional gridding method.
For over ﬁve decades, the classical Bode, Nyquist, and Nichols frequency response
plots have been of immense use in frequency domain analysis and synthesis of
linear feedback systems, see, , , and  for a thorough treatment of these
For transfer functions that have a rational form, these frequency response plots
can be computed to a fair degree of accuracy with the bode, nyquist,andnichols
routines, which are based on an automatic frequency selection procedure available
in the control systems toolbox of MATLAB , .
On the other hand, many important applications found in practice involve nonra-
tional transfer functions. For instance, systems with time delay abound in chemical
process control , hydraulic servomechanisms , , and nuclear reactor sys-
tems . However, the above mentioned automatic frequency selection procedure
is not readily applicable to nonrational transfer functions. To use the procedure
for such functions, we have to ﬁrst derive satisfactory rational approximations to
the nonrational terms, and then apply the said procedure to the resulting rational
For nonrational transfer functions involving only time delay as the nonrational
term, the well known Pade approximation  has been widely used to effect a
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