Stability of control systems in frequency domain

Unit -6
Stability Analysis in Frequency
Domain
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By: Dr.Ankita Malhotra
SVKM'S D J Sanghvi College of Engineering

Control Systems Analysis and Design
by Frequency Response Method
• By the term frequency response, we mean the steady-state response of a system to
a sinusoidal input. In frequency-response methods, we vary the frequency of the
input signal over a certain range and study the resulting response.
• An advantage of the frequency-response approach is that frequency-response tests
are, in general, simple and can be made accurately by use of readily available
sinusoidal signal generators and precise measurement equipment. Often the
transfer functions of complicated components can be determined experimentally
by frequency-response tests.
• In addition, the frequency-response approach has the advantages that a system
may be designed so that the effects of undesirable noise are negligible and that
such analysis and design can be extended to certain nonlinear control systems.
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• Obtaining Steady-State Outputs to Sinusoidal Inputs. We shall
show that the steady-state output of a transfer function system can be
obtained directly from the sinusoidal transfer function—that is, the
transfer function in which s is replaced by jω, where ωis angular
frequency.
• Suppose that the transfer function G(s) of the system can be written as
a ratio of two polynomials in s; that is
• The Laplace-transformed output Y(s) of the system is then
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• For a stable system, –s1, –s2 ,.. –sn have negative real parts. Therefore, as t
approaches infinity, the terms , approach zero.
• Thus the steady-state response becomes:
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Y(s)
The angle f may be negative, positive, or zero.
Similarly, we obtain the following
expression for G(–jv):

• Where Y=X |G(jω)|.We see that a stable, linear, time-invariant system
subjected to a sinusoidal input will, at steady state, have a sinusoidal output of
the same frequency as the input. But the amplitude and phase of the output
will, in general, be different from those of the input.
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It can be seen that for small ω, the amplitude of the steady-state output yss(t) is almost
equal to K times the amplitude of the input. The phase shift of the output is small for small
ω. For large ω, the amplitude of the output is small and almost inversely proportional to ω.
The phase shift approaches –90° as ω approaches infinity. This is a phase-lag network.
Example: Consider the given transfer function

Bode Plots
• Bode Diagrams or Logarithmic Plots. A Bode diagram consists of two graphs:
One is a plot of the logarithm of the magnitude of a sinusoidal transfer function;
the other is a plot of the phase angle; both are plotted against the frequency on a
logarithmic scale.
• Basic Factors of G( jω)H(jω). As stated earlier, the main advantage in using
the logarithmic plot is the relative ease of plotting frequency-response curves.
The basic factors that very frequently occur in an arbitrary transfer function
G(jω)H(jω) are:
• 1. Gain K
• 2. Integral and derivative factors (jω)
• 3. First-order factors (1+jωT)
• 4. Quadratic factors
Once we become familiar with the logarithmic plots of these basic factors, it is
possible to utilize them in constructing a composite logarithmic plot for any general
form of G(jω)H(jω) by sketching the curves for each factor and adding individual
curves graphically, because adding the logarithms of the gains corresponds to
multiplying them together.
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• The Gain K.
A number greater than unity has a positive value in decibels, while a number
smaller than unity has a negative value. The log-magnitude curve for a
constant gain K is a horizontal straight line at the magnitude of 20 logK
decibels. The phase angle of the gain K is zero. The effect of varying the
gain K in the transfer function is that it raises or lowers the log-magnitude
curve of the transfer function by the corresponding constant amount, but it
has no effect on the phase curve.
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• Integral and Derivative Factors ( jω). The logarithmic magnitude of 1/jω
in decibels is-
• The phase angle of 1/jω is constant and equal to –90°.
• In Bode diagrams, frequency ratios are expressed in terms of octaves or
decades. An octave is a frequency band from ω1 to 2ω1 ,where ω1 is any
frequency value. A decade is a frequency band from ω1 to 10ω1 ,where
again v1 is any frequency.
• If the log magnitude –20 logv dB is plotted against v on a logarithmic scale, it
is a straight line.
• If the transfer function contains the factor the log
magnitude becomes, respectively,
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• Quadratic Factors :Control systems often possess quadratic factors of the
form-
The magnitude and phase of the quadratic factor depend on both the corner
frequency and the damping ratio . The asymptotic frequency-response curve
may be obtained as follows:
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above.

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If |G(jw)|as a peak value at some frequency, this frequency is called the resonant
frequency. Since the numerator of is constant, a peak value of will occur when
denominator is minimum.

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Log-magnitude curves, together with the asymptotes, and
phase-angle curves of the quadratic transfer function
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• Note that when the individual asymptotic curves are added at each
frequency, the slope of the composite curve is cumulative. Below ω=√2 the
plot has the slope of –20 dB/decade. At the first corner frequency ω=√2 the
slope changes to –60 dB/decade and continues to the next corner frequency
ω=2, where the slope becomes –80 dB/decade. At the last corner frequency
ω=3, the slope changes to –60 dB/decade
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Initial magnitude of Bode plot
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Nyquist criteria
• The Nyquist stability criterion determines the stability of a closed-loop
system from its open-loop frequency response and open-loop poles.
• Let closed loop TF is
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must lie in the left-half s plane. [It is noted that, although poles and zeros of the open-loop
transfer function G(s)H(s) may be in the right-half s plane, the system is stable if all the
poles of the closed-loop transfer function (that is, the roots of the characteristic equation)
are in the left-half s plane.] The Nyquist stability criterion relates the open-loop frequency
response G(jw)H(jw) to the number of zeros and poles of 1+G(s)H(s) that lie in the
right-half s plane.

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Stability of control systems in frequency domain

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