08-05-2012, 04:42 PM
STABILITY ANALYSIS OF LINEAR SYSTEM
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THEORY POLAR PLOT
The polar plot of a sinusoidal transfer function G( jw) on polar coordinates as w is varied from zero to
infinity. Thus the polar plot is the locus of vectors G( jw) and G( jw) as w is varied from zero to infinity. The
polar plot is also called Nyquist plot.
NYQUIST STABILITY CRITERION
If G(s)H(s) contour in the G(s)H(s) plane corresponding to Nyquist contour in s-plane encircles the
point −1+ j0 in the anti – clockwise direction as many times as the number of right half s-plane ofG(s)H(s) .
Then the closed loop system is stable.
ROOT LOCUS
The root locus technique is a powerful tool for adjusting the location of closed loop poles to achieve
the desired system performance by varying one or more system parameters.
The path taken by the roots of the characteristics equation when open loop gain K is varied from 0 to
¥ are called root loci (or the path taken by a root of characteristic equation when open loop gain K is varied
from 0 to ¥ is called root locus.)
FREQUENCY DOMAIN SPECIFICATIONS
The performance and characteristics of a system in frequency domain are measured in term of frequency
domain specifications. The requirements of a system to be designed are usually specified in terms of these
specifications.
EC2259 Electrical Engineering And Control System Lab Manual
Prepared by G.Panneerselvam, Vel Tech Multi Tech
The frequency domain specifications are
1. Resonant peak r
2. Resonant Frequency r
3. Bandwidth.
4. Cut – off rate
5. Gain margin
6. Phase margin
RESONANT PEAK
The maximum value of the magnitude of closed loop transfer function is called the resonant peak r
M . A large
resonant peak corresponds to a large over shoot in transient response.
RESONANT FREQUENCY
The bandwidth is the range of frequency for which the system gain is more than −3 dB . The frequency at
which the gain is −3 dB , called cut off frequency. Bandwidth is usually defined for closed loop system and it
transmits the signals whose frequencies are less than cut-off frequency. The bandwidth is a measured of the
ability of a feedback system to produce the input signal, noise rejection characteristics and rise time. A large
bandwidth corresponds to a small rise time or fast response.
CUT-OFF RATE
The slope of the log-magnitude curve near the cut off frequency is called cut-off rate. The cut-off rate
indicates the ability of the system to distinguish the signal from noise.
GAIN MARGIN
The gain margin g
K is defined as the reciprocal of the magnitude of open loop transfer function at phase cross
over frequency. The frequency at witch the phase of open loop transfer function is 180 is called the phase
cross over frequency pc
w .
PHASE MARGIN g
The phase marging is that amount of additional phase lag at the gain cross over frequency required to bring
the system to the verge of instability, the gain cross over frequency gc
w is the frequency at which the
magnitude of open loop transfer function is unity (or it is the frequency at which the db magnitude is zero).
EC2259 Electrical Engineering And Control System Lab Manual
Prepared by G.Panneerselvam, Vel Tech Multi Tech
PROCEDURE
• Enter the command window of the MATLAB.
• Create a new M – file by selecting File – New – M – File.
• Type and save the program.
• Execute the program by either pressing F5 or Debug – Run.
• View the results.
• Analysis the stability of the system for various values of gain.