116 SAMPLED-DATA CONTROL SYSTEMS 
equation can be tested for stability by applying the standard Routh- 
Hurwitz criterion. 
While perfectly rigorous, the modified Routh-Hurwitz test is of limited 
practical utility since the effects of parameters in the system are disguised. 
More direct is the mapping method resulting in the pulse transfer locus. 
By observing the enclosure of the critical point (—1,0) in the function 
plane, the condition of stability can be readily determined. The same 
rules governing the plotting of the continuous transfer locus apply in 
sampled system, except that the unit circle is used instead of the imagi- 
nary axis. A difficulty found in shaping the pulse transfer locus is that 
the rules for cascaded elements depend on the location of samplers and 
simple techniques for shaping cannot be generally applied. 
The location of the roots of the sampled-data-system characteristic 
equation largely determine the transient response of the system. The 
root locus is useful in observing the migrations of these roots as either the 
system gain or other parameters are varied. The rules for the construc- 
tion of the root locus in the z plane are identical to those for the s plane 
since the functions governing the locations of the poles are ratios of poly- 
nomials in either z or s, respectively. The only difference is that in 
sampled systems the behavior of the root loci relative to the unit circle, 
rather than the imaginary axis, is observed. Generally, the closer the 
roots lie to the unit circle, the less damped will a sampled system be. On 
the other hand, if roots lie close to the origin of the z plane, the system will 
tend to be ‘‘dead beat.” 
Another way of representing the performance of sampled-data systems 
is by means of the frequency-response function, which relates the sinus- 
oidal envelope of the output pulse sequence to the sinusoidal input. 
Frequency-response functions fcr sampled systems are periodic at sam- 
pling frequency, a unique characteristic not found in continuous systems. 
Peaked frequency-response functions indicate that poles of the over-all 
pulse transfer function lie close to the unit circle and that the time- 
domain response is oscillatory in nature. While not as useful in the 
analysis of sampled-data systems as in continuous systems, frequency- 
response functions are valuable in relating the transient response and root 
locations. For that reason, frequency response in sampled-data systems 
is a useful concept. The combination of the three viewpoints, namely, 
transient response, root locations, and frequency response, serve to give a 
full understanding of the behavior of sampled-data systems. 
