98 SAMPLED-DATA CONTROL SYSTEMS 
excessive. It is desirable to be able to ascertain the presence or absence 
of poles outside the unit circle without actually determining their location. 
A form of modified Routh-Hurwitz or Nyquist criterion must therefore be 
established as a working tool. 
5.4 The Modified Routh-Hurwitz Criterion 
The Routh-Hurwitz criterion is a test which determines the signs of the 
real parts of the roots of a rational polynomial. This test finds direct 
application to determining the condition of stability of a linear continuous 
system by applying it to the characteristic equation of the system. The 
presence or absence of roots of this equation with positive real parts is an 
indication of instability or stability respectively. In the case of sampled- 
data systems, a direct test of this type would require the determination of 
whether or not the magnitudes of the roots of the characteristic equation 
are greater than unity, so that a direct application of the Routh-Hurwitz 
criterion is not possible. 
It is possible, however, to apply a transformation to the characteristic 
equation in z which will transform the region outside the unit circle in the 
z plane to the right half of an auxiliary plane and the region inside the unit 
circle to the left half of this plane. Such a transformation is the bilinear 
transformation which has been applied to problems in the control field by 
Oldenbourg and Sartorius.*® An auxiliary plane called the \ plane is 
defined by the following relation: 

oe (5.46) 
Pi east 
or A= eae (5.47) 
To show the relation between the z plane and the A plane, it is noted that 
both z and \ are complex, so that 
tee easy 
and A=u+yo . (5.48) 
Substituting these expressions back in (5.47) and rationalizing the result- 
ing expression, the following is obtained: 
,e@ tl, 
eS 7) Case 
In view of the definition of x and y in (5.48), it is seen that (#? + y?) is 
the magnitude squared of z, |z|?.. Thus, forall values of z whose magnitude 
is greater than unity, the real part of \ is positive, and for all values of z 
whose magnitudes is less than unity, the real part of \ is negative. Thus, 
stated differently, the entire region of the z plane which lies outside the 
(5.49) 
