100 SAMPLED-DATA CONTROL SYSTEMS 
Simplifying and clearing of fractions, this expression becomes 
0.12\? — 1.044 — 3.08 = 0 
Application of the Routh-Hurwitz criterion to this polynomial will show 
that there is one root in the left half and one root in the right half of the 
plane. Direct factoring of the characteristic equation in z would 
show that the roots are 0.4 and 1.2, a direct check. 
While simple to apply in principle, the bilinear transformation com- 
bined with the Routh-Hurwitz criterion is a fairly tedious process in the 
ease of higher-order systems. In addition, the constants of the original 
system appear in the transformed expression in a complex manner. For 
this reason, it is difficult to associate the conditions revealed by the 
application of the criterion with the constants of the original system. 
The modified Routh-Hurwitz criterion serves mainly as a check pro- 
cedure to verify results obtained with some of the mapping procedures 
outlined in the next section. 
5.5 Stability Criterion Using the Transfer Locus 
Feedback systems containing one or more samplers are characterized by 
an over-all pulse transfer function C(z)/R(z), which is called K(z) and is 
given by 
(5.50) 
for the case of an error-sampled system. For other prototypes, the form 
of K(z) changes except for one characteristic, namely, that the denomi- 
nator of K(z) contains a polynomial form 1 + F(z), where F(z) is the loop 
pulse transfer function. In the error-sampled case, F(z) is expressed as 
the z transform corresponding to the continuous transfer function 
G(s)H(s). In other cases, the relation between F(z) and the transfer 
functions of the elements comprising the loop may differ. For purposes of 
discussion, the form GH(z) will be used, though it should be remembered 
that the loop pulse transfer function, regardless of the form of the system, 
is implied. 
To determine the condition of stability for a feedback system, the roots 
of the characteristic equation obtained from the denominator polynomial 
of K(z) must be examined. This characteristic equation is given by 
1+ GH(z) =0 (5.51) 
The problem is to determine if some of the roots of this equation lie out- 
side the unit circle of the z plane by applying the same Cauchy mapping 
theorem which underlies the Nyquist criterion in continuous systems. 
It is recalled that this theorem states that if a closed contour I encloses 
poles and zeros of a function, then the number of times the map of this 
