236 SAMPLED-DATA CONTROL SYSTEMS 
1 — K(z,”) are equally spaced around a circle of radius b and are therefore 
outside the unit circle. The design would be stable if all these extraneous 
roots but the one in G(z,) are left in D(z,) because none of them involve 
pole or zero cancellation in an unstable region. The controller transfer 
function which results from the inclusion of the extra roots of 1 — K(z,”) 
in D(z,) is of very high order, however, and the device used to realize this 
transfer function is unnecessarily complicated. The design can be 
greatly simplified by including these extraneous roots from 1 — K(z,”) in 
the over-all transfer function K(z,) so that they do not appear in the con- 
troller transfer function. In terms of the design equations, the more 
practical constraints on the transfer functions for the plant with the trans- 
fer function given by (9.37) are 
Keesha ces Aen aa Bie ae ot Ae (9.40) 
and Ge") — 62) (22?) (9.41) 
Since physical realizability requires that K(z,) contain the zeros of the 
plant which are at infinity, which is certainly outside the unit circle, (9.40) 
and (9.41) actually express the constraints sufficient to ensure both sta- 
bility and physical realizability in the design. In these two equations, 
the functions A(z,) and B(z,") are general functions which include the 
design requirements other than removal of the zero at a and the pole at b 
in the plant. If there are other poles or zeros in the plant which are out- 
side the unit circle, they must, of course, be treated in the same manner 
as the two selected in (9.37). The final constraint on the transfer func- 
tion which is fairly universal is the specification of the steady-state 
performance. 
The requirement on steady-state response is most conveniently defined 
in terms of the system response to a polynomial input. As discussed in 
Chap. 7, the designer frequently requires that the system follow an input 
whose transform is 1/s* with zero error at sampling instants in the steady 
state. This requirement can be translated into a specification on K (zn) 
for the multirate system. The error of the system of Fig. 9.9 is the 
variable £;, which satisfies the relation 
Ey(2n) = Rn) — Cn) 
= R(zn) — R(éa")K (en) (9.42) 
If R(s) is of the form 1/s*, then it is easy to show that 
P(2n 
P(e," 
and Re) = gee (9.44) 
