290 SAMPLED-DATA CONTROL SYSTEMS 
This is the same formula which was derived in the example in Sec. 4.6 
from purely mathematical considerations. 
The advantage of viewing the numerical integration process from the 
physical properties described in this section is that it is possible to relate 
the accuracy of the process to the properties of the function being inte- 
grated. In the case of this particular integration formula using the poly- 
gonal data hold, the error is caused by the difference between the actual 
function to be integrated, r(t), and the reconstructed function, r;(é), 


ee) rit) 
(a) (0) 
Fic. 11.6. Sampled model of continuous feedback system. 
actually applied to the filter, in this case, a pure integrator. The differ- 
ence function is “‘scalloped”’ in shape, as can be seen from Fig. 11.2, and 
contains periodic components having a fundamental frequency equal to 
the sampling frequency. If the filter which follows the data hold has a 
frequency response which is well below the sampling frequency, the error 
in the sampled model is not excessive. In the case of pure integration, 
the error is the area contained between the actual function and the poly- 
gonal approximation. It is evident that the error can be reduced by 
sampling more frequently, which is equivalent to stating that in a 
numerical process the quadrature error is reduced by reducing the quad- 
rature interval. Some measure of the error can be obtained in more com- 
plex cases by integrating in the frequency domain the power contained in 
the error components, although an exact interpretation of the result is not 
always clear. 
11.2 Approximation of Closed-loop Continuous Systems by a Sampled 
Model 
The same general procedure which is applied to open-loop continuous 
systems can be used for closed-loop systems as well. The block diagram 
of such a system is shown in Fig. 11.6a, where G(s) is the feedforward and 
H(s) the feedback transfer functions, respectively. In converting the 
system to a sampled model, it is assumed that polygonal data reconstruc- 
tion is adequate.*° An important consideration is the location of these 
fictitious elements in the loop. As contrasted to the open-cycle case, 
where the sampler and data hold had to be placed in cascade with the con- 
