230 SAMPLED-DATA CONTROL SYSTEMS 
this example apply to any multiloop single-rate sampled-data system and 
consist essentially of the following two simple rules: 
1. Select as unknown variables all those signals at the inputs to sam- 
pling switches. This rule is based on the fact that pulse transfer func- 
tions multiply when separated by samplers. The choice of variables at 
the inputs to switches ensures that switches separate all transfer functions. 
2. Use the principle of superposition to obtain the relations between the 
chosen variables. 
Although these rules are obvious and simple to apply, they are very 
important to the success of the analysis of the system. In certain cases, 
and particularly when such ‘“‘natural”’ variables as the system output or 
the actuating error are not inputs to switches, a person with limited 
experience can rapidly become involved in enormous, complicated expres- 
sions for no reason except that he failed to follow these rules. The 
following example will illustrate both the switch decomposition method 
and the analysis method mentioned above. 
EXAMPLE 
The block diagram which must be analyzed is shown in Fig. 9.8a and 
the single-rate equivalent to the system obtained by switch decomposi- 
tion is shown in Fig. 9.8b. This system will be analyzed to find the 
response to an input step. The analysis will of necessity include 
determination of the characteristic equation so that the stability of 
the system can be checked as the analysis progresses. In Fig. 9.80 the 
variables at the inputs to the switches have been designated Hi, He, C1, 
Cs, C3, and in terms of these variables the system performance is 
described by the equations 
—sT/3 —2sT/3 
: |- e@z|S | 


Bin(@) — Al ru (S) ier 2) 5 | _ cx(ey2| — 
est/2 e78T/3esT/2 
Bee — “Get 2 C1@)z aes C2(z)Z ————_ 
C (2) z e72sT/3esT/2 
—C; (Fe 
ee : (9.24) 
Ce) Et ZG - ee E.(2z)Z —— rag 
es? /2@s8T/3 
Cx(z) = Ex(e)Z pi teer ae 
e78T/292sT/3 
Equations (9.24) are written by inspection, using superposition. The 
