Signals and Systems - Electrical Engineering

372 C H A P T E R 6: Application to Control and Communications

appropriate test input for the system; if the input applied to a system is constant or continuously

increasing, then a unit step or a ramp signal would be appropriate. Using test signals such as an

impulse, a unit-step, a ramp, or a sinusoid, mathematical and experimental analyses of systems can

be done.

When designing a control system its stability becomes its most important attribute, but there are other

system characteristics that need to be considered. The transient behavior of the system, for instance,

needs to be stressed in the design. Typically, as we drive the system to reach a desired response, the

system’s response goes through a transient before reaching the desired response. Thus, how fast the

system responds and what steady-state error it reaches need to be part of the design considerations.

First-Order Systems

As an example of a first-order system consider an RC serial circuit with a voltage sourcevi(t)=u(t)as

input (Figure 6.9), and as the output the voltage across the capacitor,vc(t). By voltage division, the

transfer function of the circuit is

H(s)=

Vc(s)

Vi(s)

=

1

1 +RCs

Considering the RC circuit, a feedback system with inputvi(t)and outputvc(t), the feedforward

transfer functionG(s)in Figure 6.9 is 1/RCs. Indeed, from the feedback system we have

E(s)=Vi(s)−Vc(s)

Vc(s)=E(s)G(s)

ReplacingE(s)in the second of the above equations, we have that

Vc(s)

Vi(s)

=

G(s)

1 +G(s)

=

1

1 + 1 /G(s)

so that the open-loop transfer function, when we compare the above equation toH(s), is

G(s)=

1

RCs

The RC circuit can be seen as a feedback system: the voltage across the capacitor is constantly com-

pared with the input voltage, and if found smaller, the capacitor continues charging until its voltage

coincides with it. How fast depends on the RC value.

FIGURE 6.9

Feedback modeling of an RC circuit in series.

vi(t) e(t)

vi(t)

vc(t)

vc(t)

+ −

G(s)=RCs^1 +

+

−

−

R

C