Signals and Systems - Electrical Engineering

6.3 Application to Classic Control 375

− 1 −0.5^0

− 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

σ

jΩ

(a)

0 10 20 30 40 50

0

0.5

1

1.5

2

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

t

t

vc

(t

)

vc

(t

)

(b)

(c)

FIGURE 6.12

(a) Clustering of poles and time responsesvc(t)of second-order feedback system for (b)

√

2 / 2 ≤ψ≤ 1 and

(c) 0 ≤ψ≤

√

2 / 2.

(a) If we plot the poles ofH(s)asψchanges from 0 (poles injaxis) to 1 (double real poles) the
responsey(t)in the steady state changes from a sinusoid shifted up by 1 to a damped signal. The
locus of the poles is a semicircle of radiusn=1. Figure 6.12 shows this behavior of the poles
and the responses.
(b) As in the first-order system, the location of the poles determines the response of the system. The
system is useless if the poles are on thejaxis, as the response is completely oscillatory and
the input will never be followed. On the other extreme, the response of the system is slow when
the poles become real. The designer would have to choose a value in between these two forψ.
(c) For values ofψbetween

√

2 /2 to 1 the oscillation is minimum and the response is relatively

fast (see Figure 6.12(b)). For values ofψfrom 0 to

√

2 /2 the response oscillates more and more,

giving a large steady-state error (see Figure 6.12(c)).

nExample 6.4

In this example we find the response of an LTI system to different inputs by using functions in the

control toolbox of MATLAB. You can learn more about the capabilities of this toolbox, or set of

specialized functions for control, by running the demorespdemoand then usinghelpto learn more

about the functionstf, impulse, step, andpzmap, which we will use here.