Modern Control Engineering

470 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

Resonant Peak Magnitude Mrand Resonant Frequency Vr. Consider the

standard second-order system shown in Figure 7–73. The closed-loop transfer function

is

(7–16)

wherezandvnare the damping ratio and the undamped natural frequency, respectively.

The closed-loop frequency response is

where

As given by Equation (7–12), for 0z0.707, the maximum value of Moccurs at

the frequency vr, where

(7–17)

The frequency vris the resonant frequency. At the resonant frequency, the value of M

is maximum and is given by Equation (7–13), rewritten

(7–18)

whereMris defined as the resonant peak magnitude. The resonant peak magnitude is

related to the damping of the system.

The magnitude of the resonant peak gives an indication of the relative stability of the

system. A large resonant peak magnitude indicates the presence of a pair of dominant

closed-loop poles with small damping ratio, which will yield an undesirable transient

response. A smaller resonant peak magnitude, on the other hand, indicates the absence

of a pair of dominant closed-loop poles with small damping ratio, meaning that the

system is well damped.

Remember that vris real only if z<0.707. Thus, there is no closed-loop resonance

ifz>0.707. [The value of Mris unity only if z>0.707. See Equation (7–14).] Since

the values of Mrandvrcan be easily measured in a physical system, they are quite useful

for checking agreement between theoretical and experimental analyses.

Mr=

1

2 z 21 - z^2

vr=vn 21 - 2 z^2

M=

1

B

a 1 -

v^2

v^2 n

b

2

+ a 2 z

v

vn

b

2

, a=-tan-^1

2 z

v

vn

1 -

v^2

v^2 n

C(jv)

R(jv)

=

1

a 1 -

v^2

v^2 n

b +j2z

v

vn

=Meja

C(s)

R(s)

=

v^2 n

s^2 + 2 zvn s+v^2 n

R(s) vn C(s)

s(s+ 2 z vn)

2

+

• Figure 7–73
  Standard second-
  order system.

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