The Stability of Linear Feedback Systems

5.3 The Relative Stability of Feedback Control Systems
219
"•
,
"
"
I
• ,
" "' ,,
'.
10
Figure 5.6. Root locations in the s-planc.
uation lie in the right halfof the s-plane. However, if the system satisfies
th_Hurwitz criterion and is absolutely stable, it is desirable to determine
;ve srabilily; that is, it is necessary to investigate the relative damping of
root ofthe characteristic equation. Thc relative stability ofa system may be
as the property that is measured by the relative settling times ofeach root
. ofroots. Therefore relative stability is represented by the real part ofeach
Thus root'! is relatively more stable than the roots,,, 'I as shown in Fig.
1be relative stability ofa system can also be defined in terms of the relative
coefficients f of each complex root pair and therefore in terms of the
ofresponse and overshoot instead ofsettling time.
Hence the investigation of the relative stability of each root is clearly necesbecause,
as we found in Chapter 4, the location of the closed-loop poles in
'"Plane determines the performance of the system. Thus it is imperative that
ioe:xamine the characteristic equation q(s) and consider several methods for
determination of relative stability.
use the relative stability of a system is dictated by the location of the
ofthe characteristic equation, a first approach using an s-plane formulation
extend the Routh-Hurwitz criterion to ascertain relative stability. This can
ply accomplished by utilizing a change of variable, which shifts the s-plane
in order to utilize the Routh-Hurwitz criterion. Examining Fig. 5.6, we notice
a ~ft of the vertical axis in the s-plane to -(11 will result in the roots 'I, 'I
on the shifted axis. The correct magnitude to shift the vertical axis
be btained on a trial-and-error basis. Then, without solving the fifth-order
iaI q(s), one may determine the real part of the dominant roots,,, 'I'
Example 5.6 Axis shift
the simple third-order characteristic equation
q(s) ~ s' + 4s' + 6s + 4. (5.17)
Y! one might shift the axis other than one unit and obtain a Routh-Hurwitz
\\>tth~ut a zero occurring in the first column. However, upon setting the
vanable Sn equal to s + I, we obtain
"
(5.18)