Page 484 - Mechanical Engineers' Handbook (Volume 2)

6 Stability 475

1. For a second-order system with no finite zeros, the transient parameters can be
approximated by Eq. (77).
2. An additional zero as in Eq. (78) in the left-half plane will increase the overshoot if
the zero is within a factor of 4 of the real part of the complex poles. A plot is given
in Fig. 26.
3. An additional zero in the right-half plane will depress the overshoot (and may cause
the step response to undershoot). This is referred to as a non-minimum-phase system.
4. An additional pole in the left-half plane will increase the rise time significantly if
the extra pole is within a factor of 4 of the real part of the complex poles. A plot is
given in Fig. 27.

6 STABILITY
We shall distinguish between two types of stabilities: external and internal stability. The
notion of external stability is concerned with whether or not a bounded input gives a bounded
output. In this type of stability we notice that no reference is made to the internal variables
of the system. The implication here is that it is possible for an internal variable to grow
without bound while the output remains bounded. Whether or not the internal variables are
well behaved is typically addressed by the notion of internal stability. Internal stability re-
quires that in the absence of an external input the internal variables stay bounded for any
perturbations of these variables. In other words internal stability is concerned with the re-
sponse of the system due to nonzero initial conditions. It is reasonable to expect that a well-
designed system should be both externally and internally stable.
The notion of asymptotic stability is usually discussed within the context of internal
stability. Specifically, if the response due to nonzero initial conditions decays to zero as-
ymptotically, then the system is said to be asymptotically stable. A LTI system is asymp-
totically stable if and only if all the system poles lie in the open left-half-plane (i.e., the
left-half s-plane excluding the imaginary axis). This condition also guarantees external sta-
bility for LTI systems. So in the case of LTI systems the notions of internal and external
stability may be considered equivalent.
For LTI systems, knowing the locations of the poles or the roots of the characteristic
equation would suffice to predict stability. The Routh–Hurwitz stability criterion is frequently
used to obtain stability information without explicitly computing the poles for LTI. This
criterion will be discussed in Section 6.1.
For nonlinear systems, stability cannot be characterized that easily. As a matter of fact,
there are many definitions and theorems for assessing stability of such systems. A discussion
of these topics is beyond the scope of this handbook. Interested reader may refer to Ref. 3.

6.1 Routh–Hurwitz Stability Criterion

This criterion allows one to predict the status of stability of a system by knowing the co-
efficients of its characteristic polynomial. Consider the characteristic polynomial of an nth-
order system:

n
P(s) as a n 1 s n 1 a n 2 s n 2 as a 0
1
n
A necessary condition for asymptotic stability is that all the coefficients {a }’s be positive.
i
If any of the coefficients are missing (i.e., are zero) or negative, then the system will have