36 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
perturbations, from the equilibrium 0 at some timet 0 , the system remains close
to it in subsequent motion. This concept of stability is due to Lyapunov, and
often referred to as ìstability in the sense of Lyapunov.î
In Figure 2.11, thefigure on the left represents stability in the sense of
Lyapunov if friction is ignored, and asymptotic stability if friction is taken into
account, whereas thefigure in the center represents instability. Thefigure on
the right represents stability, which is a local condition. In thefigure on the
left, even if friction is present, a ball would eventually return to equilibrium no
matter how large the disturbance. This is an illustration ofasymptotic stability
in the large.
2.4.1 Dampingandsystemresponse
A control system produces an output, or response, for a given input, or stim-
ulus. In a stable system, the initial responseuntilthesystemreachessteady
state is called thetransient response. After the transient response, the system
approaches its steady-state response, which is its approximation for the com-
manded or desired response. The nature and duration of the transient response
are determined by thedampingcharacteristics of the plant.
The possibility exists for a transient response that consists of damped oscilla-
tions ñ that is, a sinusoidal response whose amplitude about the steady-state
value diminishes with time. There are responses that are characterized as being
overdamped(Figure 2.12 (a)) orcritically damped(Figure 2.12 (b)). An
Figure 2.12. (a) Overdamped response (b) Critically damped responseoverdamped system is characterized by no overshoot. This occurs when there
is a large amount of energy absorption in the system that inhibits the tran-
sient response from overshooting and oscillating about the steady-state value
in response to the input. A criticallydamped response is characterized by no
overshoot and a rise time that is faster than any possible overdamped response
with the same natural frequency. Both are considered stable because there is
a steady-state value for each type of response. Stated differently, the system
is in ìequilibrium.î This equilibrium condition is achieved even if the system
is allowed to oscillate a bit before achieving steady state. Systems for which