The Stability of Linear Feedback Systems

Problems
231
A (c:edback control system has a chal1lcteristic equation:
s' + (4 + J()r + 6s + 16 + 8X - O.
...."""'K must be positive. What is the maximum value K can assume before the
beCOmes unstable When K is equal 10 the maximum value, the system oscillates.
. the frequency of oscillation.
A feedback control system has a characteristic equalion:
f' + 2s' + 5s' + 8s' + 8sl + 8s + 4 + O.
ifthe system is stable and determine the values of the roots.
The stability ofa motorcycle and rider is an important area for study because many
designs result in vehicles that are difficuh to control (10). The handling char-
• ofa motorcycle must include a model of the rider as well as one ofthe vehicle.
4J8amics of one motorcycle and rider can be represented by an open-loop tl1lnsfer
(...... P5,4)
K(r + lOs + 1125)
GH(s) - s(s + 20)(.r + lOs + I 25)(.r + 60s + 34(0) .
ID approximation, calculate the acceptable I1lnge of K for a stable system when the
polynomial (zeros) and the denominator polynomial (Sl + 60s + 34(0) are
(b) Calculate the actual range ofacceptable K accounting for all zeros and poles.
A s)'!tem has a transfer function
T( ) I
s :::: r+ 1.3.r+ 2.0s + I'
ioe ifthe system is stable. (b) Determine the roots ofthe characteristic equation.
the response of the system to a unit step input.
Problems
•
The contr~l
of the spark ignilion of an automotive engine requires constant per­
-.~ver a W1~e range ofparam.eters (21]. The control system is shown in Fig. DP5.1,
IUtoIpage With a controller gam K to be selected. The parameter p is, equal to 2 for
.... but can equal zero for high performance autos. Select a gain K that will result
Iystem for both values ofp.
~ automatically guided vehicl~ on Mars is represented by the system in Fig..
_ .5YSIem has a steerable wheelm both the front and back ofthe vehicle and the
"-tUires th I '
for .~ se ectlon of H(s) where H(s) - Ks + I. Determine (a) the value of K
10 I • ~blllty, (b) the value of K when one rool of the characteristic equation is
"-cI1he'lt, and (c) the value of the t.....o remaining roolS for the gain selected in part
response ofthe system to a step command for the gain selected in part (b).