Page 88 - Modern Control Systems

62 Chapter 2 Mathematical Models of Systems

The inverse Laplace transform of Equation (2.22) is then
1 1 -1
y(t) = 2T + £- (2.26)
5 + 1 5 + 2
Using Table 2.3, we find that
21
y(0 = 2e~' - le~ . (2.27)
Finally, it is usually desired to determine the steady-state or final value of the re-
sponse of y(t). For example, the final or steady-state rest position of the spring-mass-
damper system may be calculated. The final value theorem states that

lim y(t) lim sY(s), (2.28)
t—*oo s->0
where a simple pole of Y(s) at the origin is permitted, but poles on the imaginary
axis and in the right half-plane and repeated poles at the origin are excluded. There-
fore, for the specific case of the spring-mass-damper, we find that
lim y(t) = lim sY(s) 0. (2.29)
Hence the final position for the mass is the normal equilibrium position y - 0.
Reconsider the spring-mass-damper system. The equation for Y(s) may be writ-
ten as
(s + b/M)y Q (s + 2£co n)y 0
Y(s) = z , l (2.30)
s + (b/M)s + k/M s + 2£o) ns + cof,
where £ is the dimensionless damping ratio, and <o n is the natural frequency of the
system. The roots of the characteristic equation are
s 2 = ± w „ V r - l , (2.31)
s h -£(o n
where, in this case, co n = vk/M and £ - b/(2vkM). When £ > 1, the roots are
real and the system is overdamped; when £ < 1, the roots are complex and the sys-
tem is underdamped. When £ = 1, the roots are repeated and real, and the condi-
tion is called critical damping.
When £ < 1, the response is underdamped, and
2
s h2 = -£co n ± /o)„Vl - t . (2.32)
The s-plane plot of the poles and zeros of Y(s) is shown in Figure 2.9, where
-1
0 = cos £. As £ varies with co n constant, the complex conjugate roots follow a circular
)<»
5
ty,Vi
' ¥:• M - ?
l N N a>„
0=COS ' ^ - U
£
—o-
-#*>»
FIGURE 2.9
An s-plane plot of
7
the poles and zeros s 2 X- - A ^ ?
of Y{s).