Page 86 - Modern Control Systems

60 Chapter 2 Mathematical Models of Systems

Alternatively, the Laplace variable s can be considered to be the differential
operator so that

' = J, (2.16)

Then we also have the integral operator
1
dt. (2.17)
s
The inverse Laplace transformation is usually obtained by using the Heaviside
partial fraction expansion. This approach is particularly useful for systems analysis
and design because the effect of each characteristic root or eigenvalue can be clear-
ly observed.
To illustrate the usefulness of the Laplace transformation and the steps involved
in the system analysis, reconsider the spring-mass-damper system described by
Equation (2.1), which is
2
d y . dy
M - M
T 2 T + b^ + ky = r{t). (2.18)
dt dt
We wish to obtain the response, y, as a function of time. The Laplace transform of
Equation (2.18) is

2
M(S Y(S) - sy(Q~) - y(0") J + b(sY(s) - y(0~)) + kY(s) = R(s). (2.19)
When
dy
r(t) ~ 0, and v(0 ) = v 0 , and — = 0,
dt l=Q-
we have
2
Ms Y(s) - Msy {) + bsY(s) - by 0 + kY(s) = 0. (2.20)
Solving for Y(s), we obtain
(Ms + b)y 0 p(s)
Y(s) = = = ——. (2.21)
w 2 v
Ms + bs + k q{s)
The denominator polynomial q(s), when set equal to zero, is called the characteristic
equation because the roots of this equation determine the character of the time
response. The roots of this characteristic equation are also called the poles of the sys-
tem. The roots of the numerator polynomial p(s) are called the zeros of the system;
for example, s — —b/M is a zero of Equation (2.21). Poles and zeros are critical fre-
quencies. At the poles, the function Y(s) becomes infinite, whereas at the zeros, the
function becomes zero. The complex frequency s-plane plot of the poles and zeros
graphically portrays the character of the natural transient response of the system.
For a specific case, consider the system when k/M = 2 and b/M = 3. Then
Equation (2.21) becomes

y ( 5 ) = ( Z 2 2 )
(, + 1)(, + 2)-