Page 126 - Modern Control Systems

100 Chapter 2 Mathematical Models of Systems
The partial fraction expansion yields

^ 2 W s + ft s
Taking the inverse Laplace transform yields
AQ 2 (0 = -q 0e- w at + q,
Note that ft > 0 (see Equation (2.120)), so the term e nt approaches zero as t ap-
proaches co. Therefore, the steady-state output due to the step input of magnitude
? 0 is

We see that in the steady state, the deviation of the output mass flow rate from the
equilibrium value is equal to the deviation of the input mass flow rate from the equi-
librium value. By examining the variable ft in Equation (2.120), we find that the
larger the output port opening A 2, the faster the system reaches steady state. In
other words, as ft gets larger, the exponential term e~ nt vanishes more quickly, and
steady state is reached faster.
Similarly for the water level we have
^
= f - i - - i
AffW
w
ft \ s + ft s
Taking the inverse Laplace transform yields

AH(0 = ^ ( e - ° - - 1).

The steady-state change in water level due to the step input of magnitude q 0 is
q 0ki
A * . - fl.

Consider the sinusoidal input

AQi(0 = q Q sin ojt,
which has Laplace transform

S + ft)
Suppose the system has zero initial conditions, that is, AQ 2(0) = 0. Then from Equa-
tion (2.122) we have
, , <7 0ft)ft
AQ 2(s) = 2 2
(S + ft)(s + ft) )
Expanding in a partial fraction expansion and taking the inverse Laplace trans-
form yields
( e~ nt sin(fttf - <f>) \
A£? 2(0 = q 0&<»\ „ 2 ^ 2 + /r>2 ^ 2x1/2 '
1 1 1
\ft z + or ft)(ft z + (a ) ' J