
F
–
dH
dt
A
+F
0
= F
1
F
0
= cH
1/2
u
e
1
Controller
Tank dynamics
e
–
u
F
1
H
H
com
FIGURE E6.37
(a) The state derivative is
dH
dt
¼
f
1
(H), H 2 S
1
f
2
(H), H 2 S
2
(
Regions S
1
and S
2
are defined such that when H is in region S
1
of state space, the
controller is off, and the opposite is true when H is in region S
2
. Find expressions for S
1
and S
2
in terms of the state H.
(b) Find expressions for the state derivate functions f
1
(H) and f
2
(H).
(c) Find the discontinuity function f(H) that specifies which region the state is in based on its
sign, that is, f(H) ¼0 implies the state H is transitioning between the two regions.
(d) Use the method that finds the time of the discontinuity to simulate ...