Liquid Tank Controller

VI.7 Liquid Tank Controller

Liquid Tank Controller: Intro [Slide 212]

Simple control system maintaining the level of liquid in a tank.

A tank has an inlet pipe and an outlet pipe. The ﬂow in each pipe is controlled by a simple valve, which can be either open or closed.
A controller unit controls the valves on the inlet and outlet pipes to maintain the level of the liquid in the tank within some speciﬁed limits.

Liquid Tank Controller V1 [Slide 213]

V1 : valve on the inlet pipe; open/closed
V2 : valve on the outlet pipe; open/closed
I : rate of ﬂow of liquid in the inlet pipe
O : rate of ﬂow of liquid in the outlet pipe
L : the level of liquid in the tank
min, max : the given minimum and maximum levels

deﬁne min = 5.

deﬁne max = 10.

deﬁne closed = 0.

deﬁne open = 1.

deﬁne example() = {

exists V 1,V 2,I,O,L : {

initialise(V 1,V 2,L) and

keep{

pipe(V 1,I,0,3) and

pipe(V 2,O,1,1) and

tank(I,O,L) and

controller(V 1,V 2,L) and

format("V 1=%d,V 2=%d,I =%d,O=%d,L=%d∖n",

V 1,V 2,I,O,L)

} and len(50)

}

/* Initial state. Note valves closed imply that the ﬂows are also zero. */

deﬁne initialise(V 1,V 2,L) = {

V 1 = closed and

V 2 = closed and

L = 0

/* Deﬁnes a ﬂow rate F between minf and maxf, or zero if V is closed. */

deﬁne pipe(V,F,minf,maxf) = {

if(V = open) then

F = (Random mod (maxf −minf + 1)) + minf

else F = 0

/* The level of liquid in a unit interval. */

deﬁne tank(I,O,L) = {

L := L + I − O

/* The state of each valve in a unit interval */

deﬁne controller(V 1,V 2,L) = {

if L < max then {V 1 := open} else {V 1 := closed} and

if L > min then {V 2 := open} else {V 2 := closed}

Run of the controller:

State   0: V1=0, V2=0, I=0, O=0, L=0
...
State  13: V1=1, V2=1, I=2, O=1, L=10
State  14: V1=0, V2=1, I=0, O=1, L=11
State  15: V1=0, V2=1, I=0, O=1, L=10
...
State  27: V1=1, V2=1, I=0, O=1, L=7
State  28: V1=1, V2=1, I=2, O=1, L=6
State  29: V1=1, V2=1, I=0, O=1, L=7
...
State  42: V1=1, V2=1, I=2, O=1, L=10
State  43: V1=0, V2=1, I=0, O=1, L=11
State  44: V1=0, V2=1, I=0, O=1, L=10
...

So L does not stay between 5 and 10!

Liquid Tank Controller V2 [Slide 218]

V1 : valve on the inlet pipe; open/closed
V2 : valve on the outlet pipe; open/closed
I1 : rate of ﬂow of liquid willing to ﬂow into the inlet pipe
I : rate of ﬂow of liquid in the inlet pipe
O1 : rate of ﬂow of liquid willing to ﬂow into the outlet pipe
O : rate of ﬂow of liquid in the outlet pipe
L : the level of liquid in the tank
min, max : the given minimum and maximum levels

deﬁne min = 5.

deﬁne max = 10.

deﬁne closed = 0.

deﬁne open = 1.

deﬁne example() = {

exists V 1,V 2,I1,I,O1,O,L : {

initialise(L) and

keep{

pipe(V 1,I1,I,0,3) and

pipe(V 2,O1,O,1,1) and

tank(I,O,L) and

controller(V 1,V 2,L,I1,O1) and

format("V 1=%d,I1=%d,I =%d,

V 2=%d,O1=%d,O=%d,L=%d∖n",

V 1,I1,I,V 2,O1,O,L)

} and len(50)

}

/* Initial state. Note initial tank level is min. */

deﬁne initialise(L) = {

L = min

/* Deﬁnes a ﬂow rate F between minf and maxf,

Fw represents the willing ﬂow while

Fa represents the actually ﬂow which is either the willing ﬂow if

the valve is open and zero when valve V is closed. */

deﬁne pipe(V,Fw,Fa,minf,maxf) = {

Fw = (Random mod (maxf −minf + 1)) + minf and

if(V = open) then Fa = Fw else Fa = 0

/* The level of liquid in a unit interval. */

deﬁne tank(I,O,L) = {

L := L + I − O

/* The state of each valve in a unit interval */

deﬁne controller(V 1,V 2,I1,O1,L) = {

if L + I1 − O1 <= max then {V 1 = open} else {V 1 = closed} and

if L + I1 − O1 >= min then {V 2 = open} else {V 2 = closed}

Run of the controller:

State   0: V1=1, I1=2, I=2, V2=1, O1=1, O=1, L=5
State   1: V1=1, I1=2, I=2, V2=1, O1=1, O=1, L=6
State   2: V1=1, I1=2, I=2, V2=1, O1=1, O=1, L=7
State   3: V1=1, I1=3, I=3, V2=1, O1=1, O=1, L=8
State   4: V1=1, I1=0, I=0, V2=1, O1=1, O=1, L=10
State   5: V1=1, I1=1, I=1, V2=1, O1=1, O=1, L=9
...
...
State  44: V1=0, I1=3, I=0, V2=1, O1=1, O=1, L=10
State  45: V1=0, I1=3, I=0, V2=1, O1=1, O=1, L=9
State  46: V1=1, I1=0, I=0, V2=1, O1=1, O=1, L=8
State  47: V1=1, I1=1, I=1, V2=1, O1=1, O=1, L=7
State  48: V1=1, I1=2, I=2, V2=1, O1=1, O=1, L=7
State  49: V1=1, I1=1, I=1, V2=1, O1=1, O=1, L=8

Now L does stay between 5 and 10!

2024-08-02