PID Control of a Tank Level

proportional gain

1

integral time constant

1

differential time constant

0.1

The dynamic behavior of the tank height

h

(in meters) is governed by the following ODE:

A

dh

dt

=

F

0

-

F

1

where

A=1

is the tank area in

2

m

,

F

0

and

F

1

are the inlet and outlet flow rates (expressed in

3

m

/s

), respectively. Initially the tank height is equal to 0.1 meter.

The flow equation is given by:

F

1

=K

h

where

K=0.5

is the valve constant expressed in

3

m

/(s

0.5

m

).

The setpoint for the tank height is chosen to equal 2 meters.

The inlet flow rate is varied in order to achieve the desired setpoint value using a P, PI, or PID (proportional–integral–derivative) control:

F

0

=1+

K

p

e+

1

τ

i

edt+

τ

d

de

dt

, where

e=(2-h)

is the error,

K

p

is the proportional gain, and

τ

i

and

τ

d

are the integral and differential time constants, respectively.

For large

τ

i

and

τ

d

=0

, the control simplifies to the usual proportional control, which is usually characterized by a small offset value (i.e., the final steady-state height is not exactly equal to the setpoint value).

PI control is achieved when

τ

d

is taken to be zero. PI control can show an overshoot and dumped oscillations around the setpoint. No offset is observed and the final steady-state tank height is exactly equal to the setpoint value.

In the most general case, when

τ

d

≠0

and

τ

i

is not too large, one gets PID control of the tank height.