Proportional-Integral-Derivative (PID) Control of a Tank Level with Anti-Windup

proportional gain

1

integral time constant

1

tank height

discharge flow rate

The dynamic behavior of a tank of 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

, and

F

0

and

F

1

are the inlet and outlet flow rates (expressed in

3

m

/s

). Initially the tank height is 2 meters.

The discharge flow is given by

F

1

=maxK

h

-

K

p

e+

1

τ

i

edt,0≥0

, where

K=0.5

is the valve constant expressed in

3

m

/s

1/2

m

,

e=(3-h)

is the error,

K

p

is the proportional gain, and

τ

i

is the integral time constants. The setpoint for the tank height is chosen to be 3 meters.

The inlet flow rate is

F

0

=1.4

3

m

/s

.

The red and blue curves correspond to a controller with and without anti-windup. Anti-windup is important because it is possible that the discharge flow rate has a maximum value (taken here to be 1.5

3

m

/s

) corresponding to a fully open flow control valve. Computationally, this is achieved by setting

F

1

=minmaxK

h

-

K

p

e+

1

τ

i

edt,01.5

. When

F

1

reaches the maximal value of 1.5

3

m

/s

, the rate of change of the tank's height is constant and negative (equal to

-0.1

3

m

/s

) and the height decreases linearly versus time, as can be seen in snapshot 2.