page 19

(note: this program will be similar to all others)

4.1.3 First Order Response

•The deadbeat controller can call for extremely high gains, but this requires a high level of power not commonly found in engineered systems.

•When we want to slow down the system response so that it occurs over a number of controller steps, we can replace the deadbeat (one time step) function with a gradual first order reduction (exponential decay) over a number of time steps.

•This gradual response will use a time constant to produce the rate of response (recall that the difference between input and output is reduced 63% for each duration of the time constant).

page 20

• The general form of this relationship is seen below,
First we relate the output ‘c’ to the input ‘r’ (Note the first order differential equation).
τ		d		c + c		=	r
		----
		dt
Next, we convert this to a discrete equation using the methods (source - table) seen before,
								T
								–--
							B	1 – e τ
	G			cn		=			(from transform table)
			dr	= ----			-------------------------
				r				T
								--
					n		1 – Be– τ
				T
								actual
							desired

• Consider the example for the deadbeat controller,

page 21

Assuming the process model given below, find the general form for the control equation,

First we write the process model,

Gp( B) =

B

1-------------------– 0.5B

Next, we use this and the desired response to find the controller equation,

–T--

B 1 – e

τ

-------------------------

T

–

--

cn

1

Gdr( B)

1 – 0.5B

1 – Be

τ

Gc( B) =

=

=

----

--------------

--------------------------

-------------------

----------------------------------------

rn

Gp( B)

1 – Gdr(

B)

B

T

–-τ-

B 1 – e

1 –

T

-------------------------

–--

1 – Be

τ

–T

( 1 – 0.5B)

1 – e

–T

–

–T

B

1 – e

τ

τ

0.5 1 – e

τ

Gc( B) =

=

-------------------------------------------------------------------------

–T

–T

–T

–T

[ 1] +

B

1 – Be τ

–

B 1 – e

τ

– e

τ – 1 + e

τ

After some work.....

this should lead the control difference equation,

mn = mn – 1 + …