page 56

Given the water tank from before (with the given variables), try a variety of different values for the PID gains to see how the system responds.

egr450.1.mcd

5.3 BLOCK DIAGRAMS AND TRANSFER FUNCTIONS

•Block diagrams are the primary tool for showing process and control system models.

•In a block diagram each block has one input and one output.

•A transfer function provides functions that are a ratio of block output to block input.

•Consider the block diagram seen before, but with transfer functions (ratio of output to input) shown for the controller and process.

page 57

			Controller			Process
rn	+	en	Gc	( B)	mn	Gp	( B)	cn
		-

• Both of the transfer functions are ratios of inputs to outputs,

Gc( B)	mn	Gp( B)	cn
	= ------		= ------
	en		mn

•These transfer functions (ratios) are rarely a simple multiplication, and so we need to use an alternate representation called a ‘transform’.

•In this case we will use a backshift transform, hence the ‘B’ in the representation.

•equations for the block diagram below,

			Controller			Process
rn	+	en	Gc	( B)	mn	Gp	( B)	cn
		-

page 58

• Recall the techniques for block diagram manipulation.

5.3.1 The Backward-Shift ‘B’ Operator

•An operator is a special part of an equation - one example is the (d/dt) of calculus.

•The basic definition of the backshift operator ‘B’ is,

yn = yn

yn – 1 = Byn yn – 2 = B2yn

yn – m = Bmyn

• If we apply this operator to an equation seen before,

mn – mn – 1 = K0en + K1en – 1 + K2en – 2 + K3en – 3 + K4en – 4 + …

mn – Bmn = K0en + K1Ben + K2B2en + K3B3en + K4B4en + …

m		K		+ K	B + K	B2 + K		B3 + K	B4 + …	( B)
------	n =	---------------------------------------------------------------------------------------------	0	1	2	1 – B	3	4	= Gc
en

•This form then allow some very useful techniques that we will see later.

•Apply the ‘B’ transform to the PID expression found before,

page 59

m

= m

+ e

K

+ K

Kd

+ e

– K

– 2

Kd

+ e

Kd

n – 1

T + -----

-----

-----

n

n

p

i

T

n – 1

p

T

n – 2 T

• Apply the ‘B’ transform to the valve and water tank found before,

–T

Ko

K

Ko

K

A

A

e

-----

+ θ

i

–T -----

i

h

= h

e

+

-----

-----

n

n – 1

n – 1

Ko

Ko