Class 27: Block Diagrams

1

2

Dynamic Behavior and Stability of Closed-Loop Control Systems

• We now want to consider the dynamic behavior of

processes that are operated using feedback control.

• The combination of the process, the feedback controller,

and the instrumentation is referred to as a feedback control

loop or a closed-loop system.

Block Diagram Representation

To illustrate the development of a block diagram, we return to a

previous example, the stirred-tank blending process considered in

earlier chapters.

3

Composition control system for a stirred-tank blending process (Fig. 11.1)

CT CC

Controlled variable:

Measured variable:

Manipulated variable:

Disturbance variable:

Outlet concentration (x)Outlet concentration (x)Flow rate (w2)

Inlet concentration (x1)

(V, w1, and x2 assumed to be constant)

4

Process

In section 4.3 the approximate dynamic model of a stirred-tank

blending system was developed:

xwwxwxwdt

dxV )( 212211 +−+=ρ

( )21

22

2

1

1

212211

)1(1

)(

wwx

f

xxxxw

f

wx

f

xwwxwxwf

ss

ss

ss

+−=

∂

∂

≡−=−=

∂

∂

=

∂

∂

+−+=

variables

5

where

( )

( ) ( ) ( ) ( )sWw

xsX

w

wsXs

w

V

xwwxxwdt

xdV

211

211

11

)(1

′−

+′=′

+

′−′−+′=′

ρ

ρ

Combining partial fractions and deviation variables,

( ) ( ) ( )sWs

KsX

s

KsX 2

21

1

11′

++′

+=′

ττ

w

xKand

w

wK

w

V −===

1,, 2

11

ρτ

6

Figure 11.2 Block diagram of the process.

Disturbance

ProcessManipulated

Variable

Controlled

Variable

7

Wanted:Transfer function for each piece of equipment

+-

++

Please try to label variables, then transfer functions

8

Standard Labels

YYsp spY~

Ym

E P U Yu

+-

++

Yd

D

9

Definitions

Y = controlled variable

U = manipulated variable

D = disturbance variable (also referred

to as load variable)

P = controller output

E = error signal

Ym = measured value of Y

Ysp = set point

internal set point (used by the

controller)spY =%

Yu = change in Y due to U

Yd = change in Y due to D

Gc = controller transfer function

Gv = transfer function for final control

element (including KIP, if required)

Gp = process transfer function

Gd = disturbance transfer function

Gm = transfer function for measuring

element and transmitter

Km = steady-state gain for Gm

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Transfer Functions

YYsp spY~

Ym

E P U Yu

Km Gc Gv GP

Gd

Gm

+-

++

Yd

D

11

CT CC

Now back to our problem(Blending Tank)

Need transfer functions for:

12

Modified Block Diagram

YYsp spY~

Ym

E P U Yu

Km Gc Gv GP

Gd

Gm

+-

++

Yd

D

Gip

Pt

Notes:

1. All variables are in deviation variables except E

2. All variables are in Laplace coordinates (i.e., Y’(s))

3. Pink boxes need transfer functions

13

Composition Sensor-Transmitter (Analyzer)

We assume that the dynamic behavior of the composition sensor-

transmitter can be approximated by a first-order transfer function:

( )( )

(11-3)τ 1

m m

m

X s K

X s s

′=

′ +

Controller

Suppose that an electronic proportional plus integral controller is

used. From Chapter 8, the controller transfer function is

( )( )

11 (11-4)τ

cI

P sK

E s s

′ = +

where and E(s) are the Laplace transforms of the controller

output and the error signal e(t). Note that and e are

electrical signals that have units of mA, while Kc is dimensionless.

The error signal is expressed as

( )P s′

( )p t′ p′

Gm

Gc

14

( ) ( ) ( ) (11-5)sp me t x t x t′ ′= −%

or after taking Laplace transforms,

( ) ( ) ( ) (11-6)sp mE s X s X s′ ′= −%

The symbol denotes the internal set-point composition

expressed as an equivalent electrical current signal. This signal

is used internally by the controller. is related to the actual

composition set point by the composition sensor-

transmitter gain Km:

( )spx t′%

( )spx t′%

( )spx t′

( ) ( ) (11-7)sp m spx t K x t′ ′=%

Thus

( )( )

(11-8)sp

msp

X sK

X s

′=

′

%

Km

15

Current-to-Pressure (I/P) Transducer

Because transducers are usually designed to have linear

characteristics and negligible (fast) dynamics, we assume that the

transducer transfer function merely consists of a steady-state gain

KIP:( )( )

(11-9)t

IP

P sK

P s

′=

′

Control Valve

As discussed in Section 9.2, control valves are usually designed so

that the flow rate through the valve is a nearly linear function of

the signal to the valve actuator. Therefore, a first-order transfer

function usually provides an adequate model for operation of an

installed valve in the vicinity of a nominal steady state. Thus, we

assume that the control valve can be modeled as

( )( )

2(11-10)

τ 1

v

t v

W s K

P s s

′=

′ +

Gip

Gv

16

Block diagram for the entire blending process composition control system (Fig 11.7)

Done!

17

What about PID with derivative on measurement?

( ) ( ) ( )

( ) ( ) ( )ssYKsEs

KsP

dt

dYKteteKtP

mDc

I

c

mDC

t

I

c

ττ

ττ

−

+=′

−

+= ∫

11

1

0

G1(s) G2(s)

Ysp spY~

Ym

E P

Km G1+

-

G2

++

( ) msp YYsE ′−′=~

18

PID w/Derivative on Measurement

YYsp spY~

Ym

E P U Yu

Km GPI Gv GP

Gd

Gm

+-

++

Yd

D

Gip

Pt

Gderiv

++

19

Your Homework Problem

11.11

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Problem 11.11

c

CT

CC

Want to control c, not cTL

Disturbance

Manipulated variable

1. First identify the controlled variable,

manipulated variable, and disturbance

variable.

21

Prob. 11.11

22

Problem 11.11

c

CT

CC

Want to control c, not cTL

Set point

2. Draw a block diagram similar to the one we did in

class. My diagram has 9 boxes for transfer functions,

including the unit conversion on the set point (Km).

Also, you will have a transfer function GTL for the

transport delay in the transfer line. The time delay box

should be in the feedback loop, since it only represents

the delay in measurement.

Derivative on measurement