Closed-Loop Transfer Functions

1. Introduction

2. Stirred tank heating system

3. Closed-loop block diagrams

4. Closed-loop transfer functions

5. Simulink example

Introduction

Block diagrams

» Convenient tool to represent closed-loop systems

» Also used to represent control systems in Simulink

Closed-loop transfer functions

» Transfer function between any two signals in a

closed-loop system

» Usually involve setpoint or disturbance as the

closed-loop input and the controlled output as the

closed-loop output

» Conveniently derived from block diagram

» Can be derived automatically in Simulink

» Used to analyze closed-loop stability and compute

closed-loop responses

Stirred Tank Blending System

Control objective

» Drive outlet composition (x) to setpoint (xsp) by manipulating pure stream flow rate (w2) despite disturbances in flow rate (w1) and composition (x1) of other feed stream

Control system

» Measure x with composition analyzer (AT)

» Perform calculation with composition controller (AC)

» Convert controller output to pneumatic signal with current-pressure converter (I/P) to drive valve

Blending Process Model

Mass balances for constant volume

Linearized model

Transfer function model

),,()()()(

0

212211

2211

2121

wxxfV

xxwxxw

dt

dxwxxwxw

dt

Vxd

wwwwww

V

wxxwxw

dt

dx

'

2

'

11

' )1('

)(1

)(1

)(

1

)1()(

1

)( '

22'

11'

2

'

11' sW

s

KsX

s

KsW

sw

V

wxsX

sw

V

wwsX

Control System Components

Composition analyzer – assume first-order dynamics

Controller – assume PI controller

I/P converter – assume negligible dynamics

Control System Components cont.

Control valve – assume first-order dynamics

Entire blending system

Closed-Loop Block Diagrams

Gp(s) – process transfer function

Gd(s) – disturbance transfer function

Gv(s) – valve transfer function

Gc(s) – controller transfer function

Gm(s) – measurement transfer function

Km – measurement gain

Y(s) – controlled output

U(s) – manipulated input

D(s) – disturbance input

P(s) – controller output

E(s) – error signal

Ysp(s) – setpoint

Ym(s) – measurement

Transfer Function for Setpoint Changes

mpvc

pvcm

sp

mspmcvpcvp

mspmmsp

cvpvppudu

GGGG

GGGK

Y

Y

YGYKGGGEGGGY

YGYKYYE

EGGGPGGUGYYYY

1

~

Transfer Function for Disturbance Changes

mpvc

d

dmcvpcvp

mmmsp

dcvpdvpdpdu

GGGG

G

D

Y

DGYGGGGEGGGY

YGYYYE

DGEGGGDGPGGDGUGYYY

1

~

Simultaneous Changes

Principle of superposition

Open-loop transfer function

» Obtained by multiplying all transfer functions

in feedback loop

DGGGG

GY

GGGG

GGGKY

mpvc

dsp

mpvc

pvcm

11

DG

GY

G

GGGKY

GGGGG

OL

dsp

OL

pvcm

mpvcOL

11

General Method

Closed-loop transfer function

» Z = any variable in feedback system

» Zi = any input variable in feedback system Z and Zi

» Pf = product of all transfer functions between Z and Zi

» Pe = product of all transfer functions in feedback loop

Setpoint change

Disturbance change

e

f

iZ

Z

P

P

1

OLmpvcepvcmf GGGGGGGGK PP

OLmpvcedf GGGGGG PP

Closed-Loop Transfer Function Example

Simulink Example

>> gp=tf([6.37],[5 1]);

>> kv=0.0103;

>> kip=0.12;

>> km=50;

>> gc=tf([2.5 5],[0.5 0]);

>> gcl=gp/(1+gc*kv*gp*km)

Disturbance transfer function:

15.93 s^2 + 3.185 s

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

12.5 s^3 + 46.01 s^2 + 90.72 s + 16.4

Tank

6.37

5s+1Setpoint

0

PID Controller

PID

Level

y

Kv

0.0103

Km1

50

Km

50

Kip

0.12

Inlet flow

0.05

Add1Add