Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block...

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Closed-Loop Transfer Functions 1. Introduction 2. Stirred tank heating system 3. Closed-loop block diagrams 4. Closed-loop transfer functions 5. Simulink example

Transcript of Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block...

Page 1: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

Closed-Loop Transfer Functions

1. Introduction

2. Stirred tank heating system

3. Closed-loop block diagrams

4. Closed-loop transfer functions

5. Simulink example

Page 2: 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

Page 3: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

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

Page 4: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

Blending Process Model

Mass balances for constant volume

Linearized model

Transfer function model

),,()()()(

0

212211

2211

2121

wxxfV

xxwxxw

dt

dxwxxwxw

dt

Vxdwwwwww

V

wxxwxw

dt

dx

'2

'11

' )1('

)(1

)(1

)(1

)1()(

1)( '

22'

11'

2'1

1' sWs

KsX

s

KsW

swV

wxsX

swV

wwsX

Page 5: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

Control System Components

Composition analyzer – assume first-order dynamics

Controller – assume PI controller

I/P converter – assume negligible dynamics

Page 6: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

Control System Components cont.

Control valve – assume first-order dynamics

Entire blending system

Page 7: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

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

Page 8: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

Transfer Function for Setpoint Changes

mpvc

pvcm

sp

mspmcvpcvp

mspmmsp

cvpvppudu

GGGG

GGGK

Y

Y

YGYKGGGEGGGY

YGYKYYE

EGGGPGGUGYYYY

1

~

Page 9: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

Transfer Function for Disturbance Changes

mpvc

d

dmcvpcvp

mmmsp

dcvpdvpdpdu

GGGG

G

D

Y

DGYGGGGEGGGY

YGYYYE

DGEGGGDGPGGDGUGYYY

1

~

Page 10: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

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

Page 11: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

General Method

Closed-loop transfer function

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

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

» e = product of all transfer functions in feedback loop

Setpoint change

Disturbance change

e

f

iZ

Z

1

OLmpvcepvcmf GGGGGGGGK

OLmpvcedf GGGGGG

Page 12: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink example.

Closed-Loop Transfer Function Example

Page 13: Closed-Loop Transfer Functions 1.Introduction 2.Stirred tank heating system 3.Closed-loop block diagrams 4.Closed-loop transfer functions 5.Simulink 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