lecture-slides chapter 04

7/25/2019 Lecture-Slides CHAPTER 04

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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali


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Feedback Control System

Characteristics

Chapter 4


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Introduction

Error signal analysis

Sensitivity of control systems to variations in process

parameters

Disturbance rejection

Steady state error

Outline


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Open-loop control system

utilizes an actuating device to control the process (plant)directly without using feedback.

Closed-loop control system

uses a measurement of the output and feedback of this signal

to compare it with the desired input (reference or command).

Also called as feedback control system

Introduction


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Notation:R(s): Input signal;

U(s): Control or actuating signal;

Y(s): Output signal;

E(s): Error signal;

Gc(s): Transfer function of the controller;

G(s): Transfer function of the process or plant;

H(s): Transfer function of the sensor.

Feedback Control System

Controller Process

Sensor

Measured output

_

+ sY sR

sE sU

sG c sG

sH


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Why feedback control system:

Advantages: Decreased sensitivity of the system to variations in the parameters of

the process

Improved disturbance rejection

Improved steady-state error reduction Easy control and adjustment of the transient response of the system

Disadvantages:

Increased cost Increased system complexity

Now we examine how feedback improves system performance

Feedback Control System (FCS)


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Tracking error:

Where L(s)=Gc(s)G(s)H(s) is the loop gain

Error Signal Analysis

Controller Process

Sensor

Measured output

_

+ sY sR

sE sU

sG

c sG

sH

sRsL

sE

1

1

sRsHsGsG

sE

c

1

1

sYsHsRsE


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Sensitivity function: S(s)

Consider a unity feedback system H(s)=1:

To minimize the tracking error (for a given G(s)):

We need S(s) to be small

We need loop gain L(s) to be large, where

Error Signal Analysis continued

sL

sS

1

1

sGsGsL

sS

c

1

1

1

1

sGsGsLc


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Tracking error:

Disturbance Rejection

Controller Process

Sensor

Measured output

_

+ sY sR sE sGc

sG

1sH

++

sD

sYsHsRsE

sDsD

sYsR

sR

sYsY

sRsD 00

sD

sL

sGsR

sL

sE

11

1


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For R(s)=0; the disturbance D(s) is related to E(s) as:

To minimize the effect of disturbance (for a given G(s)):

We need loop gain L(s) to be large, where

Disturbance Rejection continued

sDsL

sGsD

sGsG

sE

c

11

1

sGsGsLc


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G(s): Transfer function of the process

G(s) includes the process parameters subjected to variationsdue to changing environment, non-exact values of

parameters, other errors, etc.

In open-loop systems, these errors and changes result in

inaccurate output Closed-loop system senses the changes in output and

attempts to correct the output (an advantage of FCS)

FCS has an ability to reduce systems sensitivity

Sensitivity of FCS to process parameters


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System sensitivity:

The ratio of the percentage change in the system transferfunction (T(s)) to the percentage change in the process

transfer function (G(s))

Where T(s) is the transfer function of the system

In the limit for small incremental changes:


sG

sG

sT

sT

S

T

G

G

T

sG

sG

sT

sT

S


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Tracking error for unity feedback system

With changes in the process parameters:

As GcG>>GcG, and for large values of L, 1+LL then


sRsGsG

sE

c

1

1

sRGGGGGG

GGsE

ccc

c

11

sRsGsGsG

sEsE

c )(1

1

sRG

G

L

sE

1


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For changes in a certain parameter K of the process:

where


T

G

G

TS

T

G

G

K

T

G

T

K SSS

G

K

K

GS

G

K


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Error of a unity feedback system

For open-loop system

For closed loop system

According to the final value theorem (properties of LT

chapter2)

Open-loop:

Closed-loop:

4.6: Steady State Error

sRsTsRsTsRsEc

1

sYsRsE

ssEtest 0

limlim

sRsGsRsGsRsEo

1

sRsGses

o

1lim0

sRsGses

c

1lim0

lecture-slides chapter 04

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