Control Sys

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Chapter 5 The Performance of Feedback Control Systems

The ability to adjust the transient and steady-state response of a feedback controlsystem is a beneficial outcome of the design of control systems.

One of the first steps in the design process is to specify the measures ofperformance.

In this chapter we introduce the common time-domain specifications such as

percent overshoot, settling time, time to peak, time to rise, and steady-state trackingerror.

We will use selected input signals such as the step and ramp to test the response ofthe control system.

The correlation between the system performance and the location of the systemtransfer function poles and eros in the s-plane is discussed.

We will develop valuable relationships between the performance specifications andthe natural fre!uency and damping ratio for second-order systems.

"elying on the notion of dominant poles, we can e#trapolate the ideas associated

with second-order systems to those of higher order.


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Introduction

Steady-State: exists a long time following any input signal initiation

Transient Response: disappears with time

Design Specifications: normally include several time-response indicesfor a specified input command as well as a desired steady-stateaccuracy.


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Test Input Signals

A unit impulse function is also useful for testsignal purposes. It s characteristics areshown to the right.


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Performance of a Second-Order System

! s" #$

s%

p s+ $ +

R s" #

! s" #n

%

s%

% n s+ n%+

& %with a unity step input

cos & ( )! s" #

n%

s%

% n s+ n%+ s

y t" # & &

e

n t sin n t +( )


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Rise Time' Tr (ea) Time' To(ercentage *vershoot' (.*.Settling Time' Ts

+ormali,ed Rise Time Tr&


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Standard performance measures are usually defined in terms ofthe step response of a system. The transient response of a systemmay e descri ed using two factors' the swiftness and thecloseness of the response to the desired response.

The swiftness of the response is measured y the rise time "Tr#and the pea) time "Tp#.

nderdamped systems: /-&//0 rise time is used

*verdamped systems: &/-1/0 rise time is used

The closeness is measured y the overshoot and settling time.sing these measurements the percent overshoot "(.*.# can e

calculated.


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(*

2 pv fv

fv &//

T s3

n

T p

n & %

2 pv & e

& %

+

(* &// e

& %


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+aturally these two performancemeasures are inopposition and acompromise must emade.


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ffects of a Third Pole and !ero on the Second-Order System

T s" #&

s%

% s+ &+( ) s &+( )


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"ample 5#$ - Parameter Selection

Select the gain $ and the parameter p so thatthe percent overshoot is less than 40 and the

settling time "within %0 of the final value#should e less than 3 seconds.


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"ample 5#$ - Parameter Selection

Ts3

n3sec

n &

When the closed-loop roots are chosen as:r & & 5 &+r % & 5 &

We have Ts 3sec and an overshoot of 4.3%.

Therefore, &%

and n&

%

T s" #6 s" #

& 6 s" #+

$

s % p s+ $ +

n%

s % % n s+ n%+

$ n%

% ( % n %


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"ample 5#% &ominant Poles of T's(

! s" #R s" #

T s" #

n%

a

s a+" #

s% % n s+ n%+( )& s+( )

For n 7 , /.&8 , and a %.4:

T s" #8%.4 s %.4+" #

s%

8 s+ %4+( )s 8.%4+" #


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"ample 5#% &ominant Poles of T's(

T s" #8%.4 s %.4+" #

s% 8 s+ %4+( )s 8.%4+" # As a first approxi ation, we ne!lect the real pole and o"tain:

T s" #&/ s %.4+" #

s% 8s+ %4+( )

We now have /.8 and n 4 for do inant poles with one

acco panyin! #ero for which a

n/.977

$sin! the previously entioned charts Fi!ure &.'3a(, we find that thepercent overshoot is &&%. We expect the settin! ti e to within )% of thefinal value to "e:

T s" #3

n3

/.8 4&.77sec

$sin! co puter si ulations the actual percent overshoot is e*ual to 3+%and the settlin! ti e is '. seconds.


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The s-Plane )oot *ocation and The Transient )esponse


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Steady-State rror of Feedback Control Systems

or

Step Input - (osition ;rror


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The Steady-State rror of +onunity Feedback Systems

or a system in

which the feed ac)is not unity " ig4.%' the units of theoutput are usuallydifferent from theoutput of the sensor.In ig. 4.%%' $& and$% convert fromrad>s to volts.


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The Steady-State rror of +onunity Feedback Systems

T s" #$ & 6 s" #

& $ & 6 s" #+

; s" # R s" # ! s" # & T s" #" # R s" # ess/s

s ; s" #lim

&

& $ & 6 /" #+

; s" # &

& $ & 6 s" #+ R s" #


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Performance Indices

A performance index is a ?uantitative measure of the performance of a system and is chosen so that emphasisis given to the important system specifications.

A system is considered an optimum control systemwhen the system parameters are ad5usted so that theindex reaches an extremum value' commonly aminimum value.


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Performance Indices

There are several performance indices:

" Integral of the s?uare of the error' IS;

"%# Integral of the a solute magnitude of the error' IA;

"7# Integral of time multiplied y a solute error' ITA;

"3# Integral of time multiplied y the s?uared error' ITS;

IS;/

T

te% t" # d

IA;/

T

te t" # d

ITA;/

T

tt e t" # d

ITS;/

T

tt e% t" # d


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System Performance ,sing .T*./ and Simulink


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"ercises and Problems

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