control systems design tp

7/26/2019 Control Systems Design TP

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Control Systems Design

Laboratory Experiment 2: Controller Design Using Root Locus

Objectives:At the conclusion of this laboratory experience, students should be able to:

Compute the angle of departure related to a pair of desired closed-loop pole locations.

Use the angle of departure to aid in the placement of compensator (controller) poles and

zeros.

esign !, !, !" and !" controllers to meet closed-loop performance specifications

including transient performance and steady-error.

Overview

"n this lab you #ill explore the use of the root locus controller design methodology. $he root

locus indicates the achie%able closed-loop locations of a system as a parameter (usually the

controller gain) %aries from zero to infinity. &or a gi%en plant it may or may not be possible to

implement a simple proportional controller (i.e, select a gain that specifies closed-loop pole

locations along the root locus) to achie%e the specified performance constraints. "n fact, in most

cases it #ill not be possible #hen this occurs, it is the control engineers 'ob to select a controller

structure (a gain and numbers of pole and zeros) to change the shape of the root locus so that for

some %alues of the controller gain, the dominant second order closed-loop poles lie #ithin the

performance region. "n this lab #e are in%estigating se%eral controller structures on indi%idual

plants and comparing the design process and performance. e #ill be using the A$*A+

sisotool toolbox to complete the root locus designs.

List o t!e E"uipment#Sotware

&ollo#ing euipment/soft#are is reuired:

A$*A+

Category 0oft-1xperiment

Deliverables

A complete lab report including the follo#ing:

&igures #ith plots of closed-loop step responses and control efforts.

Controller parameters, gain, pole(s), and zero(s), for each of the controller designs.

$n%lab & 'art (

Use the plant gi%en in (2) for this section of the lab.


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2

30 30P(s)=

(s 5)(s 6)11 30s s=

+ ++ +

)* 'roportional +', Control

etermine the root locus for this system #ith proportional control. At this point the

controller is specified as (s) 1C =

.

As k increases, the imaginary part of the closed-loop poles increases,

the percent overshoot increases and the position error decreases.

- As k=1, therefore the percent overshoot (%) is 4.1 !ith the

position error is ".#%.

3oot locus


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- As 456, therefore the percent o%ershoot (7) is 89. #ith the position error is

9.2227.

3oot locus

0ince the real part of the pole does not change once 4 is greater than about 9.99;,

the settling time remains constant at about 9.; seconds.

3oot locus


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6. "ntegral (") Control

a) Add a real pole at zero to implement the integral controller. o#e%er, for an integral

controller the pole is al#ays at zero, so lea%e it there for no#.

b)


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d) $he %alue that gi%es a response #ith settling time less than or eual to 6

seconds and has a little o%ershoot is ?58.8B. $he step response and

control effort.

3oot locus


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e) $he position error is zero. After the experiment in sisotool, #e couldnt find

a %alue for 8 second of settling time.

2. !roportional eri%ati%e (!) control

a) 1dit your compensator by remo%ing the integrator (the pole at zero) and

add a real zero. =ote that in this ! design that you can select #here you

place this real zero along the real axis. $a4e a moment to explore #hat

happens to the root locus, the step response, and the control effort as you

mo%e the zero.b) =o# mo%e the zero bet#een -8 and -D. &ind a configuration #ith a position

error less than 9.B, settling time is real zero is -6.B, #ith 452.

c) =ext mo%e the zero to -.B. $he root locus is changed. 0pecify a controller

#ith a zero in this range that produces a settling time of 9.8 seconds or less


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($9.9@@8), 45;, 0a%e the response and control effort figure and the

controller that produced it.

3oot locus

D. !roportional "ntegral(!") Control

a) 1dit your compensator by adding a real

control systems design tp

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