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Two-minute miniquiz problem

Problem 9-1

For K = 1000 determine the unit step response y(t) of the position control system.

[ H int: the polynomial s 2 + 101 s + 1100 ( s + 88 . 5 ) ( s + 12 . 5 ) ] size 12{ \[ H"int:" matrix { {} # {}} ital "the" matrix { {} # {}} ital "polynomial" matrix { {} # {}} s rSup { size 8{2} } +"101"s+"1100" approx \( s+"88" "." 5 \) \( s+"12" "." 5 \) \] } {}

Solution

Substituting K = 1000 into the system function yields

Y ( s ) = X ( s ) H ( s ) = 1 s 1000 ( s + 1 ) ( s + 100 ) + 1000 size 12{Y \( s \) =X \( s \) H \( s \) = left [ { {1} over {s} } right ] left [ { {"1000"} over { \( s+1 \) \( s+"100" \) +"1000"} } right ]} {}

The denominator polynomial can be factored and expanded in a partial fraction expansion as follows

Y ( s ) = 1000 s ( s + 88 . 5 ) ( s + 12 . 5 ) = 1000 / ( 88 . 5 ) ( 12 . 5 ) s + 100 / ( 88 . 5 ) ( 76 ) s + 88 . 5 + 1000 / ( 12 . 5 ) ( 76 ) s + 12 . 5 = 0 . 9 s + 0 . 15 s + 88 . 5 + 1 . 05 s + 12 . 5 alignl { stack { size 12{Y \( s \) = { {"1000"} over {s \( s+"88" "." 5 \) \( s+"12" "." 5 \) } } } {} #size 12{ matrix { {} # {}} = { {"1000"/ \( "88" "." 5 \) \( "12" "." 5 \) } over {s} } + { {"100"/ \( - "88" "." 5 \) \( - "76" \) } over {s+"88" "." 5} } + { {"1000"/ \( - "12" "." 5 \) \( "76" \) } over {s+"12" "." 5} } } {} # size 12{ matrix {{} # {} } = { {0 "." 9} over {s} } + { {0 "." "15"} over {s+"88" "." 5} } + { {1 "." "05"} over {s+"12" "." 5} } } {}} } {}

The step response is

y ( t ) = ( 0 . 9 + 0 . 15 e 88 . 5t 1 . 05 e 12 . 5t ) u ( t ) size 12{y \( t \) = \( 0 "." 9+0 "." "15"e rSup { size 8{ - "88" "." 5t} } - 1 "." "05"e rSup { size 8{ - "12" "." 5t} } \) u \( t \) } {}

Note that y(∞) = 0.9 which fits with the result obtained from the steady-state analysis which gives 1000/1100 ≈ 0.9.

A plot of the step response for K = 1000 along with those for several values of K are shown next.

How does the step response change as K is increased?

  • As the gain is increased, the steady-state error in position decreases.
  • As the gain is increased, the step response becomes a damped oscillation. This could be disastrous in a position control system. Suppose we designed a system for doing microsurgery on the brain or the eye!

Thus, we cannot achieve an arbitrarily small position error without causing damped oscillations with this controller design.

2/ Simple position control system with zero position error

Consider a new design in which the error is integrated.

We can use Black’s formula to find H(s) as follows

H ( s ) = K s ( s + 1 ) ( s + 100 ) 1 + K s ( s + 1 ) ( s + 100 ) = K s ( s + 1 ) ( s + 100 ) + K size 12{H \( s \) = { { { {K} over {s \( s+1 \) \( s+"100" \) } } } over {1+ { {K} over {s \( s+1 \) \( s+"100" \) } } } } = { {K} over {s \( s+1 \) \( s+"100" \) +K} } } {}

The steady-state response to a unit step is simply the response to the complex exponential x ( t ) = 1 . e 0 . t = 1 size 12{x \( t \) =" 1" "." e rSup { size 8{0 "." t} } =" 1"} {} which is y ( t ) = 1 . H ( 0 ) . e 0 . t = 1 size 12{y \( t \) =" 1" "." H \( 0 \) "." e rSup { size 8{0 "." t} } =" 1"} {} . The position error ε = 1− 1 = 0. Hence, it appears that this position control system (with integral controller) has no position error for any value of K. So how do we pick K?

3/ Simple position control system with zero position error — step response

The step response is shown for several values of K.

  • The steady-state error is zero for K<10100 (we will see how this value is determined later).
  • The step response shows oscillations that are damped for K<10100 but shows oscillations whose amplitude grows exponentially for K>10100. Such a system is called unstable. When the system becomes unstable, the steady-state position error is not zero! Furthermore, a position control system that is unstable is even more disastrous than one that exhibits damped oscillations in response to a unit step.

4/ BIBO stability

There are many ways one can define stability of a system. We shall use the following. A system for which every bounded input yields a bounded output is called BIBO stable. A feedback system with closed loop system function

H ( s ) = K ( s ) 1 + β ( s ) K ( s ) size 12{H \( s \) = { {K \( s \) } over {1+β \( s \) K \( s \) } } } {}

is BIBO stable if its poles (the natural frequencies of the closed loop system) are in the left half of the s-plane. Thus, determining the conditions for which a system is stable reduces to finding whether the zeros of 1+β(s)K(s) = 0 are in the left-half s-plane. When β(s)K(s) is a rational function, this condition is tested by determining whether the roots of the numerator polynomial of 1+β(s)K(s) are located in the left-half s-plane.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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