<< Chapter < Page Chapter >> Page >

For the resistor, if i(t) is bounded then so is v(t), but for the capacitance this is not true. Consider i(t) = u(t) then v(t) = tu(t) which is unbounded.

5/ Linear systems

for all x 1 ( t ) size 12{x rSub { size 8{1} } \( t \) } {} , x 2 ( t ) size 12{x rSub { size 8{2} } \( t \) } {} , a, and b.

6/ Time-invariant systems

for all x(t) and τ.

7/ Linear and time-invariant (LTI) systems

  • Many man-made and naturally occurring systems can be modeled as LTI systems.
  • Powerful techniques have been developed to analyze and to characterize LTI systems.
  • The analysis of LTI systems is an essential precursor to the analysis of more complex systems.

Problem — Multiplication by a time function

A system is defined by the functional description

  • Is this system linear?
  • Is this system time-invariant?

Solution — Multiplication by a time function

Let

{ y 1 ( t ) = g ( t ) x 1 ( t ) y 2 ( t ) = g ( t ) x 2 ( t ) size 12{ left lbrace matrix { y rSub { size 8{1} } \( t \) =g \( t \) x rSub { size 8{1} } \( t \) {} ##y rSub { size 8{2} } \( t \) =g \( t \) x rSub { size 8{2} } \( t \) } right none } {}

By definition the response to

x ( t ) = ax 1 ( t ) + bx 2 ( t ) size 12{x \( t \) = ital "ax" rSub { size 8{1} } \( t \) + ital "bx" rSub { size 8{2} } \( t \) } {}

Is

y ( t ) = g ( t ) ( ax 1 ( t ) + bx 2 ( t ) ) size 12{y \( t \) =g \( t \) \( ital "ax" rSub { size 8{1} } \( t \) + ital "bx" rSub { size 8{2} } \( t \) \) } {}

This can be rewritten as

y ( t ) = ag ( t ) x 1 ( t ) + bg ( t ) x 2 ( t ) size 12{y \( t \) = ital "ag" \( t \) x rSub { size 8{1} } \( t \) + ital "bg" \( t \) x rSub { size 8{2} } \( t \) } {}

y ( t ) = ay 1 + by 2 ( t ) size 12{y \( t \) = ital "ay" rSub { size 8{1} } + ital "by" rSub { size 8{2} } \( t \) } {}

Therefore, the system is linear.

Now suppose that x 1 ( t ) = x ( t ) size 12{x rSub { size 8{1} } \( t \) =" x" \( t \) } {} and x 2 ( t ) = x ( t - τ ) size 12{x rSub { size 8{2} } \( t \) =" x" \( "t - "τ \) } {} , and the response to these two inputs are y 1 ( t ) size 12{y rSub { size 8{1} } \( t \) } {} and y 2 ( t ) size 12{y rSub { size 8{2} } \( t \) } {} , respectively. Note that

y 1 ( t ) = y ( t ) = g ( t ) x ( t ) size 12{y rSub { size 8{1} } \( t \) =y \( t \) =g \( t \) x \( t \) } {}

And

y 2 ( t ) = g ( t ) x ( t τ ) y ( t τ ) size 12{y rSub { size 8{2} } \( t \) =g \( t \) x \( t - τ \)<>y \( t - τ \) } {}

Therefore, the system is time-varying.

Problem — Addition of a constant

Suppose the relation between the output y(t) and input x(t) is y(t) = x(t)+K, where K is some constant. Is this system linear?

Solution — Addition of a constant

Note, that if the input is x 1 ( t ) + x 2 ( t ) size 12{x rSub { size 8{1} } \( t \) +x rSub { size 8{2} } \( t \) } {} then the output will be y ( t ) = x 1 ( t ) + x 2 ( t ) + K y 1 ( t ) + y 2 ( t ) = ( x 1 ( t ) + K ) + ( x 2 ( t ) + K ) . size 12{y \( t \) =" x" rSub { size 8{1} } \( t \) +x rSub { size 8{2} } \( t \) +"K "<>" y" rSub { size 8{1} } \( t \) +y rSub { size 8{2} } \( t \) = \( x rSub { size 8{1} } \( t \) +K \) + \( x rSub { size 8{2} } \( t \) +K \) "." } {}

Therefore, this system is not linear.

In general, it can be shown that for a linear system if x(t) = 0 then y(t) = 0. Using the definition of linearity, choose a = b = 1 and x 2 = -x 1 ( t ) size 12{x rSub { size 8{2} } =" -x" rSub { size 8{1} } \( t \) } {} then x ( t ) = x 1 ( t ) + x 2 ( t ) = 0 size 12{x \( t \) =" x" rSub { size 8{1} } \( t \) +" x" rSub { size 8{2} } \( t \) =" 0 "} {} and y ( t ) = y 1 ( t ) + y 2 ( t ) = 0 size 12{y \( t \) =" y" rSub { size 8{1} } \( t \) +y rSub { size 8{2} } \( t \) =" 0"} {} .

Two-minute miniquiz problem

Problem 2-1

The system

y ( t ) = x 2 ( t ) size 12{y \( t \) =x rSup { size 8{2} } \( t \) } {}

is (choose one):

  1. Linear and time-invariant;
  2. Linear but not time-invariant;
  3. Not linear but time-invariant;
  4. Not linear and not time-invariant.

Solution

Note that if x 2 ( t ) = 2x 1 ( t ) size 12{x rSub { size 8{2} } \( t \) =2x rSub { size 8{1} } \( t \) } {} then y 2 ( t ) = ( 2x 1 ( t ) ) 2 = 4y 1 ( t ) size 12{y rSub { size 8{2} } \( t \) = \( 2x rSub { size 8{1} } \( t \) \) rSup { size 8{2} } =4y rSub { size 8{1} } \( t \) } {}

Hence, this system is nonlinear.

Note that if x 1 ( t ) = x ( t ) size 12{x rSub { size 8{1} } \( t \) =x \( t \) } {} and x 2 ( t ) = x ( t τ ) size 12{x rSub { size 8{2} } \( t \) =x \( t - τ \) } {} then y 1 ( t ) = y ( t ) size 12{y rSub { size 8{1} } \( t \) =y \( t \) } {}

And y 2 ( t ) = x 2 ( t τ ) = y ( t τ ) size 12{y rSub { size 8{2} } \( t \) =x rSup { size 8{2} } \( t - τ \) =y \( t - τ \) } {} . Hence, this system is time invariant.

V. LINEAR, ORDINARY DIFFERENTIAL EQUATIONS ARISE FOR A VARIETY OF SYSTEM DESCRIPTIONS

1/ Electric network

Kirchhoff’s current law yields

i ( t ) = i C ( t ) + i R ( t ) + i L ( t ) size 12{i \( t \) =i rSub { size 8{C} } \( t \) +i rSub { size 8{R} } \( t \) +i rSub { size 8{L} } \( t \) } {}

The constitutive relations for each element yield {}

i C ( t ) = C dv ( t ) dt size 12{i rSub { size 8{C} } \( t \) `=`C { { ital "dv" \( t \) } over { ital "dt"} } } {} i R ( t ) = v ( t ) R size 12{i rSub { size 8{R} } \( t \) `=` { {v \( t \) } over {R} } } {} i L ( t ) = 1 L t v ( τ ) size 12{i rSub { size 8{L} } \( t \) `=` { {1} over {L} } ` Int rSub { size 8{ - infinity } } rSup { size 8{t} } {v \( τ \) dτ} } {}

Combining KCL and the constitutive relations yields

di ( t ) dt = C d 2 v ( t ) dt 2 + 1 R dv ( t ) dt + v ( t ) L size 12{ { { ital "di" \( t \) } over { ital "dt"} } `=`C { {d rSup { size 8{2} } v \( t \) } over { ital "dt" rSup { size 8{2} } } } `+ { {1} over {R} } { { ital "dv" \( t \) } over { ital "dt"} } `+` { {v \( t \) } over {L} } } {}

2/ Mechanical system

The simplest possible model of a muscle (the linearized Hill model) consists of the mechanical network shown below which relates the rate of change of the length of the muscle v(t) to the external force on the muscle fe(t).

f c size 12{f rSub { size 8{c} } } {} is the internal force generated by the muscle, K is its stiffness and B is its damping.

The muscle velocity can be expressed as

v ( t ) = v c ( t ) + v s ( t ) size 12{v \( t \) `=`v rSub { size 8{c} } \( t \) `+`v rSub { size 8{s} } \( t \) } {}

Combining the muscle velocity equation with the constitutive laws for the elements yields

v ( t ) = f e ( t ) f c ( t ) B + 1 L df e ( t ) dt size 12{v \( t \) `=` { {f rSub { size 8{e} } \( t \) - f rSub { size 8{c} } \( t \) } over {B} } `+ { {1} over {L} } { { ital "df" rSub { size 8{e} } \( t \) } over { ital "dt"} } } {}

3/ First-order chemical kinetics

A reversible, first-order chemical reaction can be represented as follows

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
I only see partial conversation and what's the question here!
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
nanocopper obvius
Alexandre
what is the stm
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
how nano science is used for hydrophobicity
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask