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Observability is the tool we use to investigate the internal workings of a system. It lets us use what we know about the input u t and the output y t to observe the state of the system x t .

To understand this concept let's start off with the basic state-space equations describing a system: x A x B u y C x D u If we plug the general solution of the state variable, x t , into the equation for y t , we'd find the following familiar time-domain equation:

y t C A t x 0 0 t C A t B u D u t

Without loss of generality, we can assume zero input; this will significantly clarify the following discussion. This assumption can be easily justified. Based on our initial assumption above, the last two terms on the right-hand side of time-domain equation are known (because we know u t ). We could simply replace these two terms with some function of t . We'll group them together into the variable y 0 t . By moving y 0 t to the left-hand side, we see that we can again group y t y 0 t into another replacement function of t , y _ t . This result has the same effect as assuming zero input. y _ t y t y 0 t C A t x 0 Given the discussion in the above paragraph, we can now start our examination of observability based on the following formula:

y t C A t x 0

The idea behind observability is to find the state of the system based upon its output. We will accomplish this by first finding the initial conditions of the state based upon the system's output. The state equation solution can then use this information to determine the state variable x t .

base formula seems to tell us that as long as we known enough about y t we should be able to find x 0 . The first question to answer is how much is enough? Since the initial condition of the state x 0 is actually a vector of n elements, we have n unknowns and therefore need n equations to solve the set. Remember that we have complete knowledge of the output y t . So, to generate these n equations, we can simply take n 1 derivatives of base formula . Taking these derivatives is relatively straightforward. On the right-handside, the derivative operator will only act on the matrix exponential term. Each derivative of it will produce amultiplicative term of A . Then, as we're dealing with these derivatives of y t at t 0 , all of the exponential terms will go to unity( A 0 1 ). y 0 C x 0 t 1 y 0 C A x 0 t 2 y 0 C A 2 x 0 t n 1 y 0 C A n 1 x 0 This can be re-expressed in matrix notation. y 0 t 1 y 0 t 2 y 0 t n 1 y 0 C C A C A 2 C A n 1 x 0

The first term on the right-hand side is known as the observability matrix, C A :

C A C C A C A 2 C A n 1

We call the system completely observable if the rank of the observability matrix equals n . This guarantees that we'll have enough independent equations to solve for the n components of the state x t .

Whereas for controllability we talked about the system's controllable space, for observability we will talk about a system's un observable space, X unobs . The unobservable space is found by taking the kernel of the observability matrix. This makes sense because when you multiply a vector in the kernel of the observability matrix by the observability matrix, the result will be 0 . The problem is that when we get a zero result for y t , we cannot say with certainty whether the zero result was caused by x t itself being zero or by x t being a vector in the nullspace. As we cannot give a definite answer in this case, all of these vectors are said to be unobservable.

One cool thing to note is that the observability and controllability matrices are intimately related:

C A C A C

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
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