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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:


Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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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