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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

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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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