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What do we mean by the term controllability ? Simply put, we want to know if we can control the state of a system when we only have access to the inputs (i.e. when we can not directly modify the system's state). If we can "steer" a system to a certain state by controlling its inputs, we can then ask ourselves if there is a way to find the most efficient method of making this transformation.

Developing the concept of a controllable space

Say we have the following system:

x A x t B u t

Example rlc circuit

x x 1 x 2 In this case, an example controllability question could seekto know if there exists an input u such that: x 1 ms 10 V 1 A

Instead of deriving the general solution for what is called a system's controllable space, X contr , we will simply state it and then give a justification for it.

Formally, X contr is the set of all controllable states. We will define it in terms of a quantity known as the controllability matrix, C A B :

C A B B A B A 2 B A n 1 B
The controllable space can then be found by taking the image of this matrix.
X contr im C A B

To justify this expression, we begin with the formal matrix equation for a system's state and substitute inthe infinite series definition of the matrix exponential. We can then extract the A and B matrices into a larger matrix multiplication.

x A t B u I A t A 2 2 t 2 B u B u A B t 1 u A 2 B t 2 2 u B A B A 2 B A n 1 B u t u t n n u
As the second term in the multiplication is dependent on u , it can be thought of as a free variable. Therefore, the setof possible values for x is dependent on the image of first term, which can be seen to be the controllability matrix as defined above.

Continuing the example circuit started above, we can get a better feel for what controllability means. Here are the state equations: x 1 -1 R 1 C x 1 1 R 1 C u x 2 R 2 L x 2 1 L u Pulling the A and B matrices out of these equations, we can compute the controllability matrix C A B A A B . Note that as it is only a second order system, the controllability matrix is only two-dimensional. C A B 1 R 1 C -1 R 1 C 2 1 L R 2 L 2

Immediately, we can understand some things about the system by looking at the rank of the C matrix. Let's look at the determinant: C 1 L R 1 C R 2 L 1 R 1 C If the determinant of the controllability matrix is non-zero, then X contr im C 2 ; the system is completely controllable. For this to happen we'd need to ensure that R 2 L 1 R 1 C .However, if this inequality is not satisfied and the determinant of the controllability matrix is 0 , then we know that it is not full rank. If it is not full rank, then X contr will not span the entire space and the system is not completely controllable. The physical effect here is resonance in the circuit. This reduces our controllability matrix to only one dimension (the two columns are linearly dependent). X contr span 1 R 1 C 1 L

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
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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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