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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

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
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Akash Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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I'm interested in nanotube
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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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